arX

iv0

810

5128

v2 [

hep-

th]

15

May

200

9

Watching Worlds Collide

Effects on the CMB from Cosmological Bubble Collisions

Spencer Changab1 Matthew Klebana2 and Thomas S Levi a3

aCenter for Cosmology and Particle Physics

Department of Physics New York University

4 Washington Place New York NY 10003

bDepartment of Physics University of California

1 Shields Avenue Davis CA 95616

Abstract

We extend our previous work on the cosmology of Coleman-de Luccia bubble colli-sions Within a set of approximations we calculate the effects on the cosmic microwavebackground (CMB) as seen from inside a bubble which has undergone such a collisionWe find that the effects are always qualitatively similarmdashan anisotropy that dependsonly on the angle to the collision directionmdashbut can produce a cold or hot spot ofvarying size as well as power asymmetries along the axis determined by the collisionWith other parameters held fixed the effects weaken as the amount of inflation whichtook place inside our bubble grows but generically survive order 10 efolds past whatis required to solve the horizon and flatness problems In some regions of parameterspace the effects can survive arbitrarily long inflation

1e-mail spchang123gmailcom2e-mail mk161nyuedu3e-mail tl34nyuedu

Contents

1 Introduction 2

2 The Scenario 3

21 The Bubble Collision 4

22 The Reheating Surface 6

23 Observer Horizons 9

3 Finding The Reheating Surface 10

31 The Domain Wall Trajectory 11

32 Null Rays And The Structure Of H2 dS Space 11

33 General Solutions For The Scalar Field 12

34 Matching Solutions 13

4 The Redshift 15

41 Null Geodesics In Radiation Dominated Spacetime 16

42 Boosted Observers 17

43 Finding The Redshift 17

5 A Simple Model 20

51 Results 23

6 Conclusions 26

A Solving For The Scalar Field 27

B The Spherical Harmonics 28

1

ldquoThrough telescopes men of science constantly search the infinitesimal corners of our

solar system seeking new discoveries hoping to better understand the laws of the Universerdquo

ndash When Worlds Collide (The Movie)

1 Introduction

Perhaps the greatest challenge for string theory is to find a testable prediction which candifferentiate it from other potential theories of quantum gravity This has proven to beextremely difficult because the scale at which string theory effects become important is be-lieved to be far out of reach of the energies accessible to any earth-based particle acceleratorHowever cosmology provides another opportunity the early universe was arbitrarily hot anddense and remnants of the physics of that time remain in the large-scale structure of theuniverse today

It has become increasingly clear in recent years that one of the characteristic featuresof string theory is its so-called landscape [1 2 3] The landscape is a potential energyfunction of the would-be moduli scalar fields which possesses many distinct minima Theseminima are meta-stable and can decay via bubble nucleation-type tunneling or quantumthermal processes In the early universe one expects all the minima to be populated as thetemperature decreases and the fields begin to settle The minima with the highest valuesof potential energy then begin to inflate and are expected to rapidly dominate the volumeAfter a few string times the universe will be almost completely full of rapidly inflating regionsand as time continues to pass the metastable field values in these regions will begin to decayby bubble nucleation Since we live in a region with very small vacuum energy we expectthat our observable universe is contained in such a bubble

The most promising way to test this idea is to look at large scale cosmological structureIf we model the string theory landscape as a scalar field coupled to gravity with a potentialwith several minima universes inside a bubble which formed via a tunneling instanton havecertain characteristic features they are open they are curvature dominated at the ldquobig bangrdquo(rendering it non-singular) and they have a characteristic spectrum of density perturbationsgenerated during the inflationary period that takes place inside them [4 5 6 7 8 9 10]

Moreover because the bubble forms inside a metastable region other bubbles will appearin the space around it If these bubbles appear close enough to ours they will collide withit [11 12 13] The two bubbles then ldquostickrdquo together separated by a domain wall andthe force of the collision releases a pulse of energy which propagates through each of thepair This process (as seen from inside our bubble) makes the universe anisotropic affectsthe cosmic microwave background and the formation of structure A collision between twobubbles produces a spacetime which can be solved for exactly [14 15 16]

In [15] we began the study of this process We were primarily interested in the effectson the cosmic microwave background (CMB) Our solution shows that the CMB sky will bedivided into two (unequal) halves along a circle The size of the circle depends on when thecollision took place The temperature fluctuations in the disk on the side of the circle away

2

from the collision will be weakly affected (by the gravitational backreaction of the energydumped into our bubble) in a relatively simple way which we calculated The disk on theother side will be more strongly affected and in a way that depends in a more complex wayon the model which we did not calculate In this paper we will use a simplified model tocompute these latter effects

Our model consists of a collision between our bubblemdashwhich we treat as a thin-wallColeman-de Luccia bubble which undergoes a period of inflation after it formed (which wemodel as a de Sitter phase) followed by a period of radiation dominationmdashand anotherdifferent bubble the details of which are largely unimportant We assume that the tensionof the domain wall between the bubbles and the vacuum energy in the other bubble aresuch that the domain wall accelerates away from us Such collisions can produce observableeffects which are not in conflict with current data We will also assume that the inflatonis the field that underwent the initial tunneling transition so that the reheating surface isdetermined by the evolution of that field and can be affected by the collision with the otherbubble through its non-linear equation of motion and that no other scalar fields are relevant

After some number of efolds of inflation our bubble will reheat and become radiationdominated The subsequent expansion and the geometry of the reheating surface itself willbe modified by the presence of the domain wall The effect we will focus on is the change inthe reheating surface brought about by the domain wall

There are a number of anomalies in current cosmological data including a cold spot[17 18 19 20 21] hemispherical power asymmetry [22] non-Gaussianities (see eg [23])and the ldquoaxis of evilrdquo [24] While the significance of these effects is unclear there are fewif any first-principle models which can account for them Our results indicate that bubblecollisions naturally create hemispherical power asymmetries spread through the lower CMBmultipoles andor produce relatively small cold or hot spots depending on the time of thecollision (and hence the size of the disk) and can lead to measurable non-Gaussianities Thequalitative features however are always the same a collision with a single other bubbleproduces anisotropies that depend only on one anglemdashthe angle to the vector pointing fromour location to the center of the other bubblemdashand which are generically monotonic in theangle While these results are suggestive further analysis is required to determine which ofthese anomalies can be explained by bubble collisions

While we will not study structure formation in this paper it is interesting to note thatbubble collisions may also create coherent peculiar velocity flows for large-scale structuresacross large swaths of the sky This may be interesting in light of the recently publishedresults [25]

2 The Scenario

In this section we will set up the basic scenario for the bubble collision we seek to study Thedetails of the dynamics of general collisions were studied in [15] and we refer the reader therefor a more complete treatment We will focus on a collision where our bubble is dominatedby positive vacuum energy when it forms so that it undergoes a period of inflation and we

3

will assume the pressures are such that the domain wall is accelerating away from the centerof our bubble at late times This second assumption is valid for collisions between two dSbubbles if ours has a smaller cosmological constant1

Because of the difficulty in finding analytic solutions to the full problem of a scalarfield with a non-linear potential coupled to gravity we will make a number of simplifyingassumptions We enumerate them here

bull Unlike in [15] we ignore gravitational backreaction from the shell of radiation releasedinto our bubble by the collision and focus instead on the effects on the inflatonrsquosevolution and reheating

bull We approximate the inflaton potential as linear when computing its evolution insideour bubble

bull We allow the fieldrsquos derivative to be discontinuous across the edge of the lightcone ofthe collision to model the effects of non-linearities

bull We treat the wall separating our bubble from the false vacuum and the domain wallbetween our bubble and the other bubble as Dirichlet (fixed value) boundary surfacesfor the inflaton

bull We treat the walls and the edges of the collision lightcone (where the field derivativeschange) as thin

bull We assume inflationary perturbations are unaffected by the collision

bull We model the effects of the collision on the CMB by computing the redshift to thereheating surface as a function of angle rather than evolving the fluctuations in themodified spacetime

We will discuss some of these assumptions in more detail later

21 The Bubble Collision

In this section we will review the bubble collision spacetime Readers familiar with thedetails of [15] may wish to skip ahead to Sec 22

A single Coleman-de Luccia bubble in four dimensions has an SO(3 1) invariance (inher-ited from the SO(4) symmetry of the Euclidean instanton [4]) When another bubble collideswith ours a preferred direction is picked out breaking the symmetry down to SO(2 1) As aresult one can choose coordinates in such a way that the metric takes the form of a warpedproduct between two-dimensional hyperboloids (H2) and a 2D space

1It also occurs when our bubble has a larger cosmological constant or in collisions with AdS bubbles when

certain conditions involving the tension of the wall are satisfied [14 15 16 26]

4

Under the assumption that all the energy in the spacetime remains confined to thinshells or is in the form of vacuum energy one can find the solution describing the collisionby patching together general solutions to Einsteinrsquos equations in vacuum plus cosmologicalconstant with this symmetry [14 15 27] The overall result is that the spacetime inside sucha bubble is divided into two regionsmdashone that is outside the future lightcone of the collisionand can be described using an SO(3 1) invariant metric (a standard open Robertson-Walkercosmology) and one which is inside is affected by backreaction and is most convenientlydescribed using 2D hyperboloids

de Sitter space The general SO(2 1) invariant spacetime with a positive cosmologicalconstant is given by (see [15] for a full discussion of the geometry and causal structure)

ds2 =minusdt2

g(t)+ g(t) dx2 + t2dH2

2 (21)

where

g(t) = 1 +t2

ℓ2minus t0

t (22)

and

dH22 = dρ2 + sinh2 ρdϕ2 (23)

is the metric on the unit 2-hyperboloid and x ≃ x + πℓ We will find for most collisionsthat t0 ≪ t at cosmologically relevant times and thus we approximate t0 asymp 0 giving g(t) asymp1 + t2ℓ2 This is a metric for pure dS space with radius of curvature ℓ Hereafter we setℓ = 1 (it can be easily restored with dimensional analysis) These coordinates (which wedenote the H2 coordinates) are related to the SO(3 1) or H3 invariant ones by

cosh τ =radic1 + t2 cosx

sinh τ sinh ξ cos θ =radic1 + t2 sin x

(24)sinh τ cosh ξ = t cosh ρ

sinh τ sinh ξ sin θ = t sinh ρ

ϕ = ϕ

which gives a metric

ds2 = minusdτ 2 + sinh2 τdH23

(25)dH2

3 = dξ2 + sinh2 ξ(dθ2 + sin2 θdϕ2)

We display the two coordinate systems for dS space in Fig 1

Radiation domination We will also need the metric for a radiation dominated (RD)universe The general SO(3 1) invariant Friedmann-Robertson-Walker (FRW) metric isgiven by

ds2 = minusdτ 2 + a2(τ)dH23 (26)

5

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

ldquoThrough telescopes men of science constantly search the infinitesimal corners of our

solar system seeking new discoveries hoping to better understand the laws of the Universerdquo

ndash When Worlds Collide (The Movie)

1 Introduction

Perhaps the greatest challenge for string theory is to find a testable prediction which candifferentiate it from other potential theories of quantum gravity This has proven to beextremely difficult because the scale at which string theory effects become important is be-lieved to be far out of reach of the energies accessible to any earth-based particle acceleratorHowever cosmology provides another opportunity the early universe was arbitrarily hot anddense and remnants of the physics of that time remain in the large-scale structure of theuniverse today

It has become increasingly clear in recent years that one of the characteristic featuresof string theory is its so-called landscape [1 2 3] The landscape is a potential energyfunction of the would-be moduli scalar fields which possesses many distinct minima Theseminima are meta-stable and can decay via bubble nucleation-type tunneling or quantumthermal processes In the early universe one expects all the minima to be populated as thetemperature decreases and the fields begin to settle The minima with the highest valuesof potential energy then begin to inflate and are expected to rapidly dominate the volumeAfter a few string times the universe will be almost completely full of rapidly inflating regionsand as time continues to pass the metastable field values in these regions will begin to decayby bubble nucleation Since we live in a region with very small vacuum energy we expectthat our observable universe is contained in such a bubble

The most promising way to test this idea is to look at large scale cosmological structureIf we model the string theory landscape as a scalar field coupled to gravity with a potentialwith several minima universes inside a bubble which formed via a tunneling instanton havecertain characteristic features they are open they are curvature dominated at the ldquobig bangrdquo(rendering it non-singular) and they have a characteristic spectrum of density perturbationsgenerated during the inflationary period that takes place inside them [4 5 6 7 8 9 10]

Moreover because the bubble forms inside a metastable region other bubbles will appearin the space around it If these bubbles appear close enough to ours they will collide withit [11 12 13] The two bubbles then ldquostickrdquo together separated by a domain wall andthe force of the collision releases a pulse of energy which propagates through each of thepair This process (as seen from inside our bubble) makes the universe anisotropic affectsthe cosmic microwave background and the formation of structure A collision between twobubbles produces a spacetime which can be solved for exactly [14 15 16]

In [15] we began the study of this process We were primarily interested in the effectson the cosmic microwave background (CMB) Our solution shows that the CMB sky will bedivided into two (unequal) halves along a circle The size of the circle depends on when thecollision took place The temperature fluctuations in the disk on the side of the circle away

2

from the collision will be weakly affected (by the gravitational backreaction of the energydumped into our bubble) in a relatively simple way which we calculated The disk on theother side will be more strongly affected and in a way that depends in a more complex wayon the model which we did not calculate In this paper we will use a simplified model tocompute these latter effects

Our model consists of a collision between our bubblemdashwhich we treat as a thin-wallColeman-de Luccia bubble which undergoes a period of inflation after it formed (which wemodel as a de Sitter phase) followed by a period of radiation dominationmdashand anotherdifferent bubble the details of which are largely unimportant We assume that the tensionof the domain wall between the bubbles and the vacuum energy in the other bubble aresuch that the domain wall accelerates away from us Such collisions can produce observableeffects which are not in conflict with current data We will also assume that the inflatonis the field that underwent the initial tunneling transition so that the reheating surface isdetermined by the evolution of that field and can be affected by the collision with the otherbubble through its non-linear equation of motion and that no other scalar fields are relevant

After some number of efolds of inflation our bubble will reheat and become radiationdominated The subsequent expansion and the geometry of the reheating surface itself willbe modified by the presence of the domain wall The effect we will focus on is the change inthe reheating surface brought about by the domain wall

There are a number of anomalies in current cosmological data including a cold spot[17 18 19 20 21] hemispherical power asymmetry [22] non-Gaussianities (see eg [23])and the ldquoaxis of evilrdquo [24] While the significance of these effects is unclear there are fewif any first-principle models which can account for them Our results indicate that bubblecollisions naturally create hemispherical power asymmetries spread through the lower CMBmultipoles andor produce relatively small cold or hot spots depending on the time of thecollision (and hence the size of the disk) and can lead to measurable non-Gaussianities Thequalitative features however are always the same a collision with a single other bubbleproduces anisotropies that depend only on one anglemdashthe angle to the vector pointing fromour location to the center of the other bubblemdashand which are generically monotonic in theangle While these results are suggestive further analysis is required to determine which ofthese anomalies can be explained by bubble collisions

While we will not study structure formation in this paper it is interesting to note thatbubble collisions may also create coherent peculiar velocity flows for large-scale structuresacross large swaths of the sky This may be interesting in light of the recently publishedresults [25]

2 The Scenario

In this section we will set up the basic scenario for the bubble collision we seek to study Thedetails of the dynamics of general collisions were studied in [15] and we refer the reader therefor a more complete treatment We will focus on a collision where our bubble is dominatedby positive vacuum energy when it forms so that it undergoes a period of inflation and we

3

will assume the pressures are such that the domain wall is accelerating away from the centerof our bubble at late times This second assumption is valid for collisions between two dSbubbles if ours has a smaller cosmological constant1

Because of the difficulty in finding analytic solutions to the full problem of a scalarfield with a non-linear potential coupled to gravity we will make a number of simplifyingassumptions We enumerate them here

bull Unlike in [15] we ignore gravitational backreaction from the shell of radiation releasedinto our bubble by the collision and focus instead on the effects on the inflatonrsquosevolution and reheating

bull We approximate the inflaton potential as linear when computing its evolution insideour bubble

bull We allow the fieldrsquos derivative to be discontinuous across the edge of the lightcone ofthe collision to model the effects of non-linearities

bull We treat the wall separating our bubble from the false vacuum and the domain wallbetween our bubble and the other bubble as Dirichlet (fixed value) boundary surfacesfor the inflaton

bull We treat the walls and the edges of the collision lightcone (where the field derivativeschange) as thin

bull We assume inflationary perturbations are unaffected by the collision

bull We model the effects of the collision on the CMB by computing the redshift to thereheating surface as a function of angle rather than evolving the fluctuations in themodified spacetime

We will discuss some of these assumptions in more detail later

21 The Bubble Collision

In this section we will review the bubble collision spacetime Readers familiar with thedetails of [15] may wish to skip ahead to Sec 22

A single Coleman-de Luccia bubble in four dimensions has an SO(3 1) invariance (inher-ited from the SO(4) symmetry of the Euclidean instanton [4]) When another bubble collideswith ours a preferred direction is picked out breaking the symmetry down to SO(2 1) As aresult one can choose coordinates in such a way that the metric takes the form of a warpedproduct between two-dimensional hyperboloids (H2) and a 2D space

1It also occurs when our bubble has a larger cosmological constant or in collisions with AdS bubbles when

certain conditions involving the tension of the wall are satisfied [14 15 16 26]

4

Under the assumption that all the energy in the spacetime remains confined to thinshells or is in the form of vacuum energy one can find the solution describing the collisionby patching together general solutions to Einsteinrsquos equations in vacuum plus cosmologicalconstant with this symmetry [14 15 27] The overall result is that the spacetime inside sucha bubble is divided into two regionsmdashone that is outside the future lightcone of the collisionand can be described using an SO(3 1) invariant metric (a standard open Robertson-Walkercosmology) and one which is inside is affected by backreaction and is most convenientlydescribed using 2D hyperboloids

de Sitter space The general SO(2 1) invariant spacetime with a positive cosmologicalconstant is given by (see [15] for a full discussion of the geometry and causal structure)

ds2 =minusdt2

g(t)+ g(t) dx2 + t2dH2

2 (21)

where

g(t) = 1 +t2

ℓ2minus t0

t (22)

and

dH22 = dρ2 + sinh2 ρdϕ2 (23)

is the metric on the unit 2-hyperboloid and x ≃ x + πℓ We will find for most collisionsthat t0 ≪ t at cosmologically relevant times and thus we approximate t0 asymp 0 giving g(t) asymp1 + t2ℓ2 This is a metric for pure dS space with radius of curvature ℓ Hereafter we setℓ = 1 (it can be easily restored with dimensional analysis) These coordinates (which wedenote the H2 coordinates) are related to the SO(3 1) or H3 invariant ones by

cosh τ =radic1 + t2 cosx

sinh τ sinh ξ cos θ =radic1 + t2 sin x

(24)sinh τ cosh ξ = t cosh ρ

sinh τ sinh ξ sin θ = t sinh ρ

ϕ = ϕ

which gives a metric

ds2 = minusdτ 2 + sinh2 τdH23

(25)dH2

3 = dξ2 + sinh2 ξ(dθ2 + sin2 θdϕ2)

We display the two coordinate systems for dS space in Fig 1

Radiation domination We will also need the metric for a radiation dominated (RD)universe The general SO(3 1) invariant Friedmann-Robertson-Walker (FRW) metric isgiven by

ds2 = minusdτ 2 + a2(τ)dH23 (26)

5

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

from the collision will be weakly affected (by the gravitational backreaction of the energydumped into our bubble) in a relatively simple way which we calculated The disk on theother side will be more strongly affected and in a way that depends in a more complex wayon the model which we did not calculate In this paper we will use a simplified model tocompute these latter effects

Our model consists of a collision between our bubblemdashwhich we treat as a thin-wallColeman-de Luccia bubble which undergoes a period of inflation after it formed (which wemodel as a de Sitter phase) followed by a period of radiation dominationmdashand anotherdifferent bubble the details of which are largely unimportant We assume that the tensionof the domain wall between the bubbles and the vacuum energy in the other bubble aresuch that the domain wall accelerates away from us Such collisions can produce observableeffects which are not in conflict with current data We will also assume that the inflatonis the field that underwent the initial tunneling transition so that the reheating surface isdetermined by the evolution of that field and can be affected by the collision with the otherbubble through its non-linear equation of motion and that no other scalar fields are relevant

After some number of efolds of inflation our bubble will reheat and become radiationdominated The subsequent expansion and the geometry of the reheating surface itself willbe modified by the presence of the domain wall The effect we will focus on is the change inthe reheating surface brought about by the domain wall

There are a number of anomalies in current cosmological data including a cold spot[17 18 19 20 21] hemispherical power asymmetry [22] non-Gaussianities (see eg [23])and the ldquoaxis of evilrdquo [24] While the significance of these effects is unclear there are fewif any first-principle models which can account for them Our results indicate that bubblecollisions naturally create hemispherical power asymmetries spread through the lower CMBmultipoles andor produce relatively small cold or hot spots depending on the time of thecollision (and hence the size of the disk) and can lead to measurable non-Gaussianities Thequalitative features however are always the same a collision with a single other bubbleproduces anisotropies that depend only on one anglemdashthe angle to the vector pointing fromour location to the center of the other bubblemdashand which are generically monotonic in theangle While these results are suggestive further analysis is required to determine which ofthese anomalies can be explained by bubble collisions

While we will not study structure formation in this paper it is interesting to note thatbubble collisions may also create coherent peculiar velocity flows for large-scale structuresacross large swaths of the sky This may be interesting in light of the recently publishedresults [25]

2 The Scenario

In this section we will set up the basic scenario for the bubble collision we seek to study Thedetails of the dynamics of general collisions were studied in [15] and we refer the reader therefor a more complete treatment We will focus on a collision where our bubble is dominatedby positive vacuum energy when it forms so that it undergoes a period of inflation and we

3

will assume the pressures are such that the domain wall is accelerating away from the centerof our bubble at late times This second assumption is valid for collisions between two dSbubbles if ours has a smaller cosmological constant1

Because of the difficulty in finding analytic solutions to the full problem of a scalarfield with a non-linear potential coupled to gravity we will make a number of simplifyingassumptions We enumerate them here

bull Unlike in [15] we ignore gravitational backreaction from the shell of radiation releasedinto our bubble by the collision and focus instead on the effects on the inflatonrsquosevolution and reheating

bull We approximate the inflaton potential as linear when computing its evolution insideour bubble

bull We allow the fieldrsquos derivative to be discontinuous across the edge of the lightcone ofthe collision to model the effects of non-linearities

bull We treat the wall separating our bubble from the false vacuum and the domain wallbetween our bubble and the other bubble as Dirichlet (fixed value) boundary surfacesfor the inflaton

bull We treat the walls and the edges of the collision lightcone (where the field derivativeschange) as thin

bull We assume inflationary perturbations are unaffected by the collision

bull We model the effects of the collision on the CMB by computing the redshift to thereheating surface as a function of angle rather than evolving the fluctuations in themodified spacetime

We will discuss some of these assumptions in more detail later

21 The Bubble Collision

In this section we will review the bubble collision spacetime Readers familiar with thedetails of [15] may wish to skip ahead to Sec 22

A single Coleman-de Luccia bubble in four dimensions has an SO(3 1) invariance (inher-ited from the SO(4) symmetry of the Euclidean instanton [4]) When another bubble collideswith ours a preferred direction is picked out breaking the symmetry down to SO(2 1) As aresult one can choose coordinates in such a way that the metric takes the form of a warpedproduct between two-dimensional hyperboloids (H2) and a 2D space

1It also occurs when our bubble has a larger cosmological constant or in collisions with AdS bubbles when

certain conditions involving the tension of the wall are satisfied [14 15 16 26]

4

Under the assumption that all the energy in the spacetime remains confined to thinshells or is in the form of vacuum energy one can find the solution describing the collisionby patching together general solutions to Einsteinrsquos equations in vacuum plus cosmologicalconstant with this symmetry [14 15 27] The overall result is that the spacetime inside sucha bubble is divided into two regionsmdashone that is outside the future lightcone of the collisionand can be described using an SO(3 1) invariant metric (a standard open Robertson-Walkercosmology) and one which is inside is affected by backreaction and is most convenientlydescribed using 2D hyperboloids

de Sitter space The general SO(2 1) invariant spacetime with a positive cosmologicalconstant is given by (see [15] for a full discussion of the geometry and causal structure)

ds2 =minusdt2

g(t)+ g(t) dx2 + t2dH2

2 (21)

where

g(t) = 1 +t2

ℓ2minus t0

t (22)

and

dH22 = dρ2 + sinh2 ρdϕ2 (23)

is the metric on the unit 2-hyperboloid and x ≃ x + πℓ We will find for most collisionsthat t0 ≪ t at cosmologically relevant times and thus we approximate t0 asymp 0 giving g(t) asymp1 + t2ℓ2 This is a metric for pure dS space with radius of curvature ℓ Hereafter we setℓ = 1 (it can be easily restored with dimensional analysis) These coordinates (which wedenote the H2 coordinates) are related to the SO(3 1) or H3 invariant ones by

cosh τ =radic1 + t2 cosx

sinh τ sinh ξ cos θ =radic1 + t2 sin x

(24)sinh τ cosh ξ = t cosh ρ

sinh τ sinh ξ sin θ = t sinh ρ

ϕ = ϕ

which gives a metric

ds2 = minusdτ 2 + sinh2 τdH23

(25)dH2

3 = dξ2 + sinh2 ξ(dθ2 + sin2 θdϕ2)

We display the two coordinate systems for dS space in Fig 1

Radiation domination We will also need the metric for a radiation dominated (RD)universe The general SO(3 1) invariant Friedmann-Robertson-Walker (FRW) metric isgiven by

ds2 = minusdτ 2 + a2(τ)dH23 (26)

5

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

will assume the pressures are such that the domain wall is accelerating away from the centerof our bubble at late times This second assumption is valid for collisions between two dSbubbles if ours has a smaller cosmological constant1

Because of the difficulty in finding analytic solutions to the full problem of a scalarfield with a non-linear potential coupled to gravity we will make a number of simplifyingassumptions We enumerate them here

bull Unlike in [15] we ignore gravitational backreaction from the shell of radiation releasedinto our bubble by the collision and focus instead on the effects on the inflatonrsquosevolution and reheating

bull We approximate the inflaton potential as linear when computing its evolution insideour bubble

bull We allow the fieldrsquos derivative to be discontinuous across the edge of the lightcone ofthe collision to model the effects of non-linearities

bull We treat the wall separating our bubble from the false vacuum and the domain wallbetween our bubble and the other bubble as Dirichlet (fixed value) boundary surfacesfor the inflaton

bull We treat the walls and the edges of the collision lightcone (where the field derivativeschange) as thin

bull We assume inflationary perturbations are unaffected by the collision

bull We model the effects of the collision on the CMB by computing the redshift to thereheating surface as a function of angle rather than evolving the fluctuations in themodified spacetime

We will discuss some of these assumptions in more detail later

21 The Bubble Collision

In this section we will review the bubble collision spacetime Readers familiar with thedetails of [15] may wish to skip ahead to Sec 22

A single Coleman-de Luccia bubble in four dimensions has an SO(3 1) invariance (inher-ited from the SO(4) symmetry of the Euclidean instanton [4]) When another bubble collideswith ours a preferred direction is picked out breaking the symmetry down to SO(2 1) As aresult one can choose coordinates in such a way that the metric takes the form of a warpedproduct between two-dimensional hyperboloids (H2) and a 2D space

1It also occurs when our bubble has a larger cosmological constant or in collisions with AdS bubbles when

certain conditions involving the tension of the wall are satisfied [14 15 16 26]

4

Under the assumption that all the energy in the spacetime remains confined to thinshells or is in the form of vacuum energy one can find the solution describing the collisionby patching together general solutions to Einsteinrsquos equations in vacuum plus cosmologicalconstant with this symmetry [14 15 27] The overall result is that the spacetime inside sucha bubble is divided into two regionsmdashone that is outside the future lightcone of the collisionand can be described using an SO(3 1) invariant metric (a standard open Robertson-Walkercosmology) and one which is inside is affected by backreaction and is most convenientlydescribed using 2D hyperboloids

de Sitter space The general SO(2 1) invariant spacetime with a positive cosmologicalconstant is given by (see [15] for a full discussion of the geometry and causal structure)

ds2 =minusdt2

g(t)+ g(t) dx2 + t2dH2

2 (21)

where

g(t) = 1 +t2

ℓ2minus t0

t (22)

and

dH22 = dρ2 + sinh2 ρdϕ2 (23)

is the metric on the unit 2-hyperboloid and x ≃ x + πℓ We will find for most collisionsthat t0 ≪ t at cosmologically relevant times and thus we approximate t0 asymp 0 giving g(t) asymp1 + t2ℓ2 This is a metric for pure dS space with radius of curvature ℓ Hereafter we setℓ = 1 (it can be easily restored with dimensional analysis) These coordinates (which wedenote the H2 coordinates) are related to the SO(3 1) or H3 invariant ones by

cosh τ =radic1 + t2 cosx

sinh τ sinh ξ cos θ =radic1 + t2 sin x

(24)sinh τ cosh ξ = t cosh ρ

sinh τ sinh ξ sin θ = t sinh ρ

ϕ = ϕ

which gives a metric

ds2 = minusdτ 2 + sinh2 τdH23

(25)dH2

3 = dξ2 + sinh2 ξ(dθ2 + sin2 θdϕ2)

We display the two coordinate systems for dS space in Fig 1

Radiation domination We will also need the metric for a radiation dominated (RD)universe The general SO(3 1) invariant Friedmann-Robertson-Walker (FRW) metric isgiven by

ds2 = minusdτ 2 + a2(τ)dH23 (26)

5

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

Under the assumption that all the energy in the spacetime remains confined to thinshells or is in the form of vacuum energy one can find the solution describing the collisionby patching together general solutions to Einsteinrsquos equations in vacuum plus cosmologicalconstant with this symmetry [14 15 27] The overall result is that the spacetime inside sucha bubble is divided into two regionsmdashone that is outside the future lightcone of the collisionand can be described using an SO(3 1) invariant metric (a standard open Robertson-Walkercosmology) and one which is inside is affected by backreaction and is most convenientlydescribed using 2D hyperboloids

de Sitter space The general SO(2 1) invariant spacetime with a positive cosmologicalconstant is given by (see [15] for a full discussion of the geometry and causal structure)

ds2 =minusdt2

g(t)+ g(t) dx2 + t2dH2

2 (21)

where

g(t) = 1 +t2

ℓ2minus t0

t (22)

and

dH22 = dρ2 + sinh2 ρdϕ2 (23)

is the metric on the unit 2-hyperboloid and x ≃ x + πℓ We will find for most collisionsthat t0 ≪ t at cosmologically relevant times and thus we approximate t0 asymp 0 giving g(t) asymp1 + t2ℓ2 This is a metric for pure dS space with radius of curvature ℓ Hereafter we setℓ = 1 (it can be easily restored with dimensional analysis) These coordinates (which wedenote the H2 coordinates) are related to the SO(3 1) or H3 invariant ones by

cosh τ =radic1 + t2 cosx

sinh τ sinh ξ cos θ =radic1 + t2 sin x

(24)sinh τ cosh ξ = t cosh ρ

sinh τ sinh ξ sin θ = t sinh ρ

ϕ = ϕ

which gives a metric

ds2 = minusdτ 2 + sinh2 τdH23

(25)dH2

3 = dξ2 + sinh2 ξ(dθ2 + sin2 θdϕ2)

We display the two coordinate systems for dS space in Fig 1

Radiation domination We will also need the metric for a radiation dominated (RD)universe The general SO(3 1) invariant Friedmann-Robertson-Walker (FRW) metric isgiven by

ds2 = minusdτ 2 + a2(τ)dH23 (26)

5

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-15 -1 -05 0 05 1 15

02

04

06

08

1

12

14

Figure 1 Plot showing the two coordinates systems for dS space The straight (red) lines are

t x and the curvy (blue) lines are τ ξ The values on the axes are in conformal coordinates

where the vertical axis tanminus1 t runs from 0 to π2 and the horizontal axis x from minusπ2 to

π2 (with ℓ = 1)

The transformation to H2 coordinates is

a(τ) sinh ξ cos θ = z

a(τ) sinh ξ sin θ = t sinh ρ(27)

a(τ) cosh ξ = t cosh ρ

ϕ = ϕ

For radiation domination we have a(τ) =radicCτ with C a constant This gives a metric in

H2 coordinates

ds2 =minus4t4 + (C2 + 4t2)z2

C2(t2 minus z2)dt2 +

C2t2 minus 4z2(t2 minus z2)

C2(t2 minus z2)dz2

(28)+

2tz

C2(t2 minus z2)[4(t2 minus z2)minus C2]dtdz + t2dH2

2

The radiation dominated part of the spacetime will be affected by the domain wall and thecollision Because of the reduced symmetry (relative to de Sitter) we are currently unableto write down the most general radiation dominated solution with SO(2 1) invariance2

However we expect that at least in a broad class of scenarios the effects on the CMB due tobackreaction will be subleading to the other effects we will study and so we will use this formfor now It would be interesting to consider these metric effects in the future for exampleby solving the equations numerically Such a study is currently in progress [28]

22 The Reheating Surface

With these building blocks we can construct our model Our universe appears as a de Sitterbubble that collides with another bubble forming a domain wall along the interface After

2The problem is equivalent to finding the smooth metric describing a Schwarzschild black hole embedded

in an asymptotically FRW spacetime

6

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

Bubble Walls

Surfac

eReh

eatin

g

c

Radiation Shock

θ

dS

Observer

RD

t

Dom

ain Wall

Figure 2 Sketch of the collision scenario Due to the domain wall the shape of the reheating

surface deviates from that of the no collision case (displayed in the long dashed line)

the initial period of inflation the universe reheats and transitions to radiation dominationWe model this transition as sharp matching the metrics and Hubble constant across thereheating surface We ignore matter domination in calculating the effects of the collision onthe CMB and further assume that the effects on the surface of last scattering are relatedto the effects on the reheating surface simply by the redshift along ldquofree-streamingrdquo nullgeodesics In other words we assume the angular dependence of the redshift to the lastscattering surface due to the effects of the collision is given by the angular dependence of theredshift to the modified reheating surface Ignoring the evolution of perturbations betweenreheating and last scattering is a valid approximation on angular scales larger than thatof the first acoustic peak in the CMB (about one degree) On smaller scales there will bea ldquosmearingrdquo due to both this evolution and to the finite thickness of the last scatteringsurface Due to these effects we will focus on disks considerably larger than this size Wewill also ignore the effects of the collision on the perturbations generated during inflationWhile this should be justified when the overall effects are small enough to be consistent withobservation it would be interesting to study this more carefully To summarize in our setupthe main effect is to modulate the standard inflationary perturbations at last scattering bya function dependent on the angle to the collision direction We sketch the scenario in Fig2

We work in the thin wall approximation throughout treating the transition between thetwo parts of the sky (corresponding to the regions of last scattering which are inside andoutside the collision lightcone) as sharp It would be interesting to extend the present workto thick wall systems as effects such as the finite thickness of the domain walls and the shellof radiation emitted from the collision point could play interesting roles

In a standard Robertson-Walker open universe reheating occurs on a surface of almostconstant density that is SO(3 1) invariant (ie hyperbolic 3-space) During inflation the

7

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

V( )φ

(b)(a)

Figure 3 The scalar field transitions from some vacuum to ours along both the domain and

bubble walls It is possible for the field to transition from the same direction along both (a)

or from different directions for each (b)

surfaces of constant energy density are also surfaces of constant value for the inflaton fieldand the reheating surface corresponds to the point in the inflaton potential where slow rollcomes to an end This surface defines a preferred reference frame since the radiation andmatter fluids will be at rest on average with respect to it In our scenario the collisionand domain wall break this symmetry There is no longer a preferred frame and thus thestandard notion of ldquocomovingrdquo no longer holds There remains a remnant of this symmetrya two-dimensional set of comoving observers related to each other by transformations whichleave the two-dimensional hyperboloids invariant Observers outside this plane are no longerequivalent

We define the reheating surface in our scenario as a surface of constant scalar fieldφ(x t) = φ0 The presence of the domain wall will alter the shape of this surface in anon-trivial way breaking isotropy To fully solve this problem we would need to solve thenon-linear coupled scalar field and gravity system for the collision spacetime Instead wewill make an approximation We expect the scalar field to have a set of values φL in theother bubble some other set of values φR in our bubble and a third set of values in themetastable vacuum outside both bubbles φM In the thin wall limit the scalar field jumpsfrom near some other local minimum (either the false vacuum our bubble tunneled from orthe vacuum inside the bubble it collided with) to a location a bit up the ldquohillrdquo from our localminimum This transition occurs all along the domain and bubble walls so the scalar fieldis constant along each of these surfaces

Without loss of generality we choose φ = 0 along the bubble walls At the domain wallwe have two options The field could transition at the collision and domain wall from thesame direction in field space as the tunneling that initially created our bubble Fig 3(a) Inthis case to a good approximation we have φ = 0 along the domain wall as well Howeverit is also possible for the field to transition from the other side of the potential Fig 3(b) inwhich case we have φ = k 6= 0 along the domain wall3

3In the string theory landscape we expect the scalar field space to be multi-dimensional in which case

8

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

The simplest potential for φ which allows it to roll inside our bubble is linear V (φ) = microφwhich can be thought of as the first term in an expansion around the exact potential on theinflationary slope Our first task is to solve for the surfaces of constant φ in this geometryand with these boundary conditions

23 Observer Horizons

Before proceeding it is instructive to determine how much of the reheating surface is insidethe past lightcone of an observer at some given time We neglect the period of matterdomination which should not strongly affect the conclusions We wish to calculate theradius ξr of the surface defined by the intersection of the observerrsquos past light cone withreheating from which we can determine the corresponding values of the coordinates z and x(see (24) and (27)) The past lightcone of an observer at the origin ξ = 0 consists of a set ofgeodesics which in H3 coordinates remain at constant θ and ϕ In the radiation dominatedspacetime the null geodesic is given by

partτ

partξ=

radicCτ (29)

∆ξ =2radicC∆(

radicτ) (210)

We want to match this up to the dS space At reheating we need to match both the scalefactor a and Hubble constant H of the two metrics This gives

a =radic

Cτr = sinh τr (211)a

a=

1

2τr=

1

tanh τr (212)

where τr is the H3 time of reheating in the radiation dominated spacetime and τr is thesame for the dS spacetime (we could make the coordinate τ continuous at the reheatingwith a shift but it will be more convenient not to) We will use tildes for the H3 dS timethroughout this subsection the coordinates on the H3 itself are continuous The solutionsare

τr =1

2tanh τr

C = 2 sinh τr cosh τr (213)

Using the fact that temperature redshifts as 1a

eNlowast equiv TrTn = anar =radic

τnτr (214)

where n subscripts refer to the observerrsquos time and this defines Nlowast

Hence τn = e2Nlowastτr This gives

ξr =(eNlowast minus 1)

cosh τrasymp 2eNlowastminusτr = 2eNlowastminusN (215)

the field could come in from the ldquosiderdquo It would be interesting to investigate how this affects our results

9

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

where N is the number of efolds of inflation and we have used NNlowast ≫ 1 In this caseξr ≪ 1 and the flatness problem is solved (because ξ sim 1 is one curvature radius of the H3)Hence Nlowast is roughly the number of efolds required to solve the flatness problem A simplecalculation demonstrates that this same condition is required to solve the horizon problem

In [15] we calculated the location of the intersection of the future lightcone of the collisionwith the reheating surface and found that the parameter t0 can be roughly as large as t3c where tc is the time of collision and t0 is the backreaction parameter appearing in (21)We found that tc could be exponentially large and still have the collision within the ξ = 0observerrsquos past lightcone This was done in a toy model by patching a curvature dominatedopen universe onto the dS space at reheating One can see from the calculation above thatpatching on radiation domination rather than flat space dramatically changes this estimateand that tc cannot be exponentially large and still be visible to an observer at ξ = 0 Thisjustifies taking t0 sim 0 as we have done here

Because these collisions break part of the isometry group of open Robertson-Walkercosmology they define a preferred ldquocenter of massrdquo frame in which the two bubbles nucleatesimultaneously In our coordinates observers at rest in this frame move on geodesics ofconstant z (or x) The intersection of this set of observers with the set comoving insidethe bubble cosmology are geodesics with z = ξ = 0 spanned by the H2 In the analysisabove we focused on such a ldquocenter of massrdquo (COM) observer Observers who are not at restin this frame (which we will refer to as ldquoboosted observersrdquo) including those who formedfrom matter comoving with respect to the unperturbed part of the reheating surface will seeeffects which depend on their boost relative to the COM observers Those boosted towardsthe collision can see bubbles that nucleated farther away in the COM frame (and hencecollided with ours later) For a given scenario observers boosted towards the collision willsee effects enhanced relative to the COM observer (essentially because of Doppler shift) Theopposite applies to observers boosted away

We will discuss this in more detail in Sections 42 and 5

3 Finding The Reheating Surface

In this section wersquoll solve for the geometry of the reheating surface The computation involvesseveral steps First we will outline the geometry causal structure and domain wall trajectoryin our bubble These will determine the boundary conditions on the scalar field We thensolve the scalar field equation for a linear potential with boundary conditions that it isconstant on the bubble and domain walls Armed with the solution for φ we can find thereheating surface by solving φ(z t) = φ0 where φ0 is the field value where the field ceasesto slowly roll and reheating begins

10

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

31 The Domain Wall Trajectory

The trajectory of the domain wall in x t (recall that the wall is homogeneous on the H2)was found in [15]

x2 =1

g(t)

(

minus1 +t2

g(t)

)

(31)

where here a dot denotes differentiation with respect to the proper time s along the wall Atlate times the solution for t(s) is

t(s) = T0eλs (32)

T0 a constant If we assume the other bubble colliding with us is also a dS bubble withradius of curvature ℓ2 then

λ2 = 1 +1

4κ2

(

κ2 +1

ℓ22minus 1

)2

(33)

(in units of the radius of curvature of our bubble) At late times we can use g(t) asymp t2 andplugging in the solution for t(s) we find4

x(t) =

radic

1minus 1

λ2

(

1

tminus 1

tc

)

+ xc equiv α

(

1

tminus 1

tc

)

+ xc (34)

where (tc xc) are the coordinates of the collision 0 lt α lt 1 is related to the properacceleration of the wall with α = 0 corresponding to no acceleration and α = 1 to infinite(so that the trajectory with α = 1 is null)

32 Null Rays And The Structure Of H2 dS Space

In this subsection wersquoll set up the geometry and boundary conditions wersquoll need to solve thescalar field equation in the collision background For dS space the null rays are given by

dx

dt= plusmn1

grArr x(t) = plusmn tanminus1 t+ k (35)

where k is an integration constant There are three null surfaces important to us thetwo walls of the initial bubble and the lightcone of the collision We choose to orient ourcoordinates such that our initial bubble is nucleated at (t x) = (0 0) Running through eachline we find

right wall rarr x = tanminus1 t

left wall rarr x = minus tanminus1 t (36)

radiation line rarr x = tanminus1 tminus 2 tanminus1 tc

and the collision point is at (tc xc) = (tcminus tanminus1 tc)

4It is straightforward to generalize this to a collision where the other bubble is AdS results will be similar

11

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

simsimx=tanminus1t minus1t+π2

t c simsim 2tcx=minus2 tanminus1 +tanminus1t minus1t+ minusπ2

simsimx=minustanminus1t 1tminusπ2

t cx=minustanminus1

+1tcsimsim minusπ2

(t x )c c

II

I

Figure 4 Sketch of the collision scenario with the null lines labeled and approximate solutions

for late times

Wersquoll also need these rays for large t We can take the asymptotic expansions of (36) toget5

right wall rarr x = minus1

t+

π

2

left wall rarr x =1

tminus π

2 (37)

radiation line rarr x = minus1

t+

2

tcminus π

2

and the collision point is at (tc xc) = (tc1tcminus π

2) The geometry and causal structure are

summarized in Fig 4 Note that region I is outside the lightcone of the collision and thegeometry can effectively be described in H3 coordinates up to the radiation line

33 General Solutions For The Scalar Field

In this subsection wersquoll find the general solutions to the scalar field equation (with a linearpotential) in the regions before and after the collision Letrsquos begin by looking at region IHere we can make use of the H3 coordinates We have

φ = minus 1

sinh3 τpartτ

(

sinh3 τpartτφ(τ))

= micro (38)

where micro is the coefficient of the linear term in the potential We require that φ and itsderivative vanish along the bubble walls at τ = 0 The general solution satisfying these

5We have here also expanded for large tc for smaller values of tc we can convert back by exchanging

1tc harr minus tanminus1 tc + π2

12

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

boundary conditions is

φ(τ) =micro

3

[

ln4eτ

(1 + eτ )2minus 1

2tanh2 (τ2)

]

(39)

At large and small τ this becomes

φ(τ ≫ 1) asymp minusmicro

3ln sinh τ asymp minusmicro

3ln(t cosx) (310)

φ(τ ≪ 1) asymp minusmicroτ 2

8asymp minusmicro

4

(radic1 + t2 cosxminus 1

)

(311)

where we have used the coordinate transformations (24)

In region II we solve for φ using the H2 coordinates as it is in these coordinates that theboundary conditions on the domain wall and collision lightcone are simple Since the fieldis constant on the H2 we have

φ = minus 1

t2partt[

t2(1 + t2)parttφ]

+1

1 + t2part2xφ = micro (312)

The general solution is

φ(t x) = f(xminus tanminus1 t)minus 1

tf prime(xminus tanminus1 t) + g(x+ tanminus1 t) +

1

tgprime(x+ tanminus1 t)

minus micro

6ln(1 + t2) (313)

where f and g are arbitrary functions and the primes denote derivatives with respect to theargument of the respective function At large t this is given by

φ(t x) asymp f(x+ 1t)minus 1

tf prime(x+ 1t) + g(xminus 1t) +

1

tgprime(xminus 1t)minus micro

3ln t (314)

34 Matching Solutions

We have the general solution for the scalar field in the H2 symmetric background as wellas the solution in region I (310) before the collision To find the particular solution forφ in region II we make use of two boundary conditions the field must be continuous atthe radiation line and it must equal a constant at the domain wall φ|domain = k Sincereheating occurs after inflation we can use the late time approximations to the generalsolutions Carrying out these steps is somewhat technical the details are in appendix A

In Fig 5 we draw an example with Nlowast = 60 tc = 1 α = 03 micro = 1 and k = 0 Thepresence of the domain wall causes the surfaces to ldquocurl uprdquo eventually going null and thentimelike That part of the spacetime is very strongly perturbed by the collisionmdashinflationand reheating will not occur there and the linear techniques used in this paper do not applySince these regions are highly perturbed away from a standard cosmology the effects wouldbe in drastic conflict with current observations and thus we will assume that this region iswell outside our horizon

13

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-15 -1 -05 0 05 1 15

08

1

12

14

Figure 5 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 tc = 1 micro = 1

and k = 0 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall This plot is in the same coordinates as

Fig 1

-15 -1 -05 0 05 1 15

06

08

1

12

14

Figure 6 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 03 k = 0 micro = 1

and tc = 3 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

14

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-15 -125 -1 -075 -05 -025 0 025

12

13

14

15

Figure 7 Plot of surfaces of constant φ at late times for Nlowast = 60 α = 05 tc = 1 micro = 1

and k = minus1 The blue lines are the surfaces if no collision had occurred while the red lines

are the actual surfaces Black lines are the bubble walls (projected forward as if there were

no collision) and the green line is the domain wall

We can make some other general statements Increasing the collision time tc pushes thedifference between the actual surface and that of no collision farther out in x In Fig 6 wedraw the same scenario as in Fig 5 except with tc = 3 The smaller α is the more rapidlythe surfaces curl up Taking α rarr 1 in the case k = 0 removes the effects of the collisionentirely (which is a good check of our numerics)

For k lt 0 we find that the reheating surface bends down relative to the surface of theno collision scenario In Fig 7 we draw a case with Nlowast = 60 tc = 1 α = 05 k = minus1 andmicro = 1 Increasing tc causes less of the sky to be covered by the collision while the smaller αgets the more pronounced and closer to the observer the effects are Taking k gt 0 is similarto the scenarios with k = 0 except that the repulsion from the domain wall will be increased

4 The Redshift

In this section we will use our solution for the modified reheating surface to find the redshiftfrom it to observers in the bubble Using the results of Sec 23 we find the projectionof the observerrsquos past lightcone on the reheating surface and calculate the redshift as afunction of angle taking into account the doppler shift due to the fact that the reheatingsurface is not comoving We will again begin by considering COM (z = 0) observersand later generalize to boosted observers In doing this analysis we will ignore the effectsof gravitational backreaction from the collision and modified reheating surface as well asthe modification to a(τ) from spatial curvature These approximations are justified whenN ≫ 1 at least for observers away from the part of the spacetime that is strongly modified

15

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

41 Null Geodesics In Radiation Dominated Spacetime

Starting from the results of Sec 23 we can express the geodesics in H2 coordinates usingthe coordinate transformations (27)

t =

(

C

2(ξr minus ξ) +

radic

Cτr

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z =

(

C

2(ξr minus ξ) +

radic

Cτr

)

sinh ξ cos θ (41)

tanh ρ = tanh ξ sin θ

where for each geodesic θ is a constant and the set of all geodesics is given by θ taking allvalues from 0 to π with

C asymp 1

2e2N τr asymp

1

2 (42)

Since zn = 0 the observerrsquos time tn is given by

tn =radic

Cτn asympradic

1

4e2Ne2Nlowast asymp 1

2eN+Nlowast (43)

Using this we can solve for ξr by setting t(ξ = 0) = tn giving

ξr = 4eminus2N

(

tf minus1

2eN

)

asymp 2eNlowastminusN (44)

Putting this all together we get

t asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

radic

cosh2 ξ minus sinh2 ξ sin2 θ

z asymp 1

2

(

eNlowast+N + eN minus ξ

2e2N

)

sinh ξ cos θ (45)

To compute the redshift we have to take into account the relative motion of the reheatingsurface and observer To do so we need another piece of information about the geodesicsnamely their 4-momenta at both the time of emission and when they intersect the observerWe can again make use of the H3 coordinates Differentiating both sides of the z equationof (27) with respect to the affine parameter we find

z = pz =radicCτ cos θ

(

1

2τsinh ξ pτ + cosh ξ pξ

)

(46)

where pτ = τ and pξ = ξ Since these geodesics have constant θ pξ = gξξpξ is conserved

From (29) pτ = minusradicCτ pξ We can solve for pt in a similar way to get

pz = minus Cz

2(t2 minus z2)32pξ +

t cosh ρ

t2 minus z2pξ cos θ

(47)pt = minus Ct

2(t2 minus z2)32pξ +

z cosh ρ

t2 minus z2pξ cos θ

16

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

42 Boosted Observers

So far we have considered the special case of observers at rest in the frame in which thebubbles nucleated simultaneously In our coordinates these COM observers are those in thez = 0 plane and can always be taken to be at the origin of the H2 (ρ = 0) by a shift in theorigin of coordinates

An observer comoving in the open universe inside the bubble is not in general at restin this frame But since the boosts ldquoparallelrdquo to the domain wall are trivial (ie one canalways choose coordinates so any comoving geodesic lies in the t z plane and at the originof the H2) the most general comoving observer is related to an observer at z = 0 due tothe H3 isometry by a ldquoboostrdquo in the z direction only The isometry is

τ prime = τ (48)

cosh ξprime = γ (cosh ξ minus β sinh ξ cos θ)

sinh ξprime cos θprime = γ (sinh ξ cos θ minus β cosh ξ)

where γ = (1 minus β2)minus12 is a ldquoboostrdquo around the nucleation point of our bubble β gt 0corresponds to an observer closer to the collision (boosted to the left in our diagrams) InH2 coordinates this becomes

tprime cosh ρprime = γ (t cosh ρminus βz) (49)

zprime = γ (z minus βt cosh ρ)

tprime2 = t2 minus z2 + zprime2

It is clear that a boosted observer will not see the same redshift pattern from the collisionin the CMB as a COM observer even at equal comoving time To compute this effect we needto repeat the calculation we have just done only now for an observer following the boostedtrajectory We can find the boosted lightcone by acting on the lightcone of the z = 0 observerwith the boost (this works since the boost is an isometry of the unperturbed spacetime)For brevity we will not include the details here the calculation is straightforward

One should be careful to distinguish two cases structures whose comoving trajectoriesintersect the reheating surface in the region outside the lightcone of the collision and onlypass into it much later versus those that do not We expect the first class of objectsto be comoving in the unperturbed spacetime (up to peculiar velocities) but the secondclass formed in a part of the spacetime where the reheating surface was perturbed Thesestructures may be locally at rest with respect to the frame defined by the modified reheatingsurface and will possess a coherent ldquopeculiar velocityrdquo with respect to the first class Ouranalysis here is valid only for observers in the first class but in the regions of the spacewhere the effects are small it should be approximately correct

43 Finding The Redshift

Armed with the geodesics and the reheating surface we have all the information we needto compute the redshift as a function of the angle on an observerrsquos sky For a given angle

17

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-1 -05 05 1

1001

1002

1003

Figure 8 Relative redshift (rsr0) versus cos θ for the scenario (tc = 1 α = 05 N minus Nlowast =

5 k = 0 micro = 1) Relative redshift greater than one indicates higher relative temperature

θ ϕ on the observers sky we follow the null geodesic back to where it intersects the modifiedreheating surface We take the dot product of the geodesicrsquos 4-momentum with the unitnormal to the reheating surface at the intersection point and then the dot product withthe tangent vector to the observerrsquos geodesic The ratio of these two quantities gives us theredshift in the approximation we mentioned before

We first calculate the reheating surface and normals from our solution for φ in Sec 3The surface is found by solving φ(t x) = φ0 for t as a function of x and φ0 For a givenpotential V (φ) φ0 determines the number of inflationary e-folds N In the end we have themodified reheating surface for a given set of parameters (NNlowast tc α micro k)

Once we have the intersection point we can compute the vectors normal to the surfaceby taking the exterior derivative of φ(t z) (in the RD part of the spacetime) We normalizethese using the RD metric (28) The tangent 1-form along the observerrsquos trajectory isproportional to dt for an unboosted observer at z = 0 and to dτ sim tdtminuszdz in general Theredshift is given by

rs equivνnνr

=En

Er

=minus(pmicronmicro)nminus(pmicronmicro)r

(410)

where n refers to the observerrsquos position r to the point on the reheating surface and nmicro arethe unit 1-forms at each Recall if there was no collision then the redshift would be

r0 = eminusNlowast (411)

which is also the redshift for any geodesics that originate from parts of the reheating surfacethat lie outside the future lightcone of the collision In Fig 8 we show some results for thescenario (tc = 1 α = 05 N minus Nlowast = 5 k = 0 micro = 1) We plot rsr0 as a function of cos θwhere 0 lt θ lt π is the angle on the sky with θ = π pointing towards the collision (theredshifts are axially symmetric)

We can make some general observations about the redshift The modified reheatingsurface leads to bluer photons when k ge 0 the bluest coming from the angle closest to the

18

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-1 -05 05 1

0825

085

0875

09

0925

095

0975

Figure 9 Relative redshift (rsr0) versus cos θ for the scenario (tc = 105 α = 01 N minusNlowast =

3 k = minus1 micro = 1)

domain wall This makes sense as the reheating surface is closer to the observer than itwould be without the collision However the doppler shift from the motion of the reheatingsurface with respect to the observer also plays a role The effect is primarily a dipole butonly on the range of angles for which null geodesics intersect the modified reheating surfaceOnce we are viewing angles which correspond to parts of the reheating surface outside thefuture lightcone of the collision the redshift is r0 The smaller α is the more pronounced theeffects of the collision are as the domain wall is closer to the observer and accelerating awaymore slowly Raising tc means less of the sky is within the future lightcone of the collisionThus there are a wide range of values for both the maximum blueshift and the angular sizeof the part of the sky affected by the dipole piece If tc is much larger than 1 in Hubble unitsthe COM observer is likely not within the future lightcone of the collision (as we mentionedin Sec 23)

If we look at scenarios where k lt 0 we see that the modified reheating surface will leadto slightly redder photons again the reddest coming from the angle closest to the collisiondirection This is due to the reheating occurring earlier in that direction In Fig 9 wedisplay a typical scenario with (tc = 105 α = 01 N minusNlowast = 3 k = minus1 micro = 1)

Letrsquos see how this effect varies with the number of e-folds N In Fig 10 we show somesample values for a collision with tc = 1 α = 05 k = 0 micro = 1 at angle cos θ = minus09 Weplot it for 1 lt N minus Nlowast lt 11 and draw a best fit line The best fit line at this angle isrsr0 asymp 1 + 077 eNlowastminusN which decreases exponentially in N as expected from inflation Weexpect that the general behavior will go as

rsr0(α tc θ k micro) asymp 1 + ξ(α tc θ k micro)e

NlowastminusN (412)

where 0 lt |ξ| lt 1 is a function of the specifics of the collision and the angle Thus unboostedobservers can detect effects from the collision so long as the number of e-folds of inflation isnot too large

Observers boosted towards the collision will see more of the modified part of the spaceand hence the angular size of the affected spot will be larger On the other hand observers

19

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

2 4 6 8 10

105

11

115

12

125

13

Figure 10 Relative redshift (rsr0) versus N minus Nlowast for the scenario (tc = 1 α = 05 k =

0 micro = 1) at angle cos θ = minus09 The dots are actual values and the line is a best fit line with

equation rsr0 = 1 + 077 eNlowastminusN

boosted away see the opposite (smaller weaker spots) For both the angular dependencewithin the spot is similar in form to the COM observer constant away from the collisionand linear towards the collision It is interesting to note that even long inflation cannotcompletely remove the effects on the skies of some boosted observers since there are alwaysparts of the space near the domain wall However these observers are not well-approximatedby the treatment carried out here since in the limit of large boost and long inflation theyformed from strongly affected parts of the reheating surface Nevertheless it is possible thatsuch observers see a more or less standard cosmology The study of this question is beyondthe reach of this paper

tc larger than of order 1 will take the COM observer outside the future lightcone of thecollision However it is again true that a sufficiently large boost towards the collision willbring the observer back into the collisionrsquos lightcone Hence no matter what the values of tcand N are some set of observers will always be able to see the effects of the collision Howprobable we are to find ourselves in that set is again beyond the scope of this paper but see[29]

5 A Simple Model

The preceding results show that the most natural way to describe the signature of a collisionis in position space with a simple piecewise linear behavior for the redshift as a function ofcos θ However in order to make direct comparisons to previous CMB analyses we will usethis result to explore the consequences for the CMB multipoles Looking at Figures 8 and 9we see that the redshift can be approximately modeled as

r(n) = r0

(

Θ(xT minus x)(M minus 1)

xT + 1(xT minus x) + 1

)

(51)

where r0 is a constant that gives the redshift with no collision and x = cos θ The param-eter M gives the maximum value of the relative blue or redshift due to the collision The

20

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

-1 -05 05 1

2725

272501

272501

272502

Figure 11 An example of the temperature (excluding fluctuations) as a function of angle in

our toy model

parameter xT is the cosine of the angle where the forward light cone of the collision endsie for x gt xT we are looking at the unperturbed reheating surface

Including the density fluctuations at reheating the temperature today is

T (n) = T prime

0r(n)(1 + δ(n)) (52)

where T prime

0 is a constant that sets the overall temperature and δ(n) are the temperatureanisotropies from inflationary density perturbations We will take δ(n) to be Gaussianrandom variables with zero mean We plot an example of the function T prime

0r(n) in Fig 11

To find the CMB multipoles we need the fluctuations of the temperature about its averagevalue This is given by

δT (n)

T0=

T prime

0 r(n)(1 + δ(n))minus T0

T0 (53)

where

T0 =T prime

0

4π

int

d2n r(n) (54)

is the average temperature and wersquove used the fact thatint

d2n δ(n) = 0

Several authors have considered anisotropic CMB spectra where the temperature fluctu-ations are modulated by a function [30 22 31] We can parallel some of the formalism in[30] with some modifications We write

δT (n)

T0

= A(n) + f(1 + w(n))B(n) (55)

where B(n) = δ(n) f = T prime

0T0 and

w(n) = Θ(xT minus x)(xT minus x)(M minus 1)

xT + 1 (56)

21

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

In [30] A(n) is a random variable but with a distribution that differs from that of δ(n)In our case we want to find the expected CMB multipoles with fixed parameters for thecollision ie we will average over density fluctuations while holding fixed the collisionredshift function r(n) Ideally we would use a microscopic model (for example the stringtheory landscape) to compute the probability distribution for the parameters tc and βmdashwhich are determined by the location and time of nucleation of the other bubble and arerandom since the nucleation is a quantum eventmdashand hence a probability distribution forr(n) However without detailed knowledge of the probability measure for collisions weare not currently equipped to carry out this procedure In fact there could be multiplecollisions in the past lightcone of the observer each with an associated time and position forits nucleation It would interesting to study this but here we will proceed to study a singlecollision with fixed values for the parameters

A(n) is given by

A(n) =T prime

0(w(n) + 1)

T0minus 1 (57)

We can expand the fluctuations in multipoles via spherical harmonics

δT (n)

T0=

sum

lm

tlmYlm(n) (58)

For details on our conventions for the spherical harmonics see appendix B We can expandAB and w in multipoles as well

A(n) =sum

lm

almYlm(n)

B(n) =sum

lm

blmYlm(n) (59)

w(n) =sum

lm

wlmYlm(n)

Since we have chosen coordinates such that θ = π corresponds to the collision directionboth w and A have only m = 0 multipoles Other coordinates can be obtained via astraightforward rotation using Euler angles

We are now in a position to write the multipoles of the temperature fluctuations Thecoupling between w(n) and B(n) becomes a convolution in fourier space

tlm = al0δm0 + fblm + fsum

l1l2

Rl1l2lm bl2m (510)

where the coupling matrix is written in terms of Wigner 3j symbols

Rl1l2lm = (minus1)m

radic

(2l + 1)(2l1 + 1)(2l2 + 1)

4π

(

l1 l2 l0 0 0

)(

l1 l2 l0 m minusm

)

wl1 (511)

where wl = wl0

22

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

The functions A(n) and w(n) have non-zero 1-point functions

〈alm〉 = alm(512)〈wl〉 = wl

while

〈blm〉 = 0 (513)

Thus we can quickly compute the multipoles for the one-point function

〈tlm〉 = 〈alm〉 = almδm0 (514)

The only anisotropy we see at this level is a result of the collision itself Consistency withthe observed CMB dipole requires that |M minus 1| 10minus3

A more useful object is the 2-point function We have

〈alowastlmalprimem〉 = δm0alowast

l0alprime0

〈blowastlmblprimem〉 = δllprimeCbbl (515)

〈alowastlmblprimem〉 = alowastlm〈blprimem〉 = 0

Cbbl is the two point function in the absence of a collision Wersquoll discuss this in a bit

Squaring (510) and taking ensemble averages using the above rules we get

〈tlowastlmtlprimem〉 = δm0alowast

l0alprime0 + f 2δllprimeCbbl + f 2

sum

l1

Rl1lprime

lmCbblprime + (l harr lprime) + f 2

sum

ll2lprime1

Rl1l2lm R

lprime1l2

lprimemCbbl2 (516)

This determines the two-point function in terms of Cbbl and the collision redshift function

As a first approximation we can assume that Cbbl comes from primordial fluctuations that

are unaffected by the collision We use a spectrum generated with CMBFAST code [32 33]using concordance cosmology values from WMAP [34 35]

51 Results

In this section we will present our results for a few simulations of our toy model The effectsare predominantly on large angular scales so we focus on the lowest set of l modes We willlook at two examples one where the collision takes up most of the sky and another where itcovers only a small fraction For each scenario we present two figures One will display thepower in each l mode compared to a scenario where no collision occurs the other the powerasymmetry as a function of l between the two hemispheres defined by the collision direction

The Clrsquos are

Cl =1

(2l + 1)

m=lsum

m=minusl

〈tlowastlmtlm〉 (517)

23

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

10 20 30 40

10002

10004

10006

10008

Figure 12 ClC(0)l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

10 20 30 40

0996

0998

1002

1004

1006

1008

Figure 13 C leftl Cright

l vs l for the scenario M minus 1 = 2times 10minus5 xT = 08

Due to the fact that the almrsquos only contribute to the m = 0 mode of each multipole theseterms gain more power from the collision than the modes with m 6= 0 do This is due to thefact that the effects of the collision are azimuthally symmetric

We compute the power asymmetry between the left and right hemispheres using a crudeversion of the Gabor transform method [36] with a top hat window function centered ineither hemisphere There will be effects due to ringing from the edges of the top hat butthese donrsquot greatly affect the lower multipoles

For our first example we choose parameters Mminus1 = 2times10minus5 and xT = 08 The angularradius of the collision is 143 degrees Note that the total temperature difference between thetwo poles is well below the dipole due to the Earthrsquos peculiar motion In Fig 12 we plotClC

(0)l for this scenario As expected since the collision takes up a large portion of the sky

the lowest multipoles are the most affected In Fig 13 we show the hemispherical powerasymmetry

In our second scenario the collision takes up only a small part of the sky and the collisioncauses a redshift within the disk We choose M minus 1 = minus73times 10minus6 and xT = minus099 givingan angular radius of about θT = 180minus cosminus1 xT = 7 With these parameters our sky has acold spot with the coldest point about 20microK cooler than the average and the effect falling off

24

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 14 ClC(0)l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

5 10 15 20 25 30 35

10001

10002

10003

10004

Figure 15 C leftl Cright

l vs l for the scenario M minus 1 = minus73 times 10minus6 xT = minus99

with radius from the center We have chosen the angular radius and the temperature profileto approximate the WMAP cold spot smoothed over a scale of sim 5 [17 18 19 20 21]

In Fig 14 one can see that the power is shifted slightly towards higher multipoles witha first peak in excess power around l = 17 There will be a smaller secondary peak at thefirst harmonic The quadrapole receives only a small boost in power reducing it relative tonearby multipoles

In Fig 15 we show the hemispherical power asymmetry which is nearly identical tothe previous figure (since the effects of the collision are now localized wholly in the lefthemisphere) We see that collisions of this type can lead to measurable hemispherical powerasymmetries

We have presented results for two specific choices of parameters The results for generalparameters all have certain features in common most of the excess power is in the m = 0modes the temperature function is monotonic with angle within the affected disk the powerspectra are affected primarily in the l modes corresponding to the size of the disk and itsharmonics and there is a hemispherical power asymmetry with a magnitude that dependson the size and intensity of the disk

25

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

Non-Gaussianities The effects of the bubble collision induce sizable non-Gaussianitiesin the CMB temperature map However at least for the two examples presented here thenon-Gaussianity is significant only at low l In particular f local

NL is very small and f equilateralNL is

strongly dependent on l it oscillates in sign reaches a maximum near the l corresponding tothe disk size and damps rapidly (with an envelope similar to the plots of the excess power inthe 2-point function) We suspect that an analysis of the WMAP data would not constrainthese models significantly We will mention a better statistical test in the conclusions

6 Conclusions

In this paper we have computed the effects of a cosmological bubble collision on the cosmicmicrowave background using a variety of approximations Previously we focused on a rangeof parameters where gravitational backreaction significantly affects the CMB even outsidethe collision disk [15] However this effect is probably too weak to be observable in a morerealistic model Instead in this paper we concentrate on what we expect to be a strongereffect which arises from the fact that the reheating surfacemdashdefined by a constant value ofthe tunneling scalar fieldmdashis perturbed from the case with no collision Because of this theredshift back to the last scattering surface depends in a distinct and specific way on the anglefrom the collision direction (but not on the azimuthal angle) In particular our calculationsdemonstrate that this has the behavior seen in Fig 11mdashno effect for cos θ gt cos(π minus θT )and linear in cos θ for cos θ lt cos(πminusθT ) (where θT is the angular radius of the affected diskand depends on the collision parameters) This form of anisotropy is seen for all observersin the part of the spacetime where our approximations are valid but different observers willmeasure different values for θT and the temperature gradient inside the disk

To compare to the CMB maps we assume that the inflationary perturbations are unaf-fected by the collision and calculate the temperature perturbation map as well as the twopoint function for fluctuations The deviations in the Clrsquos are primarily at large scales (lowl) and depending on the transition angle affect lrsquos at harmonics of πθT Non-Gaussianitiescan be relatively large depending on the parameters and are best detected using equilateraltriangles

It is possible that this scenario can explain some of the current cosmological anomaliesbut the analysis we have done so far is not detailed enough to provide a ldquosmoking gunrdquoA good statistic for the effect described in this paper would be to pick an origin measurethe average CMB temperature in a disk of some angular size around that origin and thenfind the point and angular radius which maximizes the effect We have not performed thisanalysis but the results would be very interesting

Many open questions remain It should be possible to calculate how the inflationaryperturbations are affected by the collision and include this effect Aside from the CMBthere should be anisotropies in other cosmological observables such as large scale structureAs we mentioned in the introduction one expects coherent peculiar velocities due to thefact that the affected part of the reheating surface is not comoving in the frame of theunaffected region as observed in [25] Other potential signals could be detected using 21

26

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

cm observations CMB polarization [37] and better precision on measurements of spatialcurvature Finally it is important to estimate how many bubble collisions to expect in ourpast lightcone and what the size distribution of disks should be on the sky

Acknowledgements

We thank Hans Kristian Eriksen Ben Freivogel Lam Hui Eugene Lim Alberto NicolisStephen Shenker Lenny Susskind and especially Marc Manera Ignacy Sawicki and RomanScoccimarro for valuable discussions TSL was supported in part by NSF CAREER grantPHY-0645435 and in part by NSF grants PHY-0245068 SC was supported in part by NSFCAREER grant PHY-0449818 and DOE grant DE-FG02-06ER41417 MK is supportedby NSF CAREER grant PHY-0645435

A Solving For The Scalar Field

In this appendix we will describe how to solve for the scalar field in the collision geometryIn region I we already have the solution (39) In region II we have to match our solution tothe region I solution at the radiation line

φII(t x = 2tc minus 1tminus π2) = f(2tc minus π2)minus 1

tf prime(2tc minus π2) + g(minus2t+ 2tc minus π2)

+1

tgprime(minus2t + 2tc minus π2)minus micro

3ln t

(A1)= minusmicro

3ln tminus micro

3ln [cos (minus1t+ 2tc minus π2)]

= φI(t x = 2tc minus 1tminus π2)

This is a differential equation for g We can write u equiv minus2t + 2tc minus π2 and get

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) = minusmicro

3ln [cos (u2 + 1tc minus π4)]

(A2)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

This equation is difficult to solve due to the ln cos term We can use the fact that t ≫ 1 andexpand this term about u asymp 2tc minus π2 We find

g(u) +

(

1

tcminus π

4minus u

2

)

gprime(u) asymp minusmicro

3ln [sin(2tc)]minus

micro

6cot(2tc)

(

uminus 2

tc+

π

2

)

(A3)

minus f(2tc minus π2) +

(

1

tcminus π

4minus u

2

)

f prime(2tc minus π2)

The solution is

g(u) asymp minusf(2tc minus π2)minus micro

3ln sin(2tc)

(A4)

minus(

uminus 2

tc+

π

2

)

(

f prime(2tc minus π2) +micro

3cot(2tc)

)

+ A

(

uminus 2

tc+

π

2

)2

27

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

where A is an integration constant we will determine later

At the domain wall we have

xminus 1

t= (αminus 1)

(

1

tminus 1

tc

)

minus π

2 (A5)

x+1

t=

α + 1

tminus αminus 1

tcminus π

2 (A6)

The condition for the field to be constant along the domain wall becomes

φII(t x = α(1tminus 1tc) + 1tc minus π2 ) =

f

(

α + 1

tminus αminus 1

tcminus π

2

)

minus 1

tf prime

(

α+ 1

tminus αminus 1

tcminus π

2

)

(A7)

+g

(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

+1

tgprime(

(αminus 1)

(

1

tminus 1

tc

)

minus π

2

)

minus micro

3ln t = k

where g is given by (A4) and k is the value of the field along the wall This equation isdifficult to solve in closed form but can be solved using Mathematica We follow a similarprocedure as we did to obtain the solution for g We rewrite everything so that the argumentinside f becomes a dummy variable u In this case u = α+1

tminus αminus1

tcminus π

2 Then we re-express

all t dependence in terms of u This gives a differential equation for f(u) once we utilize(A4) for the correct argument inside of g as a function of something involving u Then onesolves the equation for f(u)

The general solution will have two integration constants one each from the differentialequations for f and g Demanding that f(2tc minus π2) is continuous at the radiation linefixes both constants to lowest order in 1t

φ(t x) asymp Θ(minusxminus 1t+ 2tc minus π2)Θ(xminus αtminus (1minus α)tc + π2)φII(t x)(A8)

+ Θ(x+ 1tminus 2tc + π2)Θ(minusxminus 1t+ π2)Θ(xminus 1t+ π2)φI(t x)

where φII is the solution from Mathematica and φI = minusmicro3ln(t cosx)

B The Spherical Harmonics

The spherical harmonics are given by

Ylm(n) =

radic

(2l + 1)

4π

(l minusm)

(l +m)Plm(cos θ)e

imφ (B1)

where

Plm(x) = (minus1)m(1minus x2)m2 dm

dxmPl(x) (B2)

28

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

are associated Legendre polynomials In particular the spherical harmonics are chosen suchthat

Y lowast

lm = (minus1)mYlminusm(B3)int

d2nYlm(n)Ylowast

lprimemprime = δllprimeδmmprime

References

[1] R Bousso and J Polchinski ldquoQuantization of four-form fluxes and dynamicalneutralization of the cosmological constantrdquo JHEP 06 (2000) 006 hep-th0004134

[2] S Kachru R Kallosh A Linde and S P Trivedi ldquoDe sitter vacua in string theoryrdquoPhys Rev D68 (2003) 046005 hep-th0301240

[3] L Susskind ldquoThe anthropic landscape of string theoryrdquo hep-th0302219

[4] S R Coleman and F De Luccia ldquoGravitational effects on and of vacuum decayrdquoPhys Rev D21 (1980) 3305

[5] J R Gott ldquoCreation of open universes from de sitter spacerdquo Nature 295 (1982)304ndash307

[6] J R Gott and T S Statler ldquoConstraints on the formation of bubble universesrdquoPhys Lett B136 (1984) 157ndash161

[7] J Garriga X Montes M Sasaki and T Tanaka ldquoSpectrum of cosmologicalperturbations in the one-bubble open universerdquo Nucl Phys B551 (1999) 317ndash373astro-ph9811257

[8] A D Linde M Sasaki and T Tanaka ldquoCmb in open inflationrdquo Phys Rev D59

(1999) 123522 astro-ph9901135

[9] B Freivogel M Kleban M Rodriguez Martinez and L Susskind ldquoObservationalconsequences of a landscaperdquo JHEP 03 (2006) 039 hep-th0505232

[10] P Batra and M Kleban ldquoTransitions between de sitter minimardquo Phys Rev D76

(2007) 103510 hep-th0612083

[11] A H Guth and E J Weinberg ldquoCould the universe have recovered from a slow firstorder phase transitionrdquo Nucl Phys B212 (1983) 321

[12] J Garriga A H Guth and A Vilenkin ldquoEternal inflation bubble collisions and thepersistence of memoryrdquo hep-th0612242

[13] A Aguirre M C Johnson and A Shomer ldquoTowards observable signatures of otherbubble universesrdquo Phys Rev D76 (2007) 063509 arXiv07043473 [hep-th]

29

[14] B Freivogel G T Horowitz and S Shenker ldquoColliding with a crunching bubblerdquoJHEP 05 (2007) 090 hep-th0703146

[15] S Chang M Kleban and T S Levi ldquoWhen Worlds Colliderdquo JCAP 0804 (2008)034 07122261

[16] A Aguirre and M C Johnson ldquoTowards observable signatures of other bubbleuniverses II Exact solutions for thin-wall bubble collisionsrdquo Phys Rev D77 (2008)123536 07123038

[17] P Vielva E Martinez-Gonzalez R B Barreiro J L Sanz and L Cayon ldquoDetectionof non-Gaussianity in the WMAP 1-year data using spherical waveletsrdquo Astrophys J

609 (2004) 22ndash34 astro-ph0310273

[18] M Cruz E Martinez-Gonzalez P Vielva and L Cayon ldquoDetection of a non-GaussianSpot in WMAPrdquo Mon Not Roy Astron Soc 356 (2005) 29ndash40 astro-ph0405341

[19] M Cruz M Tucci E Martinez-Gonzalez and P Vielva ldquoThe non-Gaussian ColdSpot in WMAP significance morphology and foreground contributionrdquo Mon Not

Roy Astron Soc 369 (2006) 57ndash67 astro-ph0601427

[20] L Cayon J Jin and A Treaster ldquoHigher Criticism Statistic Detecting andIdentifying Non- Gaussianity in the WMAP First Year Datardquo Mon Not Roy Astron

Soc 362 (2005) 826ndash832 astro-ph0507246

[21] M Cruz L Cayon E Martinez-Gonzalez P Vielva and J Jin ldquoThe non-GaussianCold Spot in the 3-year WMAP datardquo Astrophys J 655 (2007) 11ndash20astro-ph0603859

[22] H K Eriksen A J Banday K M Gorski F K Hansen and P B LiljeldquoHemispherical power asymmetry in the three-year Wilkinson Microwave AnisotropyProbe sky mapsrdquo Astrophys J 660 (2007) L81ndashL84 astro-ph0701089

[23] J D McEwen M P Hobson A N Lasenby and D J Mortlock ldquoA high-significancedetection of non-Gaussianity in the WMAP 5-year data using directional sphericalwaveletsrdquo 08032157

[24] K Land and J Magueijo ldquoThe axis of evilrdquo Phys Rev Lett 95 (2005) 071301astro-ph0502237

[25] A Kashlinsky F Atrio-Barandela D Kocevski and H Ebeling ldquoA measurement oflarge-scale peculiar velocities of clusters of galaxies results and cosmologicalimplicationsrdquo 08093734

[26] M Cvetic and H H Soleng ldquoSupergravity domain wallsrdquo Phys Rept 282 (1997)159ndash223 hep-th9604090

[27] S W Hawking I G Moss and J M Stewart ldquoBubble collisions in the very earlyuniverserdquo Phys Rev D26 (1982) 2681

30

[28] M Kleban A MacFadyen R Rahman and W Zhang ldquoTo appearrdquo

[29] B Freivogel M Kleban A Nicolis and K Sigurdson ldquoTo appearrdquo

[30] C Gordon W Hu D Huterer and T M Crawford ldquoSpontaneous Isotropy BreakingA Mechanism for CMB Multipole Alignmentsrdquo Phys Rev D72 (2005) 103002astro-ph0509301

[31] A L Erickcek M Kamionkowski and S M Carroll ldquoA Hemispherical PowerAsymmetry from Inflationrdquo 08060377

[32] U Seljak and M Zaldarriaga ldquoA Line of Sight Approach to Cosmic MicrowaveBackground Anisotropiesrdquo Astrophys J 469 (1996) 437ndash444 astro-ph9603033

[33] M Zaldarriaga U Seljak and E Bertschinger ldquoIntegral Solution for the MicrowaveBackground Anisotropies in Non-flat Universesrdquo astro-ph9704265

[34] WMAP Collaboration G Hinshaw et al ldquoFive-Year Wilkinson MicrowaveAnisotropy Probe (WMAP ) ObservationsData Processing Sky Maps amp BasicResultsrdquo 08030732

[35] WMAP Collaboration M R Nolta et al ldquoFive-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations Angular Power Spectrardquo 08030593

[36] F K Hansen K M Gorski and E Hivon ldquoGabor Transforms on the Sphere withApplications to CMB Power Spectrum Estimationrdquo Mon Not Roy Astron Soc 336

(2002) 1304 astro-ph0207464

[37] C Dvorkin H V Peiris and W Hu ldquoTestable polarization predictions for models ofCMB isotropy anomaliesrdquo Phys Rev D77 (2008) 063008 07112321

31