Quantum fluctuations in open pre-big bang cosmology

12 November 1998

Ž .Physics Letters B 440 1998 20–27

Quantum fluctuations in open pre-big bang cosmology

A. Ghosh, G. Pollifrone, G. VenezianoTheory DiÕision, CERN, 1211 GeneÕa 23, Switzerland

Received 22 July 1998Editor: R. Gatto

Abstract

Ž .We solve exactly the linear order equations for tensor and scalar perturbations over the homogeneous, isotropic, openpre-big bang model recently discussed by several authors. We find that the parametric amplification of vacuum fluctuationsŽ .i.e. particle production remains negligible throughout the perturbative pre-big bang phase. q 1998 Elsevier Science B.V.All rights reserved.

PACS: 98.80.Cq; 04.30.Db

1. Introduction

The question of whether, in the presence of spa-Ž . w xtial curvature, the pre-big bang PBB scenario 1–3

needs a very large amount of fine-tuning is still aw xsubject of debate 4–10 . Furthermore, Kaloper et al.

w x8 have argued that, even assuming that the twoŽ .classical moduli of the open KKsy1 , homoge-

w xneous, isotropic cosmological solution 11,5 liedeeply inside the perturbative region, the unavoid-able existence of vacuum quantum fluctuations mod-ifies so drastically the classical behaviour as to pre-vent the occurrence of an appreciable amount ofinflation.

In this paper, as a first step towards addressingthis second objection, we carry out a detailed studyof quantum fluctuations around the KKsy1 solu-

w x w xtion of 11,5 . It is well known 12,13 that quantumfluctuations in a non-spatially flat background areconsiderably harder to study than the correspondingones in a flat Universe. Nevertheless, somewhat toour surprise, the corresponding equations can still be

integrated exactly in terms of standard hypergeomet-ric functions. The conclusion is that particle produc-

Ž .tion i.e. the amplification of vacuum fluctuations isstrongly suppressed at very early times because of acancellation between the effect of a non-vanishingHubble parameter and the one of spatial curvature. Inother words, particle production is proportional tothe deviation of the background from its asymptoticMilne form and thus to the time variation of thebackground dilaton. As a result, particle productionremains small through the whole perturbative PBBphase and does not impede the occurrence of PBBinflation.

We will first recall the explicit form of the homo-geneous, isotropic, KKsy1 PBB background weshall be dealing with and derive the general, covari-ant form of the action to second order in the pertur-bations. We then solve, successively, the equationsfor tensor and scalar perturbations. Finally, we dis-cuss the physical implications of our results, andcomment on their possible relevance to the issue

w xraised in Ref. 8 .

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01087-9

( )A. Ghosh et al.rPhysics Letters B 440 1998 20–27 21

2. The background and the second-order action

ŽOur conventions are such that after reduction to. Ž .Ds4 the normalized string-frame action takes the

form

1y1 Ž s. 4 yf'" S s d x yG eH22 lls

= R G qG mnE fE fq . . . , 2.1Ž . Ž .Ž .m n

where G is the string-frame metric, f is themn

Ž .Ds4 dilaton, ll is the fundamental length scales

of string theory and the dots indicate other fieldsŽ .e.g. a Kalb-Ramond axion field that will be set tozero hereafter. The above action allows for classicalhomogeneous, isotropic solutions of the standard

Ž .Friedmann–Robertson–Walker FRW type

dr 22 2 2 2 2ds sa h ydh q qr dV . 2.2Ž . Ž .s 2ž /1yKKr

As usual there are both post- and pre-big bangsolutions coming from a singularity, or going to-wards it, respectively. For KKsy1, the PBB-type

w xsolution was first given in Ref. 11 and then red-w xerived and discussed in Ref. 5 . It reads:

Ž . Ž .' '1q 3 r2 1y 3 r2a h sL coshh ysinhhŽ . Ž . Ž .s

'f h sy 3 ln ytanhh qf , h-0 , 2.3Ž . Ž . Ž .in

where L and f are a dimensional and a dimension-in

less integration constant, respectively.The arbitrariness of L and f reflects the sym-in

metries of the classical problem under a constantshift of the dilaton f and a constant rescaling of themetric G . These are precisely the two parametersmn

Ž w x.to be chosen in an appropriate fine-tuned 4–10range in order to ensure a sufficient amount of PBB

Ž .inflation. Indeed, Eq. 2.3 describes a universe thatŽ . Ž .is almost trivial Milne-like from y`-h-OO y1 ,

Ž y2 .and then inflates with an initial curvature OO LŽ Ž ..and initial coupling OO exp f r2 till it meets,in

eventually, the strong curvature andror strong cou-pling regimes at h;h . The critical value h is1 1

easily determined in terms of the integration con-stants L and f :in

'1q1r 3'f r 3inyh smax e , ll rL . 2.4Ž . Ž .Ž .1 ž /s

w xIt is well known 1 that the study of perturbationsis technically simpler in the so-called Einstein frame,

Ž .defined by g sexp f yf G , and, corre-mn today mn

spondingly, by the action:

11y1 ŽE . 4 mn'" S s d x yg R g y g E fE f ,Ž .Ž .H m n222 llP

2.5Ž .

where f is the present value of the dilaton andtodayŽ .'ll ' 8p G" sexp f r2 ll ;0.1 ll is the pre-todayP s s

sent value of Planck’s length. We will computeperturbations in the Einstein frame and then convertthe results back to the original string frame for aphysical interpretation.

In the Einstein frame the background equationsfor a generic FRW universe are given by 1:

aX

X X 21HH sy f , where HHs6 a

1 X 2 XX X2HH qKKs f , f q2 HHf s0 , 2.6Ž .12

where a prime denotes differentiation with respect tothe conformal time h. For KKsy1 the solution is

Ž .just given by rewriting 2.3 in the Einstein frame:

1r2a h s ll ysinhh coshhŽ . Ž .

'f h sy 3 ln ytanhh qf , h-0 , 2.7Ž . Ž . Ž .in

where the new modulus ll , given by ll 2 s2 Ž .L exp f yf , replaces the string-frame classi-today in

cal modulus L.Ž .To estimate quantum fluctuations around 2.7 weŽ .first go over to isotropic spatial coordinates x, y, z

defined by

y1KK2 2 2 2 2rsR 1q R , where R sx qy qz ,ž /4

2.8Ž .

and by the obvious identification of the angular

1 Although we restrict our attention to the case KKsy1, wewill occasionally keep KK in the formulae for an easy comparisonwith the spatially-flat case.

( )A. Ghosh et al.rPhysics Letters B 440 1998 20–2722

coordinates. In these coordinates the FRW metrictakes the generic form

ds2 sa2 h ydh 2 qg dx idx j ,Ž . Ž .i j

y2KK2where g sd 1q R , i , js1,2,3 ,i j i j ž /4

2.9Ž .and generic perturbations are defined by

g sg Ž0.qd g , fsf Ž0.qdf , 2.10Ž .mn mn mn

Ž .where a superscript 0 denotes the background solu-tion.

Ž .We now consider the form of the action 2.5 upto second-order terms in the fluctuations. The calcu-lations are long but straightforward. After using the

Ž .background Eqs. 2.6 , and after dropping irrelevantŽ .boundary terms total divergences , the result can be

expressed covariantly in the form:

11Ž2. 4 mn a b ls'd Ss d x yg y g g gH 222 llP

= = d g = d g y= d g = d gŽ l bm s na s mn l a b

q2= d g = d g y2= d g = d g .a mn s bl l bm n as

yg mnE dfE dfqg mn g lsE fdf= d gm n l s mn

y2 g mlg nsE fdf= d gl s mn

ml nsy2 g g = E fdfd g , 2.11Ž .s l mn

where, to this order, we can replace g and f bymn

Ž .their background expression 2.7 , and all covariantderivatives are to be evaluated with respect to thebackground metric.

3. Solving the perturbation equations

3.1. Tensor perturbations

Since tensor metric perturbations are automati-cally gauge-invariant, and decouple from dilatonicperturbations, they are easier to study. They can bedefined by

d g ŽT .sdiag 0,a2 h , 3.1Ž .Ž .mn i j

where the symmetric three-tensor h satisfies thei jŽ .transverse-traceless TT conditions

=ih s0, hi s0 , 3.2Ž .i j i

with =i denoting the covariant derivative with re-

Ž . Ž .spect to g . Inserting 3.1 into Eq. 2.11 , and usingi jŽ .2.6 , we easily find:

1Ž2. ŽT . 4 2'd S s d x g aH24 llP

= hX i jhX y=lhi j

= h y2 KKhi jh .ž /i j l i j i j

3.3Ž .

For KKsy1, tensor perturbations h can be ex-i jw xpanded in TT tensor pseudospherical harmonics 14

as

` ln

h h , x s dn h h G x ,Ž . Ž . Ž .Ž .Ý ÝHi j n lm i j lmls2 msyl

3.4Ž .

Ž .nwhere the tensor harmonics G satisfy the eigen-i j lm

value equationn n2 2

= G x sy n q3 G x . 3.5Ž . Ž . Ž . Ž .Ž . Ž .i j i jlm lm

Choosing their normalization so that:Xn n3 'd x h G x G xŽ . Ž . X XŽ . Ž .H i j i jlm l m

sd nynXd X d X , 3.6Ž . Ž .l l m m

Ž . Ž .and inserting 3.4 in 3.3 , we obtain

1Ž2. ŽT . 2d S s dh dn aH24 llP

=2X 2 2h y n q1 h . 3.7Ž . Ž . Ž .Ý nlm nlm

l ,m

Introducing finally the canonical variable

u sah , 3.8Ž .nlm nlm

Ž .and using the background Eqs. 2.6 , we get:

1Ž2. ŽT .d S s dh dnH24 llP

=2X X 212 2u y n q f u ,Ž . Ž .Ý nlm nlm12

l ,m

3.9Ž .

yielding for u the simple equationnlm

XX 1 X 22u q n q f u s0 . 3.10Ž .Ž .nlm nlm12


Ž . Ž .Luckily, for the background 2.7 , Eq. 3.10 can beexactly solved in terms of the standard hypergeomet-

w xric function F' F 15 by2 1

yi nr42u h sC csch 2hŽ . Ž .N 1

=1y in 1y in 2y in

F , , ,4 4 2

2ycsch 2hŽ .

i nr42qC csch 2hŽ .2

=1q in 1q in 2q in

F , , ,4 4 2

2ycsch 2h , 3.11Ž . Ž .

Ž .where N stands for the collection of indices nlmŽ .and C are classically arbitrary integration con-1,2

stants. In order to correctly normalize the tensorŽ .perturbations, the action 3.9 has to be quantized. At

early times, n24f

X 2, and thus u is a free canonicalfield. Hence we impose 2, as h™y`,

2 llPy` yi nhu h ™u h ' e . 3.12Ž . Ž . Ž .N N 'n

w x Ž .Using F a,b,c,0 s1, Eq. 3.12 fixes the integra-'< <tion constants as C s2 ll r n ,C s0. The devia-1 2P

tion from a trivial plane-wave behaviour can easilybe computed from the small argument limit of F.We find

u h suy` h 1qa e4hyi b n , 3.13Ž . Ž . Ž .Ž .N N n

where a ,b are n-dependent constants fixed fromn n

the Taylor expansion of the hypergeometric function.We note that the correction to the vacuum amplitudedies off as e4h, i.e. as ty4 in terms of cosmic timet;yeyh.

2 Actually, both for tensor and scalar perturbations, this normal-ization is correct only for n41. For nF1 there appears to be acorrection due to the non-Cauchy character of constant Milne-time

Ž w x.hypersurfaces see e.g. 16 . We have checked that such a modifi-cation does not affect any of our conclusions, since those dependjust on the former region in n. We are grateful to E. Copeland andD. Wands for pointing out to us this subtlety.

We finally estimate the behaviour of the solutionw xnear the singularity, i.e. for h™0, using 15

G cŽ .2F a,a,c,ycsch 2h ,Ž . 1

G a G aqŽ . Ž .2

=2 a2 aq1 < < < <y2 h ln h .

3.14Ž .< <1r2Then, by virtue of the small h behaviour a, ll h

Ž .and of Eq. 3.8 , we find

2 ll npP< < < <h ,2 coth ln h . 3.15Ž .( (N ž /p ll 2

We shall come back to this result after deriving asimilar expression for scalar perturbations.

3.2. Scalar perturbations

Consider now scalar metric-dilaton perturbationsw xdefined by 12

2w = BiŽS. 2df , d g sya h .Ž .mn = B 2 cg q== Ež /Ž .i i j i j

3.16Ž .Ž . Ž .Inserting 3.16 in Eq. 2.11 , and making use of

Ž .2.6 , we find

1Ž2. ŽS . 4 2 'd S s d x a h gŽ .H22 llP

=2X X X

df y=dfP=dfq6f dfcŽ .

y2wfXdf

X y2fXdf=

2 ByEX y12cX 2Ž .

2 Xy8=wP=cq4 =c y24HHwcŽ .q12 KK w 2 yc 2 q2wcŽ .y8=c

XP=By8 HH =wP=By8 HHw=

2EX

X X X X2 2y8c = E q4 KK ByE = ByE .Ž . Ž .3.17Ž .

Ž .In 3.17 the variables B,w do not have time deriva-tives and thus act as Lagrange multipliers, whichprovide constraints. These are:

0sCC 'fXdfy4c

X y4HHwy4 KK ByEX ,Ž .B

0sCC 'fXdf

X y12 KKwq12 HHcX

w

y4 =2 q3 KK cy4HH =

2 ByEX .Ž . Ž .3.18Ž .


w xFollowing 13 , we introduce the gauge-invariantvariable C by

4X

Cs cqHH ByE , 3.19Ž . Ž .Xf

and, after inserting the constraints, we recast theŽ .action 3.17 in the convenient form

1Ž2. ŽS . 4 2 2'd S s d x a g = q3 KK CŽ .H22 llP

=X2 2E y= q2 HH qKK C . 3.20Ž . Ž .h

One can now make use of the constraints to elimi-Ž X. Ž .nate the variable ByE from the action 3.20 in

terms of w,c and df. The latter variables are notindependent either, being related by a linear combi-nation of the two constraints CC ,C . After its imple-w B

Ž .mentation the action 3.20 contains only true de-grees of freedoms.

In analogy with the case of tensor perturbations,we introduce a canonical field C and expand it asc

` l

C 'aCs dn C h Q x ,Ž . Ž .Ý ÝHc nlm nlmls0 msyl

3.21Ž .

Ž .where Q x are the scalar pseudospherical har-nlmw xmonics, satisfying 14 :

=2 Q x sy n2 q1 Q x ,Ž . Ž . Ž .nlm nlm

X3X X X X X'd x g Q x Q x sd nyn d d .Ž . Ž . Ž .H nlm n l m ll m m

3.22Ž .

Ž .As a result, 3.20 becomes

1Ž2. ŽS .d S s dh dnH22 llP

=2X X 212 2C y n y f C ,Ž . Ž .N N4

Ns nlm , 3.23Ž . Ž .2'where C ' n q4 C . The quantity C enters theN N N

action in a canonical way and therefore its vacuumfluctuations, like those of u, are easily normalized.The equation for C is simplyN

XX X 212C q n y f C s0 , 3.24Ž .Ž .N N4

so that we must impose, as h™y`,

llPy` yi nhC h ™C h ' e ,Ž . Ž .N N 'n

'ny` yi nhP h ™P h 'yi e . 3.25Ž . Ž . Ž .N N llP

As was the case for tensor perturbations, also Eq.Ž . Ž Ž ..3.24 can be transformed for the background 2.6into a hypergeometric equation. We find, specifi-cally,

& yi nr42C h sC csch 2hŽ . Ž .N 1

=y1y in 3y in 2y in

F , , ,4 4 2

& i nr42 2ycsch 2h qC csch 2hŽ . Ž .2

=y1q in 3q in 2q in

F , , ,4 4 2

2ycsch 2h , 3.26Ž . Ž .&< <where, as before, we have to take C s ll r1 P&'n ,C s0. Corrections to the free plane wave can be2

easily computed and, again, are suppressed by fourpowers of 1rt:

˜y` 4hyi b nC h sC h 1qa e , 3.27Ž . Ž . Ž .˜Ž .N N n

y` ˜Ž .where C is given by 3.25 and a ,b are n-de-˜N n n

pendent constants fixed from the expansion of thehypergeometric function.

Ž .To estimate the behaviour of 3.26 near h,0,w xwe use the formula 15

wF a,aq1,c,

G cŽ .2ycsch 2h ,Ž .

G aq1 G cyaŽ . Ž .2 aq3= y2 a aycq1Ž .

2Ž aq1. 2 a2 a< < < < < <= h ln h q2 h , 3.28Ž .and obtain:

2n q1 np< <C , ll coth( (N P ž /2p 2

=2

3r2 y1r2< < < < < <y h ln h q h . 3.29Ž .2ž /n q1


4. Discussion

In order to discuss the physical significance of ourresults it is useful to choose a convenient gauge. In

w xthe spatially flat case it was found 17 that thew xso-called off-diagonal gauge 18,17 was particularly

useful in order to suppress the large gauge artifactsw xpresent in the more commonly used 12 longitudinal

gauge. The off-diagonal gauge is defined by settingŽ .csEs0 in Eq. 3.16 . We shall now see how one

can reconstruct the scalar field fluctuation from C inthis gauge.

We first note that, in this gauge, the variables C

Ž .and B are related through 3.19 as:

4HHBCs . 4.1Ž .X

f

Ž . Ž .Using Eq. 3.24 for C , as well as 4.1 , we canN

derive the evolution equation for B:

4 KKXX X2B y= Bq 2 HHy Bž /HH

y 4HH 2 q12 KK Bs0 , 4.2Ž . Ž .w xwhich agrees with Ref. 17 for KKs0. To relate df

and C we first observe that the first of the twoŽ .constraints 3.18 provides the relation

fXdfs4 HHwqKKB , 4.3Ž . Ž .

Ž .while, eliminating df from the two constraints 3.18Ž .and using 4.2 , we arrive at a second relation

wsBX q2 HHB . 4.4Ž .Ž . Ž . Ž .Combining 4.4 and 4.3 , and making use of 4.1 ,

we are finally able to express df directly in terms ofC as

KKyHHX

XdfsC q C , 4.5Ž .

HH

implying that df represents, in this gauge, a gauge-invariant object.

It is instructive to compare the KKsy1 casewith the spatially flat one, where the relevant gauge-invariant variable, given by

HŽgi.c scq df , 4.6Ž .X

f

becomes df itself in the off-diagonal gauge. Thecanonical field, given by Õsadf, satisfies the

w xwell-known equation 12 :

zXX afX

XX 2Õ q n y Õs0, where zs . 4.7Ž .ž /z HH

Even in the presence of spatial curvature, the field Õ

still plays the role of the canonical field in the farpast, when h is large and negative. This can bechecked by computing the equation of motion for Õ

in the presence of curvature. The explicit form of theequation for Õ is given by

ÕXX qA Õ

X qA Õs0,1 2

y123 KKX 22 3 2A syKK f HH n yKKq ,1 2ž /HH

fX 2 12 KK 3 KK

2A sn q 1y y HHq A .2 12 ž /ž /12 HHHH

4.8Ž .

Thus, as long as we are interested in the early-timeregime, A is exponentially small, A ™n2, and Õ1 2

can be treated as the canonical field.Ž .Using Eq. 4.5 , the behaviour of Õ in the far past

follows directly from that of C , given in Eqs.NŽ . Ž .3.25 , 3.27 :

ll 2y inPy` yi nhÕ h ' e ,Ž . (' 2q inn

'n 2q iny` yi nhp h 'yi e , 4.9Ž . Ž .(Õ ll 2y inP

with corrections again suppressed as ty4 , i.e.

ˆy` 4hyi b nÕ h sÕ h 1qa e , 4.10Ž . Ž . Ž .ˆŽ .n

ˆwhere a ,b are n-dependent constants.ˆ n n

We can study how other variables behave nearh,0 by using their relation to C in this gauge and

Ž .the behaviour of C , Eq. 3.29 . We easily find:

2ll n q1 coth npr2Ž .P< <B , ( (N 2ll 2p n q4

=2 y1< < < < < <y h ln h q h , 4.11Ž .2ž /n q1


while

2ll n q1 coth npr2Ž .P< < < <df , ln h .( (N 2ll 2p n q4

4.12Ž .

Let us finally compare the energy contained in thequantum fluctuations of the dilaton and that in theclassical solution near the singularity. Note that the

Ž .expansion 3.28 can be trusted only up to some< <maximum n for which 1<n ;1r h . Conse-max

quently, the ratio of the kinetic energy densities near< < Ž Ž ..h ,0 up to constant prefactors of OO 1 becomes

2X3 2 2'd x g a dfŽ .H nEE ll dnmaxQ P 3s , n .H2EE nX 2 ll3 2C 'd x g a fH4.13Ž .

We can express the above result in terms of theŽ .value of the physical Hubble parameter H h 'HHra

Ž .at horizon crossing of the scale n, H n , which isHC

easily computed as

13r2H n ; h;1rn ;n rll . 4.14Ž . Ž . Ž .HC

ha

Ž .Thus 4.13 takes the suggestive form

nEE dnmaxQ 2 2s ll H n . 4.15Ž . Ž .H HCPEE nC

In general, in order to draw physical conclusions,we should transform back the results to the stringframe. However, in our case, this is hardly neces-sary. Concerning the importance of vacuum fluctua-

Ž .tions as h™0, we observe that the final result 4.15expresses the relative importance of quantum andclassical fluctuations near the singularity in terms ofa frame-independent quantity, the ratio of the effec-tive Planck length to the size of the horizon. Since,by definition of the perturbative dilaton phase, theHubble radius is always larger than the string scale,we find that the relative importance of quantumfluctuations is always bounded by the ratio ll rllP s

which is always less than one in the perturbativephase.

Let us now come to the more subtle issue of thefar-past behaviour of tensor and scalar quantum fluc-tuations. Computations may be done in either frame,since the dilaton is approximately constant in the far

Ž . Ž .past. Our results, expressed in Eqs. 3.13 and 4.10 ,show that corrections to the trivial quantum fluctua-tions are of relative order e4h ; ty4 , i.e. of order ty3

Ž .relative to the homogeneous classical perturbation.This suggests that quantum effects do not modifyappreciably classical behaviour in the far past, in

w xcontrast to the claim made in Ref. 8 . This attitude isalso supported by the structure of the superstring

Žone-loop effective-action which is well-defined.thanks to the string cutoff . Because of supersymme-

try, neither a cosmological term nor a renormaliza-tion of Newton’s constant are generated at one-loop,but only terms containing at least four derivatives.As a result, quantum corrections to early-time classi-cal behaviour are of relative order ty6 , i.e just like

Ž X X .2our corrections df rf . Note, incidentally, thatgenerating a cosmological constant by quantum cor-rections would upset completely the whole PBBscenario.

w xWe also see, however, that, as claimed in Ref. 8 ,Ž . Žthe leading free-theory fluctuations the 1’s in Eqs.

Ž . Ž ..3.13 and 4.10 dominate over the homogeneousclassical perturbation by one power of t. If taken atface value, they upset classical behaviour at early-

< < 2 w xenough times, t ) ll rll 8 . The answer to thePw xissue raised in Ref. 8 thus appears to depend on

Ž .whether zero-point, non-amplified vacuum quan-Ž .tum fluctuations in the trivial Milne background

can give physically important effects on the scale ofMilne’s Hubble radius Hy1 ; t. A complete clarifi-cation of this point would be certainly desirable.

We stress however that, irrespectively of the finalanswer to this issue, vacuum fluctuations have thesame time dependence as the typical inhomogeneous

w xclassical perturbation discussed in Refs. 7,10 , butmuch smaller amplitudes. Indeed, an initial classical

Žstate apt to give rise to a pre-big bang event i.e. to.gravitational collapse in the Einstein frame in a

region of space of size ll 4 ll must correspond,i n P

quantum mechanically, to having parametrically largew xoccupation numbers in certain quantum states 10 .

Such a quasi-classical configuration cannot be appre-Ž .ciably affected by quantum fluctuations OO 1 in

those occupation numbers.


Acknowledgements

We wish to thank Maurizio Gasperini, MassimoGiovannini and Slava Mukhanov for helpful discus-sions. We also thank Nemanja Kaloper and AndreiLinde for discussions and correspondence whichhelped understanding the relation of this work totheirs. The work of AG was supported in part byWorld Laboratory. The work of GP was supported inpart by the Angelo Della Riccia Foundation.

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