Andrea De Simone et al- Predicting the cosmological constant with the scale-factor cutoff measure

8/3/2019 Andrea De Simone et al- Predicting the cosmological constant with the scale-factor cutoff measure

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arXiv:0805

.2173v1[hep-th]

15May2008

MIT-CTP-3934

Predicting the cosmological constant with the scale-factor cutoff measure

Andrea De Simone,1 Alan H. Guth,1 Michael P. Salem,2 and Alexander Vilenkin2

1Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139

2Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155

It is well known that anthropic selection from a landscape with a flat prior distribution of cosmo-logical constant gives a reasonable fit to observation. However, a realistic model of the multiversehas a physical volume that diverges with time, and the predicted distribution of depends on howthe spacetime volume is regulated. We study a simple model of the multiverse with probabilities reg-ulated by a scale-factor cutoff, and calculate the resulting distribution, considering both positive andnegative values of . The results are in good agreement with observation. In particular, the scale-factor cutoff strongly suppresses the probability for values of that are more than about ten timesthe observed value. We also discuss several qualitative features of the scale-factor cutoff, includingaspects of the distributions of the curvature parameter and the primordial density contrast Q.

I. INTRODUCTION

The present understanding of inflationary cosmologysuggests that our universe is one among an infinite num-

ber of pockets in an eternally inflating multiverse.Each of these pockets contains an infinite, nearly homoge-neous and isotropic universe and, when the fundamentaltheory admits a landscape of metastable vacua, each maybe characterized by different physical parameters, or evendifferent particles and interactions, than those observedwithin our pocket. Predicting what physics we shouldexpect to observe within our region of such a multiverseis a major challenge for theoretical physics. (For recentreviews of this issue, see e.g. [1, 2, 3, 4, 5].)

The attempt to build a calculus for such predictions iscomplicated in part by the need to regulate the diverg-ing spacetime volume of the multiverse. A number ofdifferent approaches to this measure problem has beenexplored: a cutoff at a fixed global time [6, 7, 8]1, the so-called gauge-invariant measures [11, 12], where differ-ent cutoff times are used in different pockets in order tomake the measure approximately time-parametrizationinvariant, the pocket-based measure [13, 14, 15, 16],which avoids reference to global time by focusing onpocket abundances and regulates the diverging volumewithin each pocket with a spherical volume cutoff, andfinally the causal patch measures [17, 18], which restrictconsideration to the spacetime volume accessible to a sin-gle observer.2 Different measures make different obser-vational predictions. In order to decide which, if any, is

1 Much of the early work sought to calculate the relative vol-umes occupied by different pockets on hypersurfaces of constanttime [9, 10]. In Ref. [6] the probabilities were expressed in termsof the fluxes appearing in the Fokker-Planck equation for eternalinflation. In most (but not all) cases, this method is equivalentto imposing a cutoff at a constant time. The prescription of aglobal time cutoff was first explicitly formulated in [7].

2 We also note the recent measure proposals in Refs. [19, 20].Observational predictions of these measures have not yet beenworked out, so we shall not discuss them any further.

on the right track, one can take an empirical approach,working out the predictions of candidate measures andcomparing them with the data. In this spirit, we investi-gate one of the simplest global-time measure proposals:the scale-factor cutoff measure.

The main focus of this paper is on the prediction ofthe cosmological constant [21, 22, 23, 7, 24, 25], whichis arguably a major success of the multiverse picture.Most calculations of the distribution of in the litera-ture [24, 25, 26, 27, 28, 29] do not explicitly specify themeasure, but in fact correspond to using the pocket-basedmeasure. The distribution of positive in a causal-patchmeasure has also been considered [30]. The authors ofRef. [30] emphasize that the causal-patch measure givesa strong suppression for values of more than aboutten times the observed value, while anthropic constraintsalone might easily allow values 1000 times larger thanobserved, depending on assumptions. Here, we calculate

the distribution for in the scale-factor cutoff measure,considering both positive and negative values of , andcompare our results with those of other approaches. Wefind that our distribution is in a good agreement withthe observed value of , and that the scale-factor cutoffgives a suppression for large positive values of that isvery similar to that of the causal-patch measure.

We also show that the scale-factor cutoff measure isnot afflicted with some of the serious problems arisingin other approaches. For example, another member ofthe global time measure family the proper-time cut-off measure predicts a population of observers thatis extremely youth-dominated [31, 2, 32]. Observers whotake a little less time to evolve are hugely more numerousthan their slower-evolving counterparts, suggesting thatwe should most likely have evolved at a very early cos-mic time, when the conditions for life were rather hostile.This counter-factual prediction is known as the young-ness paradox. Furthermore, the gauge-invariant andpocket-based measures suffer from a Q catastrophe,exponentially preferring either very large or very smallvalues of the primordial density contrast Q [33, 34]. Infact, this problem is not restricted to Q there are simi-lar expectations for the gravitational constant G [35]. We
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show that the youngness bias is very mild in the scale-factor cutoff, and that there is no Q (or G) catastrophe.We also describe qualitative expectations for the distri-butions ofQ and of the curvature parameter .

This paper is organized as follows. In section II wedescribe the scale-factor cutoff, commenting on its moresalient features including its very mild youngness biasand aspects of the distributions of Q and . In sec-

tion III we compute the probability distribution of ,calculating it first for the pocket-based measure, repro-ducing previous results, and then calculating it for thescale-factor cutoff. In both cases we study positive andnegative values of . Our main results are summarizedin section IV. Finally, we include two appendices. Inappendix A we consider the possibility that the land-scape splits into several disconnected sectors, and showthat even in this situation the scale-factor cutoff measureis essentially independent of the initial state of the uni-verse. Appendix B contains an analysis of the evolutionof the collapse density threshold, along with a descriptionof the linear growth function of density perturbations.

II. THE SCALE-FACTOR CUTOFF

A. Global time cutoffs

To introduce a global time cutoff, we start with a patchof a spacelike hypersurface somewhere in the inflat-ing part of spacetime, and follow its evolution along thecongruence of geodesics orthogonal to . The spacetimeregion covered by this congruence will typically have infi-nite spacetime volume, and will include an infinite num-ber of pockets. In the global-time cutoff approach weintroduce a time coordinate t, and restrict our attentionto the finite spacetime region (, tc) swept out by thegeodesics prior to t = tc, where tc is a cutoff which istaken to infinity at the end of the calculation. The rel-ative probability of any two types of events A and B isthen defined to be

p(A)

p(B) lim

tc

n

A, (, tc)

n

B, (, tc) , (1)

where n(A, ) and n(B, ) are the number of events oftypes A and B respectively in the spacetime region .In particular, the probability Pj of measuring parame-ter values corresponding to a pocket of type j is pro-portional to the number of independent measurementsmade in that type of pocket, within the spacetime region(, tc), in the limit tc .

The time coordinate t is global in the sense thatconstant-time surfaces cross many different pockets.Note however that it does not have to be global for theentire spacetime, so the initial surface does not haveto be a Cauchy surface for the multiverse. It need notbe monotonic, either, where for nonmonotonic t we limit(, tc) to points along the geodesics prior to the firstoccurrence of t = tc.

As we will discuss in more detail in appendix A, prob-ability distributions obtained from this kind of measureare independent of the choice of the hypersurface .3

They do depend, however, on how one defines the timeparameter t. To understand this sensitivity to the choiceof cutoff, note that the eternally inflating universe israpidly expanding, such that at any time most of thevolume is in pockets that have just formed. These pock-

ets are therefore very near the cutoff surface at t = tc,which explains why distributions depend on exactly howthat surface is drawn.

A natural choice of the time coordinate t is the propertime along the geodesic congruence. But as we havealready mentioned, and will discuss in more detail inthe following subsection, this choice is plagued with theyoungness paradox, and therefore does not yield a sat-isfactory measure. Another natural option is to use theexpansion factor a along the geodesics as a measure oftime. The scale-factor time is then defined as

t ln a . (2)

The use of this time parameter for calculating probabil-ities is advocated in Ref. [36] and is studied in variouscontexts in Refs. [9], [10], [6], and [8].4 It amounts tomeasuring time in units of the local Hubble time H1,

dt = Hd . (3)

The scale-factor cutoff is imposed at a fixed value of t =tc, or, equivalently, at a fixed expansion factor ac.

The term scale factor is often used in the context ofhomogeneous and isotropic spaces, but it is easily gener-alized to spacetimes with no such symmetry. In the gen-eral case, the scale-factor time can be defined by Eq. (3)with

H = (1/3) u; , (4)

where u(x) is the four-velocity vector along thegeodesics. This definition has a simple geometric mean-ing, which can be seen by imagining that the congruenceof geodesics describes the flow of a dust of test parti-cles. If the dust of particles is assumed to have a uniform

3 Here, and in most of the paper, we assume an irreducible land-scape, where any metastable inflating vacuum is accessible fromany other such vacuum through a sequence of transitions. Alter-natively, if the landscape splits into several disconnected sectors,each sector will be characterized by an independent probabilitydistribution and our discussion will still be applicable to any ofthese sectors. The distribution in case of a reducible landscapeis discussed in appendix A.

4 The measure studied in Ref. [36] is a comoving-volume measureon surfaces of constant scale-factor time; it is different from thescale-factor cutoff measure being discussed here. In particular,the former measure has a strong dependence on the initial stateat the hypersurface . Our measure is very similar to one stud-ied in Ref. [8], which is called the pseudo-comoving volume-weighted measure.


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density 0 on the initial surface , then the four-currentof the dust can be described by j(x) = (x)u(x), where = 0 on . Conservation of the current then impliesthat u + u

; = 0, which with Eqs. (3) and (4)

implies that

D ln = u; = 3Dt , (5)where D

u is the derivative with respect to

proper time along the geodesics. The solution is then = 0e

3t. From Eq. (2) we then have a 1/3, sothe scale-factor cutoff is triggered when the density (x)of the dust in its own rest frame drops below a certainspecified level.

The divergence of geodesics during inflation or homo-geneous expansion can be followed by convergence duringstructure formation or in regions dominated by a negativecosmological constant. The scale-factor time then ceasesto be a good time variable, but this does not precludeone from using it to impose a cutoff. A geodesic is termi-nated when the scale factor first reaches the cutoff valueac. If the scale factor turns around and starts decreasing

before reaching that value, we continue the geodesic allthe way to the crunch. When geodesics cross we can stilldefine the scale factor time along each geodesic accordingto Eqs. (3) and (4); then one includes a point in (, tc)if it lies on any geodesic prior to the first occurrence oft = tc on that geodesic.

To facilitate further discussion, it will be useful to re-view some general features of eternally inflating space-times, and how they are reflected in proper time andscale-factor time slicings. Regions of an eternally inflat-ing multiverse may evolve in two distinct ways. In thecase of quantum diffusion [37, 38], inflation is driven bythe potential energy of some light scalar fields, the evo-

lution of which is dominated by quantum fluctuationsand is described by the Fokker-Planck equation (see e.g.Ref. [10]). Pockets form when the scalar field(s) fluctuateinto a region of parameter space where classical evolutiondominates, and slow-roll inflation ensues. One can de-fine spacelike hypersurfaces separating the quantum andclassical regimes (see for example Ref. [15]), which wedenote by q. In universes like ours, slow-roll inflation isfollowed by thermalization (reheating) and the standardpost-inflationary evolution. We denote the hypersurfaceof thermalization, which separates the inflationary andpost-inflationary epochs, as .

The multiverse may also (or instead) feature massivefields associated with large false-vacuum energies. Evolu-tion is then governed by bubble nucleation through quan-tum tunneling [39, 40] and can be described to good ap-proximation by a suitable master equation [41, 15]. Thetunneling may proceed into another local minimum, intoa region of quantum diffusion, or into a region of classicalslow-roll inflation. In the latter case, the bubble interiorshave the geometry of open FRW universes [42]. Bubblesof interest to us here have a period of slow-roll inflationfollowed by thermalization. The role of the hypersurfaceq is played in this case by the surface separating the ini-

tial curvature-dominated regime and the slow-roll regimeinside the bubble. The differences between quantum dif-fusion and tunneling are not important for most of thediscussion below, so we shall use notation and terminol-ogy interchangeably.

The number of objects of any type that have formedprior to some time t is proportional to et , where is thelargest eigenvalue of the physical-volume Fokker-Planck

or master equation. This is because the asymptotic be-havior is determined by the eigenstate with the largesteigenvalue. Similarly, the physical volume that thermal-izes into pockets of type j between times t and t + dt hasthe form

dVj = Cjetdt , (6)

where Cj is a constant that depends on the type ofpocket. (This was derived in Ref. [6] for models withquantum diffusion and in Refs. [43] and [32] for modelswith bubble nucleation.)

The value of in Eq. (6) is the same for all pockets,but it depends on the choice of time variable t. With a

proper-time slicing, it is given by

3Hmax (t = ) , (7)where Hmax is the expansion rate of the highest-energyvacuum in the landscape, and corrections associated withdecay rates and upward tunneling rates have been ig-nored. In this case the overall expansion of the multiverseis driven by this fastest-expanding vacuum, which thentrickles down to all of the other vacua. With scale-factor slicing, all regions would expand as a3 = e3t if itwere not for the continuous loss of volume to terminalvacua with negative or zero . Because of this loss, thevalue of is slightly smaller than 3, and the differenceis determined mostly by the rate of decay of the slowest-decaying (dominant) vacuum in the landscape [44],

3 D (t = ln a) . (8)Here,

D = (4/3) D/H4D , (9)

where D is the decay rate of the dominant vacuum perunit spacetime volume, and HD is its expansion rate. Thevacuum decay rate is typically exponentially suppressed,so for the slowest-decaying vacuum we expect it to beextremely small. Hence,

3 1 . (10)

B. The youngness bias

As we have already mentioned, the proper-time cutoffmeasure leads to rather bizarre predictions, collectivelyknown as the youngness paradox [31, 2, 32]. With propertime slicing, Eqs. (6) and (7) tell us that the growth of


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volume in regions of all types is extremely fast, so atany time the thermalized volume is exponentially domi-nated by regions that have just thermalized. With thissuper-fast expansion, observers who take a little less timeto evolve are rewarded by a huge volume factor. Thismeans most observers form closer to the cutoff, whenthere is much more volume available. Assuming thatHmax is comparable to Planck scale, as one might ex-

pect in the string theory landscape, then observers whoevolved faster than us by = 109 years would havean available thermalized volume which is larger than thevolume available to us by a factor of

e e3Hmax exp(1060) . (11)Unless the probability of life evolving so fast is sup-pressed by a factor greater than exp(1060), then theserapidly evolving observers would outnumber us by a hugefactor. Since these observers would measure the cos-mic microwave background (CMB) temperature to beT = 2.9 K, it would be hard to explain why we measureit to be T = 2.73 K. Note that because Hmax appears

in the exponent, the situation is qualitatively unchangedby considering much smaller values of Hmax or .The situation with a scale-factor cutoff is very differ-

ent. To illustrate methods used throughout this paper,let us be more precise. Let t denote the interval inscale-factor time between the time of thermalization, t,and the time when some class of observers measures theCMB temperature. A time cutoff excludes the count-ing of observers who measure the CMB temperature attimes later than tc, so the number of counted observersis proportional to the volume that thermalizes at timet < tc t. (For simplicity we focus on pockets thathave the same low-energy physics as ours.) The volumeof regions thermalized per unit time is given by Eq. (6).During the time interval t, some of this volume may de-cay by tunneling transitions to other vacua. This effectis negligible, and we henceforth ignore it. For a given t,the thermalized volume available for observers to evolve,as counted by the scale-factor cutoff measure, is

V(t) tct

et dt et . (12)

To compare with the results above, consider the rela-tive amounts of volume available for the evolution of twodifferent civilizations, which form at two different timeintervals since thermalization, t1 and t2:

V(t1)V(t2) = e(t2t1) = (a2/a1) , (13)

where ai is the scale factor at time t+ti. Thus, taking 3, the relative volumes available for observers whomeasure the CMB at the present value (T = 2.73 K),compared to observers who measure it at the value of109 years ago (T = 2.9 K), is given by

V(2.73 K)V(2.9 K)

2.73 K

2.9 K

3 0.8 . (14)

Thus, the youngness bias is very mild in the scale-factorcutoff measure. Yet, as we shall see, it can have interest-ing observational implications.

C. Expectations for the density contrast Q and the

curvature parameter

Pocket-based measures, as well as gauge-invariantmeasures, suffer from a Q catastrophe where one ex-pects to measure extreme values of the primordial densitycontrast Q. To see this, note that these measures expo-nentially prefer parameter values that generate a largenumber of e-folds of inflation. This by itself does not ap-pear to be a problem, but Q is related to parameters thatdetermine the number of e-folds. The result of this is aselection effect that exponentially prefers the observationof either very large or very small values of Q, dependingon the model of inflation and on which inflationary pa-rameters scan (i.e., which parameters vary significantlyacross the landscape) [33, 34]. On the other hand, we ob-

serve Q to lie comfortably in the middle of the anthropicrange [45], indicating that no such strong selection ef-fect is at work.5 Note that a similar story applies to themagnitude of the gravitational constant G [35].

With the scale-factor cutoff, on the other hand, this isnot a problem. To see this, consider a landscape in whichthe only parameter that scans is the number of e-folds ofinflation; all low-energy physics is exactly as in our uni-verse. Consider first the portions of the hypersurfaces qthat begin slow-roll inflation at time tq in the interval dtq.These regions begin with a physical volume proportionalto etq dtq, and those that do not decay grow by a factorof e3Ne before they thermalize at time t = tq + Ne. If

I is the transition rate out of the slow-roll inflationaryphase (as defined in Eq. (9)), then the fraction of volumethat does not undergo decay is eINe .

After thermalization at time t, the evolution is thesame in all thermalized regions. Therefore we ignore thiscommon evolution and consider the number of observersmeasuring a given value of Ne to be proportional to thevolume of thermalization hypersurfaces that appear attimes earlier than the cutoff at scale-factor time tc. Thiscutoff requires t = tq + Ne < tc. Summing over all timestq gives

P(Ne)

e(3I)Ne

tcNe

etq dtq

e(3I)Ne . (15)

Even though the dependence on Ne is exponential, thefactor

3 I D I (16)

5 Possible resolutions to this problem have been proposed inRefs. [33, 34, 46, 47].


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is exponentially suppressed. Thus we find P(Ne) is a veryweak function of Ne, and there is not a strong selectioneffect for a large number of e-folds of slow-roll inflation.In fact, since the dominant vacuum D is by definitionthe slowest-decaying vacuum, we have I > D. Thusthe scale-factor cutoff introduces a very weak selectionfor smaller values of Ne.

6

Because of the very mild dependence on Ne, we do

not expect the scale-factor measure to impose significantcosmological selection on the scanning of any inflationaryparameters. Thus, there is no Q catastrophe nor isthere the related problem for G and the distributionof Q is essentially its distribution over the states in thelandscape, modulated by inflationary dynamics and anyanthropic selection effects.

The distribution P(Ne) is also important for the ex-pected value of the curvature parameter . This is be-cause the deviation of from unity decreases during aninflationary era,

| 1| e2Ne . (17)

Hence pocket-based and gauge-invariant measures,which exponentially favor large values of Ne, predict auniverse with extremely close to unity. The distribu-tions of from a variety of models have been calculatedusing a pocket-based measure in Refs. [13] and [43].

On the other hand, as we have just described, the scale-factor cutoff measure does not significantly select for anyvalue ofNe. There will still be some prior distribution ofNe, related to the distributions of inflationary parametersover the states in the landscape, but it is not necessarythat Ne be driven strongly toward large values (in fact, ithas been argued that small values should be preferred inthe string landscape, see e.g. Ref. [48]). Thus, it appearsthat the scale-factor cutoff allows for the possibility ofa detectable negative curvature. The probability distri-bution of in this type of measure has been discussedqualitatively in Ref. [48]; a more detailed quantitativeanalysis will be given elsewhere [49].

III. THE DISTRIBUTION OF

A. Model assumptions

We now consider a landscape of vacua with the same

low-energy physics as we observe, except for an essen-tially continuous distribution of possible values of . Ac-cording to Eq. (6), the volume that thermalizes betweentimes t and t+ dt with values of cosmological constant

6 We are grateful to Ben Freivogel for pointing out to us the needto account for vacuum decay during slow-roll inflation. He hasalso suggested that this effect will lead to preference for smallervalues ofNe.

between and + d is given by

dV() = C()d etdt . (18)

The factor ofC() plays the role of the prior distribu-tion of ; it depends on the spectrum of possible valuesof in the landscape and on the dynamics of eternalinflation. The standard argument [21, 24] suggests thatC() is well approximated by

C() const , (19)because anthropic selection restricts to values that arevery small compared to its expected range of variation inthe landscape. The conditions of validity of this heuristicargument have been studied in simple landscape mod-els [44, 50, 51], with the conclusion that it does in factapply to a wide class of models. Here, we shall assumethat Eq. (19) is valid.

Anthropic selection effects are usually characterizedby the fraction of matter that has clustered in galax-ies. The idea here is that a certain average number of

stars is formed per unit galactic mass and a certain num-ber of observers per star, and that these numbers arenot strongly affected by the value of . Furthermore,the standard approach is to assume that some minimumhalo mass MG is necessary to drive efficient star forma-tion and heavy element retention. Since we regulate thevolume of the multiverse using a time cutoff, it is impor-tant for us to also track at what time observers arise. Weassume that after halo collapse, some fixed proper timelapse is required to allow for stellar, planetary, andbiological evolution before an observer can measure .Then the number of observers measuring before sometime in a thermalized volume of size V is roughly

N F(MG, )V , (20)where F is the collapse fraction, measuring the fractionof matter that clusters into objects of mass greater thanor equal to MG, at time .

Anthropic selection for structure formation ensuresthat within each relevant pocket matter dominates theenergy density before does. Thus, all thermalized re-gions evolve in the same way until well into the era ofmatter domination. To draw upon this common evolu-tion, within each pocket we define proper time withrespect to a fixed time of thermalization, . It is conve-nient to also define a reference time m such that m ismuch larger than the time of matter-radiation equalityand much less than the time of matter- equality. Thenevolution before time m is the same in every pocket,while after m the scale factor evolves as

a() =

H2/3 sinh

2/332H

for > 0

H2/3 sin

2/332

H

for < 0 .(21)

Here we have defined

H

||/3 , (22)


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and use units with G = c = 1. The prefactors H2/3

ensure that early evolution is identical in all thermalizedregions. This means the global scale factor a is relatedto a by some factor that depends on the scale-factor timet at which the region of interest thermalized.

In the case > 0, the rate at which halos accrete mat-ter decreases with time and halos may settle into galaxiesthat permit quiescent stellar systems such as ours. The

situation with < 0 is quite different. At early times,the evolution of overdensities is the same; but when theproper time reaches turn = /3H, the scale factor be-gins to decrease and halos begin to accrete matter at arate that increases with time. Such rapid accretion mayprevent galaxies from settling into stable configurations,which in turn would cause planetary systems to undergomore frequent close encounters with passing stars. Thiseffect might become significant even before turnaround,since our present environment benefits from positive slowing the collision rate of the Milky Way with othersystems.

For this reason, we use Eq. (20) to estimate the number

of observers if > 0, but for < 0 we consider twoalternative anthropic hypotheses:

A. we use Eq. (20), but of course taking account of thefact that the proper time cannot exceed crunch =2/3H; or

B. we use Eq. (20), but with the hypothesis that theproper time is capped at turn = /3H.

Here crunch refers to the proper time at which a ther-malized region in a collapsing pocket reaches its futuresingularity, which we refer to as its crunch. Anthropichypothesis A corresponds to the assumption that life can

form in any sufficiently massive collapsed halo, while an-thropic hypothesis B reflects the assumption that theprobability for the formation of life becomes negligible inthe tumultuous environment following turnaround. Simi-lar hypotheses for < 0 were previously used in Ref. [29].It seems reasonable to believe that the truth lies some-where between these two hypotheses, perhaps somewhatcloser to hypothesis B.

B. Distribution of using a pocket-based measure

Before calculating the distribution of using a scale-factor cutoff, we review the standard calculation [24, 25,26, 27, 28, 29]. This approach assumes an ensemble ofequal-size regions with a flat prior distribution of . Theregions are allowed to evolve indefinitely, without anytime cutoff, so in the case of > 0 the selection factoris given by the asymptotic collapse fraction at .For < 0 we shall consider anthropic hypotheses A andB. This prescription corresponds to using the pocket-based measure, in which the ensemble includes sphericalregions belonging to different pockets and observations

0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

FIG. 1: The normalized distribution of for > 0, with in units of the observed value, for the pocket-based measure.The vertical bar highlights the value we measure, while theshaded regions correspond to points more than one and twostandard deviations from the mean.

are counted in the entire comoving history of these re-gions. The corresponding distribution function is givenby

P()

F(MG, ) for > 0F(MG, crunch ) for < 0 (A)F(MG, turn ) for < 0 (B) ,

(23)

where, again, crunch = 2/3H is the proper time of thecrunch in pockets with < 0, while turn = /3H.

We approximate the collapse fraction F using thePress-Schechter (PS) formalism [52], which gives

F(MG, ) = erfc

c()

2 (MG, )

, (24)

where (MG, ) is the root-mean-square fractional den-sity contrast M/M averaged over a comoving scale en-closing mass MG and evaluated at proper time , whilec is the collapse density threshold. As is further ex-plained in appendix B, c() is determined by consideringa top-hat density perturbation in a flat universe, withan arbitrary initial amplitude. c() is then defined asthe amplitude reached by the linear evolution of an over-density of nonrelativistic matter m/m that has thesame initial amplitude as a top-hat density perturbationthat collapses to a singularity in proper time . c()has the constant value of 1.686 in an Einstein-de Sit-ter universe (i.e., flat, matter-dominated universe), butit evolves with time when = 0 [53, 54]. We simulatethis evolution using the fitting functions (B23), which areaccurate to better than 0.2%. Note, however, that theresults are not significantly different if one simply usesthe constant value c = 1.686.

Aside from providing the collapse fraction, the PS for-malism describes the mass function, i.e. the distribu-tion of halo masses as a function of time. N-body sim-ulations indicate that PS model overestimates the abun-dance of halos near the peak of the mass function, while


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0.30

0.35

0.01 0.1 1 10 100 1000

0.00

0.05

0.10

0.15

0.20

0.25

FIG. 2: The normalized distribution of , with in units of the observed value, for the pocket-based measure. The left columncorresponds to anthropic hypothesis A while the right column corresponds to anthropic hypothesis B. Meanwhile, the top rowshows P() while the bottom row shows P(||). The vertical bars highlight the value we measure, while the shaded regionscorrespond to points more than one and two standard deviations from the mean.

underestimating that of more massive structures [55].Consequently, other models have been developed (see e.g.

Refs. [56]), while others have studied numerical fits to N-body results [29, 57]. From each of these approaches, thecollapse fraction can be obtained by integrating the massfunction. We have checked that our results are not signif-icantly different if we use the fitting formula of Ref. [29]instead of Eq. (24). Meanwhile, we prefer Eq. (24) to thefit of Ref. [29] because the latter was performed usingonly numerical simulations with > 0.

The evolution of the density contrast is treated lin-early, to be consistent with the definition of the collapsedensity threshold c. Thus we can factorize the behaviorof (MG, ), writing

(MG, ) = (MG) G() , (25)

where G() is the linear growth function, which is nor-malized so that the behavior for small is given byG() (3H /2)2/3. In appendix B we will giveexact integral expressions for G(), and also the fit-ting formulae (B12) and (B13), taken from Ref. [29],that we actually used in our calculations. Note that for 0 the growth rate G() always decreases with time(G() < 0), while for < 0 the growth rate reachesa minimum at 0.24crunch and then starts to accel-

erate. This accelerating rate of growth is related to theincreasing rate of matter accretion in collapsed halos af-

ter turnaround, which we mentioned above in motivatingthe anthropic hypothesis B.

The prefactor (MG) in Eq. (25) depends on the scaleMG at which the density contrast is evaluated. Accordingto our anthropic model, MG should correspond to theminimum halo mass for which star formation and heavyelement retention is efficient. Indeed, the efficiency ofstar formation is seen to show a sharp transition: it fallsabruptly for halo masses smaller than MG

2

1011M,where M is the solar mass [58]. Peacock [29] showedthat the existing data on the evolving stellar density canbe well described by a Press-Schechter calculation of thecollapsed density for a single mass scale, with a best fitcorresponding to (MG, 1000) 6.74103, where 1000is the proper time corresponding to a temperature T =1000 K. Using cosmological parameters current at thetime, Peacock found that this perturbation amplitudecorresponds to an effective galaxy mass of 1.91012M.Using the more recent WMAP-5 parameters [59], as is


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8

done throughout this paper,7 we find (using Ref. [60] andthe CMBFAST program) that the corresponding effectivegalaxy mass is 1.8 1012M.

Unless otherwise noted, in this paper we set the pref-actor (MG) in Eq. (25) by choosing MG = 10

12M.Using the WMAP-5 parameters and CMBFAST, we findthat at the present cosmic time (1012M) 2.03. Thiscorresponds to (1012M, 1000) 7.35 103.

We are now prepared to display the results, plottingP() as determined by Eq. (23). We first reproduce thestandard distribution of , which corresponds to the casewhen > 0. This is shown in Fig. 1. We see that thevalue of that we measure is between one and two stan-dard deviations from the mean. Throughout the paper,the vertical bars in the plots merely highlight the ob-served value of and do not indicate its experimentaluncertainty. The quality of the fit depends on the choiceof scale MG; in particular, choosing smaller values ofMGweakens the fit [28, 61]. Note however that the value ofMG that we use is already less than that recommendedby Ref. [29].

Fig. 2 shows the distribution of for positive and neg-ative values of . We use = 5 109 years, corre-sponding roughly to the age of our solar system. Theleft column corresponds to choosing anthropic hypoth-esis A while the right column corresponds to anthropichypothesis B. To address the question of whether theobserved value of || lies improbably close to the specialpoint = 0, in the second row we plot the distribu-tions for P(||). We see that the observed value of liesonly a little more than one standard deviation from themean, which is certainly acceptable. (Another measureof the typicality of our value of has been studied inRef. [28]).

C. Distribution of using the scale-factor cutoff

We now turn to the calculation of P() using a scale-factor cutoff to regulate the diverging volume of the mul-tiverse. When we restrict attention to the evolution ofa small thermalized patch, a cutoff at scale-factor timetc corresponds to a proper time cutoff c, which dependson tc and the time at which the patch thermalized, t.Here we take the thermalized patch to be small enoughthat scale-factor time t is essentially constant over hy-persurfaces of constant . Then the various proper andscale-factor times are related by

tc t =c

H() d = ln

a(c)/a()

. (26)

Recall that all of the thermalized regions of interestshare a common evolution up to the proper time m,

7 The relevant values are = 0.742, m = 0.258, b = 0.044,ns = 0.96, h = 0.719, and 2R(k = 0.02Mpc

1) = 2.21 109.

0.01 0.1 1 10 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

FIG. 3: The normalized distribution of for > 0, with in units of the observed value, for the scale-factor cutoff. Thevertical bar highlights the value we measure, while the shadedregions correspond to points more than one and two standarddeviations from the mean.

after which they follow Eqs. (21). Solving for the proper

time cutoff c gives

c =2

3H1 arcsinh

32Hm e

3

2(tctC)

, (27)

for the case > 0, and

c =2

3H1 arcsin

32Hm e

3

2(tctC)

, (28)

for < 0. The term C is a constant that accounts forevolution from time to time m. Note that as tc t isincreased in Eq. (28), c grows until it reaches the timeof scale-factor turnaround in the pocket, turn = /3H,after which the expression is ill-defined. Physically, the

failure of Eq. (28) corresponds to when a thermalizedregion reaches turnaround before the scale-factor timereaches its cutoff at tc. After turnaround, the scale factordecreases; therefore these regions evolve without a cutoffall the way up to the time of crunch, crunch = 2/3H.

When counting the number of observers in the variouspockets using a scale-factor cutoff, one must keep in mindthe dependence on the thermalized volume V in Eq. (20),since in this case V depends on the cutoff. As statedearlier, we assume the rate of thermalization for pocketscontaining universes like ours is independent of . Thus,the total physical volume of all regions that thermalizedbetween times t and t + dt is given by Eq. (6), and is

independent of . Using Eq. (20) to count the number ofobservers in each thermalized patch, and summing overall times below the cutoff, we find

P() tc

F

MG, c(tc, t)

etdt . (29)

Note that regions thermalizing at a later time t have agreater weight et. This is an expression of the young-ness bias in the scale-factor measure. The dependenceof this distribution is implicit in F, which depends on


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9

anthropic hypothesis A anthropic hypothesis B

20 10 0 100.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

5 0 5 100.00

0.05

0.10

0.15

1.00.5 5.0 10.0 50.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.01 0.1 1 10

0.0

0.1

0.2

0.3

0.4

0.5

FIG. 4: The normalized distribution of , with in units of the observed value, for the scale-factor cutoff. The left columncorresponds to anthropic hypothesis A while the right column corresponds to anthropic hypothesis B. Meanwhile, the top rowshows P() while the bottom row shows P(||). The vertical bars highlight the value we measure, while the shaded regionscorrespond to points more than one and two standard deviations from the mean.

c(, c )/rms(, c ), and in turn on c(),which is described below.

For pockets with > 0, the cutoff on proper time c isgiven by Eq. (27). Meanwhile, when < 0, c is given byEq. (28), when that expression is well-defined. In prac-tice, the constant C of Eqs. (27) and (28) is unimportant,since a negligible fraction of structures form before theproper time m. Furthermore, for a reference time mchosen deep in the era of matter domination, the nor-malized distribution is independent of m. As mentionedabove, for sufficiently large tc t Eq. (28) becomes ill-defined, corresponding to the thermalized region reach-ing its crunch before the scale-factor cutoff. In this casewe set c = crunch or c = turn, corresponding to theanthropic hypothesis A or B described above.

To compare with previous work, we first display thedistribution of positive in Fig. 3. We have set = 3and use = 5 109 years. Clearly, the scale-factorcutoff provides an excellent fit to observation, when at-tention is limited to > 0. Note that the scale-factor-cutoff distribution exhibits a much faster fall off at large than the pocket-based distribution in Fig. 1. The rea-son is not difficult to understand. For larger values of ,the vacuum energy dominates earlier. The universe thenbegins expanding exponentially, and this quickly triggers

the scale-factor cutoff. Thus, pockets with larger valuesof have an earlier cutoff (in terms of the proper time)

and have less time to evolve observers. This tendency forthe cutoff to kick in soon after -domination may helpto sharpen the anthropic explanation [26, 62] of the oth-erwise mysterious fact that we live so close to this veryspecial epoch (matter- equality) in the history of theuniverse.

The distribution of for positive and negative values of is displayed in Fig. 4, using the same parameter valuesas before. We see that the distribution with anthropichypothesis A provides a reasonable fit to observation,with the measured value of appearing just within twostandard deviations of the mean. Note that the weightof this distribution is dominated by negative values of

, yet anthropic hypothesis A may not give the mostaccurate accounting of observers in pockets with < 0.Anthropic hypothesis B provides an alternative count ofthe number of observers in regions that crunch before thecutoff, and we see that the corresponding distributionsprovide a very good fit to observation. This is the mainresult of this work.

The above distributions all use = 5109 years andMG = 10

12M. These values are motivated respectivelyby the age of our solar system and by the mass of our


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10

10 5 0 5 100.00

0.05

0.10

0.15

0.20

0.25

5 0 5 100.00

0.05

0.10

0.15

FIG. 5: The normalized distribution of , with in units of the observed value, for anthropic hypothesis B in the scale-factorcutoff. The left panel displays curves for = 3 (solid), 5 (dashed), and 7 (dotted) 109 years, with MG = 10

12M, whilethe right panel displays curves for MG = 10

10M (solid), 1011M (dashed), and 10

12M (dotted), with = 5 109 years.

The vertical bars highlight the value of that we measure.

galactic halo, the latter being a good match to an em-pirical fit determining the halo mass scale characterizingefficient star formation [29]. Yet, to illustrate the depen-

dence of our main result on and MG, in Fig. 5 wedisplay curves for anthropic hypothesis B, using = 3,5, and 7 109 years and using MG = 1010, 1011, and1012M. The distribution varies significantly as a resultof these changes, but the fit to the observed value of remains good.

IV. CONCLUSIONS

To date, several qualitatively distinct measures havebeen proposed to regulate the diverging volume of the

multiverse. Although theoretical analysis has not pro-vided much guidance as to which of these, if any, iscorrect, the various regulating procedures make differentpredictions for the distributions of physical observables.Therefore, one can take an empirical approach, compar-ing the predictions of various measures to our observa-tions, to shed light on what measures are on the righttrack. With this in mind, we have studied some aspectsof a scale-factor cutoff measure. This measure averagesover the spacetime volume in a comoving region betweensome initial spacelike hypersurface and a final hyper-surface of constant time, with time measured in units ofthe local Hubble rate along the comoving geodesics. Atthe end of the calculation, the cutoff on scale-factor time

is taken to infinity. We shall now summarize what wehave learned about the scale-factor measure and compareits properties to those of other proposed measures.

The main focus of this paper has been on the proba-bility distribution for the cosmological constant . Al-though the statistical distribution of among states inthe landscape is assumed to be flat, imposing a scale-factor cutoff modulates this distribution to prefer smallervalues of . Combined with appropriate anthropic selec-tion effects, this gives a distribution of that is in a

good fit with observation. We have calculated the dis-tribution for positive and negative values of , as wellas for the absolute value

|

|. For > 0, we adopted

the standard assumption that the number of observers isproportional to the fraction of matter clustered in halosof mass greater than 1012M, and allowed a fixed propertime interval = 5 109 years for the evolution ofobservers in such halos. For < 0, we considered twopossible scenarios, which probably bracket the range ofreasonable possibilities. The first (scenario A) assumesthat observations can be made all the way to the bigcrunch, so we count all halos formed prior to time before the crunch. The second (scenario B) assumes thatthe contracting negative- phase is hazardous to life, sowe count only halos that formed at time or earlierbefore the turnaround.

Our results show that the observed value of is withintwo standard deviations from the mean for scenario A,and within one standard deviation for scenario B. In thelatter case, the fit is better than that obtained in thestandard calculations [24, 25, 26, 27, 28, 29], whichassume no time cutoff (this is equivalent to choosing apocket-based measure on the multiverse). The causalpatch measure also selects for smaller values of provid-ing, in the case of positive , a fit to observation similarto that of the scale-factor cutoff[30]. Note, however, thatthe approach of Ref. [30] used an entropy-based anthropicweighting (as opposed to the structure-formation-basedapproach used here) and that the distribution of negative

has not been studied in this measure.We have verified that our results are robust with re-

spect to changing the parameters MG and . Theagreement with the data remains good for MG varyingbetween 1010 and 1012M and for varying between3 109 and 7 109 years.

We have also shown that the scale-factor cutoff mea-sure does not suffer from some of the problems afflictingother proposed measures. The most severe of these is theyoungness paradox the prediction of an extremely


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11

youth-dominated distribution of observers which fol-lows from the proper-time cutoff measure. The scale-factor cutoff measure, on the other hand, predicts only avery mild youngness bias, which is consistent with obser-vation. Another problem, which arises in pocket-basedand gauge-invariant measures, is the Q catastrophe,where one expects to measure the amplitude of the pri-mordial density contrast Q to have an unfavorably large

or small value. This problem ultimately stems from anexponential preference for a large number of e-folds ofslow-roll inflation in these measures. The scale-factorcutoff does not strongly select for more inflation, and thusdoes not suffer from a Q catastrophe. An unattractivefeature of causal patch and comoving-volume measuresis that their predictions are sensitive to the assumptionsone makes about the initial conditions for the multiverse.Meanwhile, the scale-factor cutoff measure is essentiallyindependent of the initial state. This property reflectsthe attractor character of eternal inflation: the asymp-totic late-time evolution of an eternally inflating universeis independent of the starting point.

As mentioned above, a key features of the scale-factorcutoff measure is that, unlike the pocket-based or gauge-invariant measures, it does not reward large amounts ofslow-roll inflation. As a result, it allows for the possibilityof a detectable negative curvature. This issue will bediscussed in detail in Ref. [49].

With any measure over the multiverse, one must bewary that it does not over-predict Boltzmann brains observers that pop in and out of existence as a re-sult of rare quantum fluctuations [63]. This issue has notbeen addressed here, but our preliminary analysis sug-gests that, with some mild assumptions about the land-scape, the scale-factor cutoff measure does not have aBoltzmann brain problem. We shall return to this issue

in a separate publication [64].

Acknowledgments

We thank Raphael Bousso, Ben Freivogel, AndreiLinde, John Peacock, Delia Schwartz-Perlov, VitalyVanchurin, and Serge Winitzki for useful comments anddiscussions. The work of ADS is supported in part by theINFN Bruno Rossi Fellowship. ADS and AHG are sup-ported in part by the U.S. Department of Energy undercontract No. DE-FG02-05ER41360. MPS and AV aresupported in part by the U.S. National Science Founda-

tion under grant NSF 322.

APPENDIX A: INDEPENDENCE OF THE

INITIAL STATE

In section II we assumed that the landscape is irre-ducible, so that any vacuum is accessible through quan-tum diffusion or bubble nucleation from any other (deSitter) vacuum. If instead the landscape splits into sev-

eral disconnected sectors, the scale-factor cutoff can be

used to find the probability distributions P(A)j in each

of the sectors (labeled by A). These distributions aredetermined by the dominant eigenstates of the Fokker-Planck or master equation, which correspond to thelargest eigenvalues A, and are independent of the choiceof the initial hypersurfaces A that are used in imple-menting the scale-factor cutoff. But the question still

remains, how do we compare the probabilities of vacuabelonging to different sectors?

Since different sectors are inaccessible from one an-other, the probability PA of being in a given sector mustdepend on the initial state of the universe. For definite-ness, we shall assume here that the initial state is de-termined by the wave function of the universe, althoughmost of the following discussion should apply to any the-ory of initial conditions. According to both tunneling[65] and Hartle-Hawking [66] proposals for the wave func-tion, the universe starts as a 3-sphere S filled with somepositive-energy vacuum . The radius of the 3-sphere isr = H

1 , where H is the de Sitter expansion rate. The

corresponding nucleation probability is

P()nucl exp

H2

, (A1)

where the upper sign is for the Hartle-Hawking and thelower is for the tunneling wave function. Once the uni-verse has nucleated, it immediately enters de Sitter infla-tionary expansion, transitions from to other vacua, andpopulates the entire sector of the landscape to which thevacuum belongs. We thus have an ensemble of eternallyinflating universes with initial conditions at 3-surfaces Sand the probability distribution P

()nucl given by Eq. (A1).

If the landscape were not disconnected, we could ap-ply the scale factor cutoff measure to any single compo-nent of the initial wave function, and the result wouldbe the same in all cases. To generalize the scale-factorcutoff measure to the disconnected landscape, the moststraightforward prescription is to apply the scale factorcutoff directly to the initial probability ensemble. In thatcase,

Pj,A limtc

A

P()nuclN()j (tc) . (A2)

Here,

N()j (tc) = R(A) P

(A)j eAtc (A3)

is the number of relevant observations in the entire closeduniverse, starting from the hypersurface S, with a cutoff

at scale-factor time tc. The R(A) are determined by the

initial volume of the 3-surface S, and also by the effi-ciency with which this initial state couples to the leading

eigenvector of Eq. (6). In other words, the N()j (tc) arecalculated using S as the initial hypersurface A. Note

that only the overall normalization of N()j depends on


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12

the initial vacuum ; the relative probabilities of differentvacua in the sector do not. In the limit of tc , onlythe sectors corresponding to the largest of all dominanteigenvalues,

max = max{A} , (A4)

have a nonzero probability. If there is only one sector

with this eigenvalue, this selects the sector uniquely.Since the issue of initial state dependence is new, onemight entertain an alternative method of dealing withthe issue, in which the probability PA for each sector isdetermined immediately by the initial state, with

PA A

P()nucl . (A5)

Then one could calculate any probability of interestwithin each sector, using the standard scale factor cutoffmethod, and weight the different sectors by PA. How-ever, although this prescription is well-defined, we wouldadvocate the first method that we described as the nat-ural extension of the scale factor cutoff measure. First,it seems to be more closely related to the description ofthe scale-factor cutoff measure in a connected landscape:the only change is to replace the initial state by an en-semble of states, determined in principle by ones theoryof the initial wave function. Second, in a toy theory,one could imagine approaching a disconnected landscapefrom a connected one, by gradually decreasing all thecross-sector tunneling rates to zero. In that case, thelimit clearly corresponds to the first description, whereone sector is selected uniquely if it has the largest domi-nant eigenvalue.

Assuming the first of these prescriptions, the conclu-

sion is that the probability distribution (A2) defined bythe scale-factor measure is essentially independent of the

initial distribution (A1). Some dependence on P()nucl sur-

vives only in a restricted class of models where the land-scape splits into a number of sectors with strictly zeroprobability of transitions between them and, in addition,where the maximum eigenvalue max is degenerate. Eventhen, this dependence is limited to the relative probabil-ity of the sectors characterized by the eigenvalue max.

APPENDIX B: THE COLLAPSE DENSITY

THRESHOLD c

The collapse density threshold c is determined bycomparing the linearized evolution of matter perturba-tions with the nonlinear evolution of a spherical top-hatdensity perturbation, which can be treated as a closedFRW universe. The collapse density threshold c() is de-fined as the amplitude reached by the linear evolution ofan overdensity m/m that has the same initial am-plitude as a top-hat density perturbation that collapsesto a singularity in proper time . In a matter-dominated

universe with zero cosmological constant, c is a con-stant; however, it is well known that c depends on thecollapse time when is nonzero (see e.g. Refs. [53, 54]).In this appendix we first outline the calculation of thetime evolution of c, then display the results for positiveand negative , and finally describe how we apply it inour analysis of the collapse fraction F of Eq. (24).

As suggested by the definition above, both linear and

nonlinear analyses are involved at different stages of thecalculation of the collapse density. Arbitrarily small per-turbations obey linearized equations of motion, and theirevolution defines the linear growth function G():

() G() , (B1)where G() is normalized so that the behavior for small

is given by G() (3H /2)2/3, where H =||/3.

The exact nonlinear analysis is used to determine thetime at which an overdensity with a given initial ampli-tude will collapse to a singularity. For simplicity, this isworked out for the top-hat model, where the overden-sity is assumed to be uniform and spherically symmet-

ric. Such a region is embedded in a flat FRW universecontaining only non-relativistic matter and cosmologicalconstant.

By Birkhoffs theorem, the evolution of the sphericaloverdensity is equivalent to that of a closed FRW uni-verse. The Friedmann equation for a closed FRW uni-verse, with scale factor a, may be written as:

H2 = H2

sign() +

B()

a3

a2

, (B2)

where H = d ln a/d = a/a and B() is an arbitraryquantity that fixes the normalization of a. We will al-ways choose B(0) = 1, so for = 0 the scale factor isnormalized in such a way that m =

|

|at a = 1, where

= /(8) is the vacuum energy density.Let us first focus our attention on the evolution of a

linearized density perturbation in a flat FRW universewith positive cosmological constant; the case with nega-tive cosmological constant proceeds similarly. Considera closed FRW universe obtained by perturbing the flatuniverse with a small curvature term . The propertime parameter in such a universe, as a function of thescale factor, is given by an expansion with respect to theflat background: (a) = (a) + (a), where to linearorder in

(a) =

2H a

0

a

a dBd (0)

da

(1 + a3

)

3/2. (B3)

The scale factor of the closed universe is obtained byinverting the function (a):

a() = a() a() a() . (B4)As mentioned above, the evolution of this closed FRWuniverse also gives the evolution of a small density per-turbation. Using m = (3/8)H2B()/a

3, one has

=mm

= 3 aa

+dB

d(0) = 3H +

dB

d(0) , (B5)


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13

where the last equality follows from Eq. (B4). Fromhere on, unless noted otherwise, we normalize a so thatB() = 1. It is convenient to introduce the time vari-able

x ||m

= a3 , (B6)

for both choices of the sign of . To be consistent withEq. (B2), the solutions for = 0 are not normalized asin Eq. (21), but instead are given by

a() =

sinh2/3

32

H

for > 0

sin2/332H

for < 0 .

(B7)

We can then find the evolution function (x) fromEq. (B5), using Eq. (B3) and also Eq. (B2) with = 0:

(x) =1

2

1 +

1

x

x0

dy

y1/6(1 + y)3/2

=3

5

G+(x) , (B8)

where the linear growth function (for > 0),

G+(x) =5

6

1 +

1

x

x0

dy

y1/6(1 + y)3/2, (B9)

is normalized so that the behavior for small x is given byG+(x) x1/3 = a (3H /2)2/3.

In the < 0 case, the calculation proceeds along thesame steps as before and the formula (B8) is indeed validalso for negative , after replacing the growth functionwith G(x). This function now has two branches GI (x)

and GII(x), corresponding to the expanding and con-tracting phases of the universe, respectively. The firstbranch of the growth function introduces no new compli-cations, and is found to be

GI (x) =5

6

1

x 1

x0

dy

y1/6(1 y)3/2 . (B10)

For the second branch, the integration is first performedover the whole history of the universe, from x = 0 tox = 1 and back to x = 0, and then one integrates back tothe value of interest x. There is a complication, however,because for this case the denominator in Eq. (B3) is (1 a3)3/2, so the integral diverges when the upper limit isequal to 1. The cause of the problem is that for =0, amax is no longer equal to 1. A simple cure is tochoose B() = 1 + for this case, which ensures thatamax = 1 for any , and which correspondingly providesan additional term in Eq. (B3) which causes the integralto converge. After some manipulation of the integrals,the result can be written as

GII(x) =5

6

1

x 1

4

56

13

+ x0

dy

y1/6(1 y)3/2

.

(B11)The time dependence of the linear growth functions canbe made explicit by expressing x as a function of ,through Eqs. (B6) and (B7).

In practice, we carry out our calculations using fittingfunctions for the growth functions, which were devisedby Peacock [29], and which are accurate to better than0.1%. These give

G+() tanh2/332

H

1 tanh1.2732H0.82 + 1.437H2/3 1 cosh4/332H (B12)G ()

32H

2/3 1 + 0.37 ( /crunch)

2.181

1 ( /crunch)21

, (B13)

for the cases > 0 and < 0, respectively, where thelatter fitting formula is valid for both branches.

We are now prepared to set the calculation of c. Sincethe universe in Eq. (B2) can be viewed as a perturba-

tion over a flat universe with = , the time evolutionof the overdensity is described in general by

() =3

5 G() . (B14)

The quantity (3/5) a quantifies the size of the initialinhomogeneity.

In order to find the time at which the spherical over-density collapses, it is convenient to determine the timeof turnaround turn, corresponding to when H = 0. The

time of collapse is then given by 2 turn. The turnaroundtime is obtained by integrating Eq. (B2), choosing B = 1:

Hturn() =aturn()0

a da

sign() a3 a + 1 , (B15)

where the scale factor at turnaround aturn correspondsto the smallest positive solution of

sign() a3turn aturn + 1 = 0 . (B16)

For positive , the universe will collapse only if >min 3/22/3; for negative , perturbations that col-lapse before the universe has crunched have > 0.


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14

0 1 2 3 41.6

1.7

1.8

1.9

2.0

H

c

FIG. 6: The collapse density thresholds +c (for > 0) andc (for < 0), as functions of time. The solid curves rep-resent numerical evaluations of c , while the dashed curvescorrespond to the fitting functions in Eq. (B23). Note that+c decreases with time, while

c increases with time.

The numerical evaluation of the integral in Eq. (B15)allows one the extract the function turn(), which can

be inverted to give turn(), expressing the value of that leads to turnaround at time . Finally, the collapsedensity threshold as a function of the time of collapse isread from Eq. (B14):

c() =3

5turn( /2) G() . (B17)

In the limits of small and large collapse times the aboveprocedure can be carried out analytically to find the lim-iting values of c. Let us consider first the large-timeregime, corresponding to small . In the case > 0, thesmallest allowed is min; therefore

+c () =3

5 minG+ () 1.629 , (B18)

where G+() = G+() = 5(2/3)(5/6)/(3

) 1.437. The case < 0 is a little more complicated. Thecollapse time cannot exceed crunch = 2/3H, corre-sponding to = 0. At small , the integral in Eq. (B15)is expanded to give

Hturn() 12

Hcrunch 25

116

13

. (B19)On the other hand, the growth function G() in theneighborhood of crunch behaves as

G( crunch) 10

3

56

13

1(1 /crunch) . (B20)

After using Eqs. (B19) and (B20) in the general formula(B17), we simply get

c (crunch) = 2 . (B21)

In the opposite regime H 1, corresponding tolarge , the growth functions are G () a() (3H/2)2/3. The integral (B15) can be analyticallysolved in this limit: Hturn() = /(2

3/2). Combiningthese results leads to

c (0) =3

5

3

22/3

1.686 , (B22)

which is also the constant value ofc in a = 0 universe.

The time dependence of c is displayed in Fig. 6, forboth positive and negative values of . We also displaythe following simple fitting functions,

+c () = 1.629 + 0.057 e0.28H2

2

c () = 1.686 + 0.165

crunch

2.5+ 0.149

crunch

11(B23)

which are accurate to better than 0.2%. Although wechoose to include the effect of the time evolution of c,our results are not significantly changed by treating cas a constant. This is easy to understand. First of all,+c varies by only about 3%. The evolution of

c is more

significant, about 15%, and most of this happens at verylate times. But our anthropic weight in Eq. (20) neversamples c within a time of crunch.

Finally, we point out that the appearance of G()in this discussion is not needed for the calculation, andappears here primarily to make contact with other work.From Eq. (24) one sees that the collapse fraction dependsonly on the ratio ofc()/(MG, ), which from Eqs. (25)and (B17) can be seen to equal (3/5)turn( /2)/(MG).Expressed in this way, Eq. (24) becomes fairly transpar-ent. Since is a measure of the amplitude of an initialperturbation, Eq. (24) is saying that the collapse fractionat time depends precisely on the magnitude requiredfor an initial top-hat perturbation to collapse by time. In more detail, Eq. (24) is predicated on a Gaussiandistribution of initial fluctuations, where the complemen-tary error function erfc(x) is the integral of a Gaussian.The collapsed fraction at time is given by the prob-ability, in this Gaussian approximation, for the initialfluctuations to exceed the magnitude needed for collapse

at time . From a practical point of view, the use ofG() in the discussion of the collapse fraction can be ahelpful simplification if one uses the approximation thatc const. We have not used this approximation, but asdescribed above, our results would not be much differentif we had. We have maintained the discussion in terms ofG() to clarify the relationship between our work andthis approximation.


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