Four-volume cutoff measure of the

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Citation Vilenkin, Alexander and Masaki Yamada. "Four-volume cutoff measure of the multiverse." Physical Review D 101, 4 (February 2020): 043520

As Published http://dx.doi.org/10.1103/physrevd.101.043520

Publisher American Physical Society (APS)

Version Final published version

Citable link https://hdl.handle.net/1721.1/125122

Terms of Use Creative Commons Attribution 3.0 unported license

Detailed Terms http://creativecommons.org/licenses/by/3.0 PHYSICAL REVIEW D 101, 043520 (2020)

Four-volume cutoff measure of the multiverse

† Alexander Vilenkin 1,* and Masaki Yamada2, 1Institute of Cosmology, Department of Physics and Astronomy, , Medford, Massachusetts 02155, USA 2Center for , Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 13 December 2019; accepted 22 January 2020; published 14 February 2020)

Predictions in an eternally inflating multiverse are meaningless unless we specify the probability measure. The scale-factor cutoff is perhaps the simplest and most successful measure which avoids catastrophic problems such as the youngness paradox, runaway problem, and Boltzmann brain problem, but it is not well defined in contracting regions with a negative . In this paper, we propose a new measure with properties similar to the scale-factor cutoff which is well-defined everywhere. The measure is defined by a cutoff in the four volume spanned by infinitesimal comoving neighborhoods in a congruence of timelike geodesics. The probability distributions for the cosmological constant and for the curvature parameter in this measure are similar to those for the scale-factor cutoff and are in a good agreement with observations.

DOI: 10.1103/PhysRevD.101.043520

I. INTRODUCTION followed by contraction, so a starts to decrease along the geodesics. The scale-factor measure then requires that the Observational predictions in multiverse models depend on entire contracting region to the future of the turnaround one’s choice of the probability measure. Different measure point be included under the cutoff. This gives a higher prescriptions can give vastly different answers. This is the weight to regions of negative Λ, so the scale-factor measure so-called measure problem of eternal . Perhaps the tends to predict that we should expect to measure Λ < 0 simplest way to regulate the infinities of is (unless this is strongly suppressed by anthropic factors). to impose a cutoff on a hypersurface of constant global Some other measure proposals have even more severe time. One starts with a patch of a spacelike hypersurface problems with negative Λ. For example, the light cone time Σ somewhere in the inflating region of spacetime and follows cutoff [8] gives an overwhelming preference for Λ < 0 [9].1 its evolution along the congruence of geodesics orthogonal to In this paper, we introduce a new global time measure Σ. The cutoff is imposed at a hypersurface of constant time t which does not suffer from these problems. We divide the measured along the geodesics. The resulting measure, initial hypersurface Σ into infinitesimally small segments of however, depends on the choice of the time variable t. equal three-volume ϵ → 0 and follow the evolution of these An attractive choice is to use the proper time τ along the segments along the orthogonal congruence of geodesics. The geodesics [1–3]. One finds, however, that this proper time time coordinate Ω is defined as the four volume spanned by measure suffers from the youngness paradox, predicting the segment, that the should be much hotter than observed [4]. Another popular choice is the scale-factor time, t ¼ ln a, Z Z 1 pffiffiffiffiffiffi τ where a is the expansion factor along the geodesics ΩðτÞ¼ −gd4x ¼ dτ0Vð3Þðτ0Þ; ð1:1Þ [1,2,5–7]. The problem with this choice is that the scale- ϵ ð0;τÞ×ϵVð3ÞðτÞ 0 factor evolution is not monotonic. For example, in regions with a negative cosmological constant, Λ < 0, expansion is 1Local measure proposals, which sample spacetime regions around individual geodesics with subsequent averaging over an ensemble of geodesics, yield probability distributions that sensi- *[email protected] tively depend on the choice of the ensemble. This choice is † [email protected] largely arbitrary, and thus these proposals are incomplete as they now stand. The “watcher measure” of Ref. [10] follows a single Published by the American Physical Society under the terms of “eternal” geodesic, but makes the assumption that the big crunch the Creative Commons Attribution 4.0 International license. singularities in AdS bubbles lead to bounces, where contraction is Further distribution of this work must maintain attribution to followed by expansion, so that geodesics can be continued the author(s) and the published article’s title, journal citation, through the crunch regions. We do not adopt this assumption and DOI. Funded by SCOAP3. in the present paper.

2470-0010=2020=101(4)=043520(8) 043520-1 Published by the American Physical Society ALEXANDER VILENKIN and MASAKI YAMADA PHYS. REV. D 101, 043520 (2020) where ϵVð3ÞðτÞ is the three volume of the evolved segment at by different vacua in the eternally inflating part of space- proper time τ, τ is set equal to zero at Σ,andVð3Þð0Þ¼1. Ω time, assuming low transition rates between the vacua. has a clear geometric meaning and it clearly grows mono- In Secs. 3 and 4, we find, respectively, the probability tonically along the geodesics. The measure is defined by distributions for the cosmological constant and for the density parameter (or spatial curvature) under assumptions imposing a cutoff at Ωc ¼ const. If the Universe can locally be approximated as homogeneous and isotropic, we can similar to those that were used in Refs. [5,12] to calculate write Vð3ÞðτÞ¼a3ðτÞ, where aðτÞ is the scale factor with these distributions in the scale-factor measure. A formalism að0Þ¼1. Then, that can be used to determine the distributions in more Z general landscapes is outlined in Sec. V. Finally, our results τ are briefly summarized and discussed in Sec. VI. ΩðτÞ¼ dτ0a3ðτ0Þ: ð1:2Þ 0 II. VOLUME DISTRIBUTION OF VACUA We can think of the geodesics in the congruence as Consider a multiverse consisting of bubbles of de Sitter representing an ensemble of inertial observers spread uni- (dS) and terminal (anti–de Sitter [AdS] and Minkowski) formly over the initial surface Σ. The measure prescription is vacua, labeled by index j. The expansion rate of dS vacuum then that each observer samples an equal four volume ∝ Ω . c j is H and nucleation rate of bubbles of vacuum i in parent The distribution of “observers” may become rather j vacuum j per Hubble volume per Hubble time is κij. irregular in regions of structure formation. The scale factor κ ≪ 1— [or the three-volume Vð3Þ in Eq. (1.1)] comes to a halt in We shall assume that ij which is expected, since collapsed regions which have decoupled from the Hubble nucleation occurs by quantum tunneling. In this section, we flow and continues to evolve between these regions. shall calculate the three volume occupied by each dS Ω Furthermore, the geodesic congruence may develop caus- vacuum on a surface of constant in the inflating part tics where geodesics cross. One can adopt the rule that of spacetime and use the result to find the abundances of geodesics are terminated as they cross at a caustic. As it was Boltzmann brains in dS vacua. We shall not be interested in noted in Ref. [11], this does not create any gaps in the volumes occupied by terminal vacua in this section. congruence. But the resulting cutoff surface would still be rather irregular. Such dependence of the measure on details A. Relation to scale-factor cutoff of structure formation appears unsatisfactory and calls for An approximate relation between the four-volume and some sort of coarse graining, with averaging over the scale-factor cutoffs can be found if we note that the scale characteristic length scale of structure formation. This issue factor grows exponentially in the inflating regions, and was emphasized in Ref. [6] in the case of scale-factor therefore the integral in Eq. (1.2) is dominated by the measure and was further discussed in Ref. [7]. upper limit. In a region occupied by vacuum j, the scale Ω Hjτ A somewhat related problem is that even though factor is ajðτÞ¼Ce with C ¼ const., so we can write grows monotonically along geodesics of the congruence, approximately the surfaces of constant Ω are not necessarily spacelike, Z so Ω is not a good global time coordinate. As a result, an τ a3ðτÞ Ω ðτÞ ≈ 3ðτ0Þ τ0 ≈ j ð Þ event may be included under the cutoff, while some events j a d : 2:1 j 3H in its causal past are not included. A possible way to cure j Ω ¼ Ω this problem is to modify the cutoff surface c by Ω ¼ Ω ¼ 2 The cutoff surface at c const can then be excluding future light cones of all points on that surface. approximated as Then all events under the cutoff are included together with their causal past. This prescription also alleviates the a3ðτÞ problem of sensitivity of the measure to structure forma- ¼ Ω ; ð2:2Þ 3H c tion. If the characteristic scale of structure formation is j much smaller than the horizon, the modified cutoff surface so the four-volume cutoff at Ω ¼ Ω is approximately would roughly coincide with a constant Ω surface in the c equivalent to the scale-factor cutoff at background Friedmann-Robertson-Walker geometry. The implementation of the four-volume measure is 1 ¼ ð3 Ω Þ ð Þ somewhat more complicated than in the cases of proper tc 3 ln Hj c ; 2:3 time and scale-factor measures, but it becomes tractable in a number of interesting special cases. In the next section, we where the scale-factor time is defined as t ¼ ln a. use this measure to estimate the volume fraction occupied The approximations (2.1) and (2.3) are accurate, as long as the cutoff surface does not pass within a few Hubble times 2This prescription was suggested in Ref. [7] to address a of a transition from one vacuum to another (on the daughter ∼ 3 3 similar problem for the scale-factor measure. vacuum side). The correction to Eq. (2.1) is ai = Hi,where

043520-2 FOUR-VOLUME CUTOFF MEASURE OF THE MULTIVERSE PHYS. REV. D 101, 043520 (2020) X ai is the scale factor at the time when the vacuum region j ðκi − qÞsi ¼ κijsj; ð2:9Þ being considered was created from a parent vacuum i.If j Hj ≲ Hi, which is usually the case, this correction is negligible already at one Hubble time after the transition and s is the corresponding eigenvector. → ¼ ≲ −3 ≈ 1 20 i i j, when a=ai e and the correction is e = . Substituting Eq. (2.3) in Eq. (2.8), we find The correction is more significant for large upward jumps with H ≫ H . In this case, the condition for Eq. (2.3) to be j i ðΩ Þ¼ ð3 Ω Þ1−q=3 ð Þ 1=3 Vi c si Hi c : 2:10 accurate is a=ai ≳ ðHj=HiÞ ≫ 1. This would happen on some segment of the cutoff surface if it lies within a scale- factor time tji ∼ ð1=3Þ lnðHj=HiÞ of the transition from i to q is an exponentially small number, so to a good approxi- j (on the side of j). We expect such segments to be rare— mation we can write both because large upward jumps are strongly suppressed and because the interval t is much shorter than the scale- ji ViðΩcÞ ∝ siHi: ð2:11Þ factor time that geodesics typically spend in vacuum j.Thus, we expect the approximations (2.1) and (2.3) to hold for a This is the (approximate) asymptotic volume distribution generic cutoff surface. in the four-volume cutoff measure. Compared to the scale- Similar approximations should apply in spacetime factor measure, the volume of faster expanding vacua is regions where the Hubble parameter H is not constant, enhanced by a factor H . but varies on a timescale much longer that H−1 (e.g., in i The distribution (2.11) can be used to find the abundance quantum diffusion or slow-roll regions). In this case, of Boltzmann brains (BBs) in different dS vacua. Suppose Eq. (2.1) is replaced by ΓBB BBs are produced in vacuum i at a rate i per unit BB 3 spacetime volume. The number of BBs N is then a ðτÞ i ΩðτÞ ≈ : ð2:4Þ proportional to the total four volume in that vacuum. 3HðτÞ With a scale-factor cutoff at t ¼ tc, this volume is B. Volume distribution and Boltzmann brains Z Z t t ð4Þð Þ¼ c ð Þ τ ¼ −1 c ð Þ We can now find the volume distribution of different Vi tc Vi t d Hi Vi t dt vacua. We start with the volume distribution on constant 1 −1 ð3− Þ scale-factor surfaces and then rewrite the result on a ¼ H s e q tc ; ð2:12Þ 3 − q i i constant four-volume surface by using Eq. (2.3). The former distribution can be found from the rate equation (see, e.g., [7]), where we have used Eq. (2.8). Now, using Eq. (2.3) to express t in terms of Ω , we find X c c dVi ¼ 3Vi þ MijVj; ð2:5Þ dt ð4Þ − j ðΩ Þ ∝ q ð Þ Vi c siHi 2:13 where ViðtÞ is the volume occupied by vacuum i on a constant scale-factor surface t ¼ const within a region of a and fixed comoving size, t is the scale-factor time, BB ∝ ΓBB ð Þ Ni si; 2:14 Mij ¼ κij − δijκi ð2:6Þ −q ≈ 1 is the transition matrix, and where we have approximated Hi . X The difference from the scale-factor cutoff measure, κ ¼ κ ð2:7Þ BB ∝ ΓBB −1 i ri which gives [6,7] Ni i Hi si is only by a factor of Hi, r which is not exponentially large. Thus, the analysis of the is the total decay rate of vacuum i per Hubble volume per Boltzmann-brane problem in the four-volume cutoff mea- Hubble time. The late-time asymptotic solution of this sure is (almost) the same as that in the scale-factor measure. equation for dS vacua i is Since the problem can be evaded in the latter measure [6,7], we conclude that the four-volume cutoff measure may also ð3−qÞt ViðtÞ¼sie ; ð2:8Þ be free from the Boltzmann-brane problem, depending on the properties of the landscape. We expect the conditions where q>0 is the smallest solution of the eigenvalue for avoidance of the BB problem to be very similar to those equation in the scale-factor measure.

043520-3 ALEXANDER VILENKIN and MASAKI YAMADA PHYS. REV. D 101, 043520 (2020)

γt III. PROBABILITY DISTRIBUTION FOR dV ∝ e dt; ð3:5Þ COSMOLOGICAL CONSTANT

where t ¼ ln a and γ ¼ 3 − q ≈ 3. Expressing t in terms In this section, we calculate the probability distribution Ω for the cosmological constant Λ under the same assump- of ,wehave tions that were used in Ref. [5] for the scale-factor measure. V ∝ Ωγ−3 Ω ≈ Ω ð Þ Specifically, we focus on a subset of bubbles that have d d d ; 3:6 (nearly) the same physical properties as our bubble, apart from the value of Λ. We shall assume that the number of which says that thermalized volume is produced at approx- such bubble types in the landscape is very large, so the imately constant rate per unit four volume. distribution of Λ is nearly continuous. After nucleation, After thermalization, density perturbations grow, some each bubble goes through a period of slow-roll inflation, fraction of matter clusters into galaxies, and observers followed by periods of radiation and matter domination, evolve in some of these galaxies. The probability distri- Λ until Λ eventually starts to dominate. We will be interested bution for is proportional to the number of observers in Λ in the values of Λ for which this happens late in the regions with that value of . We assume that the number of observers is proportional to the number of large galaxies matter era. 12 M ≳ M ∼10 M⊙ Let a˜ ΛðτÞ be the scale factor in a region with a given with mass G ( ). Then, the probability value of Λ, where the proper time τ is measured from the distribution can be expressed as ˜ Z moment of thermalization (end of inflation) and a is Ω ˜ð0Þ¼1 c normalized so that a . We can define a reference PðΛÞ ∝ Fðτc − ΔτÞdΩ; ð3:7Þ τ τ ≪ τ ≪ τ τ 0 time m such that eq m Λ, where eq is the time of equal matter and radiation densities and τΛ is the time of Λ where FðτÞ is the fraction of matter that clusters into large domination. Then the evolution before τ is the same in all m galaxies at proper time τ after thermalization, Δτ is the time regions, while after τm the scale factor is given by required for observers to evolve, and τc is expressed in ( Ω Ω 3 −2=3 2=3 3 terms of c= from Eq. (3.4). Introducing a new variable a˜ ð HΛτ Þ sinh ð HΛτÞ for Λ > 0 m 2 m 2 X ¼ Ω=Ω , we can write a˜ ΛðτÞ¼ c 3 −2=3 2=3 3 a˜ ð HΛτ Þ sin ð HΛτÞ for Λ < 0; Z m 2 m 2 1 ðΛÞ¼ ðτ ð Þ − ΔτÞ ð Þ ð3:1Þ P N F c X dX; 3:8 0 pffiffiffiffiffiffiffiffiffiffiffi ¼ jΛj 3 ˜ ¼ ˜ðτ Þ where HΛ = . Here, am a m ; it depends on the Rwhere N is a normalization constant determined by −2=3 ðΛÞ Λ Λ ¼ 1 Λ evolution prior to τm, but the quantity a˜ mτm is indepen- P d = obs with obs being the observed value of dent of τm (and of Λ). A cutoff at Ω ¼ Ωc in a bubble cosmological constant. thermalized at Ω with a scale factor a corresponds to a In Eqs. (3.7) and (3.8), we implicitly assumed that Λ 0 cutoff at proper time τc, which can be found from > . When the landscape includes AdS vacua with Z Λ < 0, some of the AdS regions will crunch prior to the τ Ω ¼ Ω þ 3 c ˜ 3 ðτÞ τ ð Þ cutoff, and such regions should be treated separately. The c a aΛ d : 3:2 Λ 0 0 probability distribution for < should be calculated from From Eq. (2.4), we can write Z X ðΛÞ¼ crunch ðτ ð Þ − ΔτÞ P N F c Xcrunch dX 1 3 0 Ω ≈ a; ð3:3Þ Z  3H 1 þ FðτcðXÞ − ΔτÞdX ð3:9Þ X where H is the expansion rate at the end of slow-roll crunch inflation in the bubble. Hence, we can rewrite Eq. (3.2) as  ¼ ðτ − ΔτÞ  Z  N XcrunchF crunch τ c 3 Z  Ωc ≈ Ω 1 þ 3H a˜ ΛðτÞdτ : ð3:4Þ 1 0 þ FðτcðXÞ − ΔτÞdX ; ð3:10Þ Xcrunch The rest of the analysis closely follows Ref. [5], where ≡ ðτ Þ τ ≡ 2π 3 references to earlier literature can also be found. The where Xcrunch X crunch and crunch = HΛ. physical volume thermalizing in a scale-factor time interval We will be interested in regions where τc ≫ τm; then the dt in the spacetime region defined by the geodesic integral in (3.4) is dominated by the range τ ≫ τm,sowe congruence is can use Eq. (3.1) for a˜ ΛðτÞ. This gives

043520-4 FOUR-VOLUME CUTOFF MEASURE OF THE MULTIVERSE PHYS. REV. D 101, 043520 (2020)

0.06 0.35 0.30 0.05 0.25 0.04 0.20 0.03 0.15 0.02 0.10 0.01 0.05 0.00 0.00 –30 –20 –10 0 10 0.01 0.10 1 10 100

FIG. 1. Distribution of cosmological constant in the four-volume cutoff measure (solid blue curve) and the scale-factor cutoff measure (red dashed curve).R The right panel is the probability distribution Λ × PðΛÞ for Λ > 0 in the logarithmic scale. All distributions are normalized as PðΛÞdΛ=Λobs ¼ 1. The lighter (darker) blue-shaded regions represent the 1σ (2σ) ranges for the probability distribution in the four-volume cutoff measure.

8 2 ˜ 3 < Hiam ½ ð3 τ Þ − 3 τ Λ 0 effects that have not been taken into account here. After the 9H3 τ2 sinh HΛ c HΛ c for > X−1 ≈ Λ m turnaround, galaxies begin to accrete matter at a rate that : 2 ˜ 3 Hiam ½− ð3 τ Þþ3 τ Λ 0 increases with time and galactic mergers become more 9H3 τ2 sin HΛ c HΛ c for < . Λ m frequent. This may prevent galaxies from setting into stable ð3:11Þ configurations, which in turn would cause planetary sys- Note that τ is assumed to be smaller than τ ≡ tems to undergo more frequent close encounters with c crunch passing stars. Life extinctions due to nearby supernova 2π=3HΛ for Λ < 0. We use the Press-Schechter form [13,14] with a linear explosions and to gamma-ray bursts would also become perturbation theory for the collapsed fraction FðτÞ. The more frequent. Some of these effects have been discussed distribution PðΛÞ can then be found numerically from in Refs. [16,17]. With all relevant anthropic effects taken Λ Eqs. (3.8) and (3.11), as it was done in Ref. [5]. We use the into account, both distributions for are likely to be in a same parameters as the one used in the same paper (e.g., good agreement with observation. Δτ ¼ 5 × 109 years and the root-mean square fractional density contrast averaged over a comoving scale enclosing IV. PROBABILITY DISTRIBUTION FOR 12 12 SPATIAL CURVATURE mass 10 M⊙ at present σð10 M⊙Þ ≈ 2.03) while we use the updated cosmological parameters from the Planck data, In this section, we use the four-volume cutoff measure ðobsÞ ðobsÞ such as ΩΛ ¼ 0.69 and Ωm ¼ 0.31 [15]. We plot the to calculate the probability distribution for the spatial resulting probability distributions in Fig. 1, with solid blue curvature with a cosmological constant fixed at the and dashed red curves corresponding to four-volume and observed value. Again, we focus on a subset of bubbles scale-factor cutoffs, respectively. The left panel shows the that have the same physical properties as our bubble, apart full distributions, while the right panel shows the (normal- from the e-folding number of the slow-roll inflation inside ized) distributions for positive Λ in the logarithmic scale. the bubble, Ne. The lighter (darker) blue-shaded regions represent the 1σ The spacetime inside a nucleated bubble has a negative (2σ) ranges for the probability distribution in the four- spatial curvature. After a short period of curvature domi- volume cutoff measure. nation, the curvature rapidly decreases due to inflationary To plot the distribution for Λ < 0 in the scale-factor expansion and becomes completely negligible by the end of τ ¼ τ τ τ τ ≡ measure, we set c crunch for c > turn, where turn inflation. However, it may become significant again in the π=3HΛ is the turnaround time when the contracting phase late Universe and may influence structure formation. The τ ≡ 2π 3 begins and crunch = HΛ is the time of the big crunch. density parameter for the spatial curvature at present (i.e., τ τ Since crunch is twice larger than turn, this results in a at the time when the CMB temperature is the same as in our Ω ¼ 1 − ρ ρ ρ discontinuous jump of τc and in a larger probability for Universe at present), k = cr,where cr is the Λ < 0 in the scale-factor cutoff measure. We see however critical density, is related to the e-folding number Ne as −2Ne that the difference between the distributions in the two Ωk ∝ e . The proportionality constant depends on the measures is not dramatic. The total probability for Λ to be detailed history of the Universe after inflation. Since positive is 3% for the scale factor and 8% for the four- the spatial curvature depends on the reference time and volume cutoff measure. the notation for the density parameter may be confused We note that in either measure the probability of negative with the four-volume time, we use a time-independent 3 2 1=3 Λ is expected to be significantly reduced due to anthropic variable k ≡ ðjΩkj =ΩΛΩmÞ in the following calculation.

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For inflation at the grand-unified-theory scale and assuming 0.020 124−2 instantaneous reheating, k ∼ e Ne [12]. Ω 0.015 Let us define nuc as the four-volume time at bubble Ω nucleation. It is related to the time of thermalization as ) k Z ( 0.010 τ

3 3 k P Ω ¼ Ω þ τ ¼ Ω ð1 þ Ne Þ ð Þ nuc a d nuc Ce ; 4:1 0.005 τnuc where C is a constant that is universal for all bubbles. 0.000 We can neglect the factor of 1 in the parenthesis and 10–5 10–4 10–3 10–2 10–1 100 101 102 3 Ω ∝ Ne Ω obtain d e d nuc. k As we discussed in the previous section, the physical Ω FIG. 2. Distribution of spatial curvature in the four-volume volume nucleating in a four-volume interval d nuc is Ω cutoff measure (solid blue curve) and the scale-factor cutoff proportional to d nuc. After thermalization, the number 3 measure (red dashed curve). The two distributions are essentially Ne ðτ − ΔτÞ of observers is proportional to e F c , and hence the same. The shaded regions are allowed by the Planck the distribution is given by constraint. In the blue-shaded region, the spatial curvature may Z be detected in the future. Ω c 3 ð Þ ∝ ð ð ÞÞ Ne ðτ − ΔτÞ Ω P k dk Pprior Ne k dNe e F c d nuc; 0 The result is shown as the solid curve in Fig. 2.This ð Þ 4:2 distribution is almost indistinguishable from that in the scale-factor cutoff measure [12] which is shown by a ð Þ where the prior distribution Pprior Ne is determined by the dashed curve. The Planck data favor a slightly negative landscape. Generally, we expect that long inflation requires value of k [20] but is consistent with a spatially flat ð Þ fine-tuning, so Pprior Ne is a decreasing function of Ne. universe within 2σ [15]. The observationally allowed For a random Gaussian landscape, one finds [18,19] −2 range within 3σ is about jΩkj ≲ 0.01 or jkj ≲ 3 × 10 , which is indicated by shading in the figure. The proba- ð Þ ∝ −3 ð Þ Pprior Ne Ne : 4:3 bility for curvature to be in this range is about 94%. 3 A detection of curvature is probably possible in the future ¼ 0 Ω ∈ ðΩ Ne Ω Þ Noting that F for nuc c=Ce ; c and if k ≳ 3 × 10−4. The range of k where curvature satisfies ∝ 1 dNe=dk =k, we rewrite Eq. (4.2) as the observational bound and is still detectable is shown Z Ω by the blue-shaded region in the figure. The probability ð Þ ∝ −1 ð ð ÞÞ c ðτ − ΔτÞ Ω ð Þ P k k Pprior Ne k F c d : 4:4 for k to be in this range is about 7% [12,21]. 0 V. GENERAL FORMALISM The proportionalityR constant is determined by the normali- zation condition, PðkÞdk ¼ 1. Although the integral in So far, we calculated probability distributions in the four- Eq. (4.4) has the same form as Eq. (3.7), the collapsed volume cutoff measure using the approximate relation (2.4) fraction FðτÞ is different because of the effect of the spatial between the scale-factor and four-volume cutoffs. If a more curvature. Again, we use the Press-Schechter form [13,14] accurate description is needed, the analysis becomes more with a linear perturbation theory for the collapsed fraction complicated. The reason is that in order to evolve the FðτÞ, following Ref. [12]. In that paper, the collapsed distribution to larger values of Ω using dΩ ¼ a3dτ,we 3 function is expressed in terms of x ≡ ρΛ=ρ ∝ a˜ . Then, it need to know the scale factor a, which generally takes m Ω is convenient to rewrite X ≡ Ω=Ωc as different values on different parts of the constant surface. Z Z In this section, we shall introduce a formalism that can in τ c xc dz principle be used to address this issue. X−1 ∝ a˜ 3dτ ∝ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð4:5Þ 0 0 1 þ z−1 þ kz−2=3 We first consider models where eternal inflation is driven by quantum diffusion of a scalar field ϕ. Let us introduce 2 2 −1 −2=3 the distribution function fðΩ; ϕ;VÞ defined as the fraction where we use H ¼ HΛð1 þ x þ kx Þ and define xc as Ω ¼ Ω of comoving volume occupied by regions with given values the value of x at c. We can calculate Eq. (4.4) by 3 rewriting the integral in terms of x and using the collapsed of ϕ and V ¼ a on hypersurfaces of constant Ω. The function given in Ref. [12]. evolution of the multiverse can then be described by the We calculated PðkÞ numerically with the prior distribu- Fokker-Planck equation [22] tion given by Eq. (4.3). We neglect Δτ in Eq. (4.4) for ∂f ∂jϕ ∂j simplicity because it has been argued in Ref. [12] that þ þ V ¼ 0; ð5:1Þ it does not significantly affect the collapsed function. ∂Ω ∂ϕ ∂V

043520-6 FOUR-VOLUME CUTOFF MEASURE OF THE MULTIVERSE PHYS. REV. D 101, 043520 (2020) where the fluxes jϕ and jV are given by differential operator V∂=∂Ω on the left-hand side is a derivative with respect to τ. Once again, the comoving and ∂ dϕ jϕ ¼ − ðDfÞþ f; ð5:2Þ physical volume distributions of different vacua on surfaces ∂ϕ dΩ of constant Ω can be found as Z ¼ dV ð Þ ∞ jV f: 5:3 ðΩÞ¼ ðΩ ÞðÞ dΩ Fi dVf ;V 5:11 0 With dΩ ¼ Vdτ, we can express the drift velocity of ϕ as and dϕ 1 dϕ 1 dH Z ¼ ¼ − ; ð5:4Þ ∞ dΩ V dτ 4πV dϕ FiV ðΩÞ¼ dVVfðΩ;VÞ: ð5:12Þ 0 ðϕÞ¼½ð8π 3 ðϕÞ1=2 where H = U is the inflationary expan- Equations (5.7) and (5.10) are difficult to solve analyti- ðϕÞ sion rate and U is the scalar field potential. Similarly, cally, but they may be useful for a numerical analysis in we find specific models. dV ¼ 3H; ð5:5Þ dΩ VI. SUMMARY AND DISCUSSION ¼ 1 da where we have used H a dτ. The diffusion coefficient D in Eq. (5.2) can be found We have proposed a new probability measure for from the dispersion of quantum fluctuations of ϕ over eternally inflating , which regulates infinite num- proper time interval dτ, bers of events by a cutoff at a constant four-volume time Ω, defined by Eqs. (1.1) and (1.2). The main advantage of this H3 H3 hðδϕÞ2i¼ dτ ¼ dΩ ¼ 2DdΩ; ð5:6Þ measure is that it avoids the problems with contracting AdS 4π2 4π2V regions that plagued earlier measure proposals. Otherwise, its properties are similar to those of the scale-factor cutoff which gives D ¼ H3=8π2V. Combining all this, we obtain measure. With suitable assumptions about the landscape, it the following equation for fðΩ; ϕ;VÞ: does not suffer from the Boltzmann brain problem. The     ∂ ∂ 1 ∂2 1 ∂ predicted distribution for the cosmological constant Λ is þ 3 − ð 3 Þ − dH V H f 2 2 H f f similar to the scale-factor measure, but with a higher ∂Ω ∂V 8π ∂ϕ 4π ∂ϕ dϕ probability for positive values of Λ: PðΛ > 0Þ¼8% and ¼ 0: ð5:7Þ 3% in four-volume and scale-factor measures, respectively. The probability of negative Λ is likely to be greatly reduced Once the function fðΩ; ϕ;VÞ is found, the comoving and ϕ when anthropic effects in contracting regions are properly physical volume distributions of on surfaces of constant taken into account, and one expects the resulting distribu- Ω can, respectively, be found from Z tion to be in a good agreement with observation. ∞ The probability distribution for the curvature parameter FðΩ; ϕÞ¼ dVfðΩ; ϕ;VÞð5:8Þ Ω in the new measure is essentially the same as in the 0 k scale-factor measure, assuming that the cosmological con- and stant is fixed at the observed value. This distribution Z ð Þ ∞ depends on the prior distribution P Ne for the number ð Þ ∝ −3 FVðΩ; ϕÞ¼ dVVfðΩ; ϕ;VÞ: ð5:9Þ of e-foldings of slow-roll inflation. With P Ne Ne ,as 0 suggested by random Gaussian models of the landscape, Ω In models with bubble nucleation, we can define the one finds that the probability for k to be below the Ω ≲ 0 01 distribution f ðΩ;VÞ as the fraction of comoving volume observational upper bound ( k . ) and still be detect- j Ω ≳ 10−4 ∼ 7% occupied by vacuum of type j with a given value of V on able (that is, k ) is rather small, P . surfaces of constant Ω. It satisfies the equation We note finally that one could introduce a family of measure proposals with properties similar to the four-   X X ∂ ∂ volume cutoff. For example, instead of Ω one could use V þ 3H f ¼ M˜ f ¼ M H f ; ð5:10Þ ∂Ω i ∂V i ij j ij j j the “time” coordinate j j Z τ ˜ ðτÞ¼ τ0½Vð3Þp ð Þ where Mij ¼ MijHj is the proper time transition matrix and tp d ; 6:1 0 Mij is the scale-factor time transition matrix given by Eq. (2.6). The reason we have a proper time transition with p>0. The four-volume cutoff corresponds to p ¼ 1. matrix on the right-hand side of (5.10) is that the This choice may be preferred because it has a clear geometric

043520-7 ALEXANDER VILENKIN and MASAKI YAMADA PHYS. REV. D 101, 043520 (2020) meaning. One hopes however that the probability measure Science Foundation under Grant No. PHY-1820872. M. Y. will eventually be determined by the fundamental theory. was supported by JSPS Overseas Research Fellowships and the Department of Physics at MIT. M. Y. was also supported ACKNOWLEDGMENTS by the U.S. Department of Energy, Office of Science, Office WearegratefultoAlanGuthandKenOlumforuseful of High Energy Physics of U.S. Department of Energy under discussions. A. V. was supported in part by the National grant Award No. DE-SC0012567.

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