Winitzki - ion Thesis 2009

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Probability Measure in Multiverse Cosmology

Sergei Winitzki

Habilitationsschrift

des Departments fur Physik, Ludwig-Maximilians Universitat, Theresienstr.

37, 80333 Munich, fur das Fach Physik, vorgelegt November 6, 2008

von Dr. Sergei Winitzki


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Abstract

In generic models of cosmological inflation, the geometry of spacetime is highly inhomogeneouson scales of many Hubble sizes, consisting of infinitely many causally disconnected pocket uni-

verses. The values of cosmological observables and even of the low-energy coupling constantsand particle masses may vary among the pocket universes. String-theoretic landscape modelspresent a similar structure of a multiverse where an infinite number of de Sitter, asymptoti-cally flat (Minkowski), and anti-de Sitter pocket universes are nucleated via quantum tunneling.Since observers on Earth have no information about their location within the eternally inflatingmultiverse, the main question in this context has been that of obtaining statistical predictionsfor quantities observed at a random location.I discuss the long-standing technical and conceptual problems arising within this statisticalframework, known collectively as the measure problem in multiverse cosmology. After review-ing various existing approaches and mathematical techniques developed in the past two decadesfor studying these issues, I describe a new proposal for a measure in the multiverse, called the

reheating-volume (RV) measure. The RV measure is based on approximating an infinite multi-verse by a family of progressively larger but finite multiverses. Such multiverses occur seldombut are allowed by all cosmological multiverse models. I give a detailed description of the newmeasure and its applications to generic models of eternal inflation of random-walk type and tolandscape scenarios. The RV prescription is formulated differently for scenarios with eternalinflation of the random walk type and for landscape scenarios. In each case I show in a math-ematically rigorous manner that the RV measure yields well-defined results that are invariantwith respect to general coordinate transformations, independent of the initial conditions at thebeginning of inflation, and free of the youngness paradox and the Boltzmann brain problemsaffecting some of the previously proposed measures.For models of random-walk inflation, the RV cutoff considers events where one has a finite (al-

though large) total reheating volume to the future of an initial Hubble patch. I derive a generalformula for RV-regulated probability distributions that is suitable for numerical computations.Explicit analytic computations are presented in a toy model having an effective potential withan exactly flat top.For landscape scenarios, I propose to calculate the distribution of observable quantities in alandscape that is conditioned in probability to nucleate a finite total number of bubbles to thefuture of an initial bubble. A general formula for the relative number of bubbles of differenttypes can be derived. I show that the RV measure yields results independent of the choice of theinitial bubble type, as long as that type supports further bubble nucleation. As an illustration,I present explicit results for a toy landscape containing four vacuum states and for landscapeswith a single high-energy vacuum and a large number of low-energy vacua.

This dissertation is submitted in partial fulfillment of requirements for the degree of Dr. habil. inphysics. I certify that I have prepared this text through my own effort and no other sources orquotations except those listed and attributed.Selbststandigkeitserklarung

Hiermit erklare ich, dass ich die vorliegende Arbeit in allen Teilen selbstandig verfasst und keineanderen als die angegebenen Quellen und Hilfsmittel (einschlielich elektronischer Medien undOnline-Quellen) benutzt habe.Sergei Winitzki


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Contents

Preface iii

1 Introduction 1

2 Eternal inflation 52.1 Cosmological inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 The inflationary paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Inflationary models: slow roll . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Eternal inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Predictions in eternal inflation . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Physical justifications of the semiclassical picture . . . . . . . . . . . . . . 15

3 Stochastic approach to inflation 173.1 Random walk-type eternal inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Fokker-Planck equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Methods of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.3 Gauge dependence issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Self-reproduction of tunneling type . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Predictions and measure issues 294.1 Presence of eternal inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Observer-based measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Regularization for a single reheating surface . . . . . . . . . . . . . . . . . . . . . 33

4.4 Regularization for multiple types of reheating surfaces . . . . . . . . . . . . . . . 354.5 The Youngness paradox and the Boltzmann brains . . . . . . . . . . . . . . . . 37

5 A new measure for multiverse cosmology 395.1 Reheating-volume cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 RV cutoff in slow-roll inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 The RV measure for random-walk inflation 476.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2.2 Probability of finite inflation . . . . . . . . . . . . . . . . . . . . . . . . . 526.2.3 Finitely produced volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.4 Asymptotics of(V; 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.5 Distribution of a fluctuating field . . . . . . . . . . . . . . . . . . . . . . . 556.2.6 Toy model of inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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Contents

6.3.1 Positive solutions of nonlinear equations . . . . . . . . . . . . . . . . . . . 566.3.2 Nonlinear Fokker-Planck equations . . . . . . . . . . . . . . . . . . . . . . 586.3.3 Singularities ofg(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.3.4 FPRV distribution of a field Q . . . . . . . . . . . . . . . . . . . . . . . . 656.3.5 Calculations for an inflationary model . . . . . . . . . . . . . . . . . . . . 68

7 The RV measure for the landscape 757.1 Regulating the number of terminal bubbles . . . . . . . . . . . . . . . . . . . . . 777.2 Regulating the total number of bubbles . . . . . . . . . . . . . . . . . . . . . . . 827.3 A toy landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.3.1 Bubble abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.3.2 Boltzmann brains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4 A general landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.4.1 Bubble abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4.2 Example landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.4.3 Boltzmann brains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.4.4 Derivation of Eq. (7.103) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.4.5 Eigenvalues of M(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4.6 The root of0(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8 Conclusion 111

Bibliography 113

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Preface

This dissertation is based on the authors original work on the so-called measure problem inmultiverse cosmology. The measure problem has resisted solution for more than two decadessince its original formulation in the 1980s, which was in the context of cosmological inflationdriven by a scalar field. At present, the renewed interest in the measure problem is due to thediscovery of the string-theoretic landscape, which allows a very large number of metastable vacuawith different physical laws. The need to extract predictions from these models has spurred aflurry of activity resulting in the appearance of several competing measure proposals and a

deeper understanding of the issues involved.This dissertation gives an overview of the presently known approaches to the measure problem

and proceeds to describe the authors own measure proposal as developed in a series of recentpublications. Currently the measure problem in multiverse cosmology is an active area of re-search, and more progress may be achieved in the near future. Rather than trying to anticipatethe forthcoming results, the author wishes to concentrate on the exposition of concepts andmathematical methods that will remain useful for future research in this area.

Acknowledgments

The author is grateful to the Department of Physics at the Ludwig-Maximilians University in

Munich where the author has been employed for several productive years of research and to Prof.V. F. Mukhanov for guidance and ample help. Numerous conversations with Andrei Barvinsky,Martin Bucher, Cedric Deffayet, Gabriel Lopes Cardoso, Jaume Garriga, Josef Ganer, MatthewJohnson, Andrei Linde, Slava Mukhanov, Matthew Parry, Misao Sasaki, Takahiro Tanaka, VitalyVanchurin, and Alexander Vilenkin, as well as with other colleagues have been valuable andstimulated the authors thinking as well as corrected misconceptions on the authors part. Partof this work was completed on a visit to the Yukawa Institute of Theoretical Physics (Universityof Kyoto) and was supported by the Yukawa International Program for Quark-Hadron Sciences.

The entire text was typeset with the excellent LYX and TEX document preparation system ex-clusively on computers running Debian GNU/Linux. The author expresses profound gratitudeto the creators and maintainers of this outstanding free software.

Sergei Winitzki, October 2008

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1 Introduction

Cosmology is the branch of physics concerned with description of the Universe at large asmanifested by large-scale astronomical observations. Since the discovery of the expansion ofthe Universe by Hubble and the development of the general relativistic model of non-stationaryhomogeneous spacetime by Friedmann, the accepted point of view has been that the Universeis expanding and has been in an extremely different state in the distant past. How the Universeevolved to its present condition and how it began (if it had a beginning) are among the mainquestions modern cosmology hopes to answer.

Although the goal of describing the entire Universe might seem to require a theory of ev-erything, which perhaps will never be constructed but in any case is presently not available, itturns out that many important cosmological observations could be explained in the frameworkof currently available physical theories, in particular the known high-energy physics and classi-cal gravity. G. Gamow introduced a model of the Universe expanding from an extremely hotand dense state (hot Big Bang). In what concerns the evolution of the Universe after the BigBang, this scenario is in such a good agreement with observations that it is now considered to bethe standard cosmology. However, the standard cosmology leaves several important questionsunanswered. For instance, the initial hot state of the Universe turned out to be rather special,and the standard cosmology failed to adequately explain its origin.

This is why one of the central problems of cosmology today is to describe the era beforethe expansion described by the standard hot Big Bang scenario. Because of the present lack ofdetailed knowledge of high-energy physics and quantum gravity, as well as of insufficient precisionof available astrophysical observations, numerous cosmological models of the very early Universecompete on more or less equal footing.

The currently popular and observationally well-supported models of the very early Universe are

inflationary models. The scenario of inflation assumes an epoch of an extremely fast expansion(inflation) of the Universe, followed by reheating to a hot thermal state. The latter becomesthe starting point of the standard hot Big Bang model.

It was found early on that in most of these models inflation does not stop everywhere at thesame time. As a result, the universe at extremely large distance scales is divided into domains

with dramatically different properties: Some regions have already thermalized and developedmatter structures such as stars and galaxies, while other regions still undergo the inflationaryexpansion and are cold and empty. Therefore, a radical departure from homogeneity can beexpected at extremely large distance scales. Generically, at arbitrarily late times there existlarge domains that are still inflating. This phenomenon was called eternal inflation (moreprecisely one can refer to future-eternal inflation).

In some models, the various thermalized regions may also differ from each other in observablecosmological parameters or even in values of the coupling constants observed in low-energyphysics. There are two problems with such models: First, a model that allows a wide range ofparameters to be observed in various regions of the Universe has little predictive power, sincewe cannot determine which region we happen to inhabit. Second, the existence of a (typically

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1 Introduction

infinite) multitude of regions of space that are too far from us to be ever observable and/or theassumption of the a priori unobservable many-universe ensemble are not directly comparablewith the experiment.

A similar situation is found in the recently discoveredlandscape of string theory. It was foundthat string theory admits an exceedingly large number (of order 101000) of disjoint, metastablevacuum states. Different vacua have different values of the effective cosmological constant, cou-pling constants of low-energy physics, and particle masses. Transitions between these states arepossible through bubble nucleation; the interior of each bubble appears to the interior observersas an infinite homogeneous open universe (if one disregards rare bubble collisions). For thisreason, the interior of a bubble has been called a pocket universe. The presently observeduniverse is situated within a bubble where the vacuum state is, in some sense, randomly chosen.Eternal inflation occurs generically in this setting and produces an infinite number of bubbles,each containing a pocket universe in a particular vacuum state. This bewildering array of uni-verses is currently referred to as a multiverse in order to stress the fact that different pocket

universes (as well as different regions within a single pocket universe) are causally disconnectedand are seen by observers as separate universes. It appears to be impossible to predict withcertainty the values of the cosmological parameters that we will measure in our present positionin the multiverse.

To overcome these problems, one may change focus and concentrate on obtaining the proba-bility distribution of values for the measured cosmological observables, such as the cosmologicalconstant, coupling constants, or particle masses. Heuristically, one would like to compute prob-ability distributions of the cosmological parameters as measured by an observer randomly lo-cated in the spacetime. This idea, sometimes called the principle of mediocrity, has been firstclearly formulated in the mid-1990s [1, 2, 3]. One hopes that this procedure not only providesresults that are in principle testable by observation, but also indirectly confirms the existence

of the otherwise unobservable regions of space. This was the program outlined in those earlyworks on eternal inflation.

However, one runs into an immediate problem when one tries to extract statistical predictionsfor cosmological observables in this setting. The main diffuculty is due to the infinite volumeof regions where an observer may be located. Inded, an eternally inflating universe containsan infinite, inhomogeneous, and topologically complicated spacelike hypersurface (the reheatingsurface) where observers may be expected to appear with a constant density per unit 3-volume.

In the landscape scenarios, one encounters a kind of infinity that is in some sense more ill-behaved than in the random-walk inflationary scenarios. Not only each pocket universe maycontain infinitely many observers, but also the number of different pocket universes in the entirespacetime is infinite. Pocket universes of different types are not statistically equivalent to eachother because they have different rates of nucleation of other pocket universes. There seems tobe no natural ordering on the set of all pocket universes throughout the spacetime, since mostof the pocket universes are spacelike separated.

In both these contexts random-walk type models and the landscape models one canview eternal inflation as a stochastic process that generates a topologically complicated andnoncompact locus of points where observers may appear. A random location of an observerwithin that locus is a mathematically undefined concept, similarly to the concept of an integernumber uniformly chosen among all the integers, or a real number uniformly chosen amongall the reals. This is the root cause of the technical and conceptual difficulties known collectivelyas the measure problem in multiverse cosmology. Nevertheless, one may try to formulate a

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prescription for calculating observer-weighted probabilities of events. Such a prescription, alsocalled a measure, should in some sense correspond to the intuitive notion of probability ofobservation at a random location in the spacetime. These issues are discussed in Chapter 4.

Several measure prescriptions have been proposed in the literature. Below in Sec. 4.2 theseproposals will be reviewed and their contrasting features will be characterized. Almost all of theexisting prescriptions are based on cutting off the infinite volume of space by a certain geometricconstruction. Thus, one considers a finite subset of the total volume where observers may appear;the subset is characterized by a regulating parameter, such as the largest scale factor attained bythe observers or another geometric parameter. Then one gathers statistics throughout the finitepart of the volume, and finally takes the limit as the regulating parameter tends to infinity. Thelimit usually exists and yields a certain probability distribution for cosmological observables.Unfortunately, it turned out that the limiting distribution depends sensitively on the choiceof the regularizing procedure. Since a natural mathematically consistent definition of themeasure is absent, one judges a measure prescription viable if its predictions are not obviously

pathological. Possible pathologies include the dependence on choice of spacetime coordinates,the youngness paradox, and the Boltzmann brain problem, to be discussed in more detailbelow.

The main goal of this dissertation is to report on a novel class of measure prescriptions thatwere introduced in the authors recent publications, which form the basis of Chapters 5-7. Thisclass of measures is not based on regulating the spacetime by any geometric construction, butrather on manipulating the events in the probability space in order to obtain a well-definedsubensemble of finite multiverses. A finite multiverse can be generated by rare chance if infla-tion ends everywhere; the probability of this event is small but nonzero. One can characterizethe size of a finite multiverse in some way, e.g. by specifying the total volume of the reheatingregions (which will be finite), the total number of nucleated bubbles, or another such number

considered as a regulating parameter. In this way, one obtains a sequence of finite multiversesthat in a well-defined sense approximate the actual, infinite multiverse as the regulating param-eter tends to infinity. It is then expected that the probability distribution of any observable willtend to a well-defined limit. That limit is the prediction of the new measure.

I work out in detail the mathematical formalism necessary for computations in the new mea-sure, both in the case of random-walk inflation (Chapter 6) and in the case of a landscape(Chapter 7). The computations turn out to be cumbersome since the possibility of creating afinite multiverse is difficult to describe explicitly, especially in the limit of a very large size ofthe finite multiverse. Nevertheless, it is possible to obtain direct results and general proofs forvarious properties of the new measure. This is achieved by different methods in random-walketernal inflation and in landscape scenarios. In each case I derived explicit formulas for the

predictions of the RV measure so as to make the final computations more tractable if a specificmodel is given.

In the Conclusion, I summarize the results obtained in this study and discuss some problemsand possibilities for future research.

Throughout most of the exposition I use the Planck units, c = = G = 1, which correspondsto measuring energy in units of the Planck mass MP 1.2 1019GeV, time in the Planck times 1043 sec, and so on.

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2 Eternal inflation

Cosmological inflation is a currently a dominant framework in theoretical cosmology. I will brieflyoutline its origins and main postulates as far as is necessary to set the context for the main partof this work. For early reviews of inflation, see e.g. [4] and the book [5]. Eternal inflation andthe accompanying measure problem are reviewed and discussed in Refs. [6, 7, 8, 9, 10, 11].

2.1 Cosmological inflation

We begin with the story of inflation. While the hot big bang cosmological scenario has beenwidely accepted after the detection of the cosmic microwave background (CMB) radiation andexplanation of the nucleosynthesis, several poorly explained and contradictory facts remained.The main problems of the standard hot cosmology were the horizon problem and the flatnessproblem.

The horizon problem stems from the fact that the observed CMB is highly isotropic, withrelative temperature variation 105 (see, for example, [12]). The CMB radiation is comingfrom the last scattering surface, which at the time of decoupling consisted of a large numberof causally disconnected horizon-size regions, each region occupying about 2 of todays sky.However, observations of the high degree of homogeneity of the CMB temperature suggestthat all these regions had had a nearly equal temperature at the time of last scattering. Thisabsence of large-scale fluctuations is difficult to explain unless we assume rather unnatural-looking, extremely uniform initial conditions.

The flatness problem is also in a sense a problem of initial conditions: the Universe mustinitially have been unnaturally close to flat. The density parameter evolves in such a way thatany deviation from = 1 grows with time,

| 1| t2 43(1+w) a1+3w, (2.1)

which can also be expressed through the temperature T as

|

1|

T(1+3w). (2.2)

Since the value of at present is of order 1, it must have been extremely close to 1 at earlytimes. For instance, at Planck temperatures TP 1019 GeV with w = 1/3 and Tnow = 3 K oneobtains the estimate

|P 1| = |now 1|

TnowTP

2 1060. (2.3)

If one assumes a more natural initial condition for , for example, that 1 at Planck time,then one finds that the Universe would have either collapsed within a few Planck times if > 1,or cooled down to the present temperature of 3 K within 1011 sec if < 1 [13]. It is hard toexplain this exceptional fine-tuning of the initial matter density.

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2 Eternal inflation

These problems are not the only faults of the standard scenario. For instance, explanationof the origin of structure in the framework of the standard model is also problematic. Thedescription of growth of density fluctuations of matter due to gravitational instability [14, 15, 16]

would provide an explanation for the formation of stars and galaxies if the initial fluctuationshad a scale-invariant power spectrum at horizon crossing [14, 17, 18]

P (k) kn, n 1. (2.4)However, the wavelength of a galaxy-scale fluctuation must have been at early times largerthan the horizon size. Such a perturbation is difficult to explain by a causal mechanism. Thestandard model simply assumes a homogeneous initial state and neither explains how the initialfluctuations occurred nor predicts their spectrum.

These and other shortcomings of the standard hot scenario were sufficiently compelling sothat the concept of cosmological inflation was relatively quickly accepted when its advantageswere first clearly advocated by A. Guth [19].

2.1.1 The inflationary paradigm

Cosmological inflation is a regime of fast expansion of the Universe with the expansion ratea/a = H(t) given by a slowly changing function of t (so that the change of H during oneHubble time is negligible, HH1 H). Then the scale factor is approximately exponentiallygrowing with time,

a (t) exp

H(t) dt

. (2.5)

The spacetime with this scale factor is similar to the de Sitter space of constant curvatureR

12H2. More precisely, inflation is defined as a period of accelerated expansion, a(t) > 0.

This condition allows the Hubble rate H(t) to decrease as long as it does not decrease tooquickly. The earliest working proposal of an inflationary model can be found in the works ofA. Starobinsky [20, 21]. In that scenario, a period of inflation was driven by a modification ofgravity. However, this pioneering work remained relatively unappreciated until A. Guth [19]pointed out that several major problems of standard cosmology would disappear at once if anepoch of accelerated expansion were to precede the hot initial state of the standard scenario,provided that the duration of the inflationary epoch is large enough compared with the Hubbletime scale, such that the total expansion factor during inflation is a exp (60). Since sucha large expansion cools the Universe to extremely low temperatures, a reheating must occurprior to the onset of the radiation-dominated epoch. We shall now briefly explain how theproblems outlined in the previous section are solved in this modified scenario and then describe

how inflation was implemented in particular models.The evolution of during the inflationary epoch is given by

(t) 1 =

a0a (t)

2(0 1) H

20

H2 (t)(0 1) exp

2

tt0

H(t) dt

. (2.6)

Unlike the power-law expansion a t with < 1, inflation draws the value of nearer to1. If the total expansion factor a/a0 (the amount of inflation) is large enough, i.e. at leastexp (60), which is called 60 e-foldings, then for a generic initial condition 0 1 the value of at the end of inflation will be as close to 1 as the observational constraints (2.3) require. This

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2.1 Cosmological inflation

solves the flatness problem. In fact, the amount of inflation in generic models is much largerthan 60 e-foldings, and would be typically driven so close to 1 by the end of inflation thatit would not significantly deviate from = 1 afterwards, during the radiation-dominated and

matter-dominated expansion. Thus, generic models of inflation predict that we should observe 1 nowadays.

The horizon problem manifested by the observed homogeneity of the CMB is absent because,according to the inflationary scenario, the whole surface of last scattering has been before in-flation a small patch well under horizon size, and one would expect inhomogeneities in a smallregion to be small. An alternate way to express this is to say that the initial inhomogeneitieshave been inflated away. In this light, the homogeneous state at the beginning of the radiationera does not appear mysterious.

It has also been shown [22, 23, 24, 25, 26, 27] that vacuum fluctuations of matter fields duringinflation give rise to an approximately scale-invariant (n 1) spectrum of perturbations (2.4),as necessary to explain structure formation.

2.1.2 Inflationary models: slow roll

A large number of specific inflationary models have been proposed in the early 1980s. The modeloriginally proposed by Starobinsky remains viable even in view of todays experimental data. Themodel of Guth [19], sometimes called the old inflationary scenario, did not provide an adequateexplanation of the exit from inflation (the graceful exit problem). Several newscenarios weresubsequently introduced by A. Linde [28] and others. A detailed review of inflationary modelsis beyond the scope of this work; we instead concentrate on the most general common featuresof inflationary models.

The inflationary expansion must be supported either by a modification of gravity or by exotic

matter with an equation of state p < 1

3 . Since neither a detectable modification of EinsteinsGeneral Relativity nor any matter field with such an equation of state has been found in ex-periments, models of inflation necessarily hypothesize either a new field or a modified theory ofgravity. The first major type of inflationary models considered here is a class of models wherethe Hubble rate H(t) is a smooth function of time. Initially H is large (although always wellbelow the Planck scale, H 1), and inflation ends when H approaches zero.

The easiest way to model such evolution is to assume that a scalar-field with an effectivepotential V() drives inflation. We consider a prototypical model with the action

S =

d4x

g

R

16+

1

2()

2 V ()

, (2.7)

wehre the scalar field is minimally coupled to Einstein gravity. We will assume the Friedmannmetric ansatz,

ds2 = dt2 a2 (t)

dr2

1 kr2 + r2d2 + r2 sin2 d2

. (2.8)

Einsteins equations of motion for the scale factor a (t) and the field (x, t) are

a2

a2+

k

a2=

8

3

V () +

1

22

, (2.9)

+ 3a

a 1

a22 = dV ()

d. (2.10)

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2 Eternal inflation

Because of large value of the scale factor a, we can disregard the curvature term

k/a2

andthe spatial gradients of in Eqs. (2.9)(2.10). An exact treatment of the resulting equations,including a recipe of how to construct a potential V () that would yield a given evolution a (t)

of the scale factor can be found in Ref. [29].The slow roll approximation is based on the assumptions that the evolution of the field is

such that the potential V () does not change appreciably on the Hubble time scale H1 a/a.More precisely, one assumes that

1

22 V () ,

dV ()d

(2.11)and disregards the kinetic term 2/2 in Eq. (2.9) and the term in Eq. (2.10). The latterassumption means that the friction term 3H in Eq. (2.10) balances the force term V (),and the evolution of is overdamped. Then the effective equations of motion become

a2

a2 H2 () = 8

3V () , (2.12)

= 14

dH()

d. (2.13)

Here we denoted by H() the function

8V () /3, as is common in the literature. 1

Eqs. (2.12)(2.13) are the desired equations describing the slow roll regime of the evolutionof . Using Eqs. (2.12)(2.13), we can express the conditions (2.11) through H() and obtainequivalently

H

16H

2

1,

H

12H

1. (2.14)

These are the requirements on the potential V () necessary for the slow roll approximation tobe valid.2 The first of these conditions also guarantees that the relative change of V () in oneHubble time is negligible:

H1d

dtV () V () . (2.15)

For a given potential there is usually a range of for which the slow roll conditions (2.14)are satisfied. It is usually the case that reheating begins when reaches values for which theslow roll condition is violated.3 For simplicity we will assume that reheating begins at the value = such that

H(

)

16H(

). (2.16)

If the slow roll inflation starts at = 0 and ends at = , the total expansion factora (t) /a (t0) can be found from

lna (t)a (t0)

=

tt0

H(t) dt =

0

H()d

= 4

0

H()

H ()d. (2.17)

1Although in some inflationary models, notably in the open inflation, the function H() does not approximatethe Hubble expansion rate a/a, we shall keep the notation throughout this text.

2Another implied assumption is V () H()2 1, since any classical description is only valid far from thePlanck energy scales.

3A detailed theory of reheating has been worked out in Refs. [30, 31].

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2.2 Eternal inflation

There are several typical shapes of the potential V() that allow for inflation. In models ofnew inflation, the potential has a nearly flat top, and inflation ends when the field rolls downto the bottom (see Fig. 2.1). In chaotic models the potential is usually of the power-law form

V() n or of exponential form, V() e. We will not need to specify the potential in whatfollows. The considerations of eternal inflation apply generally to every such model, althoughspecific calculations of course require the knowledge of the inflaton potential.

V

0

*

Figure 2.1: The inflaton potential for the new inflationary scenario at T = 0. The neighbor-hood of the maximum of the potential at = 0 is very flat, so that the field changes very slowly. Thermalization occurs after the field finally rolls down to thethermalization point = .


Eternal inflation, or the fact that inflation never ends in the whole Universe, is a generic propertyof inflationary models. The general idea of eternally inflating spacetime was first introducedand developed in the 1980s [32, 33, 34, 35] in the context of slow-roll inflation. Let us begin by

reviewing the main features of eternal inflation, following Ref. [8].A prototypical model contains a minimally coupled scalar field (the inflaton) with an

effective potential V() that is sufficiently flat in some range of . When the field has valuesin this range, the spacetime is approximately de Sitter with the Hubble rate

a

a=

8

3V() H(). (2.18)

(We work in units where G = c = = 1.) The value of H remains approximately constant ontimescales of several Hubble times (t H1), while the field follows the slow-roll trajectorysr(t). Quantum fluctuations of the scalar field in de Sitter background grow linearly with

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2 Eternal inflation

time [36, 37, 25],

2(t + t) 2(t) = H3

42t, (2.19)

at least for time intervals t of order several H1. Due to the quasi-exponential expansion ofspacetime during inflation, Fourier modes of the field are quickly stretched to super-Hubblelength scales. However, quantum fluctuations with super-Hubble wavelengths cannot maintainquantum coherence and become essentially classical [37, 25, 38, 39, 40]; this issue is discussed inmore detail in Sec. 2.2.2 below. The resulting field evolution (t) can be visualized [27, 32, 38]as a Brownian motion with a random jump of typical step size H/(2) during a timeinterval t H1, superimposed onto the deterministic slow-roll trajectory sr(t). A statisticaldescription of this random walk-type evolution (t) is reviewed in Sec. 3.1.

One can distinguish two possible regimes of evolution depending on whether the deterministicchange of in one Hubble time is smaller or larger than a typical fluctuation. If the deterministic

change H1

dominates the fluctuations, the slow roll regime proceeds essentially unmodified.In the opposite regime, H1 , the random walk dominates the evolution of, which meansthat steps toward larger and smaller H() are almost equally probable. Once a horizon-sizedregion where is dominated by fluctuations is formed, it will expand to form several horizon-sized regions, most of which would contain the field still in the fluctuation-dominated range.One may say that a fluctuation-dominated region reproduces itself, regardless of the evolutionof the neighbor regions outside its horizon. As a result, the volume of the inflating domaingrows exponentially with time, even though any given co-moving point will eventually enter athermalized region.

In models of new inflation, the fluctuation-dominated range is near the flat top of thepotential where the deterministic change of is small. In chaotic models it is usually a

range of bounded from below by some value fluct, and from above by the Planck boundaryP, which leads to the range fluct < < P if fluct < P and to no fluctuation-dominatedrange (finite inflation) otherwise. For the power-law potential V () = n with 1, oneobtains fluct 1/(n+2) P 1/n. Generically, in both new and chaotic inflationthe regions in the self-reproducing stage expand at the fastest possible rate. We see how thefeature of self-reproduction helps solve the problem of initial conditions for inflation: Wherevera self-reproducing region is formed, it dominates the physical volume of the Universe, and allother regions (including those with initial conditions unsuitable for inflation) will occupy anexponentially small fraction of the total volume.

The jumps at points separated in space by many Hubble distances are essentially un-correlated; this is another manifestation of the well-known no-hair property of de Sitter

space [41, 42, 43]. Thus the field becomes extremely inhomogeneous on large (super-horizon)scales after many Hubble times. Moreover, in the semi-classical picture it is assumed4 thatthe local expansion rate a/a H() tracks the local value of the field (t,x) according to theEinstein equation (2.18). Here a(t,x) is the scale factor function which varies with x only onsuper-Hubble scales, a(t,x)x H1. Hence, the spacetime metric can be visualized as havinga slowly varying, locally de Sitter form (with spatially flat coordinates x),

gdxdx = dt2 a2(t,x)dx2. (2.20)

4This assumption was made in Ref. [33] and is still subject to active research. See Sec. 2.2.2 for more discussionon this issue.

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t

x

y

Figure 2.2: A qualitative diagram of self-reproduction during inflation. Shaded spacelike do-mains represent Hubble-size regions with different values of the inflaton field . The

time step is of order H1. Dark-colored shades are regions undergoing reheating( = ); lighter-colored shades are regions where inflation continues. On average,the number of inflating regions grows with time.

The deterministic trajectory sr(t) eventually reaches a (model-dependent) value signifyingthe end of the slow-roll inflationary regime and the beginning of the reheating epoch (thermaliza-tion). Since the random walk process will lead the value of away from = in some regions,reheating will not begin everywhere at the same time. Moreover, regions where remains inthe inflationary range will typically expand faster than regions near the end of inflation whereV() becomes small. Therefore, a delay of the onset of reheating will be rewarded by additionalexpansion of the proper 3-volume, thus generating more regions that are still inflating. This fea-ture is called self-reproductionof the inflationary spacetime [34]. Since each Hubble-size regionevolves independently of other such regions, one may visualize the spacetime as an ensemble ofinflating Hubble-size domains (Fig. 2.2).

The process of self-reproduction will never result in a global reheating if the probability ofjumping away from = and the corresponding additional volume expansion factors are suffi-ciently large. The corresponding quantitative conditions and their realization in typical modelsof inflation are reviewed in Sec. 4.1. Under these conditions, the process of self-reproduction ofinflating regions continues forever. At the same time, every given comoving worldline (exceptfor a set of measure zero; see Sec. 4.1) will sooner or later reach the value = and enterthe reheating epoch. The resulting situation is known as eternal inflation [34]. More precisely,the term eternal inflation means future-eternal self-reproduction of inflating regions [44].5 Toemphasize the fact that self-reproduction is due to random fluctuations of a field, one refersto this scenario as eternal inflation of random-walk type. Below we use the terms eternalself-reproduction and eternal inflation interchangeably.

Observers like us may appear only in regions where reheating already took place. Hence, itis useful to consider the locus of all reheating events in the entire spacetime; in the presentlyconsidered example, it is the set of spacetime points x there (x) = . This locus is called thereheating surface and is a noncompact, spacelike three-dimensional hypersurface [45, 3]. It is

5It is worth emphasizing that the term eternal inflation refers to future-eternity of inflation in the sensedescribed above, but does not imply past-eternity. In fact, inflationary spacetimes are generically not past-eternal [45, 46].

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2 Eternal inflation

important to realize that a finite, initially inflating 3-volume of space may give rise to a reheatingsurface having an infinite 3-volume, and even to infinitely many causally disconnected pieces ofthe reheating surface, each having an infinite 3-volume. This feature of eternal inflation is at

the root of several technical and conceptual difficulties, as will be discussed below.Everywhere along the reheating surface, the reheating process is expected to provide ap-

propriate initial conditions for the standard hot big bang cosmological evolution, includingnucleosynthesis and structure formation. In other words, the reheating surface may be visu-alized as the locus of the hot big bang events in the spacetime. It is thus natural to viewthe reheating surface as the initial equal-time surface for astrophysical observations in the post-inflationary epoch. Note that the observationally relevant range of the primordial spectrumof density fluctuations is generated only during the last 60 e-foldings of inflation. Hence, theduration of the inflationary epoch that preceded reheating is not directly measurable beyondthe last 60 e-foldings; the total number of e-foldings can vary along the reheating surface andcan be in principle arbitrarily large.6

The phenomenon of eternal inflation is also found in multi-field models of inflation [ 49, 50],as well as in scenarios based on Brans-Dicke theory [51, 52, 53], topological inflation [54, 55],braneworld inflation [56], recycling universe [57], and the string theory landscape [58]. In someof these models, quantum tunneling processes may generate bubbles of a different phase of thevacuum (see Sec. 3.2 for more details). Bubbles will be created randomly at various places andtimes, with a fixed rate per unit 4-volume. In the interior of some bubbles, additional inflationmay take place, followed by a new reheating surface. The interior structure of such bubbles issketched in Fig. 2.3. The nucleation event and the formation of bubble walls is followed by aperiod of additional inflation, which terminates by reheating. Standard cosmological evolutionand structure formation eventually give way to a -dominated universe. Infinitely many galaxiesand possible civilizations may appear within a thin spacelike slab running along the interior

reheating surface. This reheating surface appears to interior observers as an infinite, spacelikehypersurface [59]. For this reason, such bubbles are calledpocket universes,while the spacetimeis called a multiverse. Generally, the term pocket universerefers to a noncompact, connectedcomponent of the reheating surface [60].

In scenarios of this type, each bubble is causally disconnected from most other bubbles.7

Hence, bubble nucleation events may generate infinitely many statistically inequivalent, causallydisconnected patches of the reheating surface, every patch giving rise to a possibly infinitenumber of galaxies and observers. This feature significantly complicates the task of extractingphysical predictions from these models. This class of models is referred to as eternal inflationof tunneling type.

The fact that eternal inflation is generic to many scenarios significantly changes the global

picture of the Universe. It is likely that while inflation ended in our neighborhood of the Universeapproximately 1010 years ago, there still are and will always be very large domains where inflationgoes on. In a sense, eternal inflation is a reversal of the cosmological principle, yielding a pictureof the Universe which is extremely inhomogeneous on the ultra-large scale.

In the following subsections, I discuss the motivation for studying eternal inflation as well asphysical justifications for adopting the effective stochastic picture. Different techniques devel-

6For instance, it was shown that holographic considerations do not place any bounds on the total number of e-foldings during inflation [47]. For recent attempts to limit the number ofe-foldings using a different approach,see e.g. [48]. Note also that the effects of random jumpsare negligible during the last 60 e-foldings of inflation,since the produced perturbations must be of order 105 according to observations.

7Collisions between bubbles are rare [61]; however, effects of bubble collisions are observable in principle [62].

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domination

reheating

wall wall

nucleation

Figure 2.3: A spacetime diagram of a bubble interior. The infinite, spacelike reheating surfaceis shown in darker shade. Galaxy formation is possible within the spacetime regionindicated.

oped for describing eternal inflation are reviewed in Sec. 3. Section 4 contains an overview ofmethods for extracting predictions and a discussion of the accompanying measure problem.

2.2.1 Predictions in eternal inflation

The hypothesis of cosmological inflation was invoked to explain several outstanding puzzles inobservational data [19]. However, some observed quantities (such as the cosmological constant or elementary particle masses) may be expectation values of slowly-varying effective fields a.Within the phenomenological approach, we are compelled to consider also the fluctuations ofthe fields a during inflation, on the same footing as the fluctuations of the inflaton . Hence,in a generic scenario of eternal inflation, all the fields a arrive at the reheating surface = with values that can be determined only statistically. Observers appearing at different pointsin space may thus measure different values of the cosmological constant, elementary particlemasses, spectra of primordial density fluctuations, and other cosmological parameters.

It is important to note that inhomogeneities in observable quantities are created on scalesfar exceeding the Hubble horizon scale. Such inhomogeneities are not directly accessible toastrophysical experiments. Nevertheless, the study of the global structure of eternally inflatingspacetime is not merely of academic interest. Fundamental questions regarding the cosmologicalsingularities, the beginning of the Universe and of its ultimate fate, as well as the issue of thecosmological initial conditions all depend on knowledge of the global structure of the spacetimeas predicted by the theory, whether or not this global structure is directly observable (seee.g. [63, 64]). In other words, the fact that some theories predict eternal inflation influencesour assessment of the viability of these theories. In particular, the problem of initial conditionsfor inflation [65] is significantly alleviated when eternal inflation is present. For instance, itwas noted early on that the presence of eternal self-reproduction in the chaotic inflationary

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2 Eternal inflation

scenario [66] essentially removes the need for the fine-tuning of the initial conditions [67, 34].More recently, constraints on initial conditions were studied in the context of self-reproductionin models of quintessence [68] and k-inflation [69].

Since the values of the observable parameters a are random, it is natural to ask for theprobability distribution of a that would be measured by a randomly chosen observer. Un-derstandably, this question has been the main theme of much of the work on eternal inflation.Obtaining an answer to this question promises to establish a more direct contact between sce-narios of eternal inflation and experiment. For instance, if the probability distribution for thecosmological constant were peaked near the experimentally observed, puzzlingly small value(see e.g. [70] for a review of the cosmological constant problem), the smallness of would beexplained as due to observer selection effects rather than to fundamental physics. Considerationsof this sort necessarily involve some anthropic reasoning; however, the relevant assumptions areminimal. The basic goal of theoretical cosmology is to select physical theories of the early uni-verse that are most compatible with astrophysical observations, including the observation of our

existence. It appears reasonable to assume that the civilization of Planet Earth evolved near arandomly chosen star compatible with the development of life, within a randomly chosen galaxywhere such stars exist. Many models of inflation generically include eternal inflation and hencepredict the formation of infinitely many galaxies where civilizations like ours may develop. Itis then also reasonable to assume that our civilization is typical among all the civilizations thatevolved in galaxies formed at any time in the universe. This assumption is called the principleof mediocrity [3].

To use the principle of mediocrity for extracting statistical predictions from a model ofeternal inflation, one proceeds as follows [3, 71]. In the example with the fields a describedabove, the question is to determine the probability distribution for the values of a that arandom observer will measure. Presumably, the values of the fields a do not directly influence

the emergence of intelligent life on planets, although they may affect the efficiency of structureformation or nucleosynthesis. Therefore, we may assume a fixed, a-dependent mean numberof civilizations civ(a) per galaxy and proceed to ask for the probability distribution PG(a) ofa near a randomly chosen galaxy. The observed probability distribution of a will then be

P(a) = PG(a)civ(a). (2.21)

One may use the standard hot big bang cosmology to determine the average number G(a)of suitable galaxies per unit volume in a region where reheating occurred with given values ofa; in any case, this task does not appear to pose difficulties of principle. Then the computationof PG(a) is reduced to determining the volume-weighted probability distribution V(a) forthe fields a within a randomly chosen 3-volume along the reheating surface. The probabilitydistribution of a will be expressed as

P(a) = V(a)G(a)civ(a). (2.22)However, defining V(a) turns out to be far from straightforward since the reheating surfacein eternal inflation is an infinite 3-surface with a complicated geometry and topology. Thelack of a natural, unambiguous, unbiased measure on the infinite reheating surface is knownas the measure problem in eternal inflation. Existing approaches and measure prescriptionsare discussed in Sec. 4, where two main alternatives (the volume-based and worldline-basedmeasures) are presented. In Sections 4.2 and 4.4 I give arguments in favor of using the volume-based measure for computing the probability distribution of values a measured by a random

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observer. The volume-based measure has been applied to obtain statistical predictions for thegravitational constant in Brans-Dicke theories [51, 52], cosmological constant (dark energy) [72,73, 74, 75, 76, 77], particle physics parameters [78, 79, 80], and the amplitude of primordial

density perturbations [81, 74, 82, 77].The issue of statistical predictions has recently come to the fore in conjunction with the

discovery of the string theory landscape. According to various estimates, one expects to havebetween 10500 and 101500 possible vacuum states of string theory [83, 84, 58, 85, 86]. Thestring vacua differ in the geometry of spacetime compactification and have different values ofthe effective cosmological constant (or dark energy density). Transitions between vacua mayhappen via the well-known Coleman-deLuccia tunneling mechanism [59]. Once the dark energydominates in a given region, the spacetime becomes locally de Sitter. Then the tunneling processwill create infinitely many disconnected daughter bubbles of other vacua. Observers like usmay appear within any of the habitable bubbles. Since the fundamental theory does not specifya single preferred vacuum, it remains to try determining the probability distribution of vacua

as found by a randomly chosen observer. The volume-based and worldline-based measurescan be extended to scenarios with multiple bubbles, as discussed in more detail in Sec. 4.4. Somerecent results obtained using these measures are reported in Refs. [87, 7, 88, 89].

2.2.2 Physical justifications of the semiclassical picture

The standard framework of inflationary cosmology asserts that vacuum quantum fluctuationswith super-horizon wavelengths become classical inhomogeneities of the field . The calculationsof cosmological density perturbations generated during inflation [22, 23, 24, 25, 26, 27, 90, 91] alsoassume that a classicalization of quantum fluctuations takes place via the same mechanism. Inthe calculations, the statistical average 2 of classical fluctuations on super-Hubble scales issimply set equal to the quantum expectation value 0| 2 |0 in a suitable vacuum state. Whilethis approach is widely accepted in the cosmology literature, a growing body of research isdevoted to the analysis of the quantum-to-classical transition during inflation (see e.g. [92] foran early review of that line of work). Since a detailed analysis would be beyond the scope of thepresent text, I merely outline the main ideas and arguments relevant to this issue.

A standard phenomenological explanation of the classicalization of the perturbations is asfollows. For simplicity, let us restrict our attention to a slow-roll inflationary scenario with onescalar field . In the slow roll regime, one can approximately regard as a massless scalar fieldin de Sitter background spacetime [35]. Due to the exponentially fast expansion of de Sitterspacetime, super-horizon Fourier modes of the field are in squeezed quantum states withexponentially large (

eHt) squeezing parameters [93, 94, 95, 96, 97, 98]. Such highly squeezed

states have a macroscopically large uncertainty in the field value and thus quickly decoheredue to interactions with gravity and with other fields. The resulting mixed state is effectivelyequivalent to a statistical ensemble with a Gaussian distributed value of . Therefore one maycompute the statistical average

2

as the quantum expectation value 0| 2 |0 and interpret

the fluctuation as a classical noise. A heuristic description of the classicalization [35] isthat the quantum commutators of the creation and annihilation operators of the field modes,[a, a] = 1, are much smaller than the expectation values

aa

1 and are thus negligible.A related issue is the backreaction of fluctuations of the scalar field on the metric.8 According

8The backreaction effects of the long-wavelength fluctuations of a scalar field during inflation have been investi-gated extensively (see e.g. [99, 100, 101, 102, 103, 104, 105]).

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2 Eternal inflation

to the standard theory (see e.g. [90, 91] for reviews), the perturbations of the metric arising dueto fluctuations of are described by an auxiliary scalar field (sometimes called the Sasaki-Mukhanov variable) in a fixed de Sitter background. Thus, the classicalization effect should

apply equally to the fluctuations of and to the induced metric perturbations. At the same time,these metric perturbations can be viewed, in an appropriate coordinate system, as fluctuationsof the local expansion rate H() due to local fluctuations of [25, 35, 106]. Thus one arrivesat the picture of a locally de Sitter spacetime with the metric (2.20), where the Hubble ratea/a = H() fluctuates on super-horizon length scales and locally follows the value of via theclassical Einstein equation (2.18).

The picture as outlined is phenomenological and does not provide a description of the quantum-to-classical transition in the metric perturbations at the level of field theory. For instance, afluctuation of leading to a local increase of H() necessarily violates the null energy con-dition [107, 108, 109]. The cosmological implications of such semiclassical fluctuations (seee.g. the scenario of island cosmology [110, 111, 112]) cannot be understood in detail within the

framework of the phenomenological picture.A more fundamental approach to describing the quantum-to-classical transition of perturba-

tions was developed using non-equilibrium quantum field theory and the influence functionalformalism [113, 114, 115, 116]. In this approach, decoherence of a pure quantum state of intoa mixed state is entirely due to the self-interaction of the field . In particular, it is predictedthat no decoherence would occur for a free field with V() = 12 m

2. This result is at variancewith the accepted paradigm of classicalization as outlined above. If the source of the noise isthe coupling between different perturbation modes of , the typical amplitude of the noise willbe second-order in the perturbation. This is several orders of magnitude smaller than the am-plitude of noise found in the standard approach. Accordingly, it is claimed [117, 118] that themagnitude of cosmological perturbations generated by inflation is several orders of magnitude

smaller than the results currently accepted as standard, and that the shape of the perturbationspectrum depends on the details of the process of classicalization [119]. Thus, the resultsobtained via the influence functional techniques do not appear to reproduce the phenomeno-logical picture of classicalization as outlined above. This mismatch emphasizes the need fora deeper understanding of the nature of the quantum-to-classical transition for cosmologicalperturbations.

Finally, let us mention a different line of work which supports the classicalization picture.In Refs. [38, 120, 121, 122, 123, 124], calculations of (renormalized) expectation values such as2, 4, etc., were performed for field operators in a fixed de Sitter background. The resultswere compared with the statistical averages

P(, t)2d,

P(, t)4d, etc., (2.23)

where the distribution P(, t) describes the random walk of the field in the Fokker-Plankapproach (see Sec. 3.1). It was shown that the leading late-time asymptotics of the quantumexpectation values coincide with the corresponding statistical averages (2.23). These resultsappear to validate the random walk approach, albeit in a limited context (in the absence ofbackreaction).

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3 Stochastic approach to inflation

The stochastic approach to inflation is a semiclassical, statistical description of the spacetimeresulting from quantum fluctuations of the inflaton field(s) and their backreaction on the met-ric [32, 33, 125, 126, 127, 128, 129, 130, 131, 132, 133]. In this description, the spacetime remainseverywhere classical but its geometry is determined by a stochastic process. In the next sub-sections I review the main tools used in the stochastic approach for calculations in the contextof random-walk type, slow-roll inflation. Models involving tunneling-type eternal inflation areconsidered in Sec. 3.2.

3.1 Random walk-type eternal inflation

The important features of random walk-type eternal inflation can be understood by consideringa simple slow-roll inflationary model with a single scalar field and a potential V(). Theslow-roll evolution equation is

= 13H

dV

d= 1

4

dH

d v(), (3.1)

where H() is defined by Eq. (2.18) and v() is a model-dependent function describing thevelocity of the deterministic evolution of the field . The slow-roll trajectory sr(t), which

is a solution of Eq. (3.1), is an attractor [134, 135] for trajectories starting with a wide range ofinitial conditions.1

As discussed in Sec. 2.2.2, the super-horizon modes of the field are assumed to undergo arapid quantum-to-classical transition. Therefore one regards the spatial average of on scalesof several H1 as a classical field variable. The spatial averaging can be described with help ofa suitable window function,

(x)

W(x y)(y)d3y. (3.2)

It is implied that the window function W(x) decays quickly on physical distances a |x| of orderseveral H1. From now on, let us denote the volume-averaged field simply by (no other field

will be used).As discussed above, the influence of quantum fluctuations leads to random jumps superim-posed on top of the deterministic evolution of the volume-averaged field (t,x). This may bedescribed by a Langevin equation of the form [33]

(t,x) = v() + N(t,x), (3.3)

where N(t,x) stands for noise and is assumed to be a Gaussian random function with knowncorrelator [33, 136, 137, 138]

N(t,x)N(t, x)

= C(t, t, |x x| ; ). (3.4)1See Ref. [69] for a precise definition of an attractor trajectory in the context of inflation.

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An explicit form of the correlator C depends on the specific window function W used for av-eraging the field on Hubble scales [138]. However, the window function W is merely a phe-nomenological device used in lieu of a complete ab initio derivation of the stochastic inflation

picture. One expects, therefore, that results of calculations should be robust with respect to thechoice of W. In other words, any uncertainty due to the choice of the window function must beregarded as an imprecision inherent in the method. For instance, a robust result in this sense isan exponentially fast decay of correlations on time scales t H1,

C(t, t, |x x| ; ) exp 2H() t t , (3.5)which holds for a wide class of window functions [138].

For the purposes of the present consideration, we only need to track the evolution of (t,x)along a single comoving worldline x = const. Thus, we will not need an explicit form ofC(t, t, |x x| ; ) but merely the value at coincident points t = t, x = x, which is computed inthe slow-roll inflationary scenario as [33]

C(t,t, 0; ) =H2()

42. (3.6)

(This represents the fluctuation (2.19) accumulated during one Hubble time, t = H1.) Dueto the property (3.5), one may neglect correlations on time scales t H1 in the noise field.2

Thus, the evolution of on time scales t H1 can be described by a finite-difference formof the Langevin equation (3.3),

(t + t) (t) = v()t + 2D()t (t), (3.7)where

D() H3()

82(3.8)

and is a normalized random variable representing white noise,

= 0, 2 = 1, (3.9)(t)(t + t) = 0 for t H1. (3.10)

Equation (3.7) is interpreted as describing a Brownian motion (t) with the systematic driftv() and the diffusion coefficient D(). In a typical slow-roll inflationary scenario, there will

be a range of where the noise dominates over the deterministic drift,

v()t

2D()t, t H1. (3.11)

Such a range of is called the diffusion-dominated regime. For near the end of inflation,the amplitude of the noise is very small, and so the opposite inequality holds. This is thedeterministic regime where the random jumps can be neglected and the field follows theslow-roll trajectory.

2Taking these correlations into account leads to a picture of color noise [139, 140]. In what follows, we onlyconsider the simpler picture of white noise as an approximation adequate for the issues at hand.

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3.1.1 Fokker-Planck equations

A useful description of the statistical properties of (t) is furnished by the probability densityP(, t)d of having a value at time t. As in the case of the Langevin equation, the values (t)are measured along a single, randomly chosen comoving worldline x = const. The probabilitydistribution P(, t) satisfies the Fokker-Planck (FP) equation whose standard derivation weomit [141, 142],

tP = [v()P + (D()P)] . (3.12)The coefficients v() and D() are in general model-dependent and need to be calculated in eachparticular scenario. These calculations require only the knowledge of the slow-roll trajectoryand the mode functions of the quantized scalar perturbations. For ordinary slow-roll inflationwith an effective potential V(), the results are well-known expressions (6.11) and (6.12). Thecorresponding expressions for models of k-inflation were derived in Ref. [69] using the relevantquantum theory of perturbations [143].

It is well known that there exists a factor ordering ambiguity in translating the Langevinequation into the FP equation if the amplitude of the noise depends on the position. Specif-ically, the factor D() in Eq. (3.7) may be replaced by D( + t), where 0 < < 1 is anarbitrary constant. With = 0, the term (DP) in Eq. (3.12) will be replaced by a differentordering of the factors,

(DP)

D

D1P

. (3.13)

Popular choices = 0 and = 12 are called the Ito and the Stratonovich factor orderingrespectively. Motivated by the considerations of Ref. [144], we choose = 0 as shown in Eqs. (3.7)and (3.12). Given the phenomenological nature of the Langevin equation (3.7), one expectsthat any ambiguity due to the choice of represents an imprecision inherent in the stochastic

approach. This imprecision is typically of order H2 1 [145].The quantity P(, t) may be also interpreted as the fraction of the comoving volume (i.e. co-

ordinate volume d3x) occupied by the field value at time t. Another important characteristicis the volume-weighted distribution PV(, t)d, which is defined as the proper 3-volume (as op-posed to the comoving volume) of regions having the value at time t. (To avoid consideringinfinite volumes, one may restrict ones attention to a finite comoving domain in the universeand normalize PV(, t) to unit volume at some initial time t = t0.) The volume distributionsatisfies a modified FP equation [35, 128, 130],

tPV = [v()PV + (D()PV)] + 3H()PV, (3.14)

which differs from Eq. (3.12) by the term 3HPV that describes the exponential growth of 3-volume in inflating regions.3

Presently we consider scenarios with a single scalar field; however, the formalism of FP equa-tions can be straightforwardly extended to multi-field models (see e.g. Ref. [144]). For instance,the FP equation for a two-field model is

tP = (DP) + (DP) (vP) (vP) , (3.15)

where D(, ), v(, ), and v(, ) are appropriate coefficients.

3A more formal derivation of Eq. (3.14) as well as details of the interpretation of the distributions P and PV interms of ensembles of worldlines can be found in Ref. [146].

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3.1.2 Methods of solution

In principle, one can solve the FP equations forward in time by a numerical method, startingfrom a given initial distribution at t = t0. To specify the solution uniquely, the FP equationsmust be supplemented by boundary conditions at both ends of the inflating range of [132, 133].At the reheating boundary ( = ), one imposes the exit boundary conditions,

[D()P]= = 0, [D()PV]= = 0. (3.16)

These boundary conditions express the fact that random jumps are very small at the end ofinflation and cannot move the value of away from = . If the potential V() reachesPlanck energy scales at some = max (this happens generally in chaotic type inflationaryscenarios with unbounded potentials), the semiclassical picture of spacetime breaks down forregions with max. Hence, a boundary condition must be imposed also at = max. Forinstance, one can use the absorbing boundary condition,

P(max) = 0, (3.17)

which means that Planck-energy regions with = max disappear from consideration [132, 133].Once the boundary conditions are specified, one may write the general solution of the FP

equation (3.12) as

P(, t) =

CP()() et, (3.18)

where the sum is performed over all the eigenvalues of the differential operator

LP

[

v()P + (D()P)] , (3.19)

and the corresponding eigenfunctions P() are defined by

LP()() = P()(). (3.20)

The constants C can be expressed through the initial distribution P(, t0).By an appropriate change of variables z, P() F(z), the operator L may be brought

into a manifestly self-adjoint form [126, 128, 147, 129, 130, 145],

L d2

dz2+ U(z). (3.21)

Then one can show that all the eigenvalues of L are nonpositive; in particular, the (alge-braically) largest eigenvalue max < 0 is nondegenerate and the corresponding eigenfunc-tion P(max)() is everywhere positive [145, 69]. Hence, this eigenfunction describes the late-timeasymptotic of the distribution P(, t),

P(, t) P(max)() et . (3.22)

The distribution P(max)() is the stationary distribution of per comoving volume at latetimes. The exponential decay of the distribution P(, t) means that at late times most of thecomoving volume (except for an exponentially small fraction) has finished inflation and enteredreheating.

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Similarly, one can represent the general solution of Eq. (3.14) by

PV(, t) =

C

P()

V

()et, (3.23)

where

[L + 3H()]P()() = P()(). (3.24)

By the same method as for the operator L, it is possible to show that the spectrum of eigenvalues of the operator L + 3H() is bounded from above and that the largest eigenvalue max admits a nondegenerate, everywhere positive eigenfunction P()(). However, the largesteigenvalue may be either positive or negative. If > 0, the late-time behavior of PV(, t) is

PV(, t) P()()et , (3.25)

which means that the total proper volume of all the inflating regions grows with time. Thisis the behavior expected in eternal inflation: the number of independently inflating domainsincreases without limit. Thus, the condition > 0 is the criterion for the presence of eternalself-reproduction of inflating domains. The corresponding distribution P()() is called thestationary distribution [132, 133, 1, 53].

If 0, eternal inflation does not occur and the entire space almost surely (i.e. with proba-bility 1) enters the reheating epoch at a finite time.

If the potential V() is of new inflationary type [28, 148, 37, 149, 150] and has a global

maximum at say = 0, the eigenvalues and can be estimated (under the usual slow-rollassumptions on V) as [145]

V(0)

8V(0)H(0) < 0, 3H(0) > 0. (3.26)

Therefore, eternal inflation is generic in the new inflationary scenario.

Let us comment on the possibility of obtaining solutions P(, t) in practice. With the po-tential V() = 4, the full time-dependent FP equation (3.12) can be solved analytically viaa nonlinear change of variable 2 [151, 147, 152]. This exact solution, as well as an ap-proximate solution P(, t) for a general potential, can be also obtained using the saddle-point

evaluation of a path-integral expression for P(, t) [153]. In some cases the eigenvalue equationLP() = P() may be reduced to an exactly solvable Schrodinger equation. These cases includepotentials of the form V() = e, V() = 2, V() = cosh2(); see e.g. Ref. [145] forother examples of exactly solvable potentials.

A general approximate method for determining P(, t) for arbitrary potentials [154, 155, 156]consists of a perturbative expansion,

(t) = 0(t) + 1(t) + 2(t) + ..., (3.27)

applied directly to the Langevin equation. The result is (at the lowest order) a Gaussian ap-

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proximation with a time-dependent mean and variance [154],

P(, t)

122(t)

exp( 0(t))2

22(t) , (3.28)

2(t) H2(sr)

insr

H3

H3d, (3.29)

0(t) sr(t) + H

2H2(t) +

H

4

H3inH2in

H3

H2

, (3.30)

where sr(t) is the slow-roll trajectory and in is the initial value of . While methods basedon the Langevin equation do not take into account boundary conditions or volume weightingeffects, the formula (3.28) provides an adequate approximation to the distribution P(, t) in auseful range of and t [156].

3.1.3 Gauge dependence issues

An important feature of the FP equations is their dependence on the choice of the time variable.One can consider a replacement of the form

t , d T()dt, (3.31)

understood in the sense of integrating along comoving worldlines x = const, where T() > 0 isan arbitrary function of the field. For instance, a possible choice is T() H(), which makesthe new time variable dimensionless,

=

Hdt = ln a. (3.32)

This time variable is called scale factor time or e-folding time since it measures the numberof e-foldings along a comoving worldline.

The distributions P(, ) and PV(, ) are defined as before, except for considering the 3-volumes along hypersurfaces of equal . These distributions satisfy FP equations similar toEqs. (3.12)(3.14). With the replacement (3.31), the coefficients of the new FP equations aremodified as follows [145],

D() D()T()

, v() v()T()

, (3.33)

while the growth term 3HPV in Eq. (3.14) is replaced by 3HT1PV. The change in the

coefficients may significantly alter the qualitative behavior of the solutions of the FP equations.For instance, stationary distributions defined through the proper time t and the e-folding time = ln a were found to have radically different behavior [133, 1, 3]. This sensitivity to the choice ofthe time gauge is unavoidable since hypersurfaces of equal may preferentially select regionswith certain properties. For instance, most of the proper volume in equal-t hypersurfaces is filledwith regions that have gained expansion by remaining near the top of the potential V(), whilehypersurfaces of equal scale factor will under-represent those regions. Thus, a statement suchas most of the volume in the Universe has values of with high potential energy is largelygauge-dependent.

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In the early works on eternal inflation [132, 133, 1, 52], the late-time asymptotic distribution

of volume P()V () along hypersurfaces of equal proper time [see Eq. (3.25)] was interpreted

as the stationary distribution of field values in the universe. However, the high sensitivity of

this distribution to the choice of the time variable makes this interpretation unsatisfactory.Also, it was noted [157] that equal-proper time volume distributions predict an unacceptablysmall probability for the currently observed CMB temperature. The reason for this result isthe extreme bias of the proper-time gauge towards over-representing regions where reheatingoccurred very recently [158, 3]. This is a manifestation of the so-called youngness paradox [157].One might ask whether hypersurfaces of equal scale factor or some other choice of time gaugewould provide less biased answers. However, it turns out [146] that there exists no a priori choiceof the time gauge that provides unbiased equal- probability distributions for all potentialsV() in models of slow-roll inflation (see Sec. 4.3 for details).

Although the FP equations necessarily involve a dependence on gauge, they do provide auseful statistical picture of the distribution of fields in the universe. The FP techniques can also

be used for deriving several gauge-independent results. For instance, the presence of eternalinflation is a gauge-independent statement (see also Sec. 4.1): if the largest eigenvalue ispositive in one gauge of the form (3.31), then > 0 in every other gauge [159]. Using the FPapproach, one can also compute the fractal dimension of the inflating domain [160, 159] and theprobability of exiting inflation through a particular point of the reheating boundary in theconfiguration space (in case there exists more than one such point).

The exit probability can be determined as follows [144, 69]. Let us assume for simplicity thatthere are two possible exit points and E, and that the initial distribution is concentrated at = 0, i.e.

P(, t = 0) = ( 0), (3.34)where E < 0 <

. The probability of exiting inflation through = E during a time interval

[t, t + dt] isdpexit(E) = v(E)P(E, t)dt (3.35)

(note that v(E) < 0). Hence, the total probability of exiting through = E at any time is

pexit(E) =

0

dpexit(E) = v(E)

0P(E, t)dt. (3.36)

Introducing an auxiliary function F() as

F() v()

0P(, t)dt, (3.37)

one can show that F() satisfies the gauge-invariant equation,

D

vF

F

= ( 0). (3.38)

This is in accord with the fact that pexit(E) = F(E) is a gauge-invariant quantity. Equa-tion (3.38) with the boundary conditions

F() = 0, (D

vF)

=E

= 0, (3.39)

can be straightforwardly integrated and yields explicit expressions for the exit probability pexit(E)as a function of the initial value 0 [69]. The exit probability pexit() can be determined simi-larly.

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3.2 Self-reproduction of tunneling type

Until now, we considered eternal self-reproduction due to random walk of a scalar field. Another

important class of models includes self-reproduction due to bubble nucleation.4

Such scenariosof eternal inflation were studied in Refs. [161, 61, 162, 163, 164, 165].

In a locally de Sitter universe dominated by dark energy, nucleation of bubbles of false vacuummay occur due to tunneling [166, 59, 43, 167]. The bubble nucleation rate per unit 4-volumeis very small [59, 168],

= O(1)H4 expSI

H2

, (3.40)

where SI is the instanton action and H is the Hubble constant of the parent de Sitter back-ground. Hence, bubbles will generically not merge into a single false-vacuum domain [61], andinfinitely many bubbles will be nucleated at different places and times. The resulting daugh-ter bubbles may again contain an asymptotically de Sitter, infinite universe, which again gives

rise to infinitely many grand-daughter bubbles. This picture of eternal self-reproduction wascalled the recycling universe [57]. Some (or all) of the created bubbles may support a periodof additional inflation followed by reheating, as shown in Fig. 2.3.

In the model of Ref. [57], there were only two vacua which could tunnel into each other.A more recently developed paradigm of string theory landscape [84, 58, 85, 169] involves avery large number of metastable vacua, corresponding to local minima of an effective potentialin field space. A similar picture is found in the recycling universe [57] scenario. In thelandscape models, transitions between vacuum states are possible through bubble nucleation;the interior of a bubble appears as an infinite homogeneous open universe [59], if one disregardsthe small probability of bubble collisions (see Refs. [62, 170] for analyses of bubble collisions).The presently observed universe is situated within a bubble (called a pocket universe) of some

type. The value of the potential at each minimum is the effective value of the cosmologicalconstant in the corresponding vacuum. Figure 3.1 shows a phenomenologists view of thelandscape. Vacua with 0 do not allow any further tunneling5 and are called terminalvacua [175], while vacua with > 0 are called recyclable or transient since they can tunnel toother vacua with > 0 or 0. Bubbles with 0 cannot support any further tunneling andare thus called terminal bubbles. Recyclable bubbles will give rise to infinitely many nesteddaughter bubbles. Conformal diagrams of the resulting spacetime are shown in Figs. 3.2 and3.3. Of course, only a finite number of bubbles can be drawn; the bubbles actually form a fractalstructure in a conformal diagram [176].

A statistical description of the recycling spacetime can be obtained [57, 175] by consideringa single comoving worldline x = const that passes through different bubbles at different times.

(It is implied that the worldline is randomly chosen from an ensemble of infinitely many suchworldlines passing through different points x.) Let the index = 1,...,N label all the availabletypes of bubbles. For calculations, it is convenient to use the e-folding time ln a. Weare interested in the probability f() of passing through a bubble of type at time . Thisprobability distribution is normalized by

f = 1; the quantity f() can be also visualized

as the fraction of the comoving volume occupied by bubbles of type at time . Denoting by the nucleation rate for bubbles of type within bubbles of type , computed according

4Both processes may be combined in a single scenario [57], but we shall consider them separately for clarity.5Asymptotically flat = 0 vacua cannot support tunneling [171, 172, 173]; vacua with < 0 will quickly collapse

to a big crunch singularity [59, 174].

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3.2 Self-reproduction of tunneling type

X

0

Figure 3.1: A schematic representation of the landscape of string theory, consisting of a largenumber of local minima of an effective potential. The variable X collectively denotesvarious fields and is the effective cosmological constant. Arrows show possibletunneling transitions between vacua.

1

1

2

33

44

4 55

Figure 3.2: A conformal diagram of the spacetime where self-reproduction occurs via bubblenucleation. Regions labeled 5 are asymptotically flat ( = 0).

Figure 3.3: A conformal diagram of the future part of a landscape-driven spacetime (figure fromRef. [176]).

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to Eq. (3.40), and by the corresponding rate per unit time, H3 , we writethe master equation describing the evolution of f(),

df

d =

(f+ f)

Mf, (3.41)

where we introduced the auxiliary matrix M. Given a set of initial conditions f(0), one canevolve f() according to Eq. (3.41).

To proceed further, one may now distinguish the following two cases: Either terminal vacuaexist (some such that = 0 for all ), or all the vacua are recyclable. (Theory suggeststhat the former case is more probable [85].) If terminal vacua exist, then the late-time asymptoticsolution can be written as [175]

f() f(0) + seq, (3.42)where f

(0) is a constant vector that depends on the initial conditions and has nonzero components

only in terminal vacua, and s does not depend on initial conditions and is an eigenvector ofM such that

Ms = qs, q > 0. This solution shows that all comoving volume

reaches terminal vacua exponentially quickly. (As in the case of random-walk inflation, thereare infinitely many eternally recycling points x that never enter any terminal vacua, but thesepoints form a set of measure zero.)

If there are no terminal vacua, the solution f() approaches a constant distribution [177],

lim f() f

(0) ,

Mf(0) = 0, (3.43)

f(0) = H4 exp

H2 . (3.44)In this case, the quantities f

(0) are independent of initial conditions and are interpreted as the

fractions of time spent by the comoving worldline in bubbles of type .We note that the description of spacetime in terms of the distribution f() satisfying Eq. (3.41)

depends on the choice of the time variable in an essential way. A different choice (e.g. theproper time t) would lead to an inequivalent master equation and qualitatively different dis-tributions f(t) [146].

6

One may adopt another approach and ignore the duration of time spent by the worldlinewithin each bubble. Thus, one describes only the sequence of the bubbles encountered along arandomly chosen worldline [177, 178, 7]. If the worldline is initially in a bubble of type , thenthe probability of entering the bubble of type as the next bubble in the sequence after

is =

. (3.45)

(For terminal vacua , we have = 0 and so we may define = 0 for convenience.) Onceagain we consider landscapes without terminal vacua separately from terminal landscapes. Ifthere are no terminal vacua, then the matrix is normalized,

Winitzki - ion Thesis 2009

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