On adiabatic perturbations in the ekpyrotic scenario

0.15 MB·English

by  A. Linde

On adiabatic perturbations in the ekpyrotic scenario
A. Linde,1 V. Mukhanov,2,3 and A. Vikman3
1Department of Physics, Stanford University, Stanford, CA 94305, USA
2Arnold-Sommerfeld-Center for Theoretical Physics,
Department für Physik, Ludwig-Maximilians-Universität München,
Theresienstr. 37, D-80333, Munich, Germany and
3CCPP, New York University, Meyer Hall of Physics,
4 Washington Place, New York, NY 10003,USA
(Dated: January 6, 2010)
0 In a recent paper, Khoury and Steinhardt [1] proposed a way to generate adiabatic cosmological
1 perturbationswithanearlyflatspectruminacontractingUniverse. Toproducetheseperturbations
0 theyusedaregimeinwhichtheequationofstateexponentiallyrapidlychangedduringashorttime
2
interval. Leaving aside the singularity problem and the difficult question about the possibility to

n transmittheseperturbationsfromacontractingUniversetotheexpandingphase,wewillshowthat
a themethodsusedin[1]areinapplicableforthedescriptionofthecosmological evolutionandofthe
J process of generation of perturbationsin this scenario.

6

] I. INTRODUCTION wasahopethatthecosmologicalsingularityproblemwill
h
be solvedin the context of string theory, but despite the
t
- attempts of the best experts in string theory, this prob-
p Inflationarytheoryprovidesasimplesolutiontomany lem remains unsolved. A recent attempt to solve this
e
cosmological problems [2–5]. It also provides a simple problem was made in the so-called ‘new ekpyrotic sce-
h
[ mechanism for the generationof adiabatic perturbations nario’[15–18]. This scenarioinvolvedmany exoticingre-
of the metric with flat spectrum, which are responsible dients,suchasa violationofthe nullenergyconditionin
2 for large-scale structure formation and for the observed a model which combined the ekpyrotic scenario [8] with
v
CMB anisotropy [6, 7]. Many predictions of inflationary the ghost condensate theory [19]. However, this model
4
4 cosmologyare already confirmed by observations. There contained ghosts [20]. The negative sign of the higher
9 were many attempts to propose an alternative, equally derivative term in this scenario makes it hard to con-
0 compelling cosmological theory, but so far they have all structaghost-freeUVcompletionofthistheory[20]; see
. been unsuccessful. also [21] for a discussion of another attempt to improve
2
1 the new ekpyrotic scenario. Even if one eventually suc-
Nothing illustrates this statement better than the his-
9 ceeds in inventing a UV complete generalization of this
tory of the development of the ekpyrotic/cyclic scenario
0 theory,suchageneralizationisexpectedto obeythe null
[8]. This scenario was supposed to solve all cosmological
: energy condition, in which case the new ekpyrotic sce-
v problemswithoutusingastageofinflation. However,the
i nario will not work anyway.
X original version of this scenario did not work. The large
mass and entropy of the universe remained unexplained, Leaving all of these unsolved problems aside, the au-
r
a insteadofsolvingthe homogeneityproblemthis scenario thorsoftheekpyroticscenariocontinueinvestigatinggen-
onlymadeitworse,andinsteadofthebigbangexpected eration of cosmologicalperturbations in various versions
in [8], there was a big crunch [9, 10]. of this scenario. In inflationary theory, these perturba-
tionsaregeneratedafterthesingularity,butintheekpy-
As a result, this scenario was replaced by the cyclic
rotic scenario they are supposed to be generated in a
scenario, which postulated the existence of an infinite
collapsing universe before the singularity and preserved
number of periods of expansion and contraction of the
onthe waythroughthe singularity. Since the singularity
universe[11]. However,whenthisscenariowasanalyzed,
problemis not solved,one cannot really say much about
taking into account the effect of particle production in
it, unless one makes additionalassumptions. After a few
the early universe, a very different cosmological regime
years of debate, most of the experts agreed, following
was found [12, 13]. In the latest version of this scenario,
[24], that adiabatic perturbations with a flat spectrum
the homogeneity problem is supposed to be solved by
are not produced in the simplest one-field versions of
the existence of an infinitely many stages of low-scale
this theory, contrary to the original claims by the au-
inflation, each of which should last at least 56 e-folds
thors of the ekpyrotic/cyclic scenario [8, 11, 25]. There-
[14].
fore more complicated versions of the ekpyrotic scenario
The most difficult problem facing this scenario is the were proposed[26], in which the adiabatic perturbations
problemof the cosmologicalsingularity. Originallythere are produced by a mechanism similar to the inflationary
2
curvaton mechanism [27]. The simplest versions of the equation of state w = p/ρ change tremendously during
ekpyrotic scenario of this type were ruled out in [28]. the same time interval. Indeed,
That is why it is important to examine the recent pa- H˙ = 1φ˙2 2 , (5)
per by Khoury and Steinhardt [1] who have proposed a −2 ≈−c2t2
new mechanism of generation of adiabatic cosmological hence H˙ H2/c2 and H˙ c2H2. For the equa-
perturbations with a nearly flat spectrum at the stage begin ≃ 0 end ≃ 0
tion of state we have
prior to the stage of the ekpyrotic collapse. This is the
main goal of our paper. In section II, we will examine 2 H˙ 1 1 t 2 t 2
1+w= begin c2 end .
the unperturbed background cosmological solution de- −3H2 ≈ H2c2t2 ≈ c2 t ≈ t
0 (cid:18) (cid:19) (cid:18) (cid:19)
scribing the model of Ref. [1]. As we will see, this solu- (6)
tion requires investigation of the cosmological evolution At the beginning (1+w) c−2 1 and at the end
at super-planckian values of curvature, when the meth- (1+w) c2 1. begin ≈ ≪
odsusedin[1]arenotvalid. Asimilarconclusionwillbe end ≈ ≫
reachedinsectionIII wherewewillshowthatthetheory According to [1], the stage described above during
ofcosmologicalperturbationsusedin[1]isalsonotvalid. which the spectrum of perturbations is generated is fol-
lowed by a subsequent scaling ekpyrotic stage, when the
Hubble scale evolves as
II. BACKGROUND MODEL tend
H H . (7)
0
≃ t
(cid:18) (cid:19)
The authors of [1] derive the following necessary condi-
The modelrelies on investigationofa contractinguni-
tion whichmustbesatisfiedifwewanttheperturbations
verse in a theory with the scalar field potential
generatedatthe pre-ekpyroticstagetobe ontheobserv-
V =V 1 e−cφ , (1) able scales today (see equation (16) in [1]):
0
−
where c 1. To simplify a(cid:0)compari(cid:1)sonof the results, we V 10−30 H , (8)
0 end
≫ ≤ | |
willusethesamenotationsandunits(8πG=1)asin[1].
where p p
Wewillstartwiththehomogeneousbackgroundsolution.
Ignoring the evolution of the universe, the equation of tend
H H , (9)
motion for φ is end ≃ 0 t
(cid:18) ekp(cid:19)
φ¨+cV0e−cφ =0 . (2) is the Hubble constant not at t = tend, but at the end
of the ekpyrotic stage, at t = t . Taking into account
ekp
It describes evolution of the field φ when t slowly in- that H √V we obtain from (8) that this necessary
0 0
creases from large negative t. Among other solutions, condition≃c−an be rewritten as
this equation has the solution used in [1]:
1
t 10−60H−1 t 10−60 . (10)
| ekp|≤ 0 | end|≃ c2H2
2 V 0
0
φ= ln ct . (3)
c −r 2 ! HMe−re1 tim10e−i4s3es.xpOrenstshedeoitnheurnhitasndo,faPcclaonrdckingtimtoe[1t]P(se∼e
P ∼
Thissolutiondescribesaslowlycontractinguniversewith alsobelow),theamplitudeofthegeneratedperturbations
a nearly unchanging Hubble parameter H H < 0. is O(cH ). Since this amplitude should be O(10−5), the
0 0
≈
One can check that during the pre-ekpyrotic stage, at ekpyrotic stage should end at
t <t<t , where
begin end t <10−50t . (11)
ekp P
| |
1 1 t V
begin 0 In other words, we are dealing here with frequencies
t , t = , H = ,
begin ≃ H0 end ≃ H0c2 c2 0 −r 3 whichare50ordersofmagnitudegreaterthanthePlanck
(4) mass! This completely invalidates the use of the classi-
this solutiondoes notreceivesignificantcorrectionseven cal equations of motion in the investigation of the back-
if one takes into account the contraction of the universe ground solution in [1]. Moreover, any mechanism ad-
and adds the term 3Hφ˙ to equation (2). dressingthesingularityproblem(ifthismechanismexists
at all) should operate on this time scale.
Duringthetimeintervalfromt tot (4)thescale
begin end
factor does not decrease significantly and therefore one This fact was not noticed in [1] because the authors
can set it to be a 1. Note that the time variable t is concentrated on the energy density at the end of the
≃
negativeanditsmagnitudechangesfromalargenegative ekpyrotic stage,
valueatthebeginningtoasmallnegativevalueattheend
of this contracting stage. At the same time, the deriva- ρ H2 4 10100 , (12)
tive of the Hubble constant H˙ and, correspondingly, the ≃ ≃ c4t2 ≃ c4
ekp
3
in Planck units. They studied two different regimes, where Φ is the gravitational potential induced by the
c = 1028 and c = 1040. In both cases ρ 1, so perturbations of the scalar field δφ. Linear perturbation
≪
one could think that we are safely in the sub-Planckian theory is reliable only if Φ, as well as the energy density
regime. However, at the ekpyrotic stage with c2 1 perturbations δT0/T0, are simultaneously much smaller
one has p c2ρ ρ, so from the Einstein equa≫tion than unity. Othe0rwis0e the higher-order terms in pertur-
H˙ = (ρ+∼p)/2 it≫follows that bation theory dominate over linear terms, and the per-
−
turbative expansion fails.
4 10100
p 2H˙ c2H2 , (13)
≈− ∼ ≃ c2t2 ≃ c2 Before going any further, let us illustrate the state-
ekp
mentsmadeaboveusinginflationaryperturbationsasan
i.e. pressure p, as well as H˙ and the curvature R, are example. During inflation, the large-scale energy den-
morethan20ordersofmagnitudegreaterthatthePlanck sity perturbations and perturbations of metric are small
densityforbothofthevaluesoftheparametersdiscussed δT0/T0 2Φ<10−5.
0 0 ∼−
in [1].
At the transition to the post-inflationary stage, the
Thesituationwithhigher-ordercurvatureinvariantsis valueofΦgrowstoabout10−5andstabilizesatthislevel.
even worse. For instance, MeanwhiletheenergydensityperturbationsδT0/T0con-
0 0
tinue to grow. Once these perturbations become large,
1 10150
R R;α H¨ . δT0/T0 O(1),galaxiesbegintoformandseparateform
| ;α |∼ ∼ c2t3 ∼ c2 0 0 ∼
ekp thecosmologicalexpansion. Thesubsequentevolutionof
q
the large-scale structure of the universe, as well as the
This means that they exceed the Planckian values by 94
orders of magnitude for c = 1028, and by 70 orders of related evolution of Φ, cannot be described by perturba-
magnitude for c = 1040. If one considers even higher tion theory. However, this happens long after the end of
inflation,whichiswhythestandardtheoryofgeneration
invariants, the situation becomes more and more trou-
of inflationary perturbations is reliable.
blesome. This is a direct consequence of the abnormal
smallnessofthe time atthe endofthe stageofthe ekpy-
Ref. [1]didnotcontainanyinvestigationofthepertur-
rosis (11), which happens in this scenario for all values bationsδT0/T0,whichisnecessarytoexaminevalidityof
0 0
of its parameters.
theirresults. Aswewillsee,intheirmodel,theperturba-
tionsδT0/T0arelargealreadyatthepre-ekpyroticstage,
One caneasilycheckthat the time tend, when the pre- 0 0
and therefore one cannot apply perturbation theory for
ekpyroticstageendsandthestageoftheekpyrosisbegins
theinvestigationofthegenerationofperturbationsinthe
inthisscenario,isalsomanyordersofmagnitudesmaller
metric and in the energy density in this scenario.
than the Planck time, for the values of the parameters
used in [1]. Moreover,one can show that this conclusion
To quantize scalar perturbations one introduces the
is valid for all possible values of parameters compatible
canonical variable [7]
with the requirement cH 10−5 unless there is a long
0
∼
stage of inflation after the ekpyrotic big crunch.
v =a δφ+ 3(1+w)Φ , (15)
Note that we are not talking here about hypotheti-
(cid:16) p (cid:17)
calprocesseswhicharesupposedtoleadtoabounceand
which for every Fourier mode k satisfies the equation
preventthecosmologicalsingularityinthisscenario. The
pre-ekpyroticstage,aswellastheekpyroticstage,arede- z′′
scribed in [1] by standard methods of classical field the- vk′′+ k2− z vk =0 , (16)
ory and general theory of relativity, which are not valid (cid:18) (cid:19)
at t < t and at super-planckian values of curvature.
P where prime means the derivative with respect to con-
| |
Our resultsshow thereforethatthe suggestedscenariois
formal time η, and z = a 3(1+w). In the case under
completely unreliable even as a background model. In
consideration
the next section we will show that even if one ignores all p
of the problems discussed above, the calculation of the z′′ 2
, (17)
spectrumofmetricperturbationsin[1]isalsounreliable. z ≃ t2
during the time interval t < t < t . During the
begin end
pre-ekpyrotic stage a 1, but the second derivative of a
III. PERTURBATIONS ≈
can be large and hence before identifying conformal and
physicaltimewehavetoperformalldifferentiationinour
equations,andonly after thatit is safe to sett η. For
The perturbed homogeneous flat universe in the ≈
the short-wavelengthperturbations with k t 1
conformal-Newtonian coordinate system is described by
| |≫
the metric
1 1
v eikη eikt , (18)
ds2 =a2(η) (1+2Φ)dη2 (1 2Φ)δikdxidxk , (14) k ≃ √k ≃ √k
− −
(cid:2) (cid:3)
4
wheretheamplitudeisfixedbythe requirementthatthe to express δφ in terms of Φ in (15), we obtain [7]
perturbations are the minimal vacuum fluctuations. In
the long-wave limit for k t 1 the solution of (16) is v 2 Φ˙ +HΦ
| |≪ ζ = = +Φ, (28)
z 3H(1+w)
dη 1 t
k
v C z+C z , (19)
k ≃ 1 2 z2 ≃ √k t where the dot denotes the derivative with respect to the
Z
physical time t = adη. By substituting (22) and using
wherewehavefixedtheconstantofintegrationrequiring the backgroundequations of motion, according to which
the continuity of the solution at the moment of “effec- R
tive horizon” crossing, tk 1/k, and have neglected the H˙ 3
≃ = (1+w), (29)
decaying mode. It follows from here that for the long- H2 −2
wavelength perturbations the “conserved” ζ is equal
we find
v c t cH
k k 0
ζ . (20)
k ≡ z ≃ √k tb ≃ k3/2 ζ = 2 1 w˙ + u˙ + 1 Φ. (30)
3 2H(1+w)2 H(1+w)u 1+w!
Thereforeat the end of the pre-ekpyroticstage the spec-
trum It follows from here that
ζ k3/2 cH (21) 2 t
k ≃ 0 ζ Φ , (31)
≃−3t
end
isscale-invariantintherangeofwavelengths t <λ <
end
t . For cH 10−5 the amplitude of pe|rtur|bations for the perturbations with k t 1 during the pre-
begin 0
| | ≃ | | ≪
is of the right order of magnitude. This result was the ekpyroticstage. Forderivingthisresultweused(25)and
reason for the claim in [1] that the pre-ekpyrotic stage (26). Thus at the end of the pre-ekpyrotic stage ζ Φ
≃
canproduce the perturbations similarto those produced and hence the spectrum of the gravitational potential is
by inflation. In this case the flat spectrum appears be- the same as that for ζ.
cause of the very fast change of the equation of state in
The next thing to do is to determine the amplitude
a contracting universe.
of the energy density perturbations and verify that they
Letus find under whichconditionsthe resultobtained alsoremain small. Using the 0 0 Einstein equation, we
−
is valid at the end of the pre-ekpyrotic stage. With this find
purpose we will calculate the gravitationalpotential and
δT0 2k2 2
energy density perturbations. It is convenient to intro- 0 = Φ Φ˙ +HΦ =
duce the variable u relatedto the gravitationalpotential (cid:18) T00 (cid:19)k −3H2a2 k− H (cid:16) k k(cid:17)
as 2k2 w˙ 2u˙
+3(1+w) 2 Φ . (32)
Φ=H(1+w)1/2u , (22) (cid:16)−3H2a2 − − H(1+w) − Hu(cid:17) k
During the pre-ekpyrotic stage, this relation for pertur-
and satisfying the equation
bations with k >H reduces to
0
′′
u′k′+(cid:18)k2− (11//zz) (cid:19)uk =0 . (23) (cid:18)δTT0000(cid:19)k ≃(cid:18)−32Hk22 + H2t(cid:19)Φk . (33)
During the pre-ekpyrotic stage
Thisequationimpliesthatattheendofthepre-ekpyrotic
(1/z)′′ H0 stage, at t≃tend
. (24)
1/z ≃− t δT0
0 2c2Φ (34)
T0 ≃
Therefore, for perturbations with k2 t H , we have 0
0
| |≫| |
for H <k cH and
u e±ikt, (25) 0 ≪ 0
k
∝
δT0 k 2
while for k2 t H the solution is 0 c2Φ 2c2Φ , (35)
| |≪| 0| T0 ≃ k cH ≫ k
(cid:18) 0 (cid:19)k (cid:18) 0(cid:19)
u C +C t . (26)
k ≃ 1 2 for k cH0.
≫
Thus we see that for the observed value Φ 3 10−5
Using the 0 i Einstein equation,
∼ ×
− the perturbations of the energy density exceed unity un-
1 √3 less c . 102. Note that the large c approximation used
Φ˙ +HΦ= φ˙δφ= H(1+w)1/2δφ , (27) in [1] requires c 1. Eq. (35) implies that even in the
2 2
≫
5
marginal case 1 c .102 the energy density perturba- H−1. In addition, according to (12), in this case one
tionsexceedunit≪yatallscalessmallerthanO(10−2)H−1 wo0uldbeforcedtoconsidertheekpyroticstageendingat
0
forc.102.Forc&102 weexpectthatthelinearpertur- the density exceeding the Planck density by 100 orders
bation theory breaks down at all interesting scales. Let of magnitude.
usshowthatthisiswhatreallyhappensinthecaseunder
consideration.
From (27) we find that during the pre-ekpyrotic stage
IV. CONCLUSIONS
δφ cΦ , (36)
≃
forperturbationswithk t 1.Expansionoftheenergy As we have seen, the new versionof the ekpyrotic sce-
| |≪
momentumtensoraroundthebackgroundcontainsterms nariorequirescalculationstobe performedatcurvatures
of allordersin δφ. In particular,in the expansionof δT0 which are exponentially greater than the Planck curva-
0
we have ture. Moreover, the theory of generation of metric per-
turbations used in this scenario is based on calculations
1
δT0 =V δφ+ V δφ2+... (37) whichareexponentially farawayfromthe domainofap-
0 ,φ 2 ,φφ plicability of this theory. These problems appear inde-
pendently of the major problem of this scenario, which
It is clear that linear perturbation theory is applicable
is the problem of the cosmologicalsingularity.
only if the quadratic term here is small compared to the
linear term. Let us find when this may happen. Taking
Each time when a new version of the ekpyrotic/cyclic
into accountthat in the case under consideration V =
,φφ theory is proposed, the authors suggest that its predic-
cV we find that
,φ tions be compared with predictions of inflationary cos-
mology to check which of the theories is better with
V δφ2
,φφ =cδφ c2Φ . (38) respect to observations. In accordance with this tradi-
V,φδφ ≃ tion, the authors of Ref. [1] say that their new theory
“predictsnon-gaussianityandaspectrumofgravitational
This ratio does not exceed unity only if 1 c . 102,
waves that is observationally distinguishable from infla-
≪
inagreementwiththe conclusionwhichwejustobtained
tion.” Themainproblemwiththissuggestionisthatone
by a different method. Thus linear perturbation theory
can test any theory only if it is internally consistent and
is not applicable for c & 102, and it fails completely for
only if it really makes definite predictions.
c=1028 and for c=1040 studied in [1].
Note that this problem is much more severe than the
potentially curable problem of anomalously large non-
gaussianity discussed in [1]. The nongaussianity prob-
lem appearsonly for extremelysmallwavelengthsk−1 >
c−1H−1, where c can be as large as 1028 or 1040. Mean- Acknowledgments
0
while for c & 102 the problems discussed in our paper
occur at all wavelengths.
WearethankfultoIgnacySawickiforusefulcomments.
Moreover, as we already mentioned, the spectrum is The work of A.L. was supported in part by NSF grant
not satisfactory even in the marginal case 1 c . 102. PHY-0244728,bythe Alexander-von-HumboldtFounda-
≪
Eq. (35) implies, in particular, that in the limiting case tion, and by the FQXi grant RFP2-08-19. V.M. is sup-
of c = O(1), when the large c approximation used in [1] ported by TRR 33 “The Dark Universe” and the Cluster
breaks down, the spectrum of perturbations of density ofExcellence EXC 153“OriginandStructureofthe Uni-
is blue with n 5, and its amplitude becomes greater verse”. A.V. is supported by the James Arthur Fellow-
s
∼
thanunityonscalestwoordersofmagnitudesmallerthan ship.
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