University of Amsterdam

Eternal Inflation and the

Multiverse

Philip Will

6060749

Assessor:

Prof. Jan de Boer

Supervisor:

Dr. Jan-Pieter van der

Schaar

Verslag van Bachelorproject Natuurkunde en Sterrenkunde,

omvang 12 EC, uitgevoerd tussen 12-04-2012 en 26-07-2012

July 26, 2012

Abstract

In this bachelor thesis, the Coleman-De Luccia gravitational instanton is stud-ied. The vacuum transition rate calculation is explained in the thin-wall ap-proximation, assuming the existence of a single scalar field with two di↵erentclassically stable vacua. The importance of these gravitational instantons in thecontext of eternal inflation models is explained, where they would describe thequantum creation of bubble universes.

Populair-wetenschappelijke samenvatting

Stel dat iemand je vroeg waarom we de aarde waarnemen als geschikt voormenselijk leven - er zijn namelijk veel factoren die precies goed moeten zijn omintelligent leven toe te staan: de aarde heeft de goede massa om helium kwijtte raken maar stikstof vast te houden, de zon is lang genoeg stabiel, etc.

Je zou dan waarschijnlijk antwoorden dat hoewel de kans klein is dat al diefactoren tegelijk voorkomen, het feit dat er triljoenen planeten zijn ervoor zorgtdat er toch ergens leven moet zijn. En alleen leven kan waarnemen wat deomstandigheden van zijn planeet zijn.

Maar wat als die persoon je vroeg waarom we het universum waarnemenals geschikt voor leven? Hier geldt hetzelfde; je hoeft maar een klein beetje teveranderen aan de natuurconstanten en stervorming wordt onmogelijk of sterrenbranden in een paar miljoen jaar op. Een goede verklaring zou zijn dat er ookveel universa zijn, maar daar is tot nu toe geen bewijs voor.

Mijn bachelorproject gaat over een manier om een natuurkundig model temaken van het multiversum (zo heet de ruimte waarin universa zich bevinden).In dit model ontstaan universa uit het multiversum zoals belletjes in champagne- spontaan en met een zekere begingrootte - en daarna groeien ze als inktvlekkenop een nat stuk papier - het papier verandert van wit naar zwart zonder extraruimte in te nemen.

Dit fenomeen wordt beschreven door de donkere energie (dat is eigenlijk deenergiedichtheid van het vacuum) gelijk te maken aan de potentiaal van eenveld. Dit veld laat zich beschrijven alsof je voor elk punt in de ruimte eenballetje neemt dat je van een heuvel af laat rollen, en dat de hoogte van hetballetje gelijk is aan de energiedichtheid op dat punt.

Nu is het zo dat door de algemene relativiteitstheorie van Einstein, deenergiedichtheid gelijk wordt aan de constante van Hubble, de term voor deuitzetting van het heelal (als de energiedichtheid van het vacuum veel groter isdan de energiedichtheid van materie en straling). In het huidige universum zijnde balletjes overal al lang in een dal terecht gekomen, maar in het multiversumzitten ze vast in een hoger dal.

Met middelbare school-natuurkunde zou het balletje altijd boven vast zittenen de ruimte altijd keihard blijven uitzetten, maar door de quantum-mechanicakan het balletje door de heuvel heen boren. Als het balletje aan onze kant eruitkomt, trekt het de balletjes van de ruimte eromheen door de tunnel, waarna deballetjes naar beneden rollen tot ze in ons dal terecht komen.

Omdat ze nog een tijd naar beneden rollen zet de ruimte waar ze bijhorennog een tijd uit, hard genoeg om te groeien van kleiner dan een atoom naarmiljarden lichtjaren groot in minder dan een triljoenste van een seconde. Deruimte waarvan de balletjes niet door de tunnel gaan zet alleen nog harder uit,en hard genoeg om de balletjes die door de tunnel worden gezogen ver voorte blijven, waardoor de ruimte op nog veel grotere schaal hetzelfde blijft. Zoontstaat slechts een universum, maar op dezelfde manier kunnen er zich oneindigveel vormen.

En ja, dit is echte natuurkunde.

Contents

1 Introduction 2

2 Inflation 42.1 Big bang theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Big bang cosmology . . . . . . . . . . . . . . . . . . . . . 52.1.2 Problems with big bang theory . . . . . . . . . . . . . . . 62.1.3 The inflation solution . . . . . . . . . . . . . . . . . . . . 8

2.2 Classical inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Quantum perturbations . . . . . . . . . . . . . . . . . . . . . . . 102.4 Eternal inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Gravitational instantons 133.1 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Feynmann sum over histories . . . . . . . . . . . . . . . . 143.1.2 Instantons for two stable states . . . . . . . . . . . . . . . 163.1.3 Unstable states and bounces . . . . . . . . . . . . . . . . 17

3.2 Bubble nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Gravitational coupling . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Bubble nucleation . . . . . . . . . . . . . . . . . . . . . . 193.3.2 Bubble growth . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Consequences and conclusions 23

A Wick rotation 25

1

Chapter 1

Introduction

Suppose somebody asked you why the earth is so fit for intelligent life; it’s closeenough to the sun to have liquid water, but far enough to avoid a calamitousgreenhouse e↵ect, it’s heavy enough to retain an atmosphere but light enoughto lose helium and hydrogen, it has a strong magnetic field to ward o↵ cosmicradiation, the large gas giant Jupiter reduces the amount of asteroid impacts, etcetera - many factors all have to be right to allow the development of intelligentlife to take place.

Your answer is likely to be the anthropic principle: the universe is filled withan unfathomable amount of stars, many of which have planets, so even if thechance of developing life is one in a trillion, the universe would still be teemingwith life, and only on planets where sapient life has arisen is there anyone towonder why their planet is capable of life.

So what if they asked you why the universe is so fit for life. It’s a validquestion: make the strong coupling constant slightly higher and stellar fusionbecomes impossible, slightly change the Higgs field parameter and the universecurves to the size of a football. But now the anthropic principle seems to drawup empty: there’s only one universe, as far as we can see. Sure, you can saythere are many universes out there, but those have to actually contribute to thedistribution of possible universes.

Usually, at this point, the discussion veers o↵ into metaphysics, asking unan-swerable questions about untestable statements, such as whether mathematicalequations can be sentient or whether objective truth exists independently ofobservation. It would be nice if there was a way to make scientific statementsabout alternate universes, rather than philosophical or metaphysical ones.

Eternal inflation is a theory invented to do just that. It uses the inflaton fieldfrom regular inflation, and with only a few more assumptions creates a physicaldescription of a multiverse, consisting of de Sitter space in which universesform like bubbles in champagne or soda. In this bachelor thesis I will explainthe phenomenon of eternal inflation, and describe a method of quantifying theformation of universes.

In chapter 2, I will show the workings of regular inflation. First I will

2

introduce classical inflation and explain its merits. Then I’ll show what happenswhen quantum field theory is accounted for. Lastly, I show how under certainconditions the inclusion of quantum field theory allows classical inflationarymechanics to become eternal.

In chapter 3, I will show the workings of a di↵erent form of eternal inflation,which would classically be stuck in the inflationary era for an infinite time.This form of inflation lends itself more easily to exact calculation and may alsoapply in more situations. With this type of eternal inflation, a classically stablestate called a ’false vacuum’ is rendered unstable through a quantum tunnelingprocess which is called an instanton, which is the subject of the first section.I will then calculate a highly simplified form of universe formation where thee↵ects of gravity are neglected, and explain what happens when gravity is takeninto account.

As this is a purely theoretical thesis, I will finish in chapter 4 with a discus-sion of the applicability and testability of the theory of eternal inflation, andwith proposing ways for improving the accuracy of the approximation.

Units

I will be using the units c = 1 and 8⇡G = 1, and work to first order in h. Iuse the Minkowski metric in chapter 2 with a (+,-,-,-) signature, and I’ll beswitching between a Minkowski and an Euclidean signature in chapter 3. Themethod of switching between these two signatures is explained in the appendix.

3

Chapter 2

Inflation

Before we can talk about eternal inflation, we must first pin down the detailsof inflation. Inflation is a scalar field theory developed to explain some of thefailures of the predictive power of big bang theory. It holds that in the first10�33 seconds, the universe underwent exponential expansion, growing by tensof orders of magnitude. This rapid expansion would help explain some of thefailings of standard cosmological models. For example, standard cosmologycan’t explain why points outside each other’s Hubble radius (1/H

0

) appear tohave been in causal contact, or why the universe appears spatially flat.

First I will briefly explain standard cosmology. Then I will outline severalproblems with standard cosmology and how inflation fixes them. Next I willexplain the workings of the classical field theory of inflation. Last I will performa brief quantization of inflation theory, which allows for a toy model calculationof eternal chaotic inflation as well as a prediction of the shape of the cosmicmicrowave background radiation (CMBR).

This chapter draws mostly from D. Baumann’s lecture notes on the quantumorigin of structure in the inflationary universe [1].

2.1 Big bang theory

Big bang theory was developed based on a few simple observations: the universeis homogeneous, isotropic and expanding at approximately 70.2 ± 1.4 km s�1

Mpc�1 locally. [8] The Mpc�1 comes from the fact that further galaxies moveaway quicker, at a rate given by v = Hx, with x the distance.

If we make the rough assumption that this can be linearly expanded back-wards, then that means there was a moment when all galaxies in the universeoccupied the same point. The discovery of the CMBR all but confirmed the va-lidity of the downward extrapolation, and the measurement of the contributionsof various forms of energy to the universe (dark energy, radiation, and matter)suggests the universe was ever smaller the further back in time you go, at leastuntil the limits of our understanding of the laws of physics.

4

2.1.1 Big bang cosmology

Quantitatively, general relativity shows the general metric for an expanding ho-mogeneous and isotropic spacetime is the Friedmann-Robertson-Walker (FRW)metric [1]:

ds2 = dt2 � a2(t)

dr2

1 + r2+ r2(d✓2 + sin2 ✓ d�2)

�(2.1)

where a(t) is the scale factor, is �1, 0 or +1 depending on whether theuniverse is open, flat or closed and r is a comoving coordinate. The componentsof the universe behave like a perfect fluid, which is defined as having the energy-momentum tensor

Tµ⌫ = diag(⇢,�P,�P,�P ). (2.2)

Solving the Einstein equations from general relativity for this tensor resultsin the Friedmann equations:

H2 =1

3⇢�

a2(2.3)

a

a= �1

6(⇢+ 3P ) (2.4)

where H = a/a is the Hubble constant at time t and the dot marks di↵er-entiation to time. Using the first law of thermodynamics,

dU = �PdV

d(⇢a3) = �Pd(a3)

d ln ⇢

d ln a= �3(1 + w) (2.5)

⇢ / a�3(1+w) (2.6)

where w ⌘ P/⇢. Combining this with equation 2.3, we get

a / t2/3(1+w) w 6= �1 (2.7)

a / eHt w = �1 (2.8)

We also know that the main components of the universe are matter (non-relativistic w = 0, relativistic w = 1/3), radiation (w = 1/3) and dark energy(w = �1), and that the universe is almost flat within experimental accuracy.This means the universe has always been expanding, which is why big bangtheory is accepted.

5

2.1.2 Problems with big bang theory

While big bang theory is strongly supported by evidence, there are many factorsit does not manage to explain or predict. Inflation does manage to explain someof these, and these problems I will explain in detail.

Flatness problem

First, as I have mentioned before, the universe has been observed to be flatwithin experimental measurement. This is peculiar, because the curvature de-pends on how the universe is filled, so there is no particular reason for theuniverse to be flat. To express it quantitatively, equation 2.3 can be rewrittenas

1� ⌦(a) =�

(aH)2(2.9)

where

⌦(a) ⌘ ⇢

⇢crit

⌘ ⇢

3H2

(2.10)

with ⇢crit

defined as the energy density at which the universe is perfectlyflat. To see how this develops in time, it’s useful to look at the equation

d|1� ⌦(a)|d ln a

=||⇢a2(aH)4

(2 +d ln ⇢

d ln a) = (1 + 3w)⌦|1� ⌦| (2.11)

which can be derived using equations 2.3 and 2.5. The current componentsof the universe have been measured to be [8]

⌦m = 0.275± 0.016, ⌦� < 0.02 and ⌦⇤

= 0.725± 0.016 (2.12)

where the subscript m signifies the matter component, � the radiation com-ponent and ⇤ the dark energy component. Combining this with equation 2.6,it becomes clear that the universe was dominated by matter and radiation untilrecent times. That means that until recent times, 0 w 1/3. Therefore,equation 2.11 has always been larger than zero, which means ⌦ = 1 is an un-stable point: if the universe is near ⌦ = 1 now, it must have been much closerto 1 in the past.

For the observed level of curvature to occur, ⇢/⇢crit

would have to be accurateto fifty-five significant numbers or more during the grand unified theory erain standard big bang cosmology [1]. It isn’t until before this point that ourunderstanding is too limited to extrapolate with any degree of certainty.

While not impossible, it would be very peculiar if it was random chancethat caused this level of fine-tuning. It is more likely there is a cause outsidestandard big bang theory.

6

Horizon problem

Second, the universe is homogeneous on scales which could not have been incausal contact. Because nothing can travel faster than the speed of light, theedge of areas which could have been in causal contact is given by the light cone,which describes the path of a photon could take if it passes through a givenpoint. The light cone is given by the null geodesic ds2 = 0. In the FRW metric(equation 2.1), this means the light cone is given by

dt2 = a2dr2 (2.13)

It is useful to define a comoving time ⌧ as d⌧2 ⌘ a�2dt2 = dr2, because thenthe FRW metric becomes

ds2 = a(⌧)2d⌧2 � dr2

1 + r2+ r2(d✓2 + sin2 ✓ d�2)

�(2.14)

and the light cone is given by r = ⌧ . Using the definition of ⌧ , we get

⌧ ⌘Z t

0

dt0

a=

Z a

0

da0

a02H(2.15)

Now, as I have established, the universe was dominated by matter and ra-diation in the past. In the approximation of single component universes in analmost flat universe, ⌧ is given by

⌧ =

Z t

0

dt0At0�2

3(1+w) =3A(1 + w)

1 + 3wt

1+3w3(1+w) (2.16)

which is

⌧m = Bt1/3 , ⌧� = Bt1/2 (2.17)

for matter and radiation respectively, and the true evolution of ⌧ is some-where between the two. Both boundaries increase over time, so light has neverbeen able to travel further than now in standard cosmology. This means thatwhen we look at opposite ends of the CMBR, we are looking at parts of theuniverse which have never been in causal contact. However, they have almostexactly the same temperature. This is true for every region of the CMBR whichcouldn’t have been in causal contact with a di↵erent one.

While not impossible, it would be very strange indeed if the temperature ofthese independent regions was everywhere the same due to chance. Again, it ismore likely that something is missing from the cosmological models.

Missing magnetic monopoles

Third, it is an almost unavoidable consequence of trying to describe the unifica-tion of forces at high energy densities that magnetic monopoles would have beenproduced in the early universe. These would be easily observable because theycause an electrical current in any conducting coil they pass through, and would

7

have a high enough density to be detected even at the present time. However,they have never been observed.

2.1.3 The inflation solution

Inflation is a theory which works to resolve these issues. It holds that the darkenergy density is actually variable, given by the potential U of a scalar field�(t, r), and that it was much higher in the very early universe. During thisearly period, called the era of inflation, the universe expanded exponentiallyby many more orders of magnitude than it has in every cosmological era thatfollowed.

In a dark matter dominated universe (w = �1), equation 2.11 is negative, soa curved universe can become flat. Because the expansion during the inflation-ary era is many times greater than all later expansion, this solves the flatnessproblem.

Similarly, equation 2.15 becomes

⌧ =

Z a

0

da

a02H= � 1

H

1

a) a = � 1

H⌧(2.18)

which pushes the singularity back to ⌧ = �1 and solves the horizon prob-lem, because areas of space many times larger than the visible universe used tobe in causal contact, but that comoving volume decreased with time.

The monopole problem is solved by the expansion of the universe directly,because they would be formed before inflation and they decrease in density likematter.

Inflation does not manage to explain why there is more matter than an-timatter, nor the fine-tuning problem of fundamental parameters, nor variousother problems. It is not a miracle cure, but a very powerful tool which is verypopular.

2.2 Classical inflation

As mentioned above, inflation theory makes the dark energy density the poten-tial U of the inflaton field �. The workings of this field in the classical limit,assuming a homogeneous and isotropic universe, are detailed below.

First, the action of the inflaton field, coupled to gravity, is given by [1]:

S =

Zd4x

p�g

1

2gµ⌫@µ�@⌫�� U(�)� 1

2R

�(2.19)

where g = det gµ⌫ , R is the Ricci scalar, and gµ⌫ is the metric tensor ofgeneral relativity. The field equation of motion is therefore

�S

��=

1p�g@µ(

p�g@µ�) + U 0(�) = 0 (2.20)

and the stress-energy tensor is

8

Tµ⌫ = � 2p�g

�S�

�gµ⌫= @µ�@⌫�� gµ⌫

✓1

2@��@��+ U(�)

◆(2.21)

where S� is the action of equation 2.19 without the Ricci scalar term.In a flat universe, the stress-energy tensor becomes that of a perfect fluid

(equation 2.2) with

⇢� =1

2�2 + U(�) (2.22)

P� =1

2�2 � U(�) (2.23)

In order for an inflation era to exist and fit observations, the potential energy,which corresponds to the vacuum energy, has to be much larger than the kineticenergy of the scalar field. The kinetic and potential energy are the first andsecond term between the brackets in the action of equation 2.19, so

1

2�2 ⌧ U(�) (2.24)

Using equation 2.3 and the definition of ⇢, this means

H2 =1

3

✓1

2�2 + U(�)

◆⇡ U(�) ⇡ const. (2.25)

This means we have exponential expansion as per equation 2.8 in the entireuniverse. The spacetime metric that results from this is called de Sitter space.

The inflation era also has to last long enough to have enough exponentialexpansion. This puts an additional condition on the acceleration like so:

by combining the Friedmann equations, we can derive the equation

d⇢

dt= �3H(⇢+ P ) (2.26)

which, using the definition of ⇢� gives

�+ 3H�+ @�U = 0 (2.27)

This means we can demand a maximum speed of change �:

d�

dt= � 1

3H

dU

d�(2.28)

by demanding

|�| ⌧ |3H�| , |�| ⌧����dU

d�

���� . (2.29)

Equations 2.25 and 2.29 are satisfied if the slow-roll parameters ✏ and ⌘ meetthe slow roll conditions which are defined as [9]:

9

1 � ✏ ⌘ 1

2

✓@�U

U

◆2

' � H

H2

(2.30)

1 � ⌘ ⌘ @�@�U

U(2.31)

This means any universe with an inflaton potential and � such that @�Uand @�@�U are su�ciently small would expand exponentially, until the slow-rollconditions are no longer upheld.

2.3 Quantum perturbations

In the previous section, I assumed a classical and homogeneous scalar field.While this is a good first approximation, the slow-roll parameters can be sosmall that quantum e↵ects can not be neglected. Quantum fluctuations arelocalized in time and in space, so the universe will no longer inflate perfectlyhomogeneously. The e↵ect of this quantization of the inflaton field will bediscussed in this section.

The inflaton field can be decomposed into a homogenous classical backgroundand localized perturbations:

�(t,x) = �(t) + ��(t,x) (2.32)

In order to calculate the equations of motion of the perturbations, we needto expand the action (given by equation 2.19) to second order of ��. This is avery arduous calculation, for which I refer to section 10.3 of [10].

Neglecting the changes to the metric, the action of the perturbations is givenby

�2

S =1

2

Z ✓v02 � (@µv)

2 +a00

av2◆d4x (2.33)

where �2

signifies the second order approximation, the prime denotes di↵er-entiation to conformal time ⌧ and

v ⌘ a�� (2.34)

Now the quantization can begin. First we define the canonical momentum

⇡ =@(�

2

L)@v0

= v0 (2.35)

Then, v and ⇡ are turned into quantum operators in the Heisenberg picture;the operators are dependent on time and the wavefunction is time-invariant.They therefore obey equal time commutation relations:

[v(⌧,x1

), ⇡(⌧,x2

)] = ih�(3)(x1

� x2

) (2.36)

10

We can now define the mode decomposition with a Fourier transform, split-ting the operator into several operators acting on the amplitudes of specificmomenta

v(⌧,x) =

Zd3k

(2⇡)3/2vk

(⌧)a�k eik·x (2.37)

Making these modes obey the equations of motion set by the action as definedin equation 2.33 results in the equation of motion for perturbation modes:

v00k

+ (k2 � a00

a)v

k

= 0 (2.38)

In the case of a universe dominated by dark energy with a = �1/(H⌧)

a00

a=

2

⌧2= 2H2a2 (2.39)

which means the equations have an exact solution

vk

= ↵e�ik⌧

p2k

✓1� i

k⌧

◆+ �

eik⌧p2k

✓1 +

i

k⌧

◆. (2.40)

In the subhorizon limit limk⌧!�1, the equations of motion are that of anideal harmonic oscillator. In this limit, the exact solution of the lowest energyvacuum state is [1]

limk⌧!�1

vk

=e�ik⌧

p2k

(2.41)

after normalization. Therefore the exact solution for the mode functions,which are called the Bunch-Davies mode functions, is

vk

=e�ik⌧

p2k

✓1� i

k⌧

◆(2.42)

For intuitively understandable reasons, the comoving modes become fixedwhen they pass the horizon which shrinks in the comoving frame. Because theyare fixed, they’re no longer undone by future quantum fluctuations and thereforeleave a permanent mark on the value of the inflaton field. To quantify this, wecan take the superhorizon limit limk⌧!0

:

limk⌧!0

h��2k

i = limk⌧!0

|vk

|2a2

=H2

2k3(2.43)

Performing a Fourier transform gives

h��2i /Z kf

ki

dkh|��k|2i = H2 lnkfki

(2.44)

where ki and kf are modes which were at the horizon at the start and end ofinflation respectively. This means inflation predicts density perturbations which

11

are homogeneous and independent of scale at the end of inflation: if you averageover the di↵erences between many di↵erent points which are a fixed distance rapart, you will get the same answer independent of r. This is a prediction ofinflation which is satisfied in the observations of the CMBR, notwithstandingafter-e↵ects caused by post-inflationary dynamics.

2.4 Eternal inflation

This isn’t all, however. It is possible for the quantum fluctuations to be largerthan the classical change if the slow-roll conditions are upheld much strongerthan necessary for inflation.

Because it is possible for the universe to be many Hubble-radii in size, dif-ferent causally unconnected areas all have their own quantum fluctuation am-plitudes which become fixed at di↵erent values, and because an inflationaryuniverse increases in size exponentially, the area with a higher potential thanbefore can actually increase over time. This means that there will always be apoint in space where inflation is currently ongoing: eternal inflation.

Let’s show this by calculating how this can happen in the simplest inflation-ary potential there is, the chaotic potential U(�) = ↵�2. I have already derivedthe factor quantum fluctuations are proportional to in equation 2.44, but [6]gives the root mean square of the quantum fluctuations over one Hubble time(�t = 1/H) exactly, as

��2q =H2

4⇡2

) ��q =H

2⇡(2.45)

Here the slow-roll conditions are upheld for very large �. The classical changeduring one Hubble time can be found using equations 2.28 and 2.25:

��cl = � 1

3H2

dU

d�=@�U

U=

2

�(2.46)

If we set

��q � ��cl , H� � 4⇡ (2.47)

then, by Gaussian statistics, 15.9% of causally unconnected spaces will endup going against classical motion. Furthermore, because of the exponentialexpansion, space will have increased in size by a factor of e3 ⇡ 20. Since20 ⇥ 0.159 = 3.18 > 1, so the amount of space which goes against classicalmotion increases.

And that’s it: eternal inflation from nothing more than a stronger thannormal upholding of the slow-roll conditions.

In this treatment I have neglected thermal fluctuations, higher order grav-itational contributions, and various other factors, but the principle of eternalinflation remains the same.

12

Chapter 3

Gravitational instantons

In the previous chapter I have looked at what happens when quantum fluc-tuations overpower the classical slow-roll change. However, there is anotherpossibility which allows eternal inflation eternal inflation, when the classicalchange is zero. This is called the false vacuum. If U(�) has a shape like infigure 3.1, then the universe could be classically stable at �

+

, but decay due toquantum tunneling. This would lead to di↵erent dynamics than the classicallydecaying eternal inflation model.

Figure 3.1: A potential of an instanton field with one false vacuum at �+

andone true vacuum at ��. If this potential describes decay into our universe, thenU(��) = 0. Note that the intersection of the axes is neither necessarily U = 0nor necessarily � = 0.

13

It is possible for the potential to be many times more complex than theone shown in figure 3.1, and string theory among others predicts hundreds offalse vacuum solutions, but this turns out not to make the calculation any moredi�cult.

In this chapter, I will explain the quantum tunneling e↵ect which describesthe decay of the false vacuum in the presence of gravity, known as the Coleman-de Luccia instanton, in the limit of minimal potential di↵erence between falseand true vacuum. I will not take thermal fluctuations into account, and I willalways work to first order of h, so I will leave out the factor (1 +O(h)).

The Coleman-de Luccia instanton describes the sudden appearance of ahighly symmetric and clearly separated bubble in spacetime inside of whicha universe forms, with or without an era of inflation. These bubbles expand atnearly the speed of light, which means that if they form rarely enough, eternalinflation is maintained outside the bubble universes.

In this chapter, section 3.1 draws mostly from S. Coleman’s book Aspects

of Symmetry [2], and the other sections draw from Coleman and de Luccia’sarticle on gravitational e↵ects on and of vacuum decay [4].

3.1 Instantons

Before we can begin describing the nucleation of universes, we first have tofind a way to describe quantum tunneling for a scalar field. This requires wecalculate the Feynmann sum over histories of the analytic continuation of theaction to Euclidean spacetime, also known as the Wick rotation of the action(see appendix A). From now on I will be using the Euclidean metric unlessotherwise specified.

3.1.1 Feynmann sum over histories

First, I will describe the general behavior of instantons for any relevant potential.The Feynmann sum over histories is a calculation of the probability ampli-

tude that the initial state pf the universe |xii turns into a certain state |xf iafter a certain time T . It works by integrating over all possible ways that thechange can take place. If we take |xii and |xf i to be position vectors of a singleparticle, the Euclidean Feynmann sum over histories is given by

hxf |e�HT/h|xii = N

Z[dx]e�S/h (3.1)

where N is a constant, e�Ht/h is the time evolution operator derived fromthe time-dependent Euclidean Schrodinger equation:

d

dt (t) = �H

h (t) ) (t) = e�Ht/h (0) (3.2)

andR[dx] is the functional derivative over all possible ways x(t) to go from

x(0) = xi to x(T ) = xf .

14

In classical mechanics, the stationary points of S[x(t)] are the only possiblepaths. For the sake of simplicity, it is best to take the small-h limit, whichmeans the stationary points are dominant in the calculation of S, and we willalso assume there is only one stationary point, which we call x. Taking thefunctional derivative of S at x(t) gives

dS

dx=@L@x

+ @µ

✓@L

@ (@µx)

◆= �d2x

dt2+ U 0(x) = 0 (3.3)

where the prime denotes di↵erentiation with respect to x. If we define

x(t) = x+X

n

cnxn (3.4)

with the xns as a complete set of orthonormal functions, we get [7]

[dx] =Y

n

(2⇡h)�1/2dcn (3.5)

The xns can be any complete set of orthonormal functions, so we set them tobe the eigenfunctions of the second functional derivative of S:

�d2xn

dt2+ U 00(x)xn = �nxn (3.6)

Expanding the right side of equation 3.1 to second order of x, using equation3.5, gives:

hxf |e�HT/h|xii = N

Z Y

n

(2⇡h)�1/2dcn exp

✓S(x) + 0 +

1

2

��@2t + U 00(x)�(x� x)2

◆/h

�

= Ne�S(x)/h

Z Y

n

(2⇡h)�1/2dcn exp

"1

2

��@2t + U 00(x)�X

n

(cnxn)2/h

#

(3.7)

For Gaussian integrals with any operator O with positive eigenvalues �n,the following equation holds [7]:

ZeOx2

dx =

✓⇡

�n

◆1/2

�n > 0 (3.8)

Here it is assumed all eigenvalues are positive, but in the case of instan-tons negative and zero eigenvalues are also allowed, where zero values resultin multiplying by the factor (S

1

/2⇡2h)1/2, where S1

is the action of a singleinstanton [2].

Additionally, the equations xn are orthonormal, which means

15

hxf |e�HT/h|xii = Ne�S(x)/hY

n

(2⇡h)�1/2

✓2⇡h

�n

◆

= Ne�S(x)/hY

n

��1/2n

= Ne�S(x)/h⇥det(�@2t + U 00(x))

⇤�1/2(3.9)

3.1.2 Instantons for two stable states

In eternal inflation, the beginning and end state are both stable. So let’s lookat what happens when |xii and |xf i are classically stable position eigenstates.

First, we can approach the potential at their positions to second order of xby

V 00(xi,f ) = !2

i,f (3.10)

For simplicity, let’s say !i = !f ⌘ !. In [2] it is shown that in this approxi-mation, in the large T limit:

N⇥det(�@2t + !2)

⇤�1/2=⇣ !⇡h

⌘1/2

e�!T/2 (3.11)

Because both states are classically stable, it takes a relatively short amountof time, on the order of magnitude !�1, to go from |xii to |xf i. This switchof states is known as an instanton, since it is localized in both space and time.Because instantons are localized, it is possible for several instantons and anti-instantons (from |xf i to |xii) to occur after each other, which all contribute tohxf |e�HT/h|xii.

We will only be looking at cases where anti-instantons are symmetrical:where hxf |e�HT/h|xii = hxi|e�HT/h|xf i If we assume that instantons are su�-ciently separated, the action contribution and influence on the determinant areidentical for every instanton. Therefore S = nS

1

and

N⇥det(�@2t + V 00(x))

⇤�1/2=⇣ !⇡h

⌘1/2

e�!T/2Kn (3.12)

with n being the number of instantons plus the number of anti-instantonsand K a constant calculated by demanding it is correct for a single instanton.

In [2], K is derived to equal

K =

✓S0

2⇡h

◆z/2 ����det(�@2t + !2)

det0(�@2t + V 00(x))

����1/2

(3.13)

where z is the number of zero eigenvalues of [�@2t + V 00(x)], and det0 indi-cates the zero eigenvalues are omitted. There is one zero eigenvalue for everytranslation invariance allowed.

16

3.1.3 Unstable states and bounces

Though the beginning and end state are both classically stable, one is di↵erentfrom the other, so the decay into the lower state is final. Now I’ll demonstratewhat happens when a quantum unstable state decays.

Imagine a potential which decreases steadily with x, except around x = 0,where it has a local minimum. On it, we can define a single point x = � 6= 0 byU(�) = U(0). After performing a Wick rotation, it is clear an instanton can befound for h0|e�HT/h|0i, where the particle turns around at �. This is called abounce, and after Wick rotating back it turns out to be a significant solution.

In order to calculate h0|e�HT/h|0i, we have to sum over all possible amountsof instantons because all contribute to the action. We also have to integrateover the locations of the centers of the instantons in time:

Z T/2

�T/2

dt1

Z t1

�T/2

dt2

...

Z tn�1

�T/2

dtn =Tn

n!(3.14)

Therefore filling in equation 3.9 results in:

h0|e�HT/h|0i =⇣ !⇡h

⌘1/2

e�!T/2X

n

�KTe�nS0/h

�n

n!

=⇣ !⇡h

⌘1/2

e�!T/2 exp(KTe�nS0/h) (3.15)

Using the completeness of energy eigenstates we get

hxf |e�HT/h|xii =X

n

e�EnT/hhxf |nihn|xii (3.16)

which, combined, means

E0

=1

2!h+ hKe�S0/h (3.17)

It turns out that this kind of bounce has a single negative eigenvalue. Byequation 3.13, this means K is imaginary, and consequently that that contribu-tion to E

0

is imaginary as well. Imaginary energy is equivalent to instability ofthe state; the imaginary value corresponds to a certain chance that the vacuumdecays. Therefore we can derive a decay width [2]

� = h|K|e�S0/h. (3.18)

3.2 Bubble nucleation

Now we have the mathematical tools to calculate a simplified form of bubblenucleation. We can describe the decay of an unstable state through an instanton,

17

and express the decay width in the action and shape of the potential, and thiscan be applied on universe formation.

I will now describe a highly simplified form of universe formation. I willneglect the influence of gravity, and work in the limit of minimal potentialdi↵erence between the true and the false vacuum.

The Euclidean action of the scalar field, not coupled to gravity, is given by

S =

Zdx4

✓1

2(@µ�)

2 + U(�)

◆(3.19)

The simplest form of U(�) which contains barrier penetration is

U = U0

+ ✏(�� a)/2a (3.20)

where U0

is an even function which is zero at � = ±a, so the resultingfunction has two local minima at � = �± = ±a + O(✏) which di↵er in depthby ✏. Like in section 3.1.3, barrier penetration requires a bounce, which meansthat we both start and end at � = �

+

at infinite time. To prevent the actionfrom becoming infinite, it is also necessary to have � = �

+

at infinite distance.[3] proves that with these boundary conditions, a non-trivial � with extrem-

ized action exists, which possesses O(4) invariance1. This symmetry makes itconvenient to switch to radial coordinates. In these coordinates,

S =

Zd⇢ 2⇡2⇢3

✓1

2�02 + U(�)

◆(3.21)

where the prime denotes di↵erentiation to ⇢. Discounting � = �+

8⇢ astrivial, the simplest O(4) symmetric path is one where �(0) = ��, surroundedat a radius R by a spherical surface of instantons - in one dimension, if therewere an instanton at ⇢ = R.

In the small ✏ limit, this radius can be calculated by dividing the integralinto three parts: one where ⇢ is near R, one where ⇢ is significantly smaller, andone where it is larger. In the latter two regions, �0 can be neglected becausean instanton is localized. At the wall, ⇢ = R can be taken as constant, and thecontribution of ✏ to U(�) can be neglected.

U(�+

) = 0 and U(��) = �✏therefore

S = 2⇡2

Z R�⇣

0

�⇢3✏ d⇢+Z R+⇣

R�⇣

d⇢R3

✓1

2�02 + U

0

◆+

Z 1

R+⇣

0 d⇢

!(3.22)

S = 2⇡2R3S1

� 1

2⇡2R4✏ (3.23)

1O(4) invariant functions depend only on their distance to an arbitrary point in space-time.

18

where S1

is the action of the one-dimensional instanton, which depends onthe arbitrary shape of the potential and can not be calculated. The radius canthen be found by maximizing the action:

dS

dR= 0 = 2⇡2(3R2S

1

�R3✏) ) R = 3S1

✏(3.24)

Wick rotation back to Minkowski coordinates gives the following equationfor the evolution of the wall:

R2 = ⇢2 = (~x2 � t2) (3.25)

which is a hyperboloid which starts at a distance R from the center ⇢ =0, then asymptotically approaches the light cone of ⇢ = 0, and has O(3, 1)symmetry.

This describes a universe which spawns from the false vacuum as a spherewith a certain radius, and then grows outward, converting the surrounding falsevacuum into true vacuum at nearly the speed of light. This growth does notcorrespond to expansion, which is a change of the distance metric, which requirestaking gravity into account to be able to describe.

3.3 Gravitational coupling

In the previous section I neglected the e↵ects of gravity. While gravity does notfundamentally change the qualitative shape of the dynamic, it is an importantcontributing factor to the quantitative development and nucleation rate, andalso simply more accurate to include.

In this section, I’ll work under the assumption that our universe exists withina bubble and has potential U(��) = 0 everywhere. This means no inflationtakes place inside the bubble. I’ll still work in the Euclidean metric, slow-rolllimit and thin wall approximation unless otherwise indicated. I will assume thegravitational instanton with lowest action is still O(4)-invariant, although thisis not necessitated by any theorem, based on the fact that [4] uses the sameassumption.

3.3.1 Bubble nucleation

The general O(4)-invariant Euclidean metric is given by

(ds)2 = (d⇠)2 + ⇢(⇠)2(d⌦)2 (3.26)

where ⇢ is the radius of curvature of the hypersphere of constant ⇠. Theaction of equation 2.19, for a symmetric Euclidean metric thus becomes

S = 2⇡2

Zd⇠

✓⇢3(

1

2�02 + U) + 3(⇢2⇢00 + ⇢⇢02 � ⇢)

◆(3.27)

19

where the prime denotes di↵erentiation to ⇠. With this we can find theequation of motion

�00 +3⇢0

⇢�0 � dU

d�= 0 (3.28)

The Einstein equations give the additional equality [4]

⇢02 = 1 +1

3⇢2(

1

2�02 � U) (3.29)

The Euclidean action as defined here goes to infinity unless U(�+

) = 0.However, this also happens if there is no instanton present. Therefore, it’s usefulto work with the di↵erence between the false vacuum and instanton solution.The partial integration of the factor ⇢⇢00 goes to infinity in both solutions, sothe boundary term can be neglected. Entering equation 3.29 and partiallyintegrating the action, the Euclidean action becomes

S = 4⇡2

Zd⇠�⇢3U � 3⇢

�(3.30)

Because we’re comparing to the situation where � = �+

8⇢ which goes toinfinity, it is useful to define a di↵erence between the two and try and minimizethat. This di↵erence we define as

B ⌘= Sinstanton

� Sfalse vacuum

(3.31)

Now we can do the same as in section 3.2 and split the action in threecomponents. Outside the wall, obviously B

outside

= 0. At the wall itself, ⇢ = Rand the contribution of ✏ to U(�) can be neglected, so

Bwall

= 4⇡2R3

Zd⇠(U

0

(�)� U0

(�+

)) = 2⇡2R3S1

(3.32)

where S1

is, again, the action of a single bounce. Inside the wall, U(�) isconstant so

d⇠ = d⇢(1� 1

3⇢2U)�1/2 = �3⇢(1� 1

3⇢2U) (3.33)

because of the curvature caused by the energy density. Therefore Binside

isgiven by

Binside

= �12⇡2

Z R

0

⇢ d⇢

1� 1

3⇢2U(��)

�1/2

�1� 1

3⇢2U(�

+

)

�1/2!

(3.34)

Binside

= 12⇡2

⇥1� 1

3

R2U(��)⇤3/2

U(��)�⇥1� 1

3

R2U(�+

)⇤3/2

U(�+

)

!(3.35)

If we now set

20

U(�+

) = ✏ and U(��) = 0

and ignore the infinity in the Taylor expansion around U(��), then B isminimal at

R =12S

1

4✏+ 3S2

1

=R

0

1 + (R0

/2⇤)2(3.36)

where R0

= 3S1

/✏ is the bubble radius in the absence of gravity and ⇤ =(✏/3)�1/2. At this point

B =B

0

(1 + (R0

/2⇤)2)2(3.37)

where B0

= 27⇡2S4

1

/2✏3 is the action di↵erence in the absence of gravity [4].B < B

0

, so bubble nucleation becomes more likely in the presence of gravity,and the radius becomes smaller, when decaying to U = 0.

3.3.2 Bubble growth

The calculation of the bubble growth is as follows. Given we still work in thethin wall approximation with U(��) = 0, equation 3.33 still holds when ⇢ 6= R.Inside the bubble, we get d⇢ = d⇠, so the metric becomes ordinary Minkowskispace. Outside the bubble, the solution for that metric is

⇢ = ⇤ sin(⇠/⇤) (3.38)

Exponentially expanding spacetime, aka de Sitter space, can be described asa hyperboloid on a Minkowski space of one dimension higher

↵2 = �t2 + x2 + y2 + z2 + w2 (3.39)

We can define d⇢ in this metric as

ds2 = �d⇢2 � dw2 � ⇢2(d⌦)2. (3.40)

If we then take

w = ⇤ cos(⇠/⇤) (3.41)

the requirement for de Sitter space is satisfied.

3.4 Results

In section 3.1, I derived the decay width for a classically stable false vacuumstate of a single point to be

� = h|K|e�S0/h. (3.42)

21

where K is a constant given by equation 3.13. The result of this equation canbe applied to the inflaton scalar field. Doing so requires taking into account thatthe scalar field is everywhere, which means it’s useful to talk about the decaywidth density �/V . Switching to a scalar field also changes K in ways I havenot researched, so I will only talk about the proportionality. Also, because theaction of a scalar field is infinite, it is necessary to talk about the di↵erence ofthe action between the trivial state and the instanton B, as defined in equation3.32.

With this in mind, we can now express the rate at which universes form ina false vacuum, as a function of the action of the instanton:

�/V / e�B/h (3.43)

It is possible to calculate both the evolution of universes and the actionwhich determines how often they form. I have calculated these in the limit ofminimal di↵erence between the starting state and final state of the instantonfield. The switch from the starting state and final state is extremely rapid,and the resulting universe is highly symmetric, which means universes can bedescribed as bubbles forming in the multiverse, which then grow rapidly, turn-ing the surrounding multiverse into universe. Any expansion the universe ormultiverse undergo happens separately from this growth.

In section 3.2, I calculated the evolution of universes and the action whileneglecting gravity. The results are

R0

= (~x2 � t2)1/2 = 3S1

/✏ and B0

= 2⇡2R3S1

� ⇡2

2R4✏ (3.44)

where S1

is the action of a single instanton and ✏ is the di↵erence betweenvacuum states. In section 3.3, I calculated the same while including gravity, butin the specific case of decay into Minkowski space (U(��) = 0)

R =R

0

1 + (R0

/2⇤)2and B =

B0

(1 + (R0

/2⇤)2)2(3.45)

Here, the evolution of the bubble universe outside the bubble is slightlymore complex, because the multiverse itself is de Sitter space which expandsexponentially.

So there we have it: a Minkowski spacetime similar to our universe, embed-ded within an exponentially expanding de Sitter background can spontaneouslygenerate, and eternal inflation can occur, and all of this can be quantitativelydescribed with known physics. Eternal inflation is a theory.

22

Chapter 4

Consequences and

conclusions

In chapter 2 I showed that inflation theory has significant evidence to supportit; explaining why the universe appears homogeneous on super-horizon scalesand why it appears almost flat within observational accuracy, by stating there isa scalar field with a potential which corresponds to the universe energy density,which caused an era of exponential expansion in the early universe. During thisera, the field has to obey the slow-roll conditions, which makes it possible thatquantum perturbations are significant.

These perturbations create a possible test of inflation: though the shape ofthe potential is not determined by inflation theory, the quantum perturbationsshould show up in the anisotropy of the cosmic microwave background radia-tion (CMBR) before standard cosmological alterations, by an invariance in theamplitude of the di↵erence between two points regardless of angular scale.

A di↵erent e↵ect of these perturbations is classically unstable eternal infla-tion, where if the quantum perturbations are on the same order of magnitudeas the classical change, an inflating universe could keep on expanding eternally,even as the vacuum decays locally. This form of eternal inflation is di�cult tofind a test for, but requires only that the slow-roll conditions were su�cientlystrongly upheld during a period in the universe’s history.

There is a di↵erent form of eternal inflation: if the potential of the scalarfield is such that it is classically stable, it is still quantum-mechanically unstablethrough barrier penetration. In chapter 3 I discuss the Coleman-de Luccia(CdL) instanton which describes the quantum tunneling process. I calculate theradius and development of a CdL instanton in the thin-wall approximation whileneglecting gravity and thermal e↵ects, and discuss them for a CdL instantonin the thin-wall approximation, coupled to gravity, while neglecting thermale↵ects, assuming the instanton is O(4) symmetrical in Euclidean space, wheredecay takes place from de Sitter space to Minkowski space.

I find that the space where the change from the false vacuum to the true

23

vacuum takes place is localised. This edge forms half of a hyperboloid with asasymptote the future light cone of the point of symmetry in Minkowski space,starting with a radius which is directly proportional to the action of a singleinstanton. After taking gravity into account, the potential influences the geom-etry. If the initial spacetime is de Sitter and it decays into Minkowski space(which in this level of accuracy can describe the early universe), the total actionof nucleation and the radius are slightly decreased. The edge on the de Sitterspace side also asymptotically approaches a light cone.

Though the methods in this bachelor thesis use many crude approximations,we can come to the qualitative conclusion that it is possible to have a future-eternally inflating de Sitter space filled with independently forming universeswith properties similar to that of our universe, given only the existence of ascalar field with a potential whose shape meets certain conditions, and suitablestarting conditions.

A proposed method to test the validity of instanton-mediated vacuum decayis to look in the CMBR for the signature of a collision between two universebubble walls, as is attempted in [5] with inconclusive results.

Many proposed grand unified theories predict the existence of scalar fieldswith specific potentials. The veracity of these grand unified theories is thereforelinked with the veracity of eternal inflation. For certain potentials, eternalinflation o↵ers testable predictions, mostly confined to the CMBR, and thesepredictions serve as predictions for grand unified theories.

If eternal inflation is correct, it would mean the universe is but one of many,suspended like bubbles in an exponentially expanding multiverse. Each universemay have di↵erent universal parameters, which can explain even more fine-tuning problems than regular inflation, and also has significant philosophicalimplications.

This bachelor thesis is based on theoretical research originally done in andbefore the year 1980. Since that time, scientists have developed more accurateresults for the bubble nucleation rate and development of universe bubbles, forexample by including thermal fluctuations, calculating the gravitational instan-ton for decay from de Sitter space into de Sitter space with a di↵erent Hubbleconstant, or researching the e↵ects of various proposed grand unified theorieson these instantons.

On the experimental side of things, a test has been devised that the collisionof bubbles should leave distinct patterns on the cosmic microwave backgroundradiation. So far the results have been inconclusive, so this area of researchwould benefit greatly from more accurate maps of the CMBR, which can beused to test not only inflation in general, but also classically unstable eternalinflation and possibly instanton-mediated false vacuum decay.

24

Appendix A

Wick rotation

Analytic continuation is a technique of extending the domain of an analyticfunction. Wick rotation is a specific form which extends problems in Minkowskispace to Euclidean space where they acquire imaginary time, which can help tosolve them.

For example, the Minkowski action S0 of a particle in a potential U(x) attime t0 is given by the following equation:

S0 =Z

L dt0 =Z

dt0"m

2

✓dx

dt0

◆2

� U(x)

#(A.1)

Performing the Wick rotation t0 ! it gives:

S0 =Z

d(it)

"m

2

✓dx

d(it)

◆2

� U(x)

#= i

Zdt

"m

2

✓dx

dt

◆2

+ U(x)

#(A.2)

The Euclidean action also rotates as S0 ! iS

S =

Zdt

"m

2

✓dx

dt

◆2

+ U(x)

#(A.3)

Note that the Euclidean action describes a particle in a potential �U(x).This means that where at first there was no classical way to describe a particlemoving from �

+

to �� in figure 3.1, now there can be one, given the particlestarts with certain speed, moving right.

While this is unnecessary for particles because the WKB approximation issimpler, the WKB approximation can not be generalized to describe scalar fields,while instantons can.

25

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[3] Sidney Coleman, Vladimir Glaser, and Andre Martin. Action minimaamong solutions to a class of euclidean scalar field equations. Commun.

Math. Phys., 58:211–221, 1978.

[4] Sidney Coleman and Frank De Luccia. Gravitational e↵ects on and ofvacuum decay. Phys. Rev. D, 21(12):3305–3315, 1980.

[5] Stephen Feeney, Matthew Johnson, Daniel Mortlock, and Hiranya Peiris.First observational tests of inflation. Phys. Rev. Lett., 107, 2011.

[6] Alan Guth. Inflation and eternal inflation. Phys. Rep., 333-334:555–574,2000.

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[9] Andrew Liddle and David Lyth. Cobe, gravitational waves, inflation andextended inflation. Phys. Lett. B, 291:391–398, 1992.

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26