Eternal In ation The Theoretical Quest for the Multiverse

Universiteit van Amsterdam

Eternal Inflation

by

Melvin van den Bout (10545069)

Supervisor:

Dr. Jan Pieter van der Schaar

Second assessor: Prof. Dr. Daniel Baumann

Report Bachelor Project Physics and Astronomy, size 15 EC,

in the

FNWI-IoP

Institute for Theoretical Physics

In this thesis, a literature study has been conducted in the field of cosmological inflation. The primary purpose of this study is to become familiar with the mathematical and conceptual ideas of inflation and how this will lead to eternal inflation (EI). To arrive at the study of EI, we will first go through the required essential background knowledge. In doing so, we will study intensively the foundations of modern cosmology and the inflationary paradigm. We will show how models might be eternal in the future. To see how EI fits into the scientific method, we will explore how we can possibly observe consequences of EI and what problems arise in making predictions from the theory. We conclude that EI is highly speculative and our understanding of it remains incomplete. More theoretical research is needed to fully understand the implications of future eternal inflation models. Furthermore, higher precision observations of the CMB temperature fluctuations and polarization are needed to confirm or falsify EI.

Populair wetenschappelijke samenvatting

Als mensen een afstand inschatten doen ze dit over het algemeen met intu¨ıtie, of door de afstand proberen te vergelijken met een goed kenbaar object. De afstanden in de kosmologie zijn echter zo groot dat deze onze intu¨ıtie te boven gaat. Zo is bijvoorbeeld de diameter van ons observeerbaar heelal ongeveer 1026 meter. Deze afstand is alleen nog maar de grens tot hoever we kunnen kijken. De ruimte achter ons observeerbare heelal gaat nog door en wellicht is de ruimte oneinding groot voor ons heelal. Ook voor de afstanden op de allerkleinste schaal, waar de wetten van de quantum mechanica gelden op de Planck afstandschaal van 10−35meter, heeft ons brein moeite met het bevatten van de kleinheid. Het fascinerende is dat de afstandschaal van het allergrootste en het allerkleinste mogelijk aan elkaar gerelateerd zijn via kosmologische inflatie. Een uitbreiding van kosmologische inflatie voorspelt zelfs ook nog eens dat ons heelal niet de enige is. Laten we eens kijken hoe wetenschappelijk dit allemaal is..

Toen Albert Einstein in de periode tussen 1907 en 1911 de algemene relativiteitstheorie(ART) ondekte, had hij de basis gelegd voor de moderne kosmologie. Niet veel later toonde Edwin Hubble aan dat sterrenstelsels van ons af bewegen met een snelheid die propertioneel is met de afstand tot ons melkwegstelsel. Dit leidde tot de oerknal theorie, waarmee door middel van ART berekent kan worden dat 13.8 miljard jaar geleden ons heelal mogelijk oneindig klein was en vanaf toen begon te groeien. Echter bleven er enkele problemen over die niet met dit model verklaard konden worden. De fine-tuning die de theorie nog niet kon verklaren werd opgelost door een uitbreidende theorie van Alan Guth die gepubliceerd werd in 1981.

Deze theorie, kosmologische inflatie, beschrijft de knal van de oerknal en is erg succesvol in het voorspellen van waarnemingen aan onder andere de kosmische achtergrond straling (KAS). De KAS is straling die overal in ons heelal aanwezig is en informatie bevat van het vroege heelal toen het ongeveer 105 jaar oud was. Waarnemingen aan de KAS hebben bevonden dat er een patroon aanwezig is voor temperatuursverschillen op verschillende schaalgroottes. In de theorie van inflatie worden deze temperatuur fluctuaties voorspelt doormiddel van quantum fluctuaties. De ruimte op de afstandschaal van de Planck afstand is onderworpen aan de wetten van de quantum mechanica. Hieruit volgt ook het Heisenberg onzekerheidsprincipe, die onder andere toelaat dat deeltjes voor een hele korte tijd even kunnen bestaan uit het niets. Lege ruimte is dus eigenlijk nooit helemaal leeg omdat er altijd fluctuaties aanwezig zullen zijn in de achtergrond velden en de ruimte-tijd zelf. Het idee van infatie is dat wanneer het heelal een fractie van een seconde oud is, de ruimte onderhevig was aan deze quantum fluctuaties. Doordat de ruimte zich toen exponentieel snel uitbreidde, met een factor 1026tijdens een tijdsinterval van ongeveer 10−35 seconden, werden deze quantum fluctuaties zichbaar op grote schaal. Dit leidde uiteindelijk tot kleine temperatuursverschillen in de KAS.

Kosmlogische inflatie is successvol in ondere andere het verklaren van de KAS temperatuurver-schillen. Dit vertrouwen in de theorie en het belang van de quantum fluctuaties heeft er tot geleid dat men erachter kwam dat inflatie eigenlijk veel meer heelallen voorspelt in sommige gevallen. Inflatie, het exponentieel expanderen van de ruimte, stopt misschien alleen maar in plaatselijke gebieden in de ruimte, maar het process zelf van exponentiele uitbreiding stopt nooit. Inflatie is gestopt in ons gebied (heelal) zodat uiteindelijk sterrenstelsels en inteligent leven kon evolueren. Deze uitbreiding van inflatie, eeuwige inflatie, voorspelt een oneindig aan-tal heelallen met mogelijk allemaal andere natuurwetten. Dit heeft als fascinerend gevolg dat we wellicht het bestaan van intellectuele wezens kunnen verklaren door antropisch te redeneren. Eeuwige inflatie is echter nog erg speculatief en er zijn nog veel theoretische vraagstukken die we niet goed begrijpen. Om het idee van het multiversum te bevestigen of te weerleggen, is het nodig om de theorie met de wetenschappelijk methode te ontwikkelen. Dit betekent dat de theorie voorspellingen moet maken die getoetst kunnen worden door middel van een experiment. Niet alleen is het maken van voorspellingen enorm uitdagend blijkt, maar ook het bedenken en uitvoeren van deze experimenten. Andere heelallen zijn namelijk niet causal verbonden met ons heelal. Maar misschien kunnen we toch consequenties waarnemen van het bestaan van andere heelallen.

In sommige modellen voor eeuwige inflatie worden heelallen gevormd in bubbels. Op willekeurige momenten kunnen bubbels onstaan die beginnen te groeien door inflatie. Echter, de ruimte tussen deze bubbels groeit ook. Als bubbels dicht bij elkaar ontstaan is het mogelijk dat ze met elkaar botsen en mogelijk een detecteerbaar effect achterlaten in de KAS. Tot nu toe zijn er echter in de KAS nog geen overtuigende aanwijzingen gevonden die eeuwige inflatie ondersteunen. Om een bubbel botsing uit te sluiten zijn echter betere observaties nodig aan de KAS. Ook is er nog veel theoretische onduidelijkheid in de theorie. Tot op heden blijft eeuwige inflatie een speculatie. Betere precisie van metingen aan de KAS zijn nodig om de theorie te kunnen toetsen.

Acknowledgements

I would like to express my sincere gratitude to my supervisor Dr. Jan pieter van der Schaar for guiding me throughout this project. Thank you for sharing your enthusiasm, time and knowledge! I feel very fortunate to have learned from you.

In addition to my supervisor, I would like to thank Prof. Dr. Daniel Baumann for taking the time to read this thesis and being a second assessor. Moreover, your immense collection of lecture notes on cosmology and inflation were a major asset during this project.

Abstract i

Populaire samenvatting ii

Acknowledgements iv

Conventions vii

1 Introduction 1

2 Cosmology and General Relativity 3

2.1 Space-time . . . 3

2.2 Energy conservation . . . 4

2.3 The Friedmann equations . . . 5

2.4 The cosmological constant & de Sitter space . . . 6

2.5 Causality . . . 7

2.5.1 Conformal time . . . 7

2.5.2 Horizons. . . 7

2.6 Fine tuning of the initial conditions. . . 8

2.6.1 The horizon problem . . . 8

2.6.2 The flatness problem . . . 9

3 Inflation 11 3.1 Conditions for inflation . . . 11

3.2 Slow roll inflation . . . 12

3.3 Quantum fluctuations during inflation . . . 14

3.3.1 Perturbing the field . . . 14

3.3.2 Quantization of the field . . . 15

3.3.3 Quantum fluctuations in dS-space . . . 18

3.4 Connection to observations . . . 18

3.5 The energy scale of inflation . . . 20

3.6 Inflationary models . . . 21

3.6.1 Chaotic inflation . . . 21

3.6.1.1 Quadratic inflation. . . 22

4 Eternal inflation 24 4.1 Slow roll eternal inflation . . . 24

Contents vi

4.1.1 Chaotic eternal inflation . . . 26

4.1.2 Other eternal inflation models. . . 26

4.2 Remaining issues . . . 27

4.3 Observations . . . 27

4.4 Philosophical discussion . . . 29

4.4.1 The weak anthropic principle . . . 29

5 Conclusion 31

A General relativity 32

B Power spectrum 34

We use the plus metric, -+++, and work in natural units, meaning c = ~ = 1. We work with the reduced Planck mass defined as M_P−2 = 8πG. The Einstein summation convention is used whenever a Greek (µ, ν, .. = 0, 1, ..) or Latin (i, j, .. = 1, 2, ..) index is repeated.

A prime denotes a derivative with respect to conformal time (∂x_∂τ = x0), while a dot represents a derivative with respect to normal time (∂x_∂t = ˙x).

Chapter 1

Introduction

Mankind’s understanding of our place in the universe has gone trough several stages in the last millennia. One of the first models of the universe were geocentric, meaning that the earth was the center of the universe. In the 16th century, Copernicus presented the first mathematical heliocentric model. This model was soon replaced once we discovered that galaxies are moving away from us. Inevitably, we had to conclude that our location in the universe is not that special. Due to the expansion of the universe, we are confined inside our observable universe. We now know that the diameter of our observable universe is approximately 93 Gly. Furthermore, we know that space doesn’t stop beyond this boundary. Instead, our universe might be orders of magnitude bigger then our observable universe, and might even be infinite. Our picture of the universe has been growing enormously since the last centuries. But only since several decades ago, physicists have reasons to believe that our universe is one of infinitely many in the so called multiverse.

This thesis is a literature study to become familiar with the mathematical and conceptual ideas of the multiverse. This study has it roots in general relativity, which was discovered by Albert Einstein in the period between 1907 and 1911. This framework formed the basis for modern cosmology. When this theory is applied to the universe as a whole, the equations suggest that in the distance past the universe might have been infinitely small. But the initial conditions of the singularity could not be explained. Eventually, physicist became aware of several problems with the hot big bang theory, and Guth proposed the idea of cosmic inflation in 1981. This idea was developed such that eventually the problems with the hot big bang were solved, and a solid basis was established to explain the bang of the big bang. One of the great successes of inflation is that it can predict the CMB temperature fluctuations as a consequence of quantum fluctuations in the early universe. Even though there are some degrees of freedom in the inflationary paradigm, it nicely relates different phenomena to describe the early universe. The success of inflation is partly due to the connection to observations. This

allows experiments to give theoretical guidance in model building. However, inflation is not yet established well enough and is not yet generally accepted as a good description of the early universe. This is partly because the initial conditions are unknown. Furthermore, direct or indirect observations of the predicted gravitational waves from inflation are still not observed. Because of this ignorance, many models exist which all fall within observational constraints. But many models have in common that they are future-eternal, what will lead to a multiverse. The solid foundation of inflation has motivated to take research in eternal inflation serious. In order for this idea to be scientific, the theory has to make predictions that can be tested by experiments. Current research is conducted to make predictions in eternal inflation and find possible observational consequences of other universes.

The chronology of the last paragraph will also be the structure of this thesis. We will go through the most important equations and concepts that will lead to the multiverse. In chapter2we will start this by applying general relativity for the universe as a whole. This will bring us to the problems and mathematics of the hot big bang theory. In chapter 3 we will introduce cosmic inflation, derive important equations for the inflaton scalar field and quantize this scalar field. Further, we will relate these theoretical predictions to observations and discuss the energy scale and model building for inflation. These two chapters will provide some essential background knowledge for eternal inflation The main literature used for this first part of this thesis is based on [1–7]. In chapter 4 we will discuss how eternal inflation can arise from normal inflation, and how this will lead to fascinating philosophical consequences. Due to the speculative nature of eternal inflation, we will discuss how it fits into the scientific method and see how eternal inflation relates to observations and predictions. Finally, in chapter5, we will briefly summarize this thesis and conclude with prospects for eternal inflation.

Chapter 2

Cosmology and General Relativity

In this chapter we will study the foundations of modern cosmology. The underlying framework for this is the theory of general relativity(GR). A brief overview of the formulas used from GR are given in appendixA. These formulas rely heavily on the metric of space-time, which will be our staring point in this chapter. Once we obtained the metric, we will study energy conservation and will see how the energy density of the universe evolves as the universe expands. Next we will derive the Friedmann equations so that we can point out the problems for a universe without inflation.

2.1

Space-time

In the theory of special relativity one assumes that space it flat and therefore the Minkowski metric is used. In GR however, space can be curved. The Einstein equation provides a relation between the geometry of time and energy-momentum associated. The geometry of space-time is partly described by a metric which will play an important role throughout this thesis. The cosmological principle states that on a large enough scales the universe is spatially isotropic and homogeneous. This is the assumption on which the Friedmann-Lematre-Robertson-Walker (FLRW) metric is based, and is given in spherical coordinates by

ds2= gµνdxµdxν = −dt2+ a(t)2  dr2 1 − κr2 + r 2_(dθ2_{+ sin}2_θdφ)  . (2.1) In this equation a(t) is the scale factor of the universe and describes how space expands in time. The scale factor at the current time is normalized such that a0 = a(t0) = 1 and the singularity

of the big bang would correspond to a(t = 0) = 0. For an arbitrary value of a(t) we will simply write a to keep equations readable. The curvature of our universe is described by the value of κ and can take either one of three values:

κ =       

+1 closed universe (spherical) 0 spatially flat universe (euclidian) −1 open universe (hyper-spherical)

(2.2)

Measurements have shown that the universe is flat, therefore we will set κ = 0 for simplicity. Whenever κ 6= 0, this will be to explicitly show the relation between curvature and other quantities. Now, if we define γij as

γij =     (1 − κr2)−1 0 0 0 r2 ₀ 0 0 r2sin2(θ)     , (2.3)

then we can write down the metric in spherical coordinates as

gµν(x) =

−1 0 0 a(t)2γij.

!

. (2.4)

With the condition gµρgρν = δνµ, the inverse metric becomes

gµν(x) = −1 0 0 a(t)−2γij.

!

. (2.5)

Note that in flat space (κ = 0) with Cartesian coordinates, γij simply becomes δij and hence

ds2 = −dt2+ a(t)2δijdxidxj. (2.6)

Later on, for the calculation of the Ricci scalar we will use the properties of γij such that

γijγij = 3 and γji = δji.

2.2

Energy conservation

The distribution of energy and momentum in GR is described as a perfect fluid. A perfect fluid takes 2 quantities: the density and pressure. The energy-momentum tensor for a perfect fluid in its rest frame is given by

T_νµ= diag(−ρ, p, p, p). (2.7) Conservation of energy-momentum can be expressed by the vanishing of the divergence as follows:

∇_µTµν = 0. (2.8)

The divergence however, is not simple a derivative, but due to curved space we need to promote this derivative to the co-variant derivative. This results in

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∇µTµν = ∂µTµν+ Γ_µλµ Tλν+ ΓνµλTµλ= 0. (2.9)

The non-zero Christoffel symbols needed, for flat space are computed to be Γ0_ij = a ˙aγij = a ˙aδij Γi0j = Γij0=

˙a

aδij. (2.10) For ν = 0, we can obtain the continuity equation for the energy density.

∇µTµ0= 0 = ∂µTµ0+ Γµ_µλTλ0+ Γ0µλTµλ = ∂0T00+ Γµµ0T 00_{+ Γ}0 ijTij = ˙ρ + 3ρ˙a a+ 3p ˙a a (2.11) So that ˙ ρ = −3˙a a(ρ + p). (2.12)

By choosing a simple linear equation of state, p = wρ, we are left with a differential equation which we can solve to obtain

ρ ∝ a−3(1+w). (2.13)

This equation tells us how the energy density of the universe evolved for different types of contents.

2.3

The Friedmann equations

The Friedmann equations are the Einstein equations applies with a FRLW metric. The Einstein equation, including a cosmological constant (Λ), is given by

Rµν−

1

2Rgµν+ Λgµν = 8πGTµν (2.14) The Ricci tensor for µν = 00 and µν = ij can be calculated with the help of the previous calculated Christoffel symbols (2.10) to be

R00= −3 ¨ a a and Rij = (¨aa + 2 ˙a 2_{+ 2κ)γ} ij. (2.15)

The Ricci scalar, which is the trace of the Ricci tensor, is then given by

R = 6 " ¨ a a+  ˙a a 2 + κ a2 # . (2.16)

This allows us to compute the Einstein equation for µν = 00 and µν = ij. These two equations can be put in the following form

 ˙a a 2 = 8πG 3 ρ − κ a2 + Λ 3 (2.17) ¨ a a = − 8πG 3 (ρ + 3p) + Λ 3 (2.18)

and are known as the Friendmann equations. These equations, in combination with equation

2.12 will tell us how the universe evolves for different types of components. While in reality the universe consist of multiple components, often a good approximation is given by a single component universe. For a single component universe we can construct a table with properties:

matter w = 0 p = 0 ρ ∝ a−3 a(t) ∝ t2/3 radiation w = 1/3 p = −1/3ρ ρ ∝ a−4 a(t) ∝ t1/2 vacuum w = −1 p = −ρ ρ ∝ constant a(t) ∝ eHt

Table 2.1: Evolution of the energy density

The rate of expansion of the universe is given by H = _a˙a and is known as the Hubble parameter. For the purpose of this thesis, the main interest will be a vacuum dominated universe.

2.4

The cosmological constant & de Sitter space

Dark energy is commonly refereed to as anything with w < 1/3 because from equation 2.18 it follows that gravity will act as an repulsive force (¨a > 0). Whereas the cosmological constant refers to an equation of state parameter of w = −1. This value is constrained by experimental observations and is of most relevance for inflation [8].

Using the Friedmann equations for a single component universe dominated by a cosmological constant, we can solve for the scale factor:

 ˙a a 2 = Λ 3 −→ H = r Λ

3 −→ a(t) ∝ exp(Ht) = exp( r

Λ

3t) (2.19) This is commonly refereed to as a de Sitter(dS) universe, and corresponds to the dS limit P −→ −ρ. In this limit, ˙ρ = 0 (see equation 2.12) and therefore ρ = constant. This suggest that dark energy is an intrinsic property of space-time. Substituting this scale factor in the FRLW metric, we find the metric for a dS-universe as

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Additionally, dS-space is defined to have a positive cosmological constant (Λ > 0), in contrast to anti de Sitter (AdS) space (Λ < 0). In contrast to a dS-stage (P = −ρ), there can also be a quasi dS-stage (P ≈ −ρ) which will be of importance later on.

2.5

Causality

In the theory of relativity, it is not allowed to travel faster than the speed of light, otherwise causality would be violated. Causality is at the core of physics. The causality of space-time regions is often expressed in horizons. For cosmology and inflation, these principles are of significance and will be discussed including some convenient definitions.

2.5.1 Conformal time

For convenience, conformal time is defined as a(t)dτ = dt. From this definition it follows that in dS-space τ (t) = Z _dt a(t) = Z dte−Ht= −1 H e −Ht = −H−1 −→ a = − 1 Hτ. (2.21) The initial singularity of our universe corresponds to a = 0, for which it follows that τ −→ −∞. We have defined the comoving Hubble parameter as

H = aH = ˙a. (2.22)

To keep equations readable, we will denote a prime for a derivative with respect to conformal time (∂x_∂τ = x0), while a dot represents a derivative with respect to normal time (∂x_∂t = ˙x).

2.5.2 Horizons

For an isotropic universe, we can define the path for a freely traveling photon as purely radial on a null geodesic (ds2 = 0 and dθ = φ = constant). Using this for the FRLW metric, we can obtain the distance light can travel in a given time interval by

d = Z tb ta dt a(t) = Z τb τa dτ = τb− τa. (2.23)

This equation can be applied to the particle and event horizon and is simply linear in comoving time.

Particle horizon

of a photon could not have been in causal contact with the photon in the past. The particle horizon is the boundary of this past light cone.

Event horizon

The event horizon represents the boundary in space-time of the region for which an observer can never have causal contact with in the future.

Hubble radius

Using Hubble’s law, v = Hr, we can construct a radius for which the velocity (v = c = 1) is the speed of light, resulting in the Hubble radius to be H−1. Anything beyond this horizon cannot have causal influences now, since it accelerates faster than the speed of light. Note that since c = 1, the Hubble time is the same, but differs in units. The comoving Hubble radius a given by (aH)−1 = H−1.

2.6

Fine tuning of the initial conditions

The problems with the hot big bang theory so far can be encapsulated in three problems. The flatness, horizon and monopole problem. All of these problems are fine-tuning problems. The latter will not be discussed, but is nevertheless solved by inflation. The horizon problem is of most significance and will be discussed first. The cosmology problems are not violating any principle of physics, but more of less point out the fine tuning that is assumed in the theory, which will eventually be solved by inflation.

2.6.1 The horizon problem

When the universe cooled down enough, at a redshift of z ∼ 1100, recombination took place, making the universe transparent instead of opaque. Photons where able to travel freely af-terward. These same photons are the ones we observe in the Cosmic Microwave Background Radiation (CMB) and provides us with information of the early universe. It appears that this radiation is very isotropic and homogeneous with fluctuations of the order

δT T ∼ 10

−5_{. (2.24)}

It can be calculated that there are 104 causally disconnected patched of sky. The similarity of the temperatures gives rise to an important issue called the horizon problem. The CMB cannot be in thermal equilibrium because it had to be in causal contact to do so. The horizon problem will manifest itself most conveniently in a conformal diagram as show in in figure2.1.

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Figure 2.1: Since τ0  τrec, and the particle horizon is linear in comving time, we are

confronted with causcally disconnected regions for CMB photons. Image credits: [4]

Since the particle horizon is simply linear in comoving time, we just have to realize that τ0 

τrec. So if we look at two photons from opposite direction, it is clear that at the moment of

recombination they were not in causal contact with each other.

A decreasing Hubble radius in the early universe will be sufficient to solve this problem. This is equivalent to a period of positive acceleration.

2.6.2 The flatness problem

The flatness problem can be explained most conveniently if we introduce a density parameter, Ω, as follows

Ω = ρ ρcritical

and ρcritical = 3H2MP2. (2.25)

The critical energy density, ρcritical, is equal to that for a flat universe (κ = 0) containing no

cosmological constant. With these definitions we can express the Friedmann equation (2.17) as Ω − 1 = κ

H2_a2. (2.26)

From the Friedmann equation in this form it immediately follows that the curvature κ is deter-mined by Ω. The right side of the equation is not stable for a universe with different components such as matter and radiation, and will quickly diverge from zero. This is at odds with obser-vations which have determined κ = 0. Considering the current age of the universe, we must

conclude that the curvature of the universe was extremely flat at the beginning. This apparent fine tuning is known as the flatness problem in cosmology.

We can solve this if we assume that the early universe went through a period dominated by a cosmological constant. This will ensure that the curvature will be driven exponentially to zero. Since H ≈ constant and a(t) ∝ exp(Ht) if follows that during this period

Chapter 3

Inflation

In the previous chapter we gave some motivation for the need of a new theory to explain different cosmological phenomena observed in the universe. In this chapter we will discuss how a scalar field is able to describe the expansion of the early universe. A consequence of this scalar field is that it will have quantum perturbations, like any scalar field has. Quantization of the scalar field will lead us to the conclusion that quantum fluctuations can be related to the inhomogeneities in the CMB. We will show how theoretical predictions of inflation can be connected to experimental observations. This connection allows for theoretical guidance for model building. Finally, we will look at inflation models and discuss how observations can constrain the energy scale of inflation.

3.1

Conditions for inflation

Inflation is the idea that the very early universe underwent a period of very rapid expansion. The conditions for inflation can be enclosed in three equivalent statements.

¨ a > 0 ←→ ∂ ∂t(aH) −1 < 0 ←→ ρ + 3p < 0 −→ w = p ρ < − 1 3 (3.1) Accelerated expansion implies the decrease of the comoving Hubble radius. The two are equiva-lent, but are just seen from another point of view. A decreasing comoving Hubble radius nicely encapsulates the solution for the horizon problem. The 3th condition implies negative pressure and follows from the Friedmann equation (2.18) if we require positive acceleration. Finally, this would mean that a period dominated by a cosmological constant is sufficient for a period of inflation. We used this already in solving the flatness problem.

However, we need to impose a lower limit on the number of e-folds such that inflation lasts long enough to be in agreement with observations of the CMB.

N = Z tend

tstart

Hdt = H(tend− tend) (3.2)

and is directly related to the scale factor by a(t) = exp(N ). A quick calculation will show that we need at least 60 e-folds after the largest observable scales exit the horizon. The exact number depends on small corrections related to the energy scale of inflation and reheating. Therefore, this number if often approximated as N ≈ 50 − 60.

3.2

Slow roll inflation

To explain the conditions of inflation at a more fundamental level, we can make use of scalar fields. We will study a scalar field for the FRLW metric. The action for a scalar field ϕ, with potential V (ϕ), minimally coupled to gravity, is in general given by

S = Z d4x√−gL, = Z d4x√−g  − 1 2g µν_∂ µϕ∂νϕ − V  . (3.3)

For the FRLW metric in flat space, √−g ≡ p−det(g_µν) = a3. With the Euler-Lagrange equation we can obtain the equation of motion:

∂µδ( √ −gL) δ(∂µ_ϕ) − ∂(√−gL) ∂ϕ = 0 −∂µ(a3∂µϕ) − a3 ∂V ∂ϕ = 0 ¨ ϕ + 3H ˙ϕ − 1 a2∇ 2_{ϕ +}∂V ∂ϕ = 0 (3.4)

This equation of motion for the scalar field is equivalent of that for the harmonic oscillator, and we can recognize the friction term as 3H ˙ϕ. For chaotic inflation (section 3.6.1), this friction term will be of importance and will guaranty inflation.

Now we can proceed to the energy momentum tensor which is given by Tµν = √2 −g δ(√−gL) δgµν = ∂µϕ∂νϕ − δνµ  1 2∂µϕ∂νϕ + V (ϕ)  . (3.5)

Contents 13 T₀0 = −1 2ϕ˙ 2₋ 1 2a2(∇ϕ) 2_{− V (ϕ)} T_ji= δi_j 1 2ϕ˙ 2₋ 1 2a2(∇ϕ) 2_{− V (ϕ)}  T₀i= − ˙ϕ∂iϕ T_i0 = − ˙ϕ∂iϕ (3.6)

We already calculated the components of the energy momentum tensor for the FRWL metric in equation 2.7to be

T₀0= −ρ, T_ji= δ_jiP, T₀i= T_i0 = 0. (3.7)

With the assumption that the scalar field ϕ(x, t) only depends on time ϕ(t), we can neglect the divergence term. This means that the off-diagonal terms become zero. This is justified by the fact that inflation smooths out spacial variation. This allows us to write the density and pressure as ρ = 1 2ϕ˙ 2_{+ V (ϕ)} P = 1 2ϕ˙ 2_{− V (ϕ)} (3.8) w = P ρ = 1 2ϕ˙ 2_{− V (ϕ)} 1 2ϕ˙2+ V (ϕ) (3.9) This can lead to negative pressure and eventually accelerated expansion (w < −1/3) only if the kinetic energy is small compared to the potential energy. This is equivalent to a field slowly rolling down its potential. This will result in a quasi dS-stage (w ≈ −1). To ensure that inflation lasts for a sufficient amount of time, we additionally require that ¨ϕ is small compared to the slope of the potential and speed of the field. Further, we can neglect the divergence term in the equation of motion in3.4. Together, these conditions are known as the slow roll conditions and are given more quantitatively on the left side:

1 2ϕ˙ 2_{V (ϕ) −→ 3M}2 pH2≈ V (ϕ) ≈ constant | ¨ϕ|  |3H ˙ϕ|, |V,ϕ| −→ 3H ˙ϕ ≈ −V,ϕ. (3.10)

The result on the right of the slow roll conditions is the direct implication when the condition is applied to the equation of motion for the scalar field and Friendmann equation. V,ϕ denotes the

first derivative of V (ϕ) with respect to ϕ. From the slow roll conditions it follows that during inflation space-time is quasi dS space: a(t) ≈ eHt. In this thesis we will approximate this by a pure dS space as discussed in section 2.4. The slow roll conditions can be expressed in an equivalent way, known as the slow roll parameters

 = M 2 p 2  V,ϕ V 2 and η = M_p2V,ϕϕ V . (3.11)

During the slow roll regime, , |η|  1 is required. Inflation ends when (ϕend) ≈ 1. With

the use of the slow roll conditions we can also express the number of e-folds in terms of the potential: N (t) = ln aend abegin  ≈ 1 M2 p Z ϕbegin ϕend dϕ V V,ϕ ≈ 1 Mp Z ϕbegin ϕend dϕ√1 2 (3.12) Note that all the conditions for slow roll inflation can be expressed to depend only on the potential of the scalar field. The shape of the potential is one of the main differences between inflationary models. However, any mechanism with a slow roll regime can explain inflation. It is not yet known which mechanism is a good description of the early universe. We will discuss inflationary models in section 3.6.

3.3

Quantum fluctuations during inflation

At quantum scales, empty space is never really empty and will always be in the presence of fluctuating fields. Particles and anti-particles can always pop into existence and annihilate each other for a short time interval, as allowed by the Heisenberg uncertainty principle. During inflation, space expands so rapidly that during this short time interval particles and antiparticles get separated by a distance larger then the horizon H−1. Because the particles are out of causal contact, they cannot annihilate anymore and become real particles. In this section we will study these quantum perturbations and their consequences.

3.3.1 Perturbing the field

Scalar perturbations in the energy momentum tensor will give rise to metric perturbations, meaning we have to study the coupled system of equations to second order. However, it appears that after this intensive calculation one can choose a gauge transformation (spatially flat gauge) for which the metric and inflation scalar perturbations decouple. Since we are free to choose a gauge, we will study perturbations in a fixed space-time background. We will perturb the equation of motion for the scalar field and use this to obtain the second order perturbation for the action.

Using a linear perturbation, we can split the scalar field in a homogeneous background field plus a small fluctuation

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For convenience we will define the scalar field as v = aδϕ. We can now proceed to perturb the equation of motion for the scalar field (3.4). In conformal time the equation of motion is given by

¨

ϕ + 2H ˙ϕ − ∇2ϕ + a2∂V

∂ϕ = 0. (3.14)

We can transform this equation by taking the variation of it, so that we end up with the equation of motion for the perturbed scalar field. This results in

δ ¨ϕ + 2Hδ ˙ϕ − ∇2δϕ = −a2∂

2_V

∂ϕ2δϕ. (3.15)

We can neglect the term on the right due to the slow roll condition (3.10 &3.11). Now we are ready to expend the perturbed field in a Fourier decomposition

δϕ(τ, x) = Z

d3k (2π)3/2δϕke

ik·x_{. (3.16)}

Substituting this in the perturbed equation of motion, and using vk = aδϕk, we will end up

with Z d3k (2π)3/2 1 ae ik·x  v00_k+ (k2−a 00 a)vk  = 0. (3.17)

Equation3.17 can only be true if the term inside the square brackets is zero, and hence we end up with the equation of motion for the Fourier modes:

v_k00+ (k2−a 00 a ) | {z } w2 k(τ ) vk= 0. (3.18)

This equation is the equivalent to a harmonic oscillator with time dependent frequency.

3.3.2 Quantization of the field

If we had perturbed the action to second order, we would have found this normalized action δS(2) = 1 2 Z dτ d3x  (v0)2− (∂iv)2+ a00 av 2  . (3.19)

The equation of motion for the Fourier modes we obtained in equation3.18, can be derived from this second order action. This will be the starting point for the second quantization so that we can promote the scalar field to a quantum scalar field [9]. We will do this in the Heisenberg representation, meaning that operators depend on time and the basis does not change with time.

Our first task is to promote the classical fields to quantum operators, meaning v, π −→ ˆv, ˆπ. The Fourier mode decomposition, expressed in terms of creation (ak) and annihilation (a†_k)

operators, for the field is given by ˆ v(τ, x) = Z _d3_k (2π)3/2  vk(τ )akeik·x+ v∗k(τ )a † ke −ik·x  . (3.20)

The decomposition for the conjugate momentum is similar, but we only need the commutation relation between the two:

[ˆv(x), ˆπ(x0)] = iδ(3)(x − x0). (3.21)

The quantization commutation relations can be expressed in terms of creation and annihilation operators

[ak, a†_k0] = δ(3)(k − k0) and [ak, ak0] = 0 = [ak†, a †

k0]. (3.22)

If we write out equation 3.21 explicit, use that the canonical momentum is given by π = v0 and use the above commutation relations, we will obtain a boundary condition for the mode functions and can normalize this such that

hv_k, vki = (vkv∗k 0_{− v}∗

kv 0

k) = i. (3.23)

And finally our last boundary condition we need is a definition of the vacuum, given by ˆ

ak|0i = 0 ∀ k. (3.24)

The mode functions, vk(τ ), of equation 3.20 satisfy the equation of motion obtained in 3.18.

The general solution to this equation is a Bessel function, which is given by vk(τ ) = α e−ikτ √ 2k  1 − i kτ  + βe ikτ √ 2k  1 + i kτ  (3.25) During inflation, space expands such that modes leave the horizon. We will study the behaviour of the modes on different scales to see how the modes evolve. The wavenumber k is related to the wavelength as k ∝ λ−1.

Subhorizon scales

This corresponds to waves far inside the Hubble radius: λ  (aH)−1 or k  (aH). In the subhorizon limit the k2 term dominates over the a_a00 = 2H2 term. Equation3.18simplifies to

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For this time independent harmonic oscillator, a unique solution exist if we require the vacuum to be the ground state (equation3.24). The solution is given by

vk(τ ) =

e−ikτ √

2k. (3.27)

Additionally, we can use this impose a initial condition for the modes: lim

τ →−∞vk(τ ) =

e−ikτ √

2k. (3.28)

The waves on subhorizon scales behaves as normal oscillating quantum waves in locally Minkowski space.

Superhorizon scales

Here we look at waves much greater than the Hubble radius: λ  (aH)−1 or k  (aH). In the superhorizon limit, k2 a_a00 = 2H = _τ22, so that eqaution3.18 becomes

v_k00+ 2

τ2vk= 0 (3.29)

With the assumption that uk(τ ) ∝ τn, and the Hubble parameter H is constant in time,

we find that τ ∝ a−1 (see equation2.21). Using this, a general solution can be obtained:

vk = B+(k)a + B−(k)a−2 (3.30)

For the growing mode part of the general solution, the fluctuations will be constant outside the horizon:

δϕ = vk

a ∝ constant (3.31)

The Fourier modes freeze on horizon exit (k = H). All scales

Using the boundry conditions 3.28 & 3.23 we can find a unique solution for the mode functions which is valid on all scales

vk(τ ) = e−ikτ √ 2k  1 − i kτ  (3.32) The behaviour of the mode functions imply that during inflation, when the wavelength of the modes are stretched to super-horizon scales, they freeze out on horizon exit. After inflation, when the comoving horizon grows, the modes re-enter the horizon. Because the modes become constant at horizon-exit during inflation, they will stay so at super-horizon scales. This allows us the relate predictions made at horizon-exit to observables after horizon-entry.

3.3.3 Quantum fluctuations in dS-space

We are now ready to calculate the average amplitude for the quantum fluctuations in dS-space. The expectation value, hˆvi, is zero. The variance however is not zero, as we will show:

hvkvk0i = h0|ˆvkvˆk0|0i = |vk|2h0|[ak, a†_k0]|0i = |vk|2δ(k − k0)

= 1 2k  1 + 1 (kτ )2  δ(k − k0) = H 2 2k3  k H 2 + 1  δ(k − k0) (3.33)

Using vk= aδϕk, and considering the super-horizon limit k  H, the variance in Fourier space

is given by lim kHhδϕkδϕk 0i = H 2 2k3δ(k − k 0_{) =} 2π2 k3 Pδϕ(k)δ(k − k 0_{), (3.34)}

where Pδϕ(k) refers to the power spectrum, as explained in appendixB. For super-horizon scales,

we can evaluate the modes at horizon exit (k = aH) and use a Fourier transform to calculate the fluctuations in real space.

hδϕ(t)2i = h0||δ ˆϕ|2|0i = Z _d3_kd3_k0 (2π)3 h0|δϕkδϕk0|0ie i(k−k0)·x = Z kf ki dk k  H 2π 2 = H 2π 2 ln kf ki  = H 2π 2 H∆t = H 2π 2 N (3.35)

This result will be important for eternal inflation.

3.4

Connection to observations

This section provides a more complete picture of inflation and elaborates on the connection between observable quantities and theoretical predictions that can be made for the moment of horizon-exit. For the quantities that are related to observations, we must evaluate them at horizon-exit, because this is the moment of freeze-out, k = aH. Quantities with a lower-script * are related to these measurements. We only have access to the last 50-60 e-folds that happened

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after the moment of freeze-out. The results of CMB observations used in this section are found in [8].

Using the definition of a power spectrum, as explained in appendix B, we can immediately see from equation 3.34 & 3.35 that the power spectrum for the scalar field perturbations in the super-horizon limit is given by

P_δϕ(k) = H 2π 2 = k 3 2π2h|δϕ 2 k|i. (3.36)

Due to inflation, quantum fluctuations, δϕ, are stretched to super-horizon scales. At horizon exit it is convenient to switch to the comoving curvature perturbation R. Intuitively, this quantity can be seen as the curvature perturbation on comoving time slices. In the spatially flat gauge, the relationship between R and δϕ is given by

R = −H ¯ ϕ0δϕ −→ h|Rk| 2_{i =} H ¯ ϕ0 2 h|δϕ|2i. (3.37)

Therefore, we can write the power spectrum for the curvature perturbations as

PR=  H ˙ ϕ 2 Pδϕ=  H ˙ ϕ 2  H 2π 2 k=aH . (3.38)

With the assumption that the scalar field ϕ controls when to end inflation, we can quantify the density perturbations. This is known as the time delay formalism. In regions where we have a positive contribution from the scalar fluctuations, δϕ > 0, inflation will end later in contrast to regions where δϕ < 0, where inflation will end earlier. This will result in density inhomogeneities and ultimately in CMB temperature inhomogeneities because space is spread out more in some regions then others. Quantitatively, we can write this down as

δϕ ˙

ϕ = δt. (3.39)

Using all this, some algebra and appendixB, we arrive at two important formulas that connect theoretical models to experimental data

PR =  V 24π2_M4 p  k=aH −→ ns− 1 = d ln(PR) d ln(k) ≈ 2η − 6. (3.40) The parameter ns is the scalar spectral index and describes how density fluctuations vary with

scale. The power spectrum will slightly deviate from a reference scale, k∗, since H and  are

slowly-varying functions of time. The scalar power spectrum can be normalized for this scale, and is given by PR(k) = As  k k∗ ns−1 , (3.41)

where k∗ is the pivot scale and is chosen as k∗ = 0.05Mpc−1. The measured amplitude is

As= 2.2 × 10−9.

We did not elaborate in detail on gravitational waves produced during inflation. Nevertheless, we will use the result, as derived in [10], which relates gravitational perturbations from the metric to a power spectrum:

P_T =  2H2 π2_M2 P  k=aH ≈  2 3π2 V M_P4  k=aH . (3.42)

The tensor power spectrum can also be normalized such that P_T(k) = At

 k k∗

nt

. (3.43)

The ratio between the two power spectra is given by r = At/As = 16 and is known as the

tensor-to-scalar power ratio.

Experimental observations, such as the Planck satellite, can expand their observations of inho-mogeneities of the CMB in a power spectrum, allowing them to give constrains in the r − ns

plane. These constraints [11], give an indirect limit on the value of the slow roll parameters, and hence on the potential of the scalar field. These constrains are very important, because it will rule out models and give theoretical guidance.

3.5

The energy scale of inflation

Using the latest constraints on inflation, we can obtain an upper limit on the energy scale during inflation. We know from [8] that As = 2.2 × 10−9. If we plug this in to the tensor-to-scalar

power ratio, and use equation3.42, we can express the Hubble parameter as

H = 3 · 10−5  r 0.1 1/2 MP. (3.44)

Since the tensor-to-scalar power ratio is constrained by an upper limit, we can express the energy in terms of the tensor-to-scalar power ratio as

Einf = V1/4= (3H2MP2)1/4= 8 · 10 −3  r 0.1 1/4 MP. (3.45)

This result is normalized for values of r & 0.1 for which the expansion rate of inflation is approximately 10−5MP. For the upper limit on r we find that Einf 6 10−2MP ≈ 1016GeV and

Vinf 6 10−8MP4. The energy scale is approximately at the GUT energy scale. The energy scale

of inflation is, amongst other things, a promising features of inflation. With the use of clever experiments, we might be able to test our GUT theories via CMB observations.

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Additionally, we can relate the change in the field during inflation to the tensor to scalar ratio. The tensor to scalar ratio can be written down as

r = At As = 8 M_P2  ˙ ϕ H 2 = 8 M_P2  dϕ dN 2 . (3.46)

We can transform this into

∆ϕ MP = Z dN r r(N ) 8 ≈ r r 0.01. (3.47)

This result is known as the Lyth bound. B-mode polarization in the CMB due to gravitationally waves is only measurable for r & 10−3. Lyth showed that for these large values, the change in the field is super-Planckian [12]. This can be seen from equation3.47.

3.6

Inflationary models

One of the first models of inflation (old inflation) was developed by A. Guth in 1981 by using a first order phase transition for a field that tunnels from its metastable state to the true vacuum [13]. But one year later it was replaced by a new model (new inflation) by A.D. Linde, based on a second order phase transition, to solve the graceful exit problem [14]. Both of these models require the universe to be in an equilibrium state (ϕ = 0) before inflation, which is somewhat artificial and problematic [15]. In 1983, A.D. Linde developed a new model of inflation (chaotic inflation) that could arise more naturally from the quantum foam [16].

Even though there is a wide variety of inflationary models beyond this, the focus in this thesis will be on chaotic inflation because it will provide an excellent test case for eternal inflation. Inflationary models should be consistent with observations. Since observations of the CMB measure only the last stages of inflation, i.e. V (ϕ) . 10−8M_P4, there may be different models which describe all observational data equally well. The initial conditions for inflation are there-fore important. Furthermore, implementing the inflationary scenario in the context of realistic models of all fundamental interactions should be considered.

3.6.1 Chaotic inflation

In the scenario of chaotic inflation, inflation can occur for a power law potential V (ϕ) ∝ λ nM 4 P  ϕ MP n . (3.48)

For this theory, there is no need for quantum tunneling or equilibrium states to work efficiently. The slow roll regime for any power law potential will occur if the scalar field ϕ was initially large.

This is implied by the slow roll parameter for this type of potentials which yields ϕ  nMP.

The consequence of this is that H is large, and hence the friction term 3H ˙ϕ will cause the field to roll slowly to its minimum. In figure3.1, the rolling of the field in its potential is visualized.

Figure 3.1: The motion of the scalar field for a chaotic inflation potential V = m

2

2 ϕ 2 _is

visualized. The field initially starts and the Planck density and slowly rolls down in the slow roll regime towards the minimum of the potential. Inflation occurs in the region B. Inflation has ended in region C where the field oscillates in the minimum of the potential. In this region, new particles are created and the the begin of the hot big bang theory starts. In the eternal inflation chapter, we will elaborate on the consequence of a field gaining energy and moving upwards. This will happen in region A, where the quantum fluctuations will overcome the classical slope

of the field. Figure adopted from [15].

The only initial condition for this large field inflation model is that is begins at the Planck energy density scale, ρ ∼ M_P4. As the energy density lowers from this starting point we can approach space-time classically, ρ . MP4. For ρ > MP4, the Planck Era, quantum gravitational

effects become important. The fabric of space-time is subject to large quantum fluctuations at this energy density and is sometimes called ”spacetime foam”. In contrast to other inflationary models, other initials conditions for chaotic inflation are not relevant at all. Because classi-cal pieces of space-time can continually emerge from the space-time foam with random initial conditions, in some of them the conditions are just right for inflation to start.

3.6.1.1 Quadratic inflation

For the potential V = 1₂m2ϕ2 we can simple calculate the slow roll parameters (3.11) to be

(ϕ) = η(ϕ) = 2 Mp ϕ

2

. (3.49)

Inflation will end when (ϕend) ≈ 1, what will happen when ϕend ≈

√

2Mp. Using this, we can

Contents 23 N∗(ϕ) = 1 MP Z ϕ_* ϕend dϕ√1 2 = ϕ2∗ 4M2 p −1 2. (3.50)

Now we can express the scalar spectral index and tensor-to-scalar power ratio as ns− 1 ≈ 2η − 6 = −4 2N∗+ 1 N∗=60 −−−−→ ns= 0.97 (3.51) r ≈ 16 = 16 2N∗+ 1 N∗=60 −−−−→ r = 0.13. (3.52) Unfortunately, this type of model is disfavored by the latest constraints from the Planck Col-laboration. Nevertheless, the method for obtaining these parameters is generally the same for other models.

Additionally, we can obtain a value for the mass of the inflaton field. We can do this by using the power spectrum for the scalar perturbations (equation3.40), because this value is measured to be As= 2.2 · 10−9. As implied by the slow roll condition, we can approximate equation 3.50

such that N∗≈ ϕ2∗ 4M2 P . (3.53) For N∗ = 60, we obtain ϕ∗ = 2 √

N∗MP = 15.5MP. Comparing this quantity with ϕend,

we conclude that for this type of model the field displacement is Super-Planckian. With the previous approximation, we obtain for the slow roll parameters that

(ϕ) = η(ϕ) ≈ 1

2N. (3.54)

Substituting these results in to equation 3.40yields PR≈ m2_N2 ∗ 6π2_M2 P . (3.55)

Eternal inflation

In the previous chapter we studied the mechanism of inflation. Even though there are some degrees of freedom in inflation, the paradigm of inflation is very elegant in solving the cos-mological problems and explaining several phenomena related to, amongst others, the CMB inhomogeneities. This confidence in inflation has motivated to study the consequences of the quantum fluctuations of the scalar field during inflation. In this chapter we will study this and derive a condition for models to be future eternal. Many models will satisfy this condition, and hence will produce a multiverse. We will focus in detail on how eternal inflation (EI) can occur in chaotic inflation, and only briefly describe how it can occur in other models.

To see how EI relates to the scientific method, we will elaborate on what problems arise when making predictions and observations for the multiverse. Furthermore, we will discuss how EI can be falsified. The philosophical consequences of EI that one can invoke are fascinating and are briefly discussed in the last section of this chapter. The main references used for this chapter are [15,17–19].

The terminology can be somewhat confusing now that we will upgrade from one universe to infinitely many. In the context of EI we will refer to these infinitely many regions of space where inflation has ended as ”pocket”, ”mini”, ”local” or ”bubble” universes. We inhabit one of these regions. Whereas we will refer to the collection of these universes as the multiverse.

4.1

Slow roll eternal inflation

For inflation to occur, the slope of the inflation field must be very flat, as quantified in equation

3.10. To see why inflation might be eternal, we will look at the quantum fluctuations and compare these to the slope of the field. The total change in the field is given by

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∆ϕ = ∆ϕcl+ δϕ. (4.1)

For normal slow roll inflation, the field is losing energy due to ∆ϕcl. But whenever the quantum

fluctuations, δϕ, exceed this classical change in energy, the field will gain energy. The field will now move upwards in the potential instead of downwards.This is visualized for chaotic inflation in figure 3.1.

To see what the the consequence of this will be, we will look what will happen during one Hubble time ∆t = H−1 for a region of one Hubble volume H−3. During this one e-fold, 3-dimensional space will expand a factor e3 ≈ 20. At the end of one Hubble time, one Hubble volume has been inflated to 20 causally disconnected Hubble volumes. For each of these regions of space, equation

4.1applies since they will continue to evolve independently of other Hubble sized regions. So, if the probability of the field moving upwards is greater then 1/e3≈ 1/20 during one Hubble time, their will be at least one Hubble volume for which the field has moved upwards. For a Gaussian distribution function with a standard deviation of 1σ, the probability that a random variable will exceed its mean is approximately 0.159. This means that in 20 × 0.159 ≈ 3.17 regions of space, the field will move upwards instead of downwards. Since we only looked at one e-fold, this process will repeat itself and slow roll inflation will never end because there will always be regions of space where the field never reaches its minimum, and hence inflation is eternal. To quantify the probabilities needed for EI, we can calculate the probability for the quantum fluctuations to be greater then the classical change in the field using a Gaussian probability [17], which will result in

σδϕ& 0.61|∆ϕcl|. (4.2)

Using equation 3.35, we find that the standard deviation for δϕ is given by σδϕ=phδϕ2i − hδϕi2=

H

2π. (4.3)

Further, in the slow roll regime, for one e-fold, it follows from3.10 that ∆ϕcl = ˙ ϕ H = V,ϕ V M 2 p. (4.4)

Using all this, we can impose a general condition on the slow roll parameter for EI to occur as  . V (ϕ)

(0.61)2_24π2_M4 P

. (4.5)

In this condition we recognize the scalar power spectrum and conclude that PR & 1, which can

be approximated such that δρ/ρ & 1. So the EI behaviour on ultra-large scales, much larger than the current Hubble radius, may be extremely inhomogeneous. This is at odds with our assumption of large scale homogeneity and thus with the FRLW metric.

We already showed that normal inflation will happen when the slow roll parameters are satisfied (3.10). The slope of the potential for this regime is very flat. In the EI regime, the slope of the potential is even flatter. This can be seen if we rewrite the EI condition as

12(0.61)2M_P6V

2 ,ϕ

V3 . 1. (4.6)

Everything we have discussed so far about EI, implies only that inflation, once started, continues indefinitely into the future. It is currently an open question whether EI can avoid a beginning of the universe by being past-eternal.

4.1.1 Chaotic eternal inflation

For a simple chaotic inflationary model, as introduced in section 3.6.1, we can use the EI condition (4.5) to see what constraints are imposed for the model to be eternal. This will result in  MP ϕ 2 . 1 4 · 24π2 m2ϕ2 M4 P . (4.7)

This condition will give a constraint on ∆ϕ and V (ϕ) such that

∆ϕ & 1.7 · 103MP

V (ϕ) & 10−5M_P4.

(4.8)

This result indicates that the energy scale for the potential may occur at energy densities which are much smaller then the Planck density. In the 1986 paper from Linde [20], he basically obtained the same result, but uses slightly different values for the parameters.

4.1.2 Other eternal inflation models

Not only chaotic inflation is eternal in the future. False vacuum inflation models can also be eternal in the future. These models are characterized by metastable and true vacuum states. If the field in false vacuum inflation is initially located at some false vacuum state, it can tunnel quantum mechanically from one false vacuum to another if the potential barrier is low enough. While doing so, the field rolls to is minimum and creates locally a new bubble-universe. This process is known as bubble nucleation and happens for any phase transition. In this view our bubble-universe was born in a tunnelling event from a neighboring vacuum.

But, in a rapidly expanding multiverse, the rate of expanding space must be much faster then the exponentially decreasing tunneling rate. If the expansion is fast enough, the growth of inflating space will be faster than its conversion into bubbles. So bubble formation end inflation

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only locally. The space between bubbles is growing, while the bubbles themselves are also growing. For this reason, the false vacuum will never disappear and inflation will never end. Thus, inflation is eternal if the rate of bubble formation is less then one bubble per Hubble volume in one Hubble time.

EI can happen in many models, as long as the EI conditions are satisfied.

4.2

Remaining issues

In general, is it difficult to construct an inflationary model which is not eternal. EI will happen for PR & 1. The scalar power spectrum, PR ∝ V3/2/|V,ϕ|, will for most inflation potentials

grow without bound as ϕ −→ ∞. For this reason many inflationary models will allow for EI. We already hinted at some problems that arise in EI. The validity of the cosmological principle becomes doubt-full at the scale of the multiverse. But there are more problems, such as the back-reaction of quantum fluctuations on spacetime, and it is unclear if EI is past-eternal. Moreover, we do not fully understand how the quantum vacuum couples to gravity. A better understanding of quantum gravity/high energy physics is required to be more confident about the EI claims. This makes studying inflation in string theory interesting.

Besides this, for superplanckian models one needs to consider higher order corrections to the potential. One should either fine tune all these parameters, but this isn’t really an option. So this leave us with the option to invoke a symmetry requirement such that the higher order corrections become irrelevant. This can be done with a shift-symmetry.

Another problems is making predictions from an EI theory, this is known as the measure prob-lem. Any good theory can make predictions and has some way of confirming these predictions via experiments. Besides that, falsifiability should be considered to distinguish the scientific from the unscientific. However, making predictions for an eternally inflating universe raises some problems. Since EI guarantees an infinite amount of bubble-universes, each with possibly different low energy physics, anything will happen an infinite number of times. So, calculating the relative probability for the fraction of universes with a particular property will result in ∞/∞, a meaningless ratio. Some sort of method of regularization is needed to make meaningful predictions for EI.

4.3

Observations

The success of inflation is partly because it can be related to observable quantities such as the CMB. These observations do not only give guidance for theoretical models, but also make

the theory empirical. To confirm of falsify EI, we need to make predictions in the theory and related them to observable quantities. Making predictions is somewhat problematic as we have discussed. Finding observable quantities is also challenging, because causal contact is limited to the Hubble radius, our observable universe. We cannot even communicate with everything within our own bubble-universe. So naively one would think that finding observable quantities of other bubble-universes seems hopeless.

However, we might be able to observe consequences of other bubble-universe. As discussed in section 4.1.2, bubble-universes can be created via bubble nucleation. Or for chaotic EI in regions where inflation has ended. As the bubbles grow, they undergo an unlimited number of collisions with other bubbles from the same false vacuum background. Most of these bubble universes will not collide with each other, since they are to far apart and the space between them is expanding to fast. But some bubbles will be created close to one another such that they can collide as they expand. We will not elaborate on the mathematical details of bubble collisions, but instead describe the concept and leave out many mathematical conditions and details.

Depending on the type of collision, future observers can exist in the bubble and the collision will cause a perturbation in the inflaton scalar field and space-time metric. We already calculated the quantum perturbations for normal inflation and concluded that scalar field perturbations give rise to temperature fluctuations in the CMB and tensor perturbations will generate B-mode polarization of the CMB. Something similar can happen as a consequence of a bubble collision. Scalar perturbations caused by the collision may lead to a cold spot in the CMB temperature fluctuations, whereas tensor perturbations may cause E-mode polarization [21]. A visualization of a bubble collision in a conformal diagram is shown in figure 4.1.

Figure 4.1: Conformal diagram for a bubble collision. The red dot is the collision point for two expanding bubbles. The green dashed hyperbolic lines are constant slices of time in the bubble. The collision will inject energy in the inflaton field, changing the initial conditions what will lead to more e-folds for this region. Because of this, the physics in region A, B & C will change, depending on the initial conditions for this region. Eventually, for observers on earth, this will appear as a cold disk in the CMB temperature fluctuations. Image adopted from [22].

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The effects of collisions are expected to be sub-dominant, so in principle they could be observ-able. The exact details of such a collision depends upon the collision parameters and inflation potential. So it is at least theoretically possible that our universe underwent a collision in the past, which would be detectable. First observational tests have been made, but more detailed observations are necessary to rule out collision or detect them with certainty [23, 24]. This research motivates the importance of more detailed CMB observations in the temperature fluc-tuations and B and E-mode polarization. It is important to notice that not observing a bubble collision does not falsify EI. Not all EI models have bubbles collisions, and only some will result in detectable bubble collisions. Furthermore, it might be that observable quantities will only be measurable in the future.

One condition is that each bubble universe contains an open FRLW cosmology with negative spatial curvature. The curvature of a spatially negative universe (open universe) corresponds to Ω < 0. So, if we would inhabit a universe with Ω > 0, then all types of EI are falsified. However, if we inhabit a relatively non-uniform universe, one would expect small irregularities in the curvature.

4.4

Philosophical discussion

The ultimate fate of our bubble-universe is rather sad, because of the expansion it will eventu-ally lead to a heat death. Other bubble-universe might collapse in on itself. However, since EI is eternal in the future, the universe as a whole will never disappear. EI makes the universe immor-tal and possibly past-eternal such that a initial singularity can be avoided. EI will lead to some interesting conclusions regarding our place in the universe. We will explain how the multiverse might be able to explain the fine tuning problem. Note that the philosophical ideas presented in this chapter are open for discussion and subject to one’s philosophical interpretation.

4.4.1 The weak anthropic principle

In chaotic EI, the potential may wander close to the Planck energy density. At this point it may cause large quantum fluctuations in other scalar fields. This may lead to changes in elementary particle physics inside other bubble-universes. And hence, we can end up with different low energy physics inside every bubble-universe [18,20]. For false vacuum EI, each universe may also end up with different low energy physics.

Since the physical constants can take a different value in each bubble-universe, and EI will produce an infinite amount of bubble-universes, all possibilities are realized. A consequence of this reasoning is that the fine tuning problem of our universe becomes irrelevant. Since

all possibilities are realized, there will always be universes with the right physical constants such that intelligent life of our type can evolve. And it is only in those bubble-universes where questions about fine tuning are asked. This reasoning is known as the (weak) anthropic principle (WAP).

In section 2.4we already elaborated on the cosmological constant. It seems that this constant is the most fine tuned parameter in our universe. Weinberg calculated in 1987 that the value of |Λ| can only vary 2 orders of magnitude from its current value |Λ| . 10−120_M4

P for intelligent

life of our type to exist [25]. In string theory, compactification of extra dimensions will lead to different low energy physics. Because of the exponentially large variety of compactification of extra dimensions[26], the value of Λ becomes statistically typical. The value of Λ in our bubble-universe is suitable for intelligent life of our type to evolve. In other bubble-universes, where the value of Λ might be quite different, there is no intelligent life of our type to question its value. EI increased in popularity when string theorist had to deal with a large number of vacuum states in the landscape. The multiverse provided a good framework to explain this. The WAP is based on theories that lack complete understanding such as string theory, or lack empirical support such as EI. Additionally, when the WAP is adopted, we give up that there is a more fundamental theory that predict the value of physical constant. We should be cautious in adopting the WAP. As long as we do not fully understand the theories that predict the WAP, the WAP should be seen as an explanation of last resort.

Chapter 5

Conclusion

The inflationary paradigm is able to explain numerous properties of the universe such as the horizon problem and gives an explanation for the CMB anisotropy which fall in line with obser-vations. Even though there are some degrees of freedom in the inflation paradigm, it all seems to fit nicely with observations. It would be difficult to come up with an alternative theory that would be able to explain all these properties of the universe. Inflation has improved our knowledge of the early universe vastly. However, physicists do not fully understand inflation yet because the precise mechanism is unknown. Since observations of the CMB are related to inflationary models, experimental constraints will exclude models as more precise observations are made.

It is well worth the effort to keep exploring the early moments of the universe, because the early universe underwent high energies. Energies that are far from reachable in laboratories at this moment. Through clever tricks, we might be able to test GUT theories or eventually quantum gravity.

The success of describing the power spectrum of the CMB through quantum fluctuations in the inflaton scalar field has led to a more detailed study of these fluctuations and hence to slow roll eternal inflation. There is good theoretical motivation for eternal inflation, but there is lack of complete understanding and experimental support. So evidently more research has to be done to claim a confident understanding of the multiverse. To confirm or falsify eternal inflation, more detailed observations of the CMB temperature fluctuations and polarization are needed. Although eternal inflation has fascinating consequences, our understanding of it remains in-complete and speculation. But the pursue of it may lead to a deeper understanding of the universe.

General relativity

General relativity(GR) is a broad subject that studies curved space-time and its relation to gravity. For this thesis we have used the most important results from GR as derived in [1]. The definitions that where used throughout this thesis will be given.

Curvature manifest itself through the Christoffel symbol and is motivated by the need for a covariant derivative. The Christoffel symbol encodes all the information necessary to take the covariant derivative of a tensor, and is given by

Γσ_µν= 1 2g

σρ_(∂

µgνρ+ ∂νgρµ− ∂ρgµν). (A.1)

The covariant derivative for a tensor of arbitrary rank is given by ∇σTνµ11νµ22...ν...µlk = ∂σT µ1µ2...µk ν1ν2...νl + Γ µ1 σλT λµ2...µk ν1ν2...νl + Γ µ2 σλT µ1λ...µk ν1ν2...νl + ... − Γλ σν1T µ1µ2...µk λν2...νl − Γ λ σν2T µ1µ2...µk ν1λ...νl − .... (A.2)

So that the covariant derivative for a tensor of rank 2 becomes

∇µTµν = ∂µTµν+ Γ_µλµ Tλν+ ΓνµλTµλ. (A.3)

To describe the properties of space-time, we need the Ricci tensor, which is given by

Rµν = Rλµλν = ∂λΓλµν− ∂νΓλµλ + ΓλλσΓσνµ− ΓλνΓσλµ. (A.4)

The trace of the Ricci tensor is the Ricci scalar:

R = Rµ_µ= gµνRµν. (A.5)

For a given metric, we can use the above definitions to calculate the Einstein equation, which including a cosmological constant (Λ), is given by

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Rµν−

1

Power spectrum

The power spectrum is a way of quantifying the power for a given wavelength. For an arbitrary quantity f (t, x) that has a Fourier transform

f (t, x) = Z d3k (2π)3/2e ik·x_f k(t), (B.1)

the power spectrum, Pf(k) is defined as

hf_kf_k∗₀i = δ(3)(k − k0)2π

2

k3 Pf(k). (B.2)

The variance for this quantity f (t, x) is given by hf (t, x)2i = Z dk k Pf(k) = Z dln(k)Pf(k). (B.3)

The power spectrum gives the power per logarithmic momentum interval. Since the work of Harrison and Zeldovich, we know that Pf ∝ knf−1. This allows us to define the spectral index

nf(k) as follows

nf(k) = 1 +

dln(Pf(k))

dln(k) . (B.4)

A scale-invariant Harrison-Zeldovich-Peebles spectrum corresponds to nf = 1. For such a value

the power spectrum is constant.