Topics in Inflation and Eternal Inflation A

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

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

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

ii 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, Primary 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.

Stephen Shenker

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

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 in University Archives.

iii To my parents and 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 transition 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 multiverse 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 λ (φ2 v2)2 ...... 9 4 − 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 λ (φ2 v2)2 ...... 18 4 − 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 ξ 2 which is nonminimally coupled to gravity, with coupling 2 φ R [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 nonminimally 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 incompatible 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 m2 2 potential 2 φ , and the model with the standard spontaneous symmetry breaking potential λ (φ2 v2)2. Our goal is to analyze these models at the classical level, 4 − 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 λ (φ2 v2)2 with ξ < 0 in the 4 − 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 describes a scalar field φ with a canonical kinetic term but with a non-minimal coupling CHAPTER 1. 3

to gravity: 1 2 µν L = √ gJ (1 + ξφ )RJ g ∂µφ∂νφ 2VJ (φ) . (1.1) 2 − − J − The index J is to emphasize that this is a Jordan frame: the gravity part of the Lagrangian is different from the Einstein Lagrangian 1 √ gR[g] and has a local func- 2 − 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 µν 2 −1 µν gE = (1 + ξφ ) gJ and a new ﬁeld ϕ related to the ﬁeld φ as follows:

dϕ 2 1 + ξφ2 + 6ξ2φ2 1 = =: . (1.2) dφ (1 + ξφ2)2 Z(φ) In new variables, the Lagrangian, up to a total derivative, is given by

1 µν L = √ gE [RE g ∂µϕ∂νϕ 2VE(φ(ϕ))] . (1.3) 2 − − E − where the potential in the Einstein frame is

V (φ) V (φ) = J , (1.4) E (1 + ξφ2)2

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

1 V 2 V = ϕ , η = ϕϕ . (1.5) 2 V V (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

= Z , η = Zη + sgn(V 0)Z0 φ , (1.6) φ φ 2 r where prime means d/dφ. Similarly, we can ﬁnd the number of e-folds by

dϕ dφ N = . (1.7) ≈ √2 Z(φ) 2φ Z Z

p 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 1 −1 µν kinetic term Z g ∂µφ∂νφ. So when we qualitatively analyze the motion of the − 2 E field, in analogy with a particle with Lagrangian 1 mx˙ 2 V (x), we need to associate 2 − 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 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 1 √ϕ φ = e 6 . (1.8) √ξ 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 | | 1 which the effective gravitational coupling constant GN = 8π(1+ξφ2) wouldp 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 00 of the ﬁeld. CHAPTER 1. 5

One can especially clearly see it if one considers a vicinity of the would-be singularity, where 1 ξ φ2 1. In this regime − | |

1 1 − 2 ϕ φ = 1 e √ 3 . (1.9) ξ − 2 | | p 1 − 2 ϕ Therefore 1 ξ φ = e √ 3 , which means that the limit φ 1/ ξ corresponds − | | 2 → | | to the limit ϕ . These considerations will help us to understand the properties p→ ∞ p 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 supergravity 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 Kähler potential in the simplest version of this model is given by

1 χ = 3 log 1 (ΦΦ¯ + SS¯) + (Φ2 + Φ¯ 2) + ζ(SS¯)2/3 . (1.11) K − − 3 4 h i 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 = 1 + χ . The Jordan frame potential of this field is − 6 4

2 VJ = f (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

1 1 S = (s + iα) , Φ = (φ + iβ) . (1.13) √2 √2

The potential of the inﬂaton ﬁeld φ is then

2 VJ (φ) = f (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 potential, 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 trajectory 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 2 2 and α, so one should check whether mβ, ms > 0. We will impose a stronger condition, 2 2 2 mβ, ms & H , 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 = 0 is similar, but more complicated, because 6 2 2 mβ and ms depend on ξ:

4 2 2 2 2 (1 + 3ξ)f + (1 + ξφ )((∂Φf) f ∂ f) m2 = 3 − Φ , (1.15) β (1 + ξφ2)(1 + ξφ2 + 6ξ2φ2) CHAPTER 1. 7

A(φ) m2 = . (1.16) s (1 + ξφ2)2(1 + ξφ2 + 6ξ2φ2) where

2 A(φ) = ( 1 ξφ2 + 6ξ2φ2 + 6ζ(1 + 2ξφ2 + 6ξ3φ4 3 − − 2 2 2 2 2 + ξ φ (6 + φ )))f 4√2ξφ(1 + ξφ )f ∂Φf − 2 2 2 + (1 + ξφ ) (∂Φf) (1.17)

In this chapter we studied two basic models. The first one corresponds to f(Φ) = m2 2 mΦ, which leads to the quadratic inflaton potential 2 φ . The second choice is f(Φ) = √λ(Φ2 v2/2), which leads to the inflaton potential λ (φ2 v2)2. Investigation − 4 − 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 Kählerpotential. 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 Kählerpotential 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 Kählerpotential (1.11) [7]. CHAPTER 1. 8

Figure 1.1: Predictions of the chaotic inﬂation model with the potential V φ2α for diﬀerent α in the range of 0 < α < 2 are shown by two blue lines corresponding∼ to N = 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 1 m2φ2 and λ (φ2 v2)2, and extend their analysis for the case with 2 4 − ξ = 0. To put our results in a proper perspective, we will briefly remember what is 6 known about observational consequences of some closely related models. 2α −1 1) V φ , ξ = 0. For this model one has 1 ns = (1 + α)N , 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

λ 2 2 2 Figure 1.2: Predictions of the model with the potential 4 (φ v ) 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 nonminimally coupled to gravity with ξ 1 for N = 50 and N = 60 [9, 7]. The green dashed lines describe predictions of this model for v = 0 and for various values of ξ > 0 [14], for N = 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. 4 The green lines in Fig. 1.2 show ns and r for the theory with the potential φ 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 theory λ (φ2 v2)2 with arbitrary values of v and ξ. 4 − CHAPTER 1. 10

1.5 Quadratic Potential

1 2 2 We begin with the investigation of the quadratic potential VJ = 2 m φ in the Jordan frame, which leads to the Einstein frame potential

m2φ2/2 V = . (1.18) E (1 + ξφ2)2

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]. V

Figure 1.3: The quadratic potential V as a function of the canonically normalized inflaton field ϕ in the Einstein frame. The right, blue curve corresponds to ξ = 1, the left, red curve corresponds to ξ = 1. Note that the potential for the negative ξ is very steep, but it does not contain any− singularity, which appears when one expresses this 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

2(1 ξφ2)2 = − , (1.19) φ2 (1 + ξφ2 + 6ξ2φ2)

2 (12ξ4φ6 + 2ξ3φ6 36ξ3φ4 5ξ2φ4 6ξφ2 + 1) η = − − − . (1.20) φ2(1 + ξφ2 + 6ξ2φ2)2

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

3 1 ξ2φ4 1 1 ξφ2 N log − e + log − e . (1.21) ≈ 4 1 ξ2φ4 4ξ 1 ξφ2 − ∗ − ∗

ξ m2 φ2R + φ2 2 2 ξ < 0

ξ =0

ξ > 0

Figure 1.4: (ns, r) at N = 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 < 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 − . . 2 8 varying ξ from zero we deviate from the point (ns, r) = (1 , ) (corresponding to − N N 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 − × / / × −5 to obtain the observed amplitude of density perturbations δρ 5 10 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 (10−2) in ξ. O

1.6 Quartic Potential

Let us now consider the standard potential with spontaneous symmetry breaking, which looks – with non-minimal coupling – as follows

λ (φ2 v2)2 VE = − , (1.22) 4 (1 + ξφ2)2 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).

V V

j j (a) (b) Figure 1.5: Einstein frame potential (1.22) as a function of the canonically normalized ﬁeld 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 at a very large value− of the potential (the minimum at V =− 0 appears in the antigravity regime beyond the singularity), so this potential is unsuitable for the description of inﬂation in our universe.

The slow-roll parameters are now given by

8(1 + ξv2)2φ2 = , (1.23) (φ2 v2)2 (1 + ξφ2 + 6ξ2φ2) − 4A (1 + ξv2) η = , (1.24) −(φ2 v2)2 (1 + ξφ2 + 6ξ2φ2)2 − where

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

The number of e-folds is given by

3 1 + ξφ2 1 1 φ N log e + (1 + 6ξ)(φ2 φ2) + v2 log e ) . (1.26) ≈ 4 1 + ξφ2 4 (1 + ξv2) 2 ∗ − e φ ∗ ∗

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.

ξ λ 2 φ2R + φ2 v2 2 4 − ξ > 0, φ >v

v =0 v →∞

ξ →∞

Figure 1.6: (ns, r) at N = 60 for the quartic potential Eq. (1.22) with positive ξ in the case with inflation at φ > v. Each red and yellow colored line corresponds to a fixed 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 the value of v. The limiting point of this set of curves, for ξ and any v, coincides with 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+6ξ . Beyond ∆ϕ a cubic term dominates which spoils the symmetry of ξ V (ϕ) aroundq 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 (10−6). So only for very small ξ do the ends of the small- and large-φ O 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 2 , 12 ) − N N 2 ≈ (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 λ 2 φ2R + φ2 v2 2 4 − ξ > 0, φ

v →∞

v 1 ξ =0

Figure 1.7: (ns, r) at N = 60 for the quartic potential Eq. (1.22) with positive ξ in the case with inﬂation at φ < v. ξ increases from right to left and on each curve v increases 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 λv4 constant V (0) = 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.

0.4 0.4 ξ λ 2 φ2R + φ2 v2 2 4 − 0.30.3 ξ < 0

φ >v 0.20.2 ξ =0

0.10.1 φ

0.00.0 0.920.92 0.940.94 0.960.96 0.980.98 1.001.00 1.021.02

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 | | −1/2 . ξ 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. The small-φ case is represented by the curves below| | the white circle, with ξ increasing from top to bottom. The cusps correspond to moderately but not too| large| v. For example, the curve of ξ = 10−6 enters the 1σ region at v 13; then continues to − ≈ the right and for 200 . v . 800 stays at the peak of the cusp which is very close to the white circle. For rather larger ξ the curves become almost ﬂat. For example, the curve of ξ = 10−2 crosses the 1σ| region| at v 7.8 with r 0.006 and continues to − ≈ ≈ the 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 (the star). Rather unexpectedly,→ | | this point coincides with the result of the Higgs inﬂation model 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 2 12 on their right end to the same point (ns, r) = (1 , 2 ) = (0.967, 0.003). This is − N N −1/2 the limit when the minimum at φ0 = v is very close to the singularity at φ = ξ , | | 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. V V

j j (a) (b)

λ 2 2 2 Figure 1.9: The potential 4 (φ v ) 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 similarity between these two potentials. However,− physical | | interpretation of these two potentials is very different. The value of the canonically normalized field ϕ in the model with ξ < 0 typically is very large. Meanwhile the value of this field for the model with ξ > 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:

2 λ 2 V (ϕ) = 2 1 exp (ϕ0 ϕ) , (1.27) 4ξ − − 3 − ! r where ϕ0 is the location of the minimum in terms of the canonical field ϕ. One 2 can show that during inflation, in the slow-roll regime, exp (ϕ0 ϕ) 1. − 3 − Therefore by replacing ϕ0 ϕ φ one can represent this potential q during inflation − → as −2 λ 2 V = 2 1 + exp φ . (1.28) 4ξ − 3 ! r At large ϕ, this potential coincides with the inflationary potential in the Higgs inflation scenario with the potential λ (φ2 v2)2 with v 1 and ξ 1 in Ref. [9]. To 4 − illustrate this statement, one may compare the potential λ (φ2 v2)2 with ξ < 0 in 4 − 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- 2 12 ters (ns, r) = (1 , 2 ) = (0.967, 0.003) also appears in Starobinsky model [16], see − N N [17, 18]. The value of λ can be determined from the observed value of amplitude of density −5 perturbations δρ 5 10 . 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

m2 2 In this chapter we studied the simplest chaotic inflation models with potentials 2 φ and λ (φ2 v2)2, with the field φ nonminimally coupled to gravity, for arbitrary values 4 − 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 Kählerpotential. 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 model m2 φ2 with ξ > 0, and by the model λ (φ2 v2)2 in two different regimes: ξ > 0, with 2 4 − inflation at φ > v, and ξ < 0, with inflation at φ < v. In particular, Fig. 4 shows that by adding a term ξ φ2R with ξ 10−2 one can 2 ∼ suppress the tensor/scalar ratio r for the model m2 φ2 from r 0.13 0.15 (depending 2 ∼ − on N) down to r 0.01. Similarly, investigation of the model λ (φ2 v2)2, which is ∼ 4 − ξ 2 ruled out by observations for v < 1, shows that by adding to this theory a term 2 φ R 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 λ (φ2 v2)2 for ξ 1, as shown in Fig. 1.6, and of the same 4 − 2 model with ξ < 0 in the limit ξ v 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 modifications 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 interpretation 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 Kählerpotential, some of these fields may become light along the inflationary trajectory. 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 probability 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 parameterization. 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 β 3H ∆tβ by a factor e , 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 measure 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 density. 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 corresponds to the special choice β = 0 in the class of measures where expansion is re- β warded by e3H ∆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 implementation. 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 8π −3 rate 3 H0 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+

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

Φ Φ- Φ+

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 practically 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 ˙ 0 2 velocity φ = V /3Hs, where 3H = Vs is the vacuum energy density during the slow − s s 0 roll inflation and Vs 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

√3Vs(φ± φ0) ti± = − (2.1) V 0 s

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

Hst exponentially, a(t) a0 e , 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.

N± The total size of the universe at the end of inflation a(ti±) a0 e 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 ∼ −1 −1/3 during inflation, a0 H , or it may take an intermediate value a0 Hs , starting ∼ s ∼ 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 particle 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

k 1 a 3 a 4 H2 + = ρ i + ρ i + Λ , (2.3) a2 3 mi a ri a 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 k long enough to render the term a2 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 1 √3αdα tc = 2 . (2.4) Λα4 + ρ α + ρ Z0 mt rt Here α is the ratio of the scale factor top 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 π 3 thot = , (2.5) c 2 Λ s| | and for the cold universe 2π 3 tcold = . (2.6) c 3 Λ s| | k Now let us restore the term a2 in our equations and consider an open universe, CHAPTER 2. 32

ignoring ρmt and ρrt. In this case,

3 topen = π . (2.7) c Λ s| |

hot More generally, if k = 0 the collapse time is bounded from below by tc and from above by tcold. If k = 1 the upper bound is topen and the lower bound is thot. In all c − c c 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 ± of galaxies across the whole N 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+ in the eternal model. A N− 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) β = (3H 2κ)V0(τ) . (2.8) dτ 0 − 4π β−4 Here κ = 3 H0 Γ is the decay rate per unit τ. This equation has a simple exponential solution: β V0(τ) = V0(0) exp[(3H 2κ)τ]. (2.9) 0 − 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) a 3 ± 3N± g± = κe V0(τ τg±), (2.10) dτ a − i± where τg± is the time from bubble nucleation to galaxy formation:

da τ = dτ = . (2.11) g Hβa Z Z

In the special case of proper time one ﬁnds τg = tg, whereas for the scale factor time ag one gets τg = N + log . ai Noting that ag/ai is -independent and N± = Hsti± we ﬁnd for the proper time ± (β = 1) V+(τ) 3H0 2κ lim = exp 3 − (N+ N−) . (2.12) τ→∞ V (τ) − H − − β=1 s

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:

V+(τ) lim = e2κ(N+−N−), (2.13) τ→∞ V (τ) − β=0

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±, V (τ) lim + = 1 (2.14) τ→∞ V (τ) − β=0

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 β the inflationary piece, a < ai gives a contribution (N+ N−)/H . Therefore one ≈ − s finds β V+(τ) 3H0 2κ lim = exp 3 −β (N+ N−) . (2.15) τ→∞ V−(τ) " − Hs ! − # 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:

3 dV±(τ) ag± = κe3N± V (τ), (2.16) dτ a 0 i± whose solution yields: V+(τ) lim = e3(N+−N−). (2.17) τ→∞ V (τ) − stationary

All results obtained above describe the ratio V+(τ) at a given time τ, if this time V−(τ) 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):

V+(τ) limτ→∞ V (τ) = 1, i.e. the final result does not depend on the duration of the − β=0 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,

3 + ai+ N = 3 = exp[3(N+ N−)]. (2.18) − a − N i− 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+ of all galaxies N− (1) N+ in φ+ to all of those in φ− by simply computing the ratio (1) of the galaxies in N− only one φ+ domain to those in one φ− domain. Ignoring geometric factors such as (1) 3 4π/3, one can write = n±(tg±) a . As we saw earlier, ng and ag/ai are - N± g± ± 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 = t dt , and the future light cone of the point where it hits the reheating surface − tc a(t) R CHAPTER 2. 37

t dt r = . At tg < tt the smaller of these two is the latter. Thus at the time of ti a(t) galaxyR formation tg there are

3 tg dt CD = ng ag N a(t) Zti ag 3 ag ai √3dα = ng (2.19) ai Λα4 + ρ α + ρ Z1 mi ri ! p galaxies within the causal diamond. All variables appearing in this expression are

-independent and thus so is CD. Furthermore, a uniform initial distribution of ± N 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 ± CD with a -independent proportionality constant. N ∝ N ± Therefore, for the causal diamond measure we ﬁnd

+ N = 1. (2.20) − N 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

3H0 Proper Time exp[3(N+ N−)] exp (N+ N−) − − Hs − Scale Factor exp[3(N+ N−)] h 1 i − exp[3(N+ N−)] if N is small enough Causal Diamond − 1 (1 if N is large enough 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 probability 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 inflation, 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

Figure 2.3: Evolution of the scale factor in a collapsing k = 0 FRW universe with respect to the proper time (top) and the scale factor time (bottom). In each pair of plots the left one has a slightly longer stage of slow role inflation than the right one. The area after reheating is painted. Notice almost exact symmetry of behavior of the scale factor during expansion versus contraction. Stars indicate the period where galaxies are produced and life as we know it is possible. This happens at the same time t but at very different times τ for the universes experiencing longer/shorter stages 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

8π H = Gρ (3.1) Λ 3 Λ r is the de Sitter Hubble expansion rate. For the observed value of ρΛ, the de Sitter −30 temperature is extremely low, TdS = 2.3 10 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 spacetime, 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 outnumbered 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 inflating 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 highly sophisticated computer. Thus, the normal observer and the Boltzmann brains can be thought of as a set of logically equivalent computers running the same program with the same data, and hence they behave identically until they are aﬀected by further input, which might be diﬀerent. Since the computer program cannot determine whether it is running inside the brain of one of the normal observers or one of the Boltzmann brains, any intelligent probabilistic prediction that the program makes about the next observation would be based on the assumption that it is equally likely to be running 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, but we will define it below in terms of the counting of events in spacetime, so the concept of weighting will not be relevant. The term “volume-weighted” is relevant when a measure is described as a prescription for defining the probability distribution for the value of a field. In Ref. [73], this measure 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 multiverse measures, including a causal patch measure [64], several measures associated with globally defined time coordinates [73, 96, 74, 85, 83], and the pocket-based measure [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 provide 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 production 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 brain problem 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 . (3.4) ≡ ; µ 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 expanding spacetimes — such as inflating regions and thermalized regions not long after reheating — it differs dramatically from the FRW scale factor as small-scale inhomogeneities 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 approximate our intuitive notion of FRW averaging and also extend into more complicated 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 arbitrary 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 J µ for the gas of test particles, and we can define ρ as the rest frame number density, i.e. the value of J 0 in the local Lorentz frame in which J i = 0, or equivalently ρ = √J 2. 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 predictions 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],

df i = M f , (3.5) dt ij j j X where the transition matrix Mij is given by

Mij = κij δij κri , (3.6) − r X 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

−4 κij = (4π/3)Hj 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 eigenvalues 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 inﬂating vacuum 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 be degenerate, in which case the asymptotic future is dominated by a linear combination of dominant eigenvectors that is determined by the initial state. If transitions that increase the vacuum energy density are included, then the landscape can be reducible only if it splits into several disconnected sectors. That situation was discussed in Appendix A of Ref. [95], where two alternative prescriptions were described. The ﬁrst prescription (preferred by the authors) leads to initial-state dependence only if two or more sectors have the same dominant eigenvalue q, while the second prescription always leads to initial-state dependence. CHAPTER 3. 50

be written in the form6 (0) −qt fj(t) = fj + sje + ..., (3.8)

(0) where the constant term fj 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 = κij sj , (3.9) − j X where κj is the total transition rate out of vacuum j,

κj κij . (3.10) ≡ i X

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 min κj . (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 upward 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

6 Mij is not necessarily diagonalizable, but Eq. (3.8) applies in any case. It is always possible to form a complete basis from eigenvectors and generalized eigenvectors, where generalized eigenvectors satisfy (M λI)ks = 0, for k > 1. The generalized eigenvectors appear in the solution with a time dependence− given by eλt times a polynomial in t. These terms are associated with the nonleading eigenvalues omitted from Eq. (3.8), and the polynomials in t will not change the fact that they are nonleading. 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/H2 ) [101]. Here and throughout the remainder of − D 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

(3−q) t Vj(t) = V0 sj e , (3.15)

3t where V0 is the volume of the initial hypersurface Σ and e 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 observers, 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 − − BB/ NO of Boltzmann brains to normal observers, the natural logarithm of this N N ratio will always include one term with a magnitude of at least 1016. Consequently, the presence or absence of any term in ln( BB/ NO) that is small compared to 1016 N N 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 BB/ NO such factors — although N N 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 −1 curvature-dominated period ∆τ H , slow-roll inflation inside the pocket gives Ne ∼ k 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 NO density of observers on this hypersurface nik . 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

(ik) −3 3(Ne+NO) V (tnuc) = H e w(tc tnuc Ne NO) , (3.17) O k − − − CHAPTER 3. 54

where the exponential gives the volume expansion factor coming from slow-roll infla- NO −3 tion and big bang evolution to the hypersurface Σ , and H w(tc tnuc Ne NO) k − − − 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

π ξ¯(t) π w(t) = sinh2(ξ) dξ = sinh 2ξ¯(t) 2ξ¯(t) , (3.18) 2 8 − Z0 ¯ 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

(ik) 3 dnnuc (tnuc) = (3/4π)Hk κik Vk(tnuc) dtnuc

3 (3−q) tnuc = (3/4π)Hk κiksk V0 e 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

tc−Ne−NO NO NO (ik) (ik) = n V (tnuc) dn (tnuc) Nik ik O nuc Z ∞ NO (3−q) tc −(3−q) z n κiksk V0 e w(z) e dz . (3.21) ≈ ik Z0 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 (1) prefactors, and also the factor eq(Ne+NO), since q is expected to be O −qtc extraordinarily small. We have kept e , 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 (1) prefactor that we will drop. O Finally,

NO NO (3−q) tc n κik sk V0 e . (3.22) Nik ≈ ik 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 BB Boltzmann brains are produced in vacuum j at a rate Γj per unit spacetime volume. The number of Boltzmann brains BB is then proportional to the total four-volume Nj in that vacuum. Imposing the cutoﬀ at scale-factor time tc, this four-volume is

tc tc (4) −1 = Vj(t) dτ = H Vj(t) dt Vj j Z 1 Z = H−1s V e(3−q) tc , (3.23) 3 q j j 0 − where we have used Eq. (3.15) for the asymptotic volume fraction. By setting dτ = −1 Hj 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 BB spirit, we will assume that the Boltzmann brain nucleation rate Γj 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

BB BB −1 (3−q) tc Γ H sj V0 e , (3.24) Nj ≈ j j where we have dropped the (1) numerical factor. O For completeness, we may want to consider the effects of early universe evolution 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 reheating, 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 BB BB some value Γreheat,ik which persists roughly for some time interval ∆τreheat,ik, produc- BB BB ing a density of Boltzmann brains equal to Γreheat,ik ∆τreheat,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

BB,reheat BB BB (3−q) tc Γ ∆τ κik sk V0 e . (3.25) Nik ≈ reheat,ik reheat,ik CHAPTER 3. 57

Thus, the dominance of normal observers is assured if

BB BB NO Γ ∆τ κik sk n κik sk . (3.26) reheat,ik reheat,ik ik i,k i,k X X 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

BB 3 BB j Hj κj sj N NO NO , (3.27) ≈ n κik sk N Pi, k ik where the summation in the numeratorP covers only the vacua in which Boltzmann brains can arise, the summation over i in the denominator covers only anthropic BB vacua, and the summation over k includes all of their possible parent vacua. κj is BB the dimensionless Boltzmann brain nucleation rate in vacuum j, related to Γj by

Eq. (3.7). The expression can be further simpliﬁed by dropping the factors of Hj and NO ni , 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

BB BB maxj κj sj N NO { } , (3.28) ∼ maxi κik sk N { k } where the sets of j and i are restricted as forP Eq. (3.27). NO NO 3 In dropping ni , we are assuming that ni Hi is roughly of order one, as deﬁned NO 3 at the beginning of this section. It is hard to know what a realistic value for ni Hi 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 NO vacua for which ni 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 NO 3 phrase “anthropic vacuum” to refer only to those vacua for which ni Hi is roughly of order one. For any landscape that satisfies Eq. (3.8), which includes any irreducible landscape, Eq. (3.28) can be simplified by using Eq. (3.9):

BB BB maxj κj sj N NO { } , (3.29) ∼ maxi (κi q) si N { − } 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 = κIF fF − (3.31) ˙ fI = κBI fI + κIF fF . − 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 eigenvectors,

q1 = κIF , s1 = (1,C) , (3.33) q2 = κBI , s2 = (0, 1) , where the eigenvectors are written in the basis s (sF , sI ) and ≡ κ C = IF . (3.34) κBI κIF − 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

−κIF t fF (t) = e , (3.35) −κIF t −κBI t fI (t) = C (e e ) . −

The asymptotic evolution depends on whether κIF < κBI (case I) or not (case II). In case I,

−κIF t f(t ) = s1e (κIF < κBI ) , (3.36) → ∞ where s1 is given in Eq. (3.33), while in case II

−κIF t −κBI t f(t ) = e , C e (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):

BB BB BB BB κ sI κ κIF κ N NO , (3.38) ∼ κIF sF ∼ κIF κBI κIF ∼ κBI N − 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

BB κBB (κIF −κBI )t N NO e . (3.39) ∼ κIF → ∞ N 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 domination 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 + κFI sI , − − (3.41) qsI = κI sI + κIF sF , − − 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) = κIF κFI . (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 exponentially 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

κIF κFI q κI , (3.43) ≈ − κIF κI − and κIF κI sI − sF sF . (3.44) ≈ κFI 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) −qt and fF (t) have the same asymptotic time dependence, e , 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 that s κ κ κ I = IF IF IF , (3.46) sF κBI q ≈ κBI κD ≈ κBI − − 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 overwhelmingly 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

BB BB BB I κI sI κI N NO . (3.47) ∼ κIF sF ∼ κBI NI 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

s κ κ κ F = FD FD FD , (3.48) sD κF q ≈ κF κD ≈ κF − − 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

BB BB BB D κD sD κD κF N NO . (3.49) ∼ κIF sF ∼ κFD κIF NI Since we expect that the dominant vacuum has very small vacuum energy, and hence BB NO a heavily suppressed upward transition rate κFD, the requirement that / be ND NI 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

BB BB BB D κD sD κD κF N NO , (3.50) ∼ κDF sF ∼ κFD κDF ND 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 BB dimensionless rate κD 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 BB κ κFD. D

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 1 s κ s (3.52) Ii κ IiFj Fj ≈ Ii j X and 1 sFj κFj D sD , (3.53) ≈ κFj where we have assumed that q κI , κF , as we expect for vacua other than the i j 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

BB max κBB s N{Ii} i Ii Ii NO {I } ∼ i maxi κIiFj s Fj N j n o XBB 1 1 maxi κI κIiFj κFj D sD i κ j κ Ii Fj ∼ X 1 maxi κI F κF D sD j i j κ j Fj κBBX κ Ii IiFj maxi κF D j j κIi κFj X , (3.54) ∼ κIiFj maxi κF D j j κFj X 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

BB BB κ sD ND D NO ∼ {Ii} maxi κI F sF N j i j j nXκBB o D , (3.55) ∼ κIiFj maxi κF D j j κFj X and, if vacuum D is anthropic, then the ratio of Boltzmann brains in vacuum D to normal observers in vacuum D is given by

BB BB D κD N NO κ . (3.56) ∼ DFj D κF D N j j κFj X 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 complicated 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 κI F κDF κF . That i j ∼ j ∼ j 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 BB/ NO, and may be dropped. Furthermore, the N N upward transition rates from the dominant vacuum D into Fj are all comparable to one another, as can be seen by writing [101]

AF D −SD κF D e j e , (3.57) j ∼

where AFj D is the action of the instanton responsible for the transition and SD is the action of the Euclideanized de Sitter 4-sphere,

8π2 SD = 2 . (3.58) HD

But generically AF D 1/ρF . If we assume that | j | ∼ j 1 1 < 1014 (3.59) ρ − ρ Fj Fk

for every pair of vacua Fj and Fk, then κFj D = κFkD up to a factor that can be ignored because it is roughly of order one. Thus, up to subleading factors, the transition rates 7 BB NO κF D cancel out in the ratio / . j N{Ii} N Returning to Eq. (3.54) and keeping only the leading factors, we have

BB κBB N{Ii} Ii NO max , (3.60) ∼ i κI N i 7 Depending on the range of vacua Fj that are considered, the bound of Eq. (3.59) may or may not be valid. If it is not, then the simplification of Eq. (3.60) below is not justified, and the original Eq. (3.54) has to be used. Of course one should remember that there was significant arbitrariness in the choice of 1014 in the definition of “roughly of order one.” 1014 was chosen to accommodate the largest estimate that we discuss in Sec. 3.5 for the Boltzmann brain nucleation 16 14 rate, ΓBB exp( 10 ). In considering the other estimates of ΓBB, one could replace 10 by a much 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) into BB κBB D D BB SD N NO κD e , (3.61) ∼ κup ∼ N where κup κF D is the upward transition rate out of the dominant vacuum. ≡ j j Thus, theP 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 dominant 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 (3.62) ← ←→ → -& %. S A, → 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

A 8The dropping of the factor e Fj D is a more reliable approximation in this case than it was in Eq. (3.60) above. In this case the factor e−SD does not cancel between the numerator and A denominator, so the factor e Fj D can be dropped if it is unimportant compared to e−SD . We of course 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 as 122 AFj D 1/ρFj is small compared to 10 , it seems safe to drop the factor e . 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,

out κ1 = κ21 + κ1 (3.63) out κ2 = κ12 + κ2 , (3.64)

out with κ12, κ21 κ . We assume as in previous sections that the rates for large 1,2 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 '

1 2 q κ1 + κ2 (κ1 κ2) + 4κ12κ21 . (3.65) ' 2 ± − h p i out Expanding the smaller root to linear order in κ1,2 gives

out out q α1κ + α2κ , (3.66) ' 1 2 where κ12 κ21 α1 , α2 . (3.67) ≡ κ12 + κ21 ≡ κ12 + κ21

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

sD α1 sD , sD α2 sD , (3.68) 1 ≈ 2 ≈

out where sD sD + sD , and the equations hold in the approximation that κ and the ≡ 1 2 1,2 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):

s α κ + α κ Fj 1 Fj D1 2 Fj D2 . (3.69) sD ≈ κFj

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 κFj D1 and κFj D2 . 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

s q S . (3.70) sD ≈ κS q −

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 landscape, note that Eqs. (3.54) and (3.55) are modiﬁed only by the substitution

κF D κ¯F D α1 κF D + α2 κF D . (3.71) j → j ≡ j 1 j 2

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 κ¯F D. ≡ j j If, however, we allowPD1 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 rateκ ¯Fj D 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 BB D vacua were mildly anthropic, however, then the stringent requirement κ κup D BB would have to be satisfied without resort to the simple solution κD = 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, however, 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 BB/ NO we assume that factors that are roughly of N N 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 obtain BB BB i κi N NO , (3.72) ∼ κi Ni 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 BB/ NO, dropping irrelevant factors as in Eq. (3.28), and then we can iterate the Nj Ni CHAPTER 3. 74

above inequality:

BB BB BB j κj sj κj sj N NO ∼ κik sk ≤ κi si Ni k κBB s P j j ≤ (κjsj)Bj→k Bk →k Bk →i 1 1 2 ··· n κBB 1 = j , (3.76) κj Bj→k Bk →k Bk →i 1 1 2 ··· n 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 BB by Boltzmann brains in vacuum j if κ /κj 1, the same criterion found for the j 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

κBB j 1 , (3.77) Bup κj 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, BB then Bup = κup/κj, and the above criterion simplifies to κ /κup 1. Second, we j 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 BB are not anthropic, a likely method for Eq. (3.77) to be satisfied would be for κj to vanish, as would happen if the vacuum j did not support the complex structures needed to form Boltzmann brains. BB The criterion above can be summarized by saying that if κ /(Bupκj) 1, then j the Boltzmann brains in vacuum j will be overwhelmingly outnumbered by the normal 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→k Bk →k Bk →i . (3.80) ≥ 1 1 2 ··· n 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- 14 tions, Zmax = exp(10 ). 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

m+1 (κjsj) (kAsA)Z . (3.82) ≤ max

Using the bounds from Eqs. (3.80) and (3.82), the Boltzmann brain ratio is bounded by

BB BB BB j κj sj κj sj N NO ∼ κik sk ≤ κi si Ni k Zm+1 κBB P max j . (3.83) ≤ BA→k Bk →k Bk →i κj 1 1 2 ··· n 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 suppressed transitions. If Bup denotes the product of branching ratios for all the suppressed 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 BB that κ /(Bup κj) 1, where Bup is the product of all suppressed branching ratios j 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 ≥ BB the multiverse avoids domination by Boltzmann brains in vacuum i if κ /κi 1. i 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 BB dominate if κ /(Bup κi) 1. i 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 BB κ Bupκi . (3.84) i

−SD However, as pointed out in Ref. [98], Bup e (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.

BB NO 100 16 calculation of / , but Zmax = exp(10 ) can be signiﬁcant. Thus, for our choices we can justify the droppingN N of (100) factors that are roughly of order one, but not more than that. For O choices appropriate to smaller estimates of ΓBB, however, the number of factors that can be dropped will 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

κBB j 1 . (3.85) κj

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 BoltzmannP brains in vacuum j are dominated by the normal observers in vacuum i is given by κBB j 1 , (3.88) Bup κj 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 assumptions 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 normal 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 BB NO κi are comparable to each other. With these assumptions, the estimate of / N N becomes very simple. Applying Eq. (3.9) to the ﬁrst transition of Eq. (3.89),

κk sk κk D sD κupsD , (3.90) 1 1 ∼ 1 ∼ 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 κk D is replaced byκ ¯k D α`κk D , where the D` are the components 1 1 ≡ ` 1 ` of the dominant vacuum system, and theP α` are deﬁned by generalizing Eqs. (3.67) 10 and (3.68). Applying Eq. (3.9) to the next transition, k1 k2 we ﬁnd →

κk sk = Bk →k κk sk + ... κk sk , (3.91) 2 2 1 2 1 1 ∼ 1 1 10In more detail, the concept of a dominant vacuum system is relevant when there is a set of vacua ` that can have rapid transitions within the set, but only very slow transitions connecting these vacua to the rest of the landscape. As a zeroth order approximation one can neglect all transitions connecting these vacua to the rest of the landscape, and assume that κ` q, so Eq. (3.9) takes the form κ` s` = B``0 κ`0 s`0 . `0 X Here B``0 κ``0 /κ`0 is the branching ratio within this restricted subspace, where κ` = `0 κ`0` is ≡ 0 summed only within the dominant vacuum system, so ` B``0 = 1 for all ` . B``0 is nonnegative, and if we assume also that it is irreducible, then the Perron-Frobenius theorem guarantees thatP it has a P nondegenerate eigenvector v` of eigenvalue 1, with positive components. From the above equation κ` s` v`, and then ∝ s` v` α` = = 0 . 0 v` `0 s` κ` κ`0 `0 P X CHAPTER 3. 81

where we have used the fact that Bk1→k2 is roughly of order one, and that the transition is signiﬁcant. Iterating, we have

κi si κk sk κup sD . (3.92) ∼ n n ∼

Since the expression on the right is independent of i, we conclude that under these assumptions 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) ∼

NO Using Eq. (3.22) and assuming as always that nik 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:

NO NO . (3.94) Nj ∼ Ni

The dominant vacuum could conceivably be anthropic, but we begin by considering 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

BB BB BB BB j κj sj κj sj κj N NO . (3.95) ∼ κik sk ∼ κisi ∼ κj Ni k Thus, given the assumptions describedP 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

κBB j 1 . (3.96) κj 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 observers 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

BB BB BB BB D κD sD κD sD κD N NO . (3.97) ∼ κik sk ∼ κisi ∼ κup Ni k Thus, for a single dominant vacuumP 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

κBB κBB D 1 or Di 1 . (3.98) κup κup

As discussed after Eq. (3.84), probably the only way to satisfy this condition is to BB require that κD = 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 κBB = 0. D 6 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 particular, 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 factor [62, 64],

−M/TdS ΓBB e , (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.

SBB Thus ΓBB should be multiplied by the number of microstates e corresponding to the macroscopic description, where SBB is the entropy of the Boltzmann brain. Thus we expect

−M/TdS SBB −F/TdS ΓBB e e = e , (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−1 , (3.101) Λ where the ﬁrst inequality assumes that Boltzmann brains cannot be black holes. The general situation, which allows for M R H−1, will be discussed in Appendix A ∼ ∼ Λ 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,

SBB we require that TdS mn. Thus, for these Boltzmann brains the entropy factor e 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

85 ΓBB exp( 10 ) , (3.103) ∼ −

30 −1 −1 12 where we have set M 10 kg and H = (2πTdS) 10 m. This nucleation rate ∼ Λ ∼ 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 construction that is roughly similar to our own brains. If we take M 1 kg and ∼ −1 −1 H = (2πTdS) 1 m, then the corresponding Boltzmann brain nucleation rate is Λ ∼

43 ΓBB exp( 10 ) . (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 −1 8 requires HΛ & 10 m, giving a Boltzmann brain nucleation rate

51 ΓBB exp( 10 ) . (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 computer, 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 storage device as a system with N possible microstates. Smax = ln N 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 imagine 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 B entropy S = ln(N/2 ) = Smax B ln 2. Motivated by this simple model, one defines − the information content of any macroscopic state of entropy S as the difference between 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 energy M 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−1, so that the system does not “know” about the horizon. A possible Λ generalization of the Bekenstein bound for R = (H−1) was proposed in Ref. [109]; O Λ 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−1, so that we can focus on the Λ 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 2πM ΓBB exp + SBB,max IBB , (3.108) ∼ − H − Λ 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 SBB,max ΓBB exp + SBB,max IBB . (3.109) ≤ − RH − Λ −1 Since R < HΛ , the expression above is maximized by taking SBB,max equal to its smallest possible value, which is IBB. Finally, we have

IBB ΓBB exp . (3.110) ≤ −RH Λ 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 derived some incorrect conclusions, such as the statement that the largest Boltzmann brain nucleation rate ΓBB consistent with the Bekenstein bound is attained only when the radius R approaches the Schwarzschild radius RSch. This in turn led to the conclusion that the maximum rate allowed by the −2IBB 2 2 Bekenstein bound is e , which can be achieved only if M = IBB/(9πG) and HΛ = 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 if R H (see Eq. (3.110)), and otherwise grows with R/R (see Appendix A). However, once one Λ Sch is forced to consider values of R RSch, then other issues become relevant. How can the system be stabilized against the de Sitter expansion? Can the Bekenstein bound really be saturated for a system with large entropy, especially if it is dilute? In this version of the work we have added a discussion of these issues. We thank R. Bousso, B. Freivogel, and I. Yang for pointing out the error in our earlier statement of the Bekenstein bound. CHAPTER 3. 88

saturates the Bekenstein bound, with IBB = SBB,max = 2πMR. Simultaneously, we −1 should make RHΛ as large as possible, which means taking our assumption R H Λ to the boundary of its validity. Thus we write the Boltzmann brain production rate

−aIBB ΓBB e , (3.111) ≤

−1 where a (RHΛ) , 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

−SBB brain production rate, ΓBB e , 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. −1 So far, the conditions to reach the upper bound in Eq. (3.111) are R = (aHΛ) −1 ∼ (H ) and IBB = Smax,BB = SBek. However these are not enough to ensure that a O Λ Boltzmann brain of size R H−1 is stable and can actually compute. Indeed, the ∼ Λ time required for communication between two parts of a Boltzmann brain separated by a distance (H−1) is at least comparable to the Hubble time. If the Boltzmann O Λ 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 −1 R. This, together with our goal to keep R as close as possible to HΛ , leads to the following condition: M R H−1 . (3.112) ∼ ∼ Λ If the Bekenstein bound is saturated, this leads to the following relations between

IBB, HΛ, and M: −1 −2 IBB MR MH H . (3.113) ∼ ∼ Λ ∼ Λ 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]:

3 σ IBB H . (3.114) ∼ Λ

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

−aIBB −2 produced at a rate approaching e only if IBB H . For a given value of ∼ Λ 16 IBB, say IBB 10 (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 ∼ 16 Boltzmann brains with IBB 10 contained inside a domain wall in a vacuum with ∼ −120 Λ 10 , 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 in Ref. [108]. Nonetheless, if black holes can compute, our conclusions would not be changed, provided that the Bekenstein bound can be saturated for the near-black hole computers that we discuss. At this level of approximation, there would be no signiﬁcant diﬀerence between a black hole computer and 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

−aIBB production rate, ΓBB e . 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 = (1). Meanwhile, as we shall argue O 16 below, the required value of Smax should be greater than 10 . 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 16 smaller than HΛ is greater than IBB & 10 . 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 3 3 4 4/3 entropy Smax = (RT ) and energy E = R T = Smax/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:

S4/3 −E/HΛ SBB max ΓBB e e exp , (3.115) ∼ ∼ −HΛR! where we have neglected the Boltzmann brain entropy factor, since SBB Smax 4/3 −1 ≤ Smax. This probability is maximized by taking R H , which yields ∼ Λ

4/3 −Smax ΓBB . e . (3.116)

In case the full information capacity of the gas is used, one can also write

−I4/3 ΓBB . e BB . (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 ∼ 14 15 16 that its information content might be roughly IBB 10 –10 . (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 information 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):

−1016 ΓBB . e . (3.118)

14 11 11 Note that the speciﬁcation of one out of 10 neurons requires log2 10 = 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

−1021 ΓBB . e . (3.119)

Obviously, there are many uncertainties involved in the numerical estimates of the 16 required value of IBB. Our estimate IBB 10 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 amounts of information that are not part of the schematic “circuit diagram” of the brain. Thus there may be some significance to the fact that a billion years of evolution on Earth has not produced a human brain with fewer than about 1027 particles, and hence of order 1027 units of entropy. In counting the information in the synapses, for example, we counted only the information needed to specify which neurons are connected to which, but nothing about the actual path of the axons and dendrites that complete the connections. These are nothing like nearest-neighbor couplings, but instead axons from a single neuron can traverse large fractions of the brain, resulting in an extremely intertwined network [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 the neurons will in many cases form closed loops. A specification of the connections would presumably require a topological winding number for every pair of closed loops in the network. The number of bits required to specify these winding numbers would be proportional to the square of the number of closed loops, which would be proportional to the square of the number of synapses. Thus, the 28 structural information could be something like Istruct b 10 , where b is a proportionality constant that is probably a few orders of magnitude less than∼ 1.× In estimating the resulting suppression of the nucleation rate, there is one further complication: since structural information of this sort presumably has no influence on brain function, these choices would contribute to the multiplicity of Boltzmann brain microstates, thereby multiplying the nucleation rate by eIstruct . There would still −(a−1)Istruct be a net suppression, however, with Eq. (3.111) leading to ΓBB e , where a is generically greater 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 numerical 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 ΓBB ∆τ BB nNO (3.120) reheat,ik reheat,ik ik BB holds for every pair of vacua i and k, where Γreheat,ik is the peak Boltzmann brain nucleation rate in a pocket of vacuum i that forms in a parent vacuum of type k, BB NO ∆τreheat,ik is the proper time available for such nucleation, and nik 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, BB here Γreheat,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. 16 For a near-black hole computer with IBB SBB,max 10 , the minimum mass is ≈ ≈ 600 grams. If we assume that the reheat temperature is no more than the reduced 18 −6 Planck mass, mPlanck 1/√8πG 2.4 10 GeV 4.3 10 gram, we find ≡ ≈ × ≈ × BB 8 that Γ < exp √2IBB exp( 10 ). Although this is not nearly as much reheat − ∼ − 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- 4/3 ticles, and a total energy of E = Smax/R, so the Boltzmann suppression factor is 4/3 exp Smax/ (RTreheat) . The Boltzmann brain production can occur at any time dur- − ing theh reheating process,i so there is nothing wrong with considering Boltzmann brain 16 production in our universe at the present time. For Treheat = 2.7 K and Smax = 10 , 4/3 −1 this formula implies that the exponent has magnitude 1 for R = SmaxT 200 reheat ≈ 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

24π2 κ e−SD = exp . (3.121) & − V dS CHAPTER 3. 98

120 Here SD is the entropy of dS space. For our vacuum, SD 10 , 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 = 1 supergravity [131, 132, N 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 corresponds 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]:

8π2α κ exp 2 , (3.123) ∼ −3m3/2 ! 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 = (1), but it can O also be much greater or much smaller than (1). The mass m3/2 is set by the scale O of SUSY breaking, 2 4 3m3/2 = ΛSUSY , (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:

24π2α κ exp , (3.125) ∼ −Λ4 SUSY

4 Note that in the KKLT theory, ΛSUSY corresponds to the depth of the AdS vacuum CHAPTER 3. 100

before the uplifting, so that

24π2α κ exp . (3.126) ∼ − VAdS | | 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 (1) TeV O leads to the lower bound 4 5 ΛSUSY & 10 –10 GeV , (3.127) which can be reached, e.g., in the models of conformal gauge mediation [141]. This implies that for our vacuum

56 60 κour exp( 10 )– exp( 10 ) . (3.128) & − −

Using Eq. (3.99), the Boltzmann brain nucleation rate in our universe exceeds the −9 lower bound of the above inequality only if M . 10 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

m 2 M H−1 3/2 Λ 109α . (3.129) 10−2 eV 1 kg 108 m & Note that H−1 108 m corresponds to the smallest de Sitter radius for which the Λ ∼ tidal force on a 10 cm brain does not exceed the gravitational force on the surface of −2 4 the earth, while m3/2 10 eV corresponds to ΛSUSY 10 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 2 be made much less than unity, then we only require m3/2 (10 ) eV to avoid & O 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- 4 11 struct [143]. If instead of ΛSUSY 10 GeV one uses ΛSUSY 10 GeV, as in models ∼ ∼ 3 with gravity mediation, one finds m3/2 10 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 unstable [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 to D3 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 −1 as a transition from an initial state of empty de Sitter space with horizon radius HΛ 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

2GM ds2 = 1 H2 r2 dt2 − − r − Λ 2GM −1 + 1 H2 r2 dr2 + r2 dΩ2 . (A.1) − r − Λ

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

2 π + ξ RSch = cos , √3 H 3 Λ (A.2) 2 π ξ RdS = cos − , √3 H 3 Λ where

cos ξ = 3√3 GMHΛ . (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:

−1 M Mmax = (3√3GHΛ) . (A.4) ≤

Eqs. (A.2) and (A.3) can be inverted to express M and HΛ in terms of the horizon radii:

1 2 2 2 = RSch + RdS + RSchRdS (A.5) HΛ R M = dS 1 H2 R2 (A.6) 2G − Λ dS R = Sch 1 H2 R2 . (A.7) 2G − Λ Sch We relate the Boltzmann brain nucleation rate to the decrease in total entropy ∆S caused by the the nucleation process,

−∆S ΓBB e , (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 π −2 π 2 ∆S = H R + SBB . (A.9) G Λ − G dS APPENDIX A. 107

Note that for small M one can expand ∆S to ﬁnd

2πM 2 ∆S = SBB + (GM ) , (A.10) HΛ − O 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

π 1 2 π 2 S SD R = R + RSchRdS , (A.11) ≤ ≡ G H2 − dS G Sch Λ 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

π 2 S SH R . (A.12) ≤ ≡ G dS

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

π Sm RSchRdS (A.13) ≡ G 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

π S SI RSchR, (A.14) ≤ ≡ G 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 π −2 2 ∆S = H R SBB,max + IBB , (A.15) G Λ − dS − which can be combined with SBB,max SI to give ≤

π −2 2 ∆S H R RSch R + IBB , (A.16) ≥ G Λ − dS − APPENDIX A. 109

Figure A.1: Graph shows the holographic bound, the D-bound, and the m-bound for the entropy of an object that ﬁlls de Sitter space out to the horizon. The holographic and D– bounds are each shown as broken lines in the region where they are superseded by 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

π ∆S RSch (RSch + RdS R) + IBB . (A.17) ≥ G −

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

π 2 ∆S R + IBB . (A.18) ≥ G Sch

∆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

˜ −1 1 3I ξsat = 3 sin − , (A.19) p 2 ! where 2 ˜ IBB GHΛ I = IBB (A.20) ≡ SdS π 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

I˜ 1 + I˜ Msat = , (A.21) p2GHΛ 1 + I˜ 1 3I˜ RSch,sat = − − , (A.22) p 2HpΛ 1 3I˜+ 1 + I˜ RdS,sat = − . (A.23) p 2HΛp

Combining these results with Eq. (A.18), one has for this case (R = RdS) the bound

∆S 1 + I˜ 1 + I˜ 1 3I˜ . (A.24) − ˜ − IBB ≥ p 2I p

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. → → ˜ −IBB The limiting case of IBB 0, with a nucleation rate of order e , has some → peculiar features that are worth mentioning. The nucleation rate describes the nucleation 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 unsuppressed, 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 2 of freedom associated with the de Sitter entropy SdS = π/(GHΛ). As a second possibility for the radius R, we can consider the case of strong gravitational binding, R RSch, as discussed following Eq. (3.111). For this case the → bound (A.17) becomes π ∆S RSchRdS + IBB . (A.25) ≥ G

(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

˜ π −1 3I ξsat = 3 sin . (A.26) 2 − p2 !

The saturating value of the mass and the corresponding values of the Schwarzschild radius and de Sitter radius are given by

I˜ 1 I˜ − Msat = , (A.27) p 2GH Λ I˜ RSch,sat = , (A.28) pHΛ 4 3I˜ I˜ RdS,sat = − − . (A.29) p 2HΛ p Using these relations to evaluate ∆S from Eq. (A.25), one ﬁnds

∆S 4 3I˜+ I˜ = − , (A.30) IBB p 2 I˜ p p 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 considerablyp 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 for Boltzmann brains is proportional to e−∆S. All curves are based on the I-bound, as discussed in the text, but they diﬀer by their assumptions about the size R of the Boltzmann 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

GM 1/3 R = . (A.31) equil H2 Λ Boltzmann brains within this radius bound would not be pulled by the de Sitter expansion, 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 1 2A(I˜) π −1 ξsat = 3 sin − , (A.32) 2 − q 2    APPENDIX A. 113

where sin−1 1 27 I˜3 A(I˜) sin − . (A.33) ≡  3 

The saturating value of the mass and the Schwarzschild and de Sitter radii are given by

√3[1 + A(I˜)] 1 2A(I˜) Msat = − , (A.34) 9GHqΛ 1 2A(I˜) RSch,sat = − , (A.35) q √3 HΛ √3 3 3 + 2A(I˜) 1 2A(I˜) − − R = r . dS,sat 6H q Λ (A.36)

The equilibrium radius itself is given by

1/6 1/3 1 2A(I˜) 1 + A(I˜) − Requil,sat = . (A.37) h i√3HhΛ i

Using these results with Eq. (A.17), ∆S is found to be bounded by

3 1 2A(I˜) 3 + 2A(I˜) 2A(I˜) + 1 ∆S − − = r , (A.38) IBB 6I˜ 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 approximately 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 gravitationally 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

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