Boltzmann Brains and the Scale- Factor Cutoff Measure of the Multiverse

Boltzmann brains and the scale- factor cutoff measure of the multiverse

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As Published http://dx.doi.org/10.1103/PhysRevD.82.063520

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/60398

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 82, 063520 (2010) Boltzmann brains and the scale-factor cutoff measure of the multiverse

Andrea De Simone,1 Alan H. Guth,1 Andrei Linde,2,3 Mahdiyar Noorbala,2 Michael P. Salem,4 and Alexander Vilenkin4 1Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics, Stanford University, Stanford, California 94305, USA 3Yukawa Institute of Theoretical Physics, Kyoto University, Kyoto, Japan 4Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA (Received 25 July 2010; published 14 September 2010) To make predictions for an eternally inflating ‘‘multiverse,’’ one must adopt a procedure for regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has emerged: normal observers—who evolve in pocket universes cooling from hot big bang conditions—must not be vastly outnumbered by ‘‘Boltzmann brains’’—freak observers that pop in and out of existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all but vacuum energy has redshifted away, rather than the rich structure that we observe. Using the scale-factor cutoff measure, we calculate the ratio of Boltzmann brains to normal observers. We find the ratio to be finite, and give an expression for it in terms of Boltzmann brain nucleation rates and vacuum decay rates. We discuss the conditions that these rates must obey for the ratio to be acceptable, and we discuss estimates of the rates under a variety of assumptions.

DOI: 10.1103/PhysRevD.82.063520 PACS numbers: 98.80.Cq

I. INTRODUCTION defines a Boltzmann brain. The important point, however, is that is always nonzero. The simplest interpretation of the observed accelerating BB De Sitter space is eternal to the future. Thus, if the expansion of the Universe is that it is driven by a constant accelerating expansion of the Universe is truly driven by vacuum-energy density , which is about 3 times greater the energy density of a stable vacuum state, then Boltzmann than the present density of nonrelativistic matter. While brains will eventually outnumber normal observers, no ordinary matter becomes more dilute as the Universe ex- matter how small the value of [4–8] might be. pands, the vacuum-energy density remains the same, and in BB To define the problem more precisely, we use the term another 10 109 yrs or so the Universe will be completely ‘‘normal observers’’ to refer to those that evolve as a result dominated by vacuum energy. The subsequent evolution of of nonequilibrium processes that occur in the wake of the the Universe is accurately described as de Sitter (dS) space. hot big bang. If our Universe is approaching a stable It was shown by Gibbons and Hawking [1] that an de Sitter spacetime, then the total number of normal ob- observer in de Sitter space would detect thermal radiation servers that will ever exist in a fixed comoving volume of the with a characteristic temperature T ¼ H =2, where dS Universe is finite. On the other hand, the cumulative number sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of Boltzmann brains grows without bound over time, grow- 8 e3Ht H ¼ G (1) ing roughly as the volume, proportional to .When 3 extracting the predictions of this theory, such an infinite preponderance of Boltzmann brains cannot be ignored. is the de Sitter Hubble expansion rate. For the observed For example, suppose that some normal observer, at value of , the de Sitter temperature is extremely low, some moment in her lifetime, tries to make a prediction 30 TdS ¼ 2:3 10 K. Nevertheless, complex structures about her next observation. According to the theory there will occasionally emerge from the vacuum as quantum would be an infinite number of Boltzmann brains, distrib- fluctuations, at a small but nonzero rate per unit spacetime uted throughout the spacetime, that would happen to share volume. An intelligent observer, like a human, could be exactly all her memories and thought processes at that one such structure. Or, short of a complete observer, a moment. Since all her knowledge is shared with this set disembodied brain may fluctuate into existence, with a of Boltzmann brains, for all she knows she could equally pattern of neuron firings creating a perception of being likely be any member of the set. The probability that she is on Earth and, for example, observing the cosmic micro- a normal observer is then arbitrarily small, and all predic- wave background radiation. Such freak observers are tions would be based on the proposition that she is a collectively referred to as ‘‘Boltzmann brains’’ [2,3]. Of Boltzmann brain. The theory would predict, therefore, course, the nucleation rate BB of Boltzmann brains is that the next observations that she will make, if she extremely small, its magnitude depending on how one survives to make any at all, will be totally incoherent,

1550-7998=2010=82(6)=063520(30) 063520-1 Ó 2010 The American Physical Society ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) with no logical relationship to the world that she thought measure [18,27], which is an improved version of the she knew. (While it is of course true that some Boltzmann proper-time cutoff measure. However, the stationary mea- brains might experience coherent observations, for ex- sure, as well as the pocket-based measure, is afflicted with ample, by living in a Boltzmann solar system, it is easy a runaway problem, suggesting that we should observe to show that Boltzmann brains with such dressing would be extreme values (either very small or very large) of the vastly outnumbered by Boltzmann brains without any primordial density contrast Q [29,30] and the gravitational coherent environment.) Thus, the continued orderliness constant G [31], while these parameters appear to sit of the world that we observe is distinctly at odds with the comfortably in the middle of their respective anthropic predictions of a Boltzmann-brain-dominated cosmology.1 ranges [32,33]. Some suggestions to get around this issue This problem was recently addressed by Page [7], who have been described in Refs. [30,33–35]. In addition, the concluded that the least unattractive way to produce more pocket-based measure seems to suffer from the Boltzmann normal observers than Boltzmann brains is to require that brain problem. The comoving coordinate measure [11,36] our vacuum should be rather unstable. More specifically, it and the causal-patch measures [23,24] are free from these should decay within a few Hubble times of vacuum-energy problems, but have an unattractive feature of depending domination, that is, in 20 109 yrs or so. sensitively on the initial state of the multiverse. This does In the context of inflationary cosmology, however, this not seem to mix well with the attractor nature of eternal problem acquires a new twist. Inflation is generically eter- inflation: the asymptotic late-time evolution of an eternally nal, with the physical volume of false-vacuum inflating inflating universe is independent of the starting point, so it regions increasing exponentially with time, and ‘‘pocket seems appealing for the measure to maintain this property. universes’’ like ours constantly nucleating out of the false Since the scale-factor cutoff measure2 [12–14,16,17,37] vacuum. In an eternally inflating multiverse, the numbers has been shown to be free of all of the above issues [38], of normal observers and Boltzmann brains produced over we consider it to be a promising candidate for the measure the course of eternal inflation are both infinite. They can be of the multiverse. meaningfully compared only after one adopts some pre- As we have indicated, the relative abundance of normal scription to regulate the infinities. observers and Boltzmann brains depends on the choice of The problem of regulating the infinities in an eternally measure over the multiverse. This means the predicted inflating multiverse is known as the measure problem [9], ratio of Boltzmann brains to normal observers can be and has been under discussion for some time. It is crucially used as yet another criterion to evaluate a prescription to important in discussing predictions for any kind of obser- regulate the diverging volume of the multiverse: regulators vation. Most of the discussion, including the discussion in predicting that normal observers are greatly outnumbered this paper, has been confined to the classical approxima- by Boltzmann brains should be ruled out. This criterion has tion. While one might hope that someday there will be an been studied in the context of several multiverse measures, answer to this question based on a fundamental principle including a causal-patch measure [8], several measures [10], most of the work on this subject has focused on associated with globally defined time coordinates proposing plausible measures and exploring their proper- [17,18,27,39,40], and the pocket-based measure [41]. In ties. Indeed, a number of measures have been proposed this work, we apply this criterion to the scale-factor cutoff [11–27], and some of them have already been disqualified, measure, extending the investigation that was initiated in as they make predictions that conflict with observations. Ref. [17]. We show that the scale-factor cutoff measure In particular, if one uses the proper-time cutoff measure gives a finite ratio of Boltzmann brains to normal observ- [11–15], one encounters the ‘‘youngness paradox,’’ pre- ers; if certain assumptions about the landscape are valid, dicting that humans should have evolved at a very early the ratio can be small.3 cosmic time, when the conditions for life were rather The remainder of this paper is organized as follows. In hostile [28]. The youngness problem, as well as the Sec. II we provide a brief description of the scale-factor Boltzmann brain problem, can be avoided in the stationary cutoff and describe salient features of the multiverse under

1Here we are taking a completely mechanistic view of the 2This measure is sometimes referred to as the volume- brain, treating it essentially as a highly sophisticated computer. weighted scale-factor cutoff measure, but we will define it below Thus, the normal observer and the Boltzmann brains can be in terms of the counting of events in spacetime, so the concept of thought of as a set of logically equivalent computers running the weighting will not be relevant. The term ‘‘volume-weighted’’ is same program with the same data, and hence they behave relevant when a measure is described as a prescription for identically until they are affected by further input, which might defining the probability distribution for the value of a field. In be different. Since the computer program cannot determine Ref. [17], this measure is called the ‘‘pseudo-comoving volume- whether it is running inside the brain of one of the normal weighted measure.’’ observers or one of the Boltzmann brains, any intelligent proba- 3In a paper that appeared simultaneously with version 1 of this bilistic prediction that the program makes about the next obser- paper, Raphael Bousso, Ben Freivogel, and I-Sheng Yang inde- vation would be based on the assumption that it is equally likely pendently analyzed the Boltzmann brain problem for the scale- to be running on any member of that set. factor cutoff measure [42].

063520-2 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) the lens of this measure. In Sec. III we calculate the ratio of Eq. (2) can then be defined as a / 1=3, where is the Boltzmann brains to normal observers in terms of multi- density of the dust, and the cutoff is triggered when drops verse volume fractions and transition rates. The volume below some specified level. fractions are discussed in Sec. IV, in the context of toy Although the local scale-factor time closely follows landscapes, and the section ends with a general description the Friedmann-Robertson-Walker (FRW) scale factor in of the conditions necessary to avoid Boltzmann brain expanding spacetimes—such as inflating regions and ther- domination. The rate of Boltzmann brain production and malized regions not long after reheating—it differs dra- the rate of vacuum decay play central roles in our calcu- matically from the FRW scale factor as small-scale lations, and these are estimated in Sec. V. Concluding inhomogeneities develop during matter domination in uni- remarks are provided in Sec. VI. verses like ours. In particular, the local scale-factor time nearly grinds to a halt in regions that have decoupled from II. THE SCALE-FACTOR CUTOFF the Hubble flow. It is not clear whether we should impose this particular cutoff, which would essentially include the Perhaps the simplest way to regulate the infinities of entire lifetime of any nonlinear structure that forms before eternal inflation is to impose a cutoff on a hypersurface of the cutoff, or impose a cutoff on some nonlocal time constant global time [12–16]. One starts with a patch of a variable that more closely tracks the FRW scale factor.4 spacelike hypersurface somewhere in an inflating region There are a number of nonlocal modifications of scale- of spacetime, and follows its evolution along the congru- factor time that both approximate our intuitive notion of ence of geodesics orthogonal to . The scale-factor time is FRW averaging and also extend into more complicated defined as geometries. One drawback of the nonlocal approach is t ¼ lna; (2) that no single choice looks more plausible than the others. For instance, one nonlocal method is to define the factor H where a is the expansion factor along the geodesics. The in Eq. (3) by spatial averaging of the quantity HðxÞ in scale-factor time is related to the proper time by Eq. (4). A complete implementation of this approach, dt ¼ Hd; (3) however, involves many seemingly arbitrary choices regarding how to define the hypersurfaces over which where H is the Hubble expansion rate of the congruence. HðxÞ should be averaged, so here we set this possibility The spacetime region swept out by the congruence will typically expand to unlimited size, generating an infinite aside. A second, simpler method is to use the local scale- number of pockets. (If the patch does not grow without factor time defined above, but to generate a new cutoff limit, one chooses another initial patch and starts again.) hypersurface by excluding the future light cones of all The resulting four-volume is infinite, but we cut it off at points on the original cutoff hypersurface. In regions with nonlinear inhomogeneities, the underdense regions some fixed scale-factor time t ¼ tc. To find the relative probabilities of different events, one counts the numbers of will be the first to reach the scale-factor cutoff, after which such events in the finite spacetime volume between and they quickly trigger the cutoff elsewhere. The resulting cutoff hypersurface will not be a surface of constant the t ¼ tc hypersurface, and then takes the limit tc !1. The term ‘‘scale factor’’ is often used in the context of FRW scale factor, but the fluctuations of the FRW scale homogeneous and isotropic geometries; yet on very large factor on this surface should be insignificant. and on very small scales the multiverse may be very As a third and final example of a nonlocal modification inhomogeneous. A simple way to deal with this is to take of scale-factor time, we recall the description of the local the factor H in Eq. (3) to be the local divergence of the scale-factor cutoff in terms of the density of a dust of test four-velocity vector field along the congruence of particles. Instead of such a dust, consider a set of massless geodesics orthogonal to , test particles, emanating uniformly in all directions from each point on the initial hypersurface . We can then HðxÞð1=3Þu ;: (4) construct the conserved number density current J for When more than one geodesic passes through a point, the the gas of test particles, and we can define as the rest 0 scale-factor time at that point may be taken to be the frame number density, i.e. the value of J in the localpffiffiffiffiffi smallest value among the set of geodesics. In collapsing Lorentz frame in which Ji ¼ 0, or equivalently ¼ J2. regions HðxÞ is negative, in which case the corresponding Defining a / 1=3, as we did for the dust of test particles, geodesics are continued unless or until they hit a singularity. we apply the cutoff when the number density drops This ‘‘local’’ definition of scale-factor time has a simple below some specified level. Since null geodesics are barely geometric meaning. The congruence of geodesics can be perturbed by structure formation, the strong perturbations thought of as representing a ‘‘dust’’ of test particles scattered uniformly on the initial hypersurface . As one 4The distinction between these two forms of scale-factor time moves along the geodesics, the density of the dust in the was first pointed out by Bousso, Freivogel, and Yang in orthogonal plane decreases. The expansion factor a in Ref. [42].

063520-3 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) inherent in the local definition of scale-factor time are vacua, which collapse in a big crunch, and stable zero- avoided. Nonetheless, we have not studied the properties energy vacua. It was shown in Ref. [21] that all of the other of this definition of scale-factor time, and they may lead to eigenvalues of Mij have negative real parts. Moreover, the complications. Large-scale anisotropic flows in the gas of eigenvalue with the smallest (by magnitude) real part is test particles can be generated as the particles stream into pure real; we call it the ‘‘dominant eigenvalue’’ and denote expanding bubbles from outside. Since the null geodesics it by q (with q>0). Assuming that the landscape is do not interact with matter except gravitationally, these irreducible, the dominant eigenvalue is nondegenerate. anisotropies will not be damped in the same way as they In that case the probabilities defined by the scale-factor would be for photons. The large-scale flow of the gas will cutoff measure are independent of the initial state of the not redshift in the normal way, either; for example, if the multiverse, since they are determined by the dominant test particles in some region of a FRW universe have a eigenvector.5 nonzero mean velocity relative to the comoving frame, the For an irreducible landscape, the late-time asymptotic expansion of the universe will merely reduce the energies solution of Eq. (5) can be written in the form6 of all the test particles by the same factor, but will not cause ð0Þ qt the mean velocity to decrease. Thus, the detailed predic- fjðtÞ¼fj þ sje þ ...; (8) tions for this definition of scale-factor cutoff measure ð0Þ remain a matter for future study. where the constant term fj is nonzero only in terminal The local scale-factor cutoff and each of the three vacua and sj is proportional to the eigenvector of Mij nonlocal definitions correspond to different global-time corresponding to the dominant eigenvalue q, with the parametrizations and thus to different spacetime measures. constant of proportionality determined by the initial distri- In general, they make different predictions for physical bution of vacua on . It was shown in Ref. [21] that sj 0 observables; however, with regard to the relative number for terminal vacua, and sj > 0 for nonterminal vacua, as is of normal observers and Boltzmann brains, their predic- needed for Eq. (8) to describe a non-negative volume tions are essentially the same. For the remainder of this fraction, with a nondecreasing fraction assigned to any paper we refer to the generic nonlocal definition of scale- terminal vacuum. factor time, for which we take FRW time as a suitable By inserting the asymptotic expansion (8) into the dif- approximation. Note that the use of local scale-factor time ferential equation (5) and extracting the leading asymptotic would make it slightly easier to avoid Boltzmann brain behavior for a nonterminal vacuum i, one can show that domination, since it would increase the count of normal X observers while leaving the count of Boltzmann brains ði qÞsi ¼ ijsj; (9) essentially unchanged. j To facilitate later discussion, let us now describe some where j is the total transition rate out of vacuum j, general properties of the multiverse. The volume fraction X fi occupied by vacuum i on constant scale-factor time j ij: (10) slices can be found from the rate equation [43], i df X i ¼ M f ; (5) ij j 5 dt j In this work we assume that the multiverse is irreducible; that is, any metastable inflating vacuum is accessible from any other where the transition matrix Mij is given by such vacuum via a sequence of tunneling transitions. Our results, X however, can still be applied when this condition fails. In that Mij ¼ ij ij ri; (6) case the dominant eigenvalue can be degenerate, in which case r the asymptotic future is dominated by a linear combination of dominant eigenvectors that is determined by the initial state. If and ij is the transition rate from vacuum j to vacuum i per transitions that increase the vacuum-energy density are included, Hubble volume per Hubble time. This rate can also be then the landscape can be reducible only if it splits into written several disconnected sectors. That situation was discussed in Appendix A of Ref. [38], where two alternative prescriptions 4 ij ¼ð4=3ÞH ij; (7) were described. The first prescription (preferred by the authors) j leads to initial-state dependence only if two or more sectors have the same dominant eigenvalue q, while the second prescription where ij is the bubble nucleation rate per unit spacetime always leads to initial-state dependence. volume and Hj is the Hubble expansion rate in vacuum j. 6 Mij is not necessarily diagonalizable, but Eq. (8) applies in The solution of Eq. (5) can be written in terms of the any case. It is always possible to form a complete basis from eigenvectors and eigenvalues of the transition matrix Mij. eigenvectors and generalized eigenvectors, where generalized k It is easily verified that each terminal vacuum is an eigenvectors satisfy ðM IÞ s ¼ 0, for k>1. The generalized eigenvectors appear in the solution with a time dependence given eigenvector with eigenvalue zero. We here define ‘‘termi- by et times a polynomial in t. These terms are associated with nal vacua’’ as those vacua j for which ij ¼ 0 for all i. the nonleading eigenvalues omitted from Eq. (8), and the poly- Thus the terminal vacua include both negative-energy nomials in t will not change the fact that they are nonleading.

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The positivity of si for nonterminal vacua then implies where V0 is the volume of the initial hypersurface and rigorously that q is less than the decay rate of the e3t is the volume expansion factor. The volume growth in slowest-decaying vacuum in the landscape: Eq. (15) is (very slightly) slower than e3t due to the constant q minf g: (11) loss of volume from transitions to terminal vacua. Note that min j even though upward transitions from the dominant vacuum Since ‘‘upward’’ transitions (those that increase the are strongly suppressed, they play a crucial role in populat- energy density) are generally suppressed, we can gain ing the landscape [44]. Most of the volume in the asymp- some intuition by first considering the case in which all totic solution of Eq. (15) originates in the dominant vacuum upward transition rates are set to zero. (Such a landscape is D, and ‘‘trickles’’ to the other vacua through a series of reducible, so the dominant eigenvector can be degenerate.) transitions starting with at least one upward jump. In this case Mij is triangular, and the eigenvalues are precisely the decay rates i of the individual states. The III. THE ABUNDANCE OF NORMAL OBSERVERS dominant eigenvalue q is then exactly equal to min. AND BOLTZMANN BRAINS If upward transitions are included but assumed to have a very low rate, then the dominant eigenvalue q is approxi- Let us now calculate the relative abundances of mately equal to the decay rate of the slowest-decaying Boltzmann brains and normal observers, in terms of the vacuum [44], vacuum transition rates and the asymptotic volume fractions. q min: (12) Estimates for the numerical values of the Boltzmann The slowest-decaying vacuum (assuming it is unique) is brain nucleation rates and vacuum decay rates will be the one that dominates the asymptotic late-time volume of discussed in Sec. V, but it is important at this stage to the multiverse, so we call it the dominant vacuum and be aware of the kind of numbers that will be considered. denote it by D. Hence, We will be able to give only rough estimates of these rates, but the numbers that will be mentioned in Sec. V will range q D: (13) from expð10120Þ to expð1016Þ. Thus, when we calculate The vacuum decay rate is typically exponentially sup- the ratio N BB=N NO of Boltzmann brains to normal pressed, so for the slowest-decaying vacuum we expect it observers, the natural logarithm of this ratio will always to be extremely small, include one term with a magnitude of at least 1016. q 1 : (14) Consequently, the presence or absence of any term in N BB N NO 16 Note that the corrections to Eq. (13) are comparable to the lnð = Þ that is small compared to 10 is of no upward transition rate from D to higher-energy vacua, but relevance. We therefore refer to any factor f for which for large energy differences this transition rate is sup- j lnfj < 1014 (16) 2 2 pressed by the factor expð8 =HDÞ [45]. Here and throughout the remainder of this paper we use reduced as ‘‘roughly of order one.’’ In the calculation of N BB=N NO such factors—although they may be minus- Planck units, where 8G ¼ c ¼ kB ¼ 1. We shall argue in Sec. V that the dominant vacuum is likely to have a very cule or colossal by ordinary standards—can be ignored. It will not be necessary to keep track of factors of 2, ,or low-energy density, so the correction to Eq. (13) is very 8 small even compared to q. even 1010 . Dimensionless coefficients, factors of H, and A possible variant of this picture, with similar conse- factors coming from detailed aspects of the geometry are quences, could arise if one assumes that the landscape unimportant, and in the end all of these will be ignored. includes states with nearby energy densities for which We nonetheless include some of these factors in the inter- the upward transition rate is not strongly suppressed. In mediate steps below simply to provide a clearer description that case there could be a group of vacuum states that of the calculation. undergo rapid transitions into each other, but very slow We begin by estimating the number of normal observers transitions to states outside the group. The role of the that will be counted in the sample spacetime region speci- dominant vacuum could then be played by this group of fied by the scale-factor cutoff measure. Normal observers states, and q would be approximately equal to some arise during the big bang evolution in the aftermath of appropriately averaged rate for the decay of these states slow-roll inflation and reheating. The details of this evolu- to states outside the group. Under these circumstances q tion depend not only on the vacuum of the pocket in could be much less than min. An example of such a question, but also on the parent vacuum from which it situation is described in Sec. IV E. nucleated [46]. That is, if we view each vacuum as a local In the asymptotic limit of late scale-factor time t, the minimum in a multidimensional field space, then the physical volumes in the various nonterminal vacua are dynamics of inflation, in general, depend on the direction given by from which the field tunneled into the local minimum. We therefore label pockets with two indices ik, indicating ð3qÞt VjðtÞ¼V0sje ; (15) the pocket and parent vacua, respectively.

063520-5 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) Z To begin, we restrict our attention to a single tcNeNO N NO ¼ nNO VðikÞðt ÞdnðikÞðt Þ ‘‘anthropic’’ pocket—i.e., one that produces normal ik ik O nuc nuc nuc observers—which nucleates at scale-factor time tnuc. The Z 1 NO ð3qÞtc ð3qÞz internal geometry of the pocket is that of an open FRW nik ikskV0e wðzÞe dz: (21) universe. We assume that, after a brief curvature- 0 1 dominated period Hk , slow-roll inflation inside In the first expression we have ignored the (very small) the pocket gives Ne e-folds of expansion before thermal- probability that pockets of type ik may transition to other ization. Furthermore, we assume that all normal observers vacua during slow-roll inflation or during the subsequent arise at a fixed number NO of e-folds of expansion after period NO of big bang evolution. In the second line, we thermalization. (Note that Ne and NO are both measured have changed the integration variable to z ¼ tc tnuc along FRW comoving geodesics inside the pocket, which Ne NO (reversing the direction of integration) and have do not initially coincide with, but rapidly asymptote to, the dropped the Oð1Þ prefactors, and also the factor eqðNeþNOÞ, ‘‘global’’ geodesic congruence that originated outside the since q is expected to be extraordinarily small. We have pocket.) We denote the fixed-internal-time hypersurface on kept eqtc , since we are interested in the limit t !1. NO c which normal observers arise by , and call the average We have also kept the factor eqz long enough to verify that NO density of observers on this hypersurface nik . the integral converges with or without the factor, so we can NO The hypersurface would have infinite volume, due carry out the integral using the approximation q 0, to the constant expansion of the pocket, but this divergence resulting in an Oð1Þ prefactor that we will drop. is regulated by the scale-factor cutoff tc, because the global Finally, scale-factor time t is not constant over the NO hypersurface. For the pocket described above, the regulated physi- N NO NO ð3qÞtc ik nik ikskV0e : (22) cal volume of NO can be written as 3ðN þN Þ ðikÞ Note that the expansion factor e e O in Eq. (17) was V ðt Þ¼H3e3ðNeþNOÞwðt t N N Þ; (17) O nuc k c nuc e O canceled when we integrated over nucleation times, illus- where the exponential gives the volume expansion factor trating the mild youngness bias of the scale-factor cutoff coming from slow-roll inflation and big bang evolution to measure. The expansion of the Universe is canceled, so NO 3 the hypersurface , and Hk wðtc tnuc Ne NOÞ objects that form at a certain density per physical volume describes the comoving volume of the part of the NO in the early Universe will have the same weight as objects hypersurface that is underneath the cutoff. The function that form at the same density per physical volume at a later wðtÞ was calculated, for example, in Refs. [39,47], and was time, despite the naive expectation that there is more applied to the scale-factor cutoff measure in Ref. [48]. Its volume at later times. detailed form will not be needed to determine the answer To compare, we now need to calculate the number of up to a factor that is roughly of order one, but to avoid Boltzmann brains that will be counted in the sample space- mystery we mention that wðtÞ can be written as time region. Boltzmann brains can be produced in any Z anthropic vacuum, and presumably in many nonanthropic ðtÞ wðtÞ¼ sinh2ðÞd ¼ ½sinhð2ðtÞÞ 2ðtÞ; vacua as well. Suppose Boltzmann brains are produced in 2 8 BB 0 vacuum j at a rate j per unit spacetime volume. The (18) N BB number of Boltzmann brains j is then proportional to the total four-volume in that vacuum. Imposing the cutoff where ðtc tnuc Ne NOÞ is the maximum value of the Robertson-Walker radial coordinate that lies under at scale-factor time tc, this four-volume is the cutoff. If the pocket universe begins with a moderate Z t Z t ð Þ V ð4Þ c 1 c period of inflation [ exp Ne 1] with the same vacuum j ¼ VjðtÞd ¼ Hj VjðtÞdt energy as outside, then 1 1 t=2 1 ð3qÞtc ðtÞ2cosh ðe Þ: (19) ¼ H sjV0e ; (23) 3 q j Equation (17) gives the physical volume on the NO hypersurface for a single pocket of type ik, which nucleates where we have used Eq. (15) for the asymptotic volume fraction. By setting d ¼ H1dt, we have ignored the time at time tnuc. The number of ik pockets that nucleate j between time tnuc and tnuc þ dtnuc is dependence of HðÞ in the earlier stages of cosmic evolution, assuming that only the late-time de Sitter evolution ðikÞ 3 dnnucðtnucÞ¼ð3=4ÞHkikVkðtnucÞdtnuc is relevant. In a similar spirit, we will assume that the Boltzmann brain nucleation rate BB can be treated as ¼ð3=4ÞH3 s V eð3qÞtnuc dt ; (20) j k ik k 0 nuc time independent, so the total number of Boltzmann brains where we use Eq. (15)togiveVkðtnucÞ. The total number of nucleated in vacua of type j, within the sample volume, is normal observers in the sample region is then given by

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N BB BB 1 ð3qÞtc where the summation in the numerator covers only the j j Hj sjV0e ; (24) vacua in which Boltzmann brains can arise, the summation where we have dropped the Oð1Þ numerical factor. over i in the denominator covers only anthropic vacua, For completeness, we may want to consider the effects and the summation over k includes all of their possible BB of early universe evolution on Boltzmann brain production, parent vacua. j is the dimensionless Boltzmann brain BB effects which were ignored in Eq. (24). We will separate nucleation rate in vacuum j, related to j by Eq. (7). The the effects into two categories: the effects of slow-roll expression can be further simplified by dropping the NO inflation at the beginning of a pocket universe, and the factors of Hj and ni , which are roughly of order one, as effects of reheating. defined by Eq. (16). We can also replace the sum over j in To account for the effects of slow-roll inflation, we argue the numerator by the maximum over j, since the sum is at that, within the approximations used here, there is no need least as large as the maximum term and no larger than the for an extra calculation. Consider, for example, a pocket maximum term times the number of vacua. Since the universe A which begins with a period of slow-roll inflation number of vacua (perhaps 10500) is roughly of order one, during which HðÞHslow roll ¼ const. Consider also a the sum over j is equal to the maximum up to a factor that pocket universe B, which throughout its evolution has is roughly of order one. We similarly replace the sum over i H ¼ Hslow roll, and which by hypothesis has the same for- in the denominator by its maximum, but we choose to leave mation rate, Boltzmann brain nucleation rate, and decay the sum over k. Thus we can write rates as pocket A. Then clearly the number of Boltzmann brains formed in the slow-roll phase of pocket A will be BB maxfj sjg N BB j smaller than the number formed throughout the lifetime P ; (28) NO of pocket B. Since we will require that generic bubbles of N maxf k ikskg i type B do not overproduce Boltzmann brains, there will be no need to worry about the slow-roll phase of bubbles of where the sets of j and i are restricted as for Eq. (27). type A. NO NO 3 In dropping ni , we are assuming that ni Hi is roughly To estimate how many Boltzmann brains might form as of order one, as defined at the beginning of this section. It is a consequence of reheating, we can make use of the NO 3 hard to know what a realistic value for ni Hi might be, as calculation for the production of normal observers the evolution of normal observers may require some highly described above. We can assume that the Boltzmann brain improbable events. For example, it was argued in Ref. [49] nucleation rate has a spike in the vicinity of some particular that the probability for life to evolve in a region of the hypersurface in the early Universe, peaking at some value size of our observable Universe per Hubble time may be as BB which persists roughly for some time interval reheat;ik low as 101000. But even the most pessimistic estimates BB reheat;ik, producing a density of Boltzmann brains equal cannot compete with the small numbers appearing in BB BB to reheat;ikreheat;ik. This spatial density is converted into a estimates of the Boltzmann brain nucleation rate, and total number for the sample volume in exactly the same hence by our definition they are roughly of order one. way that we did for normal observers, leading to Nonetheless, it is possible to imagine vacua for which NO ni might be negligibly small, but still nonzero. We shall N BB;reheat BB BB ð3qÞtc ik reheat;ikreheat;ikikskV0e : (25) ignore the normal observers in these vacua; for the remainder of this paper we will use the phrase ‘‘anthropic NO 3 Thus, the dominance of normal observers is assured if vacuum’’ to refer only to those vacua for which ni Hi is roughly of order one. X X For any landscape that satisfies Eq. (8), which includes BB BB NO reheat;ikreheat;ikiksk nik iksk: (26) any irreducible landscape, Eq. (28) can be simplified by i;k i;k using Eq. (9):

BB If Eq. (26) did not hold, it seems likely that we would suffer maxfj sjg N BB j from Boltzmann brain problems regardless of our measure. ; (29) NO N maxfði qÞsig We leave numerical estimates for Sec. V, but we will see i that Boltzmann brain production during reheating is not a danger. where the numerator is maximized over all vacua j that Ignoring the Boltzmann brains that form during reheat- support Boltzmann brains, and the denominator is maxi- ing, the ratio of Boltzmann brains to normal observers can mized over all anthropic vacua i. be found by combining Eqs. (22) and (24), giving In order to learn more about the ratio of Boltzmann P brains to normal observers, we need to learn more about BB 3 BB N j Hj j sj P ; (27) the volume fractions sj, a topic that will be pursued in the NO NO N i;k nik iksk next section.

063520-7 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) IV. MINI-LANDSCAPES AND THE GENERAL C ¼ IF : (34) CONDITIONS TO AVOID BOLTZMANN BI IF BRAIN DOMINATION Suppose that we start in the false vacuum F at t ¼ 0, In this section we study a number of simple models of i.e. fðt ¼ 0Þ¼ð1; 0Þ. Then the solution of the FIB rate the landscape, in order to build intuition for the volume equation, Eq. (31), is fractions that appear in Eqs. (28) and (29). The reader ð Þ¼ IFt ð Þ¼ ð IFt BItÞ uninterested in the details may skip the pedagogical fF t e ;fI t C e e : (35) examples given in Secs. IVA, IV B, IV C, IV D, and The asymptotic evolution depends on whether IF <BI IV E, and continue with Sec. IV F, where we state the (case I) or not (case II). In case I, general conditions that must be enforced in order to avoid f s IFt Boltzmann brain domination. ðt !1Þ¼ 1e ðIF <BIÞ; (36) s where 1 is given in Eq. (33), while in case II, A. The FIB landscape f ðt !1Þ¼ðeIFt; jCjeBItÞð < Þ: (37) Let us first consider a very simple model of the land- BI IF scape, described by the schematic In the latter case, the inequality of the rates of decay for F ! I ! B; (30) the two volume fractions arises from the reducibility of the FIB landscape, stemming from our ignoring upward tran- where F is a high-energy false vacuum, I is a positive- sitions from I to F. energy anthropic vacuum, and B is a terminal vacuum. This For case I (IF <BI), we find the ratio of Boltzmann model, which we call the FIB landscape, was analyzed in brains to normal observers by evaluating Eq. (28) for the Ref. [21] and was discussed in relation to the abundance of asymptotic behavior described by Eq. (36): Boltzmann brains in Ref. [17]. As in Ref. [17], we assume N BB BBs BB BB that both Boltzmann brains and normal observers reside I IF ; (38) NO only in vacuum I. N IFsF IF BI IF BI Note that the FIB landscape ignores upward transitions where we drop IF compared to BI in the denominator, from I to F. The model is constructed in this way as an as we are only interested in the overall scale of the solution. initial first step, and also in order to more clearly relate our We find that the ratio of Boltzmann brains to normal analysis to that of Ref. [17]. Although the rate of upward observers is finite, depending on the relative rate of transitions is exponentially suppressed relative to the other Boltzmann brain production to the rate of decay of rates, its inclusion is important for the irreducibility of the vacuum I. Meanwhile, in case II (where < ) we find landscape, and hence the nondegeneracy of the dominant BI IF eigenvalue and the independence of the late-time asymp- N BB BB eðIFBIÞt !1: (39) NO totic behavior from the initial conditions of the multiverse. N IF The results of this subsection will therefore not always conform to the expectations outlined in Sec. II, but this In this situation, the number of Boltzmann brains over- shortcoming is corrected in the next subsection and all whelms the number of normal observers; in fact the ratio subsequent work in this paper. diverges with time. We are interested in the eigenvectors and eigenvalues of The unfavorable result of case II stems from the fact the rate equation, Eq. (5). In the FIB landscape the rate that, in this case, the volume of vacuum I grows faster than equation gives that of vacuum F. Most of this I volume is in large pockets that formed very early, and this volume dominates because _ _ f F ¼IFfF; fI ¼BIfI þ IFfF: (31) the F vacuum decays faster than I and is not replenished due to the absence of upward transitions. This leads to We ignore the volume fraction in the terminal vacuum as it Boltzmann brain domination, in agreement with the is not relevant to our analysis. Starting with the ansatz, conclusion reached in Ref. [17]. Thus, the FIB landscape f ðtÞ¼seqt; (32) analysis suggests that Boltzmann brain domination can be avoided only if the decay rate of the anthropic vacuum is we find two eigenvalues of Eqs. (31). These are, with their larger than both the decay rate of its parent false vacuum F corresponding eigenvectors, and the rate of Boltzmann brain production. Moreover, the FIB analysis suggests that Boltzmann brain domination in s q1 ¼ IF; 1 ¼ð1;CÞ;q2 ¼ BI; the multiverse can be avoided only if the first of these (33) s conditions is satisfied for all vacua in which Boltzmann 2 ¼ð0; 1Þ; brains exist. This is a very stringent requirement, since where the eigenvectors are written in the basis s ðsF;sIÞ low-energy vacua like I typically have lower decay rates and than high-energy vacua (see Sec. V). We shall see,

063520-8 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) however, that the above conditions are substantially re- in a realistic landscape this is not likely to be the case. To laxed in more realistic landscape models. see how it changes the situation to have a nonanthropic vacuum as the dominant one, we consider the model B. The FIB landscape with recycling A D $ F ! I ! B; (45) The FIB landscape of the preceding section is reducible, since vacuum F cannot be reached from vacuum I. We can which we call the ‘‘ADFIB landscape.’’ Here, D is the make it irreducible by simply allowing upward transitions, dominant vacuum and A and B are both terminal vacua. The vacuum I is still an anthropic vacuum, and the vacuum F $ I ! B: (40) F has large, positive vacuum energy. As explained in This ‘‘recycling FIB’’ landscape is more realistic than the Sec. V, the dominant vacuum is likely to have very small original FIB landscape, because upward transitions out of vacuum energy; hence we consider that at least one upward positive-energy vacua are allowed in semiclassical quan- transition (here represented as the transition to F)is tum gravity [45]. The rate equation of the recycling FIB required to reach an anthropic vacuum. landscape gives the eigenvalue system, Note that the ADFIB landscape ignores the upward transition rate from vacuum I to F; however, this is qsF ¼IFsF þ FIsI; qsI ¼IsI þ IFsF; exponentially suppressed relative to the other transition (41) rates pertinent to I and, unlike the situation in Sec. IVA, ignoring the upward transition does not significantly affect where þ is the total decay rate of vacuum I, I BI FI our results. The important property is that all vacuum as defined in Eq. (10). Thus, the eigenvalues q and q 1 2 fractions have the same late-time asymptotic behavior; correspond to the roots of this property is assured whenever there is a unique domi- ðIF qÞðI qÞ¼IFFI: (42) nant vacuum, and all inflating vacua are accessible from the dominant vacuum via a sequence of tunneling transi- Further analysis is simplified if we note that upward tions. The uniformity of asymptotic behaviors is sufficient transitions from low-energy vacua like ours are very to imply Eq. (9), which suggests immediately that strongly suppressed, even when compared to the other exponentially suppressed transition rates, i.e. sI IF IF IF FI ¼ ; (46) IF;BI. We are interested mostly in how this small cor- sF BI q BI D BI rection modifies the dominant eigenvector in the case where we used q D AD þ FD, and assumed that where BI <IF (case II), which led to an infinite ratio of Boltzmann brains to normal observers. To the lowest D BI. order in , we find This holds even if the decay rate of the anthropic FI vacuum I is smaller than that of the false vacuum F. IFFI q I (43) Even though the false vacuum F may decay rather IF I quickly, it is constantly being replenished by upward and transitions from the slowly decaying vacuum D, which overwhelmingly dominates the physical volume of the IF I sI sF sF: (44) multiverse. Note that, in light of these results, our con- FI straints on the landscape to avoid Boltzmann brain domi- The above equation is a consequence of the second of nation are considerably relaxed. Specifically, it is no longer Eqs. (41), but it also follows directly from Eq. (9), which required that the anthropic vacua decay at a faster rate than their parent vacua. Using Eq. (46) with Eq. (28), the ratio holds in any irreducible landscape. In this case fIðtÞ and qt I fFðtÞ have the same asymptotic time dependence, / e , of Boltzmann brains to normal observers in vacuum is found to be so the ratio fIðtÞ=fFðtÞ approaches a constant limit, sI=sF R. However, due to the smallness of FI, this ratio N BB BBs BB is extremely large. Note that the ratio of Boltzmann brains I I I I : (47) N NO to normal observers is proportional to R. Although it is also I IFsF BI proportional to the minuscule Boltzmann brain nucleation If Boltzmann brains can also exist in the dominant rate (estimated in Sec. V), the typically huge value of R vacuum D, then they are a much more severe problem. will still lead to Boltzmann brain domination (again, see By applying Eq. (9) to the F vacuum, we find Sec. V for relevant details). But the story is not over, since sF the recycling FIB landscape is still far from realistic. ¼ FD FD FD ; (48) sD F q F D F C. A more realistic landscape where F ¼ IF þ DF, and where we have assumed that In the recycling model of the preceding section, the D F. The ratio of Boltzmann brains in vacuum D to anthropic vacuum I was also the dominant vacuum, while normal observers in vacuum I is then

063520-9 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) N BB BB BB D D sD D F 1 : (49) sF F DsD; (53) N NO j j I IFsF FD IF Fj where we have assumed that q ; , as we expect for Since we expect that the dominant vacuum has very small Ii Fj vacuum energy, and hence a heavily suppressed upward vacua other than the dominant one. Using these results with N BB N NO Eq. (28), the ratio of Boltzmann brains in vacua I to transition rate FD, the requirement that D = I be i small could be a very stringent one. Note that compared to normal observers in vacua Ii is given by sD, both sF and sI are suppressed by the small factor FD; N BB maxfBBs g however, the ratio s =s is independent of this factor. Ii Ii I F fIig i P Since s is so large, one should ask whether Boltzmann NO D N maxf j I F sF g fIig i i j j brain domination can be more easily avoided by allowing n P o vacuum D to be anthropic. The answer is no, because the max BB 1 1 s Ii j IiFj FjD D production of normal observers in vacuum D is propor- i PIi Fj f 1 s g tional [see Eq. (22)] to the rate at which bubbles of D max j IiFj FjD D i Fj nucleate, which is not large. D dominates the spacetime BB P Ii IiFj volume due to slow decay, not rapid nucleation. If we max j FjD i nIi Fj o assume that D is anthropic and restrict Eq. (28) to vacuum P ; (54) IiFj D, we find using Eq. (48) that max j FjD i Fj N BB BBs BB D D D D F ; (50) where the denominators are maximized over the restricted N NO D DFsF FD DF set of anthropic vacua i (and the numerators are maximized without restriction). The ratio of Boltzmann brains in the so again the ratio is enhanced by the extremely small dominant vacuum (vacuum D) to normal observers in upward tunneling rate FD in the denominator. vacua Ii is given by Thus, in order to avoid Boltzmann brain domination, it seems we have to impose two requirements: (1) the N BB BB BB D PD sD n D o Boltzmann brain nucleation rate in the anthropic vacuum NO P ; (55) N f g IiFj fI g max j IiFj sFj I must be less than the decay rate of that vacuum, and i max j FjD i i Fj (2) the dominant vacuum D must either not support Boltzmann brains at all, or must produce them with a BB and, if vacuum D is anthropic, then the ratio of Boltzmann dimensionless rate D that is small even compared to brains in vacuum D to normal observers in vacuum D is the upward tunneling rate FD. If the vacuum D is an- given by thropic then it must support Boltzmann brains, so the domination by Boltzmann brains could be avoided only N BB BB BB D D by the stringent requirement . P : (56) D FD N NO DFj D j FjD Fj D. A further generalization In this case our answers are complicated by the presence The conclusions of the last subsection are robust to more of many different vacua. We can, in principle, deter- general considerations. To illustrate, let us generalize the mine whether Boltzmann brains dominate by evaluating ADFIB landscape to one with many low-vacuum-energy Eqs. (54)–(56) for the correct values of the parameters, pockets, described by the schematic but this gets rather complicated and model dependent. A D $ F ! I ! B; (51) The evaluation of these expressions can be simplified j i significantly, however, if we make some very plausible assumptions. where each high-energy false vacuum Fj decays into a set For tunneling out of the high-energy vacua Fj, one of vacua fIig, all of which decay (for simplicity) to the same terminal vacuum B. The vacua I are taken to be a large set can expect the transition rates into different channels i to be roughly comparable, so that . including both anthropic vacua and vacua that host only IiFj DFj Fj That is, we assume that the branching ratios = Boltzmann brains. Equation (9) continues to apply, so IiFj Fj

Eqs. (46) and (48) are easily generalized to this case, giving and DFj =Fj are roughly of order one in the sense of X Eq. (16). These factors (or their inverses) will therefore 1 N BB N NO s s (52) be unimportant in the evaluation of = , and may Ii IiFj Fj Ii j be dropped. Furthermore, the upward transition rates from the dominant vacuum D into Fj are all comparable to one and another, as can be seen by writing [45]

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A FjD SD Boltzmann brain nucleation rate must be less than the F D e e ; (57) j decay rate of that vacuum. where AFjD is the action of the instanton responsible for the transition and SD is the action of the Euclideanized E. A dominant vacuum system de Sitter four-sphere, In the next to last paragraph of Sec. II, we described a 82 scenario where the dominant vacuum was not the vacuum SD ¼ : (58) with the smallest decay rate. Let us now study a simple H2 D landscape to illustrate this situation. Consider the toy j j landscape But generically AFjD 1=Fj . If we assume that

1 1 14 <10 (59) Fj Fk ¼ for every pair of vacua Fj and Fk, then FjD FkD up to a factor that can be ignored because it is roughly of order one. Thus, up to subleading factors, the transition rates where as in Sec. IV D the vacua Ii are taken to include both 7 BB NO anthropic vacua and vacua that support only Boltzmann F D cancel out in the ratio N =N . j fIig brains. Vacua A and B are terminal vacua and the F have Returning to Eq. (54) and keeping only the leading j factors, we have large, positive vacuum energies. Assume that vacuum S has the smallest total decay rate. N BB BB We have in mind the situation in which D and D are fI g Ii 1 2 i max ; (60) nearly degenerate, and transitions from D to D (and vice N NO i I 1 2 i versa) are rapid, even though the transition in one direction where the index i runs over all (nondominant) vacua in is upward. With this in mind, we divide the decay rates of which Boltzmann brains can nucleate. For the dominant D1 and D2 into two parts, vacuum, our simplifying assumptions8 convert Eqs. (55) ¼ þ out; (63) and (56) into 1 21 1

BB BB ¼ þ out; N D 2 12 2 (64) D BBeSD ; (61) NO D N up out with 12;21 1;2. We assume as in previous sections P that the rates for large upward transitions (S to D or D , where is the upward transition rate out of 1 2 up j FjD and D or D to F ) are extremely small, so that we can the dominant vacuum. 1 2 j ignore them in the calculation of q. The rate equation, Thus, the conditions needed to avoid Boltzmann brain Eq. (9), then admits a solution with q ’ , but it also domination are essentially the same as what we found in D admits solutions with Sec. IV C. In this case, however, we must require that in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi any vacuum that can support Boltzmann brains, the 1 q ’ þ ð Þ2 þ 4 : (65) 2 1 2 1 2 12 21 7Depending on the range of vacua F that are considered, the j out bound of Eq. (59) may or may not be valid. If it is not, then the Expanding the smaller root to linear order in 1;2 gives simplification of Eq. (60) below is not justified, and the original out out Eq. (54) has to be used. Of course one should remember that q ’ 11 þ 22 ; (66) there was significant arbitrariness in the choice of 1014 in the definition of ‘‘roughly of order one.’’ It was chosen to accom- where modate the largest estimate that we discuss in Sec. V for the 16 Boltzmann brain nucleation rate, BB expð10 Þ. In consid- 12 21 14 ; : (67) ering the other estimates of BB, one could replace 10 by a 1 þ 2 þ much larger number, thereby increasing the applicability of 12 21 12 21 Eq. (59). 8 AF D q The dropping of the factor e j is a more reliable approxi- In principle, this value for can be smaller than D, which mation in this case than it was in Eq. (60) above. In this case the is the case that we wish to explore. factor eSD does not cancel between the numerator and denomi- In this case the vacua D and D dominate the volume A 1 2 nator, so the factor e FjD can be dropped if it is unimportant fraction of the multiverse, even if their total decay rates S 1 compared to e D . We of course do not know the value of S for and are not the smallest in the landscape. We can the dominant vacuum, but for our vacuum it is of order 10122, 2 and it is plausible that the value for the dominant vacuum is therefore call the states D1 and D2 together a dominant similar or even larger. Thus as long as 1=F is small compared vacuum system, which we denote collectively as D. A j to 10122, it seems safe to drop the factor e FjD . The rate equation [Eq. (9)] shows that

063520-11 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) pocket, and presumably much less likely to produce life. sD1 1sD;sD2 2sD; (68) We will call a vacuum in the D system ‘‘strongly þ where sD sD1 sD2 , and the equations hold in the anthropic’’ if normal observers are produced by tunnelings out approximation that 1;2 and the upward transition rates from within D, and ‘‘mildly anthropic’’ if normal observers from D1 and D2 can be neglected. To see that these vacua can be produced, but only by tunnelings from higher- dominate the volume fraction, we calculate the modiﬁed energy vacua outside D. form of Eq. (53): If either of the vacua in D were strongly anthropic, then the normal observers in D would dominate the normal sF 1F D þ 2F D j j 1 j 2 : (69) observers in the multiverse. Normal observers in the vacua s D Fj Ii would be less numerous by a factor proportional to the extremely small rate for large upward transitions. Thus the volume fractions of the F , and hence also the I FjD j j This situation would itself be a problem, however, similar and B vacua, are suppressed by the very small rate for large to the Boltzmann brain problem. It would mean that ob- upward jumps from low-energy vacua, namely, and FjD1 servers like ourselves, who arose from a hot big bang with . The volume fraction for S depends on and FjD2 AD1 energy densities much higher than our vacuum-energy

AD2 , but it is maximized when these rates are negligible, density, would be extremely rare in the multiverse. We in which case it is given by conclude that if there are any models which give a domi-

sS q nant vacuum system that contains a strongly anthropic : (70) vacuum, such models would be considered unacceptable s q D S in the context of the scale-factor cutoff measure. This quantity can, in principle, be large, if q is just a little On the other hand, if the D system included one or more smaller than S, but that would seem to be a very special mildly anthropic vacua, then the situation is very similar to case. Generically, we would expect that since q must be that discussed in Secs. IV C and IV D. In this case the smaller than S [see Eq. (11)], it would most likely be normal observers in the D system would be comparable many orders of magnitude smaller, and hence the ratio in in number to the normal observers in the vacua Ii, so they Eq. (70) would be much less than 1. There is no reason, would have no significant effect on the ratio of Boltzmann however, to expect it to be as small as the ratios that are brains to normal observers in the multiverse. If any of the D suppressed by large upward jumps. For simplicity, vacua were mildly anthropic, however, then the stringent BB however, we will assume in what follows that sS can be requirement D up would have to be satisfied without BB neglected. resort to the simple solution D ¼ 0. To calculate the ratio of Boltzmann brains to normal Thus, we find that the existence of a dominant vacuum observers in this toy landscape, note that Eqs. (54) and (55) system does not change our conclusions about the abun- are modified only by the substitution dance of Boltzmann brains, except insofar as the Boltzmann brain nucleation constraints that would apply ! þ : (71) FjD FjD 1 FjD1 2 FjD2 to the dominant vacuum must apply to every member of the Thus, the dominant vacuum transition rate is simply dominant vacuum system. Probably the most important replaced by a weighted average of the dominant vacuum implication of this example is that the dominant vacuum transition rates. If we assume that neither of the vacua, D is not necessarily the vacuum with the lowest decay rate, so 1 the task of identifying the dominant vacuum could be very or D2, are anthropic, and make the same assumptions about magnitudes used in Sec. IV D, then Eqs. (60) and (61) difficult. continue toP hold as well, where we have redefined up F. General conditions to avoid by up jF D. j Boltzmann brain domination If, however, we allow D1 or D2 to be anthropic, then new questions arise. Transitions between D1 and D2 are, by In constructing general conditions to avoid Boltzmann assumption, rapid, so they copiously produce new pockets brain domination, we are guided by the toy landscapes and potentially new normal observers. We must recall, discussed in the previous subsections. Our goal, however, however (as discussed in Sec. III), that the properties of a is to construct conditions that can be justified using only pocket universe depend on both the current vacuum and the the general equations of Secs. II and III, assuming that the parent vacuum. In this case, the unusual feature is that the landscape is irreducible, but without relying on the prop- vacua within the D system are nearly degenerate, and erties of any particular toy landscape. We will be especially hence very little energy is released by tunnelings within cautious about the treatment of the dominant vacuum and D. For pocket universes created in this way, the maximum the possibility of small upward transitions, which could be particle energy density during reheating will be only a rapid. The behavior of the full landscape of a realistic small fraction of the vacuum-energy density. Such a big theory may deviate considerably from that of the simplest bang is very different from the one that took place in our toy models.

063520-12 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010)

To discuss the general situation, it is useful to divide j ! k1 ! ...! kn ! i; (75) vacuum states into four classes. We are only interested in vacua that can support Boltzmann brains. These can be where i is an anthropic vacuum. We begin by using N BB N NO (1) anthropic vacua for which the total dimensionless Eqs. (22) and (24) to express j = i , dropping decay rate satisfies i q, irrelevant factors as in Eq. (28), and then we can iterate (2) nonanthropic vacua that can transition to anthropic the above inequality: vacua via unsuppressed transitions, (3) nonanthropic vacua that can transition to anthropic N BB BB BB BB j j sj j sj j sj vacua only via suppressed transitions, P N NO s s ð s ÞB B B (4) anthropic vacua for which the total dimensionless i k ik k i i j j j!k1 k1!k2 kn!i decay rate is q. BB 1 i ¼ j ; (76) j Bj!k Bk !k Bk !i Here q is the smallest-magnitude eigenvalue of the rate 1 1 2 n equation [see Eqs. (5)–(8)]. We call a transition ‘‘unsup- where again there is no sum on repeated indices, and pressed’’ if its branching ratio is roughly of order one in Eq. (9) was used in the second step on the first line. Each the sense of Eq. (16). If the branching ratio is smaller than inverse branching ratio on the right of the last line is greater this, it is ‘‘suppressed.’’ As before, when calculating N BB N NO than or equal to 1, but by our assumptions can be consid- = we assume that factors that are roughly of ered to be roughly of order one, and hence can be dropped. order one can be ignored. Note that Eq. (11) forbids i Thus, the multiverse will avoid domination by Boltzmann q from being less than , so the above four cases are brains in vacuum j if BB= 1, the same criterion exhaustive. j j We first discuss conditions that are sufficient to guaran- found for the first class. tee that Boltzmann brains will not dominate, postponing The third class—nonanthropic vacua that can only transition to an anthropic state via at least one suppressed until later the issue of what conditions are necessary. transition—presumably includes many states with very We begin with the vacua in the first class. Very likely all low-vacuum-energy density. The dominant vacuum of anthropic vacua belong to this class. For an anthropic our toy landscape models certainly belongs to this class, vacuum i, the Boltzmann brains produced in vacuum i cannot dominate the multiverse if they do not dominate but we do not know of anything that completely excludes the normal observers in vacuum i, so we can begin with this the possibility that the dominant vacuum might belong to comparison. Restricting Eq. (29) to this single vacuum, we the second or fourth class. That is, perhaps the dominant obtain vacuum is anthropic, or decays to an anthropic vacuum. If there is a dominant vacuum system, as described in BB BB N Sec. IV E, then i q, and the dominant vacua could i i ; (72) N NO belong to the first class, as well as to either of classes (2) i i and (3). a ratio that has appeared in many of the simple examples. To bound the Boltzmann brain production in this If this ratio is small compared to 1, then Boltzmann brains class, we consider two possible criteria. To formulate the created in vacuum i are negligible. first, we can again use Eqs. (75) and (76), but this time Let us now study a vacuum j in the second class. First the sequence must include at least one suppressed transi- note that Eq. (9) implies the rigorous inequality tion, presumably an upward jump. Let us therefore denote the branching ratio for this suppressed transition as Bup, isi ijsj ðno sum on repeated indicesÞ; (73) noting that Bup will appear in the denominator of Eq. (76). Of course, the sequence of Eq. (75) might involve more which holds for any two states i and j. [Intuitively, Eq. (73) than one suppressed transition, but in any case the product is the statement that, in a steady state, the total rate of loss of these very small branching ratios in the denominator of the volume fraction must exceed the input rate from any can be called Bup, and all the other factors can be taken as one channel.] To simplify what follows, it will be useful to roughly of order one. Thus, a landscape containing a rewrite Eq. (73)as vacuum j of the third class avoids Boltzmann brain domination if ðisiÞðjsjÞBj!i; (74) BB j 1; (77) where Bj!i ij=j is the branching ratio for the transi- Bupj tion j ! i. Suppose that we are trying to bound the Boltzmann brain in agreement with the results obtained for the dominant production in vacuum j, and we know that it can undergo vacua in the toy landscape models in the previous unsuppressed transitions subsections.

063520-13 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) A few comments are in order. First, if the only sup- Eq. (79) are significant, and that q is negligible in each pressed transition is the first, then Bup ¼ up=j, and the case, then BB above criterion simplifies to j =up 1. Second, we mþ1 should keep in mind that the sequence of Eq. (75)is ðjsjÞðkAsAÞZmax : (82) 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 Using the bounds from Eqs. (80) and (82), the Boltzmann vacua under discussion are not anthropic, a likely method brain ratio is bounded by BB for Eq. (77) to be satisfied would be for j to vanish, as BB BB BB would happen if the vacuum j did not support the complex N sj sj j P j j structures needed to form Boltzmann brains. N NO i k iksk isi The criterion above can be summarized by saying that if mþ1 BB BB Z j j =ðBupjÞ1, then the Boltzmann brains in vacuum j max : (83) will be overwhelmingly outnumbered by the normal BA!k1 Bk1!k2 Bkn!i j observers living in pocket universes that form in the decay chain starting from vacuum j. We now describe a second, But all the factors on the right are roughly of order one, alternative criterion, based on the idea that the number of except that some of the branching ratios in the denominator Boltzmann brains in vacuum j can be compared with the might be smaller, if they correspond to suppressed transi- number of normal observers in vacuum i if the two types of tions. If Bup denotes the product of branching ratios for vacua have a common ancestor. all the suppressed transitions shown in the denominator Denoting the common ancestor vacuum as A, we assume [i.e., all suppressed transitions in the sequence of Eq. (78)], that it can decay to an anthropic vacuum i by a chain of then the bound reduces to Eq. (77).9 transitions, To summarize, the Boltzmann brains in a nonanthropic vacuum j can be bounded if there is an ancestor vacuum A A ! k ! ...! k ! i; (78) 1 n that can decay to j through a chain of significant transitions and also to a Boltzmann-brain-producing vacuum j by a for which q ‘ for each vacuum, as in the sequence of chain Eq. (79), and if the same ancestor vacuum can decay to an anthropic vacuum through a sequence of transitions A ! ‘ ! ...! ‘ ! j: (79) as in Eq. (78). The Boltzmann brains will never dominate 1 m BB provided that j =ðBupjÞ1, where Bup is the product From the sequence of Eq. (78) and the bound of Eq. (74), of all suppressed branching ratios in the sequence we can infer that of Eq. (78). Finally, the fourth class of vacua consists of anthropic i ’ q ðisiÞðkAsAÞBA!k Bk !k Bk !i: (80) vacua with decay rate i , a class which could be 1 1 2 n empty. For this class, Eq. (29) may not be very useful, since the quantity ( q) in the denominator could be very To make use of the sequence of Eq. (79) we will want a i small. Yet, as in the two previous classes, this class can bound that goes in the opposite direction, for which we will be treated by using Eq. (76), where in this case the vacuum need to require additional assumptions. Starting with i can be the same as j or different, although the case i ¼ j Eq. (9), we first require q , which is plausible pro- i requires n 1. Again, if the sequence contains only un- vided that vacuum i is not the dominant vacuum. Next we suppressed transitions, then the multiverse avoids domina- look at the sum over j on the right-hand side, and we call tion by Boltzmann brains in vacuum i if BB= 1.If the transition j ! i ‘‘significant’’ if its contribution to the i i upward jumps are needed to reach an anthropic vacuum, sum is within a factor roughly of order one of the entire whether it is the vacuum i again or a distinct vacuum j, sum. (The sum over j is the sum over sources for vacuum i, then the Boltzmann brains in vacuum i will never dominate so a transition j ! i is significant if pocket universes of if BB=ðB Þ1. vacuum j are a significant source of pocket universes i up i of vacuum i.) It follows that for any significant transition 9 j ! i for which q i, Note, however, that the argument breaks down if the sequences in either Eq. (78)or(79) become too long. For the choices that we have made, a factor of Zmax is unimportant in the BB NO 100 16 ðisiÞðjsjÞZmaxBj!i ðjsjÞZmax; (81) calculation of N =N ,butZmax ¼ expð10 Þ can be significant. Thus, for our choices we can justify the dropping of Oð100Þ factors that are roughly of order one, but not more than where Zmax denotes the largest number that is roughly of that. For choices appropriate to smaller estimates of ,how- 14 BB order one. By our conventions, Zmax ¼ expð10 Þ.Ifwe ever, the number of factors that can be dropped will be many assume now that all the transitions in the sequence of orders of magnitude larger.

063520-14 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) The conditions described in the previous paragraph are The conditions described above are sufficient to guar- very difficult to meet, so if the fourth class is not empty, antee that Boltzmann brains do not dominate over normal Boltzmann brain domination is hard to avoid. These vacua observers in the multiverse, but without further assump- have the slowest decay rates in the landscape, i q,soit tions there is no way to know if they are necessary. All seems plausible that they have very low-energy densities, of the conditions that we have discussed are quasilocal, in precluding the possibility of decaying to an anthropic the sense that they do not require any global picture of the vacuum via unsuppressed transitions; in that case landscape of vacua. For each of the above arguments, the Boltzmann brain domination can be avoided if Boltzmann brains in one type of vacuum j are bounded by i BB the normal observers in some type of vacuum that is either i Bupi: (84) the same type or directly related to it through decay chains. Thus, there was no need to discuss the importance of the However, as pointed out in Ref. [42], B / eSD [see up vacua j and i compared to the rest of the landscape as a Eq. (57)] is comparable to the inverse of the recurrence whole. The quasilocal nature of these conditions, however, time, while in an anthropic vacuum one would expect the guarantees that they cannot be necessary to avoid the Boltzmann brain nucleation rate to be much faster than domination by Boltzmann brains. If two vacua j and i once per recurrence time. are both totally insignificant in the multiverse, then it To summarize, the domination of Boltzmann brains can will always be possible for the Boltzmann brains in vac- be avoided by, first of all, requiring that all vacuum states in uum j to overwhelm the normal observers in vacuum i, the landscape obey the relation while the multiverse as a whole could still be dominated by BB normal observers in other vacua. j 1: (85) We have so far avoided making global assumptions j about the landscape of vacua, because such assumptions That is, the rate of nucleation of Boltzmann brains in each are generally hazardous. While it may be possible to make vacuum must be less than the rate of nucleation, in that statements that are true for the bulk of vacua in the land- same vacuum, of bubbles of other phases. For anthropic scape, in this context the statements are not useful unless they are true for all the vacua of the landscape. Although vacua i with i q, this criterion is enough. Otherwise, the Boltzmann brains that might be produced in vacuum j the number of vacua in the landscape, often estimated at 500 must be bounded by the normal observers forming in some 10 [50], is usually considered to be incredibly large, the vacuum i, which must be related to j through decay chains. number is nonetheless roughly of order one compared to Specifically, there must be a vacuum A that can decay the numbers involved in the estimates of Boltzmann brain through a chain to an anthropic vacuum i, i.e. nucleation rates and vacuum decay rates. Thus, if a single vacuum produces Boltzmann brains in excess of re- A ! k1 ! ...! kn ! i; (86) quired bounds, the Boltzmann brains from that vacuum could easily overwhelm all the normal observers in the where either A ¼ j, or else A can decay to j through a multiverse. sequence Recognizing that our conclusions could be faulty, we

A ! ‘1 ! ...! ‘m ! j: (87) can nonetheless adopt some reasonable assumptions to see where they lead. We can assume that the multiverse is In the above sequence we insist that j q and that sourced by either a single dominant vacuum or by a domi- l q for each vacuum ‘p in the chain, and that each nant vacuum system. We can further assume that every transition must be significant, in the sense that pockets of anthropic and/or Boltzmann-brain-producing vacuum i can type ‘p must be a significant source of pockets of type be reached from the dominant vacuum (or dominant vac- ‘pþ1. [More precisely, a transition from vacuum j to i is uum system) by a single significant upward jump, with a S significant ifP it contributes a fraction that is roughly of rate proportional to e D , followed by some number of order one to jijsj in Eq. (9).] For these cases, the bound significant, unsuppressed transitions, all of which have which ensures that the Boltzmann brains in vacuum j are rates k q and branching ratios that are roughly of dominated by the normal observers in vacuum i is given by order one:

BB j 1; (88) D ! k1 ! ...! kn ! i: (89) Bupj where Bup is the product of any suppressed branching We will further assume that each nondominant anthropic ratios in the sequence of Eq. (86). If all the transitions in and/or Boltzmann-brain-producing vacuum i has a decay Eq. (86) are unsuppressed, this bound reduces to Eq. (85). rate i q, but we need not assume that all of the i are If j is anthropic, the case A ¼ j ¼ i is allowed, provided comparable to each other. With these assumptions, the that n 1. estimate of N BB=N NO becomes very simple.

063520-15 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) Applying Eq. (9) to the first transition of Eq. (89), The dominant vacuum could conceivably be anthropic, but we begin by considering the case in which it is not. In s s s ; (90) k1 k1 k1D D up D that case all anthropic vacua are equivalent, so the where we use to denote the rate of a typical transition Boltzmann brains produced in any vacuum j will either up dominate the multiverse or not, depending on whether they D ! k, assuming that they are all equal to each other up to dominate the normal observers in an arbitrary anthropic a factor roughly of order one. Here indicates equality up vacuum i. Combining Eqs. (9), (22), (24), and (93), and to a factor that is roughly of order one. If there is a omitting irrelevant factors, we find that for any nondomi- dominant vacuum system, then is replaced by P k1D k1D nant vacuum j, ‘‘k1D‘ , where the D‘ are the components of the domi- BB BB BB BB nant vacuum system, and the ‘ are defined by generaliz- N j sj j sj j 10 j P : (95) ing Eqs. (67) and (68). Applying Eq. (9) to the next N NO i k iksk isi j transition, k1 ! k2, we find Thus, given the assumptions described above, for any non- s ¼ B s þ ... s ; (91) k2 k2 k1!k2 k1 k1 k1 k1 dominant vacuum j, the necessary and sufficient condition to avoid the domination of the multiverse by Boltzmann where we have used the fact that B is roughly of k1!k2 brains in vacuum j is given by order one, and that the transition is significant. Iterating, we have BB j 1: (96) j isi kn skn upsD: (92) For Boltzmann brains formed in the dominant vacuum, Since the expression on the right is independent of i,we we can again find out if they dominate the multiverse by conclude that under these assumptions any two nondomi- determining whether they dominate the normal observers nant anthropic and/or Boltzmann-brain-producing vacua i in an arbitrary anthropic vacuum i. Repeating the above and j have equal values of s, up to a factor that is roughly analysis for vacuum D instead of vacuum j, using Eq. (92) of order one: to relate si to sD, we have

jsj isi: (93) N BB BBs BBs BB D P D D D D D : (97) N NO NO i k iksk isi up Using Eq. (22) and assuming, as always, that nik is roughly of order one, Eq. (93) implies that any two nondominant anthropic vacua i and j have comparable num- Thus, for a single dominant vacuum D or a dominant bers of ordinary observers, up to a factor that is roughly of vacuum system with members Di, the necessary and sufﬁ- order one: cient conditions to avoid the domination of the multiverse by these Boltzmann brains is given by N NO N NO: (94) j i BB BB D 1 or Di 1: (98) up up 10In more detail, the concept of a dominant vacuum system is relevant when there is a set of vacua ‘ that can have rapid As discussed after Eq. (84), probably the only way to BB transitions within the set, but only very slow transitions con- satisfy this condition is to require that D ¼ 0. necting these vacua to the rest of the landscape. As a zeroth order If the dominant vacuum is anthropic, then the conclu- approximation one can neglect all transitions connecting these sions are essentially the same, but the logic is more vacua to the rest of the landscape, and assume that ‘ q,so Eq. (9) takes the form involved. For the case of a dominant vacuum system, we X distinguish between the possibility of vacua being ‘s‘ ¼ B‘‘0 ‘0 s‘0 : ‘‘strongly’’ or ‘‘mildly’’ anthropic, as discussed in ‘0 Sec. IV E. Strongly anthropic means that normal observers Here B‘‘0 ‘‘0 =‘0 isP the branching ratio within this restricted are formed by tunneling within the dominant vacuum subspace, where ‘ ¼ ‘0 P‘0‘ is summed only within the domi- 0 system D, while mildly anthropic implies that normal nant vacuum system, so ‘B‘‘0 ¼ 1 for all ‘ . B‘‘0 is non- observers are formed by tunneling, but only from outside negative, and if we assume also that it is irreducible, then the Perron-Frobenius theorem guarantees that it has a nondegenerate D. Any model that leads to a strongly anthropic dominant eigenvector v‘ of eigenvalue 1, with positive components. From vacuum would be unacceptable, because almost all observ- the above equation ‘s‘ / v‘, and then ers would live in pockets with a maximum reheat energy s v density that is small compared to the vacuum-energy den- ¼ P ‘ ¼ P‘ : ‘ v‘0 sity. With a single anthropic dominant vacuum, or with one 0 s 0 0 ‘ ‘ ‘ ‘ 0 ‘ or more mildly anthropic vacua within a dominant vacuum

063520-16 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) system, the normal observers in the dominant vacuum S & 3M=mn; (102) would be comparable in number (up to factors roughly of order one) to those in other anthropic vacua, so they would where mn is the nucleon mass. Indeed, the actual value of have no significant effect on the ratio of Boltzmann SBB is much smaller than this upper bound because of brains to normal observers in the multiverse. An anthropic the complex organization of the Boltzmann brain. vacuum would also produce Boltzmann brains, however, Meanwhile, to prevent the Boltzmann brain from being so Eq. (98) would have to somehow be satisfied for destroyed by pair production, we require that TdS mn. BB D 0. Thus, for these Boltzmann brains the entropy factor eSBB is irrelevant compared to the Boltzmann suppression factor. To estimate the nucleation rate for Boltzmann brains, we V. BOLTZMANN BRAIN NUCLEATION AND need at least a crude description of what constitutes a VACUUM DECAY RATES Boltzmann brain. There are many possibilities. We argued A. Boltzmann brain nucleation rate in the Introduction to this paper that a theory that predicts Boltzmann brains emerge from the vacuum as large the domination of Boltzmann brains over normal observers quantum fluctuations. In particular, they can be modeled would be overwhelmingly disfavored by our continued as localized fluctuations of some mass M, in the observation of an orderly world, in which the events that thermal bath of a de Sitter vacuum with temperature we observe have a logical relationship to the events that we remember. In making this argument, we considered a class TdS ¼ H=2 [1]. The Boltzmann brain nucleation rate is then roughly estimated by the Boltzmann suppression of Boltzmann brains that share exactly the memories and factor [6,8], thought processes of a particular normal observer at some chosen instant. For these purposes the memory of the M=T BB e dS ; (99) Boltzmann brain can consist of random bits that just happen to match those of the normal observer, so there are no where our goal is to estimate only the exponent, not the requirements on the history of the Boltzmann brain. prefactor. Equation (99) gives an estimate for the nuclea- Furthermore, the Boltzmann brain need only survive long tion rate of a Boltzmann brain of mass M in any particular enough to register one observation after the chosen instant, quantum state, but we will normally describe the so it is not required to live for more than about a second. Boltzmann brain macroscopically. Thus BB should be We will refer to Boltzmann brains that meet these require- S multiplied by the number of microstates e BB correspond- ments as minimal Boltzmann brains. ing to the macroscopic description, where SBB is the While an overabundance of minimal Boltzmann entropy of the Boltzmann brain. Thus we expect brains is enough to cause a theory to be discarded, we nonetheless find it interesting to discuss a wide range of M=T S F=T BB e dS e BB ¼ e dS ; (100) Boltzmann brain possibilities. We will start with very large Boltzmann brains, discussing the minimal Boltzmann where F ¼ M TdSSBB is the free energy of the brains last. Boltzmann brain. We first consider Boltzmann brains much like us, who Equation (100) should be accurate as long as the evolved in stellar systems like ours, in vacua with low- de Sitter temperature is well defined, which will be energy particle physics like ours, but allowing for a the case as long as the Schwarzschild horizon is small de Sitter Hubble radius as small as a few astronomical compared to the de Sitter horizon radius. Furthermore, units or so. These Boltzmann brains evolved in their we shall neglect the effect of the gravitational potential stellar systems on a time scale similar to the evolution of energy of de Sitter space on the Boltzmann brain, which life on Earth, so they are in every way like us, except that, requires that the Boltzmann brain be small compared to the when they perform cosmological observations, they find de Sitter horizon. Thus we assume themselves in an empty, vacuum-dominated universe. These ‘‘Boltzmann solar systems’’ nucleate at a rate of 1 M=4

85 where the ﬁrst inequality assumes that Boltzmann brains BB expð10 Þ; (103) cannot be black holes. The general situation, which allows 1 30 1 1 for M R H , will be discussed in the Appendix and where we have set M 10 kg and H ¼ð2TdSÞ in Ref. [51]. 1012 m. This nucleation rate is fantastically small; we While the nucleation rate is proportional to eSBB , this found it, however, by considering the extravagant possi- factor is negligible for any Boltzmann brain made of atoms bility of nucleating an entire Boltzmann solar system. like those in our Universe. The entropy of such atoms is Next, we can consider the nucleation of an isolated bounded by brain, with a physical construction that is roughly similar

063520-17 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) 1 to our own brains. If we take M 1kg and H ¼ As discussed in Ref. [52], the only known substrate- 1 ð2TdSÞ 1m, then the corresponding Boltzmann independent limit on the storage of information is the brain nucleation rate is Bekenstein bound. It states that, for an asymptotically ﬂat background, the entropy of any physical system of size R 43 12 BB expð10 Þ: (104) and energy M is bounded by

If the construction of the brain is similar to ours, however, S SBek 2MR: (107) then it could not function if the tidal forces resulted in a One can use this bound in de Sitter space as well if the size relative acceleration from one end to the other that is of the system is sufﬁciently small, R H1, so that the much greater than the gravitational acceleration g on the system does not ‘‘know’’ about the horizon. A possible surface of the Earth. This requires H1 * 108 m, giving a generalization of the Bekenstein bound for R ¼ OðH1Þ Boltzmann brain nucleation rate was proposed in Ref. [53]; we will study this and other possibilities in the Appendix and in Ref. [51]. To begin, 51 BB expð10 Þ: (105) 1 however, we will discuss the simplest case, R H ,so that we can focus on the most important issues before Until now, we have concentrated on Boltzmann brains dealing with the complexities of more general results. that are very similar to human brains. However, a common Using Eq. (106), the Boltzmann brain nucleation rate of assumption in the philosophy of mind is that of substrate Eq. (100) can be rewritten as independence. Therefore, pressing onward, we study the 2M possibility that a Boltzmann brain can be any device BB exp þ SBB;max IBB ; (108) capable of emulating the thoughts of a human brain. In H other words, we treat the brain essentially as a highly which is clearly maximized by choosing M as small as sophisticated computer, with logical operations that can possible. The Bekenstein bound, however, implies that be duplicated by many different systems of hardware.11 SBB;max SBek and therefore M SBB;max=ð2RÞ. Thus With this in mind, from here out we drop the assumption that Boltzmann brains are made of the same materials as SBB;max BB exp þ SBB;max IBB : (109) human brains. Instead, we attempt to ﬁnd an upper bound RH on the probability of creation of a more generalized com- Since R

063520-18 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) assumption R H1 to the boundary of its validity. Thus 1 M R H : (112) we write the Boltzmann brain production rate If the Bekenstein bound is saturated, this leads to the aI following relations between IBB, H, and M: BB e BB ; (111) 1 2 IBB MR MH H : (113) 1 where a ðRHÞ , the value of which is of order a A second potential mechanism of Boltzmann brain sta- few. In the Appendix we explore the case in which the bilization is to surround it by a domain wall with a surface Schwarzschild radius, the Boltzmann brain radius, and the tension , which would provide pressure preventing the de Sitter horizon radius are all about equal, in which case exponential expansion of the brain. An investigation of this Eq. (111) holds with a ¼ 2. situation reveals that one cannot saturate the Bekenstein The bound of Eq. (111) can be compared to the estimate bound using this mechanism unless there is a speciﬁc of the Boltzmann brain production rate, eSBB , BB relation between IBB, H, and [51]: which follows from Eq. (2.13) of Freivogel and Lippert, 3 in Ref. [54]. The authors of Ref. [54] explained that by SBB IBBH: (114) they mean not the entropy, but the number of degrees of If is less than this magnitude, it cannot prevent the freedom, which is roughly equal to the number of particles expansion, while a larger increases the mass and there- in a Boltzmann brain. This estimate appears similar to our fore prevents saturation of the Bekenstein bound. result, if one equates S to I , or to a few times I . BB BB BB Regardless of the details leading to Eqs. (113) and (114), Freivogel and Lippert describe this relation as a lower the important point is that both of them lead to constraints bound on the nucleation rate for Boltzmann brains, on the vacuum hosting the Boltzmann brain. commenting that it can be used as an estimate of the For example, the Boltzmann brain stabilized by gravita- nucleation rate for vacua with ‘‘reasonably cooperative tional attraction can be produced at a rate approaching particle physics.’’ Here we will explore in some detail eaIBB only if I H2. For a given value of I , say the question of whether this bound can be used as an BB BB I 1016 (see the discussion below), this result applies estimate of the nucleation rate. While we will not settle BB only to vacua with a particular vacuum energy, 1016. this issue here, we will discuss evidence that Eq. (111)isa Similarly, according to Eq. (114), for Boltzmann brains valid estimate for at most a small fraction of the vacua of with I 1016 contained inside a domain wall in a vac- the landscape, and possibly none at all. BB uum with 10120, the Bekenstein bound on cannot So far, the conditions to reach the upper bound in BB be reached unless the tension of the domain wall is incredi- Eq. (111) are R ¼ðaH Þ1 OðH1Þ and I ¼ BB bly small, 10164. Thus, the maximal Boltzmann brain S ¼ S . However, these are not enough to ensure max;BB Bek production rate eaIBB saturating the Bekenstein bound that a Boltzmann brain of size R H1 is stable and can cannot be reached unless Boltzmann brains are produced on actually compute. Indeed, the time required for communi- a narrow hypersurface in the landscape. cation between two parts of a Boltzmann brain separated This conclusion by itself does not eliminate the danger O 1 by a distance ðH Þ is at least comparable to the Hubble aIBB of a rapid Boltzmann brain production rate, BB e . time. If the Boltzmann brain can be stretched by cosmo- Given the vast number of vacua in the landscape, it seems logical expansion, then after just a few operations the plausible that this bound could actually be met. If this is the different parts will no longer be able to communicate. case, Eq. (111) offers a stunning increase over previous Therefore we need a stabilization mechanism by which estimates of BB. the brain is protected against expansion. Setting aside the issue of Boltzmann brain stability, one A potential mechanism to protect the Boltzmann brain can also question the assumption of Bekenstein bound against de Sitter expansion is the self-gravity of the brain. saturation that is necessary to achieve the rather high A simple example is a black hole, which does not expand nucleation rate that is indicated by Eq. (111). Of course when the Universe expands. It seems unlikely that black 13 black holes saturate this bound, but we assume that a black holes can think, but one can consider objects of mass hole cannot think. Even if a black hole can think, it would approaching that of a black hole with radius R. This, still be an open question whether this information process- together with our goal to keep R as close as possible to 1 ing could make use of a substantial fraction of the degrees H , leads to the following condition: of freedom associated with the black hole entropy. Avariety of other physical systems are considered in Ref. [55], where 13 The possibility of a black hole computer is not excluded, the validity of SmaxðEÞ2ER is studied as a function of however, and has been considered in Ref. [52]. Nonetheless, if energy E. In all cases, the bound is saturated in a limit black holes can compute, our conclusions would not be changed, O where Smax ¼ ð1Þ. Meanwhile, as we shall argue below, provided that the Bekenstein bound can be saturated for the near- 16 black hole computers that we discuss. At this level of approxi- the required value of Smax should be greater than 10 . mation, there would be no signiﬁcant difference between a black The present authors are aware of only one example of a hole computer and a near-black hole computer. physical system that may saturate the Bekenstein bound

063520-19 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) and at the same time store sufficient information I to computer program that comes close to imitating a human emulate a human brain. This may happen if the total brain, this is not an easy question to answer. number of particle species with mass smaller than H is One way to proceed is to examine the human brain, with * 16 greater than IBB 10 . No realistic examples of such the goal of estimating its capacities based on its biological theories are known to us, although some authors have structure. The human brain contains 1014 synapses that speculated about similar possibilities [56]. may, in principle, connect to any of 1011 neurons [58], If Boltzmann brains cannot saturate the Bekenstein suggesting that its information content14 might be roughly 15 16 bound, they will be more massive than indicated in IBB 10 –10 . (We are assuming here that the logical Eq. (110), and their rate of production will be smaller functions of the brain depend on the connections among than eaIBB . neurons, and not, for example, on their precise locations, To put another possible bound on the probability of cellular structures, or other information that might be Boltzmann brain production, let us analyze a simple model necessary to actually construct a brain.) A minimal based on an ideal gas of massless particles. Dropping all Boltzmann brain is only required to simulate the workings numerical factors, we consider a box of size R filled with a of a real brain for about a second; but with neurons firing 3 gas with maximum entropy Smax ¼ðRTÞ and energy E ¼ typically at 10 to 100 times a second, it is plausible that a 3 4 4=3 substantial fraction of the brain is needed even for only 1 s R T ¼ Smax=R, where T is the temperature and we assume there is not an enormous number of particle species. of activity. Of course the actual number of required bits The probability of its creation can be estimated as follows: might be somewhat less. An alternative approach is to try to determine how much 4=3 information the brain processes, even if one does not Smax E=H SBB BB e e exp ; (115) understand much about what the processing involves. H R In Ref. [59], Landauer attempted to estimate the total where we have neglected the Boltzmann brain entropy content of a person’s long-term memory, using a variety of 4=3 experiments. He concluded that a person remembers only factor, since SBB Smax Smax. This probability is maximized by taking R H1, which yields about 2 bits=second, for a lifetime total in the vicinity of 109 bits. In a subsequent paper [60], however, he emphati- 4=3 cally denied that this number is relevant to the information & eSmax : (116) BB requirements of a ‘‘real or theoretical cognitive processor,’’ In case the full information capacity of the gas is used, one because such a device ‘‘would have so much more to do can also write than simply record new information.’’ Besides long-term memory, one might be interested in & I4=3 the total amount of information a person receives but does BB e BB : (117) not memorize. A substantial part of this information is For IBB 1, this estimate leads to a much stronger sup- visual; it can be estimated by the information stored on pression of Boltzmann brain production as compared to our high definition DVDs, watched continuously on several previous estimate, Eq. (111). monitors over the span of a hundred years. The total Of course, such a hot gas of massless particles cannot information received would be about 1016 bits. think—indeed it is not stable in the sense outlined below Since this number is similar to the number obtained Eq. (111)—so we must add more parts to this construction. above by counting synapses, it is probably as good an Yet it seems likely that this will only decrease the estimate as we can make for a minimal Boltzmann brain. Boltzmann brain production rate. As a partial test of this If the Bekenstein bound can be saturated, then the esti- conjecture, one can easily check that if instead of a gas of mated Boltzmann brain nucleation rate for the most favor- massless particles we consider a gas of massive particles, able vacua in the landscape would be given by Eq. (111): the resulting suppression of Boltzmann brain production & 1016 will be stronger. Therefore in our subsequent estimates we BB e : (118) shall assume that Eq. (117) represents our next ‘‘line of If, however, the Bekenstein bound cannot be reached for defense’’ against the possibility of Boltzmann brain domi- systems with IBB 1, then it might be more accurate to nation, after the one given by Eq. (111). One should note use instead the ideal gas model of Eq. (117), yielding that this is a rather delicate issue; see, for example, a & 1021 discussion of several possibilities to approach the BB e : (119) Bekenstein bound in Ref. [57]. A more detailed discussion of this issue will be provided in Ref. [51]. Obviously, there are many uncertainties involved in Having related BB to the information content IBB of the the numerical estimates of the required value of IBB. brain, we now need to estimate IBB. How much information storage must a computer have to be able to perform all 14Note that the specification of one out of 1011 neurons requires 11 the functions of the human brain? Since no one can write a log2ð10 Þ¼36:5 bits.

063520-20 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) 16 Our estimate IBB 10 concerns the information stored in no such physical construction exists, we are left with the the human brain that appears to be relevant for cognition. It less dangerous bound of Eq. (117), perhaps even further certainly does not include all the information that would be softened by the speculations described in Footnote 15. Note needed to physically construct a human brain, and it there- that none of these bounds is based upon a realistic model of fore does not allow for the information that might be a Boltzmann brain. For example, the nucleation of an needed to physically construct a device that could emulate actual human brain is estimated at the vastly smaller rate the human brain.15 It is also possible that extra mass might of Eq. (105). The conclusions of this paragraph apply to the be required for the mechanical structure of the emulator, to causal-patch measures [23,24] as well as the scale-factor provide the analogues of a computer’s wires, insulation, cutoff measure. cooling systems, etc. On the other hand, it is conceivable In Sec. III we discussed the possibility of Boltzmann that a Boltzmann brain can be relevant even if it has fewer brain production during reheating, stating that this process capabilities than what we called the minimal Boltzmann would not be a danger. We postponed the numerical dis- brain. In particular, if our main requirement is that the cussion, however, so we now return to that issue. Boltzmann brain is to have the same ‘‘perceptions’’ as a According to Eq. (26), the multiverse will be safe from human brain for just 1 s, then one may argue that this can Boltzmann brains formed during reheating provided that be achieved using much less than 1014 synapses. And if one BB BB NO decreases the required time to a much smaller value re- reheat;ikreheat;ik nik (120) quired for a single computation to be performed by a human brain, the required amount of information stored holds for every pair of vacua i and k, where BB is the in a Boltzmann brain may become many orders of magni- reheat;ik peak Boltzmann brain nucleation rate in a pocket of tude smaller than 1016. vacuum i that forms in a parent vacuum of type k, We ﬁnd that regardless of how one estimates the infor- BB is the proper time available for such nucleation, mation in a human brain, if Boltzmann brains can be reheat;ik and nNO is the volume density of normal observers in these constructed so as to come near the limit of Eq. (111), their ik nucleation rate would provide stringent requirements on pockets, working in the approximation that all observers vacuum decay rates in the landscape. On the other hand, if form at the same time. Compared to the previous discussion about late-time BB de Sitter space nucleation, here reheat;ik can be much

15 larger, since the temperature during reheating can be That is, the actual construction of a brainlike device would much larger than H . On the other hand, safety from presumably require large amounts of information that are not part of the schematic ‘‘circuit diagram’’ of the brain. Thus there Boltzmann brains requires the late-time nucleation rate to may be some significance to the fact that 1 109 yrs of evolu- be small compared to the potentially very small vacuum tion on Earth has not produced a human brain with fewer than decay rates, while in this case the quantity on the right- about 1027 particles, and hence of order 1027 units of entropy. In hand side of Eq. (120) is not exceptionally small. In dis- counting the information in the synapses, for example, we cussing this issue, we will consider in sequence three counted only the information needed to specify which neurons are connected to which, but nothing about the actual path of the descriptions of the Boltzmann brain: a humanlike brain, a axons and dendrites that complete the connections. These are near-black hole computer, and a diffuse computer. nothing like nearest-neighbor couplings, but instead axons from The nucleation of humanlike Boltzmann brains during a single neuron can traverse large fractions of the brain, resulting reheating was discussed in Ref. [27], where it was pointed in an extremely intertwined network [61]. To specify even the out that such brains could not function at temperatures topology of these connections, still ignoring the precise locations, could involve much more than 1016 bits. For example, the much higher than 300 K, and that the nucleation rate for 40 synaptic ‘‘wiring’’ that connects the neurons will, in many cases, a 100 kg object at this temperature is expð10 Þ. This form closed loops. A specification of the connections would suppression is clearly more than enough to ensure that presumably require a topological winding number for every Eq. (120) is satisfied. pair of closed loops in the network. The number of bits For a near-black hole computer with IBB SBB;max required to specify these winding numbers would be 16 proportional to the square of the number of closed loops, which 10 , the minimum mass is 600 g. If we assume that would be proportional to the square of the number of synapses. the reheat temperature is nopffiffiffiffiffiffiffiffiffiffi more than the reduced Thus, the structural information could be something like I 18 struct Planck mass, mPlanck 1= 8G 2:4 10pffiffiffiffiffiffiffiffiffiffiGeV b 1028, where b is a proportionality constant that is probably a 6 BB 4:3 10 g, we find that reheat < expð 2IBBÞ few orders of magnitude less than 1. In estimating the resulting 8 suppression of the nucleation rate, there is one further compli- expð10 Þ. Although this is not nearly as much suppres- cation: since structural information of this sort presumably has sion as in the previous case, it is clearly enough to no influence on brain function, these choices would contribute to guarantee that Eq. (120) will be satisfied. the multiplicity of Boltzmann brain microstates, thereby multi- For the diffuse computer, we can consider an ideal gas of I plying the nucleation rate by e struct . There would still be a net massless particles, as discussed in Eqs. (115)–(117). The suppression, however, with Eq. (111) leading to BB / ða1ÞI S e struct , where a is generically greater than 1. See the system would have approximately max particles, and a 4=3 Appendix for further discussion of the value of a. total energy of E ¼ Smax=R, so the Boltzmann suppression

063520-21 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) 4=3 factor is exp½Smax=ðRTreheatÞ. The Boltzmann brain pro- Minkowski space, one may equally well speculate about duction can occur at any time during the reheating process, spontaneous creation of inflationary universes there, each so there is nothing wrong with considering Boltzmann of which could contain infinitely many normal observers brain production in our Universe at the present time. For [62]. These issues become complicated, and we will make 16 no attempt to resolve them here. Fortunately, the estimates Treheat ¼ 2:7Kand Smax ¼ 10 , this formula implies that the exponent has magnitude 1 for R ¼ S4=3 T1 of thermal Boltzmann brain nucleation rates in de Sitter max reheat ! 200 light-years. Thus, the formula suggests that diffuse- space approach zero in the Minkowski space limit 0, so the issue of Boltzmann brains formed by quantum gas-cloud Boltzmann brains of radius 200 light-years can fluctuations in Minkowski space can be set aside for later be thermally produced in our Universe, at the present time, study. Hopefully the vague idea that these fluctuations are without suppression. If this estimate were valid, then less classical than de Sitter space fluctuations can be Boltzmann brains would almost certainly dominate the promoted into a persuasive argument that they are not Universe. relevant. 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 infor- B. Vacuum decay rates mation. The entire device has energy of order E Treheat, One of the most developed approaches to the string which is divided among approximately 1016 massless par- landscape scenario is based on the KKLT construction ticles. The mean particle energy is therefore 1016 times [63]. In this construction, one begins by finding a set of smaller than that of the thermal particles in the background stabilized supersymmetric anti–de Sitter (AdS) and radiation, and the density of Boltzmann brain particles is Minkowski vacua. After that, an uplifting is performed, 1048 times smaller than the background. To function, it e.g. by adding a D3-brane at the tip of a conifold [63]. This seems reasonable that the diffuse computer needs an uplifting makes the vacuum-energy density of some of energy per particle that is at least comparable to the back- these vacua positive (AdS ! dS), but, in general, many ground, which means that the suppression factor is vacua remain AdS, and the Minkowski vacuum corre- expð1016Þ or smaller. Thus, we conclude that for all three sponding to the uncompactified 10D space does not be- cases, the ratio of Boltzmann brains to normal observers is come uplifted. The enormous number of vacua in the totally negligible. landscape appears because of the large number of different Finally, let us also mention the possibility that topologies of the compactified space, and the large number Boltzmann brains might form as quantum fluctuations in of different fluxes and branes associated with it. stable Minkowski vacua. String theory implies at least the There are many ways in which our low-energy dS existence of a 10D decompactified Minkowski vacuum; vacuum may decay. First of all, it can always decay into Minkowski vacua of lower dimension are not excluded, but the Minkowski vacuum corresponding to the uncompacti- they require precise fine-tunings for which motivation is fied 10D space [63]. It can also decay to one of the AdS lacking. While quantum fluctuations in Minkowski space vacua corresponding to the same set of branes and fluxes are certainly less classical than in de Sitter space, they still [64]. More generally, decays occur due to the jumps be- might be relevant. The possibility of Boltzmann brains in tween vacua with different fluxes, or due to the brane-flux Minkowski space has been suggested by Page [5,6,40]. If annihilation [54,65–71], and may be accompanied by a BB is nonzero in such vacua, regardless of how small it change in the number of compact dimensions [72–74]. If might be, Boltzmann brains will always dominate in the one does not take into account vacuum stabilization, these scale-factor cutoff measure as we have defined it. Even if transitions are relatively easy to analyze [65–67]. However, Minkowski vacua cannot support Boltzmann brains, there in the realistic situations where the moduli fields are might still be a serious problem with what might be called determined by fluxes, branes, etc., these transitions involve ‘‘Boltzmann islands.’’ That is, it is conceivable that a a simultaneous change of fluxes and various moduli fields, fluctuation in a Minkowski vacuum can produce a small which makes a detailed analysis of the tunneling quite region of an anthropic vacuum with a Boltzmann brain complicated. inside it. The anthropic vacuum could perhaps even have a Therefore, we begin with an investigation of the simplest different dimension than its Minkowski parent. If such a decay modes due to the scalar field tunneling. The tran- process has a nonvanishing probability to occur, it will also sition to the 10D Minkowski vacuum was analyzed in give rise to Boltzmann brain domination in the scale-factor Ref. [63], where it was shown that the decay rate is cutoff measure. These problems would be shared by all always greater than measures that assign an infinite weight to stable 242 Minkowski vacua. There is, however, one further compli- * eSD ¼ exp : (121) V cation which might allow Boltzmann brains to form in dS Minkowski space without dominating the multiverse. If Here SD is the entropy of dS space. For our vacuum, 120 one speculates about Boltzmann brain production in SD 10 , which yields

063520-22 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010)

* eSD expð10120Þ: (122) follows we describe tunneling in one such direction. Furthermore, we assume that at least some of the AdS Because of the inequality in Eq. (121), we expect the vacua to which our dS vacuum may decay are uplifted slowest-decaying vacua to typically be those with very much less than our vacuum. This is a generic situation, small vacuum energies, with the dominant vacuum-energy since the uplifting depends on the value of the volume density possibly being much smaller than the value in our modulus, which takes different values in each vacuum. Universe. In this case the decay rate of a dS vacuum with The decay to AdS space (or, more accurately, a decay to low-energy density and broken supersymmetry can be a collapsing open universe with a negative cosmological estimated as follows [64,84]: constant) was studied in Ref. [64]. The results of Ref. [64] 2 are based on the investigation of Bogomol’nyi-Prasad- 8 exp 2 ; (123) Sommerfield (BPS) and near-BPS domain walls in string 3m3=2 theory, generalizing the results previously obtained in N ¼ 1 supergravity [75–78]. Here we briefly summarize where m3=2 is the gravitino mass in that vacuum and is a the main results obtained in Ref. [64]. quantity that depends on the parameters of the potential. Consider, for simplicity, the situation where the tunnel- Generically one can expect ¼ Oð1Þ, but it can also be ing occurs between two vacua with very small vacuum much greater or much smaller than Oð1Þ. The mass m3=2 is energies. For the sake of argument, let us first ignore the set by the scale of SUSY breaking, gravitational effects. Then the tunneling always takes 2 4 place, as long as one vacuum has higher vacuum energy 3m3=2 ¼ SUSY; (124) than the other. In the limit when the difference between the vacuum energies goes to zero, the radius of the bubble of where we recall that we use reduced Planck units, the new vacuum becomes infinitely large, R !1(the thin- 8G ¼ 1. Therefore the decay rate can also be repre- wall limit). In this limit, the bubble wall becomes flat, and sented in terms of the SUSY-breaking scale SUSY: its initial acceleration, at the moment when the bubble 242 forms, vanishes. Therefore, to find the tension of the exp 4 : (125) domain wall in the thin-wall approximation one should SUSY solve an equation for the scalar field describing a static 4 Note that in the KKLT theory, SUSY corresponds to the domain wall separating the two vacua. depth of the AdS vacuum before the uplifting, so that If the difference between the values of the scalar potential in the two minima is too small, and at least one of them 242 exp : (126) is AdS, then the tunneling between them may be forbidden jV j because of the gravitational effects [79]. In particular, all AdS supersymmetric vacua, including all KKLT vacua prior to In this form, the result for the tunneling looks very the uplifting, are absolutely stable even if other vacua with similar to the lower bound on the decay rate of a lower energy density are available [80–83]. dS vacuum, Eq. (121), with the obvious replacements It is tempting to make a closely related but opposite ! 1 and jVAdSj!VdS. statement: nonsupersymmetric vacua are always unstable. Let us apply this result to the question of vacuum decay However, this is not always the case. In order to study in our Universe. Clearly, the implications of Eq. (125) tunneling while taking account of supersymmetry (SUSY), depend on the details of SUSY phenomenology. The stan- one may start with two different supersymmetric vacua in dard requirement that the gaugino mass and the scalar two different parts of the Universe and find a BPS domain masses are Oð1Þ TeV leads to the lower bound wall separating them. One can show that if the superpo- * 4 5 tential does not change its sign on the way from one SUSY 10 –10 GeV; (127) vacuum to the other, then this domain wall plays the which can be reached, e.g., in the models of conformal same role as the flat domain wall in the no-gravity case gauge mediation [85]. This implies that for our vacuum discussed above: it corresponds to the wall of the bubble that can be formed once the supersymmetry is broken in * 56 60 our expð10 Þexpð10 Þ: (128) either of the two minima. However, if the superpotential does change its sign, then only a sufficiently large super- Using Eq. (99), the Boltzmann brain nucleation rate in our symmetry breaking will lead to the tunneling [64,75]. Universe exceeds the lower bound of the above inequality One should keep this fact in mind, but since we are only if M & 109 kg. discussing a landscape with an extremely large number On the other hand, one can imagine universes very of vacua, in what follows we assume that there is at least similar to ours except with much larger vacuum-energy one direction in which the superpotential does not change densities. The vacuum decay rate of Eq. (123) exceeds the its sign on the way from one minimum to another. In what Boltzmann brain nucleation rate of Eq. (99) when

063520-23 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) m 2 M H1 Nevertheless, the results of Refs. [54,68,69] show that the 3=2 * 109: (129) 102 eV 1kg 108 m decay rate of dS vacua in the landscape can be quite large. The rate * expð1022Þ is much greater than the 1 8 Note that H 10 m corresponds to the smallest expected rate of Boltzmann brain production given by de Sitter radius for which the tidal force on a 10 cm Eq. (105). However, it is just a bit smaller than the bosonic brain does not exceed the gravitational force on the surface gas Boltzmann brain production rate of Eq. (119) and 2 of the Earth, while m3=2 10 eV corresponds to much smaller than our most dangerous upper bound on 4 the Boltzmann brain production rate, given by Eq. (118). SUSY 10 GeV. Thus, it appears the decay rate of Eq. (123) allows for Boltzmann brain domination. However, we do not really know whether the models VI. CONCLUSIONS with low SUSY can successfully describe our world. To mention one potential problem, in models of string infla- If the observed accelerating expansion of the Universe is tion there is a generic constraint that during the last stage of driven by constant vacuum-energy density and if our & Universe does not decay in the next 20 109 yrs or so, inflation one has H m3=2 [86]. If we assume the second and third factors of Eq. (129) cannot be made much less then it seems cosmology must explain why we are ‘‘normal * O 2 observers’’—who evolve from nonequilibrium processes than unity, then we only require m3=2 ð10 Þ eV to avoid Boltzmann brain domination. While models of string in the wake of the big bang—as opposed to ‘‘Boltzmann inflation with H & 100 eV are not entirely impossible in brains’’—freak observers that arise as a result of rare the string landscape, they are extremely difficult to con- quantum fluctuations [2–4,7,8]. Put in experimental 4 terms, cosmology must explain why we observe structure struct [87]. If instead of SUSY 10 GeV one uses 11 formation in a residual cosmic microwave background, as SUSY 10 GeV, as in models with gravity mediation, 3 opposed to the empty, vacuum-energy dominated environ- one finds m3=2 10 GeV, and Eq. (129) is easily satisfied. ment in which almost all Boltzmann brains nucleate. As These arguments apply when supersymmetry violation vacuum-energy expansion is eternal to the future, the is as large as or larger than in our Universe. If supersym- number of Boltzmann brains in an initially finite comoving metry violation is too small, atomic systems are unstable volume is infinite. However, if there exists a landscape of [88], the masses of some of the particles will change vacua, then rare transitions to other vacua populate a dramatically, etc. However, the Boltzmann computers diverging number of universes in this comoving volume, described in the previous subsection do not necessarily creating an infinite number of normal observers. To weigh rely on laws of physics similar to those in our Universe the relative number of Boltzmann brains to normal observ- (in fact, they seem to require very different laws of ers requires a spacetime measure to regulate the infinities. physics). The present authors are unaware of an argument Recently, the scale-factor cutoff measure was shown to that supersymmetry breaking must be so strong that possess a number of desirable attributes, including avoid- vacuum decay is always faster than the Boltzmann brain ing the youngness paradox [28] and the Q (and G) catas- production rate of Eq. (118). trophe [29–31], while predicting the cosmological constant On the other hand, up to this point we have used the to be measured in a range including the observed value, estimates of the vacuum decay rate that were obtained in and excluding values more than about a factor of 10 larger Refs. [64,84] by investigation of the transition where and smaller than this [38]. The scale-factor cutoff does not only moduli fields changed. As we have already men- itself select for a longer duration of slow-roll inflation, tioned, the description of a more general class of transi- raising the possibility that a significant fraction of observ- tions involving the change of branes or fluxes is much more ers like us measure cosmic curvature significantly above complicated. Investigation of such processes, performed in the value expected from cosmic variance [48]. In this Refs. [54,68,69], indicates that the process of vacuum paper, we have calculated the ratio of the total number of decay for any vacuum in the KKLT scenario should be Boltzmann brains to the number of normal observers, using rather fast, the scale-factor cutoff. The general conditions under which Boltzmann brain * expð1022Þ: (130) domination is avoided were discussed in Sec. IV F, where we described several alternative criteria that can be used to The results of Refs. [54,68,69], like the results of ensure safety from Boltzmann brains. We also explored a Refs. [64,84], are not completely generic. In particular, set of assumptions that allow one to state conditions that the investigations of Refs. [54,68,69] apply to the original are both necessary and sufficient to avoid Boltzmann brain version of the KKLT scenario, where the uplifting of the domination. One relatively simple way to ensure safety AdS vacuum occurs due to D3-branes, but not to its from Boltzmann brains is to require two conditions: (1) in generalization proposed in Ref. [89], where the uplifting any vacuum, the Boltzmann brain nucleation rate must be is achieved due to D7-branes. Nor does it apply to the less than the decay rate of that vacuum, and (2) for any recent version of dS stabilization proposed in Ref. [90]. anthropic vacuum j with a decay rate j q, and for any

063520-24 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) nonanthropic vacuum j, one must construct a sequence of even if the probability of their production is not strongly transitions from j to an anthropic vacuum; if the sequence suppressed. includes suppressed upward jumps, then the Boltzmann In any case, our present understanding of the Boltzmann brain nucleation rate in vacuum j must be less than the brain problem does not rule out the scale-factor cutoff decay rate of vacuum j times the product of all the sup- measure, but the situation remains uncertain. pressed branching ratios Bup that appear in the sequence. The condition (2) might not be too difficult to satisfy, since ACKNOWLEDGMENTS it will generically involve only states with very low- We thank Raphael Bousso, Ben Freivogel, I-Sheng vacuum-energy densities, which are likely to be nearly Yang, Shamit Kachru, Renata Kallosh, Delia Schwartz- supersymmetric and therefore unlikely to support the com- Perlov, and Lenny Susskind for useful discussion. The plex structures needed for Boltzmann brains or normal work of A. D. S. is supported in part by the INFN and in observers. Condition (2) can also be satisfied if there is part by the U.S. Department of Energy (DOE) under no unique dominant vacuum, but instead a dominant Contract No. DE-FG02-05ER41360. A. H. G. is supported vacuum system that consists of a set of nearly degenerate in part by the DOE under Contract No. DE-FG02- states, some of which are anthropic, which undergo rapid 05ER41360. A. L. and M. N. are supported by the NSF transitions to each other, but only slow transitions to other Grant No. 0756174. M. P.S. and A. V. are supported in part states. Condition (1) is perhaps more difficult to satisfy. by the U.S. National Science Foundation under Grant Although nearly supersymmetric string vacua can, in prin- No. NSF 322, and A. V. is also supported in part by a grant ciple, be long-lived [63,64,75–78], with decay rates possi- from the Foundational Questions Institute (FQXi). bly much smaller than the Boltzmann brain nucleation rate, recent investigations suggest that other decay channels APPENDIX: BOLTZMANN BRAINS IN may evade this problem [54,68,69]. However, the decay SCHWARZSCHILD–DE SITTER SPACE processes studied in [54,63,64,68,69,75–78] do not describe some of the situations which are possible in the As explained in Sec. VA, Eq. (100) for the production string theory landscape, and the strongest constraints on rate of Boltzmann brains must be reexamined when the decay rate obtained in [54] are still insufficient to the Boltzmann brain radius becomes comparable to the guarantee that the vacuum decay rate is always smaller de Sitter radius. In this case we need to describe the than the fastest estimate of the Boltzmann brain production Boltzmann brain nucleation as a transition from an initial 1 rate, Eq. (118). state of empty de Sitter space with horizon radius H to a One must emphasize that we are discussing a rapidly final state in which the dS space is altered by the presence developing field of knowledge. Our estimates of the of an object with mass M. Assuming that the object can be Boltzmann brain production rate are exponentially sensi- treated as spherically symmetric, the space outside the tive to our understanding of what exactly the Boltzmann object is described by the Schwarzschild–de Sitter (SdS) brain is. Similarly, the estimates of the decay rate in the metric [91]16: landscape became possible only five years ago, and this 1 2 2GM 2 2 2 2GM 2 2 subject certainly is going to evolve. Therefore we will ds ¼ 1 Hr dt þ 1 Hr mention here two logical possibilities which may emerge r r as a result of the further investigation of these issues. dr2 þ r2d2: (A1) If further investigation will demonstrate that the Boltzmann brain production rate is always smaller than The SdS metric has two horizons, determined by the the vacuum decay rate in the landscape, the probability positive zeros of gtt, where the smaller and larger are called measure that we are investigating in this paper will be RSch and RdS, respectively. We assume the Boltzmann shown not to suffer from the Boltzmann brain problem. brain is stable but not a black hole, so its radius satisfies Conversely, if one believes that this measure is correct, the RSch

063520-25 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010)

This last equation implies that for a given value of H, of M, the tightest bound (for R ¼ RdS) is the holographic there is an upper limit on how much mass can be contained bound, which states that within the de Sitter horizon: 2 pffiffiffi S SH RdS: (A12) 1 G M Mmax ¼ð3 3GHÞ : (A4) Bousso suggests the possibility that these bounds have a Equations (A2) and (A3) can be inverted to express M and common origin, in which case one would expect that there H in terms of the horizon radii: exists a valid bound that interpolates smoothly between the two. Specifically, he points out that the function 1 ¼ R2 þ R2 þ R R ; (A5) H2 Sch dS Sch dS S R R (A13) m G Sch dS is a candidate for such a function. Fig. 1 shows a graph of RdS M ¼ ð1 H2 R2 Þ (A6) the holographic bound, the D bound, and the m bound 2G dS [Eq. (A13)] as a function of M=Mmax. While there is no reason to assume that S is a rigorous bound, it is known to R m Sch 2 2 be valid in the extreme cases where it reduces to the D and ¼ ð1 HRSchÞ: (A7) 2G holographic bounds. In between it might be valid, but in We relate the Boltzmann brain nucleation rate to the any case it can be expected to be valid up to a correction of decrease in total entropy S caused by the nucleation order one. In fact, Fig. 1 and the associated equations show process, that the worst possible violation of the m bound is at the point wherepffiffiffi the holographic and D bounds cross, at eS; (A8) BB M=Mmax ¼ 3 6p=8ffiffiffi ¼ 0:9186, where the entropy can be no where the final entropy is the sum of the entropies of the more than ð1 þ 5Þ=2 ¼ 1:6180 times as large as Sm. Boltzmann brain and the de Sitter horizon. For a Here we wish to carry the notion of interpolation one step further, because we would like to discuss in the same Boltzmann brain with entropy SBB, the change in entropy is given by formalism systems for which R RdS, where the Bekenstein bound should apply. Hence we will explore 2 2 the consequences of the bound S ¼ H RdS þ SBB : (A9) G G S S R R; (A14) Note that for small M one can expand S to find I G Sch 2M which we will call the interpolating bound. This bound O 2 S ¼ SBB þ ðGM Þ; (A10) agrees exactly with the m bound when the object is allowed H giving a nucleation rate in agreement with Eq. (100).17 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. (107), a bound which should also be applicable whenever R RdS. More general bounds in de Sitter space have been discussed by Bousso [53], who considers bounds for systems that are allowed to fill the de Sitter space out to the horizon R ¼ 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 1 2 2 S SD 2 RdS ¼ ðRSch þ RSchRdSÞ; (A11) G H G where the equality of the two expressions follows from Eq. (A5). This bound can be obtained from the principle FIG. 1. The graph shows the holographic bound, the D bound, that the total entropy cannot increase when an object and the m bound for the entropy of an object that fills de Sitter disappears through the de Sitter horizon. For larger values 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 17We thank Lenny Susskind for explaining this method to us. line, it is not.

063520-26 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) pffiffiffiffiffiffiffiffiffiffiffiffi to fill de Sitter space, with R ¼ RdS. Again we have no I~ 1 þ I~ grounds to assume that the bound is rigorously true, but we M ¼ ; (A21) sat 2GH do know that it is true in the three limiting cases where it reduces to the Bekenstein bound, the D bound, and the pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi holographic bound. The limiting cases are generally the 1 þ I~ 1 3I~ RSch;sat ¼ ; (A22) most interesting for us in any case, since we wish to explore 2H the limiting cases for Boltzmann brain nucleation. For pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi parameters in between the limiting cases, it again seems 1 3I~þ 1 þ I~ reasonable to assume that the bound is at least a valid RdS;sat ¼ : (A23) 2H estimate, presumably accurate up to a factor of order one. We know of no rigorous entropy bounds for Combining these results with Eq. (A18), one has for this de Sitter space with R comparable to RdS but not equal to case (R ¼ RdS) the bound it, so we do not see any way at this time to do better than the pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 1 þ I~ 1 þ I~ 1 3I~ interpolating bound. : (A24) Proceeding with the I bound of Eq. (A14), we can use IBB 2I~ Eq. (106) to rewrite Eq. (A9)as As can be seen in Fig. 2, the bound on S=IBB for this case 2 2 varies from 1, in the limit of vanishing I~ (or equivalently, S ¼ ðH RdSÞSBB;max þ IBB; (A15) G the limit H ! 0), to 2, in the limit RSch ! RdS. The limiting case of I~ ! 0, with a nucleation rate of which can be combined with S S to give BB BB;max I order eIBB , has some peculiar features that are worth mentioning. The nucleation rate describes the nucleation S ðH2 R2 R RÞþI ; (A16) G dS Sch BB of a Boltzmann brain with some particular memory state, so there would be an extra factor of eIBB in the sum over which can then be simplified using Eq. (A5)togive all memory states. Thus, a single-state nucleation rate of eIBB indicates that the total nucleation rate, including all S R ðR þ R RÞþI : (A17) G Sch Sch dS BB memory states, is not suppressed at all. It may seem strange that the nucleation rate could be unsuppressed, but one To continue, we have to decide what possibilities to must keep in mind that the system will function as a consider for the radius R of the Boltzmann brain, which Boltzmann brain only for very special values of the mem- is related to the question of Boltzmann brain stabilization ory state. In the limiting case discussed here, the discussed after Eq. (111). If we assume that stabilization is ‘‘Boltzmann brain’’ takes the form of a minor perturbation not a problem, because it can be achieved by a domain wall of the degrees of freedom associated with the de Sitter or by some other particle physics mechanism, then S is 2 entropy SdS ¼ =ðGHÞ. minimized by taking R at its maximum value, R ¼ RdS,so S R2 þ I : (A18) G Sch BB 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 again the I bound, one finds that saturation of the bound occurs at pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3I~ ¼ 3sin1 ; (A19) sat 2 where

2 IBB GH I~ ¼ IBB (A20) SdS is the ratio of the Boltzmann brain information to the entropy of the unperturbed de Sitter space. Note that I~ FIG. 2. The graph shows the ratio of S to IBB, where the varies from zero to a maximum value of 1=3, which occurs nucleation rate for Boltzmann brains is proportional to eS. All in the limiting case for which RSch ¼ RdS. The saturating curves are based on the I bound, as discussed in the text, but they value of the mass and the corresponding values of the differ by their assumptions about the size R of the Boltzmann Schwarzschild radius and de Sitter radius are given by brain.

063520-27 ANDREA DE SIMONE et al. PHYSICAL REVIEW D 82, 063520 (2010) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As a second possibility for the radius R, we can consider 1 2AðI~Þ the case of strong gravitational binding, R ! R ,as ¼ 3sin1 ; (A32) Sch sat 2 2 discussed following Eq. (111). For this case the bound (A17) becomes where S R R þ I : (A25) sin1ð1 27I~3Þ G Sch dS BB AðI~Þsin : (A33) 3 [Interestingly, if we take I ¼ 0 (SBB ¼ Smax) this formula agrees with the result found in Ref. [92] for black hole The saturating value of the mass and the Schwarzschild nucleation in de Sitter space.] With R ¼ R the saturation and de Sitter radii are given by Sch pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the I bound occurs at pffiffiffi 3½1 þ AðI~Þ 1 2AðI~Þ pffiffiffiffiffi M ¼ ; (A34) ~ sat 1 3I 9GH sat ¼ 3sin : (A26) 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ The saturating value of the mass and the corresponding 1 ffiffiffi 2AðIÞ RSch;sat ¼ p ; (A35) values of the Schwarzschild radius and de Sitter radius are 3H given by ffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ~ ~ I~ð1 I~Þ 3½ 3ð3 þ 2AðIÞÞ 1 2AðIÞ M ¼ ; (A27) RdS;sat ¼ : (A36) sat 6H 2GH pffiffiffi The equilibrium radius itself is given by I~ ¼ ~ 1=6 ~ 1=3 RSch;sat ; (A28) ½1 2AðIÞ ffiffiffi ½1 þ AðIÞ H Requil;sat ¼ p : (A37) 3H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 3I~ I~ Using these results with Eq. (A17), S is found to be RdS;sat ¼ : (A29) bounded by 2H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Using these relations to evaluate S from Eq. (A25), one S 3ð1 2AðI~ÞÞð3 þ 2AðI~ÞÞ 2AðI~Þþ1 ¼ ; (A38) finds I 6I~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi BB S 4 3I~þ I~ ¼ pffiffiffi ; (A30) which is also plotted in Fig. 2. As one might expect it is IBB 2 I~ intermediate between the two other cases. Like the R ¼ R case, however, the ratio S=I blows up for small I~, which is also plotted in Fig. 2. In this case (R ¼ R ) the Sch BB Sch in this case behaving as ð2=I~Þ1=4. smallest ratio S=IBB is 2, occurring at I~ ¼ 1=3, where R ¼ R . For smaller values of I~ the ratio becomes In summary, we have found that our study of tunneling Sch dS pffiffiffi in Schwarzschild–de Sitter space confirms the qualitative = I~ I~ larger, blowing up as 1 for small . Thus, the nucleation conclusions that were described in Sec. VA. In particular, R rates for this choice of will be considerably smaller than we have found that if the entropy bound can be saturated, R R those for Boltzmann brains with dS, but this case then the nucleation rate of a Boltzmann brain requiring would still be relevant in cases where Boltzmann brains information content I is given approximately by eaIBB , R R BB with dS cannot be stabilized. where a is of order a few, as in Eq. (111). The coefficient a Another interesting case, which we will consider, is to is always greater than 2 for Boltzmann brains that are small R ¼ R allow the Boltzmann brain to extend to equil, the enough to be gravitationally bound. This conclusion point of equilibrium between the gravitational attraction applies whether one insists that they be near-black holes, of the Boltzmann brain and the outward gravitational pull or whether one merely requires that they be small enough of the de Sitter expansion. This equilibrium occurs at the so that their self-gravity overcomes the de Sitter expansion. stationary point of gtt, which gives If, however, one considers Boltzmann brains whose radius GM 1=3 is allowed to extend to the de Sitter horizon, then Fig. 2 Requil ¼ 2 : (A31) shows that a can come arbitrarily close to 1. However, one H must remember that the R ¼ RdS curve on Fig. 2 can be Boltzmann brains within this radius bound would not be reached only if several barriers can be overcome. First, pulled by the de Sitter expansion, so relatively small me- these objects are large and diffuse, becoming more and chanical forces will be sufficient to hold them together. more diffuse as I~ approaches zero and a approaches 1. Again S will be minimized when the I bound is There is no known way to saturate the entropy bound for saturated, which in this case occurs when such diffuse systems, and Eq. (117) shows that an ideal gas

063520-28 BOLTZMANN BRAINS AND THE SCALE-FACTOR CUTOFF ... PHYSICAL REVIEW D 82, 063520 (2010) 3 1=3 exists with tension IBBH . Thus, it is not clear how model leads to a IBB 1. Furthermore, Boltzmann brains of this size can function only if some particle close a can come to its limiting value of 1. Finally, we physics mechanism is available to stabilize them against should keep in mind that it is not clear if any of the the de Sitter expansion. A domain wall provides a simple examples discussed in this appendix can actually be at- example of such a mechanism, but Eq. (114) indicates that tained, since black holes might be the only objects that the domain wall solution is an option only if a domain wall saturate the entropy bound for S 1.

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