Anthony Aguirre, Matthew C Johnson and Assaf Shomer- Towards observable signatures of other bubble universes

8/3/2019 Anthony Aguirre, Matthew C Johnson and Assaf Shomer- Towards observable signatures of other bubble universes

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arXiv:0704

.3473v3

[hep-th]

25Jul2007

Towards observable signatures of other bubble universes

Anthony Aguirre, Matthew C Johnson, and Assaf Shomer

SCIPP, University of California, Santa Cruz, CA 95064, USA

(Dated: February 5, 2008)

We evaluate the possibility of observable effects arising from collisions between vacuum bubblesin a universe undergoing false-vacuum eternal inflation. Contrary to conventional wisdom, we findthat under certain assumptions most positions inside a bubble should have access to a large num-

ber of collision events. We calculate the expected number and angular size distribution of suchcollisions on an observers sky, finding that for typical observers the distribution is anisotropicand includes many bubbles, each of which will affect the majority of the observers sky. After aqualitative discussion of the physics involved in collisions between arbitrary bubbles, we evaluatethe implications of our results, and outline possible detectable effects. In an optimistic sense, then,the present paper constitutes a first step in an assessment of the possible effects of other bubbleuniverses on the cosmic microwave background and other observables.

I. INTRODUCTION

Cosmological inflation never ends globally when drivenby an inflaton potential with long-lived metastable min-ima. This was discovered in the very first models of in-

flation as a failure of true (lower) vacuum bubbles in afalse vacuum background to percolate [1]. It was laterrecognized as a special case of eternal inflation in whichour observable universe would lie within a single nucle-ated bubble [2] while inflation continues forever outsideof this bubble (e.g., [1, 3]).

While important for any sufficiently complicated in-flaton potential, this issue has become prominent latelywith the realization that stabilized string theory com-pactifications appear to correspond to minima of a many-dimensional effective potential landscape [4, 5] thatwould drive just this sort of eternal inflation and thuscreate pocket or bubble universes with diverse prop-

erties. This has raised a number of very thorny questionsregarding which properties to compare to our local ob-servations (e.g. [6, 7]), as well as debates as to whetherthese other universes have any meaning if they are un-observable, as is the conventional wisdom.

But what if they are observable, so that the processesresponsible for eternal inflation can be directly probed?What is the chance we could actually see such bubbles,and how would they look on the sky? These are thequestions that the present paper begins to explore.

It would seem that for us to observe bubble collisionsin our past, three basic and successive criteria must bemet:

1. Compatibility: A bubble collision must allow stan-dard cosmological evolution including inflation andreheating and hence be potentially compatiblewith known observations in at least part of itsfuture lightcone.

Electronic address: [email protected] address: [email protected] address: [email protected]

2. Probability: Within a given observation bubble(seen as a negatively-curved Friedmann-Robertson-Walker (FRW) model by its denizens) a randomlychosen point in space should have a significantprobability of having (compatible) bubbles to its

past.3. Observability: The effects of compatible bubbles to

the past must not be diluted away by inflation intounobservability, nor affect a negligible area of theobservers sky.

Although a rigorous analysis of these issues does notyet exist, several recent studies suggest in contrast toprevious thinking that it is actually plausible that thesethree criteria may be met.

First, studies of bubble collisions boosted so thatone bubble forms much earlier than the other indi-cate that the older bubble may see the younger bub-

ble as a small perturbation that does not disrupt itsoverall structure [8], even if the younger bubble con-tains a big-crunch singularity [9]. Second, straightfor-ward arguments (see below), inspired by the results ofGarriga, Guth & Vilenkin [10] (hereafter GGV), indi-cate that a random position in the FRW space withina bubble should (with probability one) have a bubblenucleation event to its past. Third, in a complex in-flaton potential with many minima, the number of e-foldings within a randomly chosen bubble can becomea random variable with some probability distribution.Suppose that this distribution favors a small number ofe-foldings, and yet either to match our observations orfor anthropic reasons we focus only on the subsetof bubbles with > Nmin 50 60 e-foldings. Then wemight expect that our region underwent close to Nmine-foldings [11, 12]. Thus it is plausible that just enoughinflationary e-foldings occurred to explain the largenessand approximate flatness of the universe; and since theCMB perturbations on the largest scales formed Nmine-foldings before the end of inflation, perturbations atthe beginning of inflation may then be detectable.

None of these studies have actually addressed whetherbubble collisions might be observable, however, and leave
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4

0

wallX4H

1 1H X4F T X

X0X

FIG. 1: On the left is the embedding of two dS spaces ofdifferent vacuum energy in 5-D Minkowski space (three di-mensions suppressed). The construction obtained by match-ing these two hyperboloids along a plane of constant X4, asshown on the right, corresponds to the one-bubble spacetimeshown in Fig. 2 in the limit where the bubble interior is puredS. The light shaded (green) region represents the false vac-uum exterior spacetime, while the dark shaded (blue) regionrepresents the interior spacetime.

(r ,t )

o

Observation Bubble

o

n n

( , )

Initial Conditions

Begin Inflation

End Inflation

Reheating

Present

Past Light Cone

= 0

T=

T= /2

/2

T

=

co

FIG. 2: The conformal diagram for a bubble universe. Weimagine an observer at some position (o, o, o) inside of theobservation bubble, which is assumed to nucleate at t = 0and expand at the speed of light. The foliation of the bubbleinterior into constant density, negative curvature, hyperbolicslices is indicated by the solid lines. These spacelike slicesdenote epochs of cosmological evolution in the open FRWcosmology inside of the bubble. The past light cone of the ob-server is indicated by the dashed lines. There is a postulatedno-bubble surface at some time to the past of the nucleationof the observation bubble. Also shown is a ( = n, = nslice of a) colliding bubble that nucleated at some position(tn, rn, n, n), and intersects the bubble wall within the pastlight cone of the observer.

correspond to surfaces of constant- that, by the homo-geneity of the metric Eq. 4, are also surfaces of constantcurvature and density. These slices correspond to thevarious epochs of cosmological evolution inside of thebubble: the beginning of inflation (near the tunnelled-tofield value), the end of inflation at the failure of slow-roll, reheating, the recombination epoch, etc., up until

the present time.3

If the nucleation rate (per unit physical 4-volume) oftrue-vacuum bubbles is small compared to H4F, the ob-servation bubble will be one of infinitely many that formas part of what either is or approaches a steady-statebubble distribution wherein there is a foliation of thebackground dS in which the bubble distribution is statis-tically independent of both position and time (see [17],

and also [3, 18].) An infinite subset of these will actuallycollide with the observation bubble.

If we now assume that our bubble experiences whata typical bubble in the steady-state distribution does,then we can follow the strategy of GGV and considerthe bubble to exist at t = 0, model the background ashaving an initial pure false-vacuum surface at t = t0 (in-dicated in Fig. 2), then send t0 . (By doing this,GGV explicitly showed that there is a preferred framein the model of eternal inflation they treated, which co-incides with comoving observers in the steady-state fo-liation, and is related to the initial false-vacuum surface;observers with different boosts with respect to this frame

see bubble collisions at different rates.)Given an observer at time o and hyperbolic radius oinside the bubble, we can define a two-sphere by the in-tersection of the observers past lightcone (dashed lines inFig. 2) with another equal- surface (i.e. correspondingto a portion of the recombination surface or the bubblewall). The question we now wish to address is: what isthe number of bubbles observed in a given direc-tion (, ) with a given angular size on the two-sphere (the observers sky)? This quantity couldprovide the basis for a calculation of the impact on theobservers CMB of incoming bubbles that distort the re-combination (or reheating, etc.) surface.

In the next section we calculate this quantity underthe following assumptions:

1. We assume that bubbles start at zero radius andexpand at lightspeed at all times. We also assumethat the bubbles do not back-react, i.e. one bubblewill not alter the trajectory of a subsequent bubble.This may be important for directions on the skyhit with multiple bubbles, but requires a carefultreatment of bubble collisions and is reserved forfuture work.

2. We assume that no bubbles form within bubbles,and that there are no transitions from true to false

vacuum. We comment on the implications of in-cluding these features in Sec. IV

3. We assume that structure of the observation bub-ble is unaffected by the incoming bubbles, and that

3 We note that it is difficult to construct inflaton potentials (with-out considerable fine tuning) giving rise to a cosmological evolu-tion inside of the bubble similar to our own.


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the observed equal- surface is at 0, coincidingwith the bubble wall. The first rather strong assumption is discussed below in Sec. IV; the sec-ond should be reasonable insofar as we are hopingto assess the incoming bubbles impact on the firstfew e-foldings of inflation.

Within this setup, let us examine why it is plausible

for a typical observer to have one or more bubble nu-cleations within their past lightcone. Because bubblesexpand as lightcones and nucleate with some rate perunit 4-volume, the expected number of bubbles in anobservers past lightcone is just V4, where V4 is the 4-volume of the exterior spacetime contained in the pastlight cone of the observer, bounded by the initial valuesurface, the bubble wall, and the past light cone of thenucleation site of the observation bubble (which enforcesthe no bubbles-within-bubbles approximation). This 4-volume depends on the position of the observer inside ofthe bubble and the epoch of observation.

Now, the spatial volume in a coordinate interval dgoes as dV3

4 sinh2 d, thus the volume is exponen-

tially weighted towards large . If observers inside of thebubble are uniformly distributed on a given constant-surface, we would expect most of them to exist at large. But as shown by [10], on any constant- surface, the4-volume relevant for bubble nucleation diverges for large as V4 . Thus even for a tiny nucleation rate4 mostobservers have a huge 4-volume to their past and shouldtherefore expect bubbles in their past.5

We now proceed to calculate the distribution of colli-sions on our observers sky. Readers uninterested in thedetails of this calculation can proceed to Sec. IV for asummary of the results.

III. COMPUTATIONS

Consider an observer at coordinates (o, o, o) in theobservation bubble. There is nothing breaking the sym-metry in , so we are free to choose =const.

1. First, we compute the angular scale and directionobs on the sky of the triple-intersection of the ob-servers past lightcone, the bubble wall (the 0surface), and the wall of a bubble nucleated at somepoint in the background spacetime.

2. We then find the differential number (Eq. 1) of bub-bles of angular size in the direction obs by inte-grating the volume element for the exterior space-time over all available nucleation points on a sur-

4 We might expect a typical nucleation rate to be of order eSF , where SF is the entropy of the exterior de Sitter space.

5 If the interior vacuum energy is much lower than the exteriorone, this only increases the 4-volume accessible to the observer.

face of constant and obs and multiplying by thebubble nucleation rate .

Both items can be computed in two different framesthat we shall denote the unboosted and the boostedframes. In the original unboosted frame, where theobserver is at (o, o, o), we compute the locations of

triple-intersections on the 2-sphere of the observers sky,then convert these locations to an observed angle obsand angular scale on the sky (see Sec. IIIA and Ap-pendix A). While this frame is most straightforward, thecalculations are much more tractable using a trick sug-gested by GVV: given the symmetries of dS, a boost inthe embedding space changes none of the physical quanti-ties we are interested in (see below for elaboration). Thuswe can choose a boost such that the observer lies at =0,so that (a) obs coincides with the coordinate angle n atwhich the bubble nucleates, and (b) the bubbles angularscale is just given by the angular coordinate separationof the two triple-intersection points. The cost of this

simplification is that the initial false-vacuum surface isboosted into a more complicated surface. In the resultsto follow, we will employ both the boosted and unboostedviewpoints, but will focus on the boosted frame for thecalculation of the distribution function.

A. Angles according to the unboosted observer

The triple-intersection between the observation bub-

ble, the colliding bubble, and the past light cone of theobserver represent the set of events that form a boundaryto the region on the observers sky affected by the colli-sion. Working in a plane of constant , these will cor-respond to two events, and the angle between geodesicsemanating from these two events and reaching the ob-server at (o, o, o) gives the observed angle on the sky.In the particular case where the bubble interior is dSwith HT = HF, Appendix A gives the explicit solution tothis problem, although a similar (necessarily more com-plicated) procedure can be applied to the more generalcase.

Let us visualize this by focusing now on the inside ofthe observation bubble which (as discussed in Sec. II) isdescribed by an open FRW cosmology. We can use thePoincare disk representation to describe the hyperbolicequal- surfaces in this spacetime. Suppressing one ofthe spatial dimensions, the metric on a spatial slice ofEq. 4 becomes

ds2 = 4a()dz2 + z2d2

(1 z2)2 . (6)


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1

FIG. 3: A time lapse picture of the null rays reaching an ob-server from the boundary of the region affected by a collisionevent in the Poincare disk representation. The boundaries arelocated at angles 1,2 from the center of the disk, and at an-gles 1,2 from the location of the observer. The total angularscale of the collision event as recorded by the observer, whichaffects the region of the disc indicated by the double lines, isgiven by .

Where in terms of the embedding,

X0 = a()1 + z2

1 z2 (7)

X1 = a()2z cos

1 z2X2 = a()

2z sin

1 z2X3 = 0

X4 = const.,

Since there are collision events that disrupt large angular

scales, we find it useful for visualization purposes to letpolar angle assume also negative values < < and limit the range of accordingly. Scaling by a()1

gives the disk unit radius, with z = 1 corresponding tothe wall of the observation bubble, as depicted in Fig. 3.

This figure shows the time-lapse of a collision eventfrom the perspective of an interior observer on thePoincare disk. The angles 1 and 2 are the triple-intersection points. The broken lines from these pointstrace the path of null rays that reach the observer at(o, o, o = 0), where we have used the remaining sym-metry of the problem to place the observer at o = 0.

Analyzing this geometry, the angular position of anintersection from the perspective of an interior observeris given by

cos 1,2 =tanh o cos 1,2

tanh o cos 1,2 1 (8)

Notice that the denominator never vanishes unless (the boundary) where cos = 1, independent of .Using the above results, we conclude that the observerwill see a collision as having an angular scale of

= 1 2 (9)

where one has to take some care choosing the correctbranch of the cosine function in the process of solving for using Eq. 8, see Fig. 3.

Because of the hyperbolic nature of the spatial slices,an observer at large-o can record an angle that is verydifferent from . To examine this limit, transform tothe Euclidean coordinates (z, ) on the disc, and expandEq. 8 near the boundary at z = 1

cos 1,2(z, ) = 1 + 12

cot2(1,2

2)2 + O(3). (10)

Accordingly, any given angle gets mapped to = the closer we approach the boundary ( 0). On theother hand, regardless of how close to the wall we are,there are always small enough angles < that will bemapped by Eq.7 to small hyperbolic angles .

In the first case, choosing the branch of the cosine inEq. 8 determines whether the angular size is or + . Studying a few examples, it is easy to seethat in this limit intersections where 1,2 have oppositesigns get mapped to

2, and intersections where 1,2

have the same sign get mapped to 0.We will see in the following sections that most of the

phase space for bubble nucleation comes from very smallangles obs 0, typically yielding one intersection in theupper half and one in the lower half of the disk. In thisframe, we also expect the angular scale |1 2| to besmall, since the majority of colliding bubbles form at verylate times, and therefore have a tiny asymptotic comov-ing size. All of this information taken together suggeststhat typical collision events will appear to take up eithervery large or very small angular scales on the observerssky, depending on where the observer is situated insideof the bubble.

B. The boosted view

We now go on to discuss the boosted frame. We willagain exploit the symmetry of the problem to positionthe observer at o = 0, and define the following transfor-mation in the embedding space:

X0 = (X0 X1) , (11)X1 = (X1 X0) ,X2,3,4 = X2,3,4.

This is simply a boost in the X1-direction of the embed-ding space, and respects the SO(3,1) symmetry of theone-bubble spacetime, since it is in a direction perpen-dicular to the surface of pasting described in Sec. II.If = cosh o and = tanh o, the observer at o istranslated to the origin. More generally, in terms of theopen coordinates inside of the observation bubble (witharbitrary scale factor), this boost is equivalent to a trans-lation (see Appendix B for an explicit demonstration ofthis).


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Points outside of the observation bubble are also af-fected by the boost. We will be particularly concernedwith the effects on the initial value surface at t0 ,since this determines the available 4-volume to the pastof our observer. The boost will push portions of this ini-tial value surface into regions of the de Sitter manifoldnot covered by the flat slicing coordinates (see Eq. 3).It is therefore useful to employ the third foliation of dS,

into positively curved spatial sections, which cover theentire manifold. Using a conformal time variable, thesecoordinates (T , , , ) are defined by:

X0 = H1F tan T (12)

Xi = H1F

sin

cos Ti

X4 = H1F

cos

cos T,

where /2 T /2 and 0 < < , and the i arethe same as in Eq. 5. This induces the metric

ds2 =1

H2F cos2 TdT

2 + d2 + sin2 d2

2 . (13)

The transformation between the boosted and un-boosted frames in terms of the global coordinates is givenby

tan =sin sin

(sin cos sin T) (14)

tan T =

tan T sin cos

cos T

(15)

cos = cos Tcos

cos T. (16)

We now apply this transformation to the initial valuesurface at t0 . In terms of the embedding coordi-nates, we can define this (null) surface by X0 + X4 = 0(T = /2), which boosts to

X0 + X1 =

X4

. (17)

Substituting with the global coordinates, we arrive at therelation

sin T =

cos

+ sin cos

. (18)

Henceforward we will drop the prime on the boosted co-ordinates unless explicitly noted.

The boosted initial value surface Eq. 18 is a function ofthe coordinate angle, accounting for the dependence onobs of the past 4-volume for an unboosted observer. Thisis displayed for a variety of angles on the dS conformaldiagrams in the upper cell of Fig. 4.

The effects of the boost on a slice of constant (, =0) in the background spacetime is shown in the lowercell of Fig. 4. Even for this rather modest boost (herewe use o = 2), it can be seen that most of the points

observer observer

FIG. 4: The effects of the boost. The top cell shows theboosted initial value surface (at t0 in the unboostedframe) for small (left) and large (right) o for a variety ofangles (with the bottom curve (red) corresponding to =0, the top (yellow) corresponding to = , and other lines

corresponding to intermediate angles at intervals of/4). Thebottom cell shows the effects of the boost on points in theexterior spacetime on a slice of constant (, = 0). Note thateven for this very modest boost (o = 2), most of the pointsare condensed into the wedge created by the past light coneof the nucleation event and the boosted initial value surface.

in the unboosted frame are condensed into the wedgebetween the past light cone of the nucleation event andthe boosted initial value surface.

One may be worried that the presence of collidingbubbles, which break the SO(3,1) symmetry of the one-bubble spacetime, invalidates our procedure. In fact, to

calculate the quantities we are interested in, we only needa consistent description of the spacetime outside of thecolliding bubbles. We assume that the colliding bub-bles are null and since SO(3,1) symmetry transforma-tions keep points inside their light cones, it follows thatthe spacetime outside bubbles is mapped to itself. Whileit may be true that such transformation may e.g. violatecausality inside the colliding bubbles this effect does notaffect the analysis we perform here.

C. Angles according to the boosted observer

We can now calculate the angular scale of a collisionon the boosted observers sky. To do so, we must con-front the non-Euclidean geometry of spatial slices in theglobal coordinates: constant-T slices are 3-spheres of ra-dius 1/HF cos T. We can visualize a timeslice of bubbleevolution by suppressing one dimension, embedding in a3 dimensional Euclidean space, and scaling the spheres tounit radius. The polar angle on this two-sphere is givenby and the azimuthal angle by (recall that we takethe range < < ).


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A bubble wall appears as an evolving circle on theunit 2-sphere. Allowing for arbitrary bubble interiors,and continuing the global coordinate equal time slices(X0 =const. in the embedding) into them, a spatial sliceis not quite a two sphere, but rather a two sphere with di-vets and bumps describing the varying curvature of thespacetime inside of the bubbles. For colliding bubbles,these structures no matter how extreme are irrele-

vant, as we will only employ information about the bub-ble wall.

But the observation bubble requires more care, sincewe are ultimately interested in a description of collisionevents from the perspective of an inside observer. What-ever form the embedding of the bubble interior may take,by symmetry, the bubble wall will be a latitude on thebackground two-sphere. It will have = T (since it nu-cleates at T = 0), and span all from to . ForT < /2 it looks like a circle, with the bubble interiorthe portion of the sphere bounded by this circle. AtT = /2 the circle is a great circle and the bubble ex-terior a hemisphere. If we had chosen a frame in which

the observation bubble was formed at some Tn < 0, thenfor T Tn > /2 the bubble wall would again becomea small circle, with the portion of the sphere boundedby this circle corresponding to the bubble exterior. Byhomogeneity of the space a bubble nucleated elsewherewould appear similarly.

In the spherically symmetric, open FRW coordinatesthat describe the interior of the observation bubble, theboosted observer lies at the origin, which coincides withXi = 0 in the embedding space. Because of the sphericalsymmetry of this metric, radial incoming null rays fromthe bubble wall follow trajectories of constant and ,and the angle on the sky is identical to the angle wewould find if the bubble interior were replaced by a con-tinuation of the background dS. In terms of calculatingthe observed angle, we can therefore largely ignore thehyperbolic geometry of the bubble interior, and visual-ize the collision between the observation bubble and anincoming bubble as the intersection of two circles on theT = const. sphere, as shown in Fig. 5.

In analyzing the geometry it is helpful to performa stereographic projection onto a plane tangent to thenorth pole of the two-sphere ( = 0) as shown in Fig. 5.This projection maps circles on the 2-sphere to circlesin the plane, and also preserves angles since the map isconformal.

Examining the projection, there are three cases to con-sider. Colliding bubbles with an interior that does not cutout the south pole appear as filled circles in the projection(upper-left panel of Fig. 6, where the light (yellow) discrepresents the observation bubble and the dark (blue)disc represents the colliding bubble). On the time slicewhen a bubble wall intersects the south pole, the wallappears as a line in the projection, bisecting the planeinto a region inside, and outside, the bubble (upper-rightpanel of Fig. 6). If the bubble interior cuts out the southpole, it projects to a circle whose interior corresponds to

FIG. 5: A spatial slice in the global foliation of the back-ground de Sitter space, and its stereographic projection. Theobservation bubble is shaded light (yellow) and the collidingbubble is shaded dark (blue). The angle is indicated in theplane of projection.

FIG. 6: The three cases of bubble intersection in the planeof projection. The top left cell displays the case where thebubble interior does not encompass the south pole of the pro-jected two-sphere, the top right cell displays the case wherethe bubble wall intersects the south pole, and the lower celldisplays the case where the bubble interior includes the southpole.

the region outside of the bubble (see the lower panel ofFig. 6).

Now consider a bubble nucleated at arbitrary coordi-

nates (Tn, n, n). Ingoing and outgoing radial null raysfrom the center of this bubble (corresponding to the lo-cation of the bubble wall) obey:

= n (T Tn) n T. (19)

We are interested in the projection of this bubble atthe global time-slice Tco (and bubble coordinate timeco 0) when the observers past lightcone intersectsthe observation bubble wall (see Fig. 2). If we follow the


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past lightcone of the observer we find

=

o

d/a(). (20)

To determine Tco, a valid junction between the interiorand exterior spacetimes requires that the physical radiusof two-spheres (the coefficients of d2 in Eq. 4 and 13)

at the location of the wall match, and gives

Tco = arctan

HF lim

0a() sinh

o

d/a()

. (21)

In the case where the interior is pure dS (wherea() = H1T sinh HT), this works out to Tco =arctan[(HF/HT) tanh(HTo/2)]. As we send o ,it can be seen that this ranges between Tco = /4 forHT = HF and Tco = /2 for HT HF.

Viewed in the projected plane using polar coordinates(, proj), the incoming bubble has a center at = (2 +1)/2, and a radius = (2 1)/2 as shown in theupper left panel of Fig. 6. Then, since the projection ofan arbitrary point gives = 2 tan/2 (this can be seenby analyzing the geometry of Fig. 5), we can work out:

=2sin n

cos n + cos T, =

2sin Tcos n + cos T

. (22)

Finally, on the plane we can find the angle between thetwo radial null rays that come to the observer from thetwo intersection points, which is given by:

cos

2

= cot n cot Tco + cos(Tn Tco)

sin n sin Tco. (23)

At = = 0, observers at rest in the open and closed co-ordinates are in the same frame, so is the actual angularscale on the sky of the bubbles sphere of influence, asseen by the observer.

We can now foliate the background spacetime into sur-faces of constant , as shown in Fig. 7. From the symme-tries of the boosted frame, this foliation is independent of and (although the angular dependence of the boostedinitial value surface will play an important role in definingthe statistical distribution of collisions). This provides amap between the nucleation site of a colliding bubble andthe observed angular scale of the collision. The numberof collisions of a given angular scale can be found by ex-

amining how the exterior four-volume is distributed inthe causal past of the observer.

In the o limit, there is a divergent 4-volume con-taining nucleation sites that correspond to 2 andn 0 (in the corner near past null infinity enclosed bythe shaded boxes of Fig. 7, the left panel of which showsthe HT HF case). Considering the time evolution ofan observer starting from 0, most of the 4-volume inthis region will come into the observers past light coneat very early times. The observer will therefore see new

observer

observer

PLC

PLC

earlytime large scale bubbles

bubbleslatetime small scale

FIG. 7: The foliation of the exterior de Sitter space intosurfaces of constant for junctions with HT HF (left) andHT HF (right). Dark regions correspond to small andlight regions correspond to large . Superimposed on thispicture is the boosted initial value surface for various n in

the limit of large-o.

bubble collisions at a rate that is very high at first (for-mally divergent as ), and decreases with time6.

In the limit where HT HF, for all o, there is alsoa very large 4-volume containing nucleation sites thatcorrespond to 0 (in the corner near future null infin-ity enclosed by the shaded box), though the observer willnot have access to these collisions until late times. In thislate-time limit (and even for o ), the boosted ini-tial value surface cuts into the relevant phase space onlywhen obs , so the distribution is nearly isotropic.

Assembling this information, we predict that the dis-tribution function has two potentially large peaks: oneat 2 and n = 0, for large o, and one at 0and all angles, for large o; both are in complete agree-ment with the analysis of the unboosted frame. Collisionswith 2 are recorded at very early observation times,while those with 0 are recorded at very late obser-vation times. We now directly confirm these predictionsby explicitly calculating the distribution function in theboosted frame.

D. Angular distribution function

We now calculate dNdd cos obsdobs , the differential num-

ber of bubbles with an observed angular scale in adirection on the sky given by (obs, obs). In Sec. IIICwe found a mapping (Eq. 23) between the position atwhich a colliding bubble nucleates and the observed an-

6 Surfaces of constant are nearly null at early times, so this effectcan be viewed as due to time dilation in the boosted frame.


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gular scale as seen by an observer situated at the origin(for which obs = n, obs = n). We can therefore calcu-late the distribution function by determining the densityof nucleation events on surfaces of constant and n.(The symmetry in implies that the distribution is in-dependent of n.)

The differential number of bubbles nucleating in a par-cel of 4-volume somewhere to the past of the observation

bubble is:

dN = dV4 = H4F

sin2 ncos4 Tn

dTndnd(cos n)dn. (24)

A more complete analysis would include the probabil-ity that a given nucleation site is not already inside of abubble. Under our assumption that bubble walls are null,

this probability is given by fout = eVpast4 (n,Tn,n) [19],

where Vpast4 (n, Tn, n) is the 4-volume to the past of agiven nucleation point. Consider some parcel of 4-volume

from which bubbles might nucleate. At late times, inthe unboosted frame, a straightforward calculation showsthat the 4-volume to the past of any point is proportionalto t, the flat slicing time. This yields a differential num-ber of nucleated bubbles:

dN

dtdrd(cos )d= r2e(3H

4)Ht r2e3Ht , (25)

where we have used the fact that in any model of eter-nal inflation H4 1. The total number of bubbles isfound by integrating, and it can be seen (essentially forthe same reason that inflation is eternal in these models)that including fout only minutely affects both the differ-ential and total bubble counts. We will therefore neglectthis correction in our calculation.

Returning to Eq. 24, changing variables from Tn to using Eq. 23, and integrating n at constant (n, Tn),we obtain the distribution function:

dN

dd(cos obs)dobs)=

dN

dd(cos n)dn= H4F

max(o,,n)0

dnsin2 n

cos4(Tn(, n, Tco))

Tn(, n, Tco)

, (26)

with the Jacobian given by

Tn(, n, Tco) = 12 sin n sin Tco sin

2

1

cos

2

+ cot n cot Tco

2sin2 n sin

2 Tco

1/2. (27)

The lower limit of integration at n = 0 can be un-

derstood by tracing the surfaces of constant in Fig. 7and also by noting that for all and Tco, Eq. 23 yieldsTn(, n = 0, Tco) = 0. The upper limit of of integra-tion, max(o, , n), is found by determining the inter-section of the surfaces of constant- with the boostedinitial value surface; this intersection depends on n ando (due to the boosted initial value surface Eq. 4), reflect-ing the dependence of the past 4-volume on the positionof the observer.

The properties of the observation bubble enter this cal-culation through the determination ofTco via Eq. 21. Re-call that for late-time observers (o ), Tco can rangefrom 4 for HT = HF to

2 for HT

HF.

We first examine the behavior of the distribution func-tion Eq. 26 for an observer at the origin, o = 0. Inthis limit, the distribution is isotropic, and based uponthe discussion surrounding Fig. 7, we expect it to havea large peak around = 0 as Tco /2 (HT/HF 0and o ). Integrating Eq. 26, we see in Fig. 8 thatthis behavior is indeed observed. For fixed HT/HF, theamplitude of the distribution function approaches a con-stant maximum value as o (Tco approaches itsmaximum). We will see in the next section that the total

number of observable collisions at late times is bounded,

reflecting the behavior of the distribution function.From the analysis of the boosted initial value surface

in Sec. IIIB, we predicted that in the limit of large-o,the distribution function Eq. 26 should be anisotropic,peaking around n = 0. Fig. 9 shows a number ofconstant-(n, n) slices through the distribution functionfor Tco =

4 and o = 25, where we see that this be-

havior is indeed present. The peak at large , which waspredicted to arise based upon the analysis in both the un-boosted (Sec. IIIA) and boosted frames (Sec. IIIC), ispresent in this example as well. Finally, we observe thatas n 0, the distribution peaks at progressively larger. This feature can be predicted from Fig. 7 by noting

that as n 0, an increasing fraction of the 4-volumeabove the boosted initial value surface corresponds to nu-cleation sites that produce a large (the shaded box nearpast null infinity in Fig. 7).

Focusing on a slice through the distribution functionwith (n = 0, n = const.) for which the amplitude islargest we can study the effects of varying Tco and o.Fig. 10 shows the distribution function for fixed n = 0and Tco =

38 with varying o. As o increases, the am-

plitude of the peak at large increases, while the peak


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10

2

3

2

2

1

2

3

4

dN

dn dn dcos n

FIG. 8: The distribution function Eq. 26 for an observer ato = 0 with (from the blue curve on the bottom to the redcurve on top) Tco =

4, 3

8, 2

(corresponding to a varying

HT), factoring out the overall scale H4F . (This factor will

in general be astronomically small, but we choose this con-vention to more clearly display the functional behavior of thedistribution function.) This function is independent of n forthis observer. As Tco /2 (HT/HF 0), a divergent peakaround = 0 develops.

2

3

2

2

5

10

15

20

25

dN

dn dn dcos n

FIG. 9: The distribution function Eq. 26 for an observer ato = 25, with Tco =

4, for =

10, 15

, 20

, factoring out the

overall scale H4F . As n 0, the position of the peakshifts to larger , and increases in amplitude, displaying thepredicted anisotropic peak about large angular scales.

at small remains unaffected. This can be understoodfrom Figs. 4 and 7 by recognizing that as o grows, thephase space near past null infinity corresponding tonucleation points producing 2 grows, while thephase space near the intersection of the past light coneand the observation bubble wall corresponding to nu-cleation points producing 0 remains constant.

Finally, Fig. 11 shows the evolution of the distribu-tion function produced by fixing n = 0 and positiono = 2 and increasing Tco (corresponding to the actualtime-evolution of the distribution function seen by thisobserver). Here, the bimodality of the distribution be-comes apparent. Based on Fig. 7, we determined thatbubbles with large angular scales form at early (open slic-ing) observation times, and bubbles with small angularscales form at late times. This can be seen in the distribu-tion function of Fig. 11. As Tco increases, the peak near 0 becomes more and more pronounced, overtaking

2

3

2

2

5

10

15

20

25

dN

dn dn dcosn

FIG. 10: The distribution function Eq. 26 with n = 0 andco = 3

8for o = (1.5, 2, 100), factoring out the overall scale

H4F . As o gets large, the peak near 2 grows, whilethe peak near 0 remains of constant amplitude.

2

3

2

2

5

10

15

20

25

30

dN

dn dn dcosn

FIG. 11: The distribution function Eq. 26 with n = 0 ando = 2 for Tco = (

4, 3

8, 716

), factoring out the overall scale

H4F . As Tco grows, the bimodality of the distribution be-comes more and more pronounced. Both the peak about 0 and 2 grow, with the growth of the 0 peakeventually overtaking the growth of the 2 peak. Theposition of the peaks shift as well, with one peak approaching = 0 and the other = 2 as Tco

2.

the amplitude of the 2 peak, whose growth even-tually stagnates. The positions of the peaks also shift,moving towards = 0 and = 2, respectively, as Tcoincreases.

E. Behavior of the distribution near 2 and 0

Since the distribution function (as displayed in the fig-ures) is multiplied by H4F 1, it must have a verylarge amplitude for our hypothetical observer to hope tosee any collisions. We have seen that the distributionfunction is largest for 2 (corresponding to colli-sions occurring at small ) in the large-o, small-n limitas well as for 0 (corresponding to collisions occur-ring at large ) in the limit where HT HF. The originof these peaks was discussed in Sec. IIIC, but now weassess them quantitatively.


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11

10 102 103 104 105

HF

HT10

15

20

30

50

HF4

FIG. 12: A log-log plot (calculated numerically) of the totalangular area on the sky taken up by late-time collisions with 0.

1. The peak at 0

The total number of late-time collisions can be foundby evaluating times the 4-volume V04 in the exteriorspacetime corresponding to small angles. Assuming that

the bubble interior and exterior are pure dS and takingthe limit of large o with HT HF, we obtain

N0 =4

3H2TH2F

tanh2

HTo2

+ O

log

HFHT

. (28)

For fixed HT this approaches a fixed number as o , but this number can be arbitrarily large if HT 0.We see also that for N0 > 0, we require both HT HF1/2.The angular scale of late-time collisions decreases with

o, as exhibited by Fig. 11; one might then ask what totalangular area on the sky is affected. This can be found by

evaluating:

=

dV4

2 (29)

over the volume outside of the observation bubble avail-able for the nucleation of colliding bubbles, where isa function of the exterior spacetime coordinates as inEq. 23. As it turns out, the decrease in angular scalenearly cancels the growth in N0, so while the lat-ter scales as (HF/HT)2, the maximal sky fraction isnearly logarithmic in HF/HT, as shown in Fig. 12. SinceH4F 1, the total angular area is very small unlessHT is essentially zero (and o absurdly large); thus for

any realistic scenario the bubble distribution should beconsidered a set of point sources with infinitesimal totalsolid angle.

2. The peak at 2

Let us now consider the large-o, small-n limit. To doso, we take = 2 with 1 and look at Tco = /4(the amplitude of the peak would only be larger if we

were to take Tco > /4, so this gives a lower bound).Keeping terms to first order in , we can simplify thevarious objects in Eq. 26 immensely: Tn along constant surfaces is given approximately by Tn = n, and theJacobian reduces to

Tn(, n)

=

4

sin n

1 + sin(2n)(30)

yielding a distribution

dN

ddnd(cos n)=

H4F

4 (31)max

0

(tan n)3

cos n

1 + sin(2n)dn.

In the limiting case under discussion, we can solve formax from the simplified form of the initial value surface(obtained from Eq. 18)

sin max =cos max

+ sin max (32)

yielding

max = sec1

eo

1 + e2o

, (33)

where we have not yet taken o large. Integrating Eq. 31,substituting with max, and taking o 1, we obtain:

dN

ddnd(cos n)=

H4F

12e3o , (34)

which diverges as o .Integrating the distribution function over the neigh-

borhood of 2 and small n would yield the totalnumber of observed early-time collisions, which is givenby N2 4oH4F [10]. Since each of these col-lision events can in principle affect an angular scale oforder 2, only a vanishing fraction of the total an-gular area on the sky remains unaffected in the o limit (unlike the long-time limit of late-time collisionsdiscussed above).

IV. SUMMARY OF RESULTS AND

IMPLICATIONS

A. Properties of the distribution function

Given an observer at some point in their bubble definedby (o, o, o = 0), we have calculated the expected num-ber, angular size, and direction (obs, obs) of regions onthe sky affected by bubble collisions, under the assump-tion that those collisions merely perturb the observationbubble.

Three key features of this distributiondN/dd(cos obs)dobs are:


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12

For observers at o = 0 inside bubbles with HT HF, the distribution is bimodal, with peaks at 0 and 2 forming at late and early observationtimes respectively.

For early-time collisions with 2, the distri-bution is strongly anisotropic as o , with theoverwhelming majority of collision events originat-

ing from obs 0, while the distribution of collisionevents with 0 becomes isotropic at late-times. For a given HT, HF, and o, the peak at 2

diverges as exp(3o); the peak at 0 has fixedamplitude, with the total number of such collisionsbounded by N0 H4F1). If such collisions are Compat-ible (with our observations), we should therefore expectthat they exist to our past.

At very late times, observers at any position o willhave access to nearly the same distribution of collisions.We have seen that such an observer would typicallyrecord the first collision at exponentially late times (of or-der o 1/2HF), with tiny angular scale. Thereafter,the number of collisions would grow to asymptotically ap-proach H2T H2F , and the distribution would becomenearly isotropic. Note that this analysis is relevant tothe suggestion by [20, 21] that an observer residing ato = 0 inside of a bubble with HT = 0 (the census takerof[21]) could be used to define a measure over the pocketuniverses in eternal inflation; it may also be relevant forevaluating the quantum-gravitational degrees of freedomof an eternally-inflating de Sitter space [22]. In terms ofourobservations, if we fix HT to be the vacuum energy wecurrently observe, and o H1T , late-time, small angu-lar scale collisions could be observable if H4F

> 10100.While perhaps an atypically large tunneling rate, this iswell within the limit H4F


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13

cause the latter accelerate inward, and have strongly sup-pressed formation rates relative to downward-bubbles,such collisions should be extremely rare. Collisions oftype C, between nested bubbles, occur if a downward-bubble quickly nucleates within an upward-bubble, ourobservation bubble is unlikely to be such an early bubble infinitely many others will form later within the samefalse-vacuum bubble. Finally, nested upward-bubbles

may collide (cell D), but only very rarely.This general survey of two-bubble collision events, in-

dicates that the focus on situation A alone is quite justi-fied: all other possible collision events should be negligi-bly rare.

Determining the detailed aftermath of a collision eventbetween two vacuum bubbles of arbitrary vacuum energyis a very complicated problem, most likely involving nu-merical relativity. Previous numerical and analytic stud-ies have treated cases where the vacuum energy insideboth bubbles vanishes [8, 25], cases where both bubbleshave negative vacuum energy [26], and cases where a zeroand negative vacuum energy bubble collide [9].

In the absence of detailed computations, but based onthese studies, we can outline a few generic possibilities.For collisions between bubbles of the same vacuum, thedisturbed intersection region might radiate away muchof the walls energy, then be smoothed out by subse-quent inflation. For bubbles of different vacuum fieldvalue, wall energy may still radiate away (as demon-strated in [25, 26]), but a domain wall must remain, andwould presumably accelerate into the bubble of highervacuum energy.

In terms of the effect on an observation bubble, itwould seem that collisions resulting primarily in a do-main wall accelerating away from an observer are likely

to be Compatible (in the terminology of Sec. I) overa significant part of the collisions future. Even if con-siderable energy is released, it will be red-shifted by theepoch of inflation within the bubble, perhaps resultingin only a minor perturbation of the interior cosmology.On the other hand, a domain wall accelerating towardsthe observer will almost certainly be catastrophic (andhence not Compatible). In between, bubbles of the samevacuum (where there is no domain wall), or collisions re-sulting in a timelike domain wall (as in [9]), may or maynot be Compatible (for all or just a portion of the causalfuture of the collision) depending on the details of thecollision.

Returning to Fig. 13, cells A-C depict collision eventspotentially relevant to the observation bubble. In eachcase, if the vacuum energy of the observation bubble islower than the vacuum energy of both the background dSand the colliding bubble, it seems likely that the collisionis Compatible over most of its future. A or C could alter-natively be fatal if the incoming bubble (in cell A) or thebackground space (in cell C) are at lower vacuum energythan the observation bubble. However, the finer detailswill need to be studied to provide a definitive classifi-cation of these collision events and to what degree they

satisfy the Compatibility condition.

C. Observational implications

What does all of this mean for making predictionsstarting from a fundamental theory that drives eternalinflation? The above discussion of the possible results of

bubble collisions suggests a spectrum ranging from whatmight be called Fatal collisions to Perturbative ones.Fatal collisions would destroy all observers to their fu-ture, while Perturbative collisions would merely painttheir effect on the observation bubble. Realistic collisionswould fall in between these extremes.

Consider first a scenario in which Fatal (downward)bubbles can form at rate fatal and collide with our ob-servation bubble. Focusing on the = o spatial slice, onwhich we presumably exist now, we must be at a positionthat has not yet experienced such a collision. The unaf-fected volume fraction will be fOK = exp[fatalV4(o)](where V4 measures the available past 4-volume for nu-

cleations, which for o 1 is V4(o) 4oH4F ), and

as discussed in Sec. II, the 3-volume element goes asdV3 = 4H

3T sinh

2(o)do. Combining these, the dis-tribution in o, for o 1, of volume unaffected by fatalbubbles goes as

dV3 fOK exp[(2 43

fatalH4F )o] do,

For fatalH4F 1/2fatalH1F , then all of the collision events likelyto ever affect us happened in the distant past, and wewill safely inhabit our unaffected region of the observa-

tion bubble, oblivious to the fact that fatal collisions mayhave occurred elsewhere.Let us consider collisions that are Compatible but not

Fatal, so that we might exist in at least part of the colli-sions future. If this part is relatively small, or excludes

8 This analysis agrees with that of GGV, who essentially assumedthat collisions are all Fatal and then found that we are unlikelyto hit by such a bubble soon.


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14

the region that we are likely to be in, we might treatthese bubbles as Fatal, and simply assume that we arenot in the future of any of them. If, on the other hand,we might exist in essentially all of the collisions future,we might treat them as Perturbative. If a theory predictsthat at least one collision type is effectively Perturbative,then we can simply assume ourselves to be in a regionunaffected by non-Perturbative bubbles, but should still

expect to see Perturbative collisions to our past, followingour derived distribution function. Determining whethera Compatible collision is effectively Fatal or Perturbativewill be difficult, as it requires a detailed understandingof the collisions aftermath, and may also involve mea-sure issues to determine whether or not the (putative)observers in question are likely be in the perturbed or thedestroyed part of the collision result. (One cause for con-cern in this regard is that the o observers likely tosee many collisions are very highly boosted. Thereforeeven if an incoming bubble is almost perfectly Perturba-tive, this perturbation might be extremely dangerous tosuch a highly-boosted worldline. Another way to see thisis to note that most collisions observed at early times bythe boosted observer in Fig. 7 come from very earlycosmological times.)

In our analysis, we have concentrated on determiningthe region of the observers sky that is in principle af-fected by (a set of) collision events. Further, we have usedthe bubble wall as the surface upon which the observeris examining the effects of collisions. This has allowedus to avoid making any assumptions about how colli-sion products may travel inside of the observation bub-ble. However, the most relevant calculation is to deter-mine the effects of bubble collisions on the post-tunnelingequal-field surface, then in turn the observable effect onthe last-scattering surface (and therefore in the CMB).

This will necessarily involve a better understanding ofthe physics involved in bubble collisions, an investigationthat we reserve for future work.

That being said, we might speculate that the grossfeatures of the distribution function on the last scatter-ing surface will be similar to the analysis we have car-ried out, suggesting that bubble collisions would produceanisotropies and features on large angular scales in theCMB. Because of the bimodality of the distribution func-tion, the subdominant peak around 0 might alsoproduce observable effects akin to point sources, but onlyif > (HTHF)2 for some bubble type. These specu-lations must be put on much firmer ground before any

conclusions can be drawn from current or future data.

V. DISCUSSION

In Sec. I, we outlined three conditions that must bemet for there to be observable effects of bubble collisionsin false-vacuum eternal inflation: Compatibility, Prob-ability, and Observability. What do our results implyabout these?

We have not gone beyond the general arguments con-cerning Compatibility given in Sec. I, except to note thatincoming bubbles of higher vacuum energy are likely to beseparated from us by a domain wall that accelerates awayfrom us, greatly enhancing the likelihood that they willmerely perturb the observation bubble. We have not,however, actually shown that bubbles with the requisitelevel of Compatibility are expected; it will be necessary

to extend previous bubble-collision analyses [8, 9, 25, 26]to answer this question decisively, as well as to assess theresult of multiple bubble collisions affecting a single pointinside the observation bubble.

Our main result is a calculation of the statistical distri-bution of collisions coming from a direction (n, n) thatcan affect an angular scale on the 2-sphere defined bythe portion of the bubble wall causally accessible to anobserver at some instant in time, assuming that the in-coming bubbles merely perturb the observation bubble.The properties of this distribution function depend uponthe location of the observer inside of the observation bub-ble, which we have evaluated in complete generality, but

there are two limiting cases of interest.First, if we sit very far from the finite unaffectedregion near the center of the bubble (defined by o


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15

corresponding to the beginning of inflation will presum-ably be perturbed by collision products that propagateinto the bubble, and these effects translate into densityfluctuations on the surface of last scattering. Because wehave only treated the effects of collisions on the bubblewall, our analysis is only a preliminary step to answeringsuch detailed questions. Nonetheless, it seems likely thatsome basic features of the distribution function, such as

anisotropy and effects on large angular scales, will per-sist.

In some sense, bubble collisions are the most genericprediction made by false vacuum eternal inflation, inde-pendent of the properties of the fundamental theory thatmay drive it. While connecting this prediction to real ob-servational signatures will entail both difficult and com-prehensive future work (and probably no small measureof good luck), it appears worth pursuing. For a con-firmed observational signature of other universes, whilecurrently speculative even in principle, and probably far-off in practice, would surely constitute an epochal dis-covery.

Acknowledgments

The authors wish to thank T. Banks, R. Bousso, B.Freivogel, and A. Vilenkin for helpful discussions. MJthanks the ARCS Foundation and the Hierarchical Sys-tems Research Foundation for support. MJ and AAwere partially supported by a Foundational Questionsin Physics and Cosmology grant from the TempletonFoundation during the course of this work.

APPENDIX A: TRIPLE INTERSECTION IN THE

UNBOOSTED FRAME

In this appendix we solve directly for the coordinateangles denoting the boundaries of a collision on thePoincare disk. We specialize to the case HT = HF = H,where it is possible to foliate the bubble interior with theflat slicing. Working in a plane of constant-,9 we areattempting to find the triple-intersection between threecircles representing the observation bubble, the collidingbubble, and the past light cone of the observer, whose

radii are given by

robs = 1 eHt , (A1)rcoll = e

Htn eHt , (A2)rplc = e

Ht eHto. (A3)

9 As before, we work with the convention where < < tocover full circles.

Using up the remaining symmetry of the problem we canassume that the observer is at o = 0. The free parame-ters that must be specified are then the position at whichthe colliding bubble is nucleated (tn, rn, n) and the po-sition of the observer (to, ro) in terms of the flat slicingcoordinates. The transformation between the open andflat slicing location of the observer is given by

ro = H1

sinh o sinh ocosh o + cosh o sinh o

to = H1 log(cosh o + cosh o sinh o).

(A4)

The observation bubble introduces no new free parame-ters, since it is centered around the origin, and nucleatesat t = 0.

We find it useful to parameterize time with x 1 eHt (this way r = x is the observation bubble). It isstraightforward to conclude that the three light-cones arethe set of points (r(x, ), x , ) parameterized as follows:

Observation Bubble future lightcone:

(r = x, x, ) 0 x 1, (A5)

Observers past lightcone:

(ro cos

(x xo)2 r2o sin2 , x, )

x xo, || | arcsin( x xoro

)|(A6)

New bubble future lightcone:

(rn cos( n) (xn x)2 r2n sin2( n), x, )xn x, | n| | arcsin( xn x

rn)|

(A7)

The triple intersection is the set of points belongingto all three groups. Demanding first that 1 x =ro cos

(x xo)2 r2o sin2 and repeating for 1x =

rn cos( n)

(xn x)2 r2n sin2( n), then solv-ing for x() we obtain

2x = r2o x2o

ro cos xo =r2n x2n

rn cos( n) xn , (A8)

giving an equation for :

A cos + B sin + C = 0, where

A = ro

x2n r2n cos nrnx2o r2o

B = sin nrn

x2o r2o

C = xn

x2o r2o xox2n r2n.

(A9)


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There are two solutions10 to Eq. A9,

cos 1,2 =

AC BA2 + B2 C2

A2 + B2

. (A10)

One can now solve for the time of the intersection byplugging 1,2 into eq. A8. This gives the coordinatesof the two desired intersection events in the flat slicing

where the angle is measured from the origin. By spheri-cal symmetry, these angles are the same as the coordinateangles measured from the origin of the of the bubble inte-rior as described by the open slicing coordinates. We canthen use the angles 1,2 to define the angle as measuredby the observer sitting at some open slicing coordinates(o, o, o = 0) via Eq. 8.

APPENDIX B: EFFECTS OF BOOSTS ON THE

BUBBLE

In Sec. IIIB, we used the symmetries of the one-bubble

spacetime to justify performing a boost that would bringus to a frame where the observer is at the origin. Here, weexplore the effects of this boost on the interior spacetimein greater detail.

In terms of the embedding coordinates, the transfor-mation is given by Eq. 12. The first important prop-erty to note is that the X4 coordinate is invariant. Inthe open slicing, surfaces of constant X4 are surfaces ofconstant , and so we see that the boost preserves theopen slicing time. The second important property is thatthe observer at (o, o, o = 0) is translated to the origin(o = 0,

o = o,

o = 0) of the the boosted frame. From

the relation for X0 in Eq. 12,

cosh o = cosh o (cosh o tanh o sinh o) = 1, (B1)and therefore o = 0.

In Sec. III, we derived a formula for the observed an-gular scale of a collision event in both the boosted and

unboosted frames. We now establish the invariance ofthis quantity by directly applying the transformation toEq. 8. The angle in this equation corresponds to theangular position of the intersection on the null wall of theobservation bubble (as defined by the origin in the un-boosted frame), so using = T, the boosted angle fromEq. 14 is:

tan = sin (cos ) . (B2)

In this frame, can be identified as , the actual ob-served angle at which the boundary of the collision lies(which is used to find the total angular scale of the col-lision in Eq. 8). Solving for cos ,

cos =sinh o cos cosh o

sin2 + (sinh o cos cosh o)2, (B3)

and expanding into exponentials reveals that this expres-sion is in fact equal to Eq. 8, as evidenced by:

cos = cos (B4)

= 1 + 2ei + e2i + e2io 2ei+2o + e2i+2o

1 + 2ei + e2i e2io + 2ei+2o e2i+2o

In the Poincare disk representation, using the hyperboliclaw of cosines, this implies that all of the angles in thetriangle composed of (and therefore the lengths between)the observation point, the unboosted position of the ori-gin, and the edge of the collision, remain invariant under

the boost. More generally, the distance between any twopoints on the disc will be invariant under the boost (asone can check on a point-by-point basis), and so we canidentify the boost as a pure translation in the open co-ordinates.

10 The denominator A2 + B2 never vanishes because the observerand the nucleated bubble never sit on the observation bubblewall. Also, notice that the symmetry in is reflected in the fact

that the positive solution for a given n is the negative solutionfor n.

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Anthony Aguirre, Matthew C Johnson and Assaf Shomer- Towards observable signatures of other bubble universes