Properties of the scale factor measure

0.34 MB·English

by  Raphael Bousso

Properties of the scale factor measure
Raphael Bousso, Ben Freivogel and I-Sheng Yang∗
Department of Physics and Center for Theoretical Physics
University of California, Berkeley, CA 94720, U.S.A.
and
Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, U.S.A.
We show that in expanding regions, the scale factor measure can be reformulated as a local
measure: Observationsareweightedbyintegratingtheirphysicaldensityalongageodesicthatstarts
inthelongest-livedmetastablevacuum. Thisexplainswhysomeofitspropertiesaresimilartothose
of thecausal diamond measure. In particular, both measures are free of Boltzmann brains, subject
tonearlythesameconditionsonvacuumstability. However,thescalefactormeasureassignsamuch
9 smaller probability to the observed value of the cosmological constant. The probability decreases
0 further, like the inverse sixth power of the primordial density contrast, if the latter is allowed to
0 vary.
2

n
a I. INTRODUCTION flation, should eventually allow us to derive a unique
J prescription for computing probability amplitudes from

4 A. The measure problem first principles. Yet our understanding of string theory,
1 especially in cosmological settings, remains woefully in-
complete. For now, a top-down solution to the measure

] Like the cosmological constant problem [1, 2, 3], the problem seems elusive. Thus, we advocate following the
h
measure problem arises purely within the regime of va-
t traditional, phenomenologicalapproach.
- lidity of semi-classical gravity. All that is needed is a
p
long-lived vacuum state [4], or a sufficiently flat scalar
e
fieldpotential[5,6],withpositivevacuumenergy. Under
h B. Phenomenological approach
[ these conditions, a finite spatial region will inflate eter-

nally, generating an unbounded four-volume. All possi-
2 Likeanyothertheory,acompellingmeasureshouldbe
ble events will occur infinitely many times, and a cutoff
v reasonably well-defined, simple, and general. In tackling
is needed to compute probabilities.
0 a problem so vast and unfamiliar, it is natural to seek
7 Unlikethecosmologicalconstantproblem,themeasure
more specific guiding principles. For example, lessons
7 problem leaves a loophole: it is possible that the condi-
from the black hole information paradox motivated the
3 tionsforeternalinflationdonotactuallyoccurinNature.
causal diamond measure [13, 14]. We would be ill ad-
.
8 But increasing evidence indicates otherwise. vised, however, to turn our intuition into dogma, insist-
0 Slow-rollinflationisthedominantparadigmexplaining ingabsolutelyontheoretical propertiesthat“anyreason-
8 the origin of structure and the large-scale homogeneity able” measure “must” obey. Not only would we run the
0
: and flatness of our universe. If we are prepared to be- dangerofputting inbyhandthe answerswewishtoget;
v lievethatthemoderatelyfine-tunedscalarfieldpotential worse, our wishes may be misguided. Surely, for exam-
Xi necessary for driving slow-roll inflation can arise in Na- ple, anyreasonablemeasuremustrewardthe volume ex-
ture, it is hard to imagine that the more generic feature pansion during slow-roll inflation in any given vacuum?
r
a of a local minimum cannot exist. In fact, this intuitive requirement invites conflict with
Moreover, the observed universe has positive vacuum experiment [15, 16]. The causal diamond measure aban-
energy [7, 8, 9]. Unless our vacuum is unstable on a donedvolume-weighting,andtherequirementseemsnow
timescaleofordertenbillionyears—whichwouldrequire to have lost its dogmatic status [17, 18].
remarkable tuning—it alone suffices to generate eternal This brings us to the one property of a measure that
inflation. we can insist on absolutely: that its experimental pre-
Finally, the only extant explanation [10, 11, 12] of the dictions not conflict with observations. In fact, many
smallness of the observeddark energy—the cosmological innocent-looking measures do conflict with observation,
constantproblem—istheexistenceofamultidimensional and violently so. Some conflicts are extremely robust,
landscape of metastable vacua in string theory. This is arisingalmostindependently ofthe properties ofthe un-
empiricalevidenceforstringtheory,andinparticularfor derlyingvacuumlandscape. Thus,theyallowustofalsify
eternal inflation. measuresevenwhilewestillhavemuchtolearnaboutthe
A theory of everything, if it gives rise to eternal in- landscape.
For example, the proper time cutoff [19, 20, 21, 22,
23, 24] predicts a very hot universe [25, 26, 27, 28, 29,
30,31,32]withprobability1(the“Boltzmannbabies”or
∗[email protected],[email protected], [email protected] youngnessparadox). Measures that involvecounting the
2
numberofobserversperbaryon[33,34,35,36,37,38,39] (randomly chosen) congruence of timelike geodesics has
predict an empty, cold universe [40, 41] with probabil- expandedbyavolumefactorexp(3η)alongeachgeodesic.
ity 1 (the “Boltzmann brain” paradox). Such paradoxes Relativeprobabilitiesofdifferentobservationaloutcomes
are important tools for testing and eliminating measure canbe defined by computing the ratios of the number of
theories. We say “paradox”, as if there was any doubt times suchoutcomes occurin the regulatedfour-volume,
about the culprit. In fact, the above paradoxes repre- and then taking the regulator η →∞.
sent fatal failures: the measure assigns zero probability Section II contains review material and establishes
totheobservationsweactuallymake. Suchmeasuresare most of our notation. In Sec. IIA, we give a detailed
experimentally ruled out and must be discarded. definition of the scale factor measure. A separate pre-
Another useful test is the more aptly named “Q- scription is needed in regions where geodesics contract
catastrophe”[15,16]alludedtoearlier. Inmeasuresthat and intersect with each other. We review the prescrip-
reward the volume expansion during inflation, such as tion chosen by De Simone et al., who were the first to
Ref. [38], inflationary model parameters generically re- formulateonecarefully. InSec.IIB,wecollectotheruse-
ceive exponential pressure towards extreme values. In ful definitions and results; in particular, we review the
this case, anthropic constraints do not suffice to explain solution found by Garriga, Schwartz-Perlov, Vilenkin,
the moderate values we observe for, say, the primordial and Winitzki [38] for the volume distribution of differ-
density perturbation, Q. entvacua,whichalsoappliestothescalefactormeasure.
Other problems are more subtle, or depend on the de- In Section III, we determine the shape of the scale
tailed structure of the vacuum landscape. The “stag- factor cutoff hypersurface in a homogeneous, isotropic,
gering problem” can arise in the measure of Ref. [38]: open universe formed by bubble nucleation inside a par-
for some landscape models, the measure assigns such ent de Sitter vacuum. This is nontrivial, because the lo-
unequal probabilities to the cosmological production of calscalefactoralongeachgeodesic,andtheaveragescale
different vacua that most observers live in extreme en- factor in a bubble universe, are two different objects.
vironments where their existence is an unlikely fluctua-
InSectionIV, wecompute the numberofobservations
tion [42, 43, 44, 45, 46]. In particular, the cosmological
below the cutoff surface by integrating over all bubbles
constant problem cannot be solved in this case.
produced prior to the cutoff. We begin, in Sec. IVA,
Finally, the value it predicts for the cosmologicalcon-
by approximating the bubble interior metric as homo-
stant, Λ, is an important test of any measure. This par-
geneous, ignoring local gravitational collapse such as oc-
ticularobservableisspecialfortworeasons: Itsstatistical
curs during galaxy formation. We find that in this “no-
distribution in the landscape is understood well enough
collapse approximation”, the scale factor measure leads
to make detailed quantitative predictions. And its value
to a very simple result: The probability of an observa-
has been measured:
tion in some vacuum is proportional to the product of
Λ=1.5×10−123 , (1) the probability that a given geodesic in the congruence
will enter that vacuum, times the number density, per
in Planck units. physical volume, at which observations occur. It is does
At present, the causal diamond measure is the most notdependdirectlyonthetimeatwhichtheyoccur,nor
successful proposal phenomenologically. It avoids Boltz- ontheamountofvolumeexpansionsincethebubblewas
mannbabies,Boltzmannbrains,andthestaggeringprob- produced.
lem. (The absence of Boltzmann brains requires that all Our result generalizes the results of Ref. [17], encom-
vacua decay faster than they produce such rare fluctu- passinginasingleformulathefollowingimportantprop-
ations [41]—a nontrivial condition on the vacuum land- erties of the (no-collapse!) scale factor measure estab-
scape,which,however,maywellbesatisfiedbythestring lished by De Simone et al.:1
landscape [47].) It does not reward volume expansion,
thusavoidingtheQ-catastropheandraisingthepossibil- (1) No youngness paradox, since the physical density
ity of the detection of open spatial curvature by future of Boltzmann babies is negligible.
experiments [48].
The causal diamond measure predicts a value of the (2) No reward for excessive volume expansion during
cosmologicalconstantconsistentwithobservations. That slow-rollinflation: Onceinflationhasmadetheuni-
is,theobservedvalueliesclosetothe meanandishighly verseflatenoughforstructureformation,anyextra
typical in the predicted probability distribution [49].
This successful prediction is robust against variations of
Q [49, 50].
1 DeSimoneetal. appliedtheirprescriptionforcollapsingregions
tothehomogeneouscollapseofbubbleswithnegativecosmolog-
C. Summary and outline ical constant, but implicitly used a no-collapse approximation
elsewhere, ignoring the turnaround and collapse of geodesics in
structure-forming regions. This explains any discrepancies be-
In this paper, we investigate the scale factor mea-
tweenourpapers,inparticularourlessfavorableconclusioncon-
sure[17],whichcutsofftheuniverseatthetimeηwhena cerningthevalueofΛpredictedbythescalefactormeasure.
3
e-foldingswillnotaffectthe physicaldensityofob- would have had, if they had occured at the time when
servers. Thus, the scale factor measure avoids the the first structures formed that would later merge into
Q-catastrophe. the objects hosting the observations. In other words, we
can incorporate gravitational collapse by mapping each
(3) The probability distribution for Λ is in excellent observationback to the earliesttime when a geodesic on
agreement with the observed value, Eq. (1): If Λ whichitliesbegantocollapse. Ourownobservations,for
had dominated much before the time when obser- example, are thus condensedby inverse of the expansion
vations are made (i.e., now), galaxies would now factorofouruniversesincetheformationofthefirstdark
be exponentially dilute and the average density of matter halos.
observations would be highly suppressed. This re-
Since neither the inflationary era nor the early post-
sult is stable against variations of the primordial
inflationaryuniversecontaincollapsedregions,thismod-
density contrast, Q.
ification has no bearing on the results (1) and (2)
above, concerning the youngness paradox and the Q-
A simple result should have a simple explanation,
catastrophe. In Sec. V, we investigate how the inclusion
which we provide in Sec. IVB: In the no-collapse ap-
ofgravitationalcollapseaffectsthepredictionforthecos-
proximation (i.e., if all geodesics in the congruence are
mological constant.
alwaysexpanding),thescalefactormeasureisequivalent
tothefollowingprescription2: Consider a single geodesic We begin by reviewing the predictions for the cos-
that starts out in the longest-lived metastable vacuum of mological constant obtained in various measures: the
the landscape, ∗. Compute the expected number of ob- observers-per-baryonprescription (Sec. VA), the causal
servations of type µ, dhN i, occuring along a fixed phys- diamond measure (Sec. VB) and the no-collapse scale
µ
ical volume dV transverse to the geodesic. The relative factor measure (Sec. VC1). In Sec. VC2, we consider
probability of observations µ and ν is dhN i/dhN i. In thescalefactormeasure,withtheprescriptionofRef.[17]
µ ν
practice,thiscanbeaccomplishedbysummingtheprob- forcollapsedregions. We find thatthe cosmologicalcon-
abilities that the geodesic will enter each vacuum i mul- stant is not set by the timescale when observations are
tiplied by the physical density of observations of type µ made—as it is in the causal diamond measure and, ap-
in vacuumi, so this moregeneralformula reduces to our parently, in Nature. Rather, its value is controlled by
earlier result. the time scale of structure formation. This means that
We thus reformulate the no-collapse scale factor mea- the scale factor measure predicts a value that is up to
sureasalocalmeasure. Bythiswemeanameasurethat 5000 times larger than the observed value. By adding
involves averaging over the statistical ensemble defined morespecificanthropicassumptions,thediscrepancycan
by the different possible histories along a single world- be mitigated, but the observed value remains somewhat
line, without necessarily assembling them into a global atypical.
geometry. It would be natural to use the local formula- Iftheprimordialdensitycontrast,Q,isallowedtovary,
tionas the generaldefinition ofthe scalefactormeasure, we find that the preferred value of Λ scales like Q3, and
since it can be applied to collapsed regions without re- the associated probability like Q6. This means, for ex-
quiring additional rules. ample, that a universe with Q three times as large, and
The causal-diamond measure is another example of a Λ27timesaslarge,is729timesaslikely. Itisdifficultto
localmeasureinthissense. Itdiffersfromtheno-collapse seehowanthropicconstraintsorpriordistributionsforQ
scalefactormeasureonlyintwoways: (1)the transverse can overcome a pressure so strong. In Sec. VD, we dis-
volumeincludedalongthegeodesicisnotconstant,butis cuss possible modifications of the treatment of collapsed
setbythesizeofacausallyconnectedregion;and(2)ini- regionsinthe scalefactormeasure,whichmightimprove
tialconditionsarenotdeterminedbythecausaldiamond these problematic predictions.
measure,butareconsideredalogicallyindependentques- In Sec. VI, we investigate the probability for Boltz-
tion. Thisexplainsthepatternofsimilaritiesbetweenthe mann brains in the scale factor measure.3. Boltzmann
measuresthathasemerged,butitalsoclarifieshowthey brains are observers that arise from rare thermal fluctu-
differ. ations, at a superexponentially small rate per unit four-
In Sec. IVC, we go beyond the no-collapse approxi- volume. Some measures overcompensate for this sup-
mation and investigate how the De Simone et al. pre- pressionby including superexponentiallylargeregionsof
scription for collapsed regions affects the formulae for empty de Sitter space [33, 34, 35, 36, 37, 38, 39] for ev-
probabilities in the scale factor measure. We find that it ery region containing ordinary observers. Then the vast
can be incorporatedby a simple substitution: Instead of majority of observations are made by Boltzmann brains.
using the physical density of observations, what matters Since almost none of the observations made by Boltz-
isthe(potentiallyfargreater)densitythoseobservations
3 A. Vilenkinand collaborators have independently analyzed this
2 We are using here the approximation of [42]; we will be more question. We undertand that their results will appear simulta-
preciseinthebodyofthepaper. neouslyorinthenearfuture.
4
mannbrainsagreewithourobservations,thesemeasures away:
are ruled out. The causal diamond measure was shown
P(O ) N
in Ref. [41] to favor ordinary observers,if the decay rate i ≡ lim i (2)
ofeveryvacuuminthelandscapeisgreaterthantherate P(Oj) η→∞Nj
forBoltzmannbrains. Recentevidencesuggeststhatthis
A simple choice for constructing Σ would be to follow
nontrivialconditionmay be satisfiedin the stringtheory η
eachgeodesicforthesamepropertime. Buttheresulting
landscape [47].
measure is ruled out at overwhelming confidence level,
Our local formulation of the scale factor measure
and independently of the details of the underlying the-
makes it clear that similar conditions will play a role in
ory: It predicts that we should observe a much hotter
the scale factor measure. The transverse volume along
universe [25, 26, 27, 28, 29, 30, 31, 32].
the defining geodesic is the same in all vacua, whereas
in the causal diamond measure, it is set by the cosmo-
logical constant. Since the latter varies at most over an
1. The no-collapse scale factor measure
exponentially, but not a superexponentially large range,
this difference is negligible in the context of Boltzmann
A different cutoff on the congruence was recently de-
brains.
fined by De Simone et al. [17], building on earlierwork.4
Theonlyremainingpotentialdifferencearisesfromthe
In this proposal Σ is, roughly, a surface of constant lo-
initialconditionsonthe geodesic. Inthe causaldiamond η
cal scale factor. The scale factor time is defined by inte-
measure, it was reasonable to assume that these condi-
gratingthelocalexpansionraterelativetoinfinitesimally
tions do not pick out vacua with unnaturally small cos-
nearby geodesics, along each geodesic:
mologicalconstant(anecessaryconditionevenforBoltz-
mannbrains),sotheinitialvacuumreceivednoattention t θ
in the analysis of Ref. [41]. In the scale factor measure, η(x,t)≡ dt′ , (3)
3
however, the initial vacuum is the longest-lived de Sit- Z0
ter vacuum, which might well have a small cosmological where t is the proper time along the geodesic labeled by
constant. x, θ = ∇µζ is the expansion, and ζ is the 4-velocity
µ µ
Here, we refine the analysis of Ref. [41] in two re- vector field tangent to the congruence. The local scale
spects. In Sec. VIA, we include the contributions from factor is defined by
theinitialvacuumtothenumberofBoltzmannbrains. In
Sec. VIB, we demonstrate (under plausible assumptions A(x,t)≡expη(x,t) . (4)
on the structure of the landscape) that the production
Intuitively, these quantities measure the growth of a lo-
ratesofdifferent vacuawill notinvalidate ourearliercri-
calvolumeelementδV spannedbyinfinitesimallynearby
terion for the dominance of ordinary observers, and we
geodesics [55]:
find that it is augmented only by the condition that the
initial vacuum be completely unable to produce Boltz-
dη dA/dt d(δV)/dt
mannbrains(independently ofitsdecayrate). We argue θ =3 =3 = . (5)
dt A δV
that this condition is likely to be satisfied. Thus, the
scalefactormeasureandthecausaldiamondmeasureare
virtuallyequivalentforthepurposeofBoltzmannbrains.
2. Collapsed regions and other ambiguities
However, Σ cannot be defined simply as a surface of
II. DEFINITION AND RATE EQUATIONS η
constant η or A. In collapsing regions, such as pock-
ets with a negative cosmological constant, or structure
In this section we define the scale factor cutoff and forming regions, geodesics will cease to expand and be-
review some of its basic properties. gin to approacheach other. Then θ <0, and by Eq. (3),
A. Definition of the scale factor cutoff
4 Scale factor time is considered insome of the earliest literature
One approach to regulating the infinities in eternal on (slow-roll) eternal inflation, which focusses on the distribu-
tionofdifferentfieldvalues[20,21,22,23]. Themeasuredefined
inflation is through a smooth congruence of timelike
byDeSimoneetal. issimilartothe“pseudo-comovingvolume-
geodesics orthogonalto a (nearly arbitrary)finite space-
weighted measure” [24], which stops short of an explicit defini-
like surface Σ0. The idea is to use the geodesic con- tionoftheprobabilityofdifferentobservations,suchasEq.(2),
gruence to define a sequence of cutoff hypersurfaces Σ . Eq. 1 of Ref. [32] or Eq. 1 of Ref. [17], and of a prescription
η
Then one computes the number N of observations of for collapsing geodesics. It shares some properties with other
i
measures [38, 51, 52, 53, 54], in which the scale factor (in the
type i, O , in the four-volume between Σ and Σ . The
i 0 η guiseofapproximatelyequivalentformulations)regulatesbubble
relativeprobabilityoftwoobservationsisdefinedbytheir
abundances but different (or no) cutoffs regulate the number of
relative abundance in the limit where the cutoff is taken observersperbubble.
5
thescalefactortimedecreaseslocallytowardsthefuture. τ
Unlessasingularityisencountered,thefocusingtheorem
guaranteesthatgeodesicswilleventuallyencountercaus-
Σ
tics: they will intersect with infinitesimally neighboring 2
geodesics. Thereafter, they begin to expand again, and
the scale factor time increases once more.
This introduces two ambiguities: First, a given scale
factor need not be reached precisely once along each
geodesic. It may never be reached, or it may be reached Σ
1
more than once. Which (if any) occurrence defines the
actual cutoff? Second, a given event may be threaded
by morethanone geodesic. Shouldwecountsuchevents FIG. 1: Evolution of the scale factor cutoff during structure
once, or multiple times? formation. The constant scale factor time surface Σ1 lies in
To resolve these ambiguities, De Simone et al. [17] theearly,approximatelyhomogeneousuniverseandcoincides
with a surface of constant FRW time τ. As density pertur-
propose that an event should be included “if it lies on
bationsgrow, somegeodesics decouplefrom theHubbleflow,
any geodesic prior to the first occurrence” of the speci-
stopexpanding,andbecometrappedincollapsedregionssuch
fied cutoff on that geodesic. In other words, Σ is the
η asgalaxies. Ifthescalefactor cutoffexceedsthelargest scale
hypersurface that maximizes the four-volume V of the
4 factor ever reached along such geodesics, then the rule of
congruence subject to the constraint that at every point De Simone et al. requires that their entire future evolution
p ∈ V4 lies on at least one geodesic at scale factor time be included. (A similar result obtains if the local formula-
less than η. tion of the scale factor measure is used as a general defini-
This definition implies that if the cutoff is never tion; see Sec. 5.4.) Therefore, the later cutoff surface Σ2 no
reached along some geodesic, all events on the geodesic longer agrees everywherewith a constant FRW time surface;
itincludestheentirefutureofcollapsedregions(green/gray),
are included (no future cutoff). The formulation of De
whichshowupasspikessincethefigureisdrawnincomoving
Simone et al. suggests, moreover, that any event should
coordinates.
be counted at most once; we will adopt this definition
here. Note that with this definition, Σ need not be a
η
spacelike hypersurface. Rather, it becomes timelike near
collapsing geodesics, spiking up towards the future, as
ward tunneling. There is no natural way, therefore, of
shown in Fig. 1.
determining the tangent vectors of the portion of the
TheprescriptionforcollapsedregionschosenbyDeSi-
geodesic congruence entering the new bubble.
mone et al. is not the only possible one. Indeed, we will
findthatithasphenomenologicaldisadvantages(Sec.V), For the purpose of deriving the rate equations, in
and we will suggest other possible definitions. In partic- Sec. IIB, we will follow Ref. [38] and sidestep this is-
ular, the local formulation of the scale factor measure sue by making mathematically convenient assumptions.
(Sec. IVB) applies without modificationboth in collaps- Once we count observers in Sec. IV, the ambiguity will
ing andexpanding regions. However,we willnot explore not re-enter: we are only interested in the most recent
these alternatives in detail; we focus on the definition of tunneling, which is necessarily downward for producing
Ref. [17] in this paper. ordinary observers.
There is another ambiguity, however, which has yet
to be resolved. When a new vacuum with lower cos-
mological constant is formed by the Coleman-DeLuccia
process, there is a quantum region, where the geometry B. Rate equations and solutions
(in the instanton approximation) jumps sharply and the
continuation of geodesics is not well-defined. For down-
Before we can count observers in vacuum i, we will
wardtunnelings,onecanreasonablyignorethisproblem,
need to know the number n (η) of such bubbles below
i
since the initial radius of the bubble wall can be much
thecutoffη. This,inturn,requiresknowingthephysical
smallerthanthedeSitterradius,soonlyanegligiblefrac-
volume V (η) occupied by every vacuum. The evolution
i
tion of geodesics entering the new bubble pass through
and distribution of these volumes is determined by the
the bubble interior at the time of nucleation. Moreover,
rate matrix Γ describing the number of bubbles of vac-
ij
geodesicsquicklybecome comovinginregionswith slow-
uum i forming per unit four volume of vacuum j.
roll inflation, which are of the greatest interest.
For nonterminal vacua (i.e., those with positive cos-
DeSittervacuamayalsotunnelupward,tovacuawith
mological constant Λ ), the following definitions will be
larger cosmological constant. Even though upward tun- i
convenient: the expansion at late times,
nelings areverysuppressed,the scale factormeasureim-
plies that majorityofobserverslive onworldlines which
havebeenthroughupwardtunnelings[43,46]. Butthere Λi 1/2
H ≡ ; (6)
is no approximate classical geometry describing an up- i 3
(cid:18) (cid:19)
6
the total decay rate of vacuum i per unit four-volume, and loses
Γj ≡ Γij ; (7) d−V = V dη Γ 4π . (12)
i iH ji 3H3
Xi j (cid:18) i(cid:19) (cid:18) i (cid:19)
X
the dimensionless decay rate from vacuum j to i,
in physical volume per scale factor time. Combining in-
4πΓij flow and outflow yields the Fokker-Planck equation
κ ≡ ; (8)
ij 3 H4
j dV
i
=3V −κ V + κ V . (13)
the total dimensionless decay rate of vacuum j, dη i i i ij j
j
X
κ ≡ κ , (9)
j ij Ref. [38] rigorously derives the solution to this equa-
i
X tion. OnlythebehaviorofdeSittervacuawillberelevant
and the branching ratio matrix here. For generic initial conditions with some support in
deSittervacua,thesolutionapproachesattractorbehav-
β ≡κ /κ . (10)
ij ij j ior at late times. The volume in de Sitter vacuum i is
During a time interval dη, the volume V in vacuum i
i V (η)=Cs eγη . (14)
will increase due to intrinsic expansion, due to the pro- i i
duction of new bubbles of type i, and due to the expan-
Here, C is a constantwith the dimension of volume that
sion of the bubble walls into the parent vacua. It will
depends on the initial conditions but drops out in all
decrease due to decay into other vacua, and due to the
normalized probabilities;
growth of such bubble walls after they are produced.
Treating the motion of domain walls in detail is cum-
γ ≡3−q ; (15)
bersome, and it is unnecessary if all metastable vacua
arelong-lived(κi ≪1). Thisisareasonableassumption, q is the smallest-magnitude negative eigenvalue of the
since κi ≫ 1 conflicts with the notion of a vacuum, and dimensionless flow matrix κij−δijκi; and sj is the asso-
κi ∼ 1 requires fine tuning. Such vacua will be too rare ciated eigenvector:
in the landscape to play an important dynamical role.
Thus, most of the four-volume of each bubble type κijsj =(κi−q)si ≡pi , (16)
is empty de Sitter space, and we can neglect transient
wherewehavedefinedthevectorp forlaterconvenience.
effectsrightafterbubblenucleation. Onetransientisthe i
To exponentially good approximation [42], the eigen-
periodbetweenbubblecreationandvacuumdomination.
vector is dominated by the longest-lived de Sitter vac-
Therefore, the expansion θ of geodesics in vacuum j can
uum, ∗:
be approximated by the Hubble constant at late times,
θ ≈H ≡(Λ /3)1/2. Thefour-volumeinvacuumj added
j i s ≈δ ; (17)
during the scale factor time dη is thus V H−1dη. j j∗
j j
Anothertransientisthebubblewallexpansion. Abub- anditseigenvalueisequaltothedimensionlessdecayrate
ble nucleated at the time ηnuc will eventually occupy a of this vacuum,
comoving volume in the geodesic congruence that would
(in the absence ofthe decay)haveoriginatedfroma ball q ≈κ∗ ≪1 (18)
ofphysicalradiusH−1intheparentvacuumjatthetime
j
ηnuc. This asymptotic comoving size is reached,to accu- Thus, q is exponentially small in a realistic landscape.—
racyoforderexp(η−ηnuc),afteronlyafewunitsofscale By Eq.(2), this attractorsolution is all we need to com-
pute probabilities.
factor time. Thus we make a small error by anticipating
this growth: We ascribe a physical volume 4π/3H3 to A number of other results will be useful below, and
j
the new vacuum i already at the nucleation time ηnuc, we collect them here. The expected number of times a
and in exchange neglect the bubble wall growth [38].5 worldlinestartinginvacuumovacuumwillpassthrough
vacuum i can be obtained by summing over the whole
With these approximations, vacuum i gains
branching tree [13]:
dη 4π
d+V =3V dη+ V Γ . (11)
i i j (cid:18) jHj(cid:19) ij 3Hj3! eio =δio+ βiin βinin−1 ... βi1o , (19)
X paths from o to i
X
where i ...i are intermediate vacua connecting o and i,
1 n
and the branching ratios β were defined in Eq. (10).
ij
5 Thisbehaviorofthebubblewallappliesonlytodownwardtran- This can be written in matrix form as
sitions. Duringupward transitions, the behavior of the congru-
encedefiningthescalefactormeasureisnotwell-definedinsemi- ∞
classical gravity. Following Ref. [38], we will choose ad hoc to e = βn , (20)
io
usethesameruleinthiscase. (cid:18)n=0 (cid:19)io
X
7
In the scale factor measure, the initial vacuum is the ∗ t is the proper time along the geodesic passing through
vacuum, and we will denote an event, and r is the radial position the geodesic had
before entering the bubble. We will now review the co-
ei ≡ei∗ (21) ordinatetransformationbetween(τ,ξ)and(t,r),derived
in Ref. [32].
If the initial vacuum has relatively small cosmological
Without loss of generality,one can choose coordinates
constant, as one might expect for the longest-lived vac-
so that the bubble is nucleated at t=0,r=0. Consider
uum, then it is unlikely to be re-entered later in the de-
a comoving geodesic which passes through the event t=
cay chain. Then the sum in Eq. (20) converges rapidly;
0,r. If after a proper time t (as measured along the
in particular,
geodesic) it has passed into the bubble, its coordinates
(τ,ξ) will be
e ≈1 . (22)
∗
1
In Ref. [44], the vectors p and e were shown to be ξ(t,r) = −log(1−Hr)+O( )
i i Hτ
closely related:
ξ+e−ξ−1 1
τ(t,r) = t− +O( ) . (27)
∞ H Hτ
p = q βn s
i j
Note that H in these formulas refers to the constant ex-
(cid:18)n=1 (cid:19)ij
X pansion rate of the parent de Sitter space. The above
∞
≈ q βn . equations are valid for geodesics that have spent more
(cid:18)n=1 (cid:19)i∗ than a proper time H−1 inside the bubble.6 Physically,
X
this result shows that the geodesics become comoving in
≈ qe (i6=∗) (23)
i
the open coordinates after one outside Hubble time, and
Note, however,that p 6=qe . thereafter the proper time along the geodesics increases
∗ ∗
at the same rate as the open FRW time.
To construct the cutoff hypersurface, one must deter-
III. THE CUTOFF HYPERSURFACE IN A mine the scale factor time η as a function of the bubble
HOMOGENEOUS BUBBLE coordinates τ and ξ. By Eq. (14),
1 δV[r(τ,ξ),t(τ,ξ)]
In this subsection, we investigate the evolution of the η(ηnuc,τ,ξ)=ηnuc+ log , (28)
3 δV(r,0)
cutoffhypersurfaceinasinglebubbleofanarbitraryvac- (cid:18) (cid:19)
uum,nucleatedinsideaparentvacuumwithpositivecos- where ηnuc is the scale factor time at t = 0, when the
mological constant 3H2. We will drop the index i while
bubble is nucleated. The volume element δV orthogonal
discussing a single vacuum. For now, we will ignore in-
tothegeodesicsliesonahypersurfaceofconstantproper
homogeneities, such as density perturbations and local
time t [55]. By Eq. (26),
gravitationalcollapse.
Assuming that the tunneling process is very rare, the
parent vacuum will have existed for a long time before δV =
the nucleation event. This implies [56] that we can take 2 2 1/2
dξ dτ
thegeodesiccongruencetobecomovingintheflatslicing 4πa2(τ)sinh2ξ a2(τ) − δr
of de Sitter space, " (cid:18)dr(cid:19)t=const (cid:18)dr(cid:19)t=const#
ds2 =−dt2+e2Ht(dr2+r2dΩ2) . (24) 1−e−ξ 2 1/2
=4πa(τ)2sinh2ξ a(τ)2− Heξδr ,(29)
H
We can think of r as labelling a particular shell in the " (cid:18) (cid:19) #
congruence. The geodesics between r and r+δr span a
where we have used Eq. (27). We are interested in
volume element eventsmuchlaterthantheoutsideHubbletime,soa2 ≫
dV H−2, and we can drop the second term in the square
δV(t,r)= δr =4πe2Htr2δr (25) root:
dr
(cid:12)t=const
in the parent vacuum. (cid:12)(cid:12) δV =4πHa3(τ)eξsinh2ξδr (30)
(cid:12)
In the homogeneous approximation, the metric inside
the bubble is described by an open FRW geometry
ds2 =−dτ2+a2(τ)(dξ2+sinh2ξdΩ2) . (26) 6 We also assume that the cosmological constant inside is much
2
smallerthanoutside. Correctionsduetosuchapproximationare
not includedinEq.(27). However, whentheinsideandoutside
The comoving geodesics from the host vacuum continue
cosmological constants are the same, one can derive an exact
into the bubble along nontrivial trajectories, defining a
formulawhichisequivalenttoEq.(31)forallphysicalquestions
second coordinate system (t,r) inside the bubble, where weconsideredinthispaper.
8
AfterusingEq.(27)toeliminaterfromEq.(25),Eq.(28) By Eq.(31), the latestnucleationtime that allowsob-
yields the scale factor time inside the bubble: servers at radius ξ to contribute is
η(ηnuc,τ,ξ)=ηnuc+log[Ha(τ)]+23 ξ+log cosh2ξ . ηinjuc(η,ξ)=η−log Hjaoibs − 23 ξ+log cosh2ξ .
(cid:20) (cid:18) (cid:19)(3(cid:21)1) (cid:2) (cid:3) (cid:20) (cid:18) (cid:19)((cid:21)34)
Note that this time depends on the parent vacuum, j.
Hence,
IV. COUNTING ORDINARY OBSERVERS ∞ dv
N (η)= dξn ηnuc(η,ξ) ph(ξ)ρobs, (35)
i ij ij dξ i
In this section, we apply the scale factor cutoff to XJ Z0 (cid:2) (cid:3)
countingobservationsinaneternallyinflatingmultiverse where n (η′) is the total number of bubbles of type i
ij
with multiple vacua. We exclude, for now, observations nucleated inside vacuum j by the time η′.
resulting from violations of the second law (Boltzmann Using
brains,treatedinSec.VI). Initially,wewillimaginethat
all observations (if any) in a bubble of type i are made dn =Γ V dη/H (36)
ij ij j j
instantaneously7 attheFRWtimeτobs afterthevacuum
i
is produced, with number density ρobs per unit physical Eqs. (8) and (14) imply
i
volume:
3C
n (ηnuc)= H3κ s exp(γηnuc) . (37)
dN =ρobsdv . (32) ij ij 4πγ j ij j ij
i i ph
Combining the above equations, we find
We will assume, moreover,that these parameters do not
depend on the parent vacuum from which i is entered.
N(η)=
These assumptions will simplify our treatment, and it
will be easy to drop them in the end and state a more 3C ∞dξ4π sinh2ξ(expξ)−2γ/3 coshξ −2γ/3 eγη
general result, Eq. (43). 4πγ 2
"Z0 (cid:18) (cid:19) #
Our treatment will go beyond that of De Simone et
al. [17] in that we do not approximate the bubble in-
×(aobs)q Hqκ s ρobs . (38)
terior as flat and homogeneous. Our detailed treatment i  j ij j i
of collapsing regions, in Sec. IVC, has significant impli- Xj
 
cations, invalidating some of the conclusions of Ref. [17]
Theξintegralisclearlyconvergent,withsupportconcen-
(see Sec. V).
trated near ξ ∼O(1). (For small q, its value approaches
4π/3.) Physically, this shows that mainly the central
curvature volume of the infinite open bubble geometry
A. Counting observations in the no-collapse contributes in the scale factor measure. Therefore, the
approximation interesting results of Garriga, Guth, and Vilenkin [56]
regarding worldlines at large ξ are not relevant in this
Nowwewillcomputethetotalnumberofobservations, measure.
N (η), performed in vacua of type i below the cutoff, η. Since q is exponentially small, Hq and (aobs)q can
i j i
ByEq.(32),thisamountstocomputingthetotalphysical safely be neglected. By the discussion at the end of
volume of the τ = τobs hypersurfaces below the cutoff. Sec. II, κ s =qe . Thus we obtain the probability
i ij j i
In any single bubble, a shell of comoving radius ξ and
width dξ contributes a physical volume Pi ∝eiρoibs . (39)
dvph =4π(aobs)3sinh2ξdξ , (33) for observing vacuum i. The “∝” notation indicates
i i that the universal factor ≈ Cqeγη/γ has been dropped
(since, by Eq. (2), it does not affect relative probabili-
butonlyifthebubblewasnucleatedearlyenoughforthis
ties),thoughthe probabilitieshavenotbeennormalized.
volume to be included below the cutoff.
Eq. (39) immediately implies severalkey properties of
the scale factor measure:
• The probability for observing vacuum i is simply
7 In our investigation of the proper time cutoff [32], information theproductofthenumberoftimesatypicalworld-
aboutthe temporaldistributionofobservations, fi(τ), was cru- line canbe expected to enteri-bubbles,e , andthe
cialtodemonstratingtheyoungnessparadox. Inthescalefactor i
density, ρobs, of observations per physical volume
measure,however,thereisnoyoungnessparadox[17],andforthe i
purposes of this paper, it suffices to use only gi ≡ R0∞fi(τ)dτ on the homogeneous timeslice τiobs at which they
andτiobs≡gi−1R0∞fi(τ)τdτ asinputparameters. are performed.
9
• The probability of an observation depends on its η
FRWtime,τobs,andonthe FRWexpansionfactor
at that time, aobs, only through ρobs.
• As a special case, we recover an important result
of De Simone et al. [17]: Like in the causal di-
amond measure, the probability of a vacuum is
insensitive to the volume expansion factor during
slow-roll inflation. This is a desirable property,
because it avoids the “Q-catastrophe” [15, 16]—
the overwhelming pressure towards extreme (and
counterfactual) inflationary parameter values that
results when exponential volume factors are re-
warded. Moreover,this property makes it conceiv-
Σ
ablethatinflationwasshortenoughtoallowsubtle η
signatures of the preceding era to survive, such as
FIG. 2: The green (light shaded) slices are surfaces of con-
detectable curvature [48].
stant scale factor time, Ση; they have fixed comoving size
but increasing physical volume. Fat geodesics (purple, dark
• Eq. (39) also captures another important result of
shaded)havefixedphysicalwidthandthusdecreasingcomov-
Ref. [17], which we will examine in Sec. V: the
ing size. If the initial slice is chosen in the attractor regime,
probabilityofvacuawherevacuumenergycomesto
the fat-geodesics define a representative finite sample of the
dominate before τobs is exponentially suppressed,
i total four-volume. Thus, the results of the scale factor mea-
becausethematterdensity,ρobs ,willhavebeen
matter,i surecanbereproducedbyfollowing asinglegeodesicoffixed
dilutedbytheacceleratedexpansion. This,too,ap- width, starting in thelongest-lived metastable vacuum.
pears to replicate a success of the causal diamond
measure: the suppression of moderately large val-
uesofthecosmologicalconstantΛ,whicharelarger
than the observedvalue but too small to affect the overanensembleofindividualworldlines,insteadofcon-
numberofobserverspermattermass. Wewillfind, structing a particular globalspacetime. (The causaldia-
however,thatthisapparentsuccessisanartifactof mond measure is an example of a local measure.)
the no-collapse approximation. Thiscanbeseenbythefollowingargument,illustrated
inFig.2. The predictionsofthe scalefactormeasureare
Eq. (39) trivially generalizes to the case where obser- dominatedbythelate-timeattractorbehavioroftheuni-
vations are made at different different times τµ, µ = verse. (An infinite number of observations are produced
1,2,3... The total probability to observe vacuum i is during this era, while only a finite number are produced
earlier.) Thus, we may as well pick a late-time surface,
Pi ∝ei ρoibs(τµ) , (40) Ση with η verylarge,andchooseitasour initialsurface,
Xµ Σ0. In other words, we redefine η → η−∆η, ∆η ≈ ∞,
making sure we already start in the asymptotic regime.
whereρoibs(τµ)isthedensityofobservationstakingplace This cuts out irrelevant transients and leads to a very
at τµ. In the continuous case, simple picture.
The volume occupied by long-lived metastable vacua
P ∝e ∞dτ dρoibs , (41) on the late-time attractor hypersurface Σ0 is dominated
i i
dτ by empty de Sitter regions, with volume fraction s allo-
Z0 i
cated to vacuum i [38]. We can choose Σ to be as large
0
where the integrand is the number of observations made as we like, guaranteeingthat it can be allocated the cor-
in the FRW time interval (τ,τ +dτ), per unit physical rect attractor volume fractions to arbitrary accuracy.
volume of the hypersurface τ. If more differentiated ob- Now let us begin evolving forward along the congru-
servations are performed (for example, local conditions enceorthogonaltoΣ . Thescalefactormeasureinstructs
0
liketemperature,ratherthanjustadeterminationofthe us to count observations in the four-volume thus gener-
vacuum),ρicanbegivenadditionalindicesorarguments. ated, taking ratios as η → ∞. Let us instead consider a
reduced four-volume, V˜ , which is finite, by dividing the
4
initialsurfaceΣ intovolumeelements dV. As usual,we
0
B. The no-collapse scale factor measure as a local follow the geodesic orthogonal to each volume element.
measure ButwedonotincreasedV astheuniverseexpands. This
createsafamilyof“fatgeodesics”. Atthetimeη,thefat
Intheno-collapseapproximation,thescalefactormea- geodesicsoccupyonlyafractione−3η ofthetotalvolume
sure can be reformulated as a local measure. By this ofΣ . Butforeveryworldline,nomatterinwhattypeof
η
we mean that the measure can be defined by averaging regionitstarted,themissingvolumeisexactlythesame,
10
1−e−3η. It would be interesting to use Eq. (43) as the defin-
Because all fat geodesics expand by the same volume ing equationofa measure. This wouldhaveanumber of
factor, the volume they do occupy on Σ is a perfectly formal advantages. One geodesic carries much less geo-
η
faithful sample of the hypersurface. (This is the crucial metric information than a whole congruence, so we face
point—note, for example, that it would not hold in the fewer ambiguities aboutthe treatment ofupwardjumps.
attractorregimeofthe propertime measure.) Moreover, Moreover, Eq. (43) can be applied without modification
because this is true for every time interval dη, the four- in collapsing regions, whereas the formulation based on
volume sweptout by the fat geodesics outis statistically integrating the expansion along a geodesic congruence
equivalent to the four-volume between Σ and Σ . requires an additional, ad hoc, prescription to deal with
0 η
Therefore, reducing to V˜ will not affect probabilities: such regions. Finally, the use of Eq. (43) to define a
4
measureliberatesus fromthe initialconditionss picked
i
P(O ) N N (V˜ ) out by the attractor regime of the geodesic congruence.
i i i 4
≡ lim = , (42)
P(Oj) η→∞Nj Nj(V˜4) Startingthefatgeodesicwithinitialconditionsthatfavor
Planck-scale vacua, for example, would avoid the poten-
where V is obtained by following every geodesic to the tial “staggering problem” associated with the enormous
4
asymptoticfuture.8 Insteadofthinkingaboutthe actual suppressionoftheupwardjumpsfromthedominantvac-
collection of fat geodesics anchored on a large hypersur- uum ∗.
face, we may equivalently consider a statistical ensemble
of single geodesics, with initial conditions weighted by
the distribution of regions on a late-time attractor sur- C. Proper treatment of collapsed regions
face. Neglecting the rare regions occupied by recently
nucleated bubbles, this means starting out with de Sit-
At least in our own bubble, some observations are
ter vacuum i with probability s . Because the vector s
i i made in collapsed regions. The FRW metric, Eq. (26),
is dominated by the ∗ vacuum, to a good approxima-
does not capture the local geometry of such regions, but
tion[42], this meansstartingthe geodesicinthe longest-
providesonlyanaverageoverscalesonwhichtheuniverse
lived metastable vacuum, ∗.
canbe consideredhomogeneous. Hence, this metric can-
Thus,wereproducethescalefactormeasurebyfollow-
notbe usedtocompute the localscalefactor,A, andthe
ing a single geodesic starting in ∗. The worldline evolves
scale factor time, η, along geodesics entering collapsed
according to local dynamical laws, witnessing the decay
regions such as galaxies. Despite the similar names, the
ofvacuaandperhapstheproductionofobservers,untilit
FRW scale factor a and the local scale factor A are two
endsupinaterminalvacuum. We“fatten”theworldline,
completely different objects.
giving it a fixed physical cross-section, a volume element
Geodesics that end up in dark matter haloes will, by
dV orthogonalto the worldline. Finally,we compute the
definition, have ceased to expand, and decoupled from
expectation value (ensemble average) of the differential
the Hubble flow, before the halo formed. After reaching
number of observations of type µ, dN , in the resulting
µ amaximumexpansion,theywillhaveturnedaroundand
four-volume:
collapsed, with A and η decreasing during the collapse.
Thesegeodesicswillreachobservers,buttheirmaximum
P =hρ i , (43)
µ µ
scale factor A will be unrelated to the FRW scale
max
where, for a given worldline in the ensemble, factor atthe time tobs, aobs. Rather, it will be relatedto
the FRW scale factor at the time of structure formation,
dN dN
µ µ a .
ρ = = dt (44) NL
µ
dV dVdt Consider a simple, spherically symmetric model. A
Z
dark matter halo forms by the collapse of a spherical
We showed in Sec. III that geodesics quickly become
overdensity. Thenbaryonsfallintoitsgravitationalwell,
comoving after entering a new bubble. Thus, the vol-
cool, and condense in the center. Eventually the gas
ume element dV lies inside a constant-τ hypersurface of
fragments into stars.
each FRW bubble universe. With the approximations
Let us focus on the dark matter particles that end up
used in Sec. IVA, therefore, ρ is identical to the phys-
µ in the halo. Initially, each particle follows one of the
ical density of observers, ρobs, in vacuum i. The factor
i geodesics in the congruence that defines the scale factor
e is captured by averaging over different worldlines in
i cutoff. Themaximumscalefactorisachievedatthetime
the ensemble [44], so Eq. (43) reduces to Eq. (39) as a
of turnaround, when all the dark matter particles are
special case. The more general Eqs. (40) and (41), too,
momentarily stationary. Note that it is smaller than,
are special cases of Eq. (43).
but on the order of, the FRW scale factor a at the time
of turnaround.
After the turnaround, the particles fall towards the
8 Theno-collapseapproximationdoesnotallowustoconsiderneg- center of the halo. Depending on interactions, the par-
ticles will eventually stop following the geodesics, but in
ativecosmologicalconstant regions,butforthepurposesofthis
argumentwecansetΛi→0forallvacuawithΛi<0. the spherical model, the geodesics remain very simple.