Geometric origin of coincidences and hierarchies in the landscape The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bousso, Raphael et al. “Geometric origin of coincidences and hierarchies in the landscape.” Physical Review D 84.8 (2011): n. pag. Web. 26 Jan. 2012. © 2011 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.84.083517 Publisher American Physical Society (APS) Version Final published version 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, tà 3= à , 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 à 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 re- quires 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 Ã, 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 Ã,soitis d3p natural to suspect that the measure obtained from it is inapplicable to regions with à 0 [24]. (Indeed, the d logt d logt d logt obs à c causal patch does not eliminate infinities in à ¼ 0 bubbles 2 ¼ d p~  ð Þ [15]. We do not consider such regions in this paper.) M logtobs; logtc; logtà d logtÃd logtc  ð Þ logtobs; logtc; logtà : (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 ðlogtÃ; logtcÞ is produced within the region de- fined 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 logtÃ; 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; logtÃ; logtc is obs c à 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 logtÃ; logtc , so it can be computed unambiguously. The multiverse take place in negatively curved FRW universes. last factor, ðlogtobs; logtc; logtÃÞ, 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 logtÃ; 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 logtÃ; logtc; logtobs , using three differ- other regimes that in many cases we only need to know the ent measures.