M.Gasperini. Dipartimento di Fisica Teorica, Via P.Giuria 1, Turin, Italy, and INFN, Sezione di Torino, Turin, Italy. and. G.

Transcription

1 CERN-TH.778/94 Dilton Production in String Cosmology M.Gsperini Diprtimento di Fisic Teoric, Vi P.Giuri, 05 Turin, Itly, nd INFN, Sezione di Torino, Turin, Itly nd G.Venezino Theory Division, CERN, Genev, Switzerlnd Abstrct We consider the coupled evolution of density, (sclr) metric nd dilton perturbtions in the trnsition from \stringy" phse of growing curvture nd grvittionl coupling to the stndrd rdition-dominted cosmology. We show tht dilton production, with spectrum tilted towrds lrge frequencies, emerges s generl property of this scenrio. We discuss the frme-independence of the dilton spectrum nd of the intionry properties of the metric bckground by using, s model of source, pressureless gs of wekly intercting strings, which is shown to provide n pproximte but consistent solution to the full system of bckground equtions nd string equtions of motion. We combine vrious cosmologicl bounds on growing dilton spectrum with the bound on the dilton mss obtined from tests of the equivlence principle, nd we nd llowed windows comptible with universe tht is dominted, t present, by relic bckground of diltonic drk mtter. CERN-TH.778/94 Februry 994

2 . Introduction nd motivtions It is well known tht uctutions of the metric bckground re mplied in the context of intionry cosmologies, nd tht the mpliction of their trnsverse, trceless (spin ) component cn be interpreted s grviton production []. Models of the erly universe bsed on the low-energy string eective ction (wht we shll refer to, for short, s \string cosmology"), re chrcterized by the dditionl presence of dilton bckground, (t). It is nturl to expect n mpliction of the uctutions of the dilton bckground { with corresponding dilton production { to ccompny tht of the metric for suitble time-evolution of the cosmologicl elds. In this pper we discuss such dilton production in the context of the soclled \pre-big-bng" scenrio [], chrcterized by n ccelerted evolution from t, cold nd wekly coupled initil regime to nl hot, highly curved, strong coupling regime, mrking the beginning of the stndrd \post-big-bng" decelerted FRW cosmology. With this bckground, the spectrum of the produced diltons tends to grow with frequency, just s in the (previously discussed [,3]) cse of grviton production. As we shll see in Sec.3, the high-frequency prt of the spectrl distribution,, of the dilton energy density cn be prmetrized (in units of criticl density c = H =G = Mp H ) s (!; t) =! d! c d! ' GH H 4 ; > 0;! <! (:)! H where M p ' 0 9 GeV is the Plnck mss. Here H is the curvture scle evluted t the time t mrking the end of the intionry epoch (which we ssume to coincide with the beginning of rdition dominnce);! = H = is the mximum mplied proper frequency; H = _=, where, s usul, is the scle fctor of the bckground metric. The integrl over! is thus dominted by the highest frequency!, (t) = Z! d!! (!; t) ' GH H H 4 (:) nd the condition <, required to void tht the diltons overclose the universe in the rdition-dominted er ( H?= ), yields the constrint H < Mp, lredy known [,3] from the grviton spectrum. The produced diltons, however, cnnot be mssless. Lrge-distnce dilton couplings cn be estimted [4] nd turn out to be t lest of grvittionl strength. This violtes the universlity of grvity t low energy nd, in prticulr, induces corrections to the eective Newton potentil (in the sttic, wek-eld limit), which

3 my be reconciled with the present tests of the equivlence principle [5] only for dilton mss stisfying [4,6]: m > m 0 0?4 ev (:3) The expression (.) is thus vlid only until the energy density remins dominted by the reltivistic modes, with!(t) > m. But, t the present time t 0 (with H 0 0?6 M p ), the mximum frequency! is! (t 0 ) = H 0 ' 0?4 H M p = ev (:4) As H < M p, even the highest mode must then become non-reltivistic before the present epoch, becuse of the constrint (.3). At the scle H(t m ) = m the modes with!(t m ) < m begin to oscillte coherently, with frequency m, nd when they re dominnt the dilton energy density becomes non-reltivistic, with? H m (t) ' GmH H H 3 ; 0 (t) ' GmH H H 3 ; (:5) (the dependence on the spectrl index disppers for fst enough growth of the spectrum, s shown in Sec.5). Such frction of criticl density grows in time during the rdition er, while in the mtter er ( H?=3 ) it becomes xed t the mximum constnt vlue =?3 ' Gm H m ; 0 H H ' GmH H H = ; (:6) where H 0 6 H 0 is the curvture scle t the mtter{rdition trnsition. The condition then provides, for ny given intion scle H, n upper limit for the dilton mss, m <? H M 4 p H?4 =(+) ; 0 m < H = M p H?3= ; (:7)

4 vlid for m < H under the ssumption tht its lifetime is suciently long to rech the mtter-dominted er (if m > H the dilton must decy before becoming dominnt with respect to the rdition, s we shll see in Sec.5, nd the criticl density bound cnnot be pplied). In ddition to the constrint (.7), which is n unvoidble consequence of the quntum uctutions of the dilton bckground, one should lso consider, in generl, the constrints following from possible clssicl oscilltions of the bckground round the minimum of the potentil [7]. The initil mplitude of such oscilltions is, however, to lrge extent model-dependent nd, s discussed in Sec.5, we shll work under the ssumption tht clssicl oscilltions re negligible with respect to the quntum uctutions mplied by the cosmologicl evolution. This ssumption will give us the mximum llowed region in prmeter spce. In the bsence of clssicl oscilltions, the upper limit on m obtined from eq. (.7), combined with the lower limit (.3), denes for ech vlue of nd H n llowed window for the dilton mss such tht, ner the upper end of the window, the produced diltons cn close the universe. Such dilton dominnce, however, cn lst only until the energy density in diltons is dissipted into rdition, which occurs t decy scle H d '? d ' m3 M p (:8) We re thus led to the rst interesting result of this nlysis. If > 0:, the upper limit (.7) turns out to be lrger thn the lower bound (.3) even for intion scles H 0?5 M p. Moreover, lwys for H 0?5 M p, the lrgest vlue of m llowed by eq. (.7) is m ' 00 MeV (obtined for ), nd it implies H d < H0. This mens tht, for fst enough growing spectr, nd \relistic" (t lest in string cosmology context) intion scles H 0?5 M p, we cn be left tody with bckground of relic diltons possibly representing signicnt frction of the drk-mtter bckground [8]. The llowed rnges of m corresponding to this interesting possibility lie round the upper limits given in (.7) nd thus depend on nd H in complicted wy. For 0: < < 0:7 the rnge of H for which this possibility cn be relized is given by H < 0?(3?3)=(4?) M p ; (:9) while, for > 0:7, vlues of H =M p up to re possible (the cse is illustrted in Fig.). The lower bound on is imposed by the simultneous requirements H 0?5 M p nd m m 0, together with eq. (.7). As H =M p is 3

5 vried between 0?5 nd the corresponding dilton mss vries over the whole domin from 0?4 ev to 00 MeV. For lower spectrl slopes ( < 0:), the present existence of dominnt dilton bckground becomes possible only for (unrelisticlly) low intion scles, s rst pointed out in ref.[9] for the cse of sclr perturbtions with t ( = 0) spectrum. Note tht, ccording to eq. (.9), nl intion scle H exctly coinciding with the string scle itself, 0? M p, would be comptible with present light dilton dominnce only for > 0:6. The intion scle H determines lso the mplitude of sclr perturbtions of the metric itself, nd it is thus constrined by the sclr contribution to the CMBR nisotropies (the contribution of tensor perturbtions turns out to be negligible in our context, s their spectrum grows very fst with frequency [,3]). The behviour of the sclr perturbtion spectrum, s we shll see, depends in generl on the dopted model of mtter sources nd bckground evolution, nd it is fir to sy tht our present knowledge of the detils of the stringy pre-big-bng phse is too poor to mke stringent predictions on the exct vlue of the spectrl index. On the other hnd, fortuntely, the properties of mssive dilton bckground re only wekly dependent on the vlue of the spectrl index for > 0, nd rpidly become spectrum-independent s soon s. The prticulr exmple chosen in this pper (see Sec.4) to discuss dilton production, nmely three-dimensionl, isotropic, dilton-dominted bckground with negligible mtter sources, gives the sme spectrum (very fst growing, = 3), for sclr ( ), dilton () nd tensor (h ) perturbtions. However, such n exmple is chosen here for simplicity resons only, in order to develop rst qulittive sketch of the scenrio ssocited with dilton production, nd it should not be tken s prticulrly indictive for wht regrds the spectrl properties of the metric perturbtions. For phenomenologicl discussion it is better to leve open the possibility of dierent spectr for nd (possibility which is in generl llowed in this context, s we shll see in Sec.3), nd to prmetrize the sclr (metric) energy density s (!; t) ' GH with n? in generl dierent from. n?! H! H 4 (:0) The interesting question to sk t this point is whether, in the sme rnge of H which we believe to be relistic, it is possible to produce enough diltons to close the present universe nd, t the sme time, to generte sclr perturbtions with spectrum consistent with the nisotropy observed by COBE [0]. 4 This

6 mounts to requiring 0?5 < H M p < ; (t 0 ) ' ; (! 0 ; t ) ' GH!0! n? ' 0?0 ; (:) where! 0 = H 0 is the minimum mplied frequency corresponding to wve crossing tody the Hubble rdius, nd t is the time of mtter{rdition equilibrium, nerly coincident with the time of recombintion. The nswer, perhps surprisingly, is yes: the third requirement of eq. (.) is comptible with the other two, provided the sclr spectrum is lso growing, nd < n < :34 (:) This llowed rnge of n is well contined in the rnge of the spectrl index originlly determined by COBE [0], n = 0:5, nd is lso consistent with the new recent t [], which gives n = :5 0:5. It my be interesting to recll, in this context, tht growing sclr perturbtions, with n ' :5, re lso required for simultneous t of the COBE nisotropies nd of the observed bulk motion nd lrge voids structures on 50 Mpc scle []. Growing sclr spectr cn be obtined in the \hybrid intion" model proposed by Linde [3] nd recently generlized to the clss of \flse vcuum intion" [4] (see lso [5]). Note lso tht the condition (.) would not be incomptible with the lower bound on required for present dominnt dilton bckground (ccording to eq. (.9)), even in the cse of equl sclr nd dilton spectrum, = n?. Concluding this qulittive nlysis, we cn sy tht the possibility of producing dilton bckground tht sturtes the closure density, together with sclr perturbtions tht provide the observed cosmic nisotropies, seems to be nturlly ssocited with growing dilton spectrum, > 0. The fct tht such spectrum is typicl of string-bsed pre-big-bng models represents, in our opinion, n interesting spect of such models, nd motivtes the study of dilton production in the string cosmology scenrio. A requirement nlogous to eq. (.), formulted however in the context of extended intion models where the uctution spectrum of the Brns-Dicke sclr is not growing, my be stised [6] only for reheting temperture T r < 0 3 GeV, nmely for very low scles H ' T r =M p 0? M p. We note, nlly, tht the possibility of intionry production of mssless sclr prticles, ssocited with excittions of the Brns-Dicke eld, ws lso pointed out in ref. [7], nd previously discussed in ref. [8] for the mssive cse (with m < H ), but lwys in the context of exponentil intion, which is not the 5

7 nturl intionry bckground corresponding to the low-energy string eective ction. The pper is orgnized s follows. In Sec. we present the generl exct solutions (for spce-independent elds) of the system of bckground eld equtions, including clssicl string sources, following from the tree-level string eective ction t lowest order in 0. The explicit form of the solution is displyed, in prticulr, for perfect uid model of sources, in D = d + dimensions, for ny given eqution of stte. The low curvture nd lrge curvture limit of such solutions re given both in the Brns-Dicke nd in the conformlly relted Einstein frme. In Sec.3 we derive the coupled system of sclr (metric plus dilton) perturbtion equtions, including the perturbtions of the mtter sources in the perfect uid form. Such equtions re pplied to compute the sclr perturbtion spectrum for specic cse of bckground evolution motivted by model of sources (presented in Sec.4) in which the dominnt form of mtter is suciently diluted, non-intercting gs of lrge mcroscopic strings. The bckground describes phse of growing curvture nd ccelerted kinemtic (of the pre-big-bng type), which is expected to evolve towrds the stndrd, rdition-dominted cosmology. The frme-independence of the intionry properties of such bckground is lso discussed in Sec.4. The corresponding spectrum of the produced diltons is discussed in Sec.5, where it is shown tht, becuse of its fst growth with frequency, the phenomenologicl constrints leve open window comptible with the possible dominnce of relic diltons (in the hypothesis of negligible clssicl oscilltions of the dilton bckground). The min results of this pper re nlly summrized nd briey discussed in Sec.6.. Generl solution of the bckground eld equtions We will ssume the evolution of the universe to be described t curvtures below the string/plnck scle by the equtions R + 4 H H = 8G D e T (:) R? (5 ) H H = 0 ( p jgje? H ) = 0 (:3) 6

8 Such system of equtions follows from the low-energy (D-dimensionl) effective ction of closed (super)string theory [9], Z S =? p d D x jgje 6G D H H + V () +S M (:4) Here is the dilton eld nd H the eld strength of the two-index ntisymmetric (torsion) tensor B =?B. We hve included possible dilton potentil, V (), nd lso possible phenomenologicl contribution of the mtter sources represented by the ction S M, whose metric vrition produces the stress tensor T. We shll consider, in this pper, homogeneous bckgrounds tht re independent of ll spce-like coordintes (Binchi I type, with d Abelin isometries), nd for which synchronous frme exists where g 00 =, g 0i = 0 = B 0i (conventions: ; = 0; ; ::::D = d + ; i; j = ; ; ::::d). We shll ssume, moreover, tht the ction S M describes \bulk" string mtter, stisfying the clssicl string equtions of motion in the given bckground. At tree level V is constnt. In terms of the \shifted dilton" =? ln jdet(g )j (:5) the eld equtions (.){(.3) cn be written in mtrix form s [0] _?? 8 T r ( _ M)? V = 0 (:6) _ + 8 T r ( _ M)? V = e (:7) d dt (e? M M) _ = T (:8) ( dot denotes dierentition with respect to the cosmic time t, nd we hve used units in which 8G D =, so tht both nd T hve dimensions L? ). Here M is d d mtrix, G??G M =? B (:9) BG? G? BG? B where G nd B re mtrix representtion of the d d sptil prt of the metric (g ij ) nd of the ntisymmetric tensor (B ij ), in the bsis in which the O(d; d) metric is in o-digonl form, 0 I = (:0) I 0 (I is the unit d d mtrix); T is nother d d mtrix representing the sptil prt of the string stress tensor [0] (including the possible contribution of n 7

9 ntisymmetric current density, source of torsion). Finlly is relted to the energy density = T 0 0 by = q jdet(g )j (:) The three equtions (.6){(.8) correspond, respectively, to the dilton eqution (.) nd to the time nd spce prt of eqs. (.), (.3) for the homogeneous bckground tht we hve considered. Their combintion provides the covrint conservtion eqution for the source energy density, which cn be written in compct form s [0] _ + 4 T r (T M M) _ = 0 (:) The set of equtions (.6){(.8), (.) is explicitly covrint under the globl O(d; d) trnsformtion [,0]! ;! ; M! T M; T! T T (:3) where is n O(d; d) constnt mtrix stisfying T = (:4) For suitble clss of dilton potentils such system cn be solved by qudrtures, following the method presented in ref.[]. Here we shll concentrte, in prticulr, on the cse V = 0, corresponding to strings in criticl spce-time dimensions (which does not exclude, however, description of cosmology in d = 3 if we dd the right number of specttor dimensions in order to compenste the centrl chrge decit). We introduce suitble (dimensionless) time coordinte x, such tht = L dx dt (:5) (L is constnt with dimensions of length), nd we dene Z x T? = dx0 (:6) The equtions (.6){(.8) cn then be integrted rst time, with the help of eq. (.6) nd of the O(d; d) identity (M _ M) =?( _ M) ; (:7) 8

10 to give [] = e 4L D (:8) 0 =? D (x + x 0) (:9) MM 0 = 4? D (:0) where D = (x + x 0 )? T r (?) (:) ( prime denotes dierentition with respect to x, nd x 0 is n integrtion constnt). By exploiting the fct tht M is symmetric O(d; d) mtrix, MM =, nd tht, becuse of the denition of T (see ref. [0]) M? =??M; (:) eqs. (.9), (.0) cn be integrted second time to give Z dx (x) = 0? D (x + x 0) (:) Z dx M(x) = P x exp(?4 D?)M 0 (:4) where 0 nd M 0 re integrtion constnts (M 0 is symmetric O(d; d) mtrix), nd P x denotes x-ordering of the exponentil. For ny given \eqution of stte", providing reltion T = T () between the sptil prt of the stress tensor of the sources nd their energy density, eqs. (.3) nd (.4), together with (.8), represent the generl exct solution of the system of string cosmology equtions, for spce-independent bckground elds nd vnishing dilton potentil. In generl, such solution presents singulrities, for the curvture nd the eective coupling constnt e, occurring in correspondence of the zero of D(x). It is importnt to stress tht, ner the singulrity, the contribution of the mtter sources becomes negligible with respect to the curvture terms in the eld equtions (just s in generl reltivity, in the cse of Ksner's nisotropic solution). The reltive importnce of the source term is mesured indeed by the rtio (see for instnce eq. (.7)) 8e (x) =? (d? ) T r ( M) _ (:5) 9

11 (we hve normlized in such wy tht it reduces to the usul expression for the eective energy density in criticl units, = = c, when the dilton is constnt nd the metric isotropic). According to eqs. (.8), (.0) M _ M = e L? (:6) so tht, by exploiting the O(d; d) properties of M, T r ( _ M) =? e L T r (?) (:7) Therefore = D (:8) d? T r (?) goes to zero t the singulrity (D! 0). In this limit, the mtter contribution becomes negligible nd the generl solution presented here reduces to the V = 0 cse of the generl vcuum solution of the string cosmology equtions []. Denoting indeed by t c singulr point such tht D(t c ) = 0,?(t c ) 6= 0, from eq. (.6) we hve, ner this point, M _Me? = A (:9) where the constnt mtrix A stises A =?(t c) L =?AT ; MA + AM = 0 (:30) becuse of the property (.) of?. Moreover, from (.8) nd (.9) so tht, by using (.7), _ = e 4L (x + x 0) (:3) _ + 8 T r ( _ M) = e 4L D(t c) = 0 (:3) Eqs.(.9) nd (.3) correspond exctly to the equtions dening the generl vcuum solution of ref. [], for the cse of vnishing dilton potentil. Consider now the prticulr cse in which B = 0, nd we re in digonl, but not necessrily isotropic, Binchi-I type metric bckground, g 00 = ; g ij =? i (t) ij (:33) 0

12 (this is the bckground tht will we used here to discuss dilton production). The mtter sources cn be represented in perfect uid form, but with nisotropic pressure, T 0 0 = ; T i j =?p i i j ; p i = = i = const (:34) In this cse we obtin, from the previous denitions, MM 0 = 0 0 i i ij? 0 i i ij 0? = 0?i ij?? i ij 0! 0 pi ; T = ij?p i ij 0 ;? i = i x + x i D = (x + x 0 )? X i ( i x + x i ) = (x? x + )(x? x? ) (:35) where x = 8 < X : i i x i? x 0 p i = p i p jgj; " ( X i =? X i x i? x 0 ) + ( X i i i x i? x 0) # = 9 = ; (:36) nd x i, x 0 re integrtion constnts. The generl solution (.8), (.3), (.4) becomes explicitly [3] i = 0i j(x? x + )(x? x? )j i= j x? x + x? x? j i (:37) where = X i = e = e 0 j(x? x + )(x? x? )j?= j x? x + x? x? j? (:38) 4L e 0 j(x? x + )(x? x? )j (?)= j x? x + x? x? j? (:39) i i ; i = x i + i ( P i ix i? x 0 ) [( P i ix i? x 0 ) + ( P i x i? x 0 )]= (:40) nd i0, 0 re dditionl integrtion constnts. This solution hs two curvture singulrities t x = x. Ner the singulrity, the presence of mtter becomes negligible, (x) = (x? x +)(x? x? ) (d? ) P i ( ix + x i )! 0 (:4)

13 nd one recovers the nisotropic vcuum solution of string cosmology in criticl dimensions [4,5]. Indeed, for x! x, one hs jxj jtj =(P i i ), nd the solution behves like where i (t) jt? t j i ;? ln jt? t j (:4) i = x i i x x 0 + x ; X i ( i ) = : (:43) In the lrge jxj (smll curvture) limit, on the contrry, the reltion between x nd cosmic time is jxj ' jtj =(?), nd the solution (.37){(.39) behves like (for jxj! ) P i (t) jtj i=(+ ) i ;? + P i ln jtj P i? P + i ln jtj; jtj? P i =(+ P i ) (:44) The criticl density prmeter, in this limit, goes to constnt =? P i i (d? ) P i i (:45) which is obviously = for n isotropic, rdition-dominted bckground with i = =d. It is interesting to point out tht, for ny solution i ; corresponding to given set of equtions of stte, p i = i, there re corresponding \dul" solutions obtined through the reection i!? i, which leds to i ( i )! i (? i ) =? i ( i ), preserving however the vlues of nd (scle-fctor dulity [5,6]). Such dulity trnsformtion, combined with time reversl t!?t, trnsforms ny given metric describing (for i > 0) decelerted expnsion with decresing curvture, i < 0, H i > 0, _H i < 0, into new solution describing (for i < 0) superintionry expnsion with incresing curvture, i > 0, H i > 0, _H i > 0 (see lso []). It is convenient, for lter use, to write down explicitly the isotropic version of the symptotic bckgrounds (.4) nd (.44), s function of the cosmic time t nd conforml time such tht dt = d. In the (d + )-dimensionl isotropic cse, the smll curvture limit (.44) becomes, in cosmic time, (t) jtj =(+d) ; d? ln ;?d(+) (:46)

14 while in terms of we hve () jj =(?+d ) (:47) The vcuum, dilton-dominted limit (.4) becomes, in the isotropic cse, (t) jtj =pd ; p d( p d ) ln (:48) nd, in conforml time, () jj =(p d) (:49) Note tht for = =d, nd t! +, eq.(.46) describes the stndrd, rditiondominted cosmology with = const; the dul cse, =?=d nd t! 0, with t < 0, describes insted typicl pre-big-bng congurtion [], with superin- tionry expnsion driven by perfect gs of stretched strings [7]. The dul solution obtined through more generl O(d; d) trnsformtion, pplied to the rdition cse, corresponds to non-digonl metric nd n eective viscosity in the source stress tensor, nd hs been discussed in ref. [0]. We note, nlly, tht the solution presented in this section is given explicitly in the Brns-Dicke (BD) frme, whose metric coincides with the -model metric to which strings re directly coupled. The pssge to the Einstein (E) frme, dened s the frme in which the grviton nd dilton kinetic terms re digonlized, nd the ction tkes the cnonicl form Z p S E = d D x jg E j 6G D is obtined through the conforml rescling g E = g e?=(d?) ; E =?R(g E ) + g E (:50) r d? (:5) The E-trnsformed scle-fctor, E, nd the cosmic time coordinte, t E, re thus relted to the originl BD ones by E = e?=(d?) ; dt E = dte?=(d?) (:5) The symptotic limit (.46) of the previous generl solution thus becomes, in the E frme, E (t E ) jt E j ; E r (d? )(? d) ln E d?? 3

15 E?= E ; = (? ) (d? )( + d )? (d? ) where E is conformlly relted to the BD energy density by (:53) p jgj E = p jge j = e(d+)=(d?) (:54) In conforml time, E () jj?(?)=(d?)(?+d ) (:55) (note tht the conforml time coordinte is the sme in the E nd BD frme, d E = dt E E (t E ) = dt (t) = d (:56) becuse of eq.(.5)). The high curvture limit (.48) becomes, in the E frme, E (t E ) jt E j =d ; E p d(d? ) ln E (:57) nd, in conforml time, E ( E) jj =(d?) (:58) It should be stressed tht the rdition-dominted solution, with = =d, = const, is obviously the sme in both frmes, see eqs. (.53) nd (.48). We note lso tht, in the vcuum, dilton-dominted cse, the dulity trnsformtion which is represented in BD frme by the inversion of the scle fctor nd relted dilton shift,! =? ;! =? d ln (:59) becomes, in the E frme, trnsformtion between the wek coupling nd the strong coupling regime, without chnge of the metric bckground. E! E =? E ; (:60) Concluding this section, we wnt to stress tht the solutions discussed so fr describe the sitution in which the dilton potentil cn be neglected, nmely the bckground evolution t erly enough times when the eective coupling e is smll enough. Indeed, becuse of non-renormliztion theorems, the potentil is expected to pper t the non-perturbtive level only, nd hs to be extremely smll (V () exp[?exp(?)]) in the wek coupling regime. At lter times, nd lrge couplings, the min eect of the dilton potentil will be tken into ccount 4

16 in the form of dilton mss term (see Secs. 3 nd 5), which freezes the Newton constnt t its present vlue. 3. Sclr perturbtions with dilton nd perfect uid sources In order to obtin the equtions governing the clssicl evolution of sclr perturbtions, we choose to work in the Einstein frme, where the explicit form of the equtions is simpler. This is legitimte choice since, s we shll see t the end of this section, the sclr perturbtion spectrum, just like the grviton spectrum [3], is the sme in the Einstein nd Brns-Dicke frmes. In the E frme, the bckground eld equtions (with B = 0, but with non-vnishing dilton potentil V ) tke the form R? + V? (5) + T (3:) 5 + ct = 0 (3:) where c = p =(d? ) (the Einstein frme index, E, will be omitted throughout this section). The coupling of the dilton to the stress tensor of the mtter sources is xed by the conforml rescling (.5), (.54). We strt, for simplicity, with (d + )-dimensionl isotropic bckground, with perfect uid sources, g = dig(;? ij ); = (t) T = ( + p)u u? p ; u = 0 (3:3) nd we consider the pure sclr prt of the metric perturbtions, g h, together with the perturbtions of the dilton bckground,, nd of the mtter sources, ; p; u (in the liner pproximtion sclr, vector nd tensor perturbtions re decoupled, nd evolve independently). We use here for the metric the Brdeen vribles,, which re invrint under those innitesiml coordinte trnsformtions which preserve the sclr nture of the uctutions [8{30]. In the longitudinl guge we thus hve the rst-order expressions [30] h 00 = = h 00 ; h 0i = 0 h ij = ij ; h ij = ij 5

17 T 0 0 = ; T i j =? j i p; T i 0 = + p u i (3:4) These re to be inserted into the rst-order perturbtion of the Einstein equtions (3.),?h R + g R? (g R? h R ) nd of the dilton eqution @ ) + T (3:5)?h ? g (? )@ + c (? dp) = 0 (3:6) Here the covrint derivtives re to be performed with respect to the bckground metric g, nd R ;? re to be computed to rst order in h. By using the bckground eld equtions, the (i; j) component of eq. (3.5), with i 6= j, gives = (d? ) (3:7) which llows us to eliminte everywhere one of the two Brdeen's vribles. The (i; 0) component gives constrint tht cn be written, in terms of the conforml time, s i (d? ) (d? ) + 0? 0 = ( + p)u i (3:8) ( prime denotes dierentition with respect to ). The (0; 0) component provides n expression for the density perturbtion in terms of the sclr vribles nd, 5? d 0 = 0? d(d? )( 0 )? d? (d? ) 0 (d? ) (0 + ) (3:9) Finlly, the (i; i) component of eq. (3.5) nd the perturbed dilton eqution (3.6) give, respectively, 00 + (d? 3) (d? ) 00 = + (d? 4)(0 ) + d? (d? ) 0 (d? ) + p) (3:0) 6

18 00 + (d? ) ) = (d? ) 0 0 (d? ) (? dp) +? (? dp) (3:) The liner system formed by the four coupled equtions (3.8){(3.) determines the clssicl evolution of four independent perturbtion vribles ; ; nd u (n dditionl reltion between p nd is to be provided by the detiled model of mtter sources). In the bsence of dilton bckground ( = 0 = ) one recovers the usul system of equtions for hydrodynmicl perturbtions [30], while in the bsence of uid sources (T = 0 = T ) one hs the usul perturbtion system for sclr eld minimlly coupled to the metric [30]. When nd T re both non-vnishing, nd p cn be prmetrized in terms of s p = (t), we my eliminte by mens of eq. (3.9), nd the system reduces to pir of second-order dierentil equtions (3.0), (3.) for the coupled vribles nd. By introducing the bi{dimensionl vector Z = nd by prmeterizing the dilton bckground s (3:) = ln ; = const (3:3) the bove-mentioned pir of equtions cn be represented in compct form s Z 00 k + 0 AZ0 k + (k B + C)Z k = 0 (3:4) where Z k = ( k ; k ) represents the Fourier component of the perturbtion vribles, 5 Z k =?k Z k, nd? (d? 3 + d)? 4(d?) A =?(d? )[ cd(? d) + ] (d? )? c (? d) 4 C = (d? ) 00 B = 0?c(? d)(d? ) + (d? )[d? 4 + d +? (d?) ]( 0?cd(d? )(? )[d(d? )? ) 0 ]( 0 ) 0! (3:5) (we hve neglected here the possible contribution of the dilton potentil, by = 0 V=@ ). We note tht system of coupled sclr perturbtion equtions similr to (3.4) ws previously considered lso in refs. [3,3] 7

19 where, however, sclr eld model of source (\inton" mtter) ws used, insted of the uid model dopted in this pper. Without further pproximtions, nd re thus in generl non-trivilly mixed, with time-dependent mixing coecients determined by the explicit model of sources, = p=, = p=, nd by the bckground kinemtics, (t), (t), ccording to eq. (3.4). The solution of (3.4) provides in turn, for ny given bckground congurtion, unique determintion of the density contrst = through eq. (3.9), nd of the velocity perturbtion u i through eq. (3.8). Equtions (3.8){(3.) re liner in the perturbtions nd just describe their clssicl evolution without specifying their bsolute mgnitude. As clerly stressed in ref. [30] (see lso refs. [33,34]), in order to determine the bsolute mgnitude of the vcuum uctutions nd their spectrl distribution, one must express the perturbtions in terms of the correctly normlized vribles stisfying cnonicl commuttion reltions. second order in the uctutions. These cn be determined by expnding the ction to For the pure metric-sclr eld system (T = 0) such cnonicl vrible is known to be xed by the following liner combintion of nd [35{37] v = + z ; z = 0 0 : (3:6) For pure uid source ( = 0), with constnt, the cnonicl vrible is insted [38,39] w = (f? ); 6 r = 3 + p 0 where f is the velocity potentil determining the uid perturbtions s (3:7) p + p u i =? i f =? ( + p)? p 0 f + p (3:8) p (we hve ssumed d = 3 in the previous three equtions). The vribles v; w ply the role of \norml coordintes", decoupling the system of perturbtion equtions, nd reducing the ction to the free sclr eld form [30]. Only when is expressed in terms of such vribles does one get cnonicl normliztion of the Fourier modes k, nd then the correltion function for the metric uctutions Z h (x) (x 0 dk sin kr )i = k kr j (k)j (3:9) 8

20 provides the correct spectrl distribution for the metric j (k)j = k 3 j k j (3:0) nd for the dilton, (k), through eq.(3.6). If T nd re both non-vnishing, one could try perturbtive pproch to the spectrum (s in refs. [3,40]), by keeping the denitions of v nd w xed s zeroth-order pproximtion. In such cse, the constrint (3.8) gives (in d = 3) = 4 0? 4p ( + p)f (3:) By eliminting f nd in terms of v nd w through eqs. (3.6), (3.7), by using the constrint (3.) nd the bckground eld equtions, one cn then express the Fourier mode k, from eq. (3.9), s k = k (v; v 0 ; w; w 0 ; k) (3:) Moreover, the system of equtions formed by eq. (3.) nd by the combintion of eqs. (3.9) nd (3.0) obtined by eliminting, cn be written s system of two second-order dierentil equtions for the coupled modes v k nd w k. Its solutions, when inserted into eq. (3.), provide rst pproximtion to the sclr perturbtion spectrum (3.0). From eq. (3.7) one hs then the corresponding dilton spectrum, j j = k 3= j k j, nd from eqs. (3.8), (3.7) the density perturbtion spectrum j j = k 3= j(=) k j. In generl, dilton nd metric uctutions will hve dierent spectrl distributions, j j 6= j j. The coupled system of equtions is rther complicted, but it seems possible, in principle, to obtin lrge vriety of spectr s the eqution of stte nd the rtio p= re ppropritely vried [4]. In this pper we shll consider model (see Sec. 4 for its motivtions) in which the universe evolves from three-dimensionl, dilton-dominted phse of the pre-big-bng type (with negligible uid sources, T = 0 = T ), to the stndrd rdition-dominted phse (p = =3), dibtic ( = =3), nd with frozen Newton constnt ( = const). More complicted scenrios will be nlysed in future works [4]. The phse of pre-big-bng intion is ssumed to extend in time from? up to the time =? < 0, which mrks sudden trnsition to the phse of rdition dominnce. For <? the constrint (3.8) thus becomes 0 k k = 4 0 k (3:3) 9

21 where, ccording to eq. (3.6), k = v k? z k (3:4) When the constrint is inserted into eq. (3.9), nd eq. (3.4) is used in order to eliminte nd 0, we re led to reltion of the form (3.), nmely k =? 4k 0 z vk 0 z (3:4) In the bsence of mtter sources, eq. (3.) becomes equivlent to the combintion of eqs. (3.9) nd (3.0). By expressing k in terms of v k ccording to eq. (3.4), nd by eliminting 0 ; 00 through eq. (3.3), we nlly get the cnonicl perturbtion eqution [30], vlid for <?, v 00 k + (k? z00 z )v k = 0 (3:5) In the second, rdition-dominted phse ( >? ), we ssume tht the dilton cquires mss m, nd it stys frozen t the minimum of the potentil (with possible smll oscilltions round it), so tht V = = m (3:6) In this cse decouples from the metric uctutions (see eqs. (3.8){(3.)), tht re coupled now to the uid perturbtions only; the cnonicl vrible for their quntiztion is thus given by eq. (3.7). As 00 = = 0 in the rdition phse, it turns out however tht for >? both w nd stisfy the free oscilltor eqution, w 00 =w = const (prt from the dilton mss term, ssumed to be negligible t erly enough times, see Sec. 5). As consequence, nd will hve the sme spectrum (identicl, in this cse, to the tensor perturbtion spectrum), which cn be computed by dopting second quntiztion pproch, regrding the mpliction of the perturbtions s process of prticle production from the vcuum, under the ction of the cosmologicl bckground elds [30]. The Bogoliubov coecients c for such process re obtined by mtching the solution of eq. (3.5) to generl solution of the plne-wve type, vlid for >?. By ssuming, for <?, tht v k = p k (c + e?ik + c? e ik ) (3:7) (?)? ; = ln (3:8) 0

22 we hve z 00 z = 00 = ( + ) (3:9) The solution of eq. (3.5) describing oscilltions with positive frequency t =?, nd dening the initil vcuum stte, is thus given in terms of the Hnkel function of the second kind, H (), s v k = = H () (k); = + (3:30) The continuity of v k nd v 0 k t the trnsition time =? xes the Bogoliubov coecient c? (k), nd the corresponding expecttion number of prticles produced in the mode k. For k < we obtin hn(k)i = jc? (k)j ' (k )?jj? (3:3) (higher-mode production turns out to be exponentilly suppressed [3, 4], nd cn be neglected for the purpose of this pper). In terms of the proper frequency! = k=, the energy density of the produced diltons is thus chrcterized by spectrl distribution!d =d! '! 4 jc? j, which my be written, in units of criticl density c = H =G, (!; t) =! d c d! ' G!4 H!!?? = GH?! H! H 4 (3:3) where! (t) = H =(t) is the proper frequency of the highest excited mode (here we hve supposed?=). This is the sme spectrum s tht obtined in the grviton cse [43], with n intensity normlized to the nl intion scle H. It is growing for phse of superintionry pre-big-bng expnsion ( < ), t for de Sitter ( = ), nd decresing for power-lw intion ( > ). It should be stressed tht this second quntiztion pproch is convenient to discuss the squeezing properties of the produced rdition [3,34,44{47] but, s fr s the perturbtion spectrum is concerned, it is completely equivlent to the more trditionl pproch in which one computes the prmetric mpliction of the perturbtion mplitude. In this second pproch one hs indeed, ccording to the \eective potentil" z 00 =z of eq. (3.5), mode mplitude tht is constnt, jv k j ' = p k, in the initil region!? where k >> jz 00 =zj '?, nd which grows with power-like behviour in in the non-oscilltory region dened by k << jz 00 =zj (in the subsequent rdition er the solution for v is gin oscillting, with frozen mplitude). In the non-oscilltory region Z Z d 0 Z v k = c z + c z z ( 0 )? d 0 0 k z z ( 0 d 00 z ( 00 ) + O(k 4 ) (3:33) )

23 (c ; c re integrtion constnts) is the generl solution of eq. (3.5) to rst order in k (the rst sub-leding term hs been included to hve non-trivil derivtive of v=z). This gives, for the bckground (3.8) (with obvious redenition of c ; c, nd introducing further numericl constnt c 3 ), v k = c jj? + c jj +? c 3 k jj? (3:34) For > 0 (intionry expnsion) the rst term is the dominnt one in the the jj! 0 limit, nd the wve mpliction chieved in this limit cn thus be estimted s [30] v k ' v k() p k v k k'h ' p z k v k k'h [? k (? ) ] (3:35) The vrible t the denomintor is to be evluted t the time ' k?, where the mode k \hits" the eective potentil brrier z 00 =z '? (otherwise stted: t the time of rst horizon crossing, when H =!). By inserting this vlue of v into eqs.(3.4) nd (3.0), nd reclling the denition of z, we re led to j (k)j ' k 3 j k j ' k z k'h H ' _ k'h (3:36) which is the stndrd expression for the sclr perturbtion spectrum [35,48] ssocited with the intion{rdition trnsition (see Sec.4 for proof of the fct tht the sme result is recovered in the cse of contrcting bckgrounds). The sme spectrum is obtined for the dilton perturbtions, since we hve, from eq. (3.4), j j = k 3 j v k? z z k j ' k ' j j (3:37) z k'h It is importnt to stress tht this expression, when multiplied by G, exctly coincides with the spectrl energy density (3.3) (modulo numericl fctors of order unity), evluted in the rdition er. Indeed, multiplying nd dividing eq. (3.37) by H ' ( )? we hve H Gj j ' G _ k'h ' ' GH (k )? = GH G k'h!!? ' in greement with eq. (3.3) for t = H?=. ' GH k'h! d c d! rd (3:38)

24 As lredy stressed in ref. [3] for the tensor perturbtion cse, we wnt to remrk nlly tht the sclr perturbtion spectrum is the sme in the E nd BD frme, s consequence of the equlity of the two conforml time coordintes (see Sec. ). Indeed, quite independently of the computtionl method (rst or second quntiztion) the spectrl behviour of the energy density is xed by the Bessel index of the solution of eq. (3.5), which depends, in turn, on the slope of the eective potentil z 00 =z. For generic d = 3 bckground in the E frme we hve (recll eq. (.55)) () = so tht, ccording to eq. (3.9) z = z E?? + 3 (3:39) E = (? )(3? ) (3:40) E (? + 3 ) Eqution (3.5) for v k is not conformlly invrint, nd in the BD frme the eective potentil becomes z 00 z = ( BD ; BD ) =? 0 0 BD? z BD z E BD BD 00 BD BD (3:4) In this frme, however, the spectrum is determined by the conformlly trnsformed bckgrounds, nmely by the solutions (.46), (.47) of the BD eld equtions. By inserting their explicit expressions for d = 3 we get z 00 = (? )(3? ) (3:4) z BD (? + 3 ) BD which coincides with the eective E-frme potentil (3.40) becuse of the equlity BD = E. The sme result holds for dilton-driven evolution, described by the solution (.57) nd by its BD-trnsformed expressions. 4. Pre-big-bng scenrio in the Brns-Dicke nd Einstein frmes As seen in the previous section, the spectrl distribution of the perturbtions is uniquely xed by the explicit form of the bckground solution. The time evolution of the bckground elds is determined, in turn, by the prticulr model of mtter sources. As in our previous work [0, 7], our model of sources consists of 3

25 suciently diluted gs of clssicl fundmentl strings whose mutul interctions re described, in men-eld pproximtion sense, s the interction of ech single string with the bckground generted by ll the others ccording to the tree-level eective ction (.4). The source stress tensor ppering in eq. (.) is thus given by sum over ll strings (lbelled by i) of the stress-tensor of ech individul string T i, where T i (x) = 0p jgj Z dd( dx i d dx i d? dx i d dx i d )D (X? x) (4:) nd, for ech i, the coordintes X stisfy the string equtions of motion in the given bckground, d X d? d X d +? (dx d + dx d )(dx d? dx d ) = 0 g ( dx dx + dx dx d d d d ) = 0; g dx dx d d = 0 (4:) Here ( 0 )? is the string tension,? the Christoel symbol for the bckground metric g, nd the usul world-sheet time nd spce vribles (we re using the guge in which the world-sheet metric is conformlly t). The generl exct solution of the system of equtions (.){(.3), (4.), (4.) is hrd to nd nd certinly impossible to express in closed form. In some pproprite symptotic regime, however, the solution of the string equtions of motion, when inserted into the energy-momentum tensor (4.), provides n eective eqution of stte tht llows us to describe the string sources in the perfect uid pproximtion [0,7], nd to recover the generl bckground solutions of Sec.. The cosmologicl solution we re looking for is chrcterized in prticulr by hving, s initil congurtion, the string perturbtive vcuum, nmely t spcetime with vnishing torsion nd coupling constnt, H = 0, =?. In this regime, strings move freely, do not decy, nd behve s pressureless gs with n energy density. We shll ssume to be smll enough initilly so tht, s we shll see, it will represent negligible source of curvture. On the other hnd - nite is certinly sucient to mke the dilton evolve wy from the perturbtive minimum. Indeed, the negtive brnch (x x? ) of the generl bckground solution with perfect uid sources, eqs. (.37){(.39), my be written in the cse of vnishing pressure ( i = 0) s i (t) = i0 j t? T t j i ; e = 6L e? 0 jt(t? T )j ; = L dx dt = e0 4L = const 4

26 i = t i T ; T = X i t i! = ; t 0 (4:3) (t i re integrtion constnts, nd we hve performed time trnsltion to shift the singulrity from x = x? to the origin, by choosing x 0 =?T (e 0 =4L)). This bckground is certinly consistent with the solution of the string equtions of motion (4.) in the t!? limit. Indeed, in this limit, the metric is t, i = const;? ln(?t); = const (4:4) nd the solutions of eqs. (4.) re chrcterized by P i (dxi =d) = P i (dxi =d). Eqution (4.) gives then T0 0 = const, T i i = 0, nmely stress tensor describing dust-like mtter in the perfect uid pproximtion. For t?t, however, the curvture scle begins to increse, the string sources progressively enter nonoscillting unstble regime [7], nd one must then tke into ccount the fct tht the rtios i = p i = begin to evolve in time. In connection with this lst point we note tht the solution (4.3), which, for t 0, gives H i = t i t(t? T ) ; _ (t? T ) =? t(t? T ) ; e = 4 t(t? T ) = e ; (4:5) is chrcterized by two scles. One is the curvture scle (jh T j T? t t?t ) t which the trnsition from t to curved spce-time regime occurs nd intion begins. At t?t, the curvture H is of the sme order s _ or e while, much erlier, it ws negligible. By contrst, much lter thn t =?T, it is e tht becomes negligible nd one recovers the vcuum solutions; T is free phenomenologicl prmeter of the solution. The other scle is the mximl scle jh j t?, t the time t?t, fter which the solution is no longer vlid, becuse higher orders in 0 hve to be dded to the low-energy eective ction (.4). This nl scle t is thus determined by the string tension s t ' p 0 = s, where s is the fundmentl (miniml) length prmeter of string theory [49], which my be ssumed to coincide roughly with the present vlue of the Plnck length, l p = M p?. The importnt point to be stressed is tht, in ny relistic intionry scenrio, T nd s cnnot be of the sme order, s we will now show. When jtj < T, the solution describes n ccelerted evolution given symptoticlly by i (t) (?t)? i ; j ij < ; 5 X i i = (4:6)

27 nd which is of the type given in eq. (.4) (we cll \ccelerted" congurtion in which _, nd H _ hve the sme sign, positive for expnsion, negtive for contrction [, 3, 7]). In this metric, the prticle horizon long ny given sptil direction, Z t d i p(t) = i (t) dt 0? i (t 0 ) (4:7)? evolves for jtj << T like the scle fctor, d i p i, while the event horizon d i e (t) = i(t) Z 0 t dt 0? i (t 0 ) (4:8) shrinks linerly in time, d i e (?t), for t! 0. The rtio of the two proper sizes r i (t) = d i p=d i e thus grows in time, for jtj << T, s (?t)? i? (?)?. On the other hnd, the horizon problem of the stndrd cosmologicl model [50] is solved if, for every sptil direction, the growth of the rtio r i (t) when jtj is rnging from T to t, is lrge enough to compenste the decresing of the rtio in the subsequent decelerted phse down to the present time t 0. This implies, in the hypothesis tht the pre-big-bng er is followed by the stndrd rdition-dominted (until t = t ) nd mtter-dominted evolution, t T?i??i? HT > t0 ' H 0 t = t t?= t t 0?=3 = H H = H H 0 =3 ' 0 30 s H M p (4:9) (the sme condition is required to solve the tness problem, see below). For n expnding d-dimensionl isotropic bckground i = = p d (see eq. (.48)), nd the previous condition gives in prticulr, for t ' s ' M? p T > 0 30p d=( p d+) s ; H T < 0?30 p d=( p d+) M p (4:0) We shll thus ssume tht the scle T ppering in the solution (4.3) is much lrger thn the string scle s ' M? p. This fct hs n importnt consequence. In this cse the bckground (4.3) becomes in fct good zeroth-order pproximtion to the generl solution of the full system of equtions, consistent with the string equtions of motion not only in the symptotic limit t!?. By dopting n itertive pproch, let us ssume indeed the solution (4.3) to be zeroth-order pproximtion, nd let us compute the rst-order corrections by 6

28 inserting tht solution into the string equtions of motion, in order to obtin the corresponding vlue of i (t). To this im we observe tht the given bckground is chrcterized, symptoticlly, by n ccelerted metric with shrinking event horizons (see eq. (4.6)). We recll tht, in such bckground, the string equtions of motion dmit oscillting solutions, corresponding to strings with constnt proper size L s, provided L s is smller thn the size of the event horizon H? (t) (stble strings), while the solutions describe non-oscillting strings with L s (t) (t) if L s > H? (unstble strings) [7]. The evolution of network of strings with some initil distribution in bckgrounds of the type discussed bove cn be investigted [5]. One cn show tht the number n(l s ; t) of strings (per unit length) of given size L s, t time t, must stisfy in the given bckground the pproximte evolution eqution [5] where is the Heviside [nl s (L s? H? )] s Its generl solution cn be written in implicit form s [5] Ls n(l s ; (H)) = n 0 (L s )(H?? L s ) + f (L s? H? ); H? f = n 0 (H? ) ln? ln H where n 0 is the initil string distribution. The energy ssocited t time t with stble ( S ) nd unstble ( U ) strings cn be estimted s S Z H? s L s n(l s ; t)dl s ; U Z H? L s n(l s ; t)dl s (4:3) However, for perfect gs of stble strings, p S = 0, while, for unstble strings, p U = U =d, with the sign xed by the exponent i of eq. (4.6), signfp U g =?signf i g, s discussed in [7]. Therefore, the rtio = p= s function of time, for perfect gs of strings in n ccelerted metric bckground, cn be pproximted s By inserting into eqs. (t) = d U U + S (4:4) (4.3) nd (4.4) the solution (4.) expressed for our prticulr metric (4.6), with n initil string distribution n 0 (L s ) L?3, one then nds for ech sptil direction [5] s i (t) =? d sh i (t); (4:5) 7

29 where H i is given by eq.(4.5). The bove result is vlid for jh i j <? s ' M p nd is not very sensitive to the initil string distribution. We now insert this expression into the right-hnd side of the eld equtions (.8), (.9), by reclling tht, for the pressureless bckground (4.3) one hs, to zeroth-order, Then, to next order,? i = x i + Z x? =? (0) i? (0) i = x i = e 0 4L t i (4:6) i (x 0 )dx 0 = e 0 4L? s t? T dt ln t Z t i? t s d i d? i (4:7) According to our itertive pproch, the integrtion of eqs. (.8), (.9) with D(x) determined by this new expression for? i provides rst-order pproximtion to the bckground elds (t), (t). The corrections to the solution (4.3) due to non-vnishing eective pressure of the string gs re certinly negligible for jtj >> T, in the regime in which the bckground (4.3) stises H << e _. However, s clerly shown by eq. (4.7), if T >> s then the rst-order corrections remin smll lso in the t?t regime, in which H e _, nd even in the limit t!?t ' s, in which e << _ H. Within the ssumption tht T is very lrge in string units, the solution (4.3) then becomes good pproximtion to the exct solution of the system of bckground equtions nd string equtions of motion, for the whole rnge? t?t ' s. We stress tht, in this scenrio, when jtj << T the source term e becomes negligible with respect to H nd _ (see eq. (4.5)); quite independently of the exct vlue of the pressure nd of the prticulr type of eqution of stte t the scle T, the bckground rpidly converges, for jtj << T, to phse of vcuum, dilton-driven ccelerted evolution (s discussed in Sec. ), described by the metric (4.6). We re left, therefore, with two phenomenologicl possibilities. The rst is the cse in which T, nd then the temporl extension of the regime (4.6), is much lrger thn the miniml vlue xed by eq. (4.9) to secure phenomenologiclly sucient mount of intion. This mens, in conforml time, j T j >> 0 30 H M p = j j ' j 0 j (4:8) 8

30 where 0 is the time when the lrgest scle, corresponding to the minimum frequency mode! 0 = H 0, ws pushed out of the event horizon during the pre-bigbng phse. In this cse, ll tody's observble scles crossed the horizon in the dilton-driven regime (4.6), so tht the presently observed perturbtion spectrum is wholly determined by the metric behviour of tht regime, quite independently of possible erlier mtter corrections to the bckground. The second possibility is the cse of nerly \miniml" intion, corresponding to the equlity in the condition (4.9), which then implies j T j j 0 j. In this cse the lrgest scles crossed the horizon when the contribution of the string sources to the metric bckground ws of the sme order s the dilton contribution. As consequence, the low-frequency prt of the sclr perturbtion spectrum my be ected by the mtter corrections, nd my be sensitive to the prticulr type of eqution of stte. The spectrum is thus to be computed by including the nontrivil mixing induced by the source terms T nd their perturbtions, T, s discussed in Sec. 3. As ws nticipted there, in this pper we will discuss only the rst possibility. We shll ssume, in prticulr, tht the phse of ccelerted evolution responsible for the solution of the stndrd kinemtic problems, nd for the mpliction of the perturbtions (t ll presently ccessible scles), is described by threedimensionl, isotropic, dilton-dominted bckground with (t) (?t)?=p3 ; () (?)?=(p 3+) ; (3 + p 3) ln ; t?t < 0? < 0 (4:9) (ccording to eqs. (.48), (.49)). More complicted scenrios, in prticulr with higher-dimensionl, nisotropic, sourceless bckgrounds will be nlysed elsewhere [4]). The metric (4.9) describes superintionry expnsion [5]. In order to obtin the dilton spectrum, by pplying eq. (3.3), we must trnsform however the solution (4.9) into the E frme, where it tkes the form (see eqs. (.57), (.58)) E (t E ) (?t E ) =3 ; E () (?) = ; E? p ln E (4:0) This metric describes, for t! 0?, contrcting bckground. Potentilly, this represents diculty of the whole scenrio: indeed, the pproximtion of diluted string gs might be no longer vlid in contrcting bckground, s well s the 9