Spectra of primordial fluctuations in two-perfect-fluid regular bounces

Fabio Finelli*

INAF/IASF-Bologna, Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna,Istituto Nazionale di Astrofisica, via Gobetti, 101 – I-40129 Bologna – Italy,

INAF/OAB, Osservatorio Astronomico di Bologna, Istituto Nazionale di Astrofisica, via Ranzani 1, 40127 Bologna, Italy,and INFN, Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy

Patrick Peter+

Institut d’Astrophysique de Paris – GReCO, UMR 7095 CNRS, Universite Pierre & Marie Curie,98 bis boulevard Arago, 75014 Paris, France

Nelson Pinto-Neto‡

Centro Brasileiro de Pesquisas Fisicas, rua Dr. Xavier Sigaud 150, Urca 22290-180, Rio de Janeiro, RJ, Brazil(Received 19 September 2007; published 12 May 2008)

We introduce analytic solutions for a class of two components bouncing models, where the bounce is

triggered by a negative energy density perfect fluid. The equation of state of the two components are

constant in time, but otherwise unrelated. By numerically integrating regular equations for scalar

cosmological perturbations, we find that the (would-be) growing mode of the Newtonian potential before

the bounce never matches with the growing mode in the expanding stage. For the particular case of a

negative energy density component with a stiff equation of state we give a detailed analytic study, which is

in complete agreement with the numerical results. We also perform analytic and numerical calculations

for long wavelength tensor perturbations, obtaining that, in most cases of interest, the tensor spectral index

is independent of the negative energy fluid and given by the spectral index of the growing mode in the

contracting stage. We compare our results with previous investigations in the literature.

DOI: 10.1103/PhysRevD.77.103508 PACS numbers: 98.80.�k, 98.80.Cq, 98.80.Es

I. INTRODUCTION

Cosmological models with a bounce [1]—a contractionwhich reverses into an expansion—may solve the horizonproblem [2] in a noninflationary way, i.e. as the pre-bigbang [3] and Ekpyrotic [4] scenarios. In order to makebouncing models real competitors of inflation, or at leastcomplementary to it, while addressing the singularity prob-lem unavoidably present in such models, the spectrum ofdensity perturbations which results from the bounce shouldbe understood as it is in inflationary theories. Unfor-tunately, the physics of cosmological perturbations duringa bounce is much more subtle, because of the reversal ofgrowing and decaying modes before and after the bounce,so that even though the bounce duration itself may be veryshort, usual matching conditions [5,6] should be used withparticular care and verified.

These subtleties have generated many works on thesubject, in particular, after the proposal of the Ekpyroticscenario [4], which is based on a very slow contraction andneeds a bounce, as in the pre-big bang model studied in theEinstein frame. Unanimous conclusions on the resultingspectrum of metric fluctuations in the expanding stage arestill to come. Among the ongoing controversies, one con-

cerned the fate of cosmological perturbations in a hydro-dynamical radiation bounce triggered by a negative energydensity scalar field [7], generalized afterwards in [8]. Thisbounce—and its generalization—has been suggested as asimple toy model, which has the advantage of providinganalytic solutions for the background, although having acomponent which violates the null energy condition. Notethat without assuming spatial curvature and demandinggeneral relativity to hold, such a negative energy compo-nent is required at the level of an effective theory in orderfor a bounce to take place [9].The initial result in this class of two-fluid models was

that the spectrum of the Newtonian potential after thebounce was the same as that of the growing mode beforethe bounce. Such a result was obtained evolving numeri-cally [7] and analytically [7,8] a set of regular equations.This result was later challenged in [10,11], generalized inRef. [12] in which the scalar perturbations are evolvedthrough a bounce characterized by a single physical scale,arguing that the growing mode before the bounce matchedonly with the decaying mode after the bounce, a possibilitywhich has been already found [5,6,13–15]. However, theanalysis of [10,11], demanding the most general possiblesituation (the case at hand in the present work being asubset), needs to rely on a set of singular equations, a factthat could cast doubts on its accuracy had they used themdirectly; these authors, however, obtained the solution inthe form of a Born-like series containing only convergent

*finelli@iasfbo.inaf.it+peter@iap.fr‡nelsonpn@cbpf.br

PHYSICAL REVIEW D 77, 103508 (2008)

1550-7998=2008=77(10)=103508(10) 103508-1 � 2008 The American Physical Society

integrals. Nonsingular equations have also been evolved indifferent contexts, e.g. with a double scalar field bounce[16] or one with a nonlocal dilaton potential stemmingfrom string theory [17]. Note that for tensor modes wealready know that a matching between growing modesbefore and after the bounce occurs [8,18].

In this paper we consequently reanalyze the behavior ofcosmological perturbations during the radiation bounce,obtaining results in agreement with these later studies[10,11] and in contrast with our previous findings for scalarperturbations in [7,8]. In Sec. II we present the backgroundbouncing models containing two perfect fluids with linearand unrelated equations of state. We also show how todescribe the negative energy perfect fluid in terms of aK-essence scalar field. In Sec. III we propose a set ofregular equations for linear perturbations of the abovebackground models, one for the Newtonian potential andthe other for the velocity potential of the fluid responsiblefor the bounce, which can be related to the linear pertur-bations of the K-essence scalar field yielding simplerregular equations suitable for the numerical analysis pre-sented in Sec. IV (another set involving the density contrastinstead of the velocity potential is given in the appendix).In Sec. V we justify some of the numerical results throughan analytical study of approximate solutions and theirmatchings. We end in Sec. VI with discussions andconclusions.

II. BACKGROUND

We shall consider a class of bouncing universes filled bytwo noninteracting perfect fluids with parameters of statewþ, w�, constant in time [8], relating the energy densities�� to the pressures p� through p� ¼ w���. The Einsteinenergy constraint for homogeneous and isotropic solutionsis

H2 ¼�da

adt

�2 ¼ ‘2Pl

��þ

a3ð1þwþÞ ���

a3ð1þw�Þ

�; (1)

with 8�G ¼ M�2Pl ¼ 3‘2Pl, �þ, �� being constants.

Equation (1) is obtained using the background flatFriedmann-Lemaıtre-Robertson-Walker metric

d s2 ¼ a2ð�Þðd�2 � �ijdxidxjÞ; (2)

and by assuming energy conservation for each single fluidseparately in order to make explicit its dependence on thescale factor. We restrict ourselves to w� >wþ. It is clearthat the negative energy density fluid, or, in other words,��, is important only close to the bounce, in agreementwith what is required from a phenomenological model.

By introducing a new coordinate time �,

d � ¼ dt

a�; with � ¼ 3

2ð2wþ � w� þ 1Þ; (3)

we can solve Eq. (1) for the scale factor as

að�Þ ¼ a0

�1þ �2

�20

��; (4)

with

� ¼ 1

3ðw� � wþÞ ; (5)

a0 ¼����þ

��; (6)

�20 ¼4�2

‘2Pl

���2þ

: (7)

Note that the new coordinate time � makes it possible togeneralize the solution, obtained in terms of the usualconformal time found in [8] to arbitrary values of wþ,w�. Note also that this new coordinate time allowed toget general solutions for the scale factor in a universe filledby dust plus dark energy, the latter having an arbitraryconstant equation of state [19]. Finally, it is worth empha-sizing that starting with the metric of Eq. (2), one has twopossible ways of obtaining dimensionful quantities. Wechoose, in what follows, to set the scale factor to haveunits of length, so that both the comoving coordinates xi

and the conformal time � are dimensionless.One can also describe fluids in terms of velocity poten-

tials, provided we start with an action entirely based on thepressure [20] instead of the perhaps more usual energydensity. It is worth emphasizing that these two seeminglydifferent actions lead to exactly identical results if onerestricts attention to the fluid case. In the case of a perfectfluid, the velocity potential action is very simple, andidentical to that for aK-essence Lagrangian [21] simplifiedto L ¼ LðrrÞ (K-essence Lagrangians usually in-

volve an explicit dependence on the scalar field and canalso be used to produce a bounce with no curvature andonly one scalar degree of freedom [22]), namely,

S ¼Z �

� 1

2ðrrÞð1þw�Þ=ð2w�Þ

� ffiffiffiffiffiffiffi�gp

d4x; (8)

where the � sign is chosen according to whether the fluidhas positive or negative energy density. The energy-momentum tensor for reads

T� ¼ ��ð1þ w�Þ

2w�rr�ðr�r�Þð1�w�Þ=ð2w�Þ

� 1

2g�ðr�r�Þð1þw�Þ=ð2w�Þ

�; (9)

and the field equation of motion (stemming from theenergy-momentum conservation) is given by

rrþ ð1� w�Þw�

rr�r�r�g�g��

r�r�¼ 0:

(10)

FABIO FINELLI, PATRICK PETER, AND NELSON PINTO-NETO PHYSICAL REVIEW D 77, 103508 (2008)

103508-2

In the homogeneous background (2), the energy densityand pressure read

��ð0Þ � T0ð0Þ0 ¼ � 1

2w�

�’0

a

�ð1þw�Þ=w�; (11)

p�ð0Þ � �Tið0Þi3

¼ w�T0ð0Þ0 ¼ w���ð0Þ; (12)

and the equation of motion (10) reduces to

’00 þ ð3w� � 1ÞH’0 ¼ 0; (13)

where ’, ��ð0Þ, and p�ð0Þ are the homogeneous parts of ,

��, and p�, respectively. In (11) and the following equa-tions, a prime represents a derivative with respect to theconformal time � of the metric (17).

From the equation for ’ one obtains ’0 ¼ C’=að3w��1Þ,

where C’ is a constant related to �� through

�� ¼ Cð1þw�Þ=w�’

2w�: (14)

Although what we have just shown about the equiva-lence of perfect fluids and K-essence scalar fieldLagrangians with L ¼ LðrrÞ is completely gen-

eral, we shall choose to represent the negative energyperfect fluid by this K-essence scalar field and leave thepositive energy fluid with its original hydrodynamicalrepresentation in terms of the energy density (seeRef. [23] for details) in order to make contact with thenotation of Ref. [7], thus leading to the action,

S ¼ �Z �

1

16�GRþ �þ þ 1

2ðrrÞð1þw�Þ=ð2w�Þ

�

� ffiffiffiffiffiffiffi�gp

d4x; (15)

where R is the curvature scalar that takes into accountgravity. Equation (15) reduces to that of Ref. [7] in thecase where we set w� ¼ 1. Note again that one could usethe pressure instead of the energy density as a starting pointof the hydrodynamical analysis [20]. Both treatments leadto exactly equivalent results and choosing one or the othermerely amounts to choosing the most convenient approachgiven the case at hand.

The full Einstein equations then read

G� ¼ 8�GðTþ� þ T�

�Þ; (16)

where G� is the Einstein tensor, T�� are the energy-

momentum tensors of the positive and negative energyperfect fluid, respectively, with Tþ

� written in the hydro-

dynamical representation, and T�� expressed in terms of

given by Eq. (9).

Regular equations for cosmological perturbations

A set of regular equations is a necessary tool for anumerical analysis of bounce physics. We shall generalize

the treatment of Ref. [7] to the generalized class of bounc-ing models found in Sec. II. The most general form ofscalar metric perturbations on the background given byEq. (4) reads, in the longitudinal gauge,

d s2 ¼ a2ð�Þ½ð1þ 2�Þd�2 � ð1� 2�Þ�ijdxidxj�; (17)

where � is the gauge invariant Bardeen potential [23,24].For the matter fields we have

¼ ’ð�Þ þ �ðx; �Þ and �� ¼ "�ð�Þ þ ���ðx; �Þ;(18)

where �� � �T00�.

From Eq. (9) at first order we obtain

�� � ���"�

¼ 1þ w�w�

��0

’0 ��

�: (19)

One can also check from Eq. (9) that �p� � �Tii=3 ¼

w����.Using Eq. (10) to obtain a linear equation for �, and

the perturbed Einstein equations in order to obtain anequation for the Bardeen potential �, after eliminating��þ, yields the coupled set of regular equations for themodes of wave number k, namely,

�00k þ 3H ð1þ wþÞ�0

k þ�wþk2 þ 2H 0 þ ðH 2 �KÞ

� ð1þ 3wþÞ þ 3

2H 2��F

��k ¼ 3

2H 2��F

�0k

’0 ;

(20)

and

�00k þH ð3w� � 1Þ�0

k þ k2w��k

¼ ð1þ 3w�Þ’0�0k; (21)

which we wrote in full generality by including a possiblynonvanishing curvature of the homogeneous spatial sectionK, and we have set F � ðwþ � w�Þð1þ w�Þ=w�, and�� ¼ "�‘2Pla

2=H 2.

Another way one can write Eq. (20) using the back-ground equations of motion, which will be useful whendiscussing the possible spectra, reads

�00k þ 3H ð1þwþÞ�0

kþ�wþk2þðw�� 1Þ

w�H 0

þ ð1þ 3wþÞðw�� 1Þ2w�

H 2�ð1þ 3wþÞð3w�þ 1Þ2w�

K��k

¼ 3

4w�‘2Plð2w���Þ1=ð1þw�ÞF

�0k

a2: (22)

The advantage of Eqs. (20) and (21) is their general usein bounces with two hydrodynamical fluids, constituting aset of coupled regular equations for numerical analysis ofbounce physics. We should like to use this opportunity tomention the fact that attempts to write down two uncoupled

SPECTRA OF PRIMORDIAL FLUCTUATIONS IN TWO- . . . PHYSICAL REVIEW D 77, 103508 (2008)

103508-3

equations for two separate parts of the Newtonian poten-tial, as suggested in Ref. [7], are incorrect [25–27], as wellas its use [8].

Another set of regular equations for general two-fluidmodels solely in terms of hydrodynamical variables (with-out using the K-essence scalar field to describe the w�perfect fluid) can be obtained using, together with theperturbed Einstein equations, the perturbed energy-momentum tensor conservation equations

�0�k þ ð1þ w�Þð�0�k � 3�0

kÞ ¼ 0; (23)

�0�k þH ð1� 3w�Þ��k � k2�k ¼ w�k2��k

1þ w�; (24)

where ð�uk�Þi ¼ a@i��k=k2 [ð�uk�Þi is the perturbed w�

three-velocity mode], and �� � ���=��. The result reads

�00k þ 3H ð1þ wþÞ�0

k þ�wþk2 þ 2H 0

þ ðH 2 �KÞð1þ 3wþÞ þ 3

2H 2��F

��k

¼ 3

2H 2��F

�0�k þ ð1� 3w�ÞH ��k

k2; (25)

�00�k þH ð1� 3w�Þ�0�k þ ½k2w� þ ð1� 3w�ÞH 0���k

¼ k2ð1þ 3w�Þ�0k; (26)

where F � ðwþ � w�Þð1þ w�Þ=w� and �� ¼��‘2Pla

2=H 2, as before.

The relation between � and � is given by

��k ¼ k2�k

’0 ; (27)

from which the system (20) and (21) can be recovered from(25) and (26) straightforwardly.

III. NUMERICAL RESULTS

The system (20) and (21) written in terms of �k andYk ¼ ‘Pl� which evolve in the variable x ¼ �=�0 for thefamily of bounces described by (4) read

d2�k

dx2þð1þ�þ��Þ 2x

x2þ 1

d�k

dxþ�wþ~k2ð1þ x2Þ2�ð��1Þ

þ 4�ð1� 3�þ��Þð1þ x2Þ2ð2� 3�þ 2��Þ

��k

¼�2ffiffiffi2

p�ð��þ 1Þð1��þ��Þ

2� 3�þ 2��ð1þ x2Þ�ð��3Þ dYk

dx(28)

for the metric perturbation, and

d2Yk

dx2þ ð2þ ��� 3�Þ 2x

x2 þ 1

dYk

dx

þ w�~k2ð1þ x2Þ2�ð��1ÞYk ¼ 2ffiffiffi2

p�

ð1þ x2Þ3��2��� d�k

dx

(29)

with ~k ¼ k�0a��10 for the scalar field part.

In what follows below, we have solved, numerically, theset consisting of Eqs. (28) and (29), setting unnormalizedvacuum initial conditions (the fact that we do not botherabout the normalization here is because we are merelyinterested in the transmitted spectrum), reading

�k;ini ¼ x�3�ð1þwþÞffiffiffiffiffi~k3

p exp

��i

~kffiffiffiffiffiffiffiwþ

p1þ 2�ð�� 1Þ x

2�ð��1Þþ1

�;

Yk;ini ¼ x�ð1�3w�Þffiffiffi~k

p exp

��i

~kffiffiffiffiffiffiffiw�

p1þ 2�ð�� 1Þ x

2�ð��1Þþ1

�:

(30)

Figure 1 shows the time evolution for the spectrumP�=k

nS�1, for that particular case for which the theoreticalvalue for the scalar spectral index nS in the expanding stageis known and given by Eq. (51), i.e. for w� ¼ 1 and wþ ¼10�2. The plots for three different wave numbers show how

-50 0 50 100x

10-16

100

1016

|Φk| /

k n

th

-0.01 -0.005 0 0.005 0.01x

10-5

100

105

1010

|Φk| /

k n t

h

FIG. 1. Example of the time dependence of the Newtonian potential for three different wavelengths (~k ¼ 10�5, 10�6, and 10�7,respectively) as function of x ¼ �=�0. This example, for which w� ¼ 1 and wþ ¼ 10�2, is typical of most cases for which there is aconstant mode, as found in [10,11]. Note that the amplitude at the bounce can be much larger than that of the constant mode thatdominates later. The Bardeen potential is here rescaled by the predicted spectrum, which in this case is given by Eq. (51).

FABIO FINELLI, PATRICK PETER, AND NELSON PINTO-NETO PHYSICAL REVIEW D 77, 103508 (2008)

103508-4

the spectrum for the (growing mode of the) Newtonianpotential in the expanding stage is the one of curvatureperturbations in the contracting stage. Figure 2 shows thesame plot (near the bounce only) for a different value ofw�, namely, w� ¼ 1=4, again rescaled with the theoreticalprediction.

IV. ANALYTIC STUDYOF THE w� ¼ 1 BOUNCINGCOMPONENT CASE

In this section, we will analytically justify the spectra ofthe class of bounces which are driven by negative energystiff matter. Note that for w� ¼ 1 in the equation for theNewtonian potential �k (22), the ‘‘mass’’ term on the left-hand side vanishes for flat spatial sections, K ¼ 0, andinfinite wavelength, k ¼ 0. In this case, one can obtain asolution around the bounce which can be matched with thesolutions far from it to obtain the spectra for large wave-lengths. This section is a generalization of what was donein Ref. [7], which concentrated on the radiation-stiff mattercase only.

The relevant phases in the perturbations evolution

Let us first consider the asymptotic limit � ! �1,

where x / �3ð1�wþÞ=ð1þ3wþÞ and a / �2=ð1þ3wþÞ (the posi-tive energy fluid dominates). Taking into account the initialconditions (30), which in terms of � read

�k;ini / expð�ik�Þ�3ð1þwþÞ=ð1þ3wþÞ

ffiffiffiffiffik3

p ; Yk;ini / expð�ik�Þ�2=ð1þ3wþÞ ffiffiffi

kp ;

(31)

and inserting them into Eqs. (21) and (22), one can see thatthe equations effectively decouple in the far-from-the-bounce limit in the sense that the source terms are negli-

gible provided w� < 73 . Defining the variables uk �

a3ð1þwþÞ=2�k and vk � að3w��1Þ=2�k, one obtains theequations:

u00k þ�wþk2 � 6ð1þ wþÞ

ð1þ 3wþÞ2�2

�uk ¼ 0; (32)

and

v00k þ

�k2 � ðað3w��1Þ=2Þ00

að3w��1Þ=2

�vk ¼ 0: (33)

The solutions of these equations are

uk ¼ ffiffiffiffi�

p ½�ð1ÞHð1Þ� ð ffiffiffiffiffiffiffi

wþp

k�Þ þ�ð2ÞHð2Þ� ð ffiffiffiffiffiffiffi

wþp

k�Þ�; (34)

from which one derives �k, and

�k ¼ að1�3w�Þ=2 ffiffiffiffi�

p ½Xð1ÞHð1Þ ð ffiffiffiffiffiffiffi

w�p

k�Þþ Xð2ÞH

ð2Þ ð ffiffiffiffiffiffiffi

w�p

k�Þ�; (35)

where � ¼ 5þ3wþ2ð1þ3wþÞ and ¼ 3þ3wþ�6w�

2ð1þ3wþÞ . The coefficients

�ðiÞ and XðiÞ are time independent and only depend on k.Restricting now to the case w� ¼ 1, and taking into

account the initial conditions (31), one obtains that �ð1Þ ¼Xð1Þ ¼ 0, �ð2Þ ¼ 1=k, and Xð2Þ ¼ 1. In the region where

k� � 1 (we are considering long wavelengths) but still farfrom the bounce, where the source terms can still beneglected, one has

�<k ¼ A1 þ A2

�ð5þ3wþÞ=ð1þ3wþÞ þOðk2�2Þ; (36)

and

�<k ¼ B1 þ B2

�3ð1�wþÞ=ð1þ3wþÞ þOðk2�2Þ; (37)

where A1 / k3ð1�wþÞ=½2ð1þ3wþÞ�, A2 / kð�7�9wþÞ=½2ð1þ3wþÞ�,B1 / k3ð1�wþÞ=½2ð1þ3wþÞ�, and B2 / k3ðwþ�1Þ=½2ð1þ3wþÞ�.Now we have to propagate this solution through the

bounce and match it with the solution in the expandingphase. During the bounce, the source terms in Eqs. (21) and(22) cannot be neglected but now it is the terms propor-tional to k2 that are negligible. For w� ¼ 1, Eq. (22)simplifies and, upon returning to the variables Y � ‘Pl�and x ¼ �=�0, one obtains

d2�k

dx2þ 8�x

x2 þ 1

d�k

dx¼ �

ffiffiffi2

pð1þ x2Þ

dYk

dx; (38)

d2Yk

dx2þ 2x

x2 þ 1

dYk

dx¼ 8

ffiffiffi2

p�

ð1þ x2Þd�k

dx; (39)

where, as w� ¼ 1, � ¼ 1=½3ð1� wþÞ�. The solutions ofthese equations read

-10 -5 0 5 10 15 20x

10-25

10-20

10-15

10-10

10-5

100

105

1010

|Φk| /

k n th

FIG. 2. Same as Fig. 1 with a different value for wþ, namely,wþ ¼ 1=4. Again, the Bardeen potential is rescaled with thespectrum found by [10,11], which we thus independently con-firm.

SPECTRA OF PRIMORDIAL FLUCTUATIONS IN TWO- . . . PHYSICAL REVIEW D 77, 103508 (2008)

103508-5

�Bounce ¼ ~Aþ ~Bf1ðxÞ þ ~Cf2ðxÞ; (40)

and

YBounce ¼ ~Dþ ~Bf3ðxÞ þ ~Cf4ðxÞ; (41)

with ~A, ~B, ~C, and ~D arbitrary constants. The bouncefunctions fiðxÞ are found to be

f1ðxÞ � x

ð1þ x2Þ4� ; f3ðxÞ � �ffiffiffi2

pð1þ x2Þ4� ; (42)

f2ðxÞ � �8�Z x

d~x

�~x

ð1þ ~x2Þ1þ4�

�ð~x2 þ 1ÞF

�1

2;�4�

þ 1;3

2;�~x2

�þ ~xF

�� 1

2;�4�;

1

2;�~x2

���;

(43)

f4ðxÞ �Z x

d~x

�1

ð1þ ~x2Þ1þ4�F

�� 1

2;�4�;

1

2;�~x2

��;

(44)

where F denotes the hypergeometric function. For x � 1,the solutions can be written as

�Bounce � ~Aþ ~B

x8��1þ

~C

x2

¼ ~Aþ ~B

�ð5þ3wþÞ=ð1þ3wþÞ þ~C

�6ð1�wþÞ=ð1þ3wþÞ ;

(45)

and

YBounce � ~D�ffiffiffi2

p~B

x8�þ

~C

x

¼ ~Dþ ~B

�8=ð1þ3wþÞ þ~C

�3ð1�wþÞ=ð1þ3wþÞ : (46)

These solutions coincide with those obtained in Ref. [7] forwþ ¼ 1=3. One can check that the terms, absent of ourprevious analysis and appearing in Eqs. (45) and (46), arenegligible at the time at which the corresponding modesenter the potential, which is for k� � 1: given the powersin �, their ratios with respect to the other terms are pro-

portional to �3ðwþ�1Þ=ð1þ3wþÞ, which is indeed small, farfrom the bounce (j�j � 1) since wþ <w� ¼ 1. However,their relative influence increases drastically as one ap-proaches the bounce.

If we now compare Eqs. (45) and (46) with Eqs. (36) and

(37), one can obtain that ~A / k3ð1�wþÞ=½2ð1þ3wþÞ�, ~B /kð�7�9wþÞ=½2ð1þ3wþÞ�, ~D / k3ð1�wþÞ=½2ð1þ3wþÞ�, and ~C /k3ðwþ�1Þ=½2ð1þ3wþÞ�. One can also see this by noting thatthe third term in Eq. (45) is the first contribution of thesource term to �, which of course must have the k depen-dence of B2 in (37), while the second term of Eq. (46) is the

first contribution of the source term to �, which musthave the k dependence of A2 in (36).We now have to match Eqs. (45) and (46) with the

solutions in the expanding phase which are far from thebounce and where the source terms are negligible, i.e.,

�>k ¼ �A1 þ�A2

�ð5þ3wþÞ=ð1þ3wþÞ þOðk2�2Þ; (47)

and

�>k ¼ �B1 þ�B2

�3ðwþ�1Þ=ð1þ3wþÞ þOðk2�2Þ: (48)

In this region, the contributions of the terms involving thewave number k are not important, so that the point wherewe match �, � and their first derivatives turns out to beindependent of k. As a consequence, the spectrum ismostly insensitive to the precise value of the matching

point. The result is that �A1 / ~Aþ ~C, and �A2 / ~B, yielding,for the constant part of�, which determines the spectrum,

�A 1 / k3ð1�wþÞ=½2ð1þ3wþÞ� þ k3ðwþ�1Þ=½2ð1þ3wþÞ�

� k3ðwþ�1Þ=½2ð1þ3wþÞ�; (49)

because wþ < 1, and we assume the infrared, long wave-length k � 1 limit (recall k is dimensionless in our con-

-4 -2 0 2 4 6 8 10kη

10-30

10-24

10-18

10-12

10-6

100

k~-n T

℘h(η

)

FIG. 3 (color online). Example of the time dependence of thetensor perturbation for three different wavelengths (~k ¼ 10�5,10�6, and 10�7, respectively) as function of k�. This example,for which w� ¼ 1=3 and wþ ¼ 10�2, is typical, again, and is incomplete agreement with our analytical prediction. All othercases lead to identical figures, except for the actual numbers. Thecases for which the potential does not satisfy the condition thatthe potential has only one extremum at � ¼ 0 cannot becompared with theoretical expectations, since no such expecta-tion was ever obtained analytically, and so are not shown.

FABIO FINELLI, PATRICK PETER, AND NELSON PINTO-NETO PHYSICAL REVIEW D 77, 103508 (2008)

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ventions). Note that, with these parameters and initialconditions, � gets the spectrum of �.

If we now calculate the power spectrum

P � � k3

2�2j�kj2 � ASk

nS�1; (50)

we obtain

nS � 1 ¼ 12wþ1þ 3wþ

; (51)

as obtained in both our improved numerical calculationsand in Refs. [10,11]. This result was also obtained byconsiderations on the matching conditions in [5,6,13],which predict the spectral index in the expanding stageas the one of curvature perturbations in the contractingstage. Note incidentally that it coincides with the spectrumobtained in Ref. [28], where the bounce is not caused by anegative energy stiff matter but by quantum effects: thebackground and the spectrum have the same behavior.

V. GRAVITATIONALWAVES

The equation for the Fourier transforms of the amplitudeof the two polarization degrees of gravitational waves incosmology is

d2hkdt2

þ 3Hdhkdt

þ k2

a2hk ¼ 0: (52)

Once we introduce v � að3��Þ=2h and use the same coor-dinate variable � as introduced in Eq. (2), the above equa-tion becomes

€vkþ�k2a2ð��1Þþð��3Þ

2

€a

a�ð��1Þð��3Þ

4

�_a

a

�2�vk¼0;

(53)

where _f � df=d�. The very existence of an asymptoticvacuum demands the condition 2�ð1� �Þ< 1, or, in otherwords, if wþ >�1=3, which we shall therefore assume.From here on, for convenience, we also define � � 1þ2�ð�� 1Þ> 0. We also restrict towþ < 1 in order to havethe constant mode as the growing mode in the expandingstage.

The equation which we numerically evolve is

d2vk

dx2þ

�~k2ð1þ x2Þ2�ð��1Þ

þ �ð�� 3Þð1þ x2Þ2 f1� ½1þ �ð�� 3Þ�x2g

�vk ¼ 0; (54)

with, as usual, x ¼ �=�0, ~k ¼ ka��10 �0.

Finally, the initial conditions corresponding to the adia-batic vacuum are taken to be

k;ini ¼ffiffiffi3

p‘Plffiffiffik

p expð�ik�Þ ) vk;ini

¼ ffiffiffiffiffiffiffiffi3�0

p‘Pl

x�ð1��Þffiffiffi~k

p expð�ik�Þ; (55)

where k ¼ ahk, and we get rid of the prefactor since weare mostly interested in the spectral index anyway (justlike for the scalar case, the normalization here is essentially

irrelevant). Recall also that k� ¼ ~kx�=�. These equationshave been solved numerically, and some solutions as givenin Fig. 3 for a fixed set of background parameters and threedifferent wavelengths.

Analytic approximations

We are first interested in determining the matching pointbetween the short and long wavelength approximations.The potential in terms of the conformal time is

a00

a¼ a2ð1��Þ

�€a

aþ ð1� �Þ

�_a

a

�2�� Vð�Þ: (56)

Expliciting this in the � variable this is

V ¼ 2�a2ð1��Þ0 fð�Þ; (57)

where

fð�Þ � �f½1þ 2�ð�� 2Þ��2 � �20g� �4�ð��1Þ

0 ð�2 þ �20Þ�2½1þ�ð��1Þ�; (58)

so the matching point at which k2 ja00=aj is

xM ¼� ~kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2�½1þ 2�ð�� 2Þ�p��1=� � 1; (59)

where the last inequality stems from the requirement thatthere is an asymptotic vacuum, i.e. � > 0.The zeros of the first derivative of V are determined by

the equation

�f½2�ð4�� 3� 6��þ 2�þ 2��2Þ þ 1��2þ ½2�ð3� 2�Þ � 3��20g ¼ 0: (60)

We will here treat the simplest case where the potential Vhas only one extremal point, at � ¼ 0, hence imposing thatthe coefficients of �2 and �20 have the same sign.

Asymptotically far from the bounce, Eq. (54) becomes

d2vk

dx2þ

�~k2x4�ð��1Þ � ð1þ Þ

x2

�vk ¼ 0; (61)

where � �ð�� 3Þ; the above equation admits a solutionin terms of the Hankel function, in accordance with thevacuum initial conditions (55):

vk;1 ¼ Affiffiffix

pHð2Þ

�

�~kx��

�; (62)

supposed to be valid up to xM of Eq. (59), with

SPECTRA OF PRIMORDIAL FLUCTUATIONS IN TWO- . . . PHYSICAL REVIEW D 77, 103508 (2008)

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A2 ¼ 3��02�

‘2Ple�i�ð�þ1=2Þ;

and

� � þ 12

�:

Note incidentally at this point that the matching time (59)gives an argument for the Hankel function which does not

depend on ~k. We shall henceforth call Hð2Þ� ð~kx�M=�Þ ¼

Hð2Þ� ð1=�Þ � hM� .On the other hand, for long wavelengths close to the

bounce, Eq. (54) simplifies to

d2vk

dx2þ ½1� ð1þ Þx2�

ð1þ x2Þ2 vk ¼ 0: (63)

In this limit, setting vk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

pu and z ¼ ix, one gets

the Legendre equation

ð1� z2Þ d2u

dz2� 2z

du

dzþ

� ð þ 1Þ � ð1þ Þ2

1� z2

�u ¼ 0;

(64)

which in this case has, as the two independent solutions, apower law and a hypergeometric function. Summarizing,we obtain, in this second regime, the general solution

vk;2 ¼ ð1þ x2Þ� =2

�Bþ Cx2F1

�1

2;� ;

3

2;�x2

��

x�

�Bþ C

ffiffiffiffi�

p�ð� 1

2 � Þ2�ð� Þ

�þ Cx þ1

1þ 2 þ (65)

It is now a simple matter to match the solutions (62) and(65) as well as their derivatives to get

AhM� ~k�1=ð2�Þ ¼ ðBþ C�Þ~k =� þ C

1þ 2 ~kð�1� Þ=�; (66)

and

A

2~k1=ð2�ÞðhM� þ hM��1 � hM�þ1Þ

¼ ð�C�� BÞ ~kð þ1Þ=� þ C

1þ 2 ~k� =�; (67)

where we have set

� �ffiffiffiffi�

p�ð� 1

2 � Þ2�ð� Þ

for notational simplicity. The solution of this system pro-

vides B and C as functions of the reduced wave number ~k,and we shall retain in what follows the leading order terms,

which is, as we are considering wþ < 1, ~kð1þ2 Þ=ð2�Þ ¼~k3ðwþ�1Þ=½2ð3wþþ1Þ�, yielding for h � x ,

h � ~k3ðwþ�1Þ=½2ð3wþþ1Þ�ðconstþ x2 þ1Þ: (68)

The actual gravitational wave spectrum is

P h � 2k3

�2jhj2; (69)

so we end up with

P h / ~knT ; (70)

being

nT ¼ 12wþ1þ 3wþ

¼ 2�

1þ 2�ð�� 1Þ : (71)

It is worth pointing out at this stage that Eq. (71) gives thesame result as in the scalar case [Eq. (51)] for the specificcase that we could study analytically. The reason for suchsimilar results stems from the fact that the dominant termswhich match through the bounces under investigation arethe growing modes of curvature perturbation and gravita-tional waves, both satisfying the same differential equationin the single fluid regime. The above result (71) was al-ready obtained in previous investigations [8,28], althoughfor two different subsets of the family of bounces studiedhere.

VI. CONCLUSIONS

In all the early universe models which aim at solving thehorizon problem with a contraction instead of a super-luminal expansion, a deep understanding of the physicsat the bounce is crucial (and presently lacking in its fullgenerality). What we have shown here is a step towards theunderstanding of cosmological perturbations through abounce triggered by a second perfect fluid (with negativeenergy density), in the framework of flat spatial section andgeneral relativity.We have analyzed in greater detail, both numerically and

analytically, this class of two-perfect-fluid bounces withflat spatial sections, using a completely regular system ofequations, concluding indeed that the constant mode of thescalar gravitational potential after the bounce does notacquire a piece of the growing mode before the bounce.Therefore, our conclusions agree with [10,11], and are incontrast with our previous results for scalar perturbationsin [7,8]. Another important result is the unsensitivity of thescalar spectral index from the peculiarities of the bouncingcomponent in the class of models studied in this paper. Oneinteresting result is that when the negative energy fluid hasa stiff matter equation of state, the background model andthe perturbations have the same behavior as the quantumbouncing cosmological models analyzed in Ref. [28]. Ourresults are interesting for the predictions of cosmologicalalternative models. Whereas by a very slow contraction—as in Ekpyrotic/cyclic model—it seems really difficult togenerate a nearly scale-invariant spectrum of curvatureperturbations without the need of isocurvature perturba-tions or extra-dimensions, a homogeneous dust contraction[13,28] seems in agreement with observations and even

FABIO FINELLI, PATRICK PETER, AND NELSON PINTO-NETO PHYSICAL REVIEW D 77, 103508 (2008)

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free from details due to the bouncing component whichwere left open from previous investigations which focusedon w� ¼ 1 [16,28]. Note, however, as far as completemodel building is concerned, the assumption of homoge-neity may not hold close to the bounce and should thus beverified. This point is, however, out of the scope of thepresent article of which the aim was to concentrate on thepropagation of scalar and tensor perturbations through aregular, although phenomenological, bounce.

We have also performed the analytical and numericalcalculations for gravitational waves. In this case, the con-stant mode of the long wavelength tensor perturbationsafter the bounce do acquire a piece of the growing modebefore the bounce. Also in this case, the slope of the finalspectrum does not depend on the negative energy perfectfluid equation of state. This paves the way to a genericbehavior for tensor perturbations, as such a phenomeno-logical model thus does not suffer from the drawback (stillpresent for the scalar modes) of relying heavily upon thedetails of the bounce physics. Both of these results agreewith the previous investigation [8] for a restricted class ofbounces. This can be understood by noticing that thecrucial time in the evolution of the perturbations is whenthe perturbation wavelength becomes comparable with thecurvature scale of the background, when, for large wave-lengths, the Universe is still far from the bounce and hencethe effects of the negative energy fluid are negligible. Thisresult was already anticipated in Ref. [28].

ACKNOWLEDGMENTS

P. P. and N. P.-N. wish to thank CNPq of Brazil forfinancial support. We also would like to thank CAPES(Brazil) and COFECUB (France) for partial financial sup-port. F. F. is partially supported by INFN BO11 and PD51.We would also like to thank both the Institutd’Astrophysique de Paris and the Centro Brasileiro dePesquisas Fısicas, where part of this work was done, for

warm hospitality and partial support (F. F.). We very grate-fully acknowledge various enlightening conversations withRobert Brandenberger, Jerome Martin, and David Wands.Special thanks are due to Valerio Bozza and GabrieleVeneziano for their careful reading of the manuscript andconstructive remarks. We also would like to thank CAPES(Brazil) and COFECUB (France) for partial financialsupport.

APPENDIX: REGULAR EQUATIONS FORGENERAL TWO-FLUID MODELS IN TERMS OF

HYDRODYNAMICAL VARIABLES

Another possible set of regular equations uses the den-sity contrast �� � ���=�� of the fluid driving the bounceinstead of its velocity potential. The equations are

�00k þ 3H ð1þ wþÞ�0

k þ ðwþk2 þ 2H 0 þH 2

þ 3wþH 2Þ�k

¼ 32H

2����ðwþ � w�Þ (A1)

and

�00� þ ð1� 3w�ÞH�0� þ ½w�k2 � 92H

2ðwþ � w�Þ� ð1þ w�Þ�����

¼ �ð1þ w�Þf½k2ð1þ 3wþÞ þ 3½2H 0

þ ð1þ 3wþÞH 2��k þ 3½2þ 3ðwþ þ w�Þ�H�0kg:

(A2)

We note that with this new set the order of the system oflinear differential equations is increased with respect to thesystems (20) and (21) or (22) and (26): Eq. (A2) is indeedequivalent to a third order differential equation for �k

[see Eq. (19)]. As a result, solving this last set of equationsmay lead to spurious solutions and it is therefore better tostick with Eqs. (20) and (21).

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