PHYSICAL REVIEW D 67, 028301 ~2003!

Comment on ‘‘Density perturbations in the ekpyrotic scenario’’

Jerome Martin and Patrick Peter*Institut d’Astrophysique de Paris, UPR 341, CNRS, 98 boulevard Arago, 75014 Paris, France

Nelson Pinto-Neto†

Centro Brasileiro de Pesquisas Fı´sicas, Rua Dr. Xavier Sigaud 150, Urca 22290-180, Rio de Janeiro, RJ, Brazil

Dominik J. Schwarz‡

Institut fur Theoretische Physik, Technische Universita¨t Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria~Received 25 April 2002; published 15 January 2003!

In the paper by J. Khouryet al., Phys. Rev. D66, 046005~2002!, it is argued that the expected spectrum ofprimordial perturbations should be scale invariant in this scenario. Here we show that, contrary to what isclaimed in that paper, the expected spectrum depends on an arbitrary choice of matching variable. As nounderlying~microphysical! principle exists at the present time that could lift the arbitrariness, we conclude thatthe ekpyrotic scenario is not yet a predictive model.

DOI: 10.1103/PhysRevD.67.028301 PACS number~s!: 98.80.Jk, 98.80.Cq

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I. INTRODUCTION

Recently, a model called the ekpyrotic scenario@1# wasproposed as a possible competitor to the inflationary pdigm @2# on the basis that it could solve the problems of thot big bang model and also produce an almost scale invant spectrum of primordial scalar perturbations. Althoughfoundations of the model are still under debate@3#, we onlyshall comment on the controversial@4–6# claim, crucial forthe ekpyrotic scenario, that the power spectrum is scalevariant @7#.

In Ref. @7#, the complicated five-dimensional evolutionthe background is modeled by an effective four-dimensiospacetime which experiences a singular bounce ath50 (hbeing the conformal time! as the effective scale factor vanishes. The calculation of the power spectrum is a controsial issue as different authors would use different matchconditions@8–10# at the singular bounce. There, Bardeegravitational potential@11# F diverges. The authors of Re@7# suggested to work with the comoving density contrastemwhich is regular ath50 and has two linearly independemodesD and E, such thatem5e0D(h)1e2E(h). At thesingular point the equality of the coefficients before and athe bounce is assumed@7#: e0(01)5e0(02) and e2(01)5e2(02). Although this rule is given without justification~recall that the standard junction conditions follow from tEinstein equations!, we shall refer to such equalities amatching conditions hereafter.

In Ref. @12# we criticized the matching conditions suggested in Ref.@7#. We showed that they lead to an ambiguiand thus an arbitrary choice needs to be made that preday physics cannot make. Reference@7# was subsequentlymodified, see Ref.@13#, to address some critical commenthat have been made on the ekpyrotic model, one of th

*Electronic address: jmartin@iap.fr, peter@iap.fr†Electronic address: nelsonpn@cbpf.br‡Electronic address: dschwarz@hep.itp.tuwien.ac.at

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being the above mentioned ambiguity exhibited in Ref.@12#.In the new version@13#, more details about the matchinconditions are given and one claim has been added~claim 1!:‘‘ One situation of special interest is . . . where the potenis irrelevant at f→2` and there is no radiation in theincoming state. In this case,e2(02)50. In this case, wewould obtain the same final result from any matching ruwhich sete2(01)5Ae2(02), with any constant A.’’ It con-cerns a particular case that we had not considered andwhich our result supposedly does not apply. It shouldnoted, however, that the rest of Ref.@13# inconsistently relieson the general case instead of this particular one, as itthe case in the preceding version@7#. Claim 1 clearly utilizes,in an essential manner, the ideas developed in Ref.@12# andrepresents a tentativea priori response to the points Ref.@12#raised.

References@7,13# also state~claim 2!: ‘‘ The prescriptionis invariant under redefining the independent solutions, eby adding an arbitrary amount of the solution E(h) toD(h). Matching any other non-singular perturbation varable, defined to be an arbitrary linear combination ofem andem8 with coefficients which are non-singular background vaables (defined to possess power series expansions int, asabove) will, with the same prescription of matching the aplitudes of both linearly independent solutions, also yieprecisely the same result.’’ This is an important point in theekpyrotic scenario because, if true, it would support theof the variableem as a tool to define matching conditions.

Below we show that both quoted claims are incorrect.

II. IS THE EKPYROTIC SPECTRUM UNIQUE?

Let us start with claim 1, i.e. the question concerning tinvariance of the spectrum under rescaling of the parame2 and let us first define our notation. The authors of R@13# consider the equation of state close to the singubounce to bev[p/r5v01v1h1v2h21•••. Before thebounce, the kinetic energy of the scalar field dominatesthe dynamics can be approximated by that of a free sc

©2003 The American Physical Society01-1

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COMMENTS PHYSICAL REVIEW D 67, 028301 ~2003!

field for which v051 and v15v250 ~note that one canshow that the values ofv i , ; i .2, are irrelevant for thisdiscussion!. This should apply forh,0 close to the singu-larity (uhu!1) and corresponds precisely to the situation tis assumed in claim 1 and to the case where the equationRef. @12# cannot be applied~as mentioned in that article!,hence the argument of Ref.@13#. Below, we complete thearguments of Ref.@12# to include this particular case anshow that the conclusions of Ref.@12# remain unchanged.

At leading order, the behavior of the scale factor issimple power law,a(h)5,0(2h)1/2/(2A2), in agreementwith the notation of Eq.~24! in Ref. @12#. The comovingHubble rate and the sound speed areH[a8/a51/(2h) andc

S

251. The solution for the Bardeen potential@11,14# in thelong wavelength limit reads

F53

4B1~k!

Ha2

13

4B2~k!

Ha2Eh dt

u2, ~1!

with u[1/(aA11v) and the integral sign stands for thprimitive of the integration kernel. The coefficientsB1 andB2 are functions of the comoving wave numberk, to bedetermined by means of a matching with the initial vacucondition @7,12#. In the case at hand,

F523

,02h2 B11

3

8B2 . ~2!

Note that the equation of motion for the Bardeen potenfor a free scalar field readsF91(3/h)F81k2F50, whichcan be solved exactly in terms of Bessel functions,

F53

~2kh!F1

4B2J1~2kh!1

p

2

k2

,02 B1N1~2kh!G , ~3!

whose expansion in powers ofh permits us to recover thespecial solution~43!, ~44! of Ref. @7# as well as Eq.~2!.

Before the bounce the density contrast on comovingpersurfaces,em522k2F/(3H 2), is given by

em58k2

,02 B12k2h2B2 . ~4!

Comparison of Eq.~4! with Eqs.~43!, ~44! of Ref. @7# yields

e0,~k!5

8k2

,02 B1

, , e2,~k!52k2B2

, , ~5!

showing that the two ‘‘modes’’ decouple. It is interestingcompare with the casev1

,Þ0, for which one has@12# ~herewe assumev2

.50 for the sake of simplicity; this does nochange in any way the conclusion!

e0,~k!5

8k2

,02

B1,2

128 ln 2k2

9v1,2

B2, , ~6!

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-

e2,~k!5

9k2v1,2

8,02

B1,2~112 ln 2!k2B2

, . ~7!

Let us note at this point that, with the choice we made ofnormalization of the integral in Eq.~1!, the limit v1→0 issingular, as can be seen from Eqs.~6!, ~7! and as alreadymentioned in Ref.@12#. Another choice is possible as onefree to add an arbitrary constant amount ofB2 into B1, sim-ply by making the integral a definite one. In particular, ocan arrange that the constant factor (ln 2) in the above eqtions be made to vanish from the outset, and it is easyconvince oneself that such a choice does not modify athing if the correct equations are used.

In the ekpyrotic scenario, and in the long wavelenglimit, the coefficientB1

,}k23/2 is dominant with respect toB2

,}k21/2. This means that the relatione2.9v12e0/64 holds

true if v1Þ0, whereas it does not ifv150. In the lattercase, one hase252,0

2B2e0 /(8B1)}ke0Þ0, see Eq.~5!.The authors of Ref.@13# missed that the limitv1→0 does inno way implye250 exactly, as implied by claim 1, in whichthis value is said to be used to perform the matching. Moover, as demonstrated below, the fact thate2 becomes sub-dominant does not remove the ambiguity noticed in R@12#.

This is also related to the excessive claim in@13# that‘‘ there is no long wavelength contribution toz in the collaps-ing phase.’’ Inserting ~3! into the gauge-invariant variablez[(2/3)(H 21F81F)/(11v)1F ~see e.g.@9,11,14#!, gives

z51

2B2J0~2kh!1

p

4

k2

,02

B1N0~2kh!, ~8!

which, in the limith→0, yields

z;1

2B22

1

2

k2

,02

B1@ ln~2kh!1gE#, ~9!

wheregE

is Euler constant coming from the expansion of tBessel function. Since the second term is singular theterm is subdominant, but that does not imply that it vanishaltogether.

After the singular bounce (h.0), v1.Þ0 and one recov-

ers the relations@12#

e0.~k!52

8k2

,02

B1.2

128 ln 2k2

9v1.2

B2. , ~10!

e2.~k!52

9k2v1.2

8,02

B1.2~112 ln 2!k2B2

. . ~11!

The authors of Ref.@13# propose to glue the epoch after thbounce to the epoch before the bounce by imposing thae0ande2 are the same before and after the singularity. Contrto standard junction conditions, and until some more micphysics is specified, this recipe does not rest on any physprinciple and one may wonder what is the rationale behind

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COMMENTS PHYSICAL REVIEW D 67, 028301 ~2003!

The reason why one is pushed to adopt a new rule incase is linked to the fact that one is dealing with a perturtive approach around a singular background whose meais questionable@4#. Moreover, we argued in Ref.@12# thatthis prescription is ambiguous since an arbitrary rescafactor f (v,k) shows up in the final result. Therefore, thscenario contains an arbitrary function in a physically msurable quantity: until this function can be calculated unabiguously by first principles, the model cannot be falsifie

As in Ref.@12#, we rescalee2 by the completely arbitraryfactor 1/f . This gives the constant part of the gravitationpotential

B2.5B2

,f .

f ,1

9v1.2

8,02

B1, . ~12!

This is the main equation of Sec. II. It represents the equlent of Eq. ~54! of Ref. @12# for the casev1

,50. On theother hand, the decaying mode amplitude reads

B1.52~112 ln 2!B1

,216 ln 2,0

2

9v1.2

B2,

f .

f ,, ~13!

given here for the sake of completeness.Note that if the expansion of the equation of state para

eter is done to higher order inh, all the relations of thissection are only changed by inclusion ofv2 through thereplacement (3/8)v1

2→w(2)[v21(3/8)v12, in agreement

with Ref. @7,13#, while none of thev i for i .2 does contrib-ute.

Equation ~12! shows that the spectrum effectively aquires a scale invariant piece}B1

, , together with an arbi-trary piece}B2

, f ./ f ,. Indeed, as discussed in Ref.@12#, thefunctionsf , and f . can depend on the background consta~e.g.,v) and on the considered scalek in a way which is notyet given from first principles. The choicef .5 f , fixes thekdependence of the spectrum, but, without physical justifition, this remains an arbitrary choice, equivalent to assumscale invariant spectrum from the outset.

III. AN EXAMPLE

Let us now consider claim 2 according to which tchoice ofem is essentially unique. The ambiguity inherentthe method proposed in Refs.@7,13# can be most clearlyexhibited by means of an example suggested at the Econference in Annecy in 2001 by Veneziano@15#. This sec-tion relies completely upon his idea.

In Ref. @13# it is argued that the density contrastem is thequantity of interest because it is finite ath50 and we haveshown above how this quantity is used to propagate the strum through the singularity. However,em is not the onlyfinite, physically relevant, quantity. Based on the methodveloped in Ref.@16#, we could equally well consider and usthe conjugate momentumP to the conserved quantityz @9#which obeys the Hamilton like equations, valid for isentropperturbations,

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-

P5z2z8, P852k2cS

2z2z, ~14!

wherez2[a2(11v)/cS

2. It is worth pointing out that since

the quantityz diverges at the singularity, it may be moruseful to combine Eqs.~14! into the single one

P91@3~cS

22v!22#HP81k2cS2P50. ~15!

In the neighborhood of the singularity,cS

25v51, and the

solutions of Eq. ~15! are 2khJ1(2kh) and 2khN1(2kh), whereJ1 andN1 are Bessel function of the first ansecond kind, respectively. These solutions are compleregular ath50. Note that Eqs.~14! and~15! apply preciselyin the two cases of interest here, namely that of purelydrodynamical perturbations and that of a free scalar field,replacingc

S

2→1 @9# in this last case. The quantityP is re-

lated toem as

P5a2Hem. ~16!

Note also that the use ofP may be argued to be more appropriate in view of the fact that in the regular bounce ca@17#, even though admittedly a different case as the ekpyrmodel, the variableem ends up being an odd function ofh,whereasP is even. In the short-duration limit, that woulimply that em experiences a jump whereasP is continuous.

Sinceua2Hu;,02/16 is finite at the singularityh50, this

relation clearly shows that the variableP, being a linearcombination ofem andem8 ~with vanishing coefficient for thelatter!, satisfies all the requirements of claim 2. Accordingit seems that there is no convincing reason to useem ratherthanP. In the vicinity of the bounce,P reads

P5,0

2s

16 Fe023

4v1e0h1S e22

3w(2)

8e0Dh21•••G ,

~17!

where s521 before the bounce ands511 after thebounce. The proposal of Ref.@7# then would consist of assuming that the coefficients in front of the constant termthe one hand and in front of theh2 term on the other handare the same before and after the bounce. In the precontext, this reduces to

2e0,5e0

. , 2e2,5e2

.23w(2).

8e0

. , ~18!

sincew(2),50 by definition. From these relations, it is eato establish that

e0.52

8k2

,02

B1, , e2

.5k2B2,2

3w(2).k2

,02

B1, . ~19!

Finally, the computation ofB2. shows that the spectrum

no longer contains a scale invariant piece,

B2.52B2

,}k21/2, ~20!

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COMMENTS PHYSICAL REVIEW D 67, 028301 ~2003!

but gives a spectral index equal to 3. It should be noticthat the same calculation goes through, with the same reing spectral index if, in Eq.~16!, one replaces the factora2Hby its magnitudeua2Hu, which is not only regular but alsosymmetric across the bounce. Therefore, the choice ofvariable used to apply the rule proposed in Ref.@7# plays acrucial role. In the absence of an underlying reason to choa variable rather than another at the present stage,P seemsto be as relevant asem, and we are led to the conclusion thclaim 2 is definitely erroneous. Moreover,P is not the onlywell-defined variable that leads to such a conclusion: theran infinite number of such possible combinations.

IV. CONCLUSIONS

To conclude, let us mention yet another possibility.Ref. @10# it was suggested that the condition used in Ref.@7#may be cast in the form of a matching of the energy perbation as seen by an observer comoving with the fluid,that such a matching is ‘‘at least as natural as matching thenergy in the longitudinal gauge.’’ Firstly, one should noticethat no rigorous proof exists that the conditions of Ref.@10#are equivalent to the prescription utilized in Ref.@7#. Sec-ondly, we have shown in Ref.@12# that, in the well knowncase of radiation to matter transition for which one can copare with the exact solution, this matching condition yiean incorrect result. A similar conclusion has been reacheRef. @5# in the case of the reheating transition in which tequation of state also jumps. However, it can be argueda bouncing situation is in no way comparable to a jumpthe equation of state and that the usual matching conditmay then not apply. This is indeed what was found in R@17# for the case of a non-singular bounce. The case forekpyrotic scenario is still more involved, as it was also

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gued that the bounce must be singular if the brane collisioto produce radiation@18#. In any case, matching through aunspecified bounce is still an open question as there isgeometrical or physical argument imposing some particuchoice, and no conclusion can be made about the pospectrum of perturbations in general, unless some nsingular bounce is specified, as is the case of Ref.@17#.

Finally, we would like to insist on the fact that the overagauge-invariant perturbation theory may be completmeaningless when a singularity is reached, since some gtransformations that are admissible in any other situatcould turn out to be singular. In such a situation, a ‘‘gauginvariant’’ variable will, at the singular point, becom‘‘gauge-dependent,’’ and its use rather than the use ofother variable becomes an arbitrary, physically meaninglchoice. In fact, the transformation between the zero sh~conformal Newtonian! slicing and the comoving slicing isindeed singular at the bounce, which actually means thator both slices are unphysical. Lyth@4# has shown that there ino slicing on which the density contrast and the intrincurvature perturbation are finite simultaneously. In suchsituation, linear perturbation theory becomes meaningles

ACKNOWLEDGMENTS

We would like to thank Robert Brandenberger, Ruth Drer, Justin Khoury, David Lyth and Raymond Schutz fmany enlightening discussions. We are especially indebteGabriele Veneziano for proposing the variableP of Sec. IIIand numerous other discussions. N.P.N. would like toknowledge IAP for warm hospitality during the time thwork was being done, and CNRS and CNPq for financsupport. D.J.S. acknowledges financial support from the Atrian Academy of Sciences.

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