TOPICS IN INFLATION AND ETERNAL INFLATION

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Mahdiyar Noorbala

August 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/bd819xz1063

© 2011 by Mahdiyar Noorbala Tafti. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Andrei Linde, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Shenker

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Leonard Susskind

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

To my parentsand Leila

iv

Preface

The theory of inflation was invented in early 1980’s to solve a number of puzzles in

cosmology, most notably the horizon, flatness, homogeneity and monopoles problems.

The idea was an exponential phase of expansion in the early universe that could

stretch out any primordial inhomogeneities, dilute the monopoles that we don’t see

today and make a very flat universe. In addition, different corners of the universe that

are not in causal contact today, were within a causal horizon in that stage and hence

the observed long-range correlations do not violate relativistic causality. Inflation

has ever since become the standard paradigm in modern cosmology for solving these

problems and for many other robust predictions that are confirmed by observation.

There are many inflationary models that share the same robust features but differ in

other details that in principle can be used to rule out some of them. In fact there is

ongoing research to pinpoint a particular inflationary model arising from fundamental

theories, study its detailed predictions and compare them with observations.

The first chapter of this thesis is along this line of investigations. The model we

study features nonminimal coupling between gravity and the scalar field (inflaton)

and is embedded within the framework of supergravity. The model is characterized

by a potential which itself depends on some parameters, so in fact we have a family

of models and correspondingly we have a family of predictions. The key observable is

the tensor-to-scalar ratio of density perturbations. We find that with a suitable choice

of the parameters of the model we can reproduce a wide range of possible outcomes

in future observations by, for example, Planck satellite.

Soon after inflation was proposed, it was realized that in almost all models inflation

is eternal. This means that there are always parts of the universe, outside our horizon,

v

that undergo inflation. So while inflation eventually ends at any single point there

are always many other points where it’s not yet terminated. When a neighborhood of

points exit the inflationary phase a universe like ours emerges whose future inhabitants

will never be able to communicate with the other points outside their own universe.

This picture is known as the multiverse and the individual universes like ours are called

pocket, bubble or baby universes. Within decades the idea that string theory has a

landscape of solutions offered a suitable arena for eternal inflation and multiverse. It

is to be emphasized that regardless of details of individual models, the basic behavior

of eternal inflation is a direct result of some widely accepted principles of physics,

namely, general relativity and semi-classical approximation to quantum mechanics of

a field in a fairly generic potential.

This inevitable eternal inflation has both blessings and curses. On the one hand

it extends the scope of questions that we may hope to answer: If we live in only one

pocket universe out of many in the multiverse, then some of the fundamental constants

of our universe can be merely environmental variables that take the values they take

only due to anthropic reasons. These are usually constants that are otherwise very

hard to predict or explain (e.g., the cosmological constant which is extremely fine-

tuned). The rest of the constants cannot be constrained anthropically but statistical

statements can be made about them. Therefore, a particular theory can be falsified

at some confidence level by comparing its statistical prediction with the observed

values of the fundamental constants. On the other hand, there is an ambiguity in

calculating the statistical distribution associated with the fundamental constants and

other physical observables. This is known as “the measure problem” and is the subject

of the last two chapters of this thesis.

The ambiguity in defining probabilities in eternal inflation is due to the infiniteness

of events occurring in spacetime. In a stochastic approach1 one can study random

realizations of spacetime geometry that are weighted by some probability distribution.

Almost all of these realizations have an infinite 4-volume and any given event of any

kind occurs in them infinitely many times. It is the comparison of these infinities

that poses the ambiguity problem. There are several competing prescriptions, called

1That is, a semi-classical approach that does not take into account quantum interferences.

vi

measures, that yield different results. It is hard to justify any of these measures based

on any known first principle; hence a prevalent approach has been to discriminate

between them based on their predictions. This is possible because some of these

measures have predictions that are already inconsistent, by a large confidence level,

with known experimental results (and in fact, sometime with known facts of life).

In the second chapter we compare the predictions of various measures as we tran-

sition from the regime of eternal to non-eternal inflation. We find that except for

“the stationary measure” the other three measures we study undergo discontinuities

in their predictions. Finally in the third chapter we study the issue of “Boltzmann

brains” for “the scale factor cutoff measure.” We find that the success of this measure

highly depends on unknown details of the landscape of string theory, although with

our current knowledge the situation is marginally acceptable.

vii

Acknowledgements

I would like to thank my adviser Andrei Linde. He introduced me to research in this

field and it has been a pleasure to learn from the wonderful physical intuition of a

pioneer. Andrei has also been a great support during the past couple of years. I have

benefited from many others at Stanford Institute for Theoretical Physics (SITP),

most notably Lenny Susskind from whom I’ve learned a lot. I’m also grateful to

Savas Dimopoulos, Shamit Kachru, Renata Kallosh, Steve Shenker, Eva Silverstein

and Jay Wacker for what they taught me in classes and conversations. I’d like to

thank my non-Stanford collaborators Andrea De Simone and two prominent experts

in the field, Alan Guth and Alex Vilenkin.

I have enjoyed the company of my nice friends and office mates: Xi Dong, Daniel

Harlow and Dusan Simic. We had wonderful discussions about physics as well as many

other topics. I’m also grateful to postdocs at SITP, Alex Westphal, Mike Salem and

particularly Vitaly Vanchurin. I spent so many hours in Vitaly’s office talking about

various issues of eternal inflation. I should especially thank my two Iranian friends at

SITP, Masoud Soroush and Siavosh Rezvan Behbahani who created a warm Persian

environment. Masoud was a great support in my first months at Stanford and in fact

in all subsequent years. I also feel so happy to be acquainted with the greater Iranian

community at Stanford who were like my family abroad. Thank you all guys!

Finally thanks to The Mellam Family Foundation for a year of scholarship support.

* * *

Above all I am indebted to my parents for pretty much everything I have. I

viii

appreciate that they always valued education, encouraged me and inspired me with

their love and affection. I am deeply grateful for all that they have provided for me

in every stage of my life. I’d also like to thank my sisters, Maryam and Fateme, who

were a great source of advice, kindness and friendship. At last but not least, I’m

grateful to my beloved wife Leila without whose endless support, love and sacrifice I

couldn’t complete this work. I have been blessed by the joy of her company in every

moment of life. I love you all!

I cannot finish without mentioning our lovely baby daughter Zaynab who has

brought delight to our life since the moment we are gifted with her. Thanks God!

ix

Contents

Preface v

Acknowledgements viii

1 Observational Consequences of Chaotic Inflation with Nonminimal

Coupling to Gravity 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Canonical Field in Einstein Frame . . . . . . . . . . . . . . . . . . . . 2

1.3 Supergravity perspective . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Basic models and their extensions . . . . . . . . . . . . . . . . . . . . 8

1.5 Quadratic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Quartic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Measure Problem for Eternal and Non-Eternal Inflation 22

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Eternal and Non-eternal Inflation: A Toy Model . . . . . . . . . . . . 27

2.3 The Eternal Inflation Model . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 The Non-eternal Inflation Model . . . . . . . . . . . . . . . . . . . . . 35

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Boltzmann brains and the scale-factor cutoff measure of the multi-

verse 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

x

3.2 The Scale Factor Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 The Abundance of Normal Observers and Boltzmann Brains . . . . . 52

3.4 Mini-landscapes and the General Conditions to Avoid Boltzmann Brain

Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1 The FIB Landscape . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2 The FIB Landscape with Recycling . . . . . . . . . . . . . . . 61

3.4.3 A More Realistic Landscape . . . . . . . . . . . . . . . . . . . 63

3.4.4 A Further Generalization . . . . . . . . . . . . . . . . . . . . . 65

3.4.5 A Dominant Vacuum System . . . . . . . . . . . . . . . . . . 68

3.4.6 General Conditions to Avoid Boltzmann Brain Domination . . 72

3.5 Boltzmann Brain Nucleation and Vacuum Decay Rates . . . . . . . . 83

3.5.1 Boltzmann Brain Nucleation Rate . . . . . . . . . . . . . . . . 83

3.5.2 Vacuum Decay Rates . . . . . . . . . . . . . . . . . . . . . . . 97

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A Boltzmann Brains in Schwarzschild–de Sitter Space 105

Bibliography 115

xi

List of Tables

2.1 Predictions of the four measures in the eternal and non-eternal models. 38

xii

List of Figures

1.1 Predictions with the potential V ∼ φ2α . . . . . . . . . . . . . . . . . 8

1.2 Predictions with the potential λ4(φ2 − v2)2 . . . . . . . . . . . . . . . 9

1.3 The quadratic potential . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Predictions with the quadratic potential . . . . . . . . . . . . . . . . 11

1.5 The quartic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Predictions with the quartic potential, ξ > 0 and φ > v . . . . . . . . 14

1.7 Predictions with the quartic potential, ξ > 0 and φ < v . . . . . . . . 16

1.8 Predictions with the quartic potential and ξ < 0 . . . . . . . . . . . . 17

1.9 The potential λ4(φ2 − v2)2 . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 The potential for the eternal inflation model . . . . . . . . . . . . . . 28

2.2 The potential for the non-eternal inflation model . . . . . . . . . . . . 28

2.3 Evolution of the scale factor in a collapsing universe . . . . . . . . . . 40

A.1 The holographic bound, the D-bound, and the m-bound . . . . . . . . 109

A.2 The ratio of ∆S to IBB . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xiii

Chapter 1

Observational Consequences of

Chaotic Inflation with Nonminimal

Coupling to Gravity

1.1 Introduction

The chaotic inflation scenario [1] is one of the simplest and most general versions

of inflationary cosmology. It can be realized in many theories with sufficiently flat

potentials, including the models with V ∼ φn. However, for a long time it seemed

very difficult to implement chaotic inflation in superstring theory and in supergravity.

This situation changed when a simple realization of the chaotic inflation scenario in

supergravity was proposed in [2], and a class of chaotic inflation models was developed

in the context of string theory [3]. In this class of models, one could have an inflaton

potential ∼ φ2/3, or φ4/5, or φ, predicting a much smaller value of r.

Recently a broad class of models of chaotic inflation in supergravity was proposed

in [4]. In these models one can obtain chaotic inflation with an arbitrary inflaton

potential V (φ). In these models one can also implement the curvaton scenario [5]

with a controllable degree of non-gaussianity [6].

Moreover, this new class of supergravity models can describe an inflaton field

which is nonminimally coupled to gravity, with coupling ξ2φ2R [7, 4]. Recently there

1

CHAPTER 1. 2

was a wave of interest to such models after the suggestion to implement chaotic

inflation in the standard model of electroweak interactions, with the Higgs field non-

minimally coupled to gravity playing the role of the inflaton field [8, 9], see also [10].

However, the traditional textbook formulation of supergravity did not provide any

description of scalar fields nonminimally coupled to gravity. This could suggest that

the models with nonminimal coupling to gravity are outdated, because they are in-

compatible with modern developments in particle physics based on supersymmetry.

Fortunately, this problem was solved in [7, 4], where a consistent generalization of

supergravity describing scalar fields nonminimally coupled to gravity was developed

and a supersymmetric generalization of the inflationary model of Refs. [8, 9] was

proposed.

These developments prompted us to return to an investigation of various versions

of chaotic inflation with the inflaton field nonminimally coupled to gravity. In this

chapter we will consider two basic models: the simplest model with a quadratic

potential m2

2φ2, and the model with the standard spontaneous symmetry breaking

potential λ4(φ2 − v2)2. Our goal is to analyze these models at the classical level,

describe the way to incorporate them in supergravity, and study the stability of the

inflationary trajectory in these theories. We will also calculate the spectral index ns

and the tensor-to-scalar ratio r for various values of the parameter ξ describing the

nonminimal coupling of the inflaton field to gravity in these models. We will show

that by changing parameters of these models one can cover a significant part of the

space of parameters (ns, r) allowed by recent observational data. Finally, we will

show that the inflationary predictions of the model λ4(φ2 − v2)2 with ξ < 0 in the

limit |ξ|v2 → 1 coincide with the predictions of the Higgs inflation scenario for ξ 1

and v 1.

1.2 Canonical Field in Einstein Frame

The Lagrangian of the inflaton field in the models discussed in the Introduction de-

scribes a scalar field φ with a canonical kinetic term but with a non-minimal coupling

CHAPTER 1. 3

to gravity:

L =1

2

√−gJ[(1 + ξφ2)RJ − gµνJ ∂µφ∂νφ− 2VJ(φ)

]. (1.1)

The index J is to emphasize that this is a Jordan frame: the gravity part of the

Lagrangian is different from the Einstein Lagrangian 12

√−gR[g] and has a local func-

tion 1 + ξφ2 in front. It is possible to proceed with this form of the Lagrangian but

we will be able to use the familiar equations of general relativity, the inflationary

solutions and the standard slow-roll analysis if we switch to the Einstein frame and

use variables that show minimal coupling between the scalar field and gravity.

To this end, we change variables from gJ and φ to a conformally related metric

gµνE = (1 + ξφ2)−1gµνJ and a new field ϕ related to the field φ as follows:

(dϕ

dφ

)2

=1 + ξφ2 + 6ξ2φ2

(1 + ξφ2)2=:

1

Z(φ). (1.2)

In new variables, the Lagrangian, up to a total derivative, is given by

L =1

2

√−gE [RE − gµνE ∂µϕ∂νϕ− 2VE(φ(ϕ))] . (1.3)

where the potential in the Einstein frame is

VE(φ) =VJ(φ)

(1 + ξφ2)2, (1.4)

The canonically normalized Einstein-Hilbert term of the action shows that gE is the

metric in the Einstein frame. Therefore we can use all results of the standard slow-roll

analysis.

In doing the slow-roll analysis we should note that it is ϕ and not φ that has a

canonical kinetic term. Therefore the slow-roll parameters are

ε =1

2

(VϕV

)2

, η =VϕϕV. (1.5)

(From now on, whenever we use V without a subscript or the word potential we mean

the Einstein frame potential VE.) The same quantities can be formally defined for

CHAPTER 1. 4

the field φ but are not of physical meaning. However, they are easier to compute and

are related to the physical slow-roll parameters via

ε = Zεφ, η = Zηφ + sgn(V ′)Z ′√εφ2, (1.6)

where prime means d/dφ. Similarly, we can find the number of e-folds by

N ≈∣∣∣∣∫

dϕ√2ε

∣∣∣∣ =

∣∣∣∣∣

∫dφ

Z(φ)√

2εφ

∣∣∣∣∣ . (1.7)

This enables us to do almost all computations in terms of φ without having to solve

Eq. (1.2) to find ϕ. But we should not forget the fact that φ has a non-canonical

kinetic term −12Z−1gµνE ∂µφ∂νφ. So when we qualitatively analyze the motion of the

field, in analogy with a particle with Lagrangian 12mx2 − V (x), we need to associate

a variable ‘rolling mass’ Z−1(φ) to the field φ.1 This doesn’t change the direction

of rolling but modifies the velocity of the field such that with heavier rolling mass

(smaller Z) the motion becomes slower (smaller ε and η) as indicated by Eq. (1.6).

In the small field limit, φ = ϕ, but at large values of the fields their relation to each

other is more complicated. In particular, for ξ > 0 and φ 1/√ξ (or, equivalently,

ϕ 1), one has

φ =1√ξeϕ√6 . (1.8)

Therefore if the potential V (φ) is a polynomial of φ, at φ 1/√ξ (i.e. at ϕ 1) it

depends on the canonically normalized field exponentially.

For negative ξ, the potential becomes singular when φ approaches 1/√|ξ|, beyond

which the effective gravitational coupling constant GN = 18π(1+ξφ2)

would change sign

(antigravity regime). However, in terms of the canonically normalized field ϕ, the

singularity is never reached, because it corresponds to infinitely large values of ϕ.

That is why in this scenario one should not worry about the possibility to cross the

boundary between gravity and antigravity, or even to approach this boundary [11, 12].

1This ‘rolling mass’ should not be confused with the usual mass V ′′ of the field.

CHAPTER 1. 5

One can especially clearly see it if one considers a vicinity of the would-be singu-

larity, where 1− |ξ|φ2 1. In this regime

φ =1√|ξ|

(1− 1

2e−√

23ϕ

). (1.9)

Therefore 1−√|ξ|φ = 1

2e−√

23ϕ, which means that the limit φ→ 1/

√|ξ| corresponds

to the limit ϕ → ∞. These considerations will help us to understand the properties

of the inflaton potential in the Einstein frame.

Before investigating cosmological implications of this class of models, we would

like to make a short detour and discuss implementation of these models in the context

of supergravity. The results obtained in this chapter do not depend on the possibility

to implement the inflationary models with such potentials in supergravity, but super-

gravity allows to look at such models and evaluate them from an entirely different

perspective. The readers interested only in observational consequences of the models

with nonminimal coupling to gravity may skip the next section and go straight to

section 1.5.

1.3 Supergravity perspective

The new class of chaotic inflation models in supergravity [4] describes two fields, S

and Φ, with the superpotential

W = S f(Φ), (1.10)

where f(Φ) is a real holomorphic function such that f(Φ) = f(Φ). The Kahler

potential in the simplest version of this model is given by

K = −3 log[1− 1

3(ΦΦ + SS) +

χ

4(Φ2 + Φ2) + ζ(SS)2/3

]. (1.11)

The term ζ(SS)2/3 is added for stabilization of the inflationary trajectory, to ensure

that the fields S and Im Φ vanish during inflation: S = Im Φ = 0. The field φ =√2Re Φ plays the role of the inflaton field in this scenario. According to [7, 4], the

CHAPTER 1. 6

evolution of this field is described by the theory of the scalar field φ nonminimally

interacting with gravity, with ξ = −16

+ χ4. The Jordan frame potential of this field is

VJ = f 2(Re Φ) . (1.12)

It is convenient to present the result in terms of the real fields s, α, φ and β, related

to the complex fields through

S =1√2

(s+ iα) , Φ =1√2

(φ+ iβ) . (1.13)

The potential of the inflaton field φ is then

VJ(φ) = f 2(φ/√

2) , (1.14)

and the Einstein frame potential is given by Eq. (1.4).

Thus, in this context one has a functional freedom of choice of the inflaton poten-

tial, determined by the function f(Φ). However, this scenario works only if one can

actually ignore the cosmological evolution of the fields S and Im Φ, by fixing them at

their values S = Im Φ = 0 during inflation. Therefore we must check whether the tra-

jectory S = Im Φ = 0 corresponds to the minimum of the potential in the directions

S and Im Φ. In our scenario, the potential has the same curvature in the directions s

and α, so one should check whether m2β,m

2s > 0. We will impose a stronger condition,

m2β,m

2s & H2, to avoid production of inflationary perturbations of any fields except

for the inflaton field φ.

These conditions can be analyzed along the lines of [4], where it was shown that

for the theories with ξ = 0 and arbitrary function f(Φ) one can always stabilize the

inflationary trajectory S = Im Φ = 0 by a proper choice of the parameter ζ.

An investigation of this issue for ξ 6= 0 is similar, but more complicated, because

m2β and m2

s depend on ξ:

m2β =

43(1 + 3ξ)f 2 + (1 + ξφ2)((∂Φf)2 − f ∂2

Φf)

(1 + ξφ2)(1 + ξφ2 + 6ξ2φ2), (1.15)

CHAPTER 1. 7

m2s =

A(φ)

(1 + ξφ2)2(1 + ξφ2 + 6ξ2φ2). (1.16)

where

A(φ) =2

3(−1− ξφ2 + 6ξ2φ2 + 6ζ(1 + 2ξφ2 + 6ξ3φ4

+ ξ2φ2(6 + φ2)))f 2 − 4√

2ξφ(1 + ξφ2)f ∂Φf

+ (1 + ξφ2)2(∂Φf)2 (1.17)

In this chapter we studied two basic models. The first one corresponds to f(Φ) =

mΦ, which leads to the quadratic inflaton potential m2

2φ2. The second choice is

f(Φ) =√λ(Φ2−v2/2), which leads to the inflaton potential λ

4(φ2−v2)2. Investigation

of stability in the supergravity versions of these models is straightforward but quite

involved. Here we will skip the details and briefly summarize our main results.

First of all, we verified that the inflationary trajectory is stable with respect to

generation of the field β ∼ Im Φ. However, the situation with the field S is more

complicated. Just as in the models studied in [7], stabilization requires the existence

of the stabilizing term ζ(SS)2/3 in the Kahler potential. For ξ > 0, stabilization can

be achieved by taking 0 < ζ . O(1). Similar conclusion is valid for ξ < 0 as well,

with one caveat.

As we discussed in the previous section, the point |ξ|φ2 = 1 for ξ < 0 corre-

sponds to a singularity, where the effective gravitational constant becomes infinite

and changes its sign. When |ξ|φ2 increases towards 1, which corresponds to the

vicinity of this dangerous point, or to the infinitely large values of the canonically

normalized inflaton field ϕ, stabilization of the field S requires ζ 1. This is not

particularly attractive, but appearance of large parameters in the Kahler potential

is not a novelty in this scenario. Indeed, the Higgs inflation scenario with ξ 1 [9]

also requires introduction of a very large coefficient χ in the Kahler potential (1.11)

[7].

CHAPTER 1. 8

Figure 1.1: Predictions of the chaotic inflation model with the potential V ∼ φ2α fordifferent α in the range of 0 < α < 2 are shown by two blue lines corresponding toN = 60 and N = 50. Our results are shown on top of the WMAP results.

1.4 Basic models and their extensions

In this chapter we will concentrate on observational consequences of two basic models,

with potentials 12m2φ2 and λ

4(φ2 − v2)2, and extend their analysis for the case with

ξ 6= 0. To put our results in a proper perspective, we will briefly remember what is

known about observational consequences of some closely related models.

1) V ∼ φ2α, ξ = 0. For this model one has 1 − ns = (1 + α)N−1, r = 8α/N ,

where N is the number of e-folds. In this case 1−ns = 1/N + r/8 [3]. The results are

shown by two blue straight lines in Fig. 1.1. Not surprisingly, these lines go through

the circles corresponding to the theories with potentials proportional to φ2 and φ4;

these lines continue going down when α decreases.

2) V ∼ (φ2 − v2)2, ξ = 0. The results depend on the value of v, and also on

the initial conditions for the field φ. If inflation occurs at φ > v, and v is not too

large, the results coincide with those in the model with the potential φ4, see Fig. 1.2.

For v > O(10), the last stages of inflation occur not far from the minimum of the

potential, where it looks quadratic. Therefore in this regime ns and r coincide with

CHAPTER 1. 9

Figure 1.2: Predictions of the model with the potential λ4(φ2−v2)2 are bounded by the

two blues lines corresponding to the number of e-foldings N = 50 and N = 60 [13].The blue stars correspond to this model with v = 0 and the inflaton nonminimallycoupled to gravity with ξ 1 for N = 50 and N = 60 [9, 7]. The green dashed linesdescribe predictions of this model for v = 0 and for various values of ξ > 0 [14], forN = 50 and N = 60.

their value for the quadratic potential. This property remains valid for v > O(10)

even if the field falls from φ < v. However, inflation is also possible for φ < v and

1 . v . O(10), in which case ns and r can be much smaller, as shown on the lower

branch of the blue line in Fig. 1.2, below the white circle.

The green lines in Fig. 1.2 show ns and r for the theory with the potential φ4 with

ξ > 0. This potential coincides with our potential ∼ (φ2−v2)2 in the limit v = 0 [14].

The investigation which we are going to perform should generalize the previously

obtained results for the theory m2φ2/2 with nonminimal coupling ξ, and for the theoryλ4(φ2 − v2)2 with arbitrary values of v and ξ.

CHAPTER 1. 10

1.5 Quadratic Potential

We begin with the investigation of the quadratic potential VJ = 12m2φ2 in the Jordan

frame, which leads to the Einstein frame potential

VE =m2φ2/2

(1 + ξφ2)2. (1.18)

This equation immediately reveals an unusual behavior of the potential. At small φ,

the potential behaves as the original quadratic potential. However, for ξ > 0 it has a

maximum at ξφ2 = 1, see the blue line in Fig. (1.3). If the field begins its evolution

beyond this maximum, it continues rolling forever, as in the theory of quintessence.

The normal inflationary regime which ends by oscillations near φ = 0 occurs only if

the evolution occurs to the left of the maximum, i.e., at ξφ2 < 1. However, inflation

may begin close to the maximum of this potential Ref. [15].

j

V

Figure 1.3: The quadratic potential V as a function of the canonically normalizedinflaton field ϕ in the Einstein frame. The right, blue curve corresponds to ξ = 1, theleft, red curve corresponds to ξ = −1. Note that the potential for the negative ξ isvery steep, but it does not contain any singularity, which appears when one expressesthis potential in terms of the original scalar field φ as in Eq. (1.18).

Meanwhile for ξ < 0, there is a singularity of the potential at ξφ2 = −1. Once

again, one can ignore the regime that begins at large φ, beyond the singularity, because

it describes the universe in the antigravity regime. One can especially clearly see it if

one plots VE not as a function of φ, but as a function of the canonically normalized

CHAPTER 1. 11

field ϕ, see the red line in Fig. 1.3.

The physical slow-roll parameters are

ε =2(1− ξφ2)

2

φ2 (1 + ξφ2 + 6ξ2φ2), (1.19)

η =2 (12ξ4φ6 + 2ξ3φ6 − 36ξ3φ4 − 5ξ2φ4 − 6ξφ2 + 1)

φ2(1 + ξφ2 + 6ξ2φ2)2 . (1.20)

The number of e-folds that it takes for the field to roll from φ∗ (when the primordial

fluctuations were formed) to φe (when inflation ends) is

N ≈ 3

4log

1− ξ2φ4e

1− ξ2φ4∗

+1

4ξlog

1− ξφ2e

1− ξφ2∗. (1.21)

ξ = 0

ξ

2φ2R +

m2

2φ2

ξ < 0

ξ > 0

Figure 1.4: (ns, r) atN = 60 from the quadratic potential Eq. (1.18). The ξ = 0 point,corresponding to the simple m2φ2/2 potential with minimal gravitational coupling,separates the ξ < 0 (red, top) and ξ > 0 (green, bottom) cases.

Although we have analytic results for slow-roll parameters, it is not easy to solve

them to obtain φ∗ at which the observables ns and r should be evaluated. Instead

CHAPTER 1. 12

we proceed numerically by finding φe (the point where either ε < 1 or η < 1 fails)

and then going N = 60 e-folds back to obtain φ∗ while making sure that the slow-roll

parameters remain small in this range of φ. The result is that inflationary trajectories

via slow roll approximation are feasible only for −10−2 . ξ . 10−1. Note that by

varying ξ from zero we deviate from the point (ns, r) = (1− 2N, 8N

) (corresponding to

the m2φ2/2 potential) and obtain a one-dimensional curve (corresponding to the one

parameter ξ) in the ns-r plane; see Fig. (1.4). For large and negative ξ the potential

in Fig. 1.3 (the red line) becomes like an exponential barrier within a distance ∆ϕ ∼ 1

from the origin. Hence N & 60 is accommodated only for |ξ| 10−2. Even when

N & 60 is possible the predicted ns and r may be well outside the current bounds.

In fact, for N = 60 the part of the curve in Fig. (1.4) that lies within the WMAP

2σ data corresponds to an even smaller range −7 × 10−4 / ξ / 7 × 10−3. In order

to obtain the observed amplitude of density perturbations δρ ≈ 5× 10−5 one should

increase the mass from m = 3 × 10−6 at ξ = 7 × 10−3 to m = 6 × 10−6 at ξ = 0,

followed by an insignificant decrease in m to reach ξ = −7 × 10−4 (in fact, only at

ξ = −2.6 × 10−3 which is far outside the observed region, we do need m = 3 × 10−6

again). In either case, the mass remains within the same order of magnitude of the

minimal coupling case. We see that negative ξ is almost ruled out at 1σ but a range of

positive values of ξ is available to account for forthcoming data down to r ∼ 5×10−3,

albeit with a fine-tuning of at least O(10−2) in ξ.

1.6 Quartic Potential

Let us now consider the standard potential with spontaneous symmetry breaking,

which looks – with non-minimal coupling – as follows

VE =λ

4

(φ2 − v2)2

(1 + ξφ2)2, (1.22)

in the Einstein frame.

To have a better intuitive understanding of the properties of the potential in

the Einstein frame, we plot it as a function of the canonically normalized field ϕ in

CHAPTER 1. 13

Figs. 1.5a and 1.5b (with ξ = 1 and −1, respectively) for two different values of v

(v < 1, blue line; and v > 1, red line).

j

V

(a)

j

V

(b)

Figure 1.5: Einstein frame potential (1.22) as a function of the canonically normalizedfield ϕ. The blue lines correspond to v < 1, and the red lines correspond to v > 1. (a)ξ = 1. (b) ξ = −1. Notice that the potential with ξ = −1, v > 1 has a minimum ata very large value of the potential (the minimum at V = 0 appears in the antigravityregime beyond the singularity), so this potential is unsuitable for the description ofinflation in our universe.

The slow-roll parameters are now given by

ε =8(1 + ξv2)

2φ2

(φ2 − v2)2 (1 + ξφ2 + 6ξ2φ2), (1.23)

η = − 4A (1 + ξv2)

(φ2 − v2)2 (1 + ξφ2 + 6ξ2φ2)2 , (1.24)

where

A = 12ξ3φ6 + 2ξ2φ6−24ξ3v2φ4−4ξ2v2φ4−12ξ2φ4− ξφ4−3ξv2φ2−3φ2 +v2 . (1.25)

The number of e-folds is given by

N ≈∣∣∣∣3

4log

1 + ξφ2e

1 + ξφ2∗

+1

4 (1 + ξv2)

[1

2(1 + 6ξ) (φ2

∗ − φ2e) + v2 log

φeφ∗

)

]∣∣∣∣ . (1.26)

CHAPTER 1. 14

Like the quadratic potential there are different behaviors depending on the sign of

ξ. Let us first consider ξ > 0 where there are two possibilities: ‘large-φ’ and ‘small-φ’

corresponding to φ > v and φ < v; see Fig. (1.5a). In both cases the field rolls to

φ = v but in the small-φ case from left and in the large-φ case from right. Since

neither the potential nor the rolling mass Z−1 are symmetric around φ = v we don’t

expect to obtain the same (ns, r). In fact, on the ns-r plane we find that for any

fixed ξ there are two branches on each of which v runs from zero to infinity. This is

depicted in Fig. (1.6) and 1.7.

v = 0

v →∞

ξ →∞

ξ

2φ2R +

λ

4

φ2 − v2

2

ξ > 0, φ > v

Figure 1.6: (ns, r) at N = 60 for the quartic potential Eq. (1.22) with positive ξ inthe case with inflation at φ > v. Each red and yellow colored line corresponds to afixed value of ξ. For all ξ, the colored lines begin at the green dashed line v = 0,which was found in [14]. Along each colored line v grows from 0 on left to∞ on right.For ξ ' 0.0027 each line segment lies entirely within the 1σ region, regardless of thevalue of v. The limiting point of this set of curves, for ξ → ∞ and any v, coincideswith the result of the Higgs inflation model with ξ 1 and v 1 studied in [9].This point is shown by a gray star.

For ξ = 0 the two branches are joined at v →∞ by the point corresponding to the

CHAPTER 1. 15

m2φ2/2 potential, since in this limit the potential near its minimum is indistinguish-

able from a quadratic one which is symmetric when approached from right or left.

This is not the case for larger ξ because the rolling mass Z−1 ∝ φ−2 is not symmetric

around φ = v so the field rolls with different velocities on the right and the left. This

can also be seen by looking directly at the potential V (ϕ) written in terms of the field

with canonical kinetic term: If we expand V (ϕ) at large v around its minimum, we

find that we can approximate it by a quadratic polynomial only within an interval

∆ϕ √

1+6ξξ

. Beyond ∆ϕ a cubic term dominates which spoils the symmetry of

V (ϕ) around its minimum. Of course, as ξ tends to zero the size of this symmetric

interval grows. In order to have an interval of ϕ that extends ∼ 15 (in Planck units

Mp = 1) on either side of the minimum and still remains at least 90% symmetric (i.e.,

the ratio of the potential at the two ends of the interval is greater than 0.9) we need a

ξ as small as O(10−6). So only for very small ξ do the ends of the small- and large-φ

branches meet. This is in marked contrast with the potential V (ϕ) ∝ (ϕ2 − v2)2 for

which ∆ϕ ∝ v and hence is always symmetric, for large v, over an interval of size

> 2× 15. In addition, unlike the ξ = 0 case, the v →∞ limit of neither branch falls

on the curve of Fig. (1.4) (corresponding to the quadratic potential) since Z−1 near

φ = 0 and φ = v are different.

We see that as ξ increases from 0 the small-φ branch covers the region above and

to the left of the ξ = 0 curve in Fig. 1.7. Most of this area is already ruled out, yet the

remaining part covers a significant subset of the allowed region. Meanwhile the large-

φ branches cover a smaller triangular area which shrinks to the point (1− 2N, 12N2 ) ≈

(0.967, 0.003) in the limit ξ →∞, independently of the value of v. This result matches

the result obtained in Ref. [9] for ξ 1 and the special case v 1.

Let us now turn to the case of negative ξ. Again we encounter a singularity at

φ = |ξ|−1/2 like the quadratic potential. But this is more complicated since depending

on whether |ξ|−1/2 is smaller or larger than v the singularity in V (φ) is to the left or

right of φ = v. In the either case, we ignore the right side of the singularity because

it leads to anti-gravity. Therefore if we plot V (ϕ) we only see the minimum V = 0

in the |ξ|−1/2 > v case and not in the |ξ|−1/2 < v case. These correspond to the blue

and red line in Fig. (1.5b), respectively.

CHAPTER 1. 16

ξ

2φ2R +

λ

4

φ2 − v2

2

ξ > 0, φ < v

v →∞

ξ = 0v 1

Figure 1.7: (ns, r) at N = 60 for the quartic potential Eq. (1.22) with positive ξ inthe case with inflation at φ < v. ξ increases from right to left and on each curve vincreases to ∞ as we move away from the bottom-left corner.

Let us first consider the red line of Fig. (1.5b) where |ξ|v2 > 1. In this case,

the only minimum of the potential is at ϕ = 0 (ignoring the minimum V = 0 at

large φ in the antigravity regime). This is a minimum with a huge cosmological

constant V (0) = λv4

4, so this regime is unsuitable for inflation describing our part of

the universe.

A more interesting possibility |ξ|v2 < 1 is illustrated by the blue line of Fig. (1.5b).

We still have ‘small-φ’ and ‘large-φ’ on the left of the singularity, now defined as

0 < φ < v and v < φ < |ξ|−1/2 respectively. With N = 60 the result of large- and

small-φ cases are shown in Fig. (1.8). Unless |ξ| is extremely small, the large-φ case

gives predictions that are incompatible with the WMAP results, see the set of lines

above the white circle in Fig. (1.8). The small-φ case is more realistic. It covers a

CHAPTER 1. 17

rather small but observationally important area with ns / 0.97 and the tensor-to-

scalar ratio r as tiny as ∼ 10−3. The motion along the lines from left to right in

Fig. (1.8) shows what happens in this model, for a given value of ξ < 0, when v

grows. These curves have some interesting features.

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4ξ

2φ2R +

λ

4

φ2 − v2

2

ξ < 0

φ > v

φ < v

ξ = 0

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4

Figure 1.8: (ns, r) at N = 60 for the quartic potential Eq. (1.22) with negative ξ and|ξ|v2 < 1 (the blue line of Fig. (1.5b)). On each curve v increases from 0 on left to. |ξ|−1/2 on right. The large-φ case is shown by the curves above the white circle.The curves of constant ξ are shown with |ξ| increasing from bottom to top. Thesmall-φ case is represented by the curves below the white circle, with |ξ| increasingfrom top to bottom. The cusps correspond to moderately but not too large v. Forexample, the curve of ξ = −10−6 enters the 1σ region at v ≈ 13; then continues tothe right and for 200 . v . 800 stays at the peak of the cusp which is very close tothe white circle. For rather larger |ξ| the curves become almost flat. For example, thecurve of ξ = −10−2 crosses the 1σ region at v ≈ 7.8 with r ≈ 0.006 and continues tothe right almost horizontally to v = 9.9 with ns ≈ 0.972 and r ≈ 0.0069. Eventually,in the limit v → |ξ|−1/2, all curves bend to the left and meet at a single point (thestar). Rather unexpectedly, this point coincides with the result of the Higgs inflationmodel with ξ 1 studied in [9] (the star in Fig. 6(a)).

CHAPTER 1. 18

First of all, there are cusps, i.e., maxima just before the right end of each ξ = const

curve. This feature can be qualitatively understood as follows: When φ < v |ξ|−1/2, it doesn’t matter whether ξ is positive or negative. In other words, φ = v

(where the potential vanishes) is far away from φ = |ξ|−1/2 where effects of ξ begin

to show up. So the left ends of the curves below the line ξ = 0 of Fig. (1.8) are

almost the same as those of Fig. (1.7). But when v grows and starts approaching

|ξ|−1/2, then the potential differs for positive and negative ξ, since for the latter case

a singularity develops. This is the place when the curves of constant ξ in Fig. (1.7) and

Fig. (1.8) depart; the former go up (because they approach the large-field-excursion-

type potential of m2φ2/2) and the latter go down (because they approach a small-

field-excursion-type potential similar to new inflation).

The second feature is that, regardless of the value of ξ, all of these curves converge

on their right end to the same point (ns, r) = (1 − 2N, 12N2 ) = (0.967, 0.003). This is

the limit when the minimum at φ0 = v is very close to the singularity at φ = |ξ|−1/2,

i.e., when |ξ|v2 approaches 1 from below. We now show that it corresponds to a

new-inflation type potential with an extremely flat plateau.

j

V

(a)

j

V

(b)

Figure 1.9: The potential λ4(φ2−v2)2 as a function of the canonically normalized field

ϕ in the Einstein frame. (a) ξ < 0, 1 − |ξ|v2 1. (b) ξ > 0. Notice the similaritybetween these two potentials. However, physical interpretation of these two potentialsis very different. The value of the canonically normalized field ϕ in the model withξ < 0 typically is very large. Meanwhile the value of this field for the model withξ > 0 can be very small, which is the basis of the Higgs inflation scenario [9].

In this limit we can use Eq. (1.9) to approximate the form of the potential in the

CHAPTER 1. 19

vicinity of its minimum at φ0 = v:

V (ϕ) =

(λ

4ξ2

)(1− exp

(−√

2

3(ϕ0 − ϕ)

))2

, (1.27)

where ϕ0 is the location of the minimum in terms of the canonical field ϕ. One

can show that during inflation, in the slow-roll regime, exp(−√

23(ϕ0 − ϕ)

) 1.

Therefore by replacing ϕ0 − ϕ → φ one can represent this potential during inflation

as

V =λ

4ξ2

(1 + exp

(−√

2

3φ))−2

. (1.28)

At large ϕ, this potential coincides with the inflationary potential in the Higgs infla-

tion scenario with the potential λ4(φ2 − v2)2 with v 1 and ξ 1 in Ref. [9]. To

illustrate this statement, one may compare the potential λ4(φ2 − v2)2 with ξ < 0 in

the limit |ξ|v2 → 1 with the potential in the Higgs inflation model [9] with ξ > 0 and

v < 1, for the same values of parameters λ and |ξ|, see Figs. 1.9a and 1.9b. The results

of our calculations of the parameters ns and r and of the amplitude of perturbations

of metric in this model coincide with the corresponding results of Ref. [9].

The relation between the inflationary regime with φ < v and ξ < 0 in the limit

|ξ|v2 → 1 and Higgs inflation with φ > v and ξ 1 is quite interesting and even

somewhat mysterious, as if there is some hidden duality between these two classes of

models with opposite signs of ξ. Moreover, the same set of the cosmological parame-

ters (ns, r) = (1− 2N, 12N2 ) = (0.967, 0.003) also appears in Starobinsky model [16], see

[17, 18].

The value of λ can be determined from the observed value of amplitude of density

perturbations δρ ≈ 5×10−5. For the two observationally interesting cases the situation

is as follows: For the large-φ case with positive ξ, λ decreases along the constant-ξ

lines. At ξ = 0 we have λ ≈ 2× 10−13 and for large v we have λ ≈ 4× 10−11/v2 (this

follows from the fact that for large v the quartic potential near its minimum looks like

a quadratic one with mass squared m2 = λv2). For large ξ we have λ ≈ 5× 10−10ξ2

independently of v, as in Ref. [9]. For the small-φ case with negative ξ the behavior

is similar. For small |ξ| and large v (but not too large to pass the cusp peak) we have

CHAPTER 1. 20

λ ≈ 4× 10−11/v2. In the limit |ξ|v2 → 1 the potential is the same as that of Ref. [9]

and hence again we have λ ≈ 5× 10−10ξ2. Thus we see that λ can be parametrically

large or small.

1.7 Conclusions

In this chapter we studied the simplest chaotic inflation models with potentials m2

2φ2

and λ4(φ2−v2)2, with the field φ nonminimally coupled to gravity, for arbitrary values

of the parameters v and ξ. We demonstrated that these models can be implemented

in the context of supergravity, along the lines of [4], and the inflationary trajectory

can be stabilized by a proper choice of the Kahler potential.

We found that by changing parameters of these models one can cover a significant

part of the space of the observable parameters (ns, r) allowed by recent observational

data. As one can see from Figs. 4, 1.6 and 1.8, the best fit is achieved by the modelm2

2φ2 with ξ > 0, and by the model λ

4(φ2 − v2)2 in two different regimes: ξ > 0, with

inflation at φ > v, and ξ < 0, with inflation at φ < v.

In particular, Fig. 4 shows that by adding a term ξ2φ2R with ξ ∼ 10−2 one can

suppress the tensor/scalar ratio r for the model m2

2φ2 from r ∼ 0.13−0.15 (depending

on N) down to r ∼ 0.01. Similarly, investigation of the model λ4(φ2 − v2)2, which is

ruled out by observations for v < 1, shows that by adding to this theory a term ξ2φ2R

with ξ ' 0.0027 one makes it consistent with observational data within 1σ (red area

in Fig. 1.6), for all values of v. In other words, in order to make this model consistent

with observations, there is no need to go to the limit ξ 1; a very small positive

value of ξ is quite sufficient. Further increase of ξ leads to a rapid convergence of the

predictions to a single point ns = 0.967 and r = 0.003, independently of the value of

v, see the gray star in Fig. 1.6.

One of our most unexpected results is the coincidence between the inflationary

predictions of the model λ4(φ2− v2)2 for ξ 1, as shown in Fig. 1.6, and of the same

model with ξ < 0 in the limit |ξ|v2 → 1, Fig. 1.8. In both cases one finds ns = 0.967

and r = 0.003.

Thus, the non-minimal coupling of the inflaton field to gravity may significantly

CHAPTER 1. 21

modify predictions of the simplest inflationary models. In several cases, these modi-

fications suppress the amplitude of tensor modes to the level below r ∼ 10−2, which

may make tensor modes difficult to observe. In addition, a proper theoretical inter-

pretation of the cosmological data may become difficult because one can obtain the

same set of parameters (ns, r) in different models. This is a general problem discussed

e.g. in [19]. The degeneracy between the predictions of different models in certain

cases may be removed if one takes into account additional scalar fields which may

exist in the supergravity-based versions of these models. For a certain choice of the

Kahler potential, some of these fields may become light along the inflationary trajec-

tory. This may lead to an additional source of inflationary perturbations for some of

these models, which can be very non-gaussian [6]. Thus, this class of models has some

unique features, which provide new interesting possibilities for a proper interpretation

of upcoming observational data.

Chapter 2

Measure Problem for Eternal and

Non-Eternal Inflation

2.1 Introduction

Inflationary cosmology provides a simple mechanism which explains the observed

homogeneity of our world: Inflation takes a tiny domain of the universe and rapidly

expands it to the size which may exceed by many orders of magnitude the size of

the observable part of the universe. This stretching removes all previously existing

inhomogeneities and renders our world uniform. However, this mechanism does not

make the universe globally uniform. If the universe from the very beginning consisted

of different parts with different properties (e.g. the scalar fields occupying different

minima of their potential energy), then the post-inflationary universe becomes divided

into many exponentially large parts with different properties and even with different

laws of the low-energy physics operating in each of them [20, 21]. Moreover, even if

initially the universe was represented by a single homogeneous domain, inflationary

quantum fluctuations may divide it into many exponentially large parts with different

properties. In effect, an inflationary universe becomes a multiverse consisting of

exponentially large “universes.” This process leads to most profound consequences if

inflation is eternal [22, 67, 24, 25, 26, 27, 28], but similar effects may occur even if

inflation is not eternal [29].

22

CHAPTER 2. 23

These observations provided a simple scientific justification for the use of anthropic

principle in inflationary cosmology. One should be much less surprised to see that

various parameters of the theory of elementary particles take non-generic, fine-tuned

values if this is what makes our life possible. Our life is also non-generic, so if the

universe provides us with the choice of generic vacua where we cannot live and non-

generic ones where we can live, the choice is obvious. However, in order to use this

argument to its full potential, and to go from “possible” to “probable,” one should

learn how to compare the probabilities to live in different parts of the multiverse.

The main problem here is that in an eternally inflating universe the total volume

occupied by all, even absolutely rare types of the “universes,” is indefinitely large.

Therefore comparison of different types of vacua involves comparison of infinities. As

emphasized already in the first papers on the probability measure in eternal inflation

[25, 26, 27], such a comparison is inherently ambiguous and depends on tshe choice

of the cutoff, which is required to regularize the infinities. However, as we are going

to see shortly, the measure problem may appear even if inflation is not eternal and

the universe is finite.

Historically, the first probability distribution considered in the literature was the

function Pc(φ, t) [93, 31]. It described the probability to find a given scalar field at

a given time at a given point. One can equivalently interpret Pc(φ, t) as the proba-

bility distribution in the comoving coordinates, which do not reflect the exponential

expansion of the universe during inflation.

This probability distribution is inconvenient for the description of eternal inflation,

which occurs because of the exponential growth of the parts of the universe remaining

at the stage of inflation. Eternal inflation occurs even when it could seem improbable

in terms of Pc(φ, t). As a result, an investigation of eternal inflation initially was

performed with the help of a different probability distribution P (φ, t), which rewarded

vicinity of each inflationary point by a factor reflecting the growth of its volume

[67, 31]. An important advantage of this probability measure was the stationarity

(time-independence) of the distribution P (φ, t) in the limit of large t [25, 26].

However, the probability distribution P (φ, t) depends on the choice of time pa-

rameterization. If t is the usual proper time, which is measured by the usual clock,

CHAPTER 2. 24

then the vicinity of each point during inflation grows like e3H∆t during each small

time interval ∆t. This rewards expansion of the universe with large H. But one

can also measure time in terms of expansion of space τ ∼ a, where a is the scale

factor [93, 67, 31, 25, 26], or, equivalently, in terms of η ∼ log τ = log a. In this

case, vicinity of each point is uniformly rewarded by the H-independent factor e3∆η.

More generally, one can introduce a family of measures where expansion is rewarded

by a factor e3Hβ∆tβ , where β is some constant depending on choice of time slicing tβ.

Each of these choices is quite legitimate, but the results obtained by this method are

exponentially sensitive to the choice of the cutoff, i.e. to the choice of β [25, 26].

Since that time, dozens of different candidates for the role of the probability mea-

sure have been proposed, most of them giving different predictions. It is impossible

to give a full list of different proposals here, a partial list can be found e.g. in [32].

We will mention only few of them, which are often discussed now, and briefly discuss

their advantages and problems.

Out of all of these measures, the original proper time cutoff measure P (φ, t) is

the simplest. However, this measure suffers from the youngness problem [33], which

was especially clearly formulated in [84]: This measure exponentially rewards parts

of the universe staying as long as possible at the highest values of energy density. As

a result, this measure exponentially favors life appearing in the parts of the universe

with an extremely large temperature, which blatantly contradicts the observational

data.

The scale factor cutoff measure P (φ, τ), which corresponds to the choice β = 0,

does not suffer from the youngness problem because it does not give exponentially

large rewards to the parts of the universe spending extra time at a large energy den-

sity. Therefore this measure, as well as its various modifications and generalizations,

became quite popular lately, see e.g. [35, 36, 37, 38, 39].

However, this advantage occurs because the scale factor cutoff measure corre-

sponds to the special choice β = 0 in the class of measures where expansion is re-

warded by e3Hβ∆tβ . For all values of β > 0, these measures suffer from the youngness

problem, and for β < 0 they suffer from the opposite problem, which can be called

“oldness” problem: Life is predicted to exist mostly in cold empty space. This is

CHAPTER 2. 25

equivalent to the so-called Boltzmann brain problem. It appears in a certain class

of measures predicting that typical observers should be created not as a result of

the usual cosmological evolution, but because of quantum fluctuations in an empty

post-inflationary universe [40].

Thus the choice β = 0 must be made with an incredible precision. An optimist may

consider it as an indication that the special choice β = 0 is preferable. A pessimist

may counter it by saying that the scale factor cutoff measure provides an ultimate

example of an exponential instability of predictions with respect to the choice of time

parameterization, which seems unphysical.

Another possibility is to consider the causal diamond measure [41, 42] which,

unlike the measures discussed so far, is not global and does not involve any time

cutoff. This measure cuts from the spacetime inside any given vacuum a finite four-

volume subset (the causal diamond) formed by the intersection of the future light

cone of the point where “an observer” crosses the hypersurface of reheating and

the past light cone of the point where he/she leaves that vacuum. This makes all

regularized quantities finite within the causal diamond. The measure then assigns

(i) prior probabilities to vacua of each type based on the number of times when an

observer appears in a given vacuum and (ii) weights to each vacuum according to the

entropy produced in its causal diamond. The net probability is given by the product

of these two numbers. The predictions obtained by this method depend on the initial

probability distribution of the vacua. If, for example, one evaluates the probability of

initial conditions using the Hartle-Hawking wave function, then the causal diamond

measure suffers from a severe Boltzmann brain problem [43]. Under many alternative

assumptions concerning the probability distribution of initial conditions, the causal

diamond measure gives practically the same results as the scale factor cutoff measure

and the recently proposed measure based on considerations related to holography and

conformal invariance [38]. When using the causal diamond measure in this chapter

we assume that the prior probabilities are uniform on different vacua. We also take

the number of galaxies in a causal diamond as a proxy for the amount of entropy

produced there.

All of these measures share certain vulnerability with respect to the Boltzmann

CHAPTER 2. 26

brain problem. Roughly speaking, in order to avoid this problem for these measures,

the decay rate of each of the vacua in the landscape must be greater than the rate of

the Boltzmann brain production there. For a more exact formulation of this condition

see [36, 37]. This condition may be satisfied for a rather broad class of vacua in

string theory landscape [44], but the answer for generic string theory vacua is not

known yet. And here lies a potential problem: If there is a 50% probability that the

required condition is satisfied in each of the exponentially large number of the as yet

unexplored stringy vacua (we made the 50% probability assumption simply because

we do not really know the answer), then the probability that it is satisfied in all of

the unexplored vacua is exponentially small.

Potential difficulties of this class of measures are not as severe as the youngness

problem plaguing the proper time cutoff measure P (φ, t). This problem was addressed

in a modified version of this measure, which was called the stationary measure [45, 46].

The main observation of Ref. [45] was that the stationary regime, in which P (φ, t)

becomes time-independent, is established at different times for different processes.

Therefore it does not make sense to compare all processes at the same time. Instead of

that one should compare different processes starting at the time when the stationarity

is first achieved for each of them separately.

The corresponding procedure is quite complex and requires a very careful imple-

mentation. This is not really surprising given the complexity of the eternal inflation

in string theory landscape. One of the advantages of this measure is the absence of

the exponential sensitivity of predictions to the choice of the time parameterization

[45]. An investigation performed in [46] suggests that this measure does not suffer

from the youngness problem and the Boltzmann brain problem.

Stationary measure does not reward us for growth of volume during a purely

de Sitter stage in a metastable vacuum state (i.e. in false vacuum). However, the

probabilities are proportional to the growth of volume during the stage of the slow

roll inflation. This property explains flatness of the universe. There is a potential

danger that the exponential sensitivity of the total volume of the universe to the

choice of the inflationary parameters may make the total number of observers in the

universe exponentially large, but simultaneously make the probability of emergence

CHAPTER 2. 27

of life in any finite volume extremely small. This is the essence of the problem which

is sometimes called Q catastrophe [28, 47, 48]. Possible solutions of this problem were

proposed in [90, 91, 51].

This brief discussion illustrates the general situation with the probability measure

for eternal inflation. In this chapter we will discuss the situations when inflation is

not eternal and the universe is compact. As we will see, even in such situations one

may come to different conclusions with respect to probabilities. Therefore it may

make sense to temporarily suppress our ambitions and try to understand non-eternal

inflation, which at the first glance could seem quite trivial, and then return again

to the investigation of eternal inflation. We will see that the stationary probability

measures lead to similar predictions for eternal and non-eternal inflation. However,

all other measures discussed above give very different predictions for models with

eternal and non-eternal inflation. While this discontinuity does not necessarily mean

that such measures are problematic, we think that this fact requires certain attention.

2.2 Eternal and Non-eternal Inflation: A Toy Model

We consider inflation driven by a scalar field in models with two different potentials.

In one model (which we call the eternal model; see Fig. 2.1), the scalar field starts

at φ0. Inflation proceeds via tunneling through the potential barriers to the right

or left with an equal rate Γ per proper volume. Bubbles of new vacua nucleate and

undergo a subsequent slow roll inflation along the fairly flat shoulders of the potential

in either side. The value of the potential Vs on these shoulders is the same but the

field excursion along them is different. Eventually inflation ends and the scalar field

takes on the value φ+ or φ− in the corresponding part of the universe. We assume, for

simplicity, that these are vacua with the same particle physics and the same vacuum

energy densities Λ < 0, which collapse in a finite time. As long as the expansion

rate 3H0 of the volume populated by false vacuum is greater than its total decay

rate 8π3H−3

0 Γ (which is true in most realistic situations), inflation is eternal. The

space-time takes a fractal structure with infinite 4-volume and a measure is required

to make any statistical statement.

CHAPTER 2. 28

Φ- Φ+

Φ

V0

Vs

V

Figure 2.1: The scalar field potential for the eternal inflation model.

Φ- Φ+

Φ

V0

Vs

V

Figure 2.2: The scalar field potential for the non-eternal inflation model.

In the other model (the non-eternal model; see Fig. 2.2) the scalar field starts

from φ0 again but there is no potential barrier and, instead of tunneling, the field

just falls down to either right or left (again with equal probabilities since the shape

of the potential is symmetric). This part of the potential is assumed to be steep

enough that quantum fluctuations cannot cause an upward jump and hence inflation

is non-eternal. Finally there is a subsequent slow roll inflation, like in the eternal

model, ending in one of the collapsing vacua.

In both cases, we will make a simplifying assumption that the different parts of

the universe experience a long stage of inflation, so that each such part becomes

CHAPTER 2. 29

exponentially large, and all interesting processes in each of these parts occur practi-

cally independently, as if they were separate universes. Of course, this picture is only

approximately correct, and one should be careful not to use it beyond the limits of

its applicability (we will discuss the situation where this assumption breaks in Sec-

tion 2.4). We hope that this approximate picture will be sufficient to identify some

important differences which appear when one tries to describe models of eternal and

non-eternal inflation.

Even if inflation is not eternal, one may still face the problem of regularizing

infinities if, for example, we discuss an open or flat spatially infinite universe. There

are two ways to avoid it. The most obvious way is to consider a closed universe with

Λ ≤ 0. It has finite size and it collapses in finite time. Another, less trivial possibility

is to consider a compact open or flat universe, the simplest example being a flat

universe with periodic boundary conditions, represented by a compact k = 0 3-torus.

Whereas this possibility may seem a bit unusual, an investigation of the probability of

quantum creation of such universes shows that their formation is exponentially more

probable than formation of closed universes [52, 53, 54]. Therefore in this chapter we

will concentrate on the models describing a flat or open compact universe. However,

all results will remain valid for the close universe case if the duration of inflation in a

close universe is long enough to make it effectively flat.

The evolution of the scale factor after the field falls down the hill but before it

begins the slow roll stage is the same on the right and left shoulders of the potential

since it is symmetric in the vicinity of φ0. This stage occurs in a similar way for

eternal and non-eternal inflation. In the eternal model the field tunnels from the false

vacuum V0 to an escape point close to the top of the potential, and then it starts

falling down the hill with zero initial velocity. In the non-eternal model the field just

falls down from the same height V0 in the potential of a very similar shape, with zero

initial velocity. In both cases it takes the same time for the field to reach the terminal

velocity φ = −V ′s/3Hs, where 3H2s = Vs is the vacuum energy density during the slow

roll inflation and V ′s is the slope of the potential during this stage. For simplicity we

assume that V0 and Vs have the same order of magnitude, the scalar potential does

not change much during inflation, and the slope of the potential is very small and

CHAPTER 2. 30

nearly constant.

The field then rolls for a time

ti± =

∣∣∣∣√

3Vs(φ± − φ0)

V ′s

∣∣∣∣ (2.1)

until it reaches the right or left minimum, respectively. The stage of slow roll inflation

and the subsequent reheating is the same for the two models. The only difference

is that in the eternal model the spatial geometry of the bubble interior is a k = −1

hyperboloid (open inflation) while in the non-eternal model the newly formed domain

can be either open or flat [52, 53, 54]. This difference is rather inessential since soon

after the beginning of inflation the universe becomes flat. Thus, for the purposes of

our work one can assume that the universe from the very beginning was expanding

exponentially, a(t) ∼ a0 eHst, and the total number of e-folds of inflation is given by

N± = Hsti±, where 0 < t < ti± is the time duration of inflation on right and left,

respectively.

The total size of the universe at the end of inflation a(ti±) ∼ a0 eN± depends on

the size of the universe a0 at the beginning of inflation. We will assume that the size

of the universe at the beginning of inflation was very small. It can be as small as

the Planck length, a0 ∼ 1, or it may be of the order of the inverse Hubble constant

during inflation, a0 ∼ H−1s , or it may take an intermediate value a0 ∼ H

−1/3s , starting

from which the classical description of a compact flat universe becomes possible [54].

However, even a very large difference between various possible values of a0 can be

compensated by a slight change of N±. For the purposes of our work we will simply

take a0 = 1, and treat N± as input parameters (instead of the details of the potential).

At the end of the slow roll inflation, reheating takes place, which we assume to

be instantaneous. It produces radiation and matter with respective densities ρri and

ρmi in a universe with a small negative cosmological constant, |Λ| Vs. Energy

conservation at t = ti requires that

Vs = ρmi + ρri + Λ. (2.2)

CHAPTER 2. 31

The ratio ρmi/ρri of produced matter to produced radiation is determined by the par-

ticle physics which is the same on right and left. Since Vs and Λ are fixed throughout

this chapter, we conclude that ρmi and ρri are also constant parameters independent

of model (i.e., of eternal or non-eternal nature of the model) and of ± (i.e., of right

or left minimum of the potential). Such a universe will reach a maximum size at a

turning time t = tt, then begins contracting and finally collapses to a singularity at

the finite time t = tc. To see this we note from the FRW equation

H2 +k

a2=

1

3

[ρmi

(aia

)3

+ ρri

(aia

)4

+ Λ

], (2.3)

that for Λ < 0 and ρm, ρr ≥ 0 there is always a turning point Ht = 0 after which a

phase of contraction starts leading to a singularity at a = 0.

To estimate the total lifetime of the universe, let us first assume that inflation was

long enough to render the term ka2 irrelevant. (This term does not appear for the flat

compact universe anyway.) Ignoring the duration of the period of inflation (ti tc),

one can show that the total lifetime of the universe from its creation to its collapse

is given by

tc = 2

∫ 1

0

√3αdα√

Λα4 + ρmtα + ρrt. (2.4)

Here α is the ratio of the scale factor to its value at the turning point, and ρmt and ρrt

are the densities of matter and radiation at the turning point, where ρmt+ρrt+Λ = 0.

It is instructive to consider two separate cases: hot universe, ρmt = 0, ρrt + Λ = 0,

and cold universe, ρrt = 0, ρmt + Λ = 0. Analytic integration of this equation shows

that for the hot universe

thotc =

π

2

√3

|Λ| , (2.5)

and for the cold universe

tcoldc =

2π

3

√3

|Λ| . (2.6)

Now let us restore the term ka2 in our equations and consider an open universe,

CHAPTER 2. 32

ignoring ρmt and ρrt. In this case,

topenc = π

√3

|Λ| . (2.7)

More generally, if k = 0 the collapse time is bounded from below by thotc and from

above by tcoldc . If k = −1 the upper bound is topen

c and the lower bound is thotc . In all

of these cases the lifetime of the universe does not depend on the duration of inflation

and is given by c|Λ|−1/2, where the coefficient c = O(1) depends on the matter

contents of the universe. The only exception from this simple rule appears if one

considers a closed universe with a short stage of inflation and an exponentially small

cosmological constant. Then the universe collapses at a time shorter than O(|Λ|−1/2).

As we already mentioned, in this chapter we will concentrate, for simplicity, on open or

flat compact universes, or on closed universes with a sufficiently long stage of inflation,

when the rule described above holds. However, most of the qualitative conclusions of

this chapter will remain valid for closed universes with a short inflationary stage.

We are interested in counting the total number N± of galaxies across the whole

spacetime, which are located inside a φ± vacuum. In a part of the universe where

inflation has ended, galaxies are formed at a time tg ti when the scale factor is

ag = a(tg). As long as Ω ≈ 1, tg must be independent of the inflationary history

of universe and in particular of the number of e-folds (as long as the galaxies are

much smaller than the total size of the universe). The time tg, and, consequently, the

ratio ag/ai, and the number density of galaxies n(tg) can depend only on the particle

physics of the vacuum, so they are ±-independent.

2.3 The Eternal Inflation Model

In this section we calculate the relative probability N+

N− in the eternal model. A

careful analysis of the problem within the three cutoff-based measures requires solving

a master equation which incorporates both the tunneling process and the diffusion

equation that describes the stochastic behavior of the field during slow roll inflation

[55, 43, 45]. We will not get into the details of this analysis, rather take a shortcut

CHAPTER 2. 33

to the answer and explain our simplifying assumptions.

Let us consider a time cutoff τ that is related to the proper time t via dτ = H1−βdt.

The total volume V0 in the false vacuum state φ0 grows because of the exponential

expansion of de Sitter space, but also slightly decreases due to the false vacuum

decay. Ignoring the probability of jumps back to the original state φ0, one can write

an equation for V0:dV0(τ)

dτ= (3Hβ

0 − 2κ)V0(τ) . (2.8)

Here κ = 4π3Hβ−4

0 Γ is the decay rate per unit τ . This equation has a simple exponential

solution:

V0(τ) = V0(0) exp[(3Hβ0 − 2κ)τ ]. (2.9)

The second half of the process is the slow roll from the escape point(s) of tunneling

to φ±. We ignore the diffusion due to the quantum fluctuations and only consider the

classical roll. Then the volume V± of the part of the universe that has reached the

time of galaxy formation on right/left is determined by

dV±(τ)

dτ= κe3N±

(ag±ai±

)3

V0(τ − τg±), (2.10)

where τg± is the time from bubble nucleation to galaxy formation:

τg =

∫dτ =

∫da

Hβa. (2.11)

In the special case of proper time one finds τg = tg, whereas for the scale factor time

one gets τg = N + log agai

.

Noting that ag/ai is ±-independent and N± = Hsti± we find for the proper time

(β = 1)

limτ→∞

V+(τ)

V−(τ)

∣∣∣∣β=1

= exp

[(3− 3H0 − 2κ

Hs

)(N+ −N−)

]. (2.12)

The term 3H0/Hs dominates the first factor in the exponent and leads to favoring

of shorter inflation: the probability to be in φ+ is exponentially suppressed [45]. For

CHAPTER 2. 34

the scale factor time (β = 0), however, we find:

limτ→∞

V+(τ)

V−(τ)

∣∣∣∣β=0

= e2κ(N+−N−), (2.13)

indicating a mild favoring of longer inflation (which is actually the source of the mild

Boltzmann brain problem in this measure [37]). But since κ is exponentially small,

for the purposes of our comparison we can just consider the limit κ→ 0 which gives

equal likelihood to φ±,

limτ→∞

V+(τ)

V−(τ)

∣∣∣∣β=0

= 1 (2.14)

To find the result for arbitrary β one can break the integral in Eq. (2.11) into two

pieces: a < ai and a > ai. The post inflation piece, a > ai, is ±-independent (similar

to the integrals appearing in Eqs. (2.4) and (2.19)) so it cancels out in the ratio while

the inflationary piece, a < ai gives a contribution ≈ (N+ − N−)/Hβs . Therefore one

finds

limτ→∞

V+(τ)

V−(τ)= exp

[(3− 3Hβ

0 − 2κ

Hβs

)(N+ −N−)

]. (2.15)

In the stationary measure one synchronizes the exponential growth of the volume

of the two sides based on the time they reach the stationarity regime. This amounts

to modifying Eq. (2.10) to read:

dV±(τ)

dτ= κe3N±

(ag±ai±

)3

V0(τ), (2.16)

whose solution yields:

limτ→∞

V+(τ)

V−(τ)

∣∣∣∣stationary

= e3(N+−N−). (2.17)

All results obtained above describe the ratio V+(τ)V−(τ)

at a given time τ , if this time

is large enough. The same results describe the ratio of all galaxies which ever existed

in the universe until the cut-off time τ , in the limit τ → ∞. This observation will

play an important role when we will discuss possible generalizations of these results

for the non-eternal inflation.

CHAPTER 2. 35

Finally for the causal diamond measure one finds the same result as in (2.14):

limτ→∞V+(τ)V−(τ)

∣∣∣β=0

= 1, i.e. the final result does not depend on the duration of the

slow roll inflation. Indeed, in our model the total lifetime of the universe, and,

consequently, the total size of the causal diamond and the entropy produced there,

do not depend on the duration of inflation. In the next section we will explain this

result in a more detailed way and show that it remains valid for the non-eternal

inflation as well, but only if inflation is long enough.

2.4 The Non-eternal Inflation Model

In the non-eternal model the total proper volume of all collapsing universes at their

time of galaxy formation is finite, and so is the total number of galaxies. This makes

the calculation of the total number of galaxies simple and unambiguous. However,

one immediately realizes some additional problems which were swept under the carpet

in the eternal inflation scenario.

Indeed, in the eternal inflation scenario, galaxies of a given type will always exist

in some parts of the universe at any time τ , and the ratio of their number in different

vacua asymptotically becomes time-independent. This is not the case for the non-

eternal inflation scenario. One can see it most clearly in the lower panel of Fig. 3,

which shows two “diamonds” describing the evolution of the universe with respect

to the scale factor time. Whereas the proper time required for the galaxy formation

in these universes is approximately the same, the corresponding scale factor time is

dramatically different. As a result, it simply does not make any sense to compare the

number of galaxies at the same scale factor time τ .

This problem does not appear if one uses an analogue of the stationary measure,

where the comparison occurs not at a given cosmological time, but at the time when

certain physical processes (i.e galaxy formation) take place. The corresponding result

in the stationary measure is proportional to the total volume of the universe at the

time of the galaxy existence, which, in its turn, is proportional to the growth of

CHAPTER 2. 36

volume during the slow roll inflation,

N+

N−=a3i+

a3i−

= exp[3(N+ −N−)]. (2.18)

This result coincides with the result which we obtained using the stationary measure

in the eternal inflation case, see Eq. (2.17).

An alternative approach is to compare the total number of galaxies which may be

formed over the whole lifetime of the universe. This method works not only for the

stationary measure, but for the proper time measure and for the scale factor measure.

On average, φ± domains are formed at the same time and each occupies half of the

volume of the original false vacuum. Thus we can find the ratio N+

N− of all galaxies

in φ+ to all of those in φ− by simply computing the ratioN (1)

+

N (1)−

of the galaxies in

only one φ+ domain to those in one φ− domain. Ignoring geometric factors such as

4π/3, one can write N (1)± = n±(tg±) a3

g±. As we saw earlier, ng and ag/ai are ±-

independent. Therefore, the ratio of the total number of galaxies which ever existed

in the φ+ vacuum to those in the φ− vacuum is again given by Eq. (2.18). Once

again, we recovered the answer which was obtained using the stationary measure for

the eternal inflation scenario.

As already mentioned, the spacetime 4-volume is finite in this model and there

is no need for a cutoff to regularize infinities. Thus Eq. (2.18) is an unambiguous

result of counting of all galaxies in the universe. However, if one decides to impose an

additional cutoff, which is not necessary for regularization of infinities in a compact

universe, one may come to a different conclusion.

In particular, the causal diamond measure does not take into account the parts of

the universe that do not belong to the causal diamond. Let us find the ratio N+/N−given by this measure. Consider an observer whose worldline crosses the reheating

surface and hits the singularity far away from the walls, so that an FRW description

with the observer at the center of coordinates is valid. We need to count galaxies

formed at tg but only those which lie inside the causal diamond. The diamond has two

boundaries: the past light cone of the point where the worldline hits the singularity

r = −∫ ttc

dta(t)

, and the future light cone of the point where it hits the reheating surface

CHAPTER 2. 37

r =∫ tti

dta(t)

. At tg < tt the smaller of these two is the latter. Thus at the time of

galaxy formation tg there are

NCD = ng

(ag

∫ tg

ti

dt

a(t)

)3

= ng

(agai

∫ agai

1

√3dα√

Λα4 + ρmiα + ρri

)3

(2.19)

galaxies within the causal diamond. All variables appearing in this expression are

±-independent and thus so is NCD. Furthermore, a uniform initial distribution of

worldlines implies that on average an equal number of worldlines enter the φ± domains

and since they are far away from the walls they will stay in their domain until they fall

into the singularity. Thus N± ∝ NCD with a ±-independent proportionality constant.

Therefore, for the causal diamond measure we find

N+

N−= 1. (2.20)

Thus the causal diamond measure, unlike all other measures discussed above, does

not reward us for the exponential growth of volume during a long stage of slow roll

inflation. This result is valid for eternal and non-eternal inflation, but only if the

stage of inflation is very long.

Indeed, if inflation is very long, any observer will be able to see (and to influence)

only a tiny fraction of the universe determined by the size of the causal diamond,

which does not depend on the duration of inflation. However, if inflation is short

and the universe is small enough, an observer will be able to see all galaxies in the

universe. In this case the total number of galaxies accessible to observations will

be proportional to the total volume of the universe, and instead of Eq. (2.20) one

should use Eq. (2.18), which shows that the probability depends exponentially on the

duration of inflation, if it is short enough.

CHAPTER 2. 38

Measure Non-Eternal Model Eternal Model

Proper Time exp[3(N+ −N−)] exp[−3H0

Hs(N+ −N−)

]

Scale Factor exp[3(N+ −N−)] 1

Causal Diamond

exp[3(N+ −N−)] if N is small enough

1 if N is large enough1

Stationary exp[3(N+ −N−)] exp[3(N+ −N−)]

Table 2.1: Predictions of the four measures in the eternal and non-eternal models.

2.5 Conclusions

We have investigated four different measures in an eternal and a non-eternal toy

model. Table 2.1 summarizes the predictions of each measure for the relative prob-

ability of being in φ+ versus being in φ−. In terms of rewarding for a long stage of

inflation we have found that:

• The proper time (standard volume weighting) measure exponentially favors in-

flation in the non-eternal case but exponentially disfavors it in the eternal case.

The two exponential behaviors are different: in the non-eternal case only the

difference in the number of e-folds appears in the exponent while in the latter

the exponent contains, in addition to N+ − N−, a large factor H0/Hs coming

from the high energy of the false vacuum relative to the energy scale of inflation.

• The scale factor cutoff measure behaves the same as the proper time measure in

the non-eternal case. But in eternal inflation case, it does not care much about

the duration of the slow roll inflation.

• The causal diamond measure is like the scale factor cutoff in the eternal case: no

reward for inflation. But in non-eternal inflation it has an interesting behavior

being sensitive to the number of e-folds only up to a critical number. Beyond

that the observer doesn’t distinguish longer stages of inflation and hence the

measure is insensitive to N .

• The prediction of the stationary measure coincides with the unambiguous count-

ing of galaxies in non-eternal inflation. This is shared by the previous two

CHAPTER 2. 39

measures as well. But the stationary measure gives the same result in both

eternal and non-eternal case. It is the only measure among the four we studied

here that produces a result that continuously varies as one goes from eternal to

non-eternal inflation.

It is quite interesting that all of these measures, being applied to non-eternal infla-

tion, tell us that the probability to live in a given part of the universe is proportional

to the exponential growth of volume during the slow-roll inflation, or at least during

a certain part of inflationary expansion of the universe. Surprisingly, the first three

of the measures in Table 2.1 lost this universal property when applied to the eternal

inflation scenario.

These results do not necessarily disfavor the first three measures since they have

been invented for eternal inflation rather than for the non-eternal inflation. Never-

theless, dramatic discontinuity of predictions during the transition from non-eternal

to eternal inflation is quite intriguing. We believe that at least with respect to the

first two measures in Table 2.1 this discontinuity can be traced back to the use of the

asymptotic stationary distributions at the stage when the stationarity is not reached

for some of the processes. Once this problem is taken care of [45, 46], the transition

from the non-eternal to eternal inflation becomes continuous. It would be interesting

to see whether a similar modification can restore the continuity of predictions of other

probability measures.

CHAPTER 2. 40

t t

a a

a a

τ

τ

Figure 2.3: Evolution of the scale factor in a collapsing k = 0 FRW universe withrespect to the proper time (top) and the scale factor time (bottom). In each pairof plots the left one has a slightly longer stage of slow role inflation than the rightone. The area after reheating is painted. Notice almost exact symmetry of behaviorof the scale factor during expansion versus contraction. Stars indicate the periodwhere galaxies are produced and life as we know it is possible. This happens at thesame time t but at very different times τ for the universes experiencing longer/shorterstages of inflation.

Chapter 3

Boltzmann brains and the

scale-factor cutoff measure of the

multiverse

3.1 Introduction

The simplest interpretation of the observed accelerating expansion of the universe is

that it is driven by a constant vacuum energy density ρΛ, which is about three times

greater than the present density of nonrelativistic matter. While ordinary matter

becomes more dilute as the universe expands, the vacuum energy density remains the

same, and in another ten billion years or so the universe will be completely dominated

by vacuum energy. The subsequent evolution of the universe is accurately described

as de Sitter space.

It was shown by Gibbons and Hawking [56] that an observer in de Sitter space

would detect thermal radiation with a characteristic temperature TdS = HΛ/2π, where

HΛ =

√8π

3GρΛ (3.1)

is the de Sitter Hubble expansion rate. For the observed value of ρΛ, the de Sitter

temperature is extremely low, TdS = 2.3× 10−30 K. Nevertheless, complex structures

41

CHAPTER 3. 42

will occasionally emerge from the vacuum as quantum fluctuations, at a small but

nonzero rate per unit spacetime volume. An intelligent observer, like a human, could

be one such structure. Or, short of a complete observer, a disembodied brain may

fluctuate into existence, with a pattern of neuron firings creating a perception of being

on Earth and, for example, observing the cosmic microwave background radiation.

Such freak observers are collectively referred to as “Boltzmann brains” [57, 58]. Of

course, the nucleation rate ΓBB of Boltzmann brains is extremely small, its magnitude

depending on how one defines a Boltzmann brain. The important point, however, is

that ΓBB is always nonzero.

De Sitter space is eternal to the future. Thus, if the accelerating expansion of

the universe is truly driven by the energy density of a stable vacuum state, then

Boltzmann brains will eventually outnumber normal observers, no matter how small

the value of ΓBB [59, 62, 60, 63, 64] might be.

To define the problem more precisely, we use the term “normal observers” to refer

to those that evolve as a result of non-equilibrium processes that occur in the wake

of the hot big bang. If our universe is approaching a stable de Sitter spacetime, then

the total number of normal observers that will ever exist in a fixed comoving volume

of the universe is finite. On the other hand, the cumulative number of Boltzmann

brains grows without bound over time, growing roughly as the volume, proportional to

e3HΛt. When extracting the predictions of this theory, such an infinite preponderance

of Boltzmann brains cannot be ignored.

For example, suppose that some normal observer, at some moment in her lifetime,

tries to make a prediction about her next observation. According to the theory there

would be an infinite number of Boltzmann brains, distributed throughout the space-

time, that would happen to share exactly all her memories and thought processes at

that moment. Since all her knowledge is shared with this set of Boltzmann brains,

for all she knows she could equally likely be any member of the set. The probability

that she is a normal observer is then arbitrarily small, and all predictions would be

based on the proposition that she is a Boltzmann brain. The theory would predict,

therefore, that the next observations that she will make, if she survives to make any

at all, will be totally incoherent, with no logical relationship to the world that she

CHAPTER 3. 43

thought she knew. (While it is of course true that some Boltzmann brains might

experience coherent observations, for example by living in a Boltzmann solar system,

it is easy to show that Boltzmann brains with such dressing would be vastly outnum-

bered by Boltzmann brains without any coherent environment.) Thus, the continued

orderliness of the world that we observe is distinctly at odds with the predictions of

a Boltzmann-brain-dominated cosmology.1

This problem was recently addressed by Page [63], who concluded that the least

unattractive way to produce more normal observers than Boltzmann brains is to

require that our vacuum should be rather unstable. More specifically, it should decay

within a few Hubble times of vacuum energy domination; that is, in 20 billion years

or so.

In the context of inflationary cosmology, however, this problem acquires a new

twist. Inflation is generically eternal, with the physical volume of false-vacuum in-

flating regions increasing exponentially with time and “pocket universes” like ours

constantly nucleating out of the false vacuum. In an eternally inflating multiverse,

the numbers of normal observers and Boltzmann brains produced over the course of

eternal inflation are both infinite. They can be meaningfully compared only after one

adopts some prescription to regulate the infinities.

The problem of regulating the infinities in an eternally inflating multiverse is

known as the measure problem [65], and has been under discussion for some time.

It is crucially important in discussing predictions for any kind of observation. Most

of the discussion, including the discussion in this chapter, has been confined to the

classical approximation. While one might hope that someday there will be an answer

to this question based on a fundamental principle [66], most of the work on this

subject has focussed on proposing plausible measures and exploring their properties.

1Here we are taking a completely mechanistic view of the brain, treating it essentially as a highlysophisticated computer. Thus, the normal observer and the Boltzmann brains can be thought of asa set of logically equivalent computers running the same program with the same data, and hencethey behave identically until they are affected by further input, which might be different. Since thecomputer program cannot determine whether it is running inside the brain of one of the normalobservers or one of the Boltzmann brains, any intelligent probabilistic prediction that the programmakes about the next observation would be based on the assumption that it is equally likely to berunning on any member of that set.

CHAPTER 3. 44

Indeed, a number of measures have been proposed [67, 68, 69, 70, 71, 72, 73, 74, 75,

76, 77, 78, 79, 80, 81, 82, 83], and some of them have already been disqualified, as

they make predictions that conflict with observations.

In particular, if one uses the proper-time cutoff measure [67, 68, 69, 70, 71], one

encounters the “youngness paradox,” predicting that humans should have evolved at

a very early cosmic time, when the conditions for life were rather hostile [84]. The

youngness problem, as well as the Boltzmann brain problem, can be avoided in the

stationary measure [74, 83], which is an improved version of the proper-time cutoff

measure. However, the stationary measure, as well as the pocket-based measure, is

afflicted with a runaway problem, suggesting that we should observe extreme values

(either very small or very large) of the primordial density contrast Q [86, 87] and

the gravitational constant G [88], while these parameters appear to sit comfortably

in the middle of their respective anthropic ranges [89, 88]. Some suggestions to

get around this issue have been described in Refs. [90, 87, 91, 92]. In addition,

the pocket-based measure seems to suffer from the Boltzmann brain problem. The

comoving coordinate measure [93, 67] and the causal-patch measures [79, 80] are free

from these problems, but have an unattractive feature of depending sensitively on

the initial state of the multiverse. This does not seem to mix well with the attractor

nature of eternal inflation: the asymptotic late-time evolution of an eternally inflating

universe is independent of the starting point, so it seems appealing for the measure to

maintain this property. Since the scale-factor cutoff measure2 [68, 69, 70, 72, 73, 94]

has been shown to be free of all of the above issues [95], we consider it to be a

promising candidate for the measure of the multiverse.

As we have indicated, the relative abundance of normal observers and Boltzmann

brains depends on the choice of measure over the multiverse. This means the predicted

ratio of Boltzmann brains to normal observers can be used as yet another criterion

2This measure is sometimes referred to as the volume-weighted scale-factor cutoff measure, butwe will define it below in terms of the counting of events in spacetime, so the concept of weightingwill not be relevant. The term “volume-weighted” is relevant when a measure is described as aprescription for defining the probability distribution for the value of a field. In Ref. [73], thismeasure is called the “pseudo-comoving volume-weighted measure.”

CHAPTER 3. 45

to evaluate a prescription to regulate the diverging volume of the multiverse: regu-

lators that predict normal observers are greatly outnumbered by Boltzmann brains

should be ruled out. This criterion has been studied in the context of several mul-

tiverse measures, including a causal patch measure [64], several measures associated

with globally defined time coordinates [73, 96, 74, 85, 83], and the pocket-based mea-

sure [97]. In this work, we apply this criterion to the scale-factor cutoff measure,

extending the investigation that was initiated in Ref. [73]. We show that the scale-

factor cutoff measure gives a finite ratio of Boltzmann brains to normal observers; if

certain assumptions about the landscape are valid, the ratio can be small.3

The remainder of this chapter is organized as follows. In Section 3.2, we pro-

vide a brief description of the scale-factor cutoff and describe salient features of the

multiverse under the lens of this measure. In Section 3.3 we calculate the ratio of

Boltzmann brains to normal observers in terms of multiverse volume fractions and

transition rates. The volume fractions are discussed in Section 3.4, in the context

of toy landscapes, and the section ends with a general description of the conditions

necessary to avoid Boltzmann brain domination. The rate of Boltzmann brain pro-

duction and the rate of vacuum decay play central roles in our calculations, and these

are estimated in Section 3.5. Concluding remarks are provided in Section 3.6.

3.2 The Scale Factor Cutoff

Perhaps the simplest way to regulate the infinities of eternal inflation is to impose a

cutoff on a hypersurface of constant global time [68, 69, 70, 71, 72]. One starts with

a patch of a spacelike hypersurface Σ somewhere in an inflating region of spacetime,

and follows its evolution along the congruence of geodesics orthogonal to Σ. The

scale-factor time is defined as

t = ln a , (3.2)

3In a paper that appeared simultaneously with the original paper corresponding to this chapter,Raphael Bousso, Ben Freivogel, and I-Sheng Yang independently analyzed the Boltzmann brainproblem for the scale-factor cutoff measure [98].

CHAPTER 3. 46

where a is the expansion factor along the geodesics. The scale-factor time is related

to the proper time τ by

dt = H dτ , (3.3)

where H is the Hubble expansion rate of the congruence. The spacetime region swept

out by the congruence will typically expand to unlimited size, generating an infinite

number of pockets. (If the patch does not grow without limit, one chooses another

initial patch Σ and starts again.) The resulting four-volume is infinite, but we cut it

off at some fixed scale-factor time t = tc. To find the relative probabilities of different

events, one counts the numbers of such events in the finite spacetime volume between

Σ and the t = tc hypersurface, and then takes the limit tc →∞.

The term “scale factor” is often used in the context of homogeneous and isotropic

geometries; yet on very large and on very small scales the multiverse may be very

inhomogeneous. A simple way to deal with this is to take the factor H in Eq. (3.3)

to be the local divergence of the four-velocity vector field along the congruence of

geodesics orthogonal to Σ,

H(x) ≡ (1/3)uµ;µ . (3.4)

When more than one geodesic passes through a point, the scale-factor time at that

point may be taken to be the smallest value among the set of geodesics. In collapsing

regions H(x) is negative, in which case the corresponding geodesics are continued

unless or until they hit a singularity.

This “local” definition of scale-factor time has a simple geometric meaning. The

congruence of geodesics can be thought of as representing a “dust” of test particles

scattered uniformly on the initial hypersurface Σ. As one moves along the geodesics,

the density of the dust in the orthogonal plane decreases. The expansion factor a in

Eq. (3.2) can then defined as a ∝ ρ−1/3, where ρ is the density of the dust, and the

cutoff is triggered when ρ drops below some specified level.

Although the local scale-factor time closely follows the FRW scale factor in ex-

panding spacetimes — such as inflating regions and thermalized regions not long after

reheating — it differs dramatically from the FRW scale factor as small-scale inhomo-

geneities develop during matter domination in universes like ours. In particular, the

CHAPTER 3. 47

local scale-factor time nearly grinds to a halt in regions that have decoupled from the

Hubble flow. It is not clear whether we should impose this particular cutoff, which

would essentially include the entire lifetime of any nonlinear structure that forms

before the cutoff, or impose a cutoff on some nonlocal time variable that more closely

tracks the FRW scale factor.4

There are a number of nonlocal modifications of scale factor time that both ap-

proximate our intuitive notion of FRW averaging and also extend into more compli-

cated geometries. One drawback of the nonlocal approach is that no single choice

looks more plausible than the others. For instance, one nonlocal method is to define

the factor H in Eq. (3.3) by spatial averaging of the quantity H(x) in Eq. (3.4). A

complete implementation of this approach, however, involves many seemingly arbi-

trary choices regarding how to define the hypersurfaces over which H(x) should be

averaged, so we here set this possibility aside. A second, simpler method is to use

the local scale-factor time defined above, but to generate a new cutoff hypersurface

by excluding the future lightcones of all points on the original cutoff hypersurface.

In regions with nonlinear inhomogeneities, the underdense regions will be the first

to reach the scale-factor cutoff, after which they quickly trigger the cutoff elsewhere.

The resulting cutoff hypersurface will not be a surface of constant FRW scale factor,

but the fluctuations of the FRW scale factor on this surface should be insignificant.

As a third and final example of a nonlocal modification of scale factor time, we

recall the description of the local scale-factor cutoff in terms the density ρ of a dust

of test particles. Instead of such a dust, consider a set of massless test particles,

emanating uniformly in all directions from each point on the initial hypersurface Σ.

We can then construct the conserved number density current Jµ for the gas of test

particles, and we can define ρ as the rest frame number density, i.e. the value of

J0 in the local Lorentz frame in which J i = 0, or equivalently ρ =√J2. Defining

a ∝ ρ−1/3, as we did for the dust of test particles, we apply the cutoff when the number

density ρ drops below some specified level. Since null geodesics are barely perturbed

by structure formation, the strong perturbations inherent in the local definition of

4The distinction between these two forms of scale-factor time was first pointed out by Bousso,Freivogel, and Yang in Ref. [98].

CHAPTER 3. 48

scale factor time are avoided. Nonetheless, we have not studied the properties of

this definition of scale factor time, and they may lead to complications. Large-scale

anisotropic flows in the gas of test particles can be generated as the particles stream

into expanding bubbles from outside. Since the null geodesics do not interact with

matter except gravitationally, these anisotropies will not be damped in the same way

as they would be for photons. The large-scale flow of the gas will not redshift in

the normal way, either; for example, if the test particles in some region of an FRW

universe have a nonzero mean velocity relative to the comoving frame, the expansion

of the universe will merely reduce the energies of all the test particles by the same

factor, but will not cause the mean velocity to decrease. Thus, the detailed predictions

for this definition of scale-factor cutoff measure remain a matter for future study.

The local scale-factor cutoff and each of the three nonlocal definitions correspond

to different global-time parameterizations and thus to different spacetime measures.

In general they make different predictions for physical observables; however with

regard to the relative number of normal observers and Boltzmann brains, their pre-

dictions are essentially the same. For the remainder of this chapter we refer to the

generic nonlocal definition of scale factor time, for which we take FRW time as a

suitable approximation. Note that the use of local scale factor time would make

it slightly easier to avoid Boltzmann brain domination, since it would increase the

count of normal observers while leaving the count of Boltzmann brains essentially

unchanged.

To facilitate later discussion, let us now describe some general properties of the

multiverse. The volume fraction fi occupied by vacuum i on constant scale-factor

time slices can be found from the rate equation [99],

dfidt

=∑

j

Mijfj , (3.5)

where the transition matrix Mij is given by

Mij = κij − δij∑

r

κri , (3.6)

CHAPTER 3. 49

and κij is the transition rate from vacuum j to vacuum i per Hubble volume per

Hubble time. This rate can also be written

κij = (4π/3)H−4j Γij , (3.7)

where Γij is the bubble nucleation rate per unit spacetime volume and Hj is the

Hubble expansion rate in vacuum j.

The solution of Eq. (3.5) can be written in terms of the eigenvectors and eigen-

values of the transition matrix Mij.

It is easily verified that each terminal vacuum is an eigenvector with eigenvalue

zero. We here define “terminal vacua” as those vacua j for which κij = 0 for all

i. Thus the terminal vacua include both negative-energy vacua, which collapse in a

big crunch, and stable zero-energy vacua. It was shown in Ref. [77] that all of the

other eigenvalues of Mij have negative real parts. Moreover, the eigenvalue with the

smallest (by magnitude) real part is pure real; we call it the “dominant eigenvalue”

and denote it by −q (with q > 0). Assuming that the landscape is irreducible, the

dominant eigenvalue is nondegenerate. In that case the probabilities defined by the

scale-factor cutoff measure are independent of the initial state of the multiverse, since

they are determined by the dominant eigenvector.5

For an irreducible landscape, the late-time asymptotic solution of Eq. (3.5) can

5In this work we assume that the multiverse is irreducible; that is, any metastable inflatingvacuum is accessible from any other such vacuum via a sequence of tunneling transitions. Our results,however, can still be applied when this condition fails. In that case the dominant eigenvalue can bedegenerate, in which case the asymptotic future is dominated by a linear combination of dominanteigenvectors that is determined by the initial state. If transitions that increase the vacuum energydensity are included, then the landscape can be reducible only if it splits into several disconnectedsectors. That situation was discussed in Appendix A of Ref. [95], where two alternative prescriptionswere described. The first prescription (preferred by the authors) leads to initial-state dependenceonly if two or more sectors have the same dominant eigenvalue q, while the second prescriptionalways leads to initial-state dependence.

CHAPTER 3. 50

be written in the form6

fj(t) = f(0)j + sje

−qt + . . . , (3.8)

where the constant term f(0)j is nonzero only in terminal vacua and sj is proportional

to the eigenvector of Mij corresponding to the dominant eigenvalue −q, with the

constant of proportionality determined by the initial distribution of vacua on Σ. It

was shown in Ref. [77] that sj ≤ 0 for terminal vacua, and sj > 0 for nonterminal

vacua, as is needed for Eq. (3.8) to describe a nonnegative volume fraction, with a

nondecreasing fraction assigned to any terminal vacuum.

By inserting the asymptotic expansion (3.8) into the differential equation (3.5)

and extracting the leading asymptotic behavior for a nonterminal vacuum i, one can

show that

(κi − q)si =∑

j

κij sj , (3.9)

where κj is the total transition rate out of vacuum j,

κj ≡∑

i

κij . (3.10)

The positivity of si for nonterminal vacua then implies rigorously that q is less than

the decay rate of the slowest-decaying vacuum in the landscape:

q ≤ κmin ≡ minκj . (3.11)

Since “upward” transitions (those that increase the energy density) are generally

suppressed, we can gain some intuition by first considering the case in which all up-

ward transition rates are set to zero. (Such a landscape is reducible, so the dominant

eigenvector can be degenerate.) In this case Mij is triangular, and the eigenvalues

are precisely the decay rates κi of the individual states. The dominant eigenvalue q

6Mij is not necessarily diagonalizable, but Eq. (3.8) applies in any case. It is always possible toform a complete basis from eigenvectors and generalized eigenvectors, where generalized eigenvectorssatisfy (M − λI)ks = 0, for k > 1. The generalized eigenvectors appear in the solution with a timedependence given by eλt times a polynomial in t. These terms are associated with the nonleadingeigenvalues omitted from Eq. (3.8), and the polynomials in t will not change the fact that they arenonleading.

CHAPTER 3. 51

is then exactly equal to κmin.

If upward transitions are included but assumed to have a very low rate, then the

dominant eigenvalue q is approximately equal to the decay rate of the slowest-decaying

vacuum [100],

q ≈ κmin . (3.12)

The slowest-decaying vacuum (assuming it is unique) is the one that dominates the

asymptotic late-time volume of the multiverse, so we call it the dominant vacuum

and denote it by D. Hence,

q ≈ κD . (3.13)

The vacuum decay rate is typically exponentially suppressed, so for the slowest-

decaying vacuum we expect it to be extremely small,

q ≪ 1 . (3.14)

Note that the corrections to Eq. (3.13) are comparable to the upward transition rate

from D to higher-energy vacua, but for large energy differences this transition rate is

suppressed by the factor exp(−8π2/H2D) [101]. Here and throughout the remainder of

this chapter we use reduced Planck units, where 8πG = c = kB = 1. We shall argue

in Section 3.5 that the dominant vacuum is likely to have a very low energy density,

so the correction to Eq. (3.13) is very small even compared to q.

A possible variant of this picture, with similar consequences, could arise if one

assumes that the landscape includes states with nearby energy densities for which

the upward transition rate is not strongly suppressed. In that case there could be

a group of vacuum states that undergo rapid transitions into each other, but very

slow transitions to states outside the group. The role of the dominant vacuum could

then be played by this group of states, and q would be approximately equal to some

appropriately averaged rate for the decay of these states to states outside the group.

Under these circumstances q could be much less than κmin. An example of such a

situation is described in Subsection 3.4.5.

In the asymptotic limit of late scale-factor time t, the physical volumes in the

CHAPTER 3. 52

various nonterminal vacua are given by

Vj(t) = V0 sj e(3−q) t , (3.15)

where V0 is the volume of the initial hypersurface Σ and e3t is the volume expansion

factor. The volume growth in Eq. (3.15) is (very slightly) slower than e3t due to the

constant loss of volume from transitions to terminal vacua. Note that even though

upward transitions from the dominant vacuum are strongly suppressed, they play a

crucial role in populating the landscape [100]. Most of the volume in the asymptotic

solution of Eq. (3.15) originates in the dominant vacuum D, and “trickles” to the

other vacua through a series of transitions starting with at least one upward jump.

3.3 The Abundance of Normal Observers and Boltz-

mann Brains

Let us now calculate the relative abundances of Boltzmann brains and normal ob-

servers, in terms of the vacuum transition rates and the asymptotic volume fractions.

Estimates for the numerical values of the Boltzmann brain nucleation rates and

vacuum decay rates will be discussed in Section 3.5, but it is important at this stage

to be aware of the kind of numbers that will be considered. We will be able to give

only rough estimates of these rates, but the numbers that will be mentioned in Section

3.5 will range from exp (−10120) to exp (−1016). Thus, when we calculate the ratio

NBB/NNO of Boltzmann brains to normal observers, the natural logarithm of this

ratio will always include one term with a magnitude of at least 1016. Consequently,

the presence or absence of any term in ln(NBB/NNO) that is small compared to 1016

is of no relevance. We therefore refer to any factor f for which

| ln f | < 1014 (3.16)

as “roughly of order one.” In the calculation of NBB/NNO such factors — although

they may be minuscule or colossal by ordinary standards — can be ignored. It

CHAPTER 3. 53

will not be necessary to keep track of factors of 2, π, or even 10108. Dimensionless

coefficients, factors of H, and factors coming from detailed aspects of the geometry

are unimportant, and in the end all of these will be ignored. We nonetheless include

some of these factors in the intermediate steps below simply to provide a clearer

description of the calculation.

We begin by estimating the number of normal observers that will be counted in

the sample spacetime region specified by the scale-factor cutoff measure. Normal

observers arise during the big bang evolution in the aftermath of slow-roll inflation

and reheating. The details of this evolution depend not only on the vacuum of the

pocket in question, but also on the parent vacuum from which it nucleated [102]. That

is, if we view each vacuum as a local minimum in a multidimensional field space, then

the dynamics of inflation in general depend on the direction from which the field

tunneled into the local minimum. We therefore label pockets with two indices, ik,

indicating the pocket and parent vacua respectively.

To begin, we restrict our attention to a single “anthropic” pocket — i.e., one that

produces normal observers — which nucleates at scale-factor time tnuc. The internal

geometry of the pocket is that of an open FRW universe. We assume that, after a brief

curvature-dominated period ∆τ ∼ H−1k , slow-roll inflation inside the pocket gives Ne

e-folds of expansion before thermalization. Furthermore, we assume that all normal

observers arise at a fixed number NO of e-folds of expansion after thermalization.

(Note that Ne and NO are both measured along FRW comoving geodesics inside the

pocket, which do not initially coincide with, but rapidly asymptote to, the “global”

geodesic congruence that originated outside the pocket.) We denote the fixed-internal-

time hypersurface on which normal observers arise by ΣNO, and call the average

density of observers on this hypersurface nNOik .

The hypersurface ΣNO would have infinite volume, due to the constant expansion

of the pocket, but this divergence is regulated by the scale-factor cutoff tc, because

the global scale-factor time t is not constant over the ΣNO hypersurface. For the

pocket described above, the regulated physical volume of ΣNO can be written as

V(ik)O (tnuc) = H−3

k e3(Ne+NO) w(tc − tnuc −Ne −NO) , (3.17)

CHAPTER 3. 54

where the exponential gives the volume expansion factor coming from slow-roll infla-

tion and big bang evolution to the hypersurface ΣNO, and H−3k w(tc− tnuc−Ne−NO)

describes the comoving volume of the part of the ΣNO hypersurface that is underneath

the cutoff. The function w(t) was calculated, for example, in Refs. [103] and [85], and

is applied to scale-factor cutoff measure in Ref. [104]. Its detailed form will not be

needed to determine the answer up to a factor that is roughly of order one, but to

avoid mystery we mention that w(t) can be written as

w(t) =π

2

∫ ξ(t)

0

sinh2(ξ) dξ =π

8

[sinh

(2ξ(t)

)− 2ξ(t)

], (3.18)

where ξ(tc − tnuc − Ne − NO) is the maximum value of the Robertson-Walker radial

coordinate ξ that lies under the cutoff. If the pocket universe begins with a moderate

period of inflation (exp(Ne) 1) with the same vacuum energy as outside, then

ξ(t) ≈ 2 cosh−1(et/2). (3.19)

Eq. (3.17) gives the physical volume on the ΣNO hypersurface for a single pocket

of type ik, which nucleates at time tnuc. The number of ik-pockets that nucleate

between time tnuc and tnuc + dtnuc is

dn(ik)nuc (tnuc) = (3/4π)H3

kκik Vk(tnuc) dtnuc

= (3/4π)H3kκiksk V0 e

(3−q) tnuc dtnuc , (3.20)

where we use Eq. (3.15) to give Vk(tnuc). The total number of normal observers in

the sample region is then

NNOik = nNO

ik

∫ tc−Ne−NOV

(ik)O (tnuc) dn

(ik)nuc (tnuc)

≈ nNOik κiksk V0 e

(3−q) tc∫ ∞

0

w(z) e−(3−q) z dz . (3.21)

In the first expression we have ignored the (very small) probability that pockets of type

ik may transition to other vacua during slow-roll inflation or during the subsequent

CHAPTER 3. 55

period NO of big bang evolution. In the second line, we have changed the integration

variable to z = tc − tnuc −Ne −NO (reversing the direction of integration) and have

dropped the O(1) prefactors, and also the factor eq(Ne+NO), since q is expected to be

extraordinarily small. We have kept e−qtc , since we are interested in the limit tc →∞.

We have also kept the factor eqz long enough to verify that the integral converges with

or without the factor, so we can carry out the integral using the approximation q ≈ 0,

resulting in an O(1) prefactor that we will drop.

Finally,

NNOik ≈ nNO

ik κik sk V0 e(3−q) tc . (3.22)

Note that the expansion factor e3(Ne+NO) in Eq. (3.17) was canceled when we inte-

grated over nucleation times, illustrating the mild youngness bias of the scale-factor

cutoff measure. The expansion of the universe is canceled, so objects that form at a

certain density per physical volume in the early universe will have the same weight

as objects that form at the same density per physical volume at a later time, despite

the naive expectation that there is more volume at later times.

To compare, we now need to calculate the number of Boltzmann brains that will

be counted in the sample spacetime region. Boltzmann brains can be produced in any

anthropic vacuum, and presumably in many non-anthropic vacua as well. Suppose

Boltzmann brains are produced in vacuum j at a rate ΓBBj per unit spacetime volume.

The number of Boltzmann brains NBBj is then proportional to the total four-volume

in that vacuum. Imposing the cutoff at scale-factor time tc, this four-volume is

V(4)j =

∫ tc

Vj(t) dτ = H−1j

∫ tc

Vj(t) dt

=1

3− qH−1j sj V0 e

(3−q) tc , (3.23)

where we have used Eq. (3.15) for the asymptotic volume fraction. By setting dτ =

H−1j dt, we have ignored the time-dependence of H(τ) in the earlier stages of cosmic

evolution, assuming that only the late-time de Sitter evolution is relevant. In a similar

spirit, we will assume that the Boltzmann brain nucleation rate ΓBBj can be treated

as time-independent, so the total number of Boltzmann brains nucleated in vacua of

CHAPTER 3. 56

type j, within the sample volume, is given by

NBBj ≈ ΓBB

j H−1j sj V0 e

(3−q) tc , (3.24)

where we have dropped the O(1) numerical factor.

For completeness, we may want to consider the effects of early universe evolu-

tion on Boltzmann brain production, effects which were ignored in Eq. (3.24). We

will separate the effects into two categories: the effects of slow-roll inflation at the

beginning of a pocket universe, and the effects of reheating.

To account for the effects of slow-roll inflation, we argue that, within the approx-

imations used here, there is no need for an extra calculation. Consider, for example,

a pocket universe A which begins with a period of slow-roll inflation during which

H(τ) ≈ Hslow roll = const. Consider also a pocket universe B, which throughout its

evolution has H = Hslow roll, and which by hypothesis has the same formation rate,

Boltzmann brain nucleation rate, and decay rates as pocket A. Then clearly the

number of Boltzmann brains formed in the slow roll phase of pocket A will be smaller

than the number formed throughout the lifetime of pocket B. Since we will require

that generic bubbles of type B do not overproduce Boltzmann brains, there will be

no need to worry about the slow-roll phase of bubbles of type A.

To estimate how many Boltzmann brains might form as a consequence of re-

heating, we can make use of the calculation for the production of normal observers

described above. We can assume that the Boltzmann brain nucleation rate has a

spike in the vicinity of some particular hypersurface in the early universe, peaking at

some value ΓBBreheat,ik which persists roughly for some time interval ∆τBB

reheat,ik, produc-

ing a density of Boltzmann brains equal to ΓBBreheat,ik ∆τBB

reheat,ik. This spatial density is

converted into a total number for the sample volume in exactly the same way that

we did for normal observers, leading to

NBB,reheatik ≈ ΓBB

reheat,ik ∆τBBreheat,ik κik sk V0 e

(3−q) tc . (3.25)

CHAPTER 3. 57

Thus, the dominance of normal observers is assured if

∑

i,k

ΓBBreheat,ik ∆τBB

reheat,ikκik sk ∑

i,k

nNOik κik sk . (3.26)

If Eq. (3.26) did not hold, it seems likely that we would suffer from Boltzmann brain

problems regardless of our measure. We leave numerical estimates for Section 3.5,

but we will see that Boltzmann brain production during reheating is not a danger.

Ignoring the Boltzmann brains that form during reheating, the ratio of Boltzmann

brains to normal observers can be found by combining Eqs. (3.22) and (3.24), giving

NBB

NNO≈∑

j H3j κ

BBj sj∑

i, k nNOik κik sk

, (3.27)

where the summation in the numerator covers only the vacua in which Boltzmann

brains can arise, the summation over i in the denominator covers only anthropic

vacua, and the summation over k includes all of their possible parent vacua. κBBj is

the dimensionless Boltzmann brain nucleation rate in vacuum j, related to ΓBBj by

Eq. (3.7). The expression can be further simplified by dropping the factors of Hj and

nNOi , which are roughly of order one, as defined by Eq. (3.16). We can also replace

the sum over j in the numerator by the maximum over j, since the sum is at least as

large as the maximum term and no larger than the maximum term times the number

of vacua. Since the number of vacua (perhaps 10500) is roughly of order one, the

sum over j is equal to the maximum up to a factor that is roughly of order one. We

similarly replace the sum over i in the denominator by its maximum, but we choose

to leave the sum over k. Thus we can write

NBB

NNO∼ maxjκBB

j sjmaxi

∑k κik sk

, (3.28)

where the sets of j and i are restricted as for Eq. (3.27).

In dropping nNOi , we are assuming that nNO

i H3i is roughly of order one, as defined

at the beginning of this section. It is hard to know what a realistic value for nNOi H3

i

might be, as the evolution of normal observers may require some highly improbable

CHAPTER 3. 58

events. For example, it was argued in Ref. [105] that the probability for life to evolve

in a region of the size of our observable universe per Hubble time may be as low as

∼ 10−1000. But even the most pessimistic estimates cannot compete with the small

numbers appearing in estimates of the Boltzmann brain nucleation rate, and hence

by our definition they are roughly of order one. Nonetheless, it is possible to imagine

vacua for which nNOi might be negligibly small, but still nonzero. We shall ignore the

normal observers in these vacua; for the remainder of this chapter we will use the

phrase “anthropic vacuum” to refer only to those vacua for which nNOi H3

i is roughly

of order one.

For any landscape that satisfies Eq. (3.8), which includes any irreducible land-

scape, Eq. (3.28) can be simplified by using Eq. (3.9):

NBB

NNO∼ maxjκBB

j sjmaxi(κi − q) si

, (3.29)

where the numerator is maximized over all vacua j that support Boltzmann brains,

and the denominator is maximized over all anthropic vacua i.

In order to learn more about the ratio of Boltzmann brains to normal observers,

we need to learn more about the volume fractions sj, a topic that will be pursued in

the next section.

3.4 Mini-landscapes and the General Conditions

to Avoid Boltzmann Brain Domination

In this section we study a number of simple models of the landscape, in order to build

intuition for the volume fractions that appear in Eqs. (3.28) and (3.29). The reader

uninterested in the details may skip the pedagogical examples given in Subsections

3.4.1–3.4.5, and continue with Subsection 3.4.6, where we state the general conditions

that must be enforced in order to avoid Boltzmann brain domination.

CHAPTER 3. 59

3.4.1 The FIB Landscape

Let us first consider a very simple model of the landscape, described by the schematic

F → I → B , (3.30)

where F is a high-energy false vacuum, I is a positive-energy anthropic vacuum, and

B is a terminal vacuum. This model, which we call the FIB landscape, was analyzed

in Ref. [77] and was discussed in relation to the abundance of Boltzmann brains in

Ref. [73]. As in Ref. [73], we assume that both Boltzmann brains and normal observers

reside only in vacuum I.

Note that the FIB landscape ignores upward transitions from I to F . The model

is constructed in this way as an initial first step, and also in order to more clearly

relate our analysis to that of Ref. [73]. Although the rate of upward transitions

is exponentially suppressed relative the other rates, its inclusion is important for

the irreducibility of the landscape, and hence the nondegeneracy of the dominant

eigenvalue and the independence of the late-time asymptotic behavior from the initial

conditions of the multiverse. The results of this subsection will therefore not always

conform to the expectations outlined in Section 3.2, but this shortcoming is corrected

in the next subsection and all subsequent work in this chapter.

We are interested in the eigenvectors and eigenvalues of the rate equation, Eq. (3.5).

In the FIB landscape the rate equation gives

fF = −κIFfFfI = −κBIfI + κIFfF .

(3.31)

We ignore the volume fraction in the terminal vacuum as it is not relevant to our

analysis. Starting with the ansatz,

f(t) = s e−qt , (3.32)

CHAPTER 3. 60

we find two eigenvalues of Eqs. (3.31). These are, with their corresponding eigenvec-

tors,

q1 = κIF , s1 = (1, C) ,

q2 = κBI , s2 = (0, 1) ,(3.33)

where the eigenvectors are written in the basis s ≡ (sF , sI) and

C =κIF

κBI − κIF. (3.34)

Suppose that we start in the false vacuum F at t = 0, i.e. f(t = 0) = (1, 0). Then

the solution of the FIB rate equation, Eq. (3.31), is

fF (t) = e−κIF t ,

fI(t) = C (e−κIF t − e−κBI t) .(3.35)

The asymptotic evolution depends on whether κIF < κBI (case I) or not (case II). In

case I,

f(t→∞) = s1e−κIF t (κIF < κBI) , (3.36)

where s1 is given in Eq. (3.33), while in case II

f(t→∞) =(e−κIF t , |C|e−κBI t

)(κBI < κIF ) . (3.37)

In the latter case, the inequality of the rates of decay for the two volume fractions

arises from the reducibility of the FIB landscape, stemming from our ignoring upward

transitions from I to F .

For case I (κIF < κBI), we find the ratio of Boltzmann brains to normal observers

by evaluating Eq. (3.28) for the asymptotic behavior described by Eq. (3.36):

NBB

NNO∼ κBBsIκIF sF

∼ κBB

κIF

κIFκBI − κIF

∼ κBB

κBI, (3.38)

where we drop κIF compared to κBI in the denominator, as we are only interested in

CHAPTER 3. 61

the overall scale of the solution. We find that the ratio of Boltzmann brains to normal

observers is finite, depending on the relative rate of Boltzmann brain production to

the rate of decay of vacuum I. Meanwhile, in case II (where κBI < κIF ) we find

NBB

NNO∼ κBB

κIFe(κIF−κBI)t →∞ . (3.39)

In this situation, the number of Boltzmann brains overwhelms the number of normal

observers; in fact the ratio diverges with time.

The unfavorable result of case II stems from the fact that, in this case, the volume

of vacuum I grows faster than that of vacuum F . Most of this I-volume is in large

pockets that formed very early; and this volume dominates because the F -vacuum

decays faster than I, and is not replenished due to the absence of upward transitions.

This leads to Boltzmann brain domination, in agreement with the conclusion reached

in Ref. [73]. Thus, the FIB landscape analysis suggests that Boltzmann brain domi-

nation can be avoided only if the decay rate of the anthropic vacuum is larger than

both the decay rate of its parent false vacuum F and the rate of Boltzmann brain

production. Moreover, the FIB analysis suggests that Boltzmann brain domination

in the multiverse can be avoided only if the first of these conditions is satisfied for

all vacua in which Boltzmann brains exist. This is a very stringent requirement,

since low-energy vacua like I typically have lower decay rates than high-energy vacua

(see Section 3.5). We shall see, however, that the above conditions are substantially

relaxed in more realistic landscape models.

3.4.2 The FIB Landscape with Recycling

The FIB landscape of the preceding section is reducible, since vacuum F cannot

be reached from vacuum I. We can make it irreducible by simply allowing upward

transitions,

F ↔ I → B . (3.40)

This “recycling FIB” landscape is more realistic than the original FIB landscape,

because upward transitions out of positive-energy vacua are allowed in semi-classical

CHAPTER 3. 62

quantum gravity [101]. The rate equation of the recycling FIB landscape gives the

eigenvalue system,

−qsF = −κIF sF + κFIsI ,

−qsI = −κIsI + κIF sF ,(3.41)

where κI ≡ κBI + κFI is the total decay rate of vacuum I, as defined in Eq. (3.10).

Thus, the eigenvalues q1 and q2 correspond to the roots of

(κIF − q)(κI − q) = κIFκFI . (3.42)

Further analysis is simplified if we note that upward transitions from low-energy

vacua like ours are very strongly suppressed, even when compared to the other expo-

nentially suppressed transition rates, i.e. κFI κIF , κBI . We are interested mostly

in how this small correction modifies the dominant eigenvector in the case where

κBI < κIF (case II), which led to an infinite ratio of Boltzmann brains to normal

observers. To the lowest order in κFI , we find

q ≈ κI −κIFκFIκIF − κI

, (3.43)

and

sI ≈κIF − κIκFI

sF sF . (3.44)

The above equation is a consequence of the second of Eqs. (3.41), but it also follows

directly from Eq. (3.9), which holds in any irreducible landscape. In this case fI(t)

and fF (t) have the same asymptotic time dependence, ∝ e−qt, so the ratio fI(t)/fF (t)

approaches a constant limit, sI/sF ≡ R. However, due to the smallness of κFI , this

ratio is extremely large. Note that the ratio of Boltzmann brains to normal observers

is proportional to R. Although it is also proportional to the minuscule Boltzmann

brain nucleation rate (estimated in Section 3.5), the typically huge value of R will

still lead to Boltzmann brain domination (again, see Section 3.5 for relevant details).

But the story is not over, since the recycling FIB landscape is still far from realistic.

CHAPTER 3. 63

3.4.3 A More Realistic Landscape

In the recycling model of the preceding section, the anthropic vacuum I was also the

dominant vacuum, while in a realistic landscape this is not likely to be the case. To

see how it changes the situation to have a non-anthropic vacuum as the dominant

one, we consider the model

A← D ↔ F → I → B , (3.45)

which we call the “ADFIB landscape.” Here, D is the dominant vacuum and A

and B are both terminal vacua. The vacuum I is still an anthropic vacuum, and

the vacuum F has large, positive vacuum energy. As explained in Section 3.5, the

dominant vacuum is likely to have very small vacuum energy; hence we consider that

at least one upward transition (here represented as the transition to F ) is required

to reach an anthropic vacuum.

Note that the ADFIB landscape ignores the upward transition rate from vacuum

I to F ; however this is exponentially suppressed relative the other transition rates

pertinent to I and, unlike the situation in Subsection 3.4.1, ignoring the upward

transition does not significantly affect our results. The important property is that

all vacuum fractions have the same late-time asymptotic behavior, and this property

is assured whenever there is a unique dominant vacuum, and all inflating vacua are

accessible from the dominant vacuum via a sequence of tunneling transitions. The

uniformity of asymptotic behaviors is sufficient to imply Eq. (3.9), which implies

immediately thatsIsF

=κIF

κBI − q≈ κIFκBI − κD

≈ κIFκBI

, (3.46)

where we used q ≈ κD ≡ κAD + κFD, and assumed that κD κBI .

This holds even if the decay rate of the anthropic vacuum I is smaller than that

of the false vacuum F .

Even though the false vacuum F may decay rather quickly, it is constantly being

replenished by upward transitions from the slowly-decaying vacuum D, which over-

whelmingly dominates the physical volume of the multiverse. Note that, in light of

CHAPTER 3. 64

these results, our constraints on the landscape to avoid Boltzmann brain domination

are considerably relaxed. Specifically, it is no longer required that the anthropic vacua

decay at a faster rate than their parent vacua. Using Eq. (3.46) with Eq. (3.28), the

ratio of Boltzmann brains to normal observers in vacuum I is found to be

NBBI

NNOI

∼ κBBI sIκIF sF

∼ κBBI

κBI. (3.47)

If Boltzmann brains can also exist in the dominant vacuum D, then they are a

much more severe problem. By applying Eq. (3.9) to the F vacuum, we find

sFsD

=κFDκF − q

≈ κFDκF − κD

≈ κFDκF

, (3.48)

where κF = κIF + κDF , and where we have assumed that κD κF . The ratio of

Boltzmann brains in vacuum D to normal observers in vacuum I is then

NBBD

NNOI

∼ κBBD sDκIF sF

∼ κBBD

κFD

κFκIF

. (3.49)

Since we expect that the dominant vacuum has very small vacuum energy, and hence

a heavily suppressed upward transition rate κFD, the requirement that NBBD /NNO

I be

small could be a very stringent one. Note that compared to sD, both sF and sI are

suppressed by the small factor κFD; however the ratio sI/sF is independent of this

factor.

Since sD is so large, one should ask whether Boltzmann brain domination can be

more easily avoided by allowing vacuum D to be anthropic. The answer is no, because

the production of normal observers in vacuum D is proportional (see Eq. (3.22)) to the

rate at which bubbles of D nucleate, which is not large. D dominates the spacetime

volume due to slow decay, not rapid nucleation. If we assume that D is anthropic

and restrict Eq. (3.28) to vacuum D, we find using Eq. (3.48) that

NBBD

NNOD

∼ κBBD sDκDF sF

∼ κBBD

κFD

κFκDF

, (3.50)

so again the ratio is enhanced by the extremely small upward tunneling rate κFD in

CHAPTER 3. 65

the denominator.

Thus, in order to avoid Boltzmann brain domination, it seems we have to impose

two requirements: (1) the Boltzmann brain nucleation rate in the anthropic vacuum

I must be less than the decay rate of that vacuum, and (2) the dominant vacuum

D must either not support Boltzmann brains at all, or must produce them with a

dimensionless rate κBBD that is small even compared to the upward tunneling rate

κFD. If the vacuum D is anthropic then it must support Boltzmann brains, so the

domination by Boltzmann brains could be avoided only by the stringent requirement

κBBD κFD.

3.4.4 A Further Generalization

The conclusions of the last subsection are robust to more general considerations. To

illustrate, let us generalize the ADFIB landscape to one with many low-vacuum-

energy pockets, described by the schematic

A← D ↔ Fj → Ii → B , (3.51)

where each high energy false vacuum Fj decays into a set of vacua Ii, all of which

decay (for simplicity) to the same terminal vacuum B. The vacua Ii are taken to be a

large set including both anthropic vacua and vacua that host only Boltzmann brains.

Eq. (3.9) continues to apply, so Eqs. (3.46) and (3.48) are easily generalized to this

case, giving

sIi ≈1

κIi

∑

j

κIiFj sFj (3.52)

and

sFj ≈1

κFjκFjD sD , (3.53)

where we have assumed that q κIi , κFj , as we expect for vacua other than the

dominant one. Using these results with Eq. (3.28), the ratio of Boltzmann brains in

CHAPTER 3. 66

vacua Ii to normal observers in vacua Ii is given by

NBBIi

NNOIi

∼ maxiκBBIisIi

maxi

∑jκIiFj sFj

∼maxi

κBBIi

1

κIi

∑jκIiFj

1

κFjκFjD sD

maxi

∑jκIiFj

1

κFjκFjD sD

∼maxi

κBBIi

κIi

∑j

κIiFjκFj

κFjD

maxi

∑j

κIiFjκFj

κFjD

, (3.54)

where the denominators are maximized over the restricted set of anthropic vacua i

(and the numerators are maximized without restriction). The ratio of Boltzmann

brains in the dominant vacuum (vacuum D) to normal observers in vacua Ii is given

by

NBBD

NNOIi

∼ κBBD sD

maxi

∑jκIiFj sFj

∼ κBBD

maxi

∑j

κIiFjκFj

κFjD

, (3.55)

and, if vacuum D is anthropic, then the ratio of Boltzmann brains in vacuum D to

normal observers in vacuum D is given by

NBBD

NNOD

∼ κBBD∑

j

κDFjκFj

κFjD. (3.56)

In this case our answers are complicated by the presence of many different vacua.

We can in principle determine whether Boltzmann brains dominate by evaluating

Eqs. (3.54)–(3.56) for the correct values of the parameters, but this gets rather com-

plicated and model-dependent. The evaluation of these expressions can be simplified

CHAPTER 3. 67

significantly, however, if we make some very plausible assumptions.

For tunneling out of the high-energy vacua Fj, one can expect the transition rates

into different channels to be roughly comparable, so that κIiFj ∼ κDFj ∼ κFj . That

is, we assume that the branching ratios κIiFj/κFj and κDFj/κFj are roughly of order

one in the sense of Eq. (3.16). These factors (or their inverses) will therefore be

unimportant in the evaluation of NBB/NNO, and may be dropped. Furthermore, the

upward transition rates from the dominant vacuum D into Fj are all comparable to

one another, as can be seen by writing [101]

κFjD ∼ eAFjDe−SD , (3.57)

where AFjD is the action of the instanton responsible for the transition and SD is the

action of the Euclideanized de Sitter 4-sphere,

SD =8π2

H2D

. (3.58)

But generically |AFjD| ∼ 1/ρFj . If we assume that

∣∣∣∣1

ρFj− 1

ρFk

∣∣∣∣ < 1014 (3.59)

for every pair of vacua Fj and Fk, then κFjD = κFkD up to a factor that can be ignored

because it is roughly of order one. Thus, up to subleading factors, the transition rates

κFjD cancel out7 in the ratio NBBIi/NNO.

Returning to Eq. (3.54) and keeping only the leading factors, we have

NBBIi

NNO∼ max

i

κBBIi

κIi

, (3.60)

7Depending on the range of vacua Fj that are considered, the bound of Eq. (3.59) may ormay not be valid. If it is not, then the simplification of Eq. (3.60) below is not justified, andthe original Eq. (3.54) has to be used. Of course one should remember that there was significantarbitrariness in the choice of 1014 in the definition of “roughly of order one.” 1014 was chosen toaccommodate the largest estimate that we discuss in Sec. 3.5 for the Boltzmann brain nucleationrate, ΓBB ∼ exp(−1016). In considering the other estimates of ΓBB, one could replace 1014 by amuch larger number, thereby increasing the applicability of Eq. (3.59).

CHAPTER 3. 68

where the index i runs over all (non-dominant) vacua in which Boltzmann brains can

nucleate. For the dominant vacuum, our simplifying assumptions8 convert Eqs. (3.55)

and (3.56) intoNBBD

NNO∼ κBB

D

κup

∼ κBBD eSD , (3.61)

where κup ≡∑

j κFjD is the upward transition rate out of the dominant vacuum.

Thus, the conditions needed to avoid Boltzmann brain domination are essentially

the same as what we found in Subsection 3.4.3. In this case, however, we must

require that in any vacuum that can support Boltzmann brains, the Boltzmann brain

nucleation rate must be less than the decay rate of that vacuum.

3.4.5 A Dominant Vacuum System

In the next to last paragraph of Section 3.2, we described a scenario where the dom-

inant vacuum was not the vacuum with the smallest decay rate. Let us now study a

simple landscape to illustrate this situation. Consider the toy landscape

Fj → Ii → B

A← D1 ←→ D2 → A

S → A ,

(3.62)

where as in Subsection 3.4.4 the vacua Ii are taken to include both anthropic vacua

and vacua that support only Boltzmann brains. Vacua A and B are terminal vacua

and the Fj have large, positive vacuum energies. Assume that vacuum S has the

smallest total decay rate.

We have in mind the situation in which D1 and D2 are nearly degenerate, and

8The dropping of the factor eAFjD is a more reliable approximation in this case than it wasin Eq. (3.60) above. In this case the factor e−SD does not cancel between the numerator anddenominator, so the factor eAFjD can be dropped if it is unimportant compared to e−SD . We ofcourse do not know the value of S for the dominant vacuum, but for our vacuum it is of order 10122,and it is plausible that the value for the dominant vacuum is similar or even larger. Thus as long as1/ρFj is small compared to 10122, it seems safe to drop the factor eAFjD .

CHAPTER 3. 69

transitions from D1 to D2 (and vice versa) are rapid, even though the transition in

one direction is upward. With this in mind, we divide the decay rates of D1 and D2

into two parts,

κ1 = κ21 + κout1 (3.63)

κ2 = κ12 + κout2 , (3.64)

with κ12, κ21 κout1,2 . We assume as in previous sections that the rates for large

upward transitions (S to D1 or D2, and D1 or D2 to Fj) are extremely small, so that

we can ignore them in the calculation of q. The rate equation, Eq. (3.9), then admits

a solution with q ' κD, but it also admits solutions with

q ' 1

2

[κ1 + κ2 ±

√(κ1 − κ2)2 + 4κ12κ21

]. (3.65)

Expanding the smaller root to linear order in κout1,2 gives

q ' α1κout1 + α2κ

out2 , (3.66)

where

α1 ≡κ12

κ12 + κ21

, α2 ≡κ21

κ12 + κ21

. (3.67)

In principle this value for q can be smaller than κD, which is the case that we wish

to explore.

In this case the vacua D1 and D2 dominate the volume fraction of the multiverse,

even if their total decay rates κ1 and κ2 are not the smallest in the landscape. We

can therefore call the states D1 and D2 together a dominant vacuum system, which

we denote collectively as D. The rate equation (Eq. (3.9)) shows that

sD1 ≈ α1 sD , sD2 ≈ α2 sD , (3.68)

where sD ≡ sD1 + sD2 , and the equations hold in the approximation that κout1,2 and the

upward transition rates from D1 and D2 can be neglected. To see that these vacua

CHAPTER 3. 70

dominate the volume fraction, we calculate the modified form of Eq. (3.53):

sFjsD≈ α1 κFjD1 + α2 κFjD2

κFj. (3.69)

Thus the volume fractions of the Fj, and hence also the Ij and B vacua, are suppressed

by the very small rate for large upward jumps from low energy vacua, namely κFjD1

and κFjD2 . The volume fraction for S depends on κAD1 and κAD2 , but it is maximized

when these rates are negligible, in which case it is given by

sSsD≈ q

κS − q. (3.70)

This quantity can in principle be large, if q is just a little smaller than κS, but that

would seem to be a very special case. Generically, we would expect that since q must

be smaller than κS (see Eq. (3.11)), it would most likely be many orders of magnitude

smaller, and hence the ratio in Eq. (3.70) would be much less than one. There is no

reason, however, to expect it to be as small as the ratios that are suppressed by large

upward jumps. For simplicity, however, we will assume in what follows that sS can

be neglected.

To calculate the ratio of Boltzmann brains to normal observers in this toy land-

scape, note that Eqs. (3.54) and (3.55) are modified only by the substitution

κFjD → κFjD ≡ α1 κFjD1 + α2 κFjD2 . (3.71)

Thus, the dominant vacuum transition rate is simply replaced by a weighted average

of the dominant vacuum transition rates. If we assume that neither of the vacua

D1 nor D2 are anthropic, and make the same assumptions about magnitudes used in

Subsection 3.4.4, then Eqs. (3.60) and (3.61) continue to hold as well, where we have

redefined κup by κup ≡∑

j κFjD.

If, however, we allow D1 or D2 to be anthropic, then new questions arise. Tran-

sitions between D1 and D2 are by assumption rapid, so they copiously produce new

pockets and potentially new normal observers. We must recall, however (as discussed

in Section 3.3), that the properties of a pocket universe depend on both the current

CHAPTER 3. 71

vacuum and the parent vacuum. In this case, the unusual feature is that the vacua

within the D system are nearly degenerate, and hence very little energy is released

by tunnelings within D. For pocket universes created in this way, the maximum

particle energy density during reheating will be only a small fraction of the vacuum

energy density. Such a big bang is very different from the one that took place in our

pocket, and presumably much less likely to produce life. We will call a vacuum in the

D system “strongly anthropic” if normal observers are produced by tunnelings from

within D, and “mildly anthropic” if normal observers can be produced, but only by

tunnelings from higher energy vacua outside D.

If either of the vacua in D were strongly anthropic, then the normal observers

in D would dominate the normal observers in the multiverse. Normal observers in

the vacua Ii would be less numerous by a factor proportional to the extremely small

rate κFjD for large upward transitions . This situation would itself be a problem,

however, similar to the Boltzmann brain problem. It would mean that observers like

ourselves, who arose from a hot big bang with energy densities much higher than

our vacuum energy density, would be extremely rare in the multiverse. We conclude

that if there are any models which give a dominant vacuum system that contains

a strongly anthropic vacuum, such models would be considered unacceptable in the

context of the scale-factor cutoff measure.

On the other hand, if the D system included one or more mildly anthropic vacua,

then the situation is very similar to that discussed in Subsections 3.4.3 and 3.4.4.

In this case the normal observers in the D system would be comparable in number

to the normal observers in the vacua Ii, so they would have no significant effect on

the ratio of Boltzmann brains to normal observers in the multiverse. If any of the

D vacua were mildly anthropic, however, then the stringent requirement κBBD κup

would have to be satisfied without resort to the simple solution κBBD = 0.

Thus, we find that the existence of a dominant vacuum system does not change

our conclusions about the abundance of Boltzmann brains, except insofar as the

Boltzmann brain nucleation constraints that would apply to the dominant vacuum

must apply to every member of the dominant vacuum system. Probably the most

important implication of this example is that the dominant vacuum is not necessarily

CHAPTER 3. 72

the vacuum with the lowest decay rate, so the task of identifying the dominant vacuum

could be very difficult.

3.4.6 General Conditions to Avoid Boltzmann Brain Domi-

nation

In constructing general conditions to avoid Boltzmann brain domination, we are

guided by the toy landscapes discussed in the previous subsections. Our goal, how-

ever, is to construct conditions that can be justified using only the general equations

of Sections 3.2 and 3.3, assuming that the landscape is irreducible, but without rely-

ing on the properties of any particular toy landscape. We will be especially cautious

about the treatment of the dominant vacuum and the possibility of small upward

transitions, which could be rapid. The behavior of the full landscape of a realistic

theory may deviate considerably from that of the simplest toy models.

To discuss the general situation, it is useful to divide vacuum states into four

classes. We are only interested in vacua that can support Boltzmann brains. These

can be

(1) anthropic vacua for which the total dimensionless decay rate satisfies κi q,

(2) non-anthropic vacua that can transition to anthropic vacua via unsuppressed

transitions,

(3) non-anthropic vacua that can transition to anthropic vacua only via suppressed

transitions,

(4) anthropic vacua for which the total dimensionless decay rate is κi ≈ q.

Here q is the smallest-magnitude eigenvalue of the rate equation (see Eqs. (3.5)–(3.8)).

We call a transition “unsuppressed” if its branching ratio is roughly of order one in

the sense of Eq. (3.16). If the branching ratio is smaller than this, it is “suppressed.”

As before, when calculating NBB/NNO we assume that factors that are roughly of

order one can be ignored. Note that Eq. (3.11) forbids κi from being less than q, so

the above four cases are exhaustive.

CHAPTER 3. 73

We first discuss conditions that are sufficient to guarantee that Boltzmann brains

will not dominate, postponing until later the issue of what conditions are necessary.

We begin with the vacua in the first class. Very likely all anthropic vacua belong

to this class. For an anthropic vacuum i, the Boltzmann brains produced in vacuum

i cannot dominate the multiverse if they do not dominate the normal observers in

vacuum i, so we can begin with this comparison. Restricting Eq. (3.29) to this single

vacuum, we obtainNBBi

NNOi

∼ κBBi

κi, (3.72)

a ratio that has appeared in many of the simple examples. If this ratio is small

compared to one, then Boltzmann brains created in vacuum i are negligible.

Let us now study a vacuum j in the second class. First note that Eq. (3.9) implies

the rigorous inequality

κi si ≥ κij sj (no sum on repeated indices) , (3.73)

which holds for any two states i and j. (Intuitively, Eq. (3.73) is the statement that,

in steady state, the total rate of loss of volume fraction must exceed the input rate

from any one channel.) To simplify what follows, it will be useful to rewrite Eq. (3.73)

as

(κisi) ≥ (κjsj)Bj→i , (3.74)

where Bj→i ≡ κij/κj is the branching ratio for the transition j → i.

Suppose that we are trying to bound the Boltzmann brain production in vacuum

j, and we know that it can undergo unsuppressed transitions

j → k1 → . . .→ kn → i , (3.75)

where i is an anthropic vacuum. We begin by using Eqs. (3.22) and (3.24) to express

NBBj /NNO

i , dropping irrelevant factors as in Eq. (3.28), and then we can iterate the

CHAPTER 3. 74

above inequality:

NBBj

NNOi

∼ κBBj sj∑k κik sk

≤ κBBj sj

κi si

≤ κBBj sj

(κjsj)Bj→k1Bk1→k2 · · ·Bkn→i

=κBBjκj

1

Bj→k1Bk1→k2 · · ·Bkn→i, (3.76)

where again there is no sum on repeated indices, and Eq. (3.9) was used in the last

step on the first line. Each inverse branching ratio on the right of the last line is

greater than or equal to one, but by our assumptions can be considered to be roughly

of order one, and hence can be dropped. Thus, the multiverse will avoid domination

by Boltzmann brains in vacuum j if κBBj /κj 1, the same criterion found for the

first class.

The third class — non-anthropic vacua that can only transition to an anthropic

state via at least one suppressed transition — presumably includes many states with

very low vacuum energy density. The dominant vacuum of our toy landscape models

certainly belongs to this class, but we do not know of anything that completely ex-

cludes the possibility that the dominant vacuum might belong to the second or fourth

classes. That is, perhaps the dominant vacuum is anthropic, or decays to an anthropic

vacuum. If there is a dominant vacuum system, as described in Subsection 3.4.5, then

κi q, and the dominant vacua could belong to the first class, as well as to either of

classes (2) and (3).

To bound the Boltzmann brain production in this class, we consider two possible

criteria. To formulate the first, we can again use Eqs. (3.75) and (3.76), but this time

the sequence must include at least one suppressed transition, presumably an upward

jump. Let us therefore denote the branching ratio for this suppressed transition as

Bup, noting that Bup will appear in the denominator of Eq. (3.76). Of course, the

sequence of Eq. (3.75) might involve more than one suppressed transition, but in any

case the product of these very small branching ratios in the denominator can be called

Bup, and all the other factors can be taken as roughly of order one. Thus, a landscape

CHAPTER 3. 75

containing a vacuum j of the third class avoids Boltzmann brain domination if

κBBj

Bup κj 1 , (3.77)

in agreement with the results obtained for the dominant vacua in the toy landscape

models in the previous subsections.

A few comments are in order. First, if the only suppressed transition is the first,

then Bup = κup/κj, and the above criterion simplifies to κBBj /κup 1. Second, we

should keep in mind that the sequence of Eq. (3.75) is presumably not unique, so other

sequences will produce other bounds. All the bounds will be valid, so the strongest

bound is the one of maximum interest. Finally, since the vacua under discussion

are not anthropic, a likely method for Eq. (3.77) to be satisfied would be for κBBj

to vanish, as would happen if the vacuum j did not support the complex structures

needed to form Boltzmann brains.

The criterion above can be summarized by saying that if κBBj /(Bupκj) 1, then

the Boltzmann brains in vacuum j will be overwhelmingly outnumbered by the nor-

mal observers living in pocket universes that form in the decay chain starting from

vacuum j. We now describe a second, alternative criterion, based on the idea that

the number of Boltzmann brains in vacuum j can be compared with the number of

normal observers in vacuum i if the two types of vacuum have a common ancestor.

Denoting the common ancestor vacuum as A, we assume that it can decay to an

anthropic vacuum i by a chain of transitions

A→ k1 → . . .→ kn → i , (3.78)

and also to a Boltzmann-brain-producing vacuum j by a chain

A→ `1 → . . .→ `m → j . (3.79)

From the sequence of Eq. (3.78) and the bound of Eq. (3.74), we can infer that

(κisi) ≥ (kAsA)BA→k1Bk1→k2 · · ·Bkn→i . (3.80)

CHAPTER 3. 76

To make use of the sequence of Eq. (3.79) we will want a bound that goes in the

opposite direction, for which will need to require additional assumptions. Starting

with Eq. (3.9), we first require q κi, which is plausible provided that vacuum i is

not the dominant vacuum. Next we look at the sum over j on the right-hand side,

and we call the transition j → i “significant” if its contribution to the sum is within

a factor roughly of order one of the entire sum. (The sum over j is the sum over

sources for vacuum i, so a transition j → i is “significant” if pocket universes of

vacuum j are a significant source of pocket universes of vacuum i.) It follows that for

any significant transition j → i for which q κi,

(κisi) ≤ (κjsj)ZmaxBj→i ≤ (κjsj)Zmax , (3.81)

where Zmax denotes the largest number that is roughly of order one. By our conven-

tions, Zmax = exp(1014). If we assume now that all the transitions in the sequence of

Eq. (3.79) are significant, and that q is negligible in each case, then

(κjsj) ≤ (kAsA)Zm+1max . (3.82)

Using the bounds from Eqs. (3.80) and (3.82), the Boltzmann brain ratio is bounded

by

NBBj

NNOi

∼ κBBj sj∑k κik sk

≤ κBBj sj

κi si

≤ Zm+1max

BA→k1Bk1→k2 · · ·Bkn→i

κBBj

κj. (3.83)

But all the factors on the right are roughly of order one, except that some of the

branching ratios in the denominator might be smaller, if they correspond to sup-

pressed transitions. If Bup denotes the product of branching ratios for all the sup-

pressed transitions shown in the denominator (i.e., all suppressed transitions in the

sequence of Eq. (3.78)), then the bound reduces to Eq. (3.77).9

9Note, however, that the argument breaks down if the sequences in either of Eqs. (3.78) or(3.79) become too long. For the choices that we have made, a factor of Zmax is unimportant in the

CHAPTER 3. 77

To summarize, the Boltzmann brains in a non-anthropic vacuum j can be bounded

if there is an ancestor vacuum A that can decay to j through a chain of significant

transitions for which q κ` for each vacuum, as in the sequence of Eq. (3.79), and

if the same ancestor vacuum can decay to an anthropic vacuum through a sequence

of transitions as in Eq. (3.78). The Boltzmann brains will never dominate provided

that κBBj /(Bup κj) 1, where Bup is the product of all suppressed branching ratios

in the sequence of Eq. (3.78).

Finally, the fourth class of vacua consists of anthropic vacua i with decay rate

κi ' q, a class which could be empty. For this class Eq. (3.29) may not be very

useful, since the quantity (κi − q) in the denominator could be very small. Yet, as

in the two previous classes, this class can be treated by using Eq. (3.76), where in

this case the vacuum i can be the same as j or different, although the case i = j

requires n ≥ 1. Again, if the sequence contains only unsuppressed transitions, then

the multiverse avoids domination by Boltzmann brains in vacuum i if κBBi /κi 1.

If upward jumps are needed to reach an anthropic vacuum, whether it is the vacuum

i again or a distinct vacuum j, then the Boltzmann brains in vacuum i will never

dominate if κBBi /(Bup κi) 1.

The conditions described in the previous paragraph are very difficult to meet, so

if the fourth class is not empty, Boltzmann brain domination is hard to avoid. These

vacua have the slowest decay rates in the landscape, κi ≈ q, so it seems plausible

that they have very low energy densities, precluding the possibility of decaying to

an anthropic vacuum via unsuppressed transitions; in that case Boltzmann brain

domination can be avoided if

κBBi Bupκi . (3.84)

However, as pointed out in Ref. [98], Bup ∝ e−SD (see Eq. (3.57)) is comparable to

the inverse of the recurrence time, while in an anthropic vacuum one would expect

the Boltzmann brain nucleation rate to be much faster than once per recurrence time.

calculation of NBB/NNO, but Z100max = exp(1016) can be significant. Thus, for our choices we can

justify the dropping of O(100) factors that are roughly of order one, but not more than that. Forchoices appropriate to smaller estimates of ΓBB, however, the number of factors that can be droppedwill be many orders of magnitude larger.

CHAPTER 3. 78

To summarize, the domination of Boltzmann brains can be avoided by first of all

requiring that all vacuum states in the landscape obey the relation

κBBj

κj 1 . (3.85)

That is, the rate of nucleation of Boltzmann brains in each vacuum must be less than

the rate of nucleation, in that same vacuum, of bubbles of other phases. For anthropic

vacua i with κi q, this criterion is enough. Otherwise, the Boltzmann brains that

might be produced in vacuum j must be bounded by the normal observers forming in

some vacuum i, which must be related to j through decay chains. Specifically, there

must be a vacuum A that can decay through a chain to an anthropic vacuum i, i.e.

A→ k1 → . . .→ kn → i , (3.86)

where either A = j, or else A can decay to j through a sequence

A→ `1 → . . .→ `m → j . (3.87)

In the above sequence we insist that κj q and that κl q for each vacuum `p in the

chain, and that each transition must be “significant,” in the sense that pockets of type

`p must be a significant source of pockets of type `p+1. (More precisely, a transition

from vacuum j to i is “significant” if it contributes a fraction that is roughly of order

one to∑

j κijsj in Eq. (3.9).) For these cases, the bound which ensures that the

Boltzmann brains in vacuum j are dominated by the normal observers in vacuum i

is given byκBBj

Bup κj 1 , (3.88)

where Bup is the product of any suppressed branching ratios in the sequence of

Eq. (3.86). If all the transitions in Eq. (3.86) are unsuppressed, this bound reduces

to Eq. (3.85). If j is anthropic, the case A = j = i is allowed, provided that n ≥ 1.

The conditions described above are sufficient to guarantee that Boltzmann brains

CHAPTER 3. 79

do not dominate over normal observers in the multiverse, but without further as-

sumptions there is no way to know if they are necessary. All of the conditions that we

have discussed are quasi-local, in the sense that they do not require any global picture

of the landscape of vacua. For each of the above arguments, the Boltzmann brains in

one type of vacuum j are bounded by the normal observers in some type of vacuum

i that is either the same type, or directly related to it through decay chains. Thus,

there was no need to discuss the importance of the vacua j and i compared to the

rest of the landscape as a whole. The quasi-local nature of these conditions, however,

guarantees that they cannot be necessary to avoid the domination by Boltzmann

brains. If two vacua j and i are both totally insignificant in the multiverse, then it

will always be possible for the Boltzmann brains in vacuum j to overwhelm the nor-

mal observers in vacuum i, while the multiverse as a whole could still be dominated

by normal observers in other vacua.

We have so far avoided making global assumptions about the landscape of vacua,

because such assumptions are generally hazardous. While it may be possible to make

statements that are true for the bulk of vacua in the landscape, in this context the

statements are not useful unless they are true for all the vacua of the landscape.

Although the number of vacua in the landscape, often estimated at 10500 [106], is

usually considered to be incredibly large, the number is nonetheless roughly of order

one compared to the numbers involved in the estimates of Boltzmann brain nucleation

rates and vacuum decay rates. Thus, if a single vacuum produces Boltzmann brains

in excess of required bounds, the Boltzmann brains from that vacuum could easily

overwhelm all the normal observers in the multiverse.

Recognizing that our conclusions could be faulty, we can nonetheless adopt some

reasonable assumptions to see where they lead. We can assume that the multiverse is

sourced by either a single dominant vacuum, or by a dominant vacuum system. We

can further assume that every anthropic and/or Boltzmann-brain-producing vacuum

i can be reached from the dominant vacuum (or dominant vacuum system) by a single

significant upward jump, with a rate proportional to e−SD , followed by some number

of significant, unsuppressed transitions, all of which have rates κk q and branching

CHAPTER 3. 80

ratios that are roughly of order one:

D → k1 → . . .→ kn → i . (3.89)

We will further assume that each non-dominant anthropic and/or Boltzmann-brain-

producing vacuum i has a decay rate κi q, but we need not assume that all of the

κi are comparable to each other. With these assumptions, the estimate of NBB/NNO

becomes very simple.

Applying Eq. (3.9) to the first transition of Eq. (3.89),

κk1 sk1 ∼ κk1D sD ∼ κupsD , (3.90)

where we use κup to denote the rate of a typical transition D → k , assuming that

they are all equal to each other up to a factor roughly of order one. Here ∼ indicates

equality up to a factor that is roughly of order one. If there is a dominant vacuum

system, then κk1D is replaced by κk1D ≡∑

` α`κk1D` , where the D` are the components

of the dominant vacuum system, and the α` are defined by generalizing Eqs. (3.67)

and (3.68).10 Applying Eq. (3.9) to the next transition, k1 → k2 we find

κk2 sk2 = Bk1→k2κk1sk1 + . . . ∼ κk1 sk1 , (3.91)

10In more detail, the concept of a dominant vacuum system is relevant when there is a set ofvacua ` that can have rapid transitions within the set, but only very slow transitions connectingthese vacua to the rest of the landscape. As a zeroth order approximation one can neglect alltransitions connecting these vacua to the rest of the landscape, and assume that κ` q, so Eq. (3.9)takes the form

κ` s` =∑

`′

B``′ κ`′ s`′ .

Here B``′ ≡ κ``′/κ`′ is the branching ratio within this restricted subspace, where κ` =∑`′ κ`′` is

summed only within the dominant vacuum system, so∑`B``′ = 1 for all `′. B``′ is nonnegative, and

if we assume also that it is irreducible, then the Perron-Frobenius theorem guarantees that it has anondegenerate eigenvector v` of eigenvalue 1, with positive components. From the above equationκ` s` ∝ v`, and then

α` =s`∑`′ s`′

=v`

κ`∑

`′

v`′

κ`′

.

CHAPTER 3. 81

where we have used the fact that Bk1→k2 is roughly of order one, and that the tran-

sition is significant. Iterating, we have

κi si ∼ κkn skn ∼ κup sD . (3.92)

Since the expression on the right is independent of i, we conclude that under these as-

sumptions any two non-dominant anthropic and/or Boltzmann-brain-producing vacua

i and j have equal values of κs, up to a factor that is roughly of order one:

κjsj ∼ κisi . (3.93)

Using Eq. (3.22) and assuming as always that nNOik is roughly of order one, Eq. (3.93)

implies that any two non-dominant anthropic vacua i and j have comparable numbers

of ordinary observers, up to a factor that is roughly of order one:

NNOj ∼ NNO

i . (3.94)

The dominant vacuum could conceivably be anthropic, but we begin by consid-

ering the case in which it is not. In that case all anthropic vacua are equivalent, so

the Boltzmann brains produced in any vacuum j will either dominate the multiverse

or not depending on whether they dominate the normal observers in an arbitrary

anthropic vacuum i. Combining Eqs. (3.22), (3.24), (3.9), and (3.93), and omitting

irrelevant factors, we find that for any non-dominant vacuum j

NBBj

NNOi

∼ κBBj sj∑k κik sk

∼ κBBj sj

κisi∼ κBB

j

κj. (3.95)

Thus, given the assumptions described above, for any non-dominant vacuum j the

necessary and sufficient condition to avoid the domination of the multiverse by Boltz-

mann brains in vacuum j is given by

κBBj

κj 1 . (3.96)

CHAPTER 3. 82

For Boltzmann brains formed in the dominant vacuum, we can again find out if

they dominate the multiverse by determining whether they dominate the normal ob-

servers in an arbitrary anthropic vacuum i. Repeating the above analysis for vacuum

D instead of vacuum j, using Eq. (3.92) to relate si to sD, we have

NBBD

NNOi

∼ κBBD sD∑k κik sk

∼ κBBD sDκisi

∼ κBBD

κup

. (3.97)

Thus, for a single dominant vacuum D or a dominant vacuum system with members

Di, the necessary and sufficient conditions to avoid the domination of the multiverse

by these Boltzmann brains is given by

κBBD

κup

1 orκBBDi

κup

1 . (3.98)

As discussed after Eq. (3.84), probably the only way to satisfy this condition is to

require that κBBD = 0.

If the dominant vacuum is anthropic, then the conclusions are essentially the

same, but the logic is more involved. For the case of a dominant vacuum system, we

distinguish between the possibility of vacua being “strongly” or “mildly” anthropic, as

discussed in Subsection 3.4.5. “Strongly anthropic” means that normal observers are

formed by tunneling within the dominant vacuum system D, while “mildly anthropic”

implies that normal observers are formed by tunneling, but only from outside D. Any

model that leads to a strongly anthropic dominant vacuum would be unacceptable,

because almost all observers would live in pockets with a maximum reheat energy

density that is small compared to the vacuum energy density. With a single anthropic

dominant vacuum, or with one or more mildly anthropic vacua within a dominant

vacuum system, the normal observers in the dominant vacuum would be comparable

in number (up to factors roughly of order one) to those in other anthropic vacua,

so they would have no significant effect on the ratio of Boltzmann brains to normal

observers in the multiverse. An anthropic vacuum would also produce Boltzmann

brains, however, so Eq. (3.98) would have to somehow be satisfied for κBBD 6= 0.

CHAPTER 3. 83

3.5 Boltzmann Brain Nucleation and Vacuum De-

cay Rates

3.5.1 Boltzmann Brain Nucleation Rate

Boltzmann brains emerge from the vacuum as large quantum fluctuations. In partic-

ular, they can be modeled as localized fluctuations of some mass M , in the thermal

bath of a de Sitter vacuum with temperature TdS = HΛ/2π [56]. The Boltzmann

brain nucleation rate is then roughly estimated by the Boltzmann suppression fac-

tor [62, 64],

ΓBB ∼ e−M/TdS , (3.99)

where our goal is to estimate only the exponent, not the prefactor. Eq. (3.99) gives

an estimate for the nucleation rate of a Boltzmann brain of mass M in any particular

quantum state, but we will normally describe the Boltzmann brain macroscopically.

Thus ΓBB should be multiplied by the number of microstates eSBB corresponding to

the macroscopic description, where SBB is the entropy of the Boltzmann brain. Thus

we expect

ΓBB ∼ e−M/TdSeSBB = e−F/TdS , (3.100)

where F = M − TdS SBB is the free energy of the Boltzmann brain.

Eq. (3.100) should be accurate as long as the de Sitter temperature is well-defined,

which will be the case as long as the Schwarzschild horizon is small compared to the

de Sitter horizon radius. Furthermore, we shall neglect the effect of the gravitational

potential energy of de Sitter space on the Boltzmann brain, which requires that the

Boltzmann brain be small compared to the de Sitter horizon. Thus we assume

M/4π < R H−1Λ , (3.101)

where the first inequality assumes that Boltzmann brains cannot be black holes. The

general situation, which allows for M ∼ R ∼ H−1Λ , will be discussed in Appendix A

and in Ref. [107].

While the nucleation rate is proportional to eSBB , this factor is negligible for any

CHAPTER 3. 84

Boltzmann brain made of atoms like those in our universe. The entropy of such atoms

is bounded by

S . 3M/mn , (3.102)

where mn is the nucleon mass. Indeed, the actual value of SBB is much smaller

than this upper bound because of the complex organization of the Boltzmann brain.

Meanwhile, to prevent the Boltzmann brain from being destroyed by pair production,

we require that TdS mn. Thus, for these Boltzmann brains the entropy factor eSBB

is irrelevant compared to the Boltzmann suppression factor.

To estimate the nucleation rate for Boltzmann brains, we need at least a crude

description of what constitutes a Boltzmann brain. There are many possibilities. We

argued in the introduction to this chapter that a theory that predicts the domination

of Boltzmann brains over normal observers would be overwhelmingly disfavored by our

continued observation of an orderly world, in which the events that we observe have

a logical relationship to the events that we remember. In making this argument, we

considered a class of Boltzmann brains that share exactly the memories and thought

processes of a particular normal observer at some chosen instant. For these purposes

the memory of the Boltzmann brain can consist of random bits that just happen

to match those of the normal observer, so there are no requirements on the history

of the Boltzmann brain. Furthermore, the Boltzmann brain need only survive long

enough to register one observation after the chosen instant, so it is not required to

live for more than about a second. We will refer to Boltzmann brains that meet these

requirements as minimal Boltzmann brains.

While an overabundance of minimal Boltzmann brains is enough to cause a theory

to be discarded, we nonetheless find it interesting to discuss a wide range of Boltzmann

brain possibilities. We will start with very large Boltzmann brains, discussing the

minimal Boltzmann brains last.

We first consider Boltzmann brains much like us, who evolved in stellar systems

like ours, in vacua with low-energy particle physics like ours, but allowing for a de

Sitter Hubble radius as small as a few astronomical units or so. These Boltzmann

brains evolved in their stellar systems on a time scale similar to the evolution of life on

CHAPTER 3. 85

Earth, so they are in every way like us, except that, when they perform cosmological

observations, they find themselves in an empty, vacuum-dominated universe. These

“Boltzmann solar systems” nucleate at a rate of roughly

ΓBB ∼ exp(−1085) , (3.103)

where we have set M ∼ 1030 kg and H−1Λ = (2πTdS)−1 ∼ 1012 m. This nucleation rate

is fantastically small; we found it, however, by considering the extravagant possibility

of nucleating an entire Boltzmann solar system.

Next, we can consider the nucleation of an isolated brain, with a physical con-

struction that is roughly similar to our own brains. If we take M ∼ 1 kg and

H−1Λ = (2πTdS)−1 ∼ 1 m, then the corresponding Boltzmann brain nucleation rate is

ΓBB ∼ exp(−1043) . (3.104)

If the construction of the brain is similar to ours, however, then it could not function

if the tidal forces resulted in a relative acceleration from one end to the other that is

much greater than the gravitational acceleration g on the surface of the Earth. This

requires H−1Λ & 108 m, giving a Boltzmann brain nucleation rate

ΓBB ∼ exp(−1051) . (3.105)

Until now, we have concentrated on Boltzmann brains that are very similar to

human brains. However a common assumption in the philosophy of mind is that of

substrate-independence. Therefore, pressing onward, we study the possibility that a

Boltzmann brain can be any device capable of emulating the thoughts of a human

brain. In other words, we treat the brain essentially as a highly sophisticated com-

puter, with logical operations that can be duplicated by many different systems of

hardware.11

With this in mind, from here out we drop the assumption that Boltzmann brains

11Note that the validity of the assumption of substrate-independence of mind is not entirely self-evident. We are skeptical of identifying human consciousness with operations of a generic substrate-independent computer, but accept it as a working hypothesis for the purpose of this work.

CHAPTER 3. 86

are made of the same materials as human brains. Instead, we attempt to find an

upper bound on the probability of creation of a more generalized computing device,

specified by its information content IBB, which is taken to be comparable to the

information content of a human brain.

To clarify the meaning of information content, we can model an information stor-

age device as a system with N possible microstates. Smax = lnN is then the maximum

entropy that the system can have, the entropy corresponding to the state of complete

uncertainty of microstate. To store B bits of information in the device, we can imag-

ine a simple model in which 2B distinguishable macroscopic states of the system are

specified, each of which will be used to represent one assignment of the bits. Each

macroscopic state can be modeled as a mixture of N/2B microstates, and hence has

entropy S = ln(N/2B) = Smax − B ln 2. Motivated by this simple model, one defines

the information content of any macroscopic state of entropy S as the difference be-

tween Smax and S, where Smax is the maximum entropy that the device can attain.

Applying this definition to a Boltzmann brain, we write

IBB = SBB,max − SBB , (3.106)

where IBB/ln 2 is the information content measured in bits.

As discussed in Ref. [108], the only known substrate-independent limit on the

storage of information is the Bekenstein bound. It states that, for an asymptotically

flat background, the entropy of any physical system of size R and energyM is bounded

CHAPTER 3. 87

by12

S ≤ SBek ≡ 2πMR . (3.107)

One can use this bound in de Sitter space as well if the size of the system is sufficiently

small, R H−1Λ , so that the system does not “know” about the horizon. A possible

generalization of the Bekenstein bound for R = O(H−1Λ ) was proposed in Ref. [109];

we will study this and other possibilities in Appendix A and in Ref. [107]. To begin,

however, we will discuss the simplest case, R H−1Λ , so that we can focus on the

most important issues before dealing with the complexities of more general results.

Using Eq. (3.106), the Boltzmann brain nucleation rate of Eq. (3.100) can be

rewritten as

ΓBB ∼ exp

(−2πM

HΛ

+ SBB,max − IBB

), (3.108)

which is clearly maximized by choosing M as small as possible. The Bekenstein

bound, however, implies that SBB,max ≤ SBek and therefore M ≥ SBB,max/(2πR).

Thus

ΓBB ≤ exp

(−SBB,max

RHΛ

+ SBB,max − IBB

). (3.109)

Since R < H−1Λ , the expression above is maximized by taking SBB,max equal to its

smallest possible value, which is IBB. Finally, we have

ΓBB ≤ exp

(− IBB

RHΛ

). (3.110)

Thus, the Boltzmann brain production rate is maximized if the Boltzmann brain

12In an earlier version of this work we stated an incorrect form of this bound, and from it derivedsome incorrect conclusions, such as the statement that the largest Boltzmann brain nucleationrate ΓBB consistent with the Bekenstein bound is attained only when the radius R approaches theSchwarzschild radius RSch. This in turn led to the conclusion that the maximum rate allowed by theBekenstein bound is e−2IBB , which can be achieved only if M2 = IBB/(9πG) and H2

Λ = π/(3GIBB).While these relations hold in the regime we considered, they are not necessary in the general case.With the corrected bound, we find that the maximum nucleation rate is independent of R/RSch ifR HΛ (see Eq. (3.110)), and otherwise grows with R/RSch (see Appendix A). However, once oneis forced to consider values of R RSch, then other issues become relevant. How can the systembe stabilized against the de Sitter expansion? Can the Bekenstein bound really be saturated for asystem with large entropy, especially if it is dilute? In this version of the work we have added adiscussion of these issues. We thank R. Bousso, B. Freivogel, and I. Yang for pointing out the errorin our earlier statement of the Bekenstein bound.

CHAPTER 3. 88

saturates the Bekenstein bound, with IBB = SBB,max = 2πMR. Simultaneously, we

should make RHΛ as large as possible, which means taking our assumption R H−1Λ

to the boundary of its validity. Thus we write the Boltzmann brain production rate

ΓBB ≤ e−aIBB , (3.111)

where a ≡ (RHΛ)−1, the value of which is of order a few. In Appendix A we explore

the case in which the Schwarzschild radius, the Boltzmann brain radius, and the de

Sitter horizon radius are all about equal, in which case Eq. (3.111) holds with a = 2.

The bound of Eq. (3.111) can be compared to the estimate of the Boltzmann

brain production rate, ΓBB ∼ e−SBB , which follows from Eq. (2.13) of Freivogel and

Lippert, in Ref. [110]. The authors of Ref. [110] explained that by SBB they mean

not the entropy, but the number of degrees of freedom, which is roughly equal to the

number of particles in a Boltzmann brain. This estimate appears similar to our result,

if one equates SBB to IBB, or to a few times IBB. Freivogel and Lippert describe this

relation as a lower bound on the nucleation rate for Boltzmann brains, commenting

that it can be used as an estimate of the nucleation rate for vacua with “reasonably

cooperative particle physics.” Here we will explore in some detail the question of

whether this bound can be used as an estimate of the nucleation rate. While we will

not settle this issue here, we will discuss evidence that Eq. (3.111) is a valid estimate

for at most a small fraction of the vacua of the landscape, and possibly none at all.

So far, the conditions to reach the upper bound in Eq. (3.111) are R = (aHΛ)−1 ∼O(H−1

Λ ) and IBB = Smax,BB = SBek. However these are not enough to ensure that a

Boltzmann brain of size R ∼ H−1Λ is stable and can actually compute. Indeed, the

time required for communication between two parts of a Boltzmann brain separated

by a distance O(H−1Λ ) is at least comparable to the Hubble time. If the Boltzmann

brain can be stretched by cosmological expansion, then after just a few operations

the different parts will no longer be able to communicate. Therefore we need a

stabilization mechanism by which the brain is protected against expansion.

A potential mechanism to protect the Boltzmann brain against de Sitter expansion

is the self-gravity of the brain. A simple example is a black hole, which does not

CHAPTER 3. 89

expand when the universe expands. It seems unlikely that black holes can think,13

but one can consider objects of mass approaching that of a black hole with radius

R. This, together with our goal to keep R as close as possible to H−1Λ , leads to the

following condition:

M ∼ R ∼ H−1Λ . (3.112)

If the Bekenstein bound is saturated, this leads to the following relations between

IBB, HΛ, and M :

IBB ∼MR ∼MH−1Λ ∼ H−2

Λ . (3.113)

A second potential mechanism of Boltzmann brain stabilization is to surround it

by a domain wall with a surface tension σ, which would provide pressure preventing

the exponential expansion of the brain. An investigation of this situation reveals that

one cannot saturate the Bekenstein bound using this mechanism unless there is a

specific relation between IBB, HΛ, and σ [107]:

σ ∼ IBB H3Λ . (3.114)

If σ is less than this magnitude, it cannot prevent the expansion, while a larger σ

increases the mass and therefore prevents saturation of the Bekenstein bound.

Regardless of the details leading to Eqs. (3.113) and (3.114), the important point

is that both of them lead to constraints on the vacuum hosting the Boltzmann brain.

For example, the Boltzmann brain stabilized by gravitational attraction can be

produced at a rate approaching e−aIBB only if IBB ∼ H−2Λ . For a given value of

IBB, say IBB ∼ 1016 (see the discussion below), this result applies only to vacua

with a particular vacuum energy, Λ ∼ 10−16. Similarly, according to Eq. (3.114), for

Boltzmann brains with IBB ∼ 1016 contained inside a domain wall in a vacuum with

Λ ∼ 10−120, the Bekenstein bound on ΓBB cannot be reached unless the tension of

the domain wall is incredibly small, σ ∼ 10−164. Thus, the maximal Boltzmann brain

13The possibility of a black hole computer is not excluded, however, and has been considered inRef. [108]. Nonetheless, if black holes can compute, our conclusions would not be changed, providedthat the Bekenstein bound can be saturated for the near-black hole computers that we discuss. Atthis level of approximation, there would be no significant difference between a black hole computerand a near-black hole computer.

CHAPTER 3. 90

production rate ∼ e−aIBB saturating the Bekenstein bound cannot be reached unless

Boltzmann brains are produced on a narrow hypersurface in the landscape.

This conclusion by itself does not eliminate the danger of a rapid Boltzmann brain

production rate, ΓBB ∼ e−aIBB . Given the vast number of vacua in the landscape, it

seems plausible that this bound could actually be met. If this is the case, Eq. (3.111)

offers a stunning increase over previous estimates of ΓBB.

Setting aside the issue of Boltzmann brain stability, one can also question the

assumption of Bekenstein bound saturation that is necessary to achieve the rather

high nucleation rate that is indicated by Eq. (3.111). Of course black holes saturate

this bound, but we assume that a black hole cannot think. Even if a black hole can

think, it would still be an open question whether this information processing could

make use of a substantial fraction of the degrees of freedom associated with the black

hole entropy. A variety of other physical systems are considered in Ref. [111], where

the validity of Smax(E) ≤ 2πER is studied as a function of energy E. In all cases,

the bound is saturated in a limit where Smax = O(1). Meanwhile, as we shall argue

below, the required value of Smax should be greater than 1016.

We are aware of only one example of a physical system that may saturate the

Bekenstein bound and at the same time store sufficient information I to emulate

a human brain. This may happen if the total number of particle species with mass

smaller than HΛ is greater than IBB & 1016. No realistic examples of such theories are

known to us, although some authors have speculated about similar possibilities [112].

If Boltzmann brains cannot saturate the Bekenstein bound, they will be more

massive than indicated in Eq. (3.110), and their rate of production will be smaller

than e−aIBB .

To put another possible bound on the probability of Boltzmann brain production,

let us analyze a simple model based on an ideal gas of massless particles. Dropping

all numerical factors, we consider a box of size R filled with a gas with maximum

entropy Smax = (RT )3 and energy E = R3T 4 = S4/3max/R, where T is the temperature

and we assume there is not an enormous number of particle species. The probability

CHAPTER 3. 91

of its creation can be estimated as follows:

ΓBB ∼ e−E/HΛ eSBB ∼ exp

(−S

4/3max

HΛR

), (3.115)

where we have neglected the Boltzmann brain entropy factor, since SBB ≤ Smax S

4/3max. This probability is maximized by taking R ∼ H−1

Λ , which yields

ΓBB . e−S4/3max . (3.116)

In case the full information capacity of the gas is used, one can also write

ΓBB . e−I4/3BB . (3.117)

For IBB 1, this estimate leads to a much stronger suppression of Boltzmann brain

production as compared to our previous estimate, Eq. (3.111).

Of course, such a hot gas of massless particles cannot think — indeed it is not

stable in the sense outlined below Eq. (3.111) — so we must add more parts to

this construction. Yet it seems likely that this will only decrease the Boltzmann

brain production rate. As a partial test of this conjecture, one can easily check that

if instead of a gas of massless particles we consider a gas of massive particles, the

resulting suppression of Boltzmann brain production will be stronger. Therefore in

our subsequent estimates we shall assume that Eq. (3.117) represents our next “line

of defense” against the possibility of Boltzmann brain domination, after the one given

by Eq. (3.111). One should note that this is a rather delicate issue; see for example a

discussion of several possibilities to approach the Bekenstein bound in Ref. [113]. A

more detailed discussion of this issue will be provided in Ref. [107].

Having related ΓBB to the information content IBB of the brain, we now need to

estimate IBB. How much information storage must a computer have to be able to

perform all the functions of the human brain? Since no one can write a computer

program that comes close to imitating a human brain, this is not an easy question to

answer.

CHAPTER 3. 92

One way to proceed is to examine the human brain, with the goal of estimating

its capacities based on its biological structure. The human brain contains ∼ 1014

synapses that may in principle connect to any of ∼ 1011 neurons [114], suggesting

that its information content14 might be roughly IBB ∼ 1015–1016. (We are assuming

here that the logical functions of the brain depend on the connections among neurons,

and not for example on their precise locations, cellular structures, or other information

that might be necessary to actually construct a brain.) A minimal Boltzmann brain

is only required to simulate the workings of a real brain for about a second, but with

neurons firing typically at 10 to 100 times a second, it is plausible that a substantial

fraction of the brain is needed even for only one second of activity. Of course the

actual number of required bits might be somewhat less.

An alternative approach is to try to determine how much information the brain

processes, even if one does not understand much about what the processing involves.

In Ref. [115], Landauer attempted to estimate the total content of a person’s long-

term memory, using a variety of experiments. He concluded that a person remembers

only about 2 bits/second, for a lifetime total in the vicinity of 109 bits. In a subsequent

paper [116], however, he emphatically denied that this number is relevant to the

information requirements of a “real or theoretical cognitive processor,” because such

a device “would have so much more to do than simply record new information.”

Besides long-term memory, one might be interested in the total amount of informa-

tion a person receives but does not memorize. A substantial part of this information

is visual; it can be estimated by the information stored on high definition DVDs

watched continuously on several monitors over the span of a hundred years. The

total information received would be about 1016 bits.

Since this number is similar to the number obtained above by counting synapses, it

is probably as good an estimate as we can make for a minimal Boltzmann brain. If the

Bekenstein bound can be saturated, then the estimated Boltzmann brain nucleation

rate for the most favorable vacua in the landscape would be given by Eq. (3.111):

ΓBB . e−1016

. (3.118)

14Note that the specification of one out of 1011 neurons requires log2

(1011

)= 36.5 bits.

CHAPTER 3. 93

If, however, the Bekenstein bound cannot be reached for systems with IBB 1, then

it might be more accurate to use instead the ideal gas model of Eq. (3.117), yielding

ΓBB . e−1021

. (3.119)

Obviously, there are many uncertainties involved in the numerical estimates of the

required value of IBB. Our estimate IBB ∼ 1016 concerns the information stored in the

human brain that appears to be relevant for cognition. It certainly does not include

all the information that would be needed to physically construct a human brain, and

it therefore does not allow for the information that might be needed to physically

construct a device that could emulate the human brain. 15 It is also possible that

extra mass might be required for the mechanical structure of the emulator, to provide

the analogues of a computer’s wires, insulation, cooling systems, etc. On the other

hand, it is conceivable that a Boltzmann brain can be relevant even if it has fewer

capabilities than what we called the minimal Boltzmann brain. In particular, if our

main requirement is that the Boltzmann brain is to have the same “perceptions” as

a human brain for just one second, then one may argue that this can be achieved

15That is, the actual construction of a brain-like device would presumably require large amountsof information that are not part of the schematic “circuit diagram” of the brain. Thus there may besome significance to the fact that a billion years of evolution on Earth has not produced a humanbrain with fewer than about 1027 particles, and hence of order 1027 units of entropy. In countingthe information in the synapses, for example, we counted only the information needed to specifywhich neurons are connected to which, but nothing about the actual path of the axons and dendritesthat complete the connections. These are nothing like nearest-neighbor couplings, but instead axonsfrom a single neuron can traverse large fractions of the brain, resulting in an extremely intertwinednetwork [117]. To specify even the topology of these connections, still ignoring the precise locations,could involve much more than 1016 bits. For example, the synaptic “wiring” that connects theneurons will in many cases form closed loops. A specification of the connections would presumablyrequire a topological winding number for every pair of closed loops in the network. The number ofbits required to specify these winding numbers would be proportional to the square of the numberof closed loops, which would be proportional to the square of the number of synapses. Thus, thestructural information could be something like Istruct ∼ b×1028, where b is a proportionality constantthat is probably a few orders of magnitude less than 1. In estimating the resulting suppression ofthe nucleation rate, there is one further complication: since structural information of this sortpresumably has no influence on brain function, these choices would contribute to the multiplicity ofBoltzmann brain microstates, thereby multiplying the nucleation rate by eIstruct . There would stillbe a net suppression, however, with Eq. (3.111) leading to ΓBB ∝ e−(a−1)Istruct , where a is genericallygreater than 1. See Appendix A for further discussion of the value of a.

CHAPTER 3. 94

using much less than 1014 synapses. And if one decreases the required time to a much

smaller value required for a single computation to be performed by a human brain,

the required amount of information stored in a Boltzmann brain may become many

orders of magnitude smaller than 1016.

We find that regardless of how one estimates the information in a human brain,

if Boltzmann brains can be constructed so as to come near the limit of Eq. (3.111),

their nucleation rate would provide stringent requirements on vacuum decay rates in

the landscape. On the other hand, if no such physical construction exists, we are

left with the less dangerous bound of Eq. (3.117), perhaps even further softened by

the speculations described in Footnote 15. Note that none of these bounds is based

upon a realistic model of a Boltzmann brain. For example, the nucleation of an actual

human brain is estimated at the vastly smaller rate of Eq. (3.105). The conclusions of

this paragraph apply to the causal patch measures [79, 80] as well as the scale-factor

cutoff measure.

In Section 3.3 we discussed the possibility of Boltzmann brain production during

reheating, stating that this process would not be a danger. We postponed the nu-

merical discussion, however, so we now return to that issue. According to Eq. (3.26),

the multiverse will be safe from Boltzmann brains formed during reheating provided

that

ΓBBreheat,ik ∆τBB

reheat,ik nNOik (3.120)

holds for every pair of vacua i and k, where ΓBBreheat,ik is the peak Boltzmann brain

nucleation rate in a pocket of vacuum i that forms in a parent vacuum of type k,

∆τBBreheat,ik is the proper time available for such nucleation, and nNO

ik is the volume

density of normal observers in these pockets, working in the approximation that all

observers form at the same time.

Compared to the previous discussion about late-time de Sitter space nucleation,

here ΓBBreheat,ik can be much larger, since the temperature during reheating can be

much larger than HΛ. On the other hand, safety from Boltzmann brains requires the

late-time nucleation rate to be small compared to the potentially very small vacuum

decay rates, while in this case the quantity on the right-hand side of Eq. (3.120) is

CHAPTER 3. 95

not exceptionally small. In discussing this issue, we will consider in sequence three

descriptions of the Boltzmann brain: a human-like brain, a near-black hole computer,

and a diffuse computer.

The nucleation of human-like Boltzmann brains during reheating was discussed in

Ref. [83], where it was pointed out that such brains could not function at temperatures

much higher than 300 K, and that the nucleation rate for a 100 kg object at this

temperature is ∼ exp(−1040). This suppression is clearly more than enough to

ensure that Eq. (3.120) is satisfied.

For a near-black hole computer with IBB ≈ SBB,max ≈ 1016, the minimum mass is

600 grams. If we assume that the reheat temperature is no more than the reduced

Planck mass, mPlanck ≡ 1/√

8πG ≈ 2.4 × 1018 GeV ≈ 4.3 × 10−6 gram, we find

that ΓBBreheat < exp

(−√2IBB

)∼ exp(−108). Although this is not nearly as much

suppression as in the previous case, it is clearly enough to guarantee that Eq. (3.120)

will be satisfied.

For the diffuse computer, we can consider an ideal gas of massless particles, as

discussed in Eqs. (3.115)–(3.117). The system would have approximately Smax par-

ticles, and a total energy of E = S4/3max/R, so the Boltzmann suppression factor is

exp[−S4/3

max/ (RTreheat)]. The Boltzmann brain production can occur at any time dur-

ing the reheating process, so there is nothing wrong with considering Boltzmann brain

production in our universe at the present time. For Treheat = 2.7 K and Smax = 1016,

this formula implies that the exponent has magnitude 1 for R = S4/3maxT

−1reheat ≈ 200

light-years. Thus, the formula suggests that diffuse-gas-cloud Boltzmann brains of

radius 200 light-years can be thermally produced in our universe, at the present time,

without suppression! If this estimate were valid, then Boltzmann brains would almost

certainly dominate the universe.

We argue, however, that the gas clouds described above would have no possibility

of computing, because the thermal noise would preclude any storage or transfer of

information. The entire device has energy of order E ≈ Treheat, which is divided

among approximately 1016 massless particles. The mean particle energy is therefore

1016 times smaller than that of the thermal particles in the background radiation, and

the density of Boltzmann brain particles is 1048 times smaller than the background. To

CHAPTER 3. 96

function, it seems reasonable that the diffuse computer needs an energy per particle

that is at least comparable to the background, which means that the suppression

factor is exp(−1016) or smaller. Thus, we conclude that for all three cases, the ratio

of Boltzmann brains to normal observers is totally negligible.

Finally, let us also mention the possibility that Boltzmann brains might form

as quantum fluctuations in stable Minkowski vacua. String theory implies at least

the existence of a 10D decompactified Minkowski vacuum; Minkowski vacua of lower

dimension are not excluded, but they require precise fine tunings for which motivation

is lacking. While quantum fluctuations in Minkowski space are certainly less classical

than in de Sitter space, they still might be relevant. The possibility of Boltzmann

brains in Minkowski space has been suggested by Page [62, 60, 96]. If ΓBB is nonzero

in such vacua, regardless of how small it might be, Boltzmann brains will always

dominate in the scale-factor cutoff measure as we have defined it. Even if Minkowski

vacua cannot support Boltzmann brains, there might still be a serious problem with

what might be called “Boltzmann islands.” That is, it is conceivable that a fluctuation

in a Minkowski vacuum can produce a small region of an anthropic vacuum with a

Boltzmann brain inside it. The anthropic vacuum could perhaps even have a different

dimension than its Minkowski parent. If such a process has a nonvanishing probability

to occur, it will also give rise to Boltzmann brain domination in the scale-factor

cutoff measure. These problems would be shared by all measures that assign an

infinite weight to stable Minkowski vacua. There is, however, one further complication

which might allow Boltzmann brains to form in Minkowski space without dominating

the multiverse. If one speculates about Boltzmann brain production in Minkowski

space, one may equally well speculate about spontaneous creation of inflationary

universes there, each of which could contain infinitely many normal observers [118].

These issues become complicated, and we will make no attempt to resolve them here.

Fortunately, the estimates of thermal Boltzmann brain nucleation rates in de Sitter

space approach zero in the Minkowski space limit Λ → 0, so the issue of Boltzmann

brains formed by quantum fluctuations in Minkowski space can be set aside for later

study. Hopefully the vague idea that these fluctuations are less classical than de

Sitter space fluctuations can be promoted into a persuasive argument that they are

CHAPTER 3. 97

not relevant.

3.5.2 Vacuum Decay Rates

One of the most developed approaches to the string landscape scenario is based on

the KKLT construction [119]. In this construction, one begins by finding a set of

stabilized supersymmetric AdS and Minkowski vacua. After that, an uplifting is

performed, e.g. by adding a D3 brane at the tip of a conifold [119]. This uplifting

makes the vacuum energy density of some of these vacua positive (AdS → dS), but

in general many vacua remain AdS, and the Minkowski vacuum corresponding to the

uncompactified 10d space does not become uplifted. The enormous number of the

vacua in the landscape appears because of the large number of different topologies of

the compactified space, and the large number of different fluxes and branes associated

with it.

There are many ways in which our low-energy dS vacuum may decay. First of all,

it can always decay into the Minkowski vacuum corresponding to the uncompactified

10d space [119]. It can also decay to one of the AdS vacua corresponding to the

same set of branes and fluxes [120]. More generally, decays occur due to the jumps

between vacua with different fluxes, or due to the brane-flux annihilation [121, 122,

123, 124, 125, 126, 110, 127], and may be accompanied by a change in the number

of compact dimensions [128, 129, 130]. If one does not take into account vacuum

stabilization, these transitions are relatively easy to analyze [121, 122, 123]. However,

in the realistic situations where the moduli fields are determined by fluxes, branes,

etc., these transitions involve a simultaneous change of fluxes and various moduli

fields, which makes a detailed analysis of the tunneling quite complicated.

Therefore, we begin with an investigation of the simplest decay modes due to the

scalar field tunneling. The transition to the 10d Minkowski vacuum was analyzed in

Ref. [119], where it was shown that the decay rate κ is always greater than

κ & e−SD = exp

(−24π2

VdS

). (3.121)

CHAPTER 3. 98

Here SD is the entropy of dS space. For our vacuum, SD ∼ 10120, which yields

κ & e−SD ∼ exp(−10120

). (3.122)

Because of the inequality in Eq. (3.121), we expect the slowest-decaying vacua to

typically be those with very small vacuum energies, with the dominant vacuum energy

density possibly being much smaller than the value in our universe.

The decay to AdS space (or, more accurately, a decay to a collapsing open universe

with a negative cosmological constant) was studied in Ref. [120]. The results of

Ref. [120] are based on investigation of BPS and near-BPS domain walls in string

theory, generalizing the results previously obtained in N = 1 supergravity [131, 132,

133, 134]. Here we briefly summarize the main results obtained in Ref. [120].

Consider, for simplicity, the situation where the tunneling occurs between two

vacua with very small vacuum energies. For the sake of argument, let us first ignore

the gravitational effects. Then the tunneling always takes place, as long as one vacuum

has higher vacuum energy than the other. In the limit when the difference between

the vacuum energies goes to zero, the radius of the bubble of the new vacuum becomes

infinitely large, R →∞ (the thin-wall limit). In this limit, the bubble wall becomes

flat, and its initial acceleration, at the moment when the bubble forms, vanishes.

Therefore to find the tension of the domain wall in the thin wall approximation one

should solve an equation for the scalar field describing a static domain wall separating

the two vacua.

If the difference between the values of the scalar potential in the two minima is

too small, and at least one of them is AdS, then the tunneling between them may be

forbidden because of the gravitational effects [135]. In particular, all supersymmetric

vacua, including all KKLT vacua prior to the uplifting, are absolutely stable even if

other vacua with lower energy density are available [136, 137, 138, 139].

It is tempting to make a closely related but opposite statement: non-supersymmetric

vacua are always unstable. However, this is not always the case. In order to study

tunneling while taking account of supersymmetry (SUSY), one may start with two

different supersymmetric vacua in two different parts of the universe and find a BPS

CHAPTER 3. 99

domain wall separating them. One can show that if the superpotential does not

change its sign on the way from one vacuum to the other, then this domain wall plays

the same role as the flat domain wall in the no-gravity case discussed above: it corre-

sponds to the wall of the bubble that can be formed once the supersymmetry is broken

in either of the two minima. However, if the superpotential does change its sign, then

only a sufficiently large supersymmetry breaking will lead to the tunneling [131, 120].

One should keep this fact in mind, but since we are discussing a landscape with an

extremely large number of vacua, in what follows we assume that there is at least one

direction in which the superpotential does not change its sign on the way from one

minimum to another. In what follows we describe tunneling in one such direction.

Furthermore, we assume that at least some of the AdS vacua to which our dS vacuum

may decay are uplifted much less than our vacuum. This is a generic situation, since

the uplifting depends on the value of the volume modulus, which takes different values

in each vacuum.

In this case the decay rate of a dS vacuum with low energy density and broken

supersymmetry can be estimated as follows [120, 140]:

κ ∼ exp

(− 8π2α

3m23/2

), (3.123)

where m3/2 is the gravitino mass in that vacuum and α is a quantity that depends

on the parameters of the potential. Generically one can expect α = O(1), but it can

also be much greater or much smaller than O(1). The mass m3/2 is set by the scale

of SUSY breaking,

3m23/2 = Λ4

SUSY , (3.124)

where we recall that we use reduced Planck units, 8πG = 1. Therefore the decay rate

can be also represented in terms of the SUSY-breaking scale ΛSUSY:

κ ∼ exp

(−24π2α

Λ4SUSY

), (3.125)

Note that in the KKLT theory, Λ4SUSY corresponds to the depth of the AdS vacuum

CHAPTER 3. 100

before the uplifting, so that

κ ∼ exp

(−24π2α

|VAdS|

). (3.126)

In this form, the result for the tunneling looks very similar to the lower bound on the

decay rate of a dS vacuum, Eq. (3.121), with the obvious replacements α → 1 and

|VAdS| → VdS.

Let us apply this result to the question of vacuum decay in our universe. Clearly,

the implications of Eq. (3.125) depend on the details of SUSY phenomenology. The

standard requirement that the gaugino mass and the scalar masses are O(1) TeV

leads to the lower bound

ΛSUSY & 104–105 GeV , (3.127)

which can be reached, e.g., in the models of conformal gauge mediation [141]. This

implies that for our vacuum

κour & exp(−1056)– exp(−1060) . (3.128)

Using Eq. (3.99), the Boltzmann brain nucleation rate in our universe exceeds the

lower bound of the above inequality only if M . 10−9 kg.

On the other hand, one can imagine universes very similar to ours except with

much larger vacuum energy densities. The vacuum decay rate of Eq. (3.123) exceeds

the Boltzmann brain nucleation rate of Eq. (3.99) when

(m3/2

10−2 eV

)2(M

1 kg

)(H−1

Λ

108 m

)& 109α . (3.129)

Note that H−1Λ ∼ 108 m corresponds to the smallest de Sitter radius for which the

tidal force on a 10 cm brain does not exceed the gravitational force on the surface of

the earth, while m3/2 ∼ 10−2 eV corresponds to ΛSUSY ∼ 104 GeV. Thus, it appears

the decay rate of Eq. (3.123) allows for Boltzmann brain domination.

However, we do not really know whether the models with low ΛSUSY can suc-

cessfully describe our world. To mention one potential problem: in models of string

CHAPTER 3. 101

inflation there is a generic constraint that during the last stage of inflation one has

H . m3/2 [142]. If we assume the second and third factors of Eq. (3.129) cannot

be made much less than unity, then we only require m3/2 & O(102) eV to avoid

Boltzmann brain domination. While models of string inflation with H . 100 eV are

not entirely impossible in the string landscape, they are extremely difficult to con-

struct [143]. If instead of ΛSUSY ∼ 104 GeV one uses ΛSUSY ∼ 1011 GeV, as in models

with gravity mediation, one finds m3/2 ∼ 103 GeV and Eq. (3.129) is easily satisfied.

These arguments apply when supersymmetry violation is as large or larger than

in our universe. If supersymmetry violation is too small, atomic systems are unsta-

ble [144], the masses of some of the particles will change dramatically, etc. However,

the Boltzmann computers described in the previous subsection do not necessarily

rely on laws of physics similar to those in our universe (in fact, they seem to require

very different laws of physics). We are unaware of an argument that supersymmetry

breaking must be so strong that vacuum decay is always faster than the Boltzmann

brain production rate of Eq. (3.118).

On the other hand, up to this point we have used the estimates of the vacuum

decay rate that were obtained in Refs. [120, 140] by investigation of the transition

where only moduli fields changed. As we have already mentioned, the description of

a more general class of transitions involving the change of branes or fluxes is much

more complicated. Investigation of such processes, performed in Refs. [124, 125, 110],

indicates that the process of vacuum decay for any vacuum in the KKLT scenario

should be rather fast,

κ & exp(−1022) . (3.130)

The results of Refs. [124, 125, 110], like the results of Refs. [120, 140], are not

completely generic. In particular, the investigations of Refs. [124, 125, 110] apply to

the original version of the KKLT scenario, where the uplifting of the AdS vacuum

occurs due toD3 branes, but not to its generalization proposed in Ref. [145], where the

uplifting is achieved due to D7 branes. Neither does it apply to the recent version of dS

stabilization proposed in Ref. [146]. Nevertheless, the results of Refs. [124, 125, 110]

show that the decay rate of dS vacua in the landscape can be quite large. The rate

CHAPTER 3. 102

κ & exp(−1022) is much greater than the expected rate of Boltzmann brain production

given by Eq. (3.105). However, it is just a bit smaller than the bosonic gas Boltzmann

brain production rate of Eq. (3.119) and much smaller than our most dangerous upper

bound on the Boltzmann brain production rate, given by Eq. (3.118).

3.6 Conclusions

If the observed accelerating expansion of the universe is driven by constant vacuum

energy density and if our universe does not decay in the next 20 billion years or

so, then it seems cosmology must explain why we are “normal observers” — who

evolve from non-equilibrium processes in the wake of the big bang — as opposed

to “Boltzmann brains” — freak observers that arise as a result of rare quantum

fluctuations [57, 58, 59, 63, 64]. Put in experimental terms, cosmology must explain

why we observe structure formation in a residual cosmic microwave background, as

opposed to the empty, vacuum-energy dominated environment in which almost all

Boltzmann brains nucleate. As vacuum-energy expansion is eternal to the future,

the number of Boltzmann brains in an initially-finite comoving volume is infinite.

However, if there exists a landscape of vacua, then rare transitions to other vacua

populate a diverging number of universes in this comoving volume, creating an infinite

number of normal observers. To weigh the relative number of Boltzmann brains to

normal observers requires a spacetime measure to regulate the infinities.

Recently, the scale-factor cutoff measure was shown to possess a number of de-

sirable attributes, including avoiding the youngness paradox [84] and the Q (and G)

catastrophe [86, 87, 88], while predicting the cosmological constant to be measured in

a range including the observed value, and excluding values more than about a factor

of ten larger and smaller than this [95]. The scale-factor cutoff does not itself select

for a longer duration of slow-roll inflation, raising the possibility that a significant

fraction of observers like us measure cosmic curvature significantly above the value

expected from cosmic variance [104]. In this chapter, we have calculated the ratio of

the total number of Boltzmann brains to the number of normal observers, using the

scale-factor cutoff.

CHAPTER 3. 103

The general conditions under which Boltzmann brain domination is avoided were

discussed in Subsection 3.4.6, where we described several alternative criteria that

can be used to ensure safety from Boltzmann brains. We also explored a set of

assumptions that allow one to state conditions that are both necessary and sufficient

to avoid Boltzmann brain domination. One relatively simple way to ensure safety from

Boltzmann brains is to require two conditions: (1) in any vacuum, the Boltzmann

brain nucleation rate must be less than the decay rate of that vacuum, and (2) for

any anthropic vacuum j with a decay rate κj ≈ q, and for any non-anthropic vacuum

j, one must construct a sequence of transitions from j to an anthropic vacuum; if the

sequence includes suppressed upward jumps, then the Boltzmann brain nucleation

rate in vacuum j must be less than the decay rate of vacuum j times the product of

all the suppressed branching ratios Bup that appear in the sequence. The condition

(2) might not be too difficult to satisfy, since it will generically involve only states with

very low vacuum energy densities, which are likely to be nearly supersymmetric and

therefore unlikely to support the complex structures needed for Boltzmann brains or

normal observers. Condition (2) can also be satisfied if there is no unique dominant

vacuum, but instead a dominant vacuum system that consists of a set of nearly

degenerate states, some of which are anthropic, which undergo rapid transitions to

each other, but only slow transitions to other states. The condition (1) is perhaps

more difficult to satisfy. Although nearly-supersymmetric string vacua can in principle

be long-lived [131, 132, 133, 134, 119, 120], with decay rates possibly much smaller

than the Boltzmann brain nucleation rate, recent investigations suggest that other

decay channels may evade this problem [124, 125, 110]. However, the decay processes

studied in [131, 132, 133, 134, 119, 120, 124, 125, 110] do not describe some of

the situations which are possible in the string theory landscape, and the strongest

constraints on the decay rate obtained in [110] are still insufficient to guarantee that

the vacuum decay rate is always smaller than the fastest estimate of the Boltzmann

brain production rate, Eq. (3.118).

One must emphasize that we are discussing a rapidly developing field of knowledge.

Our estimates of the Boltzmann brain production rate are exponentially sensitive to

our understanding of what exactly the Boltzmann brain is. Similarly, the estimates of

CHAPTER 3. 104

the decay rate in the landscape became possible only five years ago, and this subject

certainly is going to evolve. Therefore we will mention here two logical possibilities

which may emerge as a result of the further investigation of these issues.

If further investigation will demonstrate that the Boltzmann brain production

rate is always smaller than the vacuum decay rate in the landscape, the probability

measure that we are investigating in this chapter will be shown not to suffer from the

Boltzmann brain problem. Conversely, if one believes that this measure is correct,

the fastest Boltzmann brain production rate will give us a rather strong lower bound

on the decay rate of the metastable vacua in the landscape. We expect that similar

conclusions with respect to the Boltzmann brain problem should be valid for the

causal-patch measures [79, 80].

On the other hand, if we do not find a sufficiently convincing theoretical reason to

believe that the vacuum decay rate in all vacua in the landscape is always greater than

the fastest Boltzmann brain production rate, this would motivate the consideration

of other probability measures where the Boltzmann brain problem can be solved even

if the probability of their production is not strongly suppressed.

In any case, our present understanding of the Boltzmann brain problem does not

rule out the scale-factor cutoff measure, but the situation remains uncertain.

Appendix A

Boltzmann Brains in

Schwarzschild–de Sitter Space

As explained in Subsection 3.5.1, Eq. (3.100) for the production rate of Boltzmann

brains must be reexamined when the Boltzmann brain radius becomes comparable to

the de Sitter radius. In this case we need to describe the Boltzmann brain nucleation

as a transition from an initial state of empty de Sitter space with horizon radius H−1Λ

to a final state in which the dS space is altered by the presence of an object with

mass M . Assuming that the object can be treated as spherically symmetric, the space

outside the object is described by the Schwarzschild–de Sitter (SdS) metric [147]:1

ds2 = −(

1− 2GM

r−H2

Λr2

)dt2

+

(1− 2GM

r−H2

Λr2

)−1

dr2 + r2 dΩ2 . (A.1)

The SdS metric has two horizons, determined by the positive zeros of gtt, where the

smaller and larger are called RSch and RdS, respectively. We assume the Boltzmann

brain is stable but not a black hole, so its radius satisfies RSch < R < RdS. The radii

1We restore G = 1/8π in this Appendix for clarity.

105

APPENDIX A. 106

of the two horizons are given by

RSch =2√

3HΛ

cos

(π + ξ

3

),

RdS =2√

3HΛ

cos

(π − ξ

3

),

(A.2)

where

cos ξ = 3√

3GMHΛ . (A.3)

This last equation implies that for a given value of HΛ, there is an upper limit on

how much mass can be contained within the de Sitter horizon:

M ≤Mmax = (3√

3GHΛ)−1 . (A.4)

Eqs. (A.2) and (A.3) can be inverted to express M and HΛ in terms of the horizon

radii:

1

H2Λ

= R2Sch +R2

dS +RSchRdS (A.5)

M =RdS

2G

(1−H2

ΛR2dS

)(A.6)

=RSch

2G

(1−H2

ΛR2Sch

). (A.7)

We relate the Boltzmann brain nucleation rate to the decrease in total entropy

∆S caused by the the nucleation process,

ΓBB ∼ e−∆S , (A.8)

where the final entropy is the sum of the entropies of the Boltzmann brain and the

de Sitter horizon. For a Boltzmann brain with entropy SBB, the change in entropy is

given by

∆S =π

GH−2

Λ −( πGR2

dS + SBB

). (A.9)

APPENDIX A. 107

Note that for small M one can expand ∆S to find

∆S =2πM

HΛ

− SBB +O(GM2) , (A.10)

giving a nucleation rate in agreement with Eq. (3.100).2

To find a bound on the nucleation rate, we need an upper bound on the entropy

that can be attained for a given size and mass. In flat space the entropy is believed

to be bounded by Bekenstein’s formula, Eq. (3.107), a bound which should also be

applicable whenever R RdS. More general bounds in de Sitter space have been

discussed by Bousso [109], who considers bounds for systems that are allowed to fill

the de Sitter space out to the horizon R = RdS of an observer located at the origin.

For small mass M , Bousso argues that the tightest known bound on S is the D-bound,

which states that

S ≤ SD ≡π

G

(1

H2Λ

−R2dS

)=π

G

(R2

Sch +RSchRdS

), (A.11)

where the equality of the two expressions follows from Eq. (A.5). This bound can be

obtained from the principle that the total entropy cannot increase when an object

disappears through the de Sitter horizon. For larger values of M , the tightest bound

(for R = RdS) is the holographic bound, which states that

S ≤ SH ≡π

GR2

dS . (A.12)

Bousso suggests the possibility that these bounds have a common origin, in which

case one would expect that there exists a valid bound that interpolates smoothly

between the two. Specifically, he points out that the function

Sm ≡π

GRSchRdS (A.13)

is a candidate for such a function. Fig. (A.1) shows a graph of the holographic bound,

the D-bound, and the m-bound (Eq. (A.13)) as a function of M/Mmax. While there

2We thank Lenny Susskind for explaining this method to us.

APPENDIX A. 108

is no reason to assume that Sm is a rigorous bound, it is known to be valid in the

extreme cases where it reduces to the D– and holographic bounds. In between it

might be valid, but in any case it can be expected to be valid up to a correction

of order one. In fact, Fig. (A.1) and the associated equations show that the worst

possible violation of the m-bound is at the point where the holographic and D–

bounds cross, at M/Mmax = 3√

6/8 = 0.9186, where the entropy can be no more

than (1 +√

5)/2 = 1.6180 times as large as Sm.

Here we wish to carry the notion of interpolation one step further, because we

would like to discuss in the same formalism systems for which R RdS, where the

Bekenstein bound should apply. Hence we will explore the consequences of the bound

S ≤ SI ≡π

GRSchR , (A.14)

which we will call the interpolating bound. This bound agrees exactly with the m-

bound when the object is allowed to fill de Sitter space, with R = RdS. Again we have

no grounds to assume that the bound is rigorously true, but we do know that it is true

in the three limiting cases where it reduces to the Bekenstein bound, the D-bound,

and the holographic bound. The limiting cases are generally the most interesting

for us in any case, since we wish to explore the limiting cases for Boltzmann brain

nucleation. For parameters in between the limiting cases, it again seems reasonable

to assume that the bound is at least a valid estimate, presumably accurate up to a

factor of order one. We know of no rigorous entropy bounds for de Sitter space with

R comparable to RdS but not equal to it, so we don’t see any way at this time to do

better than the interpolating bound.

Proceeding with the I-bound of Eq. (A.14), we can use Eq. (3.106) to rewrite

Eq. (A.9) as

∆S =π

G

(H−2

Λ −R2dS

)− SBB,max + IBB , (A.15)

which can be combined with SBB,max ≤ SI to give

∆S ≥ π

G

(H−2

Λ −R2dS −RSch R

)+ IBB , (A.16)

APPENDIX A. 109

Figure A.1: Graph shows the holographic bound, the D-bound, and the m-bound forthe entropy of an object that fills de Sitter space out to the horizon. The holographicand D– bounds are each shown as broken lines in the region where they are supersededby the other. Although the m-bound looks very much like a straight line, it is not.

which can then be simplified using Eq. (A.5) to give

∆S ≥ π

GRSch (RSch +RdS −R) + IBB . (A.17)

To continue, we have to decide what possibilities to consider for the radius R of the

Boltzmann brain, which is related to the question of Boltzmann brain stabilization

discussed after Eq. (3.111). If we assume that stabilization is not a problem, because

it can be achieved by a domain wall or by some other particle physics mechanism,

then ∆S is minimized by taking R at its maximum value, R = RdS, so

∆S ≥ π

GR2

Sch + IBB . (A.18)

∆S is then minimized by taking the minimum possible value of RSch, which is the

value that is just large enough to allow the required entropy, SBB,max ≥ IBB. Using

APPENDIX A. 110

again the I-bound, one finds that saturation of the bound occurs at

ξsat = 3 sin−1

(√1− 3I

2

), (A.19)

where

I ≡ IBB

SdS

=GH2

Λ

πIBB (A.20)

is the ratio of the Boltzmann brain information to the entropy of the unperturbed de

Sitter space. Note that I varies from zero to a maximum value of 1/3, which occurs

in the limiting case for which RSch = RdS. The saturating value of the mass and the

corresponding values of the Schwarzschild radius and de Sitter radius are given by

Msat =I√

1 + I

2GHΛ

, (A.21)

RSch,sat =

√1 + I −

√1− 3I

2HΛ

, (A.22)

RdS,sat =

√1− 3I +

√1 + I

2HΛ

. (A.23)

Combining these results with Eq. (A.18), one has for this case (R = RdS) the bound

∆S

IBB

≥ 1 + I −√

1 + I√

1− 3I

2I. (A.24)

As can be seen in Figure A.2, the bound on ∆S/IBB for this case varies from 1, in the

limit of vanishing I (or equivalently, the limit HΛ → 0), to 2, in the limit RSch → RdS.

The limiting case of IBB → 0, with a nucleation rate of order e−IBB , has some

peculiar features that are worth mentioning. The nucleation rate describes the nucle-

ation of a Boltzmann brain with some particular memory state, so there would be an

extra factor of eIBB in the sum over all memory states. Thus, a single-state nucleation

rate of e−IBB indicates that the total nucleation rate, including all memory states, is

not suppressed at all. It may seem strange that the nucleation rate could be unsup-

pressed, but one must keep in mind that the system will function as a Boltzmann

APPENDIX A. 111

brain only for very special values of the memory state. In the limiting case discussed

here, the “Boltzmann brain” takes the form of a minor perturbation of the degrees

of freedom associated with the de Sitter entropy SdS = π/(GH2Λ).

As a second possibility for the radius R, we can consider the case of strong grav-

itational binding, R → RSch, as discussed following Eq. (3.111). For this case the

bound (A.17) becomes

∆S ≥ π

GRSchRdS + IBB . (A.25)

(Interestingly, if we take I = 0 (SBB = Smax) this formula agrees with the result

found in Ref. [148] for black hole nucleation in de Sitter space.) With R = RSch the

saturation of the I-bound occurs at

ξsat =π

2− 3 sin−1

(√3I

2

). (A.26)

The saturating value of the mass and the corresponding values of the Schwarzschild

radius and de Sitter radius are given by

Msat =

√I(

1− I)

2GHΛ

, (A.27)

RSch,sat =

√I

HΛ

, (A.28)

RdS,sat =

√4− 3I −

√I

2HΛ

. (A.29)

Using these relations to evaluate ∆S from Eq. (A.25), one finds

∆S

IBB

=

√4− 3I +

√I

2√I

, (A.30)

which is also plotted in Figure A.2. In this case (R = RSch) the smallest ratio ∆S/IBB

is 2, occurring at I = 1/3, where RSch = RdS. For smaller values of I the ratio becomes

larger, blowing up as 1/√I for small I. Thus, the nucleation rates for this choice

of R will be considerably smaller than those for Boltzmann brains with R ≈ RdS,

APPENDIX A. 112

Figure A.2: Graph shows the ratio of ∆S to IBB, where the nucleation rate forBoltzmann brains is proportional to e−∆S. All curves are based on the I-bound, asdiscussed in the text, but they differ by their assumptions about the size R of theBoltzmann brain.

but this case would still be relevant in cases where Boltzmann brains with R ≈ RdS

cannot be stabilized.

Another interesting case, which we will consider, is to allow the Boltzmann brain

to extend to R = Requil, the point of equilibrium between the gravitational attraction

of the Boltzmann brain and the outward gravitational pull of the de Sitter expansion.

This equilibrium occurs at the stationary point of gtt, which gives

Requil =

(GM

H2Λ

)1/3

. (A.31)

Boltzmann brains within this radius bound would not be pulled by the de Sitter ex-

pansion, so relatively small mechanical forces will be sufficient to hold them together.

Again ∆S will be minimized when the I-bound is saturated, which in this case

occurs when

ξsat =π

2− 3 sin−1

√1− 2A(I)

2

, (A.32)

APPENDIX A. 113

where

A(I) ≡ sin

sin−1(

1− 27 I3)

3

. (A.33)

The saturating value of the mass and the Schwarzschild and de Sitter radii are given

by

Msat =

√3[1 + A(I)]

√1− 2A(I)

9GHΛ

, (A.34)

RSch,sat =

√1− 2A(I)√

3HΛ

, (A.35)

RdS,sat =

√3

[√3(

3 + 2A(I))−√

1− 2A(I)

]

6HΛ

.

(A.36)

The equilibrium radius itself is given by

Requil,sat =

[1− 2A(I)

]1/6 [1 + A(I)

]1/3

√3HΛ

. (A.37)

Using these results with Eq. (A.17), ∆S is found to be bounded by

∆S

IBB

=

√3(

1− 2A(I))(

3 + 2A(I))− 2A(I) + 1

6I, (A.38)

which is also plotted in Figure A.2. As one might expect it is intermediate between

the two other cases. Like the R = RSch case, however, the ratio ∆S/IBB blows up for

small I, in this case behaving as (2/I)1/4.

In summary, we have found that our study of tunneling in Schwarzschild–de Sit-

ter space confirms the qualitative conclusions that were described in Subsection 3.5.1.

In particular, we have found that if the entropy bound can be saturated, then the

nucleation rate of a Boltzmann brain requiring information content IBB is given ap-

proximately by e−aIBB , where a is of order a few, as in Eq. (3.111). The coefficient a

APPENDIX A. 114

is always greater than 2 for Boltzmann brains that are small enough to be gravita-

tionally bound. This conclusion applies whether one insists that they be near-black

holes, or whether one merely requires that they be small enough so that their self-

gravity overcomes the de Sitter expansion. If, however, one considers Boltzmann

brains whose radius is allowed to extend to the de Sitter horizon, then Figure A.2

shows that a can come arbitrarily close to 1. However, one must remember that the

R = RdS curve on Figure A.2 can be reached only if several barriers can be overcome.

First, these objects are large and diffuse, becoming more and more diffuse as I ap-

proaches zero and a approaches 1. There is no known way to saturate the entropy

bound for such diffuse systems, and Eq. (3.117) shows that an ideal gas model leads

to a ∼ I1/3BB 1. Furthermore, Boltzmann brains of this size can function only if some

particle physics mechanism is available to stabilize them against the de Sitter expan-

sion. A domain wall provides a simple example of such a mechanism, but Eq. (3.114)

indicates that the domain wall solution is an option only if a domain wall exists with

tension σ ∼ IBBH3Λ. Thus, it is not clear how close a can come to its limiting value of

1. Finally, we should keep in mind that it is not clear if any of the examples discussed

in this appendix can actually be attained, since black holes might be the only objects

that saturate the entropy bound for S 1.

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Mahdiyar Noorbala

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Andrei Linde) Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Leonard Susskind)

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Stephen Shenker)

Approved for the University Committee on Graduate Studies