Geometric Origin of Coincidences and Hierarchies in the Landscape

Geometric origin of coincidences and hierarchies in the landscape

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As Published http://dx.doi.org/10.1103/PhysRevD.84.083517

Publisher American Physical Society (APS)

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Citable link http://hdl.handle.net/1721.1/68666

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 84, 083517 (2011) Geometric origin of coincidences and hierarchies in the landscape

Raphael Bousso,1,2 Ben Freivogel,3 Stefan Leichenauer,1,2 and Vladimir Rosenhaus1,2 1Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720-7300, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720-8162, USA 3Center for Theoretical Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 2 August 2011; published 14 October 2011) We show that the geometry of cutoffs on eternal inﬂation strongly constrains predictions for the time scales of vacuum domination, curvature domination, and observation. We consider three measure proposals: the causal patch, the fat geodesic, and the apparent horizon cutoff, which is introduced here for the ﬁrst time. We impose neither anthropic requirements nor restrictions on landscape vacua. For vacua with positive cosmological constant, all three measures predict the double coincidence that most observers live at the onset of vacuum domination and just before the onset of curvature domination. The hierarchy between the Planck scale and the cosmological constant is related to the number of vacua in the landscape. These results require only mild assumptions about the distribution of vacua (somewhat stronger assumptions are required by the fat geodesic measure). At this level of generality, none of the three measures are successful for vacua with negative cosmological constant. Their applicability in this regime is ruled out unless much stronger anthropic requirements are imposed.

DOI: 10.1103/PhysRevD.84.083517 PACS numbers: 98.80.Cq

I. INTRODUCTION keeps a fixed physical volume surrounding the geodesic; in simple situations, it is equivalent to the global scale String theory appears to contain an enormous landscape factor time cutoff [30].1 We also introduce a new measure, of metastable vacua [1,2], with a corresponding diversity of which restricts to the interior of the apparent horizon low-energy physics. The cosmological dynamics of this surrounding the geodesic. theory is eternal inflation. It generates a multiverse in From little more than the geometry of these cutoffs, we which each vacuum is produced infinitely many times. are able to make remarkable progress in addressing cos- In a theory that predicts a large universe, it is natural to mological coincidence and hierarchy problems. Using each assume that the relative probability for two different out- comes of an experiment is the ratio of the expected number measure, we will predict three time scales: the time when of times each outcome occurs. But in eternal inflation, observations are made,pffiffiffiffiffiffiffiffiffiffiffiffitobs, the time of vacuum energy j j every possible outcome happens infinitely many times. domination, t 3= , and the time of curvature domi- 2 The relative abundance of two different observations is nation, tc. We work in the approximation that observations ambiguous until one defines a measure: a prescription for occur in nearly homogeneous Friedmann-Robertson- regulating the infinities of eternal inflation. Walker (FRW) universes so that these time scales are Weinberg’s prediction [3] of the cosmological constant well defined. [4,5] was a stunning success for this type of reasoning. In We will allow all vacuum parameters to vary simulta- hindsight, however, it was based on a measure that was ill- neously. Parameters for which we do not compute proba- suited for a landscape in which parameters other than bilities are marginalized. We will not restrict attention to can vary. Moreover, the measure had severe phenomeno- vacua with specific features such as baryons or stars. We logical problems [6]. This spurred the development of will make some weak, qualitative assumptions about the more powerful measure proposals in recent years [7–27]. Surprisingly, some of these measures do far more than to 1Local measures are sensitive to initial conditions and the resolve the above shortcomings. As we shall see in this equivalences hold if the initial conditions specify starting in the longest lived vacuum in the landscape. paper, they obviate the need for Weinberg’s assumption 2 that observers require galaxies, and they help overcome the In regions where curvature will never dominate, such as our own vacuum, tc is defined as the time when curvature would limitation of fixing all parameters but one to their observed come to dominate if there were no vacuum energy. Since our values. observations are consistent with a flat universe, we can only In this paper we will analyze three different measure place a lower bound on the observed value of tc. We include tc in proposals. Each regulates the infinite multiverse by re- our analysis because bubble universes in the multiverse are stricting attention to a finite portion. The causal patch naturally very highly curved, so the absence of curvature requires an explanation. Moreover, we shall see that some mea- measure [15] keeps the causal past of the future endpoint sures select for high curvature in vacua with negative of a geodesic; it is equivalent to a global cutoff known as cosmological constant. This effect is partly responsible for the light-cone time [24,28]. The fat geodesic measure [29] problems encountered in this portion of the landscape.

1550-7998=2011=84(8)=083517(16) 083517-1 Ó 2011 American Physical Society BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) prior distribution of parameters in the landscape. We will what extent early curvature domination disrupts the for- assume that most observers are made of something that mation of observers. What is clear from our analysis is that redshifts faster than curvature. (This includes all forms of all three measures are in grave danger of predicting that radiation and matter but excludes, e.g., networks of domain most observers see negative , in conflict with observa- walls.) But we will not impose any detailed anthropic tion, unless the anthropic factor takes on rather specific requirements, such as the assumption that observers re- forms. Assuming that this is not the case, we must ask quire galaxies or complex molecules; we will not even whether the measures can be modified to agree with assume that they are correlated with entropy production observation. Both the causal patch measure and the fat [15,31,32]. Thus we obtain robust predictions that apply to geodesic measure are dual [28,29] to global time cutoffs, essentially arbitrary observers in arbitrary vacua. the light-cone time and the scale factor time cutoff, re- The probability distribution over all three variables can spectively. These global cutoffs, in turn, are motivated by be decomposed into three factors, as we will explain fur- an analogy with the UV/IR relation in AdS/CFT[19]. But ther in Sec. II: this analogy really only applies to positive ,soitis d3p natural to suspect that the measure obtained from it is inapplicable to regions with 0 [24]. (Indeed, the d logt d logt d logt obs c causal patch does not eliminate infinities in ¼ 0 bubbles 2 ¼ d p~ ð Þ [15]. We do not consider such regions in this paper.) M logtobs; logtc; logt d logtd logtc ð Þ logtobs; logtc; logt : (1.1) Outline and summary of results Here p~ is the probability density that a bubble with pa- In Sec. II, we will describe in detail our method for counting observations. We will derive an equation for the rameters ðlogt; logtcÞ is produced within the region defined by the cutoff. Its form can be estimated reliably probability distribution over the three variables ð Þ enough for our purposes from our existing knowledge logt; logtc; logtobs . We will explain how simple qualita- ð Þ tive assumptions about ðlogt ; logt ; logt Þ, the number about the landscape. The factor M logtobs; logt; logtc is obs c of observations per unit mass per unit logarithmic time the mass inside the cutoff region at the time tobs. This is completely determined by the geometry of the cutoff and interval, allow us to compute probabilities very generally the geometry of a FRW bubble with parameters for all measures. ð Þ We work in the approximation that observations in the logt; logtc , so it can be computed unambiguously. The multiverse take place in negatively curved FRW universes. last factor, ðlogtobs; logtc; logtÞ, is the one we know the least about. It is the number of observations per unit mass In Sec. III, we will obtain solutions for their scale factor, in per logarithmic time interval, averaged over all bubbles the approximation where the matter-, vacuum-, and possi- ð Þ with the given values logt; logtc . bly the curvature-dominated regime are widely separated in A central insight exploited in this paper is the following. time. In bubbles with positive cosmological constant, the calcu- In Secs. IV, V, and VI, we will compute the probability ð Þ lable quantity M so strongly suppresses the probability in distribution over logt; logtc; logtobs , using three differ- other regimes that in many cases we only need to know the ent measures. For each measure we consider separately the form of in the regime where observers live before vacuum cases of positive and negative cosmological constant. As & energy or curvature become important, tobs t, tc. Under described above, the results for negative cosmological very weak assumptions, must be independent of t and tc constant are problematic for all three measures. We now in this regime. This is because neither curvature nor vacuum summarize our results for the case > 0. energy play a dynamical role before observers form, so that In Sec. IV, we show that the causal patch measure predicts neither can affect the number of observers per unit mass. the double coincidence logtc logt logtobs.Wefind Thus, for positive cosmological constant, is a function that the scale of all three parameters is related to the number only of tobs in the only regime of interest. The success of the of vacua in the landscape. This result is compatible with measures in explaining the hierarchy and coincidence of the current estimates of the number of metastable string vacua. three time scales depends on the form of this function. We Such estimates are not sufficiently reliable to put our pre- will find that the causal patch and apparent horizon cutoff diction to a conclusive test, but it is intriguing that the size of succeed well in predicting the three time scales already the landscape may be the origin of the hierarchies we under very weak assumptions on . The fat geodesic cutoff observe in nature (see also [15,32–36]). We have previously requires somewhat stronger assumptions. reported the result for this subcase in more detail [37]. For negative cosmological constant, however, the geo- Unlike the causal patch, the new ‘‘apparent horizon metric factor M favors the regime where tobs is not the measure’’ (Sec. V) predicts the double coincidence logtc shortest time scale. Thus, predictions depend crucially on logt logtobs for any fixed value tobs. When all parame- understanding the form of in this more complicated ters are allowed to scan, its predictions agree with those of regime. For example, we need to know on average to the causal patch, with mild assumptions about the function

083517-2 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011) . The apparent horizon measure is significantly more dNi ð Þ ð Þ involved than the causal patch: it depends on a whole M logtobs; logtc; logt i logtobs ; (2.2) d logtobs geodesic, not just on its endpoint, and on metric information, rather than only causal structure. If our assumptions where M is the mass inside the cutoff region, and i is the about are correct, this measure offers no phenomenologi- number of observations per unit mass per logarithmic time cal advantage over its simpler cousin and may not be interval inside a bubble of type i. In this decomposition, M worthy of further study. contains all of the information about the cutoff procedure. The fat geodesic cutoff, like the apparent horizon cutoff, For a given cutoff, M depends only on the three parameters predicts the double coincidence logtc logt logtobs of interest. Because we are considering geometric cutoffs, for any fixed value of tobs. However, it favors small values the amount of mass that is retained inside the cutoff region of all three time scales unless either (a) an anthropic cutoff does not depend on any other details of the vacuum i.On is imposed or (b) it is assumed that far more observers form the other hand, i depends on details of the vacuum, such at later times, on average, than at early times, in vacua as whether observers form, and when they form, but it is where curvature and vacuum energy are negligible at the independent of the cutoff procedure. time tobs. (Qualitatively, the other two measures also re- Since we are interested in analyzing the probability quire such an assumption, but quantitatively, a much distribution over three variables, we now want to average ð Þ weaker prior favoring late observers suffices.) In the latter i over the bubbles with a given logt; logtc , to get the case (b), the results of the previous two measures are average number of observations per unit mass per logarith- reproduced, with all time scales set by the size of the mic time . With this decomposition, the full probability landscape. In the former case (a), the fat geodesic would distribution over all three variables is predict that observers should live at the earliest time com- 3 patible with the formation of any type of observers (in any d p kind of vacuum). It is difficult to see why this minimum d logtobsd logtd logtc value of t would be so large as to bring this prediction 2 obs ¼ d p~ ð Þ into agreement with the observed value, t 1061. M logtobs; logtc; logt obs d logtd logtc ð Þ logtobs; logtc; logt : (2.3) II. COUNTING OBSERVATIONS To recap, p~ is the probability for the formation of a bubble In this section, we explain how we will compute with parameters ðlogt ; logt Þ inside the cutoff region; ð t ; c the trivariate probability distribution over log obs Mðlogt ; logt ; logt Þ is the mass inside the cutoff region t ; t Þ obs c log c log . We will clarify how we isolate geometric at the FRW time t in a bubble with parameters effects, which can be well computed for each cutoff, obs ðlogt; logtcÞ; and ðlogtobs; logtc; logtÞ is the number from anthropic factors, and we explain why very few of observations per unit mass per logarithmic time interval, assumptions are needed about the anthropic factors once ð Þ averaged over all bubbles with parameters logt; logtc . the geometric effects have been taken into account. This is a useful decomposition because the mass M Imagine labeling every observation within the cutoff inside the cutoff region can be computed exactly, since it ð t ; t ; t Þ region by log obs log c log . We are interested in count- just depends on geometrical information. We will assume ing the number of observations as a function of these 2 that d p=d~ logtd logtc can be factorized into contribution parameters. It is helpful to do this in two steps. First, we from t and a contribution from t . Vacua with 1 can ð t ; t Þ c count bubbles, which are labeled by log c log to get the be excluded since they contain only a few bits of informa- ‘‘prior’’ probability distribution tion in any causally connected region. In a large landscape, by Taylor expansion about ¼ 0, the prior for the cos- d2p~ : (2.1) mological constant is flat in for 1, dp=d~ ¼ d logtd logtc const. Transforming to the variable logt, we thus have This p~ is the probability density to nucleate a bubble with d2p~ t2gðlogt Þ: (2.4) the given values of the parameters inside the cutoff region. d logt d logt c The next step is to count observations within the bub- c bles. A given bubble of vacuum i will have observations at The factor gðlogtcÞ encodes the prior probability distribu- a variety of FRW times. In the global description of eternal tion over the time of curvature domination. We will assume inflation, each bubble is infinite and contains an infinite that g decreases mildly, like an inverse power of logtc. number of observations if it contains any, but these local (Assuming that slow-roll inflation is the dominant mecha- measures keep only a finite portion of the global spacetime nism responsible for the delay of curvature domination, and hence a finite number of observations. We parameter- logtc corresponds to the number of e-foldings. If g de- ize the probability density for observations within a given creased more strongly, like an inverse power of tc, then bubble as inflationary models would be too rare in the landscape to

083517-3 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) explain the observed flatness.) The detailed form of the d3p ¼ t2gðlogt ÞMðlogt ;logt ;logt Þ prior distribution over logtc will not be important for our dlogt dlogt dlogt c obs c results; any mild suppression of large logt will lead to obs c c ð Þ similar results. logtobs : (2.8) With these reasonable assumptions, the probability dis- This is the formula that we will analyze in the rest of the tribution becomes paper. The only quantity that depends on all three variables is the mass, which we can compute reliably for each cutoff d3p using the geometry of open bubble universes, to which we d logtobsd logtd logtc turn next. ¼ 2 ð Þ ð Þ t M logtobs; logtc; logt g logtc ð Þ logtobs; logtc; logt : (2.5) III. OPEN FRW UNIVERSES WITH COSMOLOGICAL CONSTANT Because depends on all three variables, it is very difficult to compute in general. However, it will turn out that we can In this section, we find approximate solutions to the make a lot of progress with a few simple assumptions about scale factor, for flat or negatively curved vacuum bubbles . First, we assume that in the regime, t t , is with positive or negative cosmological constant. The land- obs scape of string theory contains a large number of vacua that independent of t. Similarly, we assume that in the regime, form a ‘‘discretuum,’’ a dense spectrum of values of the tobs tc, is independent of tc. By these assumptions, in the regime where the observer time is the shortest time cosmological constant [1]. These vacua are populated by & Coleman-DeLuccia bubble nucleation in an eternally in- scale, tobs t, tc, the anthropic factor will only depend on logt : flating spacetime, which produces open FRW universes obs [38]. Hence, we will be interested in the metric of FRW ð Þ ð Þ & universes with open spatial geometry and nonzero cosmo- logtobs;logtc;logt logtobs for tobs t;tc: (2.6) logical constant . The metric for an open FRW universe is These assumptions are very weak. Because curvature and 2 ¼ 2 þ ð Þ2ð 2 þ 2 2Þ ds dt a t d sinh d2 : (3.1) are not dynamically important before tc and t, respectively, they cannot impact the formation of observers at The evolution of the scale factor is governed by the such early times. One could imagine a correlation between Friedmann equation: the number of e-foldings and the time of observer forma- a_ 2 t 1 1 tion even in the regime t t , for example, if each one ¼ c þ obs c 3 2 2 : (3.2) is tied to the supersymmetry breaking scale, but this seems a a a t highly contrived. In the absence of a compelling argument pffiffiffiffiffiffiffiffiffiffiffiffi Here t ¼ 3=jj is the time scale for vacuum domina- to the contrary, we will make the simplest assumption. tion, and t is the time scale for curvature domination. The Second, we assume that when either t t or c obs c term t =a3 corresponds to the energy density of t t , is not enhanced compared to its value when m c obs pressureless matter. (To consider radiation instead, we t is the shortest time scale. This is simply the statement obs would include a term t2=a4; this would not affect that early curvature and vacuum domination does not help rad c any of our results qualitatively.) The final term is the the formation of observers. This assumption, too, seems vacuum energy density, ; the‘‘þ’’ sign applies when rather weak. With this in mind, let us for the time being > 0, and the‘‘’’ sign when < 0. overestimate the number of observers by declaring to be We will now display approximate solutions for the scale completely independent of logt and logt : c factor as a function of FRW time t. There are four cases, ð Þ ð Þ which are differentiated by the sign of and the relative logtobs; logtc; logt logtobs : (2.7) size of tc and t. We will compute all geometric quantities This is almost certainly an overestimate of the number of in the limit where the three time scales t, tc, and t are well observations in the regime where tobs is not the shortest separated, so that some terms on the right-hand side of time scale. However, we will show that the predictions for Eq. (3.2) can be neglected. In this limit we obtain a piece- positive cosmological constant are insensitive to this sim- wise solution for the scale factor. We will not ensure that plification because the geometrical factor M, which we can these solutions are continuous and differentiable at the compute reliably, suppresses the contribution from this crossover times. This would clutter our equations, and it regime. We will return to a more realistic discussion of would not affect the probability distributions we compute when we are forced to, in analyzing negative cosmological later. Up to order-one factors, which we neglect, our for- constant. mulas are applicable even in crossover regimes. With these assumptions and approximations, the three- If tc t, curvature never comes to dominate. (One can variable probability distribution takes a more tractable still define tc geometrically, or as the time when curvature form, would have dominated in a universe with ¼ 0.) In this

083517-4 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011) limit the metric can be well approximated as that of a IV. THE CAUSAL PATCH CUTOFF perfectly flat FRW universe, and so becomes independent With all our tools lined up, we will now consider each of t . We implement this case by dropping the term t =a3 in c c measure in turn and derive the probability distribution. We Eq. (3.2). will treat positive and negative values of the cosmological constant separately. After computing M, we will next A. Positive cosmological constant calculate the bivariate probability distribution over logt We begin with the case > 0 and tc t. By solving and logtc, for fixed logtobs. In all cases this is a sharply Eq. (3.2) piecewise, we find peaked function of the two variables logt and logtc,sowe 8 make little error in neglecting the probability away from > 1=3 2=3 <> tc t ;t t; tc begin with the causal patch measure [15,39], which re- : t=t 1 te ;t

B. Negative cosmological constant 0 Z t ð Þ¼ f dt For < 0, the scale factor reaches a maximum and then CP t 0 : (4.1) t aðt Þ begins to decrease. The universe ultimately collapses at a time tf, which is of order t: If < 0, tf is the time of the big crunch (see Sec. III). For tf t: (3.5) long-lived metastable de Sitter vacua ( > 0), the causal patch coincides with the event horizon. It can be computed The evolution is symmetric about the turnaround time, as if the de Sitter vacuum were eternal (t !1), as the f tf=2 t=2. correction from late-time decay is negligible. Again, we consider the cases t tc and t tc separately. For tc t, the scale factor is 8 A. Positive cosmological constant > 1=3 2=3 <> tc t ;t 0, tc c c :> t1=3ðt0Þ2=3;t0 f >1þlogðt =t Þþ3½1ðt =t Þ1=3;t 1 log t=tobs ;tc : tobs=t yields the scale factor e ;t

083517-5 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) 8 > 2 We will ﬁrst analyze this probability distribution for > t =tc;tobs ttc=tobs;tc for the time being overestimate the number of observers : 3t =t tce obs ;tc 0, t 1 > ;tobs tc ð Þ > horizontal, vertical, or diagonal arrows in the logtc; logt > tc ;t t2 c obs plane (Fig. 1). We ignore the prior gðlogt Þ for now since it > obs c <> is not exponential. tc 3tobs g t2 exp t ;tc > t obs c c > 1 3tobs > exp ;t t t > on t. This is shown by a left-pointing horizontal arrow in :> 1 t ;tobs

log t log t f

I I II

II V

log tobs V log 2tobs III IV III log tobs IV

log tc log tc log tobs log tobs

FIG. 1. The probability distribution over the time scales of curvature and vacuum domination at ﬁxed observer time scale logtobs, before the prior distribution over logtc and the ﬁniteness of the landscape are taken into account. The arrows indicate directions of increasing probability. For > 0 (a), the distribution is peaked along the degenerate half-lines forming the boundary between regions I and II and the boundary between regions IV and V. For < 0 (b), the probability distribution exhibits a runaway toward the small tc, large t regime of region II. The shaded region is excluded because tobs >tf ¼ t is unphysical.

083517-6 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011) Since the exponential dominates, the force goes toward ð Þ dp g logtobs ð Þ larger logt ; logt is pushed up as well. In region IV logtobs : (4.8) c d logtobs tobs (t 0: (4.9) along which the probability force is zero. These are the boundaries between regions IV/V and I/II, respectively. (We will justify this assumption at the end of this section, They are shown by thick lines in Fig. 1. The fact that the where we will also describe what happens if it is not distribution is flat along these lines indicates a mild run- satisfied.) Then the maximum is at the largest value of away problem: at the crude level of approximation we have logtobs subject to the constraint defining regions I and II, used thus far, the probability distribution is not integrable. logtobs < logt. It follows that the maximum of the three- Let us consider each line in turn to see if this problem variable probability distribution is at persists in a more careful treatment. max ¼ logtobs logtc logt logt : (4.10) Along the line logt logtobs, the prior distribution gðlogt Þ will suppress large logt and make the probability c c Therefore, in a landscape with p>0 and vacua with integrable, rendering a prediction for logt possible. (With c > 0, the causal patch predicts that all three scales are a plausible prior, the probability of observing a departure ultimately set by logtmax and, thus, by the (anthropic) from flatness in a realistic future experiment is of order 10% [41,42]; see also [36]. See [43,44] for different vacuum with smallest cosmological constant. This, in priors.) We will find the same line of stability for the other turn, is set by the discretuum limit, i.e., by the number of two measures in the following section, and it is lifted by the vacua in the landscape [15,31,32,35], according to same argument. max N 1=2 ¼ t : (4.11) The line logtc logtobs looks more serious. The prior on logt follows from very general considerations [3] and This is a fascinating result [37]. It implies not only that cannot be modified. In fact, the probability distribution is the remarkable scales we observe in nature, such as the rendered integrable if, as we will suppose, the landscape vacuum energy and the current age of the universe, are contains only a finite number N of vacua. This implies mutually correlated, but that their absolute scale can be that there is a ‘‘discretuum limit,’’ a finite average gap explained in terms of the size of the landscape. If current between different possible values of . This, in turn, estimates of the number of vacua [1,45] hold up, i.e., implies that there is a smallest positive in the landscape, 3 if log N is of order hundreds, then Eq. (4.10) may of order 10 well prove to be in agreement with the observed value t 1061. 1=N : (4.7) min Let us go somewhat beyond our order-of-magnitude estimates and determine how precisely logt and logt This argument, however, only renders the distribution in- obs can be expected to agree. To that end, we will now calcu- tegrable; it does not suffice to bring it into agreement with late p ðf1t logt .) case that after averaging over the times when observers obs could live most observers see logtobs logt. To address 3 N this question, we need to allow logtobs to vary. For fixed The number of anthropic vacua, , may be smaller by logt , the maximum of the probability with respect to dozens or even hundreds of orders of magnitude than the total obs number of vacua, N , for low-energy reasons that are unrelated logtc and logt is obtained along the boundary between to the cosmological constant or curvature and so are not included N ð Þ region I and region II, as discussed above. Along this line, in our analysis. Hence, log10 O 1000 may be compatible the probability is with Eq. (4.10).

083517-7 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) max The probability density in region V is to the probability for larger t , so the approximation gets max max ¼ 60 1þp better as t increases. However, even for t 10 the dp tobs / gðlogtcÞ: (4.12) result is only off by a few percent compared to a numerical d logtobsd logtcd logt t integration. We will further restrict to t >tmax, which is reasonable if Let us now return to discussing our assumption, Eq. (4.9). c If p were not positive, that is, if increased at most linearly tobs is pushed to large values and gðlogtcÞ does not strongly with tobs, then the maximum of the probability distribution prefer small values of logtc. Since we are computing a probability marginalized over logt , this restriction on the would be located at the smallest value of tobs compatible c with observers. In this case the causal patch would predict range of logtc means that the exact form of g will not affect the answer. The quantity t tobs. This would be in conflict with observation Z except under the extremely contrived assumption that 1 tmax tmin. d logtcgðlogtcÞ (4.13) obs max logt However, the assumption that p>0 is quite plausible [37]. Recall that we are only discussing the form of in the will factor out of our computations, and hence we will & regime where tobs is the shortest time scale, tobs tc, t,so ignore it. Having eliminated the logtc dependence, we we do not have to worry that later observations may be continue by computing the normalization factor Z for disrupted by curvature or vacuum energy. Recall, more- logt > logtobs: over, that is defined by averaging over many vacua, so Z Z 1þp logtmax logtmax t ðtmaxÞp we must consider only how this average depends on tobs.In ¼ obs Z dlogtobs dlogt ð þ Þ: particular, this means that we should not imagine that in 0 logtobs t p 1 p moving from one value of tobs to another, we need to hold (4.14) fixed the vacuum, or even restrict to only one or two In the Eq. (4.14) we have dropped terms negligible for parameters of particle physics and cosmology. Typical max vacua with most observers at one value of tobs are likely t 1. Now we will calculate the unnormalized probabilty for to differ in many details from vacua in which most observ- 1 ers arise at a different time. f tobs 3 2 log tc=2t 3 1 t=tc ;t ð Þþ ð 0 Þ 0 ð 1 Þ 1p 3 logtan t=2t logtan tc=2t ;tc > 0 1=3 ð Þ : 3 t ;t0

083517-8 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011) ð Þ Recall that a prime denotes time remaining before independent of logtc; logt .However,iftc tobs 0 the crunch: t tf t. To better show the structure of (t tobs), then curvature (or vacuum energy) could the above expressions, we have kept some order-one dynamically affect the processes by which observers factors and subleading terms that will be dropped below. form. One would expect that such effects are generally ð 0 Þ¼ ð Þ We will approximate logtan tc=2t logtan tc=2t detrimental. ð Þ log tc=2t . Here, for the first time, we find a distribution that peaks The mass inside the causal patch at the time tobs is in a regime where tc tobs. This means that the detailed dependence of on logt is important for understanding ¼ 3 ½ ð Þ c MCP a VCP CP tobs tcVCP: (4.23) the prediction and must be included. We do not know this We will again approximate the comoving volume inside a function except for special classes of vacua. ! sphere of radius by 3 for & 1 and by e2 for * 1, Instead of letting logtc 0 so that tc becomes min giving Planckian, we will only allow tc to fall as low as tc .We 8 do this because it does not make our analysis any more > 4 3 <> t=tc;tobs c obs c obs c positive quantity. :> 0 0 tobs;tc max > 3 ;tobs tc > logtmin, logt logtmin, with maximum probability > 1 0 obs c c > 2ð Þ ;tc tctan tobs=2t t =tobs logtobs g logtc =tc . (Here we have intro- < 0 min min t 0 obs duced logtobs in an analogous way to logtc . The point g t2 ;tc > here is that typical observers live at the earliest possible > t0 > obs ;t=2 t2 f obs c > Do these predictions conflict with observation? Not so : 1 ;tobs 0, so the t ð Þ relevant probability distribution over logt; logtc is the The analysis of the probability ‘‘forces’’ proceeds as in one computed in the previous subsection. This led to * the positive cosmological constant case discussed in the the predictions that logt logtobs and logtc logtobs, previous subsection, by identifying and following the di- both of which agree well with observation, and that the rections along which the probability grows in each distinct scale of logt is controlled by the number of vacua in the ð Þ region of the logt; logtc plane. The result, however, is landscape, which is not ruled out. rather different [Fig. 1(b)]. For fixed logtobs, the unnormal- However, we do get a conflict with observation if we ized probability density diverges in the direction of small ask about the total probability for each sign of the cosmo- 2 1 logtc and large logt (region II) like ttc . The discrete logical constant. The total probability for positive cosmo- spectrum of bounds logt from above, and the Planck logical constant is approximately given by the value of scale is a lower limit on logtc. Recall that, so far, we have the distribution of the maximum. With our assumption ð maxÞ ð maxÞ max approximated the rate of observations per unit mass as about [Eq. (4.9)], this is pþ g logt logt =t .

083517-9 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) The total probability for negative is also controlled by the probability density at the maximum of the distribution; as mentioned earlier, it is given by ð maxÞ ð minÞ min ð max minÞ2 logt g logtc =tc if p>1, and by t =tobs ð minÞ ð minÞ min logtobs g logtc =tc for p<1. Dividing these, we ﬁnd that a negative value of is favored by a factor

p tmaxgðlogtminÞ ¼ c for p>1: (4.30) pþ min ð maxÞ tc g logt

max We know that t must be at least as large as the observed FIG. 2 (color online). The causal patch can be characterized as max 61 value of t, which is of order tobs: t >tobs 10 . the union of all past light cones (all green lines, including ð maxÞ ð minÞ dashed) of the events along a worldline (vertical line). The Furthermore, we expect that g logt 0. Its surface area is ent horizon cutoff entails a further restriction: it consists given by [50]

083517-10 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011)

FIG. 3 (color online). Conformal diagrams showing the apparent horizon cutoff region. The boundary of the causal patch is shown as the past light cone from a point on the conformal boundary. The domain wall surrounding a bubble universe is shown as the future light cone of the bubble nucleation event. The region selected by the cutoff is shaded. For > 0 (a), the boundary of the causal patch is always exterior to the apparent horizon. For < 0 (b), the apparent horizon diverges at a ﬁnite time. Because the apparent horizon cutoff is constructed from light cones, however, it remains ﬁnite. The upper portion of its boundary coincides with that of the causal patch.

3 the apparent horizon is M ¼ V t V =a3. A ðtÞ¼ ; (5.1) AH m AH c AH AH 2ðtÞ Combining the above results, we ﬁnd 8 from which its comoving radius can easily be deduced. The <> tobs;tobs tct=tobs;ti 0, t tc, the mass can be obtained by where the apparent horizon is necessarily contained within setting tc ti t in the above result: the causal patch [51]. In universes with < 0, there is a FRW time t when the apparent horizon coincides with the t ;t 0, tc t. The scale factor 8 tobs aðtÞ is given by Eq. (3.3). The energy density of the > t2 ;tobs vacuum, t2, begins to dominate over the den- > > tc ;t t2 i obs sity of matter, m tc=a , at the intermediate time > obs <> 1=3 2=3 tc tobs 2 exp 3 1 ;ti > > Note that tc ti t if tc and t are well separated. > 1 tobs > exp 3 1 ;t t t Thus we can approximate Eq. (5.1)by :> t obs 1ð Þ t2 ;tobs ti: (5.6) 2 The comoving area of the apparent horizon, AAH=a ,is The probability forces are shown in Fig. 4. The boundary ¼ initially small and grows to about one at the time tc.It between regions I and II is given by logti logtobs, which ¼ 3 1 remains larger than unity until the time t and then corresponds to logt 2 logtobs 2 logtc. In region I, the becomes small again. The proper volume within the ap- probability is proportional to t2, corresponding to a force parent horizon is VAH aAAH when the comoving area toward smaller logt. In region II there is a force toward 3=2 is large and VAH AAH when it is small. The mass within large logtc. In region III, the exponential dominates the

083517-11 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011)

log t log t f

I II I

II V VII log t log 2t obs obs III VI log 2tobs III IV V log t IV obs

log tc log tc log tobs log tobs

FIG. 4. The probability distribution from the apparent horizon cutoff. The arrows indicate directions of increasing probability. For > 0 (a), the probability is maximal along the boundary between regions IV and V before a prior distribution over logtc is included. Assuming that large values of tc are disfavored, this leads to the prediction logt logtc logtobs.For < 0 (b), the distribution is dominated by a runaway toward small tc and large t along the boundary between regions II and III.

logt dependence, giving a preference for large logt; the when the positive matter density is sufficiently dilute to be 2 tc prefactor provides a force towards large logtc. In regions overwhelmed by the negative vacuum energy, t . IV and V the probabilities are independent of logtc except As discussed in Sec. VA, the apparent horizon exists ð Þ for the prior g logtc . The force is towards large logt in on the FRW time slice t only if the total density at that region IV, while in region V small logt is preferred. time is positive. By Eq. (5.1), the apparent horizon diverges Following the gradients in each region, we find that the when the density vanishes. Slightly earlier, at the time t ¼ distribution peaks on the boundary between regions IVand ð1 Þti, the apparent horizon intersects the boundary of V. Along this line, the probability density is constant ex- the causal patch. For tt, the causal patch does (see Fig. 3). acy is lifted by a realistic prior that mildly disfavors large To compute t, notice that tc t and Eq. (5.10) imply values of logtc. Thus, the apparent horizon cutoff predicts tc ti t. This implies that the scale factor can be well ð Þ ð Þ the double coincidence approximated by a t t sin t=t t in a neighbor- hood of t . This range includes the time t if is small. logt logt logt : (5.7) i obs c We will assume this approximation for now, and we will This is in good agreement with observation. find that 1 follows self-consistently. By Eq. (5.1), the ð Þ What if the observer time scale is allowed to vary? After proper area of the apparent horizon at the time ti 1 is ð Þ¼ 2 optimizing logt and logtc, the probability distribution AAH t t=2. From Eq. (4.22), we find that the causal 3 4 2 þ ð 2Þ over logtobs is patch has proper area 16e t=tc O . Equating these expressions, we find dp ðlogt Þ gðlogt Þ obs : (5.8) obs 1 t2 d logtobs tobs ¼ c 3 2 ; (5.11) 32e t We have argued in Sec. IVA that grows faster than tobs; under this assumption, all three time scales are driven to which is much less than unity. the discretuum limit: For times tt we use the results for the causal patch from Sec. IV B. 8 >tobs;tobs > 3 1 < tobs We turn to the case < 0, tc t. The scale factor is tobs 1 ;tc (5.12) > 2 1 2 0 at the intermediate time >t t tan ðt =2t Þ;t c obs obs c 0 0 1=3 2=3 tobs;tc

083517-12 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011)

Finally, we consider the case < 0, tf=2 t we use our formulas c from the causal patch in the previous section to obtain In either case, we are driven to tc tobs. 8 0 Note that, as in the case of the causal patch cutoff with > t ;tf=2 f obs f c tobs. So there is some uncertainty in our result coming from flat the dependence of on logt when logt < logt .Wedo tobs;tobs tobs > t2 ;tobs > logtmax if increases faster than quadratically with t . > tobs obs > t ;t t2 ½1ð obsÞ3 c obs In this case we predict logt logt .If grows more > ti obs > > 1 0 slowly with logtobs, then we predict logtobs logt. > 2ðtobsÞ ;t tctan Let us compare the total probability for negative to the <> 2t 0 t 0 total probability for positive , assuming the form (4.9) for g obs ;t t2 c obs f . For negative , we will assume that the large logt > obs > 0 > t regime is the relevant one, so that the correct probability > obs > t2 ;tf=2 distribution over tobs is (5.18). Note that this is the same as > > 1 flat (4.29), the result for negative in the causal patch. > ;t t > Additionally, (5.8) is identical to (4.8), the result for posi- : tobs flat t2 ;tobs 1; (5.20) pþ min ð maxÞ haps at the boundary with region II. Although the formula tc g logt in region III is already reasonably simple, there is a simpler and a similar result for p<1. form that is correct at the same level of approximation as the rest of our analysis, VI. THE FAT GEODESIC CUTOFF t2 ðlogt Þgðlogt Þ: (5.16) t t2 obs obs In this section, we compute probabilities using the fat c obs geodesic cutoff, which considers a ﬁxed proper volume V This is a good approximation for tobs t, but it is only near a timelike geodesic [29]. To compute probabilities, wrong by an order-one factor througout region III, so we one averages over an ensemble of geodesics orthogonal to will go ahead and use this. an initial hypersurface whose details will not matter. As For ﬁxed logtobs, it is clear that logt wants to be as large discussed in the previous section, geodesics quickly be- as possible, and logtc as small as possible, but we must come comoving upon entering a bubble of new vacuum. By

083517-13 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) log t log t f

I II II I V

log to IV log 2t III obs III IV log tobs

log tc log tc log to log tobs

FIG. 5. The probability distribution computed from the scale factor (fat geodesic) cutoff. The arrows indicate directions of increasing probability. For > 0 (a), the probability distribution is maximal along the boundary between regions IV and V; with a mild prior favoring smaller logtc, this leads to the prediction of a nearly ﬂat universe with logtc logt logtobs.For < 0 (b), the probability distribution diverges as the cosmological constant increases to a value that allows the observer time scale to coincide with the big crunch.

8 the symmetries of open FRW universes, we may pick a fat > 2 > 1=t ;tobs c obs c obs :> cutoff region is large compared to the scale of inhomoge- ð 3 Þ 3tobs=t tc=t e ;tc 0, tc >t), we obtain inﬁnitesimal. Thus, in this section, we shall neglect the 8 < 1=t2 ;t 1 ;t t2 t2 obs c > obs > > tc ;t t3 t2 c obs The fat geodesic cutoff is closely related to the scale > obs <> factor time cutoff, but it is more simply deﬁned and easier tc tobs g t5 exp 3 t ;tc > ence of timelike geodesics orthogonal to some initial > > 1 tobs > t4 exp 3 t ;t > proper time along each geodesic and 3H is the local :> 1 2 2 ;tobs 0, tc t. Combining Eqs. (6.1) and (3.3), we obtain in good agreement with observation.

083517-14 GEOMETRIC ORIGIN OF COINCIDENCES AND ... PHYSICAL REVIEW D 84, 083517 (2011) ð ð Þ What if we allow logtobs to scan? Optimizing logt; t;tc because the distribution near the maximum was Þ logtc , we ﬁnd the probability distribution over logtobs: exponential. Here, we will get different answers for the two procedures, so we choose to marginalize over dp ðlogt Þgðlogt Þ obs obs ðlogt ; logt Þ, leaving logt to scan. The resulting distri- 4 : (6.6) c obs d logtobs tobs bution is

The denominator provides a strong preference for logtobs to dp 3 ð Þ be small. To agree with observation, must grow at least tobs logtobs : (6.10) d logtobs like the fourth power of tobs for values of tobs smaller than 61 the observed value tobs 10 . We cannot rule this out, but There is no geometric pressure on logtc in region III, where it is a much stronger assumption than the ones needed for Eq. (6.7) peaks, so the value of logtc will be determined by the causal patch and apparent horizon cutoffs. the prior distribution and anthropic selection. Assuming The preference for early logtobs in the fat geodesic cutoff that the prior favors small values of logtc, it seems likely can be traced directly to the fact that the probability is that the expected value of logtc is much less than logtobs. proportional to the matter density. This result has an inter- As in the apparent horizon and causal patch measures for esting manifestation [29] in the more restricted setting of < 0, this complicates the computation of . However, universes similar to our own: it is the origin of the strong the situation here is not the same. The difference is that preference for large initial density contrast, =, which here we have observers forming late in the recollapse phase allows structure to form earlier and thus at higher average of a crunching universe, where the dominant contribution density. to the energy density actually comes from matter. The fact that the universe is in a recollapse phase makes it very hard B. Negative cosmological constant to say what the form of will be, whether or not there is an era of curvature domination. For < 0, t t , we use Eq. (3.6) for the scale factor. c Regardless of the form of , the ﬁrst factor in Eq. (6.10) The mass in the cutoff region is 8 has a preference for logtobs to be small. If grows faster 3 > 2 than t , then it is favorable for logtobs to be large and > 1=t ;tobs tc=t sin tobs=t ;tc means that some anthropic boundary determines the ex- : 02 0 1=tobs;tc tf=2, we use Eq. (3.7) for the scale factor and ﬁnd ment with the observed value of we need to assume grows like a fourth power of t . Then for both positive M t2sin2ðt =t Þ;t=2 1 > 2 2 ;tobs ttobs > Finally, for < 0 it is worth noting the behavior of the > tc 0 < 5 3ð Þ ;tc 1 0 for instance, Eq. (6.8) and neglecting for simplicity the > 2 ð 0 Þ2 ;tc t tobs > factor . Depending on whether tobs is larger or smaller > 1 min0 : than tf=2, logtobs will be driven either to logt or to t4 sin2ðt =t Þ ;tf=2

083517-15 BOUSSO et al. PHYSICAL REVIEW D 84, 083517 (2011) no conflict with observation. There are two reasons for this ACKNOWLEDGMENTS discrepancy. First, the fat geodesic measure differs from We thank Roni Harnik and Yasunori Nomura for inter- the detailed prescription given in [30] in the recollapsing esting discussions. This work was supported by the region. Second, the analysis of [30] made an unjustified Berkeley Center for Theoretical Physics, by the National approximation [29], computing the scale factor time in the Science Foundation (0855653), by fqxi Grant No. RFP2- approximation of a homogeneous FRW universe. It re- 08-06, and by the U.S. Department of Energy under mains to be seen if there is a precise definition of the scale Contract No. DE-AC02-05CH11231. V.R. is supported factor cutoff that will give the result computed in [30]. The by the NSF. fat geodesic is our best attempt to define a simple measure in the spirit of [30].

[1] R. Bousso and J. Polchinski, J. High Energy Phys. 06 [27] D. N. Page, J. Cosmol. Astropart. Phys. 03 (2011) 031. (2000) 006. [28] R. Bousso and I.-S. Yang, Phys. Rev. D 80, 124024 (2009). [2] S. Kachru, R. Kallosh, A. Linde, and S. P. Trivedi, Phys. [29] R. Bousso, B. Freivogel, and I.-S. Yang, Phys. Rev. D 79, Rev. D 68, 046005 (2003). 063513 (2009). [3] S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987). [30] A. De Simone, A. H. Guth, M. P. Salem, and A. Vilenkin, [4] A. G. Riess et al. (Supernova Search Team Collaboration), Phys. Rev. D 78, 063520 (2008). Astron. J. 116, 1009 (1998). [31] R. Bousso, R. Harnik, G. D. Kribs, and G. Perez, Phys. [5] S. Perlmutter et al. (Supernova Cosmology Project Rev. D 76, 043513 (2007). Collaboration), Astrophys. J. 517, 565 (1999). [32] R. Bousso and R. Harnik, Phys. Rev. D 82, 123523 (2010). [6] R. Bousso and B. Freivogel, J. High Energy Phys. 06 [33] J. Polchinski, arXiv:hep-th/0603249. (2007) 018. [34] R. Bousso, Gen. Relativ. Gravit. 40, 607 (2007). [7] A. Linde and A. Mezhlumian, Phys. Lett. B 307,25 [35] R. Bousso, L. J. Hall, and Y. Nomura, Phys. Rev. D 80, (1993). 063510 (2009). [8] A. Linde, D. Linde, and A. Mezhlumian, Phys. Rev. D 49, [36] R. Bousso and S. Leichenauer, Phys. Rev. D 81, 063524 1783 (1994). (2010). [9] J. Garcıá-Bellido, A. Linde, and D. Linde, Phys. Rev. D [37] R. Bousso, B. Freivogel, S. Leichenauer, and V. 50, 730 (1994). Rosenhaus, Phys. Rev. Lett. 106, 101301 (2011). [10] J. Garcıá-Bellido and A. D. Linde, Phys. Rev. D 51, 429 [38] S. Coleman and F. D. Luccia, Phys. Rev. D 21, 3305 (1995). (1980). [11] J. Garcıá-Bellido and A. Linde, Phys. Rev. D 52, 6730 [39] R. Bousso, B. Freivogel, and I.-S. Yang, Phys. Rev. D 74, (1995). 103516 (2006). [12] J. Garriga, D. Schwartz-Perlov, A. Vilenkin, and S. [40] L. J. Hall and Y. Nomura, Phys. Rev. D 78, 035001 (2008). Winitzki, J. Cosmol. Astropart. Phys. 01 (2006) 017. [41] B. Freivogel, M. Kleban, M. Rodriguez Martinez, and L. [13] V. Vanchurin and A. Vilenkin, Phys. Rev. D 74, 043520 Susskind, J. High Energy Phys. 03 (2006) 039. (2006). [42] A. De Simone and M. P. Salem, Phys. Rev. D 81, 083527 [14] V. Vanchurin, Phys. Rev. D 75, 023524 (2007). (2010). [15] R. Bousso, Phys. Rev. Lett. 97, 191302 (2006). [43] J. March-Russell and F. Riva, J. High Energy Phys. 07 [16] A. Linde, J. Cosmol. Astropart. Phys. 01 (2007) 022. (2006) 033. [17] A. Linde, J. Cosmol. Astropart. Phys. 06 (2007) 017. [44] B. Bozek, A. Albrecht, and D. Phillips, Phys. Rev. D 80, [18] D. N. Page, J. Cosmol. Astropart. Phys. 10 (2008) 025. 023527 (2009). [19] J. Garriga and A. Vilenkin, J. Cosmol. Astropart. Phys. 01 [45] F. Denef and M. R. Douglas, J. High Energy Phys. 05 (2009) 021. (2004) 072. [20] S. Winitzki, Phys. Rev. D 78, 043501 (2008). [46] M. P. Salem, Phys. Rev. D 80, 023502 (2009). [21] S. Winitzki, Phys. Rev. D 78, 063517 (2008). [47] L. Susskind and E. Witten, arXiv:hep-th/9805114. [22] S. Winitzki, Phys. Rev. D 78, 123518 (2008). [48] J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). [23] A. Linde, V. Vanchurin, and S. Winitzki, J. Cosmol. [49] R. Bousso, Rev. Mod. Phys. 74, 825 (2002). Astropart. Phys. 01 (2009) 031. [50] R. Bousso, J. High Energy Phys. 07 (1999) 004. [24] R. Bousso, Phys. Rev. D 79, 123524 (2009). [51] R. Bousso, J. High Energy Phys. 06 (1999) 028. [25] R. Bousso, B. Freivogel, S. Leichenauer, and V. [52] R. Bousso and I.-S. Yang, Phys. Rev. D 75, 123520 Rosenhaus, Phys. Rev. D 82, 125032 (2010). (2007). [26] D. Schwartz-Perlov and A. Vilenkin, J. Cosmol. Astropart. [53] D. Schwartz-Perlov and A. Vilenkin, J. Cosmol. Astropart. Phys. 06 (2010) 024. Phys. 06 (2006) 010.

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