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Four-Volume Cutoff Measure of the Multiverse Four-volume cutoff measure of the multiverse The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. 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, Tufts University, Medford, Massachusetts 02155, USA 2Center for Theoretical Physics, 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 cosmological constant. 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 inflation. Perhaps the tends to predict that we should expect to measure Λ < 0 simplest way to regulate the infinities of eternal inflation 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 Universe 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.
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