Axion perturbation spectra in string cosmologies

10 December 1998

Ž .Physics Letters B 443 1998 97–103

Axion perturbation spectra in string cosmologies

E.J. Copeland a, James E. Lidsey a,b, David Wands c

a Centre for Theoretical Physics, UniÕersity of Sussex, Brighton, BN1 9QJ, UKb Astronomy Centre, UniÕersity of Sussex, Brighton, BN1 9QJ, UK

c School of Computer Science and Mathematics, UniÕersity of Portsmouth, Portsmouth PO1 2EG, UK

Received 18 September 1998Editor: P.V. Landshoff

Abstract

We discuss the semi-classical perturbation spectra produced in the massless fields of the low energy string action in a preŽ .big bang type scenario. Axion fields may possess an almost scale-invariant Dnf0 spectrum on large scales dependent

upon the evolution of the dilaton and moduli fields to which they are coupled. As an example we calculate the spectra forthree axion fields present in a truncated type IIB model and show that they are related with at least one of the fields having a

Ž .scale-invariant or red Dn-0 perturbation spectrum. In the simplest pre big bang scenario this may be inconsistent withthe observed isotropy of the microwave background. More generally the relations between the perturbation spectra in lowenergy string cosmologies should reflect the symmetries of the theory. q 1998 Elsevier Science B.V. All rights reserved.

Superstring theory is at present the best candidatefor a theory uniting gravity with the other fundamen-tal forces. If it is the correct description of ourphysical world, it must have important consequencesfor models of the very early universe. Much theoreti-cal work is currently devoted to building models ofcosmological inflation driven by slow-rolling, self-interacting scalar fields in the context of supergravitymodels. However, there are problems inherent insuch an approach due to the large masses for thescalar fields that are generally introduced by super-

w xgravity terms 1 . A radically different cosmologicalscenario has been proposed by Gasperini andVeneziano based on the low-energy vacuum solu-tions derived from the generic superstring effective

w xaction 2 . Inflation in this pre big bang model isdriven by the kinetic energy of the fast-rolling dila-ton field rather than any interaction potential. A

number of new problems appear in such a scenario,w xmost notably the graceful exit problem 3 . There are

also concerns about the specific initial conditionsw xrequired 4 .

The key test of all these models of the very earlyuniverse is the spectrum of perturbations that theypredict. In conventional slow-roll inflation the onlyperturbation spectra usually generated are the gravi-tational wave background and perturbations in asingle quasi-massless inflaton field. This naturallyproduces a nearly scale-invariant spectrum of adia-batic density perturbations. By contrast there arepotentially many massless scalar fields in a pre bigbang string cosmology which each produce theirown spectrum of perturbations. The fast-rolling dila-ton and moduli fields can only yield a steep blue

w xspectrum 5 . However, it was recently realised thataxion fields, which are always present in the low-en-

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01322-7

( )E.J. Copeland et al.rPhysics Letters B 443 1998 97–10398

w xergy effective action 6 , may have significantly dif-ferent spectral slopes due to their explicit coupling to

w xthe dilaton and moduli fields 7,8 .For instance, the pseudo-scalar axion field, s ,1

whose gradient is dual to the NeveurSchwarz-Ž .NeveurSchwarz NS-NS three-form field strength

w xin four-dimensions 9 , can have a spectral tilt in the'range from 3y2 3 ,y0.46FDn F3, depending1

upon the evolution of the dilaton and moduli fieldsw x7,8 . In highly symmetrical cases the spectrum be-

w xcomes scale-invariant, Dn s0 10 . Durrer et al.1w x11 have noted that such a spectrum may provide anovel scenario for structure formation induced byseed perturbations. The massless axion field canyield an almost scale-invariant spectrum of densityperturbations at horizon crossing whose amplitude isfixed by the string coupling constant at the end ofthe pre big bang era, drrrfew f10y2 . Hence aslightly ‘‘blue’’ spectrum, with Dn )0, may be1

consistent with observations of the microwave back-ground anisotropies. More detailed analyses are re-quired to examine whether this model of structureformation can compete with the successful model

w xbased around conventional inflation 12 , but thisexample emphasises the potential challenge to thestandard picture raised by the pre big bang scenarioas well as the importance of perturbation spectra intesting these ideas.

Thus far, calculations of axion perturbations havebeen restricted to simple axion-dilaton systemsw x7,8,10,13,14 . In this paper we investigate anotherimportant feature of the pre big bang model whichhas received little attention to date. The variousŽ .pseudo– scalar axion fields present in low energyeffective actions have different perturbation spectradue to their different couplings to the dilaton andmoduli fields. In general, however, these numerousfields are coupled to the same dilaton and moduliwhich will lead to distinctive relations between thecorresponding perturbation spectra. As an example,we explore such a relation within the context of atriple axion system derived from the type IIB super-

w xstring reduced to four-dimensions 15 . This modelhas recently been studied in the context of homoge-

w xneous cosmological backgrounds 16 .The type IIB superstring contains a dilaton, a

graviton and a two–form potential in the NS–NSsector of the theory, together with a second two–form

potential and an axion field, s , in the Ramond–3Ž . ŽRamond RR sector. There is also a four–form in

the RR sector, but this can be consistently set to.zero . A 4-D action may be derived by compactify-

ing the 10-D spacetime on a 6-D Ricci-flat internalspace so that

'2 w Ž x . m n yŽ x .r 3 a bds se g x dx dx qe g dX dX ,Ž .˜10 mn ab

1Ž .

where w is the 4-D dilaton and y describes thevolume of the internal space and is the only modulusfield considered. We have included the conformalfactor ew in our definition of the 4-D external metricg in order to work in the 4-D Einstein frame,˜mn

where the dilaton field is minimally coupled to grav-ity. In four dimensions the three-form field strengthsfrom the NS-NS and RR sectors are dual to thegradients of two pseudo-scalar axion fields s and1

s . The third axion field is the RR axion already2

present in the 10-D theory. In this dual formulation itcan be shown that the equations of motion for the

w xfields follow from an effective action 15 :

1 2 21 14 ˜ ˜ ˜'Ss d x yg Ry =w y = y˜ Ž . Ž .H 2 222k

2 21 1' '3 yqw y 3 yqw˜ ˜y e =s y e =sŽ . Ž .3 22 2

21 2 w ˜ ˜y e =s ys =s . 2Ž .ž /1 3 22

where k 2 s8prm2 . This describes a non-linearPl

sigma model in Einstein gravity where the scalarŽ . Ž . w xfields parametrise an SL 3,R rSO 3 coset 15 . The

Ž .global symmetries of the action include the SL 2,Zw x‘S-duality’ of the original 10-D action 17 and a Z2

‘T-duality’ corresponding to invariance under y™w xyy 18 .

We assume that the external four dimensionalspacetime is described by a flat Friedmann-Robert-

Ž .son-Walker FRW metric with the line element

ds2 sa2 h ydh 2 qd dx idx j , 3Ž . Ž .˜ ˜ � 4i j

where h is the conformally invariant time coordinateŽ .and a h is the scale factor. FRW solutions with˜

w xnon-zero spatial curvature can also be found 19,16 .The familiar field equations of general relativityapply in the 4-D Einstein frame and the combinedstress-energy tensor for homogeneous massless fields

( )E.J. Copeland et al.rPhysics Letters B 443 1998 97–103 99

reduces to that for a perfect fluid with a maximallyw xstiff equation of state 20,21 , i.e., with pressure

equal to energy density. This leads to the simplesolution for the scale factor

< <1r2asa h . 4Ž .˜ ˜)

The dilaton-moduli-vacuum solutions are mono-tonic power-law solutions

'w w 3 cos j) < <e se h , 5Ž .

'y y 3 sin j) < <e se h , 6Ž .

where the integration constant j determines the rela-tive rate of change of the effective dilaton andinternal volume respectively. If stable compactifica-tion has occurred and the volume of the internalspace is fixed, we have sinjs0.

In order to understand the perturbation spectraproduced in different fields it is revealing to look atconformally related metrics, g ™V 2 g . If themn mn

conformal factor V 2 is itself homogeneous, thetransformed metric remains a FRW metric but withscale factor a™V a and proper time t™HV dt. Afinite proper time interval in one frame does notnecessarily coincide with a finite proper time inanother frame and, in particular, we shall see thatwhat seems to be a singular evolution in one framemay appear to be non-singular in another frame.

In the original string frame the scale factor evolvesas

.'Ž1q 3 cos j r2w r2 < <a'e asa h , 7Ž .˜)

and there is an accelerated expansion in this frame'for cosj-y1r 3 if h-0. In the Einstein frame

we see that h™0y always corresponds to a collaps-ing universe with a™0. However, for any value of˜j this fulfils one definition of inflation, namely, that

Ž < <y1 < X <˜the comoving Hubble length dardt s ara s˜ ˜ ˜< <. w x2 h decreases with time 22 . A given comoving

scale that starts arbitrarily far within the Hubblescale in either conformal frame at h™y` in-evitably becomes larger than the Hubble scale in thatframe as h™0y. This allows one to produce pertur-bations in the fields on scales much larger than thepresent Hubble scale from quantum fluctuations inflat-spacetime at earlier times.

Because the dilaton and moduli are both mini-mally coupled to the Einstein metric, the field equa-

tions for their linearised scalar perturbations are givenw xby 5,23,24

XX ˜ X 2dw q2hdw qk dws0 , 8Ž .XX ˜ X 2d y q2hd y qk d ys0 , 9Ž .

where a prime denotes differentiation with respect toconformal time, the comoving Hubble rate in the

˜ XEinstein frame is given by h'a ra and k is the˜ ˜Žcomoving wavenumber. Perturbations in the gravita-

w x.tional field obey a similar field equation 25 . Thesingular evolution of the metric as h™0y impliesthat their perturbation spectra grow dramatically onshorter wavelengths that leave the Hubble radiusclose to the singularity. This leads to steep blue

w xspectra with spectral tilt Dns3 5 which leaveseffectively no perturbations in these fields on largeŽ .astronomically observable scales in our presentuniverse.

On the other hand, axion fields’ kinetic termsretain a non-minimal coupling to the dilaton or mod-uli fields in the Einstein frame. This non-minimalcoupling can be eliminated by a conformal transfor-mation to an alternative conformally related metric,which we will refer to as the corresponding axion

w xframe. For the NS-NS axion this is given by 72 wg se g and hence˜Ž1.mn mn

wa 'e a . 10Ž .˜1

The NS-NS axion field is a minimally coupled mass-less scalar field in this frame and thus axionic parti-cles follow null geodesics with respect to this metric.

More generally, for the three axion fields in theŽ .truncated type IIB action given in Eq. 2 we can

define

2 2 2a sV a , 11Ž .˜i i

where the conformal factor

° 2 we for s1

'wy 3 y2 ~e for sV s 12Ž .2i

'wq 3 y¢e for s3

reflects the differing couplings of the axion fields tothe dilaton and moduli in the Lagrangian.

Although conformally related to the string andEinstein frames, the metric ‘‘seen’’ by the axionsmay behave very differently. In terms of conformal


time, the axionic scale factors for the dilaton-Ž . Ž .moduli-vacuum solutions given by Eqs. 5 and 7

evolve as

r qŽ1r2.i< <a sa h , 13Ž .i ) i

where

°'3 cosj for s1

~'r s 143 cos jqpr3 for s Ž .Ž .i 2¢'3 cos jypr3 for sŽ . 3

The proper time in the axion frame is given by

r qŽ3r2.i< <t ' a dh; h , 15Ž .Hi i

so it takes an infinite proper time to reach hs0 forr Fy3r2 and the scalar curvature for the axioni

y2metric, R ; t , vanishes as h™0. However, thesei i

same dilaton-moduli-vacuum solutions then reachh™"` in a finite proper time where R diverges.i

< <Because the conformal factor diverges as h ™0 itstretches out the curvature singularity in the stringmetric into a non-singular evolution in the axionframe. But since V 2 sew ™0 as h™"` the non-singular asymptotic behaviour in this limit in theEinstein or string frames gets compressed into acurvature singularity in the axion frame.

In terms of the proper time in the axion frame wehave

Ž . Ž .1q2 r r 3q2 ri itia sa . 16Ž .i ) i ž /t

) i

For r -y3r2 we have conventional power-lawiŽ .inflation not pole-inflation with lna ;p ln t , wherei i iw Ž .xp s1 q 2r y2 r y3 )1. This has importanti i

consequences for the tilt of the power spectrum ofsemi-classical perturbations in the axion field pro-duced on large scales.

The field equations for the linearised scalar per-w xturbations in the axion fields are 7

XX X 2ds q2h ds qk ds s0 , 17Ž .i i i i

where the comoving Hubble rate in the axion framefor each field is given by

X X Xa a V˜i ih ' s q . 18Ž .i a a V˜i i

The canonically normalised axion field perturbationsw xare given by 26,7

1Õ ' a ds 19Ž .i i i'2 k

Ž .and the equation of motion given in Eq. 17 can bere-written in terms of Õ asi

XXaiXX 2Õ q k y Õ s0 . 20Ž .i iž /ai

w xIn the terminology of Ref. 14 , the pump field S forthe perturbations in each axion field is given by thesquare of the scale factor in the corresponding con-

2formal frame, S sa . After inserting the power-lawi i

solution for the axion frame scale factor given in Eq.Ž .13 , we find that these equations give the generalsolutions

1r2 Ž . Ž .1 2< < < < < <Õ s kh Õ H kh qÕ H kh ,Ž . Ž .i q m y mi i

21Ž .Ž j. < <where H are Hankel functions of order m s r .m i ii

For pre big bang solutions, i.e., h-0, we cannormalise modes on small scales at early times whenall the modes are far inside the Hubble scale, k4

< <y1h . They can be assumed to be in the flat-space-time vacuum 1. Allowing only positive frequencymodes in the flat-spacetime vacuum state at earlytimes requires that

eyi kh

Õ ™ 22Ž .i '2k

w xas kh™y` 27 . This gives

'piŽ2 m q1.p r4iÕ se , Õ s0 23Ž .q y'2 k

and hence we have

'p ykhiŽ2 m q1.p r4 Ž1.ids sk e H ykh .Ž .(i m i2k a

24Ž .Just as in conventional inflation, this produces

perturbations on scales far outside the horizon,

1 It is interesting to note that in conventional inflation we haveto assume that this result for a quantum field in a classicalbackground holds at the Planck scale. Here, however, the normali-sation is done in the zero-curvature limit in the infinite past.


< <y1 yk< h , at late times, h™0 . The power spec-trum for perturbations is conventionally denoted by

k 32< <PP ' d x , 25Ž .d x 22p

Žand thus for modes far outside the horizon ykh™. 20 we have

2 2C m kŽ .i 1y2 m2 iPP s2k ykh , 26Ž . Ž .ds 2i ž /2p a

where the numerical coefficient

2 m iG mŽ .iC m ' 27Ž . Ž .i 3r22 G 3r2Ž .approaches unity for m s3r2.i

The expression for the axion power spectrum canbe written in terms of the field perturbation when

Ž .each mode crosses outside the horizon kh sy1 :c

22C m HŽ .i i2PP s2k , 28Ž .ds i c ž /r q 1r2 2pŽ .i c

3<where H is the Hubble rate in the axion frameci

when kh sy1. This is the power spectrum for ac

massless scalar field during power-law inflation4 <which approaches the famous result PP scds i

2 2Ž .2k H r2p as r ™y3r2, this critical case aris-i c i

ing when the expansion in the axion frame becomesexponential.

The amplitude of the power spectra at the end ofthe pre big bang phase can be written as

22 3y2 m iC m H kŽ .i i2PP s2k ,ds i s ž / ž /r q 1r2 2p kŽ .i ss

29Ž .where k is the comoving wavenumber of the scales

just leaving the Hubble radius at the end of the pre

2 When r s0 the dilaton remains constant and the axion frameand Einstein frame coincide, up to a constant factor. Thus, theaxion spectra behave in the same way as those of the dilaton andmoduli fields. The late time evolution in this case is logarithmic

w xwith respect to y kh 5 .3 ŽThe Hubble rate in the axion frame can be written as H s 1i

y1 ˜ 2. Ž .q2 r V H, where the conformal factor V is given in Eq. 12i i i˜and H is the expansion parameter in the Einstein frame.

4 The factor 2k 2 arises due to our dimensionless definitionof s .i

big bang phase, k h sy1. The subsequent evolu-s s

tion of these perturbations may depend upon theŽ .nature of the exit from the pre big bang h-0 to

Ž . 5the post big bang h)0 . The simplest assumptionis that the modes remain frozen-in on large scalesŽ < < .kh <1 in which case the massless axion pertur-bations contribute an energy density

2 3y2 m2 2 i˜PPk k H kds i s2 2r ;V sC m ,Ž .˜i i i2 2 2 ž /ž /2p ka 2k a˜ ˜ ss

30Ž .

in the Einstein frame. We note that although theamplitude of the perturbations in each axion fielddepends upon the conformal factor V 2, the effectivei

energy density in the Einstein frame for perturba-tions with k;k of a massless axion field ares

independent of V 2. Thus, the amplitude of densityi

perturbations on larger scales depends only upon thespectral tilt. Durrer et al. have noted that a spectrumfor a massless axion, slightly tilted towards smallerscales, may be consistent with the observed ampli-tude of anisotropies in the cosmic microwave back-

2 ˜ 2Ž . Žground with DTrT ; r rr ; k H kr˜i ksa H crit s3y2 m i 2 ˜ 2 w y2. w xk for k H ;e ;10 11 .s s

The spectral tilt of the perturbation spectra isgiven by

dln PPds iDn ' 31Ž .i dlnk

The spectral tilt for each of the fields follows fromŽ .Eq. 26 , and are shown in Fig. 1. They take the

values

' < <Dn s3y2m s3y2 3 cos jyj 32Ž . Ž .i i i

where

0 for s° 1

~ypr3 for sj s 33Ž .2i ¢pr3 for s3

The tilts depend crucially upon the value of m . Thei

spectrum becomes scale-invariant spectrum in the

5 This simple assumption was verified in the specific scenariosw xinvestigated in Ref. 10 .


Fig. 1. Spectral tilts Dn for three axion fields’ perturbationi

spectra in truncated type IIB action as a function of integrationconstant j in pre big bang solutions. The solid line corresponds toDn , the dotted line to Dn and the dashed line to Dn .1 2 3

limit m ™3r2. The lowest possible value of thei

spectral index for any of the axion fields is 3y'2 3 ,y0.46. Requiring conventional power-law

inflation, rather than pole inflation, in the axionŽframe, guarantees a negatively tilted spectrum Dni

. 6-0 .All three spectral indices for the axion fields in

the truncated type IIB model which we have consid-ered are determined by the single integration con-stant j . In particular, we find that one of the axion

Ž .fields always has a red spectrum Dn -0 while theiŽ .other two spectra are blue Dn )0 , except in thei'< <critical case cosj s 3 r2, where two of the spec-

tra are scale-invariant and only one is blue. Thisprovides an example of the important phenomeno-logical role that the RR sector of string theory can

w xplay in cosmological solutions 28 .More generally, the axion perturbation spectra can

have different spectral indices, but in a given stringmodel there is a specific relationship between them.This follows as a direct consequence of the symme-tries of the effective action. These symmetries relatethe coupling parameters between the various fieldsand are manifested in the spectra. Such perturbationspectra could provide distinctive signatures of theearly evolution of our universe. The analysis pre-

6 Note that although the power spectrum for axion perturbationsdiverges on large scales for Dn -0, the energy density for modesi

outsde the horizon is proportional to k 2 PP and this remainsds i

finite.

sented above should be applicable to a wide class ofnon-linear sigma models coupled to gravity. In suchmodels, the couplings between the massless scalarfields are specified by the functional form of thetarget space metric. These couplings determine theappropriate conformal factors analogous to those in

Ž .Eq. 12 that leave the fields minimally coupled andit is the evolution of these couplings that directlydetermine the scale dependence of the perturbationspectra.

Because large symmetry groups which includeŽ .SL 3,R are ubiquitous in supergravity theories ob-

tained from compactification of higher dimensionalw xtheories 29 , our result raises a serious challenge for

the pre big bang scenario. We have shown that atŽ .least one massless axion field in an SL 3,R non-lin-

ear sigma model will have a negatively tilted spec-w xtrum. In the scenario considered by Durrer et al 11 .

the amplitude of density perturbations at horizoncrossing is determined by the string scale at the endof the pre big bang era, drrr;ew. Only a posi-tively tilted spectrum can be consistent with theusual string scale, ew ;10y2 if density perturbationson larger scales are to remain compatible with theisotropy of the microwave background sky. Thisassumes that the axion remains massless. One wouldnaıvely expect that the introduction of a mass for the¨axion field would only make matters worse. Onepossible way out, would be for the axion to developa periodic potential in which case the axion mightcontribute a large fraction of the dark matter in ouruniverse, but the large field fluctuations might lead

w xto only small density fluctuations 30 .

Acknowledgements

We are grateful to Andrew Liddle for usefuldiscussions, and to Gabriele Veneziano for drawingour attention to the problems posed by the negativelytilted axion spectra. EJC and JEL are supported bythe Particle Physics and Astronomy Research Coun-

Ž .cil PPARC , UK.

References

w x1 D.H. Lyth, A. Riotto, hep-phr9807278.w x Ž .2 G. Veneziano, Phys. Lett. B 265 1991 287; M. Gasperini,


Ž .G. Veneziano, Astropart. Phys. 1 1993 317, hep-thr9211021.

w x Ž .3 R. Brustein, G. Veneziano, Phys. Lett. B 329 1994 429,hep-thr9403060; N. Kaloper, R. Madden, K.A. Olive, Nucl.

Ž .Phys. B 452 1995 677, hep-thr9506027; Phys. Lett. B 371Ž .1996 34, hep-thr9510117; R. Easther, K. Maeda, D. Wands,

Ž .Phys. Rev. D 53 1996 4247, hep-thr9509074; R. Easther,Ž .K. Maeda, Phys. Rev. D 54 1996 7572, hep-thr9605173;

M. Gasperini, M. Maggiore, G. Veneziano, Nucl. Phys. BŽ .494 1997 315, hep-thr9611039.

w x Ž .4 M.S. Turner, E.J. Weinberg, Phys. Rev. D 56 1997 4604,hep-thr9705035; J. Maharana, E. Onofri, G. Veneziano,

Ž .J. High. En. Phys. 4 1998 4, hep-thr9802001; D. Clancy,J.E. Lidsey, R. Tavakol, Phys. Rev. D 58, 044017, gr-qcr9802052; N. Kaloper, A. Linde, R. Bousso, hep-thr9801073; A. Buonanno, T. Damour, G. Veneziano, hep-thr9806230.

w x5 R. Brustein, M. Gasperini, M. Giovannini, V.F. Mukhanov,Ž .G. Veneziano, Phys. Rev. D 51 1995 6744, hep-

thr9501066.w x Ž .6 E.S. Fradkin, A.A. Tseytlin, Phys. Lett. B 158 1985 316;

C.G. Callan, D. Friedan, E.J. Martinec, M.J. Perry, Nucl.Ž .Phys. B 262 1985 593; C. Lovelace, Nucl. Phys. B 273

Ž .1986 413.w x7 E.J. Copeland, R. Easther, D. Wands, Phys. Rev. D 56

Ž .1997 874, hep-thr9701082.w x8 E.J. Copeland, J.E. Lidsey, D. Wands, Nucl. Phys. B 506

Ž .1997 407, hep-thr9705050.w x9 A. Shapere, S. Trivedi, F. Wilczek, Mod. Phys. Lett. A 6

Ž . Ž .1991 2677; A. Sen, Mod. Phys. Lett. A 8 1993 2023.w x10 A. Buonanno, K. Meissner, C. Ungarelli, G. Veneziano, J.

Ž .High En. Phys. 1 1998 4, hep-thr9710118.w x11 R. Durrer, M. Gasperini, M. Sakellariadou, G. Veneziano,

gr-qcr9804076; astro-phr9806015.w x Ž .12 A.R. Liddle, D.H. Lyth, Phys. Rep. 231 1993 1.w x Ž .13 R. Brustein, M. Hadad, Phys. Rev. D 57 1998 725, hep-

thr9709022.

w x14 R. Brustein, M. Gasperini, G. Veneziano, Phys. Lett. B 431Ž .1998 277, hep-thr9803018.

w x15 E.J. Copeland, J.E. Lidsey, D. Wands, Phys. Rev. D 58Ž .1998 043503, hep-thr9708153.

w x16 E.J. Copeland, J.E. Lidsey, D. Wands, Phys. Rev. D 57Ž .1998 625, hep-thr9708154.

w x Ž .17 C. Hull, P. Townsend, Nucl. Phys. B 438 1995 109,Ž .hep-thr9410167; J. Maharana, Phys. Lett. B 402 1997 64,

hep-thr9703009; S. Roy, hep-thr9705016.w x Ž .18 K. Meissner, G. Veneziano, Phys. Lett. B 267 1991 33; A.

Ž .Sen, Phys. Lett. B 271 1991 295; J. Maharana, J.H.Ž .Schwarz, Nucl. Phys. B 390 1993 3, hep-thr9207016.

w x Ž .19 E.J. Copeland, A. Lahiri, D. Wands, Phys. Rev. D 50 19944868, hep-thr9406216.

w x Ž .20 R. Tabensky, A.H. Taub, Comm. Math. Phys. 29 1973 61.w x Ž .21 J.P. Mimoso, D. Wands, Phys. Rev. D 51 1995 477,

Ž .gr-qcr9405025; 52 1995 5612, gr-qcr9501039.w x Ž .22 M. Gasperini, G. Veneziano, Mod. Phys. Lett. A 8 1993

3701, hep-thr9309023.w x Ž .23 J. Hwang, Astropart. Phys. 8 1998 201, gr-qcr9612065.w x24 M. Gasperini, M. Giovannini, Class. Quantum Grav. 14

Ž .1997 739, gr-qcr9604002; M. Giovannini, Phys. Rev. D 55Ž .1997 595, hep-thr9610179.

w x25 R. Brustein, M. Gasperini, M. Giovannini, G. Veneziano,Ž .Phys. Lett. B 361 1995 45, hep-thr9507017.

w x Ž .26 V.F. Mukhanov, Sov. Phys. JETP 68 1988 1297.w x27 N.D. Birrel, P.C.W. Davies, Quantum Fields in Curved

Space, Cambridge University Press, 1982.w x Ž .28 A. Lukas, B.A. Ovrut, D. Waldram, Phys. Lett. B 393 1997

Ž .65, hep-thr9608195; Nucl. Phys. B 495 1997 365, hep-Ž .thr9610238; N. Kaloper, Phys. Rev. D 55 1997 3394,

hep-thr9609087; H. Lu, S. Mukherji, C.N. Pope, K.-W. Xu,¨Ž .Phys. Rev. D 55 1997 7926, hep-thr9610107; R. Poppe, S.

Ž .Schwager, Phys. Lett. B 393 1997 51, hep-thr9610166.w x Ž .29 E. Cremmer, B. Julia, Phys. Lett. B 80 1978 48; Nucl.

Ž .Phys. B 159 1979 141.w x30 R. Brustein, M. Hadad, hep-phr9806202.

Documents

Axion perturbation spectra in string cosmologies