PHYSICAL REVIEW D 71, 063505 (2005)

Perturbations in bouncing cosmologies: Dynamical attractor versus scale invariance

Paolo Creminelli,1 Alberto Nicolis,1 and Matias Zaldarriaga1,2

1Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, USA2Center for Astrophysics, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 15 December 2004; published 7 March 2005)

1In the pretraction is drivno potential.case of the ekdraw are diffe

1550-7998=20

For bouncing cosmologies such as the ekpyrotic/cyclic scenarios we show that it is possible to makepredictions for density perturbations which are independent of the details of the bouncing phase. This canbe achieved, as in inflationary cosmology, thanks to the existence of a dynamical attractor, which makeslocal observables equal to the unperturbed solution up to exponentially small terms. Assuming that thephysics of the bounce is not extremely sensitive to these corrections, perturbations can be evolved even atthe nonlinear level. The resulting spectrum is not scale invariant and thus incompatible with experimentaldata. This can be explicitly shown in synchronous gauge where, contrary to what happens in thecommonly used Newtonian gauge, all perturbations remain small going towards the bounce and theexistence of the attractor is manifest.

DOI: 10.1103/PhysRevD.71.063505 PACS numbers: 98.80.Cq

I. INTRODUCTION

From cosmological observations, we know that the cur-rent Universe is to a good approximation flat, homogene-ous and isotropic on large scales. It is well known that inthe standard big bang cosmology this requires an enormousamount of fine-tuning on the initial state emerging from thePlanck era. This problem is usually solved postulating aperiod of inflation which acts like a dynamical attractor forthe cosmological evolution. From a generic initial condi-tion sufficiently close to a flat, homogeneous and isotropicstate the Universe evolves towards this symmetric configu-ration. Another logical possibility that has been explored isthat a dynamical attractor is present in a contracting phasewhich precedes the present expansion. After this phase, theUniverse would reach a state of high curvature and thenbounce and start expanding. Although string-inspired, theekpyrotic/cyclic scenarios [1,2] are fully described in thecontracting phase well before the bounce by a 4D effectivefield theory. The only light degree of freedom (besides thegraviton) is a scalar field which moves along a steepnegative potential and drives the cosmic contraction.1

Even in the presence of perturbations, if the potential issteep enough, inhomogeneities and anisotropies becomeless and less relevant going towards the bounce and thecosmic evolution gets closer and closer to the unperturbedone.

The crucial question is whether this class of models cangive rise to an approximately scale-invariant spectrum ofadiabatic density perturbations as clearly required by ex-periments. The answer to this question is not straightfor-ward because there is no explicit model of a bouncing

-big bang scenario [3] the (Einstein frame) con-en by the kinetic energy of the dilaton, which has

As we will discuss, this can be seen as a limitingpyrotic/cyclic models and the conclusions we willrent.

05=71(6)=063505(12)$23.00 063505

phase; it is just assumed that at sufficiently high energyUV corrections will stop the contraction and lead to anexpanding phase. It is therefore important to understand ifone can make predictions which are robust, i.e., indepen-dent of the details of the unknown bouncing phase. Ininflationary cosmology, we face a closely related issuebecause perturbations, that are created during inflation,are observed much later and there are cosmological phasesin between, like reheating, whose details are completelyunknown. It is well understood that in models with a singlefluctuating field all predictions are independent of the de-tails of the unknown cosmological phases. There are tworeasons why this is true. First of all, we are interested inmodes which are well outside the horizon during the un-known regimes of the cosmological evolution. This wouldnot be enough however. The situation is further simplifiedby the absence of isocurvature perturbations: Every ob-server will eventually go through the same cosmologicalhistory. Following the dynamical attractor, after a modegoes outside the horizon, the Universe locally approachesthe unperturbed solution up to exponentially small terms.This ‘‘parallel Universes’’ approach [4] allows to followperturbations through the unknown cosmological regimesat any order in perturbation theory. On the contrary, if weadd another field into the game, separate regions of theUniverse will be characterized by different values of thisnew field and will thus undergo a different evolution, sothat the final state depends on the details of the wholecosmological history. The fate of the isocurvature compo-nent can be very different: It can be washed away bythermal equilibrium or, on the contrary, become the leadingsource of adiabatic perturbations as in the curvaton andvariable decay width scenarios.

The purpose of this paper is to understand whether thesame kind of arguments used for single field inflation canbe applied to evolve density perturbations through theunknown bouncing phase of the ekpyrotic/cyclic scenario.

-1 2005 The American Physical Society

CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

In Newtonian gauge the most generic scalar perturbationis (at linear order) a superposition of the two independentsolutions for the Newtonian potential �. It has beenpointed out in the context of the ekpyrotic/cyclic scenario[5] that, in the limit of a very steep potential, one of thesetwo solutions has an approximately scale-invariant spec-trum in the contracting phase. Unfortunately the mostnaive evolution from the contracting to the expandingphase suggests that this scale-invariant mode is orthogonalto the ‘‘growing mode’’ in the expanding phase, the onerelevant for observations. However, it has been claimedthat a generic mixing of the two solutions at the bouncewould induce an observationally viable spectrum. In thispaper (Sec. III) we show that this generality cannot bedefended in any way. For example, if we change gaugeand describe the most generic perturbation in terms of thevariable � (proportional to the curvature of comovingsurfaces), none of the two independent modes is scaleinvariant. This tells us that for a contracting cosmologythe concept of scale invariance of a particular variable hasno physical meaning before one specifies how this variableis related to what we will eventually observe.

As we discussed, in the presence of a dynamical attrac-tor in the contracting phase, close to the bounce the cos-mological solution is locally (i.e., on regions smaller thanthe Hubble radius) homogeneous, isotropic and flat, andthe scalar follows its unperturbed evolution, up to expo-nentially small terms. This leads us to make a simpleassumption (Sec. IV), which is implicitly made also inthe case of an inflationary Universe when evolving pertur-bations through unknown phases. If the physics of thebounce is not tremendously sensitive to these exponentiallysmall corrections, every observer will go through thebounce in the same way, following the unperturbed evolu-tion. This allows to predict, not only at linear level but atany order in the perturbation, the statistical properties ofthe fluctuations in the expanding phase. Unfortunately, theconclusion is that the spectrum of density fluctuations is notscale invariant and thus it is ruled out by experiments. Thiscan be explicitly shown in synchronous gauge where, con-trary to what happens in the commonly used Newtoniangauge, all perturbations remain small going towards thebounce and the existence of the attractor is manifest. If ourassumptions are not satisfied, which is a logical possibility,then all predictions, like the scale invariance of the 2-pointfunction or the level of non-Gaussianity of the perturba-tions, strongly depend on the details of the bounce and norobust prediction can be made.

We stress that we are not giving any prescription tomatch perturbations across the bounce (for example, weare not claiming that � must be continuous [6–9], althoughthe final result for density perturbations in the expandingphase is the same). Given a generic initial condition beforethe bounce there is no way to evolve it through the un-known phase without specifying all the details. The point

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here is that we do not have a generic initial condition but avery special one, because of the presence of the dynamicalattractor. Locally the solution is exponentially close to theunperturbed solution and this is a powerful simplification.Again the situation is similar to inflationary cosmology;nobody knows how to evolve a generic initial state throughreheating because we do not know what reheating lookslike, but there is no need for this as inflation leads to a veryspecific state, locally (but not on large scales) indistin-guishable from the unperturbed solution.

II. INTUITIVE ARGUMENT ANDITS LIMITATIONS

In this section we want to approach the problem by a(critical) review of the basic heuristic argument for a scale-invariant spectrum of density perturbations in a bouncingcosmology [5,10,11]. This will show that there are quali-tative differences with respect to an inflationary scenariothat must be taken into account. For the time being weneglect how perturbations will evolve through the bounce.The argument is an estimate of the fluctuations of the scalarfield �, which is driving the contraction of the Universetowards the bounce. As the field is pulled by a steepnegative potential, it seems natural to assume that gravityis negligible in the dynamics of the fluctuations of �.Extending the discussion of Refs. [6,7], we will arguethat this is not the case.

To simplify the algebra it is useful to approximate thepotential over some range of � with the exponential form[5]

V��� � �V0e�

������2=p

p��=MP�; MP � �8G��1=2: (1)

In the ekpyrotic/cyclic scenario the potential is very steep,i.e., p � 1. The Friedmann equations and the equation ofmotion for a homogeneous � configuration,

��� 3H _�� V 0��� � 0; (2)

are exactly solved by the background

a�t� � �t=t0�p; H�t� �

pt;

�0�t� � MP

������2p

plog

tt0;

(3)

where t0 � �����������������������p�1� 3p�

pMP=

������V0

p. The scalar � is

moving from �1 to �1 and t is negative and runningtowards 0 which corresponds to the unknown bouncingphase. This solution has a constant, large pressure-to-energy density ratio, w � �2=3p� � 1 � 1.

It can be shown that for p < 1=3 this solution is adynamical attractor, in the sense that the system rapidlyapproaches it starting from a generic homogeneous andisotropic initial condition [12]. Roughly speaking, in thepresence of a very steep potential the kinetic energy of thescalar becomes bigger and bigger in the evolution, so that

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PERTURBATIONS IN BOUNCING COSMOLOGIES: . . . PHYSICAL REVIEW D 71, 063505 (2005)

the asymptotic state does not depend on the initial kineticenergy. In the following, we will explicitly see that thesame conclusion remains true in the presence of (suffi-ciently small) departures from isotropy and homogeneity.As discussed, this property will turn out to be crucial forour argument.

Since in the limit p � 1 the scale factor changes veryslowly with time, one could be tempted to say that thegravitational backreaction is small and to study perturba-tions in the scalar � on the fixed gravity background. Theequation of motion on a fixed Friedmann-Robertson-Walker (FRW) metric would be

� ��~k � 3H� _�~k �

�k2

a2� V00���

���~k � 0; (4)

where ~k is the comoving wave vector. For the backgroundabove we get

� ��~k �3pt� _�~k �

�k2

a2�

2�1� 3p�

t2

���~k � 0: (5)

It is evident that the contraction of the Universe gives asubleading contribution for p � 1. Moreover, for small tthe gradient term can be neglected so that the mode freezes.The equation reduces to � ��~k �

2t2 ��~k � 0, which gives

the two behaviors ��~k t�1 and t2. Therefore the leadingsolution for small t, properly normalized to match the usualMinkowski limit for large negative t, is

��~k ’1���k

p 1

kt: (6)

This implies a scale-invariant spectrum in position space ash���x�2i

Rd logkk3j��~kj

2.It is however easy to realize that the argument above is

not physically motivated, even at the heuristic level. In fact,there is no limit in which the fluctuations of � can bedecoupled from the metric perturbations since � is whatdrives the evolution of the Universe and its potential issteep. Taking the naive decoupling limit MP ! 1, oneobviously ends up with a trivial background. The onlycorrect decoupling limit would be to have a flat potentialfor �; in this case Eq. (4) is correct because fluctuations of� do not gravitate except for gradient terms. This is thereason why in slow-roll inflation the intuitive argumentabove gives a correct estimate of the density perturbationsup to slow-roll corrections.

In our case, there is no sense in which the dynamics ofperturbations is dominated by the potential term: Gravity isa crucial ingredient and must be taken into account. Let usclarify this statement in a gauge which is quite close to theintuition of keeping �� as the dynamical variable charac-terizing scalar perturbations,

��� ~x; t� � �� ~x; t� ��0�t�; gij � a2�t��ij: (7)

Unless otherwise specified we concentrate on scalar

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modes, setting tensor modes to zero. In this gauge gravi-tational effects are suppressed in inflation by slow-rollparameters, so that it explicitly realizes the decouplinglimit discussed above.

The other components of the metric, g00 and g0i, areLagrange multipliers and as such are determined as afunction of �� through the constraint Einstein equations.Substituting them into the action for gravity and the scalarfield and specializing to our background solution, we get aLagrangian for �� of the form [13] (see Appendix A forthe derivation of this result)

S � �1

2

Zd3xdt

��������g

p�@����2: (8)

This is the Lagrangian for a scalar in the given backgroundbut without any potential term, so that from this point ofview it is difficult to argue that the potential is driving thedynamics of the perturbations. Of course the equation ofmotion deriving from this Lagrangian is exactly Eq. (2)without the mass term. The effect of metric perturbations isso important that it exactly cancels out the potential. Notethat the very simple Lagrangian above is exact only in ascaling solution a�t� / tp, like the collapsing backgrounddescribed above (p � 1) or inflation with an exponentialpotential (p � 1). In both cases the action will receivesmall corrections proportional to the variation of the equa-tion of state in one Hubble time, i.e., subleading, respec-tively, in the fast-roll or slow-roll expansion.

From the action above, the properly normalized solu-tions for small t are now ��~k 1=

���k

pand ��~k

���k

pt.

None of these implies a scale-invariant spectrum.To get further insight into the problem, in the next

section we will perform a detailed analysis of scalar per-turbations in two different gauges.

III. STANDARD GAUGES AND THE APPARENTRAPID GROWTH OF PERTURBATIONS

The study of scalar perturbations can be performed indifferent gauges. In each one, after satisfying the constraintequations, the perturbation is parametrized by a singlescalar function, which satisfies, at linear order, a secondorder linear differential equation. The most generic scalarfluctuation will therefore be described by a linear combi-nation of the two independent solutions of this equation.Usually one of the solutions dominates over the other atlate times (‘‘growing’’ and ‘‘decaying’’ mode). However,in some cases the same physical perturbation is describedin one gauge as growing mode and in another as decaying(we will see an example of this in the following).Neglecting the decaying mode is therefore quite mislead-ing, especially in a contracting background like the one athand, and thus will be avoided.

To make contact with the literature, in this section we aregoing to perform the calculation in two commonly usedgauges. Doing so we will stress some subtle point which

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CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

has been usually overlooked. However, it will turn out thatin both gauges perturbation theory breaks down for t ! 0in the background under study. In the next section we willtherefore move to synchronous gauge in which the pertur-bativity of the calculation remains manifest.

The analysis of perturbations in inflation is greatly sim-plified by using the variable � introduced in Ref. [14],because it is conserved outside the horizon under simpleassumptions on the stress energy tensor. We therefore startour analysis in the �-gauge defined by

�� ~x; t� � �0�t�; gij � �1� 2�� ~x; t��a2�t��ij: (9)

In this gauge the scalar field is unperturbed so that therelation with the gauge introduced in the previous sectionis simply a time reparametrization

t ! t���_�0

) � � �H_�0

��: (10)

In our scaling background solution �� and � are simplyrelated by a constant factor

� � �

����p2

r1

MP�� (11)

so that the Lagrangian for � is again very simple:

S � �M2

P

p

Zd3xdt

��������g

p�@���2: (12)

The other components of the metric are fixed as functionsof � through the constraint equations. At linear order theyare (see Appendix A for details)

g00 � �1� 2_�H

� �1�2tp

_� (13)

g0i � �@i

��H

�_�20

2H2M2P

a2

@2_��

� �@i

�tp� �

1

p�t=t0�2p

@2_��; (14)

where, as usual, 1=@2 is defined in Fourier transform. Notethat � is related to the Ricci scalar of the spatial metricinduced on comoving surfaces (i.e., spatial hypersurfacesof constant �) simply by

�3�R � 4k2

a2� ~k; (15)

as in this gauge comoving surfaces are surfaces of constantt.

The equation of motion for � deriving from its actionabove is

�� ~k �3pt

_� ~k �k2

�t=t0�2p� ~k � 0; (16)

which can be integrated in terms of J and Y Bessel func-

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tions with argument k , where is the conformal timedefined as usual by dt � a�t�d . The full solution is

� ~k� � �1

a

�����������k

p�A~kJ�1�3p�=�2�2p���k �

� B~kY�1�3p�=�2�2p���k ��; (17)

where A~k and B~k are integration constants, to be deter-mined below. It has been claimed that the variable � is notsuitable for discussing density perturbations in a contract-ing phase, because it shows no ‘‘classical instability’’ andtherefore is not ‘‘amplified’’ [5,6]. The meaning of this isunclear since the two modes for � above parametrize themost generic scalar perturbation of the system.

Following the standard procedure [15], we then quantizethe system by writing the quantum field operator as� ~k� � � � ~k� �a ~k � ��~k � �a

y

� ~k, where � ~k� � is the above

classical solution, and the creation-annihilation operatorsobey the standard commutation rule �a ~k; a

y~k0� �

�2�3�3� ~k� ~k0�. Since for large negative the Universereduces to Minkowski space-time, we impose that theclassical solution � ~k� � asymptotes to the standardMinkowski wave function / 1����

2kp e�ik of a massless scalar

field. Also, we assume that the initial state is the usualMinkowski vacuum. The proper normalization can be readfrom the action Eq. (12) written in conformal time,

� ~k� � !����p

p���2

pMP

1

a

1�����2k

p e�ik for ! �1: (18)

This fixes the linear combination of the two independentsolutions to be

� ~k� � �1

a1�����2k

p

����

p

2MP

����p

p �����������k

p�J�1�3p�=�2�2p���k �

� iY�1�3p�=�2�2p���k ��: (19)

At small t we recover the two behaviors that solve Eq. (16)neglecting gradient terms

� ~k 1

MP�ik��1=2��ptp0 � k�1=2��pt1�3pt2p0 �; (20)

where we expanded the exponents up to linear order in p.The first term comes from the Y function while the secondcomes, for generic p, both from Y and J. Even though �remains finite for t ! 0, the off-diagonal term g0i inEq. (14) diverges like t�p, so that linear theory cannot bejustified in this gauge for small t.

The late time spectrum of � is

h� ~k�t�� ~k0 �t�i � �2�3�3� ~k� ~k0�j� ~k�t�j2; (21)

with

j� ~k�t�j2

1

M2P

�k�1�2pt2p0 � k1�2pt2�6pt4p0 �; (22)

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2Note that even though this seems to validate the naive argu-ment discussed in the previous section, the time dependence ofthe second solution, � � const, is not a solution of Eq. (5). Thesame happens for �� in Newtonian gauge which has the sametime dependence as � [see Eq. (29)].

3To be precise, only the anisotropic piece of the extrinsiccurvature in Eq. (24) is recovered in this way. To get the isotropiccomponent proportional to _� , we have to keep further terms ( t1�3p) in the small t expansion of � in addition to those ofEq. (28). This again tells us that it is not obvious to get thephysical importance of a term by simply looking at its contri-bution to �; a term diverging as t�1�p and one decaying as t1�3p

give comparable extrinsic curvatures.

PERTURBATIONS IN BOUNCING COSMOLOGIES: . . . PHYSICAL REVIEW D 71, 063505 (2005)

which does not contain any scale-invariant component.This remains true even if one includes all further terms inthe k expansion of the complete solution Eq. (19) as theyonly add positive powers of k. This result can be written ina more transparent form as a function of the physical wavevector kph � k=a,

h��x�2i Z

d logkphpH2

M2P

��kph=H�2�2p � �kph=H�4�2p�:

(23)

We stress again that the complete solution for � de-scribes the most generic scalar fluctuation of the system.But, as discussed in Sec. I, a scale-invariant spectrum isobtained for the Newtonian potential � so that the situationis at first puzzling.

Before analyzing the relation between the results ob-tained for � and �, it is important to appreciate that thesecond term in Eq. (20) cannot be simply disregarded asdecaying. Although it drops off in time like t1�3p, thusgiving a negligible contribution to the intrinsic curvature ofcomoving surfaces [see Eq. (15)], it dominates the pertur-bation in the extrinsic curvature of comoving surfaces. Infact, at linear order the extrinsic curvature is

Kij �

1

2gik� _gkj � @kg0j � @jgk0�

� �H� _���ij �

1

p

@i@j@2

_� �tp

@i@ja2

�: (24)

The leading contribution comes from the term � ~k t1�3p

in Eq. (20), which gives a perturbation in Kij diverging as

t�3p towards the bounce. Note that this is however a smallperturbation with respect to the unperturbed extrinsic cur-vature which diverges faster, as H t�1. We see that thetwo terms in Eq. (20) are both physically relevant as theydominate, respectively, the intrinsic and extrinsic curva-ture. Further terms in the small t expansion of Eq. (19) areinstead subleading in their contribution to physicalcurvatures.

Let us now see how these perturbations are reinterpretedin the commonly employed Newtonian gauge. This gaugeis defined by

ds2 � ��1� 2��dt2 � �1� 2��a2�t�d~x2: (25)

The most generic scalar perturbation can be described bythe Newtonian potential �. All other variables are relatedto � through the constraint Einstein equations. In particu-lar, � � � at linear order in the absence of anisotropicstress, as in our case. The equation of motion for � in ourbackground is (see, for example, Ref. [5])

�� ~k �2� pt

_� ~k �k2

�t=t0�2p� ~k � 0: (26)

Following the usual procedure as done for � we get theproperly normalized solution

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� ~k� � �1

a1�����2k

p

����

p

2MP

p1� p

1�����������k

p �J�1�p�=�2�2p���k �

� iY�1�p�=�2�2p���k ��; (27)

with late time behavior

� ~k 1

MP�k�3=2�pt�1�p � ik�1=2�ptp0 �: (28)

We see that the first term, which diverges approaching thebounce (thus casting doubt on the linear approximation),has an approximately scale-invariant dependence on k asfirst pointed out in Ref. [5].2 But how is it possible that herewe have an almost scale-invariant spectrum, while there isno sign of scale invariance in �? After all, we said that themost generic perturbation can be described using either �or �. It is straightforward to verify that the scale-invariantterm in the expression for � dominates the extrinsic cur-vature of comoving surfaces. In fact, from the relationbetween �� and � in this gauge [5]

�� �2M2

P_�0

� _��H��; (29)

we obtain the contribution of the first term in Eq. (28) to theextrinsic curvature of the �� � 0 surfaces

Kij H�1

kikja2

� ~k / t�3pk1=2; (30)

which is the same t and k dependence we obtained in theprevious gauge, Eq. (24).3 Notice that the different depen-dence on k of the two variables � and � corresponds to thefact that their relation with physical quantities like curva-tures can involve a different number of derivatives. Thistells us that the scale invariance of a scalar fluctuation hasno well-defined meaning before specifying which variableis relevant for observation. The scale-invariant term inEq. (28) gives a vanishing contribution to the intrinsiccurvature equation (15). This is clear from the explicitrelation between � and �

� �2

3�1� w�1

addt

��

H=a

�; (31)

that is why there is no sign of scale invariance in � . The fact

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CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

that � vanishes for the first term in Eq. (28) could suggestthat we are losing it in going to the variable � , but as wesaw its geometric effect in perturbing the extrinsic curva-ture is completely captured by � . This is also a niceexample of how misleading the description of a solutionas decaying or growing can be; the behavior � t�1�p

would be considered as growing in Newtonian gauge, butwe saw that the same physical fluctuation is decaying in �gauge, � t1�3p.

4The synchronous gauge does not fix completely the repara-metrization invariance and one should get rid of the remaininggauge modes when writing the equations of motion in this gauge.However, the solutions for � are here derived in a differentgauge.

IV. SYNCHRONOUS GAUGE: MAKING THEDYNAMICAL ATTRACTOR MANIFEST

We are now going to discuss the relation between thescalar perturbations produced in the contracting phase andwhat is observed today. The behavior at late times of scalarfluctuations in an expanding phase can be obtained fromour Eqs. (20) and (28) (with generic coefficients), remem-bering that now t is positive and going towards �1. Notethat now the time-independent piece in both � and �dominates each variable, with the time-dependent piecegoing to zero (now p � 2=3 in matter dominance and p �1=2 in radiation dominance).

In the present expanding phase, all the ambiguity de-scribed above about the different intuition in the twogauges and the distinction between growing and decayingmodes is absent. This is because once the fluctuationreenters the horizon all physical quantities depend (apartfrom exponentially small corrections) only on the � � const mode; the concept of growing and decayingmodes has thus a clear physical meaning. To have a viablemodel, the constant mode must have an approximatelyscale-invariant spectrum. Even though the time-independent term of � in Eq. (28) does not have a scale-invariant spectrum in the contracting phase, it has beenargued that the bounce can induce a generic ‘‘mixing’’between the two solutions, so that the scale-invariant spec-trum of the � t�1�p piece before the bounce is inheritedalso by the � � const solution in the expanding phase. It ishowever clear from the discussion above that it is difficultto defend this point of view, as in the �-gauge there is noterm with a scale-invariant spectrum. It is therefore crucialto identify the relevant variable in which this mixing, ifany, happens.

The first issue one should address in studying the evo-lution of the perturbations towards the bounce is the valid-ity of perturbation theory. In Newtonian gauge both � and�� blow up as t�1�p; in the �-gauge, even though �remains finite all the way to the bounce, the g0i compo-nents of the metric Eq. (14) diverge as t�p. These factsindicate that a fully nonlinear description might be re-quired to approach the bounce in these gauges. Note,however, that as t ! 0 the physical curvatures inducedby the perturbations are smaller and smaller with respectto H, which sets the scale of the unperturbed curvatures.Similarly to the inflationary scenario, the Universe be-

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comes closer and closer to the unperturbed solution, asimplied by the existence of an attractor. Thus, it is obvi-ously useful to look for a gauge in which the smallness ofperturbations remains manifest approaching the bounce.

This is achieved in synchronous gauge defined by g00 ��1, g0i � 0. Performing a gauge transformation from the�-gauge and specializing to our scaling background, we getfor the metric and the scalar field4

ds2 � �dt2 � a2�t�

" 1� 2� � 2H

Z t

0dt0

_�H

!�ij

�2

p

@i@j@2

� �O�� k2t2�2p�

#dxidxj (32)

�� � �

������2p

pMP

t

Z t

0dt0

_�H: (33)

These expressions are explicitly derived in Appendix B. Inthe anisotropic piece of the metric only the time-dependentpart of � is relevant as a constant term can be set to zero bya spatial gauge transformation. The terms in brackets areeither constant or decaying in time, making manifest thatperturbations stay small and we are approaching the un-perturbed solution; perturbations are becoming locally un-observable. After a mode goes outside the horizon a localobserver will still feel a residual curvature, with a conse-quent deviation from the unperturbed FRW solution. Acurvature term blueshifts as a�2 t�2p, so that it becomesrapidly irrelevant with respect to H2 t�2; this effect isdescribed by terms going as t2�2p in the solution above. Inthe metric above there are also terms decaying as t1�3p,i.e., slower than curvature terms. They describe the effectof anisotropies, which blueshift as a�6. Their time depen-dence is compatible with the fact that they become irrele-vant as long as the unperturbed solution has w> 1 (so thatits energy density blueshifts faster than a�6) as we areassuming. The physical significance of the terms that wehave sketched will be much more transparent when we willdescribe the full nonlinear solution close to the bounce inthe next section. Finally, also �� goes to zero in this gauge,as t1�3p; again this agrees with the existence of an attractorfor p < 1=3. This shows that the divergence of theNewtonian potential is just a gauge artifact; the deviationsfrom the unperturbed solution are getting small.

We now have a good description of the perturbed metricbefore the bounce, in which linear theory does not breakdown. We can finally address the issue of how perturba-tions evolve through the bounce to the expanding phase.We are not going to specify anything about the transition

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PERTURBATIONS IN BOUNCING COSMOLOGIES: . . . PHYSICAL REVIEW D 71, 063505 (2005)

between the contracting and expanding regimes; our pur-pose is to understand at what level the final predictions areindependent of the details of the bounce. The crucialsimplification comes from the fact that we are interestedonly in modes with huge wavelength compared to theHubble radius before we enter in the unknown bouncingphase. From the expression above, the contribution to anylocal observable O of a given mode with physical wavevector kph is suppressed with respect to the time-independent part of � (which we call �const) by

�O

O �const

�kphH

�1�2p

: (34)

If we assume that the steep potential phase lasts at least 60e-folds, we obtain that the modes of interest have anincredibly small effect for an observer entering the bounc-ing phase. The ratio �kph=H�1�2p is of order e�60 and itdescribes the fact that the local effect of a mode is verysmall in the presence of the dynamical attractor. Moreover,the initial departures from the unperturbed solution whenthe mode leaves the horizon are of order

����p

pHcrossing=MP

[see Eq. (23)]. This gives the size of �const in the equationabove. For the modes of interest this gives a further ex-ponential suppression e�60.

In the concrete proposed models, the scaling solution weare describing ceases to be a good approximation evenbefore we enter the bouncing phase, because the potentialis modified at large negative �. The scaling solution ishowever valid when all cosmologically relevant modes exitthe horizon; this ensures that their physical local effect getsexponentially small.

These considerations lead us to a simple assumption,which makes the predictions of this class of models inde-pendent of the details of the bounce. We assume that theunknown phase is not sensitive to these very tiny effects;every comoving observer goes through exactly the samehistory (whatever it is) apart from exponentially smalldeviations. With this assumption we can neglect all termsgoing to zero in the metric above

ds2 � �dt2 � a2�t��1� 2�const� ~x��d~x2; (35)

where �const� ~x� is the time-independent term of �� ~x; t�. It isimportant to realize that �const� ~x� has an exponentiallysmall local effect as, neglecting gradients, it can be locallyreabsorbed with a rescaling of the spatial coordinates. Nowthe crucial point is that the form of the metric Eq. (35) willhold also in the expanding phase after the bounce, with thesame time-independent rescaling of the spatial coordinates�const� ~x�. Assuming that we have a continuous historythrough the bounce, any observer is going through thesame unperturbed history �H�t�; ��t�� once we have ne-glected the exponentially small corrections. But at eachpoint we have in the metric Eq. (35) a different normaliza-tion of the scale factor �1� �const� ~x��a�t�. This implies that�const� ~x� cannot change as it is locally an integration con-

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stant of the unperturbed solution. The reader could beworried that our conclusions depend on the validity of a4D metric description of the bounce with a continuoussolution for the scale factor a�t�. Actually it is easy torealize that the result holds independently of the unknownphysics of the bounce, even if it cannot be described in a 4-dimensional field theory language. Given an unperturbedsolution, every comoving observer will follow it (apartfrom exponentially small deviations), so that the measuredredshift factor between a given comoving time t before thebounce and a given time after must be the same for allobservers; �const� ~x� therefore cannot be modified in theunknown phase.

Our assumption therefore implies that the time-independent term of � is the same before and after thebounce. The spectrum of density perturbations at horizonreentering ��&=&�� in the present expanding phase is thusgiven by the first term in Eq. (22),

h��&=&�2�i h��i / k�1�2p: (36)

The spectrum is not scale invariant and therefore incom-patible with observations.

We want to stress that our assumption is not trivial. TheHubble constant must change sign during the bounce; onecan therefore envisage a model in which the bounce issufficiently ‘‘slow’’ so that H remains small for a while andall interesting modes come back into the horizon and startoscillating before freezing again [16]. From all we said itshould be clear that whether or not we obtain a scale-invariant spectrum does now depend on the unknownphysics at the bounce and it is not a generic prediction.For example, we saw that � contains no scale-invariantterm, so that it is not possible, without specifying themodel, to state that a generic mixing of modes will givea viable prediction.

Another possibility to get around our conclusion withoutallowing all the modes to come back into the horizon is toassume that the physics at the bounce is sensitive to theexponentially small corrections we have neglected. Thelocal effect of the perturbations, which has been exponen-tially suppressed during the contraction as a consequenceof the attractor behavior, could be amplified and ‘‘resur-rected.’’ This is physically hard to believe and it wouldrequire a certain amount of fine-tuning to pump up atremendously small effect and stop at the desired 10�5

level before entering the nonlinear regime. Anyway alsothis possibility does not generically give a scale-invariantspectrum.

To avoid confusion it is important to underline that ourapproach does not tell us how to evolve a generic scalarperturbation to the expanding phase. A generic fluctuationwill be characterized by two constants associated with thetwo independent modes. Different comoving observerswill follow different histories; for example, at the samevalue of H they will see a different value of the scalar field.How the bounce occurs will depend on these differences

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CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

and to know the final state we must obviously specify thedetails of the bounce. Fortunately the existence of anattractor solution drives every observer to see the samehistory, which is locally indistinguishable from the unper-turbed one: At the same value of H we have the same valueof � and _� apart from exponentially small terms. With thiscrucial simplification the evolution of perturbationsthrough the bounce is fixed by the unperturbed solution.

An approximately scale-invariant 2-point function is notthe only constraint a model of the early Universe mustsatisfy. Present data constrain the non-Gaussianity of theperturbations to be below 10�3 [17] and this also needs tobe explained. In the next section we extend our procedureto the nonlinear case, so that perturbations can be followedto our present expanding phase also at nonlinear level. Ifour assumption of neglecting the exponentially small termsis disregarded, the predictions for the higher moments ofthe perturbations will again depend on the unknown phys-ics at the bounce.

V. NONLINEARLY THROUGH THE BOUNCE

Given our assumptions, we have shown in the last sec-tion that a bouncing model is not compatible with data. Itseems therefore useless to proceed with our analysis at thenonlinear level. We do it for several reasons. First of all, itclarifies the physical meaning of our argument, making itclear that it is not based on any linear approximation.Second, it shows the analogy with the similar problem offollowing perturbations at the nonlinear level in inflation,for example, to calculate the higher moments of the spec-trum. Finally, it is interesting to understand how predic-tions can be made even if the cosmological evolution goesthrough a high curvature and strong coupling regime at thebounce, which at first sight seems to jeopardize any per-turbative approach.

We have already stressed many times that after a modeleaves the horizon its effect becomes rapidly negligible fora local observer, analogously to what happens in inflation.A perturbation induces an intrinsic curvature of comovingsurfaces. This effect blueshifts quite slowly, as a�2, so thatit becomes rapidly irrelevant with respect to the unper-turbed contraction: k2=�a2H2� t2�2p. Once the curvaturehas become negligible, a local observer will see a homo-geneous, flat but anisotropic Universe: Perturbations leav-ing the horizon have left an anisotropic initial condition. Insynchronous gauge, the metric of an anisotropic homoge-neous, flat space can be put in the form

ds2 � �dt2 � a2�t�Xi

e2'i�t�dxidxi;Xi

'i � 0; (37)

where 'i�t� describe the anisotropic expansion. Their evo-lution is simply given by

_' i � cia�3; (38)

and their effect in the Friedmann equation blueshifts as

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a�6,

3M2PH

2 � 3M2P

�_aa

�2� &�

1

2a6�c21 � c22 � c23�: (39)

These equations together with the equation of motion forthe scalar field, ��� 3H _�� V 0��� � 0, are all we need todescribe the anisotropic evolution of the Universe. It isstraightforward to check that Eqs. (32) and (33) are alinearized solution of the equations above. The given non-linear description of an anisotropic Universe has been usedin Refs. [12,18] to show that a contracting phase with w>1 has an attractor towards which nearby solutions flow. Allcurvatures and anisotropies get diluted away, the fluctua-tions in the scalar drop to zero and we locally get back tothe unperturbed solution up to exponentially small terms.In fact, even though it is not easy to get explicit solutionsbecause the perturbation induces also fluctuations in theequation of state w, we see that the qualitative features wegot in linear analysis remain valid. For instance, in Eq. (39)the unperturbed energy density goes as & a�2=p, so thatthe anisotropic terms give a relative correction going likea�6�2=p t2�6p, which becomes irrelevant for p < 1=3.We therefore expect that the unperturbed solution is recov-ered up to corrections that drop off like powers of t1�3p, inclose analogy with the linear case.

Following the same argument we used in the previoussection, we can neglect the exponentially small terms.Although distant regions of the Universe experience thesame unperturbed history, as in the linear case their localmetric is characterized by different, locally unobservable,integration constants. In general these constants describethe rescalings of the three spatial coordinates (along axesthat can rotate from point to point) and thus the metrictakes the form

ds2 � �dt2 � e2�� ~x�e2)ij� ~x�a2�t�dxidxj; )ii � 0:

(40)

The anisotropic piece )ij comes from the fact that we havenot restricted the analysis to scalar perturbations, as weinstead did in the previous sections. Given a generic scalar-tensor initial configuration, going towards the bounce thesystem is driven to the form as Eq. (40) up to exponentiallysmall terms. The prescription to follow the long-wavelength perturbations to the expanding phase is nowclear: As in the linear case, the metric will have the sameform as Eq. (40) after the bounce with the same space-dependent constants. They cannot change between thecontracting and expanding phase because their variationwould imply observable deviations from the unperturbedsolution, while we know that every comoving observer isfollowing the unperturbed history up to exponentiallysmall terms.

In the long-wavelength approximation, i.e., for modeswell outside the horizon, the variables � and ) are non-linearly conserved independently of the phases the

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PERTURBATIONS IN BOUNCING COSMOLOGIES: . . . PHYSICAL REVIEW D 71, 063505 (2005)

Universe goes through. This property is crucial to followthe perturbations created during inflation and to make thenonlinear observables, for instance the 3-point function ofdensity perturbations, independent of the unknown physicsof reheating [4,13]. The same property holds also in ourcase, always assuming that the unknown bouncing phase isnot sensitive to the exponentially small deviations from theunperturbed solution. Notice that the same assumption isimplicitly made in the case of reheating.

VI. CONCLUSIONS

The standard cosmological model has been extremelysuccessful at explaining the evolution of the structures weobserve in the Universe, from the early epochs we measureusing the anisotropies in the cosmic microwave back-ground to the large scale structures in the distribution ofgalaxies we see locally. However, this model requires amechanism to create initial perturbations that correlatepoints with separations larger than the horizon. Such initialseeds will then grow under the action of gravity to form thestructures we observe.

The natural assumption is to postulate that the initialseeds were created during an early phase in the history ofthe Universe and somehow ‘‘stretched’’ outside the hori-zon. The ‘‘stretching’’ of perturbations can be accom-plished in two different ways. One can have a period ofaccelerated expansion in which the length scale of pertur-bations grows rapidly, becoming larger than the horizonwhich evolves only slowly. This is done in inflationarymodels. The other possibility is to have the Universe con-tract slowly so that the horizon shrinks faster than does thescale of perturbations.

One fact is certain, whatever the scenario, acceleratedexpansion or slow contraction, this period has to end andgive rise to the standard hot big bang phase. In inflationaryscenarios the transition epoch is called reheating and couldinvolve rather complicated physics. The beauty of inflationlays on the fact that the predictions of the model areindependent of the details of reheating. The insensitivityto reheating stems from the fact that the wavelengths ofinterest to astronomy were extremely large compared tothe horizon at the time and to the presence of a dynamicalattractor. If the details of reheating were crucial to deter-mining the predictions of inflation the scenario would notbe that appealing.

If perturbations were created during a contracting phase,with the Universe evolving towards a big crunch, theconnection between the contracting phase and subsequenthot big bang phase may seem more problematic. TheUniverse has to go through a bounce with curvaturesdiverging as it approaches. This has been considered prob-lematic by many detractors of such scenarios; in our mind,however, such criticisms are a bit unfair. Just as for infla-tionary models the important point is that their predictions

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are insensitive to the details of reheating, the real questionfor bouncing models is not how the bounce happened butwhether or not one can show that their predictions areindependent of the yet unknown UV physics responsiblefor the bounce. If one could argue that predictions areindependent of the details of the bounce itself then bounc-ing models would be in almost as good a shape as inflation.

In this paper we have studied whether the predictions fordensity perturbations in the ekpyrotic/cyclic scenario arerobust, i.e., independent of the details of the unknownbouncing phase. The presence of a dynamical attractor inthe contracting phase leads to a simple and natural assump-tion, namely, that the physics of the bounce is independentof the exponentially small deviations from the unperturbedsolution present as the bounce approaches. Even thoughseldom stated, this is the exact same assumption one makesin inflationary models about reheating. This assumptionallowed us to calculate all the statistical properties of thefluctuations in the expanding phase. Unfortunately, thespectrum is not scale invariant and thus incompatiblewith the data.

If this assumption is evaded, all predictions (scale in-variance of the spectrum, level of non-Gaussianities, etc.)strongly depend on the physics in the high curvature re-gime of the bounce. In this case, predictions cannot bemade without a full resolution of the singularity. In a senseour conclusions are more pessimistic than just the state-ment that in order to make bouncing models viable oneneeds to understand the details of the bounce. We haveshown that from the perspective of a local observer thebounce has to be exponentially sensitive to the initial state.

In the literature many prescriptions have been proposedto match the two independent modes in the contractingphase to those in the expanding one [6–9,19,20]. Theseprescriptions usually involve variables (like the Newtonianpotential �) which diverge going towards the bounce andthus hide both the perturbativity of fluctuations and theexistence of a dynamical attractor, features which are in-stead manifest in the synchronous gauge. Moreover, asdifferent variables like � and � behave very differentlyin the contracting phase, one is lead to different ‘‘natural’’prescriptions depending on which variable and gauge isused. We think that these mathematical prescriptions arenot based on well motivated physical assumptions, and weconsider the attractor as the only physical guide across thebounce. In fact it should be now clear that prescriptionsgiving a scale-invariant spectrum describe a bouncingphase which is sensitive to exponentially small departuresfrom the unperturbed solution, or in which all relevantmodes come back into the horizon. In both cases resultsare anyway completely dependent on the unknown UVphysics.

For example, some of the proponents of the ekpyrotic/cyclic scenario have developed a matching procedurebased on analytic continuation in the full 5D setup [21]

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CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

(see also Ref. [22]). The result is that � has a jump at thebounce proportional to the comoving energy density per-turbation *m, �� *m=�kphL�2, where L is a UV lengthscale characterizing the physics of the bounce. The vari-able *m goes to zero during the contracting phase as t1�3p

and it is therefore exponentially small at the bounce.Therefore from the 4D perspective this prescription de-scribes a bounce which is exponentially sensitive to theincoming state; *m is in fact multiplied by a hugek-dependent term. This extreme sensitivity to initial con-ditions is clearly problematic; for instance, it amplifies alsoall preexisting anisotropies which were diluted away dur-ing the contracting phase thanks to the presence of theattractor.

We conclude stressing that the situation is slightly differ-ent for pre-big bang scenarios [3]. In this case there is nopotential for the scalar (dilaton) and only its kinetic energyis driving the (Einstein frame) contracting phase; thiscorresponds to the limit p ! 1=3 in our solutions ofSec. II. The kinetic energy blueshifts as a�6, exactly likethe contribution of anisotropies to the Friedmann equa-tion (39). This implies that we are somewhat borderlinewith respect to the existence of the attractor. In fact theanisotropic perturbation in synchronous gauge and thefluctuation in the scalar field, instead of going to zero,diverge logarithmically. The situation now resembles aninflationary cosmology in the presence of isocurvatureperturbations; separated regions of the Universe are char-acterized by a different mixture of two independent modesand therefore go through the bounce in a different waydepending on the relative size of the two solutions. Ourconclusions cannot be applied and it is necessary to specifythe details of the high curvature regime to evolve thefluctuations across the bounce.

ACKNOWLEDGMENTS

We thank Nima Arkani-Hamed, Massimo Porrati,Leonardo Senatore, Toby Wiseman, and especially JustinKhoury and Paul Steinhardt for useful discussions. M. Z. issupported by NSF Grant No. AST 0098606 and by theDavid and Lucille Packard Foundation Fellowship forScience and Engineering and by the Sloan Foundation.

APPENDIX A: THE QUADRATIC ACTION FORSCALAR FLUCTUATIONS

The complete action for gravity and a minimallycoupled scalar field � is

S �1

2

Zd4x

��������g

p�R� �@���2 � 2V����: (A1)

(For notational simplicity we are setting MP � 1; at theend of the computation MP can be restored by dimensionalanalysis.) In order to find an action for the scalar fluctua-tions of this coupled system around a given background it

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is convenient to work in the Arnowitt-Deser-Misner(ADM) formalism. This has been done in Ref. [13], andin this section we briefly review the results derived there. Interms of the ADM variables the metric is

ds2 � �N2dt2 � hij�dxi � Nidt��dxj � Njdt�; (A2)

and the action Eq. (A1) reads

S �1

2

Zd4x

���h

p�NR�3� � 2NV��� � N�1�EijE

ij � E2�

� N�1� _�� Ni@i��2 � Nhij@i�@j��: (A3)

Here R�3� is the intrinsic curvature of spatial slices, whileEij is related to the extrinsic curvature,

Eij �12�

_hij �riNj �rjNi�; Kij � N�1Eij; (A4)

and E � Eii. From the action above it is clear that N and Ni

are just Lagrange multipliers. One can then solve for themthe constraint equations, express them in terms of the otherdegrees of freedom, and substitute their value back in theaction. At this stage it is useful to pick a gauge. We choosethe �-gauge, defined at linear order in the fluctuations by

�� ~x; t� � �0�t�; hij � a2�t��1� 2���ij; (A5)

where �0�t� and a�t� are the background solutions for thescalar and the scale factor, and we have neglected tensormodes. In this gauge the constraint equations read

ri�N�1�Eij � �i

jE�� � 0; (A6)

R�3� � 2V��0� � N�2�EijEij � E2� � _�2

0 � 0; (A7)

whose solution at linear order is

N � 1�_�H;

Ni � @i

��

�

a2H� 0

�; @i@i0 �

_�20

2H2_�:

(A8)

Plugging this into the action above and expanding up tosecond order, one gets a quadratic action for the only scalardegree of freedom � . After integrating by parts and usingthe background equations of motion (i.e., the Friedmannequation and the equation for the scalar), one finally gets

S �1

2

Zdtd3x

_�20

H2 �a3 _�2 � a�@i��

2�

� �1

2

Zd4x

���g

p _�20

H2 �g�1@��@1��; (A9)

where g�1 is the background FRW metric. For the scalingsolution discussed in the text the quantity _�2

0=H2 is a

constant, and the action above reduces simply to Eq. (12).In the alternative gauge Eq. (7), which keeps �� as the

scalar dynamical variable, the result for the action is not assimple as in the �-gauge. This is because the relation

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PERTURBATIONS IN BOUNCING COSMOLOGIES: . . . PHYSICAL REVIEW D 71, 063505 (2005)

between � and �� is in general time dependent, � �

��H= _�0���, so that time derivatives acting on � in theaction above also produce time derivatives of the factorH= _�0. However, for the scaling solution we are interestedin such a factor is time independent, so that again thequadratic action takes the very simple form in Eq. (8).

APPENDIX B: FROM � -GAUGE TOSYNCHRONOUS GAUGE

In this section we explicitly derive the gauge transfor-mation that goes from �-gauge to synchronous gauge. In�-gauge the metric for a general scalar fluctuation is atlinear order (see the previous section)

g00 � �1� 2_�H; (B1)

g0i � �@i

�H

�_�20

2H2

a2

@2_��� �@i�; (B2)

gij � a2�1� 2���ij: (B3)

Under a gauge transformation of parameter 2�, the metrictransforms as g�1 ! g�1 �r�21 �r12�, where the r’sare the covariant derivatives associated to the unperturbedFRW metric. The nonzero Christoffel symbols in a FRWgeometry are

�0ij � a2H�ij; �i

0j � H�ij; (B4)

so that the explicit transformation of the metric compo-nents is

g00 ! g00 � 2 _20; (B5)

g0i ! g0i � _2i � @i20 � 2H2i � g0i � a2ddt

2ia2

� @i20;

(B6)

gij ! gij � @i2j � @j2i � 2a2H�ij20: (B7)

We want to end up in synchronous gauge, which is definedby g00 � �1 and g0i � 0. The explicit form of g00 com-bined with its transformation law tells us that we must

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choose

_2 0 �_�H

) 20 �Z t

0

_�Hdt0: (B8)

To be completely general we should also add to 20 ageneric function of ~x, but it will turn out that it is notnecessary, so we set it to zero.

Then we want to set to zero the space-time componentsg0i. In order to do so 2i must satisfy

ddt

2ia2

�1

a2@i

"��

Z t

0

_�Hdt0#; (B9)

so that

2i � a2@iZ t

0dt0

1

a2

"��

Z t0

0

_�Hdt00

#: (B10)

The last step is to combine Eqs. (B3) and (B7) and read offthe spatial metric in synchronous gauge,

gij ! a2(

1� 2�� 2HZ t

0

_�Hdt0!�ij

� 2@i@jZ t

0dt0"�

_�20

2H2

1

@2_��

1

a2

�H�Z t0

0

_�Hdt00

!#);

(B11)

where we used the definition of �, Eq. (B2). Specializingto our scaling background a / tp, H � p=t, _�0 �

������2p

p=t

we get precisely Eq. (32).Finally we want to express the scalar �� ~x; t� in synchro-

nous gauge. Under a gauge transformation

��x� ! ��x� � @���x� 2�: (B12)

Since in �-gauge the scalar is unperturbed, �� ~x; t� ��0�t�, in synchronous gauge we simply have

�� � �� ~x; t� ��0�t� � _�020 � � _�020

� � _�0

Z t

0

_�Hdt0: (B13)

For our scaling solution this reduces to Eq. (33).

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CREMINELLI, NICOLIS, AND ZALDARRIAGA PHYSICAL REVIEW D 71, 063505 (2005)

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[16] C. Gordon and N. Turok, Phys. Rev. D 67, 123508 (2003).[17] E. Komatsu et al., Astrophys. J. Suppl. Ser. 148, 119

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