Table Of ContentPredictions in eternal inflation
Sergei Winitzki
Department of Physics, Ludwig-Maximilians University, 80333 Munich, Germany
Ingenericmodelsofcosmologicalinflation,quantumfluctuationsstronglyinfluencethespacetime
metric and produce infinitely many regions where the end of inflation (reheating) is delayed until
arbitrarily late times. The geometry of the resulting spacetime is highly inhomogeneous on scales
ofmanyHubblesizes. Therecentlydevelopedstring-theoreticpictureofthe“landscape”presentsa
similar structure,where an infinitenumberof deSitter,flat,and anti-deSitteruniversesarenucle-
atedviaquantumtunneling. SinceobserversontheEarthhavenoinformation abouttheirlocation
withintheeternallyinflatinguniverse,themainquestioninthiscontextistoobtainstatisticalpre-
dictions for quantities observed at a random location. I describe the problems arising within this
statistical framework, such as the need for a volume cutoff and the dependence of cutoff schemes
on time slicing and on the initial conditions. After reviewing different approaches and mathemat-
ical techniques developed in the past two decades for studying these issues, I discuss the existing
7 proposals for extractingpredictions and give examplesof theirapplications.
0
0
2 I. ETERNAL INFLATION of de Sitter space [12, 13, 14]. Thus the field φ becomes
n extremelyinhomogeneousonlarge(super-horizon)scales
a The general idea of eternally inflating spacetime was aftermanyHubbletimes. Moreover,inthesemi-classical
J first introducedand developedin the 1980s[1, 2, 3, 4] in picture it is assumed [2] that the local expansion rate
2 thecontextofslow-rollinflation. Letusbeginbyreview- a˙/a≡H(φ) tracks the local value of the field φ(t,x) ac-
ing the main features of eternalinflation, following these cording to the Einstein equation (1). Here a(t,x) is the
3
early works. scale factor function which varies with x only on super-
v
4 A prototypical model contains a minimally coupled Hubble scales, a(t,x)∆x & H−1. Hence, the spacetime
6 scalar field φ (the“inflaton”) with an effective potential metric can be visualized as having a slowly varying,“lo-
1 V(φ) that is sufficiently flat in some range of φ. When cally de Sitter”form (with spatially flat coordinatesx),
2 the field φ has values in this range, the spacetime is ap- g dxµdxν =dt2−a2(t,x)dx2. (3)
1 µν
proximately de Sitter with the Hubble rate
6
The deterministic trajectory φ (t) eventually reaches
0 a˙ 8π sr
/ = V(φ)≡H(φ). (1) a (model-dependent) value φ∗ signifying the end of the
c a r 3 slow-rollinflationaryregimeandthe beginning ofthe re-
q
(WeworkinunitswhereG=c=~=1.) ThevalueofH heating epoch (thermalization). Since the random walk
-
r remains approximately constant on timescales of several process will lead the value of φ away from φ = φ∗ in
g
Hubble times (∆t & H−1), while the field φ follows the some regions, reheating will not begin everywhere at
:
v slow-roll trajectory φ (t). Quantum fluctuations of the the same time. Moreover, regions where φ remains in
sr
i scalar field φ in de Sitter background grow linearly with the inflationary range will typically expand faster than
X
time [5, 6, 7], regions near the end of inflation where V(φ) becomes
r
a H3 small. Therefore, a delay of the onset of reheating will
hφˆ2(t+∆t)i−hφˆ2(t)i= ∆t, (2) be rewarded by additional expansion of the proper 3-
4π2 volume, thus generating more regions that are still in-
atleastfortimeintervals∆toforderseveralH−1. Dueto flating. This feature is called“self-reproduction”of the
the quasi-exponential expansion of spacetime during in- inflationary spacetime [3]. Since each Hubble-size region
flation,Fouriermodesofthefieldφarequicklystretched evolves independently of other such regions,one may vi-
to super-Hubble length scales. However, quantum fluc- sualizethespacetimeasanensembleofinflatingHubble-
tuations with super-Hubble wavelengths cannot main- size domains (Fig. 1).
tain quantum coherence and become essentially classi- The process of self-reproduction will never result in
cal[6,7,8,9,10];thisissueisdiscussedinmoredetailin a global reheating if the probability of jumping away
Sec. IB below. The resulting field evolution φ(t) can be from φ = φ and the corresponding additional volume
∗
visualized[1,8,11]asaBrownianmotionwitha“random expansion factors are sufficiently large. The correspond-
jump”of typical step size ∆φ ∼ H/(2π) during a time ingquantitativeconditionsandtheirrealizationintypical
interval ∆t ∼ H−1, superimposed onto the determinis- modelsofinflationarereviewedinSec.IIIA. Underthese
tic slow-roll trajectory φ (t). A statistical description conditions,theprocessofself-reproductionofinflatingre-
sr
of this“random walk”-type evolution φ(t) is reviewed in gions continues forever. At the same time, every given
Sec. IIA. comovingworldline(exceptforasetofmeasurezero;see
The “jumps” at points separated in space by many Sec. IIIA) will sooner or later reach the value φ = φ
∗
Hubble distances are essentially uncorrelated; this is an- andenter the reheatingepoch. The resultingsituationis
othermanifestationofthewell-known“no-hair”property knownas“eternalinflation”[3]. Moreprecisely,the term
2
disconnected pieces of the reheating surface, each hav-
ing an infinite 3-volume (see Fig. 2 for an illustration).
This feature of eternal inflation is at the root of several
t technical and conceptual difficulties, as will be discussed
below.
x Everywherealong the reheating surface, the reheating
y
process is expected to provide appropriate initial condi-
tionsforthe standard“hotbigbang”cosmologicalevolu-
tion, including nucleosynthesis and structure formation.
Figure 1: A qualitative diagram of self-reproduction during
In other words, the reheating surface may be visualized
inflation. ShadedspacelikedomainsrepresentHubble-sizere-
gions with different values of the inflaton field φ. The time as the locus of the “hot big bang”events in the space-
step is of order H−1. Dark-colored shades are regions under- time. It is thus natural to view the reheating surface as
going reheating (φ = φ∗); lighter-colored shades are regions the initial equal-time surface for astrophysical observa-
whereinflationcontinues. Onaverage,thenumberofinflating tions in the post-inflationary epoch. Note that the ob-
regions grows with time. servationally relevant range of the primordial spectrum
of density fluctuations is generated only during the last
60 e-foldings of inflation. Hence, the duration of the in-
flationary epoch that preceded reheating is not directly
measurablebeyondthe last60 e-foldings;the totalnum-
berofe-foldingscanvaryalongthereheatingsurfaceand
can be in principle arbitrarily large.2
The phenomenon of eternal inflation is also found in
multi-field models of inflation [22, 23], as well as in sce-
nariosbasedonBrans-Dicketheory[24,25,26],topologi-
calinflation[27,28], braneworldinflation[29],“recycling
universe”[30], and the string theory landscape [31]. In
some of these models, quantum tunneling processes may
generate“bubbles”ofadifferentphaseofthevacuum(see
Figure 2: A 1+1-dimensional slice of the spacetime struc- Sec. IIE for more details). Bubbles will be created ran-
ture in an eternally inflating universe (numerical simulation domly at various places and times, with a fixed rate per
in Ref. [19]). Shades of different color represent different, unit4-volume. Intheinteriorofsomebubbles,additional
causallydisconnectedregionswherereheatingtookplace. The inflationmaytakeplace,followedbyanewreheatingsur-
reheating surface is the line separating the white (inflating)
face. Theinteriorstructureofsuchbubblesissketchedin
domain and theshaded domains.
Fig. 3. The nucleation event and the formation of bub-
ble walls is followed by a period of additional inflation,
which terminates by reheating. Standard cosmological
“eternalinflation”means future-eternalself-reproduction
evolutionandstructureformationeventuallygivewayto
ofinflatingregions[15].1 Toemphasizethefactthatself-
a Λ-dominated universe. Infinitely many galaxies and
reproductionisduetorandomfluctuationsofafield,one
possible civilizations may appear within a thin spacelike
referstothisscenarioas“eternalinflationofrandom-walk
slab running along the interior reheating surface. This
type.”Belowweusetheterms“eternalself-reproduction”
reheating surface appears to interior observers as an in-
and“eternal inflation”interchangeably.
finite, spacelike hypersurface [32]. For this reason, such
Observers like us may appear only in regions where
bubblesarecalled“pocketuniverses,”whilethespacetime
reheating already took place. Hence, it is useful to con-
iscalleda“multiverse.”(Generally,theterm“pocketuni-
siderthe locusofallreheatingeventsinthe entirespace-
verse”refers to a noncompact, connected component of
time; in the presently considered example, it is the set
the reheating surface [33].)
of spacetime points x there φ(x) = φ . This locus is
∗
In scenarios of this type, each bubble is causally dis-
called the reheating surface and is a noncompact, space-
connected from most other bubbles.3 Hence, bubble nu-
like three-dimensionalhypersurface [16, 18]. It is impor-
tant to realize that a finite, initially inflating 3-volume
of space may give rise to a reheating surface having an
infinite 3-volume, and even to infinitely many causally
2 For instance, it was shown that holographic considerations do
not place any bounds on the total number of e-foldings during
inflation [20]. For recent attempts to limit the number of e-
foldings usinga different approach, see e.g. [21]. Note also that
1 It is worth emphasizing that the term“eternal inflation”refers the effects of“random jumps”are negligible during the last 60
to future-eternity of inflation in the sense described above, but e-foldingsofinflation,sincetheproducedperturbationsmustbe
doesnotimplypast-eternity. Infact,inflationaryspacetimesare oforder10−5 accordingtoobservations.
genericallynot past-eternal[16,17]. 3 Collisionsbetweenbubblesarerare[34];however,effectsofbub-
3
rectlyaccessibletoastrophysicalexperiments. Neverthe-
Λ domination less, the study of the global structure of eternally inflat-
ing spacetimeis notmerelyofacademicinterest. Funda-
mentalquestionsregardingthecosmologicalsingularities,
the beginning of the Universe andofits ultimate fate, as
wall wall well as the issue of the cosmological initial conditions
all depend on knowledge of the global structure of the
spacetimeaspredictedbythetheory,whetherornotthis
nucleation
global structure is directly observable (see e.g. [37, 38]).
reheating In other words, the fact that some theories predict eter-
nal inflation influences our assessment of the viability of
these theories. In particular, the problem of initial con-
Figure 3: A spacetime diagram of a bubble interior. The in-
ditions for inflation [39] is significantly alleviated when
finite, spacelike reheating surface is shown in darker shade.
eternal inflation is present. For instance, it was noted
Galaxy formation is possible within the spacetime region in-
early on that the presence of eternal self-reproduction
dicated.
in the“chaotic”inflationary scenario [40] essentially re-
moves the need for the fine-tuning of the initial condi-
cleationeventsmaygenerateinfinitelymanystatistically tions [3, 41]. More recently, constraints on initial condi-
inequivalent,causallydisconnectedpatchesofthereheat- tions were studied in the context of self-reproduction in
ing surface, every patch giving rise to a possibly infinite models of quintessence [42] and k-inflation [43].
number of galaxies and observers. This feature signifi- Since the values of the observable parameters χ are
a
cantlycomplicatesthetaskofextractingphysicalpredic- random, it is natural to ask for the probability distribu-
tions from these models. This class of models is referred tionofχ thatwouldbemeasuredbyarandomlychosen
a
to as“eternal inflation of tunneling type.” observer. Understandably, this question has been the
In the following subsections, I discuss the motivation main theme of much of the work on eternal inflation.
forstudyingeternalinflationaswellasphysicaljustifica- Obtaining an answer to this question promises to estab-
tionsforadoptingtheeffectivestochasticpicture. Differ- lish a more direct contact between scenarios of eternal
ent techniques developed for describing eternal inflation inflationandexperiment. Forinstance,ifthe probability
are reviewed in Sec. II. Section III contains an overview distributionforthecosmologicalconstantΛ werepeaked
ofmethods forextractingpredictionsanda discussionof near the experimentallyobserved,puzzlingly smallvalue
the accompanying“measure problem.” (see e.g. [44] for a review of the cosmological constant
problem), the smallness of Λ would be explained as due
to observer selection effects rather than to fundamen-
A. Some motivation tal physics. Considerations of this sort necessarily in-
volve some anthropic reasoning; however, the relevant
assumptions are minimal. The basic goal of theoretical
The hypothesis of cosmological inflation was invoked
cosmology is to select physical theories of the early uni-
to explain several outstanding puzzles in observational
versethataremostcompatible withastrophysicalobser-
data[36]. However,someobservedquantities(suchasthe
vations,includingtheobservationofourexistence. Itap-
cosmological constant Λ or elementary particle masses)
pearsreasonabletoassumethatthecivilizationofPlanet
may be expectation values of slowly-varying effective
Earth evolved near a randomly chosen star compatible
fieldsχ . Withinthephenomenologicalapproach,weare
a
with the development of life, within a randomly chosen
compelled to consider also the fluctuations of the fields
galaxy where such stars exist. Many models of inflation
χ during inflation, on the same footing as the fluctua-
a
genericallyincludeeternalinflationandhencepredictthe
tions of the inflaton φ. Hence, in a generic scenario of
formation of infinitely many galaxies where civilizations
eternal inflation, all the fields χ arrive at the reheating
a
like ours may develop. It is then also reasonable to as-
surface φ = φ with values that can be determined only
∗
sume that our civilization is typical among all the civi-
statistically. Observers appearing at different points in
lizations that evolved in galaxies formed at any time in
space may thus measure different values of the cosmo-
the universe. This assumption is called the“principle of
logical constant, elementary particle masses, spectra of
mediocrity”[18].
primordial density fluctuations, and other cosmological
parameters. To use the“principleof mediocrity”for extractingsta-
Itisimportanttonotethatinhomogeneitiesinobserv- tisticalpredictions from a model of eternalinflation, one
able quantities are created on scales far exceeding the proceeds as follows [18, 45]. In the example with the
Hubble horizon scale. Such inhomogeneities are not di- fields χa described above, the question is to determine
the probability distribution for the values of χ that a
a
random observer will measure. Presumably, the values
of the fields χ do not directly influence the emergence
a
blecollisionsareobservableinprinciple[35]. of intelligent life on planets, although they may affect
4
the efficiency of structure formation or nucleosynthesis. to scenarios with multiple bubbles, as discussed in more
Therefore, we may assume a fixed, χ -dependent mean detail in Sec. IIID. Some recent results obtained using
a
number of civilizations ν (χ ) per galaxy and proceed these measures are reported in Refs. [61, 62, 63].
civ a
to ask for the probabilitydistribution P (χ ) ofχ near
G a a
arandomlychosengalaxy. The observedprobabilitydis-
tribution of χ will then be
a B. Physical justifications of the semiclassical
picture
P(χ )=P (χ )ν (χ ). (4)
a G a civ a
The standard frameworkof inflationary cosmologyas-
One may use the standard“hot big bang”cosmology to
serts that vacuum quantum fluctuations with super-
determine the average number ν (χ ) of suitable galax-
G a
horizon wavelengths become classicalinhomogeneities of
ies per unit volume in a regionwhere reheating occurred
the fieldφ. The calculationsofcosmologicaldensity per-
with given values of χ ; in any case, this task does not
a
turbations generated during inflation [7, 11, 64, 65, 66,
appear to pose difficulties of principle. Then the compu-
67, 68] also assume that a“classicalization”of quantum
tation of P (χ ) is reduced to determining the volume-
G a
fluctuations takes place via the same mechanism. In the
weighted probability distribution V(χ ) for the fields χ
a a
calculations,thestatisticalaverage δφ2 ofclassicalfluc-
within a randomly chosen 3-volume along the reheating
tuationsonsuper-Hubblescalesissimplysetequaltothe
surface. The probability distribution of χ will be ex-
pressed as a quantum expectation value h0|φˆ2|(cid:10)0i in(cid:11) a suitable vac-
uum state. While this approach is widely accepted in
P(χ )=V(χ )ν (χ )ν (χ ). (5) the cosmology literature, a growing body of research is
a a G a civ a
devotedto the analysis of the quantum-to-classicaltran-
However,definingV(χ )turnsouttobefarfromstraight- sition during inflation (see e.g. [69] for an early review).
a
forwardsince the reheating surface in eternal inflation is Since a detailed analysis would be beyond the scope of
an infinite 3-surface with a complicated geometry and the presenttext, I merely outline the main ideas and ar-
topology. The lack of a natural, unambiguous, unbi- guments relevant to this issue.
ased measure on the infinite reheating surface is known Astandardphenomenologicalexplanationofthe“clas-
as the “measure problem”in eternal inflation. Existing sicalization”of the perturbations is as follows. For sim-
approaches and measure prescriptions are discussed in plicity, let us restrict our attention to a slow-roll infla-
Sec.III,wheretwomainalternatives(the“volume-based” tionary scenario with one scalar field φ. In the slow roll
and“worldline-based”measures) are presented. In Sec- regime, one can approximately regard φ as a massless
tions IIIB and IIID I give arguments in favor of using scalarfieldindeSitterbackgroundspacetime[4]. Dueto
the volume-based measure for computing the probabil- the exponentially fast expansion of de Sitter spacetime,
ity distribution of values χa measured by a random ob- super-horizonFouriermodesofthefieldφareinsqueezed
server. The volume-based measure has been applied to quantumstateswith exponentiallylarge(∼eHt)squeez-
obtain statistical predictions for the gravitational con- ing parameters [70, 71, 72, 73, 74, 75]. Such highly
stant in Brans-Dicke theories [24, 25], cosmological con- squeezedstates havea macroscopicallylarge uncertainty
stant (dark energy) [46, 47, 48, 49, 50, 51], particle in the field value φ and thus quickly decohere due to
physics parameters [52, 53, 54], and the amplitude of interactions with gravity and with other fields. The re-
primordial density perturbations [48, 51, 55, 56]. sulting mixed state is effectively equivalent to a statis-
Theissueofstatisticalpredictionshasrecentlycometo tical ensemble with a Gaussian distributed value of φ.
the fore in conjunction with the discovery of the string Thereforeonemaycomputethestatisticalaverage δφ2
theory landscape. According to various estimates, one asthequantumexpectationvalueh0|φˆ2|0iandinterpret
expects to have between 10500 and 101500 possible vac- (cid:10) (cid:11)
the fluctuation δφ as a classical“noise.” A heuristic de-
uum states of string theory [31, 57, 58, 59, 60]. The
scription of the“classicalization”[4] is that the quantum
stringvacuadifferinthegeometryofspacetimecompact-
commutators of the creation and annihilation operators
ification and have different values of the effective cosmo- ofthe field modes,[aˆ,aˆ†]=1,are muchsmallerthan the
logical constant (or“dark energy”density). Transitions expectation values a†a ≫1 and are thus negligible.
betweenvacuamayhappenviathewell-knownColeman-
A related issue is the backreaction of fluctuations of
deLuccia tunneling mechanism [32]. Once the dark en- the scalar field φ on(cid:10)the(cid:11)metric.4 According to the stan-
ergydominatesina givenregion,the spacetimebecomes
dard theory (see e.g. [68, 84] for reviews), the perturba-
locally de Sitter. Then the tunneling process will cre-
tions of the metric arising due to fluctuations of φ are
ate infinitely many disconnected “daughter”bubbles of
described by an auxiliary scalar field (sometimes called
other vacua. Observers like us may appear within any
of the habitable bubbles. Since the fundamental theory
does not specify a single“preferred”vacuum, it remains
to try determining the probability distribution of vacua
4 The backreaction effects of the long-wavelength fluctuations of
as found by a randomly chosen observer. The“volume-
ascalarfieldduringinflationhavebeeninvestigated extensively
based”and“worldline-based”measures can be extended (seee.g.[76,77,78,79,80,81,82,83]).
5
the“Sasaki-Mukhanovvariable”)inafixeddeSitterback- Sec. IIA). It was shown that the leading late-time
ground. Thus, the“classicalization”effect should apply asymptotics of the quantum expectation values coincide
equallytothefluctuationsofφandtotheinducedmetric with the corresponding statistical averages (6). These
perturbations. At the same time, these metric perturba- results appear to validate the“random walk”approach,
tionscanbeviewed,inanappropriatecoordinatesystem, albeit in a limited context (in the absence of backreac-
as fluctuations of the local expansion rate H(φ) due to tion).
local fluctuations of φ [4, 7, 85]. Thus one arrives at the
picture of a“locally de Sitter”spacetime with the met-
ric (3), where the Hubble rate a˙/a=H(φ) fluctuates on II. STOCHASTIC APPROACH TO INFLATION
super-horizon length scales and locally follows the value
of φ via the classical Einstein equation (1).
The stochastic approach to inflation is a semiclassi-
The picture as outlined is phenomenological and does
cal,statisticaldescriptionofthespacetimeresultingfrom
not provide a description of the quantum-to-classical
quantum fluctuations of the inflaton field(s) and their
transitioninthemetricperturbationsattheleveloffield
backreactionon the metric [1, 2, 104, 105, 106, 107, 108,
theory. For instance, a fluctuation of φ leading to a lo-
109, 110, 111, 112]. In this description, the spacetime
cal increase of H(φ) necessarily violates the null energy
remains everywhere classical but its geometry is deter-
condition [86, 87, 88]. The cosmological implications of
mined by a stochastic process. In the next subsections I
such“semiclassical”fluctuations (see e.g. the scenario of
reviewthemaintoolsusedinthestochasticapproachfor
“island cosmology” [89, 90, 91]) cannot be understood
calculationsinthecontextofrandom-walktype,slow-roll
in detail within the framework of the phenomenological
inflation. Models involving tunneling-type eternal infla-
picture.
tion are considered in Sec. IIE.
A more fundamental approach to describing the
quantum-to-classical transition of perturbations was de-
veloped using non-equilibrium quantum field theory and
A. Random walk-type eternal inflation
theinfluencefunctionalformalism[92,93,94,95]. Inthis
approach,decoherenceofa pure quantumstate of φ into
amixedstateisentirelydue tothe self-interactionofthe The important features of random walk-type eternal
field φ. Inparticular,it is predicted that no decoherence inflationcanbeunderstoodbyconsideringasimpleslow-
would occur for a free field with V(φ) = 1mφ2. This roll inflationary model with a single scalar field φ and a
2
resultisatvariancewiththe acceptedparadigmof“clas- potential V(φ). The slow-rollevolution equation is
sicalization”asoutlinedabove. Ifthesourceofthe“noise”
1 dV 1 dH
isthecouplingbetweendifferentperturbationmodesofφ, φ˙ =− =− ≡v(φ), (7)
the typical amplitude of the“noise”will be second-order 3H dφ 4π dφ
in the perturbation. This is several orders of magnitude
smaller than the amplitude of“noise”found in the stan- where H(φ) is defined by Eq. (1) and v(φ) is a model-
dard approach. Accordingly, it is claimed [96, 97] that dependent function describing the“velocity”φ˙ of the de-
the magnitude of cosmological perturbations generated terministic evolution of the field φ. The slow-roll trajec-
by inflation is several orders of magnitude smaller than tory φsr(t), which is a solution of Eq. (7), is an attrac-
the results currently accepted as standard, and that the tor [113, 114] for trajectories starting with a wide range
shape of the perturbation spectrum depends on the de- of initial conditions.5
tails of the process of“classicalization”[98]. Thus, the As discussed in Sec. IB, the super-horizon modes of
results obtained via the influence functional techniques the field φ are assumed to undergo a rapid quantum-
do not appear to reproduce the phenomenological pic- to-classical transition. Therefore one regards the spatial
ture of “classicalization” as outlined above. This mis- average of φ on scales of several H−1 as a classical field
match emphasizes the need for a deeper understanding variable. The spatial averaging can be described with
of the nature of the quantum-to-classical transition for help of a suitable window function,
cosmologicalperturbations.
Finally, let us mention a different line of work which hφ(x)i≡ W(x−y)φ(y)d3y. (8)
supports the “classicalization”picture. In Refs. [8, 99,
Z
100,101,102,103],calculationsof(renormalized)expec-
tation values such as hφˆ2i, hφˆ4i, etc., were performed for It is implied that the window function W(x) decays
quickly on physical distances a|x| of order several H−1.
field operators φˆ in a fixed de Sitter background. The
From now on, let us denote the volume-averaged field
results were compared with the statistical averages
simply by φ (no other field φ will be used).
P(φ,t)φ2dφ, P(φ,t)φ4dφ, etc., (6)
Z Z
where the distribution P(φ,t) describes the “random 5 SeeRef.[43]foraprecisedefinitionofanattractortrajectoryin
walk”of the field φ in the Fokker-Plank approach (see thecontext ofinflation.
6
As discussed above, the influence of quantum fluctu- and ξ is a normalized random variable representing
ations leads to random“jumps”superimposed on top of “white noise,”
the deterministic evolution of the volume-averaged field
φ(t,x). This may be described by a Langevin equation hξi=0, ξ2 =1, (15)
of the form [2] hξ(t)ξ(t+∆t)i=0 fo(cid:10)r (cid:11)∆t&H−1. (16)
φ˙(t,x)=v(φ)+N(t,x), (9) Equation (13) is interpreted as describing a Brownian
motionφ(t)withthesystematic“drift”v(φ)andthe“dif-
where N(t,x) stands for“noise”and is assumed to be a fusion coefficient”D(φ). In a typical slow-roll inflation-
Gaussianrandomfunctionwithknowncorrelator[2,115, ary scenario, there will be a range of φ where the noise
116, 117] dominates over the deterministic drift,
N(t,x)N(t˜,x˜) =C(t,t˜,|x−x˜|;φ). (10) v(φ)∆t≪ 2D(φ)∆t, ∆t≡H−1. (17)
An explici(cid:10)t form of the co(cid:11)rrelatorC depends on the spe- Such a range of φ pis called the “diffusion-dominated
cific window function W used for averaging the field φ regime.” For φ near the end of inflation, the amplitude
on Hubble scales [117]. However, the window function of the noise is very small, and so the opposite inequality
W is merely a phenomenologicaldevice used in lieu of a holds. This is the“deterministic regime”where the ran-
complete ab initio derivation of the stochastic inflation dom jumps can be neglected and the field φ follows the
picture. One expects, therefore, that results of calcula- slow-rolltrajectory.
tions should be robust with respect to the choice of W.
In other words, any uncertainty due to the choice of the
window function must be regarded as an imprecision in- B. Fokker-Planck equations
herent in the method. For instance, a robust result in
this sense is an exponentially fast decay of correlations A useful description of the statistical properties of
on time scales ∆t&H−1, φ(t) is furnished by the probability density P(φ,t)dφ
of having a value φ at time t. As in the case of the
C(t,t˜,|x−x˜|;φ)∝exp −2H(φ) t−t˜ , (11)
Langevinequation, the values φ(t) are measured along a
single, randomly chosen comoving worldline x = const.
(cid:0) (cid:12) (cid:12)(cid:1)
which holds for a wide class of window(cid:12)funct(cid:12)ions [117]. The probability distribution P(φ,t) satisfies the Fokker-
For the purposes of the present consideration,we only
Planck (FP) equation whose standard derivation we
need to track the evolution of φ(t,x) along a single co-
omit [120, 121],
moving worldline x = const. Thus, we will not need an
explicitformofC(t,t˜,|x−x˜|;φ)butmerelythevalueat ∂ P =∂ [−v(φ)P +∂ (D(φ)P)]. (18)
t φ φ
coincident points t=t˜, x=x˜, which is computed in the
slow-rollinflationary scenario as [2] The coefficients v(φ) and D(φ) are in general model-
dependent and need to be calculated in each particular
H2(φ) scenario. These calculations require only the knowledge
C(t,t,0;φ)= . (12)
4π2 of the slow-rolltrajectoryand the mode functions of the
quantized scalar perturbations. For ordinary slow-roll
(This represents the fluctuation (2) accumulated during inflation with an effective potential V(φ), the results are
one Hubble time, ∆t=H−1.) Due to the property (11), well-known expressions (7) and (14). The correspond-
onemayneglectcorrelationsontime scales∆t&H−1 in ing expressions for models of k-inflation were derived in
the“noise”field.6 Thus,theevolutionofφontimescales Ref. [43] using the relevantquantum theory of perturba-
∆t&H−1 canbe describedby afinite-differenceformof tions [122].
the Langevin equation (9), It is well known that there exists a“factor ordering”
ambiguity in translating the Langevin equation into the
φ(t+∆t)−φ(t)=v(φ)∆t+ 2D(φ)∆tξ(t), (13) FP equation if the amplitude of the“noise”depends on
the position. Specifically, the factor D(φ) in Eq. (13)
p
where may be replaced by D(φ+θ∆t), where 0 < θ < 1 is an
arbitrary constant. With θ 6= 0, the term ∂ (DP) in
H3(φ) φφ
D(φ)≡ (14) Eq. (18) will be replaced by a different ordering of the
8π2
factors,
∂ (DP)→∂ Dθ∂ D1−θP . (19)
φφ φ φ
6 Takingthesecorrelationsintoaccountleadstoapictureof“color Popular choices θ = 0 and(cid:2)θ = 1(cid:0)are calle(cid:1)d(cid:3) the Ito and
noise”[118, 119]. In what follows, we only consider the simpler 2
the Stratonovich factor ordering respectively. Motivated
picture of“white noise”as an approximation adequate for the
issuesathand. by the considerations of Ref. [123], we choose θ = 0 as
7
shown in Eqs. (13) and (18). Given the phenomenolog- aboundaryconditionmustbeimposedalsoatφ=φ .
max
ical nature of the Langevin equation (13), one expects For instance,one canuse the absorbingboundarycondi-
that any ambiguity due to the choice of θ represents an tion,
imprecision inherent in the stochastic approach. This
imprecision is typically of order H2 ≪1 [124]. P(φmax)=0, (23)
The quantity P(φ,t) may be also interpreted as the
which means that Planck-energy regions with φ = φ
fraction of the comoving volume (i.e. coordinate volume max
d3x) occupied by the field value φ at time t. Another disappear from consideration [111, 112].
Once the boundary conditions are specified, one may
importantcharacteristicisthevolume-weighteddistribu-
write the general solution of the FP equation (18) as
tion P (φ,t)dφ, which is defined as the proper 3-volume
V
(as opposed to the comoving volume) of regions having
the value φ at time t. (To avoid considering infinite vol- P(φ,t)= CλP(λ)(φ)eλt, (24)
umes, one may restrictone’s attention to a finite comov- λ
X
ingdomainintheuniverseandnormalizeP (φ,t)tounit
V where the sum is performed over all the eigenvalues λ of
volume at some initial time t = t .) The volume distri-
0 the differential operator
bution satisfies a modified FP equation [4, 107, 109],
LˆP ≡∂ [−v(φ)P +∂ (D(φ)P)], (25)
φ φ
∂ P =∂ [−v(φ)P +∂ (D(φ)P )]+3H(φ)P , (20)
t V φ V φ V V
andthecorrespondingeigenfunctionsP(λ) aredefinedby
which differs from Eq. (18) by the term 3HP that de-
V
scribes the exponential growth of 3-volume in inflating LˆP(λ)(φ)=λP(λ)(φ). (26)
regions.7
Presently we consider scenarios with a single scalar The constants C can be expressed through the initial
λ
field; however, the formalism of FP equations can be distribution P(φ,t ).
0
straightforwardly extended to multi-field models (see By an appropriate change of variables φ→z, P(φ)→
e.g. Ref. [123]). For instance, the FP equationfor a two- F(z), the operator Lˆ may be brought into a manifestly
field model is self-adjoint form [105, 107, 108, 109, 124, 126],
∂tP =∂φφ(DP)+∂χχ(DP)−∂φ(vφP)−∂χ(vχP), (21) Lˆ → d2 +U(z). (27)
dz2
whereD(φ,χ),v (φ,χ),andv (φ,χ)areappropriateco-
φ χ
efficients. ThenonecanshowthatalltheeigenvaluesλofLˆarenon-
positive; in particular, the (algebraically) largest eigen-
value λ ≡ −γ < 0 is nondegenerate and the cor-
max
C. Methods of solution responding eigenfunction P(λmax)(φ) is everywhere pos-
itive [43, 124]. Hence, this eigenfunction describes the
Inprinciple,onecansolvetheFPequationsforwardin late-time asymptotic of the distribution P(φ,t),
timebyanumericalmethod,startingfromagiveninitial
distribution at t = t . To specify the solution uniquely, P(φ,t)∝P(λmax)(φ)e−γt. (28)
0
the FP equations must be supplemented by boundary
conditions at both ends of the inflating range of φ [111, The distribution P(λmax)(φ) is the“stationary”distribu-
tion of φ per comoving volume at late times. The expo-
112]. At the reheating boundary (φ = φ ), one imposes
∗
nential decay of the distribution P(φ,t) means that at
the“exit”boundary conditions,
late times most of the comoving volume (except for an
exponentially small fraction) has finished inflation and
∂ [D(φ)P] =0, ∂ [D(φ)P ] =0. (22)
φ φ=φ∗ φ V φ=φ∗ entered reheating.
Theseboundaryconditionsexpressthe factthatrandom Similarly, one can represent the general solution of
jumps are very small at the end of inflation and cannot Eq. (20) by
move the value of φ away from φ = φ . If the potential
V(φ)reachesPlanckenergyscalesatso∗meφ=φ (this P (φ,t)= C P(λ˜)(φ)eλ˜t, (29)
max V λ˜ V
happensgenerallyin“chaotic”typeinflationaryscenarios Xλ˜
with unbounded potentials), the semiclassical picture of
spacetimebreaksdownforregionswithφ∼φ . Hence, where
max
[Lˆ+3H(φ)]P(λ˜)(φ)=λ˜P(λ˜)(φ). (30)
By the same method as for the operator Lˆ, it is possible
7 AmoreformalderivationofEq.(20)aswellasdetailsofthein-
terpretationofthedistributionsP andPV intermsofensembles to show that the spectrum of eigenvalues λ˜ of the op-
ofworldlinescanbefoundinRef.[125]. erator Lˆ +3H(φ) is bounded from above and that the
8
largesteigenvalue λ˜ ≡γ˜ admits a nondegenerate,ev- where φ (t) is the slow-rolltrajectory and φ is the ini-
max sr in
erywhere positive eigenfunction P(γ˜)(φ). However, the tial value of φ. While methods based on the Langevin
largest eigenvalue γ˜ may be either positive or negative. equation do not take into account boundary conditions
If γ˜ >0, the late-time behavior of P (φ,t) is orvolumeweightingeffects,theformula(34)providesan
V
adequate approximation to the distribution P(φ,t) in a
P (φ,t)∝P(γ˜)(φ)eγ˜t, (31) useful range of φ and t [137].
V
which means that the total proper volume of all the in-
flating regions grows with time. This is the behavior ex-
D. Gauge dependence issues
pected in eternal inflation: the number of independently
inflatingdomainsincreaseswithoutlimit. Thus,thecon-
An important feature of the FP equations is their de-
dition γ˜ > 0 is the criterion for the presence of eternal
pendence on the choice of the time variable. One can
self-reproduction of inflating domains. The correspond-
consider a replacement of the form
ing distribution P(γ˜)(φ) is called the“stationary”distri-
bution [26, 111, 112, 127].
t→τ, dτ ≡T(φ)dt, (37)
Ifγ˜ ≤0,eternalinflationdoesnotoccurandtheentire
space almost surely (i.e. with probability 1) enters the
understood in the sense of integrating along comoving
reheating epoch at a finite time.
worldlines x = const, where T(φ) > 0 is an arbitrary
If the potential V(φ) is of“new”inflationary type [6,
function of the field. For instance, a possible choice is
128, 129, 130, 131] and has a global maximum at say
T(φ) ≡ H(φ), which makes the new time variable di-
φ=φ , the eigenvaluesγ andγ˜ canbe estimated(under
0 mensionless,
the usual slow-rollassumptions on V) as [124]
V′′(φ0) τ = Hdt=lna. (38)
γ ≈ H(φ )<0, γ˜ ≈3H(φ )>0. (32)
0 0
8πV(φ ) Z
0
This time variable is called “scale factor time” or “e-
Therefore, eternal inflation is generic in the“new”infla-
folding time”since it measures the number of e-foldings
tionary scenario.
along a comoving worldline.
Let us comment on the possibility of obtaining solu-
The distributions P(φ,τ) and P (φ,τ) are defined as
tions P(φ,t) in practice. With the potential V(φ) = V
before, except for considering the 3-volumes along hy-
λφ4, the full time-dependent FP equation (18) can be
persurfaces of equal τ. These distributions satisfy FP
solved analytically via a nonlinear change of variable
equations similar to Eqs. (18)–(20). With the replace-
φ → φ−2 [126, 132, 133]. This exact solution, as well
ment (37), the coefficients of the new FP equations are
as an approximate solution P(φ,t) for a general poten-
modified as follows [124],
tial, can be also obtained using the saddle-point evalu-
ation of a path-integral expression for P(φ,t) [134]. In
D(φ) v(φ)
some cases the eigenvalue equation LˆP(λ) = λP(λ) may D(φ)→ , v(φ)→ , (39)
T(φ) T(φ)
be reduced to an exactly solvable Schr¨odinger equation.
These cases include potentials of the form V(φ)=λeµφ,
while the “growth”term 3HP in Eq. (20) is replaced
V(φ) = λφ−2, V(φ) = λcosh−2(µφ); see e.g. [124] for V
by 3HT−1P . The change in the coefficients may sig-
other examples. V
nificantly alter the qualitative behavior of the solutions
AgeneralapproximatemethodfordeterminingP(φ,t)
of the FP equations. For instance, stationary distribu-
for arbitrary potentials [135, 136, 137] consists of a per-
tionsdefinedthroughthepropertimetandthee-folding
turbative expansion,
time τ = lna were found to have radically different be-
havior[18,112,127]. Thissensitivitytothechoiceofthe
φ(t)=φ (t)+δφ (t)+δφ (t)+..., (33)
0 1 2 “timegauge”τ isunavoidablesincehypersurfacesofequal
τ may preferentially select regions with certain proper-
applied directly to the Langevin equation. The result is
ties. For instance, most of the proper volume in equal-
(at the lowest order) a Gaussian approximation with a
t hypersurfaces is filled with regions that have gained
time-dependent mean and variance [135],
expansion by remaining near the top of the potential
1 (φ−φ (t))2 V(φ),whilehypersurfacesofequalscalefactorwillunder-
P(φ,t)≈ exp − 0 , (34) representthoseregions. Thus,astatementsuchas“most
2πσ2(t) " 2σ2(t) # of the volume in the Universe has values of φ with high
σ2(t)≡ Hp′2(φsr) φin H3dφ, (35) V(Iφn)”thiseleaargrleylywgoarukgseo-dnepeetenrdneanlt.inflation [25, 111, 112,
π H′3
Zφsr 127], the late-time asymptotic distribution of volume
φ (t)≡φ (t)+ H′′ σ2(t)+ H′ Hi3n − H3 , (36) P(γ˜)(φ) along hypersurfaces of equal proper time [see
0 sr 2H′ 4π (cid:20)Hi′n2 H′2(cid:21) EVq. (31)] was interpreted as the stationary distribution
9
of field values in the universe. However, the high sensi- boundary conditions
tivity of this distribution to the choice of the time vari-
able makes this interpretation unsatisfactory. Also, it D
F(φ )=0, ∂ ( F) =0, (45)
∗ φ
was noted [138] that equal-proper time volume distribu- v
(cid:12)φ=φE
tions predict an unacceptably small probability for the (cid:12)
(cid:12)
currently observed CMB temperature. The reason for can be straightforwardly integrat(cid:12)ed and yields explicit
thisresultistheextremebiasoftheproper-timegaugeto- expressionsfortheexitprobabilitypexit(φE)asafunction
wardsover-representingregionswherereheatingoccurred ofthe initialvalue φ0 [43]. The exitprobabilitypexit(φ∗)
veryrecently[18,139]. Onemightaskwhetherhypersur- can be determined similarly.
faces of equal scale factor or some other choice of time
gauge would provide less biased answers. However, it
turns out[125] thatthere existsno a priori choiceofthe E. Self-reproduction of tunneling type
time gauge τ that provides unbiased equal-τ probability
distributionsforallpotentialsV(φ)inmodelsofslow-roll Untilnow,weconsideredeternalself-reproductiondue
inflation (see Sec. IIIC for details). torandomwalkofascalarfield. Anotherimportantclass
Although the FP equations necessarily involve a de- ofmodelsincludesself-reproductionduetobubblenucle-
pendence on gauge, they do provide a useful statisti- ation.8 Such scenarios of eternal inflation were studied
cal picture of the distribution of fields in the universe. in Refs. [34, 142, 143, 144, 145, 146].
The FP techniques can also be used for deriving sev- In a locally de Sitter universe dominated by dark en-
eral gauge-independent results. For instance, the pres- ergy, nucleation of bubbles of false vacuum may occur
enceofeternalinflationisagauge-independentstatement due to tunneling [14, 32, 147, 148]. Since the bubble nu-
(see also Sec. IIIA): if the largest eigenvalue γ˜ is posi- cleation rate κ per unit 4-volume is very small [32, 149],
tive in one gauge of the form (37), then γ˜ > 0 in ev-
π
ery other gauge [140]. Using the FP approach, one can κ=O(1)H−4exp −S − , (46)
also compute the fractal dimension of the inflating do- I H2
h i
main [140, 141] and the probability of exiting inflation
where S is the instanton action and H is the Hubble
I
through a particular point φ of the reheating boundary
∗ constantofthe deSitter background,bubbleswillgener-
intheconfigurationspace(incasethereexistsmorethan
ically not merge into a single false-vacuum domain [34].
one such point).
Hence, infinitely many bubbles will be nucleated at dif-
The exit probability can be determined as follows [43,
ferent places and times. The resulting“daughter”bub-
123]. Let us assume for simplicity that there are two
bles may again contain an asymptotically de Sitter, in-
possible exit points φ and φ , and that the initial dis-
∗ E finite universe, which again gives rise to infinitely many
tribution is concentrated at φ=φ , i.e.
0 “grand-daughter”bubbles. This picture of eternal self-
reproduction was called the “recycling universe” [30].
P(φ,t=0)=δ(φ−φ ), (40)
0 Some(orall)ofthecreatedbubblesmaysupportaperiod
ofadditionalinflationfollowedby reheating,asshownin
whereφ <φ <φ . Theprobabilityofexitinginflation
E 0 ∗
Fig. 3.
through φ=φ during a time interval [t,t+dt] is
E
In the model of Ref. [30], there were only two vacua
dp (φ )=−v(φ )P(φ ,t)dt (41) which could tunnel into each other. A more recently de-
exit E E E
veloped paradigm of “string theory landscape” [31] in-
(note that v(φ ) < 0). Hence, the total probability of volves a very large number of metastable vacua, corre-
E
exiting through φ=φ at any time is spondingtolocalminimaofaneffectivepotentialinfield
E
space. The value of the potential at each minimum is
∞ ∞
the effective value of the cosmological constant Λ in the
p (φ )= dp (φ )=−v(φ ) P(φ ,t)dt.
exit E exit E E E
corresponding vacuum. Figure 4 shows a phenomenolo-
Z0 Z0
(42) gist’s view of the“landscape.” Vacua with Λ≤ 0 do not
Introducing an auxiliary function F(φ) as allow any further tunneling9 and are called “terminal”
vacua [154], while vacua with Λ > 0 are called “recy-
∞
F(φ)≡−v(φ) P(φ,t)dt, (43) clable”since they can tunnel to other vacua with Λ > 0
Z0 or Λ ≤ 0. Bubbles of recyclable vacua will give rise to
infinitely many nested“daughter”bubbles. A conformal
one can show that F(φ) satisfies the gauge-invariant
equation,
D
∂φ ∂φ F −F =δ(φ−φ0). (44) 8 Both processes may be combined in a single scenario [30], but
v
(cid:20) (cid:18) (cid:19) (cid:21) weshallconsiderthemseparatelyforclarity.
9 AsymptoticallyflatΛ=0vacuacannot supporttunneling[150,
This is in accord with the fact that pexit(φE) = F(φE) 151,152];vacuawithΛ<0willquicklycollapsetoa“bigcrunch”
is a gauge-invariant quantity. Equation (44) with the singularity[32,153].
10
Λ such that κ = 0 for all α), or all the vacua are re-
αβ
cyclable. (Theory suggests that the former case is more
probable[59].) Ifterminalvacuaexist,thenthelate-time
asymptotic solution can be written as [154]
f (τ)≈f(0)+s e−qτ, (48)
α α α
X wherefα(0)isaconstantvectorthatdependsontheinitial
conditions andhas nonzero components only in terminal
0
vacua, and s does not depend on initial conditions and
α
is an eigenvector of M such that M s = −qs ,
αβ α βα α β
q > 0. This solution shows that all comoving volume
Figure 4: A schematic representation of the “landscape of P
reaches terminal vacua exponentially quickly. (As in the
string theory,”consisting of a large number of local minima
case of random-walk inflation, there are infinitely many
of an effective potential. The variable X collectively denotes
various fields and Λ is the effective cosmological constant. “eternallyrecycling”pointsxthatneverenteranytermi-
Arrows show possible tunnelingtransitions between vacua. nal vacua, but these points form a set of measure zero.)
If there are no terminal vacua, the solution f (τ) ap-
α
5 14 3 5 3 proaches a constant distribution [156],
4 4 lim f (τ)≈f(0), M f(0) =0, (49)
2 α α βα α
τ→∞
β
X
1 π
f(0) =H4exp . (50)
α α H2
(cid:20) α(cid:21)
Figure 5: A conformal diagram of the spacetime where self- Inthis case,thequantitiesf(0) areindependentofinitial
α
reproduction occurs via bubble nucleation. Regions labeled conditions and are interpreted as the fractions of time
“5”are asymptotically flat (Λ=0). spent by the comoving worldline in bubbles of type α.
One may adopt another approach and ignore the du-
ration of time spent by the worldline within each bub-
diagram of the resulting spacetime is outlined in Fig. 5. ble. Thus, one describes only the sequence of the
Ofcourse,only afinite numberofbubblescanbe drawn; bubbles encountered along a randomly chosen world-
the bubbles actuallyforma fractalstructurein a confor- line[62,156,157]. Iftheworldlineisinitiallyinabubble
mal diagram [155]. oftypeα,thentheprobabilityµ ofenteringthebubble
βα
A statistical description of the “recycling” spacetime of type β as the next bubble in the sequence after α is
can be obtained [30, 154] by considering a single comov-
κ
ing worldline x = const that passes through different µ = βα . (51)
βα
κ
bubbles at different times. (It is implied that the world- γ γα
line is randomly chosen from an ensemble of infinitely
(For terminal vacua α, we hPave κ = 0 and so we may
γα
many such worldlines passing through different points
define µ =0 for convenience.) Once againwe consider
βα
x.) Let the index α = 1,...,N label all the available
landscapes without terminal vacua separately from ter-
types of bubbles. For calculations, it is convenient to
minal landscapes. If there are no terminal vacua, then
use the e-folding time τ ≡lna. We are interested in the
the matrix µ is normalized, µ = 1, and is thus
probability f (τ) of passing through a bubble of type α αβ β βα
α a stochastic matrix [158] describing a Markov process of
at time τ. This probability distribution is normalized P
choosing the next visited vacuum. The sequence of vis-
by f = 1; the quantity f (τ) can be also visual-
α α α ited vacua is infinite, so one can define the mean fre-
ized as the fraction of the comoving volume occupied by
bubPbles of type α at time τ. Denoting by κ the nucle- quencyfα(mean) ofvisitingbubblesoftypeα. Iftheprob-
αβ
ability distribution for the first element in the sequence
ationratefor bubbles oftypeα withinbubbles oftypeβ
is f , then the distribution of vacua after k steps is
[computed according to Eq. (46)], we write the“master (0)α
given (in the matrix notation) by the vector
equation”describing the evolution of f (τ),
α
f =µkf , (52)
df (k) (0)
β
= (−κ f +κ f )≡ M f , (47)
dτ αβ β βα α βα α where µk means the k-th power of the matrix µ ≡ µ .
αβ
α α
X X Therefore, the mean frequency of visiting a vacuum α is
where we introduced the auxiliary matrix Mαβ. Given computedasanaverageoff(k)α overnconsecutivesteps
a set of initial conditions f (0), one can evolve f (τ) in the limit of large n:
α α
according to Eq. (47).
n n
1 1
To proceed further, one may now distinguish the fol- f(mean) = lim f = lim µkf . (53)
lowing two cases: Either terminal vacua exist (some β n→∞n (k) n→∞n (0)
k=1 k=1
X X
