The Measure Problem in Eternal Inflation by Andrea De Simone Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of MASSACHUSETTS INSTITUTE Doctor of Philosophy OF TECHN'*OLOGY at the N)V 8 210 MASSACHUSETTS INSTITUTE OF TECHNOLOGY LIBPARIES June 2010 ARCHNES @ Massachusetts Institute of Technology 2010. All rights reserved. 11 1. t( \ - A uthor ....................... Department of Physics May 4, 2010 Certified by.......... Alan H. Guth Victor F. Weisskopf Professor of Physics Thesis Supervisor / Accepted by ............... ............ Krishn'aaja opal Professor of Physics Associate Department Head for Education The Measure Problem in Eternal Inflation by Andrea De Simone Submitted to the Department of Physics on May 4, 2010, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Inflation is a paradigm for the physics of the early universe, and it has received a great deal of experimental support. Essentially all inflationary models lead to a regime called eternal inflation, a situation where inflation never ends globally, but it only ends locally. If the model is correct, the resulting picture is that of an infinite number of "bubbles", each of them containing an infinite universe, and we would be living deep inside one of these bubbles. This is the multiverse. Extracting meaningful predictions on what physics we should expect to observe within our bubble encounters severe ambiguities, like ratios of infinities. It is therefore necessary to adopt a prescription to regulate the diverging spacetime volume of the multiverse. The problem of defining such a prescription is the measure problem, which is a major challenge of theoretical cosmology. In this thesis, I shall describe the measure problem and propose a promising candidate solution: the scale-factor cutoff measure. I shall study this measure in detail and show that it is free of the pathologies many other measures suffer from. In particular, I shall discuss in some detail the "Boltzmann brain" problem in this measure and show that it can be avoided, provided that some plausible requirements about the landscape of vacua are satisfied. Two interesting applications of the scale-factor cutoff measure are investigated: the probability distributions for the cosmological constant and for the curvature pa- rameter. The former turns out to be such that the observed value of the cosmological constant is quite plausible. As for the curvature parameter, its distribution using the scale-factor measure predicts some chance to detect a nonzero curvature in the future. Thesis Supervisor: Alan H. Guth Title: Victor F. Weisskopf Professor of Physics Acknowledgments This thesis is the result of years of studies and it would have never been possible without the presence and the help of several people I have been fortunate to be in contact with. I would like to express my deep gratitude to my advisor, Prof. Alan Guth, for his wise guidance, tireless patience and constant support. Then I wish to thank my family who has always been a firm point of reference and a safe harbor. Many thanks go to my friends for their reliable company and the joyful moments spent together. Finally, I am indebted to Elisa, to whom this thesis is dedicated, for incessantly injecting enthusiasm and happiness into my life. A Elisa Contents 1 Introduction 8 1.1 Inflation ..... .. 8 1.1.1 Standard cosmology and its shortcomings 8 1.1.2 The inflationary Universe ...... ... ... 10 1.2 Eternal Inflation and the Measure Problem . ........ 11 1.3 Structure of the thesis ..... ....... ... 14 2 The Scale-factor Cutoff Measure 2.1 Some proposed measures ... ........ 2.2 The Scale-factor cutoff measure ....... 2.2.1 Definitions ............... 2.2.2 General features of the multiverse .. 2.2.3 The youngness bias ........ .. .. ... 2.2.4 Expectations for the density contrast Q and the curvature pa- ram eter Q . ......... .... 23 2.3 Conclusions .................. ....... ....... 25 3 The Distribution of the Cosmological Constant 26 3.1 Model assumptions .................... ...... 26 3.2 Distribution of A using a pocket-based measure . .. ........ 28 3.3 Distribution of A using the scale-factor cutoff measure . .. ... ... 32 3.4 Distribution of A using the "local" scale-factor cutoff measure 3.5 Conclusions ............................ 4 The Distribution of the Curvature Parameter 4.1 Background.. .. ........... ............ .. 4.1.1 The Geometry of pocket nucleation ...... ........ 4.1.2 Structure formation in an open FRW Universe ....... 4.2 The Distribution of Qk.. ....... ....... ........ 4.3 Anthropic considerations and the "prior" distribution of Ne.... .. 4.4 The distribution of Qk using the "local" scale-factor cutoff measure 4.5 Conclusions... ........ ............ ........ 5 The Boltzmann Brain problem 62 5.1 The abundance of normal observers and Boltzmann brains . 65 5.2 Toy landscapes and the general conditions to avoid Boltzmann brain domination. ..................... ... 69 5.2.1 The FIB landscape ........................ 69 5.2.2 The FIB landscape with recycling ................ 71 5.2.3 A more realistic landscape ....... ............. 72 5.2.4 A further generalization ..................... 74 5.2.5 A dominant vacuum system ................... 76 5.2.6 General conditions to avoid Boltzmann brain domination .. 79 5.3 Boltzmann brain nucleation and vacuum decay rates ......... 87 5.3.1 Boltzmann brain nucleation rate............ ... 87 5.3.2 Vacuum decay rates........ ......... 96 5.4 Conclusions................ ........ ... 100 6 Conclusions 102 A Independence of the Initial State 104 B The Collapse Density Threshold 6, 106 C Boltzmann Brains in Schwarzschild-de Sitter Space 112 Bibliography 120 List of Figures 1-1 Schematic form of a potential giving rise to false-vacuum eternal inflation. 12 1-2 Schematic form of a potential giving rise to slow-roll eternal inflation. 13 3-1 The normalized distribution of A for A > 0, for the pocket-based measure. 30 3-2 The normalized distribution of A for the pocket-based measure. .. 31 3-3 The normalized distribution of A for A > 0, for the scale-factor cutoff m easure. ................................. 33 3-4 The normalized distribution of A, for the scale-factor cutoff measure. 34 3-5 The normalized distribution of A, for the scale-factor cutoff measure, for different Ar and MG- .... ...... -------------. 35 3-6 The normalized distribution of A for A > 0, for the "local" scale-factor cutoff m easure. ..................... ......... 37 3-7 The normalized distribution of A, for the "local" scale-factor cutoff measure. ........... ....................... 38 3-8 The normalized distribution of A, for the "local" scale-factor cutoff measure, for different Ar and MG .. .. - -----. .... 39 4-1 The potential V(<p) describing the parent vacuum, slow-roll inflation potential, and our vacuum. ........................ 44 4-2 The geometry near the bubble boundary. ................ 45 4-3 The collapse fraction Fe(Qk) and mass function Fm(Qk). ....... 49 4-4 The relevant portion of the distribution of Q0, along with a simple approximation. ........... .................. 52 4-5 The distribution dP(k)/dk. ....................... 55 4-6 The distribution dP(Ne)/dN assuming a flat "prior" for N ...... 56 4-7 The distribution dP(k)/dk using scale-factor times t' and t. 59 B-i The collapse density thresholds 6 (for A > 0) and 6- (for A < 0), as functions of tim e. .................. 110 C-1 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 ........ 115 C-2 The ratio of AS to IBB . .......... ------------. 117 Chapter 1 Introduction This Chapter is devoted to introduce the problem which is the subject of the present thesis. After briefly recalling some basic material of standard cosmology (see e.g. Ref. [1] for a comprehensive account), we shall introduce inflation and review its successes in Section 1.1, and then we shall turn to discuss a generic feature of infla- tionary models known as "eternal inflation" (Section 1.2). Finally, in Section 1.3 we shall describe how the rest of the thesis is organized. 1.1 Inflation Inflation [2, 3, 4] is one of the basic ideas of modern cosmology and has become a paradigm for the physics of the early universe. In addition to solving the shortcomings of the standard Big Bang theory, inflation has received a great deal of experimental support, for example it provided successful predictions for observables like the mass density of the Universe and the fluctuations of the cosmic microwave background ra- diation. Before discussing inflation in more detail, let us first review some background material about standard cosmology, which serves also to introduce the notation, and outline its major shortcomings. 1.1.1 Standard cosmology and its shortcomings The standard cosmology is based upon the spatially homogeneuos and isotropic Uni- verse described by the Friedmann-Robertson-Walker (FRW) metric: 2 2 2 2 ds = dt - a(t) 2 (d02 + sin 2d(2) where a(t) is the cosmic scale factor and the curvature signature is k = -1, 0, 1 for spaces of constant negative, zero, positive spatial curvature, respectively. The evolution of the scale factor is governed by the Einstein equations, where the energy- momentum tensor is assumed to be that of a perfect fluid with total energy density Ptot and pressure p. One of the Einstein equations takes the simple form (known as the Friedmann equation): 2 2 8,rG k H _ Ptot 2 , (1.2) a