COURSES t\ND PLE:-.'ARY TALKS REVISTA MEXICANA DE FÍSICA 49 SUPLEMEl'TO 1, 30-37

Sorne aspects of pre Big Bang cosrnology

A. FeinsteinDepartamento de Fúica Teórica, Universidad del PaÍS Vasco,

Apartado 644, ¡';-48080, Bilbao, Spain

Recibido el 11 de abril de 2(X)l; aceptado el 19 de mayo de 2001

JUNIO 2003

This is a summary of a coursc givcn at the Fourth Mexican School on Gravitation and MaLhematical Physics on sorne aspccts of pre Hig Hang(PBB) cosmology. After introductory remarks lhe lecLures concentrate on sorne amusing cOllsequences dcrivcd from Lhesyrnmetries of lhestring theory with respcct to such classical concepts as isotropy and homogeneity. The symmetries ofthe M-Lheory and extra dimensions arefurther applied to show that the classical singularities might be just physically irrelcvant. In the finallecture a model univcrse is "produced"from "almost nothing" and íl is argued that initial plane waves are thermodinamically natural state for the universe to emerge from.

Keyword'i: Pre big bang cosmology; M theory

Este es el resumen de un curso ofrecido en la Cuarta Escuela Mexicana de Gravitación y Física Matemática sobre algunos aspectos dc lacosmología antes de la gran explosión (PHB). Después eJealgunos comentarios de canícler introductorio, el curso se concentra en algunasconsecuencias inlcresentes que se derivan de las simetrías de la teoría de cuerdas con respecto a los conceptos clásicos de isotropía yhomogeneidad. Las simetrías de la teoría rvl y dimensiones adicionales se aplican para demostrar que las singularidades clásicas podrían serirrelevantes desde el punto de vista físico. En la última ponencia se "genera" un modelo de universo de "casi nada" y se argumenta que lasondas planas iniciales conforman un estado natural del cual puede aparecer el universo.

Descriptores: Cosmología antes de la gran explosión; teoría

PACS: 11.25.Yb; 98.80ÜI

1. Introduction

First I would Iikc to thank lhe Organizcrs roc giving us lheopporlunily lo give leclure Course in such a bcauliful sur-rounding.

As rar as my course is concemed, I will be concenlralingon OUf reccnl works in Slring Cosmology done mostly in col-laboral ion with Kerslin Kunze and Miguel Vázquez-Mozo. Iwill firsl brieOy review some general aspects of String Cos-mology, ami in lhe following days wc wil! louch lhe InitialCondilions, and lhe Singularily Problem. Mosl of Ihe refer-cnccs may be found in a niccly organizcd M. Gasperini WEBpage: http://www.to.infn.it/ gaspcrin.

Thc ccmral problems of modem tilcorctical cosmologyare: the initial conditions, the singularity problcm and lhedimcnsionality of lhe univcrsc. The firsllO begin madcm dis-cussions of the initial condilions problem was Ch. Misner [11in his Chao/ic Cosmology Pmgram. The idea was to allow Iheuniverse to slarl from an arbitrary complex initial state ami toidentify lhe mcchanism to smoolhen oul the irregularilics, tofinally explain as 10 why lhe universe is homogeneous amiisolropic to such a high extent. Unfarlunately, even if one ro-cuses 00 homogelleous hut anisotropic cosmological models,lhe so-called Biallchi models, only those which include theFRW models as particular cases would isotropisc, and thcsewe say are of measure O.

Injlation [2], sornehow brought Misner's idea back tolife, in lhe sCllse thal perturbalions around FRW backgroundswere lO disappear if lhe lIniverse lo lIndergo a period of ac-

ceJcraled expansiono Nevertheless, geIlerically non-linear in-hornogeneilies such as slrong shock primordial waves etc. re-main difficull to deal with.

A solution lo the i.c. problem may come from rather dif-ferent a direclion, such as thcrmodynamics, foc exampJc, ifsome idea as lo how lo assign cntropy to gravilalional ficldwere available. In lhe 70's Pemose 131 pul forward a spec-ulation lhal one should assign a O gravitational entropy togcometries with O \Veyl tensor, so that the universe musthave been FR\V lo start with. Yel, lhis notion of gravitn-tional cntropy is apparently unrelated to the notion of cn-tropy even in the cases we know of, such a'i Black Holes,or CTllropyfoc high frcqucncy gravity waves ctc. Moreover,the idea of assigning O enlropy to FR\V geometrics is inf1u-cnccd slrongly by a classical view on lhe initial slale of theuniverse. Taking for example "stringy" approach lo the earlyuniverse one would have rather thoughtlhat the backgroundgeomctry should represenl a larget space of sorne exact Con-formal Field Theory, and FRW doesn'l seem lo be an exactCFr. Also lherc is ralhcr a differenl physical view on maUerssuch as isotropy ami homogencity in the contcxt of the stringIheory. I will be arguing laler thal certain kinds of inhomo-gcncitics ami anisotropies do not rcally matter whcn symmc-tries of the low energy fltring theory arc considered-thus itcould be that OUT notions of isometries must be changed.

The pre Big Bang (PBB) [4] cosmology is ralher differ-cnt. First, it pushcs the i.c. into the "cold radiatÍvc" past wherephysics is believed lo be known to us. The hot regime, nearlhe singularity, now becomes the intermcdiate phase and il

SOME ASPECrS OF PRE BIG BANG COSMOLOGY 31

is believed that !he symmetries of the non-perturbative sIringtheory may help lO deal with the singularity. To summarizelhe PBB scenario in Dne paragraph:

Thc univcrsc starl'i a'i a pcrturbation in an otherwisc Oatbackground 15,12,16,14] this perturbalion collapses fromlhe point of vicw of an Einstein observcr. It cxpands and in.flates in the so-called String frame. The inflalion drives lheunivcrsc towards lhe "intcrmcdiatc" singularity mcanwhilcgctting rid of all lhe inhomogcncitics and anisotropies andlhe string symmetrics act lo squcczc lhe univcrsc unharrncdthrough !he hot regime. Finally, emerges the observed uni-verse.

Thc ¡nnation in thc PBR univcrsc is drivcn by lhe kincticenergy of ma"lcss dilaton ficld which is the fundamenlal in-grcdient of any string theory, and there are subslanliaI differ-ences between lhe standard inflalion and Ihe PBB phase. Instandard inflation lhe i.e. are prcscribcd in lhe rcgime Orle

knows vcry linlc about physics, lhe i"¡tial curvaturc scalcmay be vcry largc ami rcmaios constant oc dccrcascs UPOTl cx-pansioo, 00 lhe othcr hand, the ¡n¡tial coupling is arbitrarilywcak in PBB, the curvalure grows while the initial curvatureis practically O [14J.

Now, people orteTl talk about String Cosmology whcntalking about PBB. What does one really mean when SIringCosmology is quoted" Basically we laek a complete non-perturbalive string theory valid at Plank lime. So really lhegame is resIrictcd lo the energies ~ 1019 GeV ~ CUT~ 1016 GeV. Thcrefore, one may write the low energy ef-fective aClion for!he theory which reduces 10 GR + a bunchof extra massless fields depending on Ihe type of lhe stringtheory. Thc qucstions, however, the cosmologists ask rcmainthe same: why !he universe is homogeneous, what happcnswith initial singularity, can we solve the horizon and flatnessproblem, ando .. docs the universe we speculate about corre-sponds to the one we observe?

On general grounds, Ihe ma"less bosonie sector of su-perstrings includes the gravitational field G.v, the dilalon </J,with vacuum expectation value dctermining the string cou-pling constant, and the antisymmctric rank.two tensor B1~.There are more ma"less fields depending on Ihe particularsuperstring model. The lowest order effective action [or thema~slessfields can be written as

1 /dDxvae-2•(a')!!? •

x [R+4(iJ</J)2 - 112(Hl)2] +3md,

where Hl = dBl is !he field strength associated wilh theNS-NS two-form and 3md is a model-dependent part whiehincludes olher massless degrees of freedom.

When D < 10 sorne of these ma"less fields correspondto gauge ami moduli fields associated with the spccific com-pactification choscn, and the dilaton 4>appcaring in the aboyeequalion is re¡ated to lhe ten-dimensional dilaton </J1O by2</J = 21>10- lag VlO-u, where VlO-D is the volume of theinlernal manifold measurcd in unit' ofN.

Let us resIriet the attention lOgenerie degrccs of freedom,leaving aside the intemal component~of the ten-dimensionalfields. In the heterotic string case, Smd contains the Yang-Milis aetion for the background gauge fields A~. In theca,e of lhe modeI-dependent part of the type-I1B supersIringthings are more involved; among the massless degrees of[reedom in the R-R sector we find, along with a pseudo-scalarX (the axion) and a rank.two antisymmetric tensor B12J, arank four self-dual fonn A~dvO">..' The presence of this self-dual form spoils the covariance of the effective action for themasslcss fields, since there is no way of imposing the self-duality condilion in a generalIy covarianl way. Bul, if we selAsd to zero, wc can write a covariant action for the remainingfields with

Slln = _ 1 /dlJx vamd (a')¥

NOI;ce Ihat Ihe R-R fields do nol couple directly to Ihe dila-ton. Thus, the Iower dimensional (D < 10) R-R JiekIs Xand Bt:;} are obtained from the ten-dimensional ones throllghX = J1I¡O-UXIO ami B(2) = JVlO_lJB¡~)

By combining the dilaton ami the axion o[ the ten-dimensionaltype-TIB sllperstring inlO a single complex field).= XlO + ie-lPtO, it is possible to check that the bosonic e[-[eclive action is invariant under the SL(2, R) transfonnation,\ --t (a,\ + b)/(c,\ + d), G.v --t Id+ dIG.v. Wriling Ihistransformation in terms of four-dimensional fields (D = 4)we find that the four-dimensional fields aC(juire an "anoma-lous weight" due lo lhe factthat V(j, as measured in the stringframe, does lransform under SL(2, R) (see Ihe 3rd referellcein Ref. 5):

bd + aee-2q"[(eX4 + d)2 + e2e-2Ó,]!

aex~ + (ad + be)x,+ ----------[(CX, + d)2 + e2e-2.,]! 'e-lP.¡

e-lP~ = --- _[(ex, + d)2 + C2e-2",]!'

H(1)' = dH(1) - eH(2),

H(2)' = [(eX4 + d)2 + e2e-2q,,]~ (_ bH(l) + aH(2)),

G~v = [(ex, + d)2 + e2e-2q,']!G.v

2. T.duality

The T-dualily symmelry is probably one of Ihe most inlcr-esting consequetlces of the string theory. We often hear peo-pie say that the transformation R --t 1/ R, where R is thestring compactification scale, leaves the energy spectrum ¡n-variant. The existence of a compact dircction is essential for

Rev. Mex. Fú. 49 SI (2002) 30-37

32 A. fEINSTEIN

eoupled to a dilaton {icld

Here k is the spatial eurvature (k = -1, O,1)and

a' + {32 = 2.

I:;xcercise 2 [715tart wilh a 4 dimensional model

k=!k=Ok =-1.{

sin II¡'(lI) = II

sinh II

This is an exaet solution of dilaton gravity provided theconstants Q' and {3 satisfy

1 (2,,)1>= 4 ({3 - a) log T - k .

(2 )-0 (2 )13ds2 = - ¡" - k dI' + t" - k dx2

(2 ) 1-0+ t" - k ¡2[d02 + fklll)2d<p2]

the model witb an inverted seale factor. \Vhat is tbe metriein the Einstein frame? \Vhat happens witb tbe sealar field?Find the kincmatical quantities of the model ( aceeleration,eurvature seale etc). \Vhat happens if you doalise this modeljust wrt a single Killing direction, say 8x? Are the T-dualitytransformations invariant under diffcomorphism?

Identify all the Killing vectors. Show that only dualis-ing wrt ~1= 8" keeps all isometries intael. Dualising wrt6 = 8~, for example, tbe bomogeneity is lost, yet the stringtheory remains invariant. This signals that the hornogeneity,and isotropy are not fundamental coneepts of slring eosOlo1-ogy. Identify the Sehwarzsehild solution among the above (ithas a eonsLont sealar field) and spceulate about the possiblefate of the blaek hole applying tbe duality transfonnation.

Thcre are sorne subtle differenees between scale factorduality and T-duality. The inversion of one ur more scalefaetors in the metrie (tbe sea le factor duality) leaves thelow-energy equations of motion invariant. T-duality, is ratherstronger a tool, it is a transformation of the metric, not ncc-essarily only of the seale faetors, leading to a new solutionof tbe low-energy field equations whieb eorresponds to ancquivalent string theory. In the metrie of Exereise 2 we havefour Killing veetors, and the degree of homogeneity may beredueed by just performing a T-duality transformation along1> direetion. Consequently, for this modcl deseribed by tbeline elernent with a cyclic x coordinate, the only T-dualitytransformation lcaving thc spatial symmetry group ¡ntaet isthe one pcrforrned along lhe compactified isomelric dircctionx. In this case it is formally identieal to the seale factor du-ality of the rnetric. For seale factor duality, however, lhere isno necd 10 impose the cornpaetness of the x-coordinate sincethe "dua!" model does not neeessarily have to be equivalentto the origiualone as in complete striug tbeories.

this symmetry. I will not go here into great detai!s as to howthis comes, just will make several rcmarks as applicd lo cos-mology.

Suppose we probe by strings a 10 dimensional geometry,using two different string theories. The topology of the 10dimensional spaee is R9 x SI. The T-duality [61 identifiesthese two string theories and, if a' = 1/2"T, where T is thelcnsion in both lhe theories, then lhe compactification radiisatisfy R1R, = a'. Note that if one of the radii shrinks to Othe other diverges.

Compaclification on a c¡rele implics quantization of mo-mellla whicb now comes in quantas of l/R. These mo-menta aprear as ma<.;scs fOf slates which arc masslcss inhighcr dimcnsions. In addition thefe CXiS1S YCl anolhcr ex-citation typc, lhe windings. If m is the number the stringwinds around the cirele, the cncrgy of these cxcitations ism x 27rRT = mR/a', thcrcforc ir lhe compactification radiiare intcrchangcd al the sume time as the momcnla arc intcr-changed with windings, the energy spcetrum of the theoryrcmains invarianl. Now, this is what happeos OIl lhe levelof the string theory. To relate this to the background spaee-lime whcrc lhe strings propagatc, ofle must take jmo accountthat the slrings propagate on a 10 dimensional targel space,and in tum, the target spaee must satisfy that the so ealled {3fUnClioTlsare to vanish (somelhing roughly similar to Einsteinequations, sec for example Ref. 20).

If the target spaee happens to have a eompaet Killing di-rcction then the actions we where ~'llking about aboye are in-variant undcr the following "duality" transfonnation: takingadapted coordinates in which XO denotes the coordinate alonglhc Killing vector choscn to dualize, we find ncw values fm[

(1)G"", 1>, B""]

E.xercise 1Start wilh the solution of spatially nal FR\V modcl drivenby a massless scalar field. Write il both in Einstein and inthe String frame. Use the T-dualily transfonuations to obtain

- -1900 =900'

9 Oi = 9 oc} B 01,

9 'j = 9 'j -9001 (9 'o 90j -B iO BOj),- -1B 'j = B 'j -900 (9'0 BOJ-B ,090j),- -1BOi =900 90i,

- 11> = 1> - 21og900'

The PB B cosmologieal models are based on the so-ealled seale factor duality wbieb uses the T-duality symmetryof thc string theory to invert the scale of expansion as in9 00 = 9001

. In Ihe case of FR\V model, one may invertthe overall seale factor due to isotropy of tbe mode!. \Vhathappens, then, is that one may have a prolonged inOationarypcriod in the invertcd-scale model, while the standard modclis dccelerating.

Rev. Mex. Fís. 49 SI C2(X)2) 30---37

SOME ASPECrS 01; PRE BIG BAi'\G COSMOLOGY 33We have jusI only slarlcd lOplay lhe game amI already on

this level see lhat there are sorne interesting a'ipeCISlo stringcosmology, for if for example the symmetries of the stringtheory are used a'i a guide. we may conclude lhat two quiteinequivalent backgrounds may host equivalent string theoriesand thus from me point of view of string propagalioTl he iTl-distinguishable. Strings propagating on lwo hackgrounds re-laled by me T-dualily transformation would not nOlice lhedifference and wouldn't care of such aspecIs as homogcncityor isolropy.

Now, we have scen mat the inversion of the scalc factoris pcrmilled by lhe slring symmelries. BUl whm happcns alt = O' lhere are differenl approaches lOlhis problem, knownin me lilerature as an "exit problem", In the next scction I willlry lo surnmarize our work on this problem.

3. Regularization oC singularities

We all bclieve that me emergence of s-t singularities in GRsuggesls !he breakdown of lhe lheory m nalUral scales of lhetheory. Allhese scales one expccL'i lhe quantllm corrcctionslO lake a lcading role aod save lhe silualion. While StringlM-theories are the best candidates to quantize gmvity, near thesingularity one can not use the pcrturbative approach, bUlra!her a fulI-nedged M lheory. Unfortunalely we lack sucha meory. Diffcrcnl approaches lO regularize singularity andto find a way out to solve the exit problem are discussed inlhe papers qUOled in 18J.One of lhe ways of dealing wilh lhe problem, mighl be

more qualilalive an approach, trying lo use the symmetriesand dualilies of lhe M-lheory (c.f Ref 9) lo map singularbackgrounds to nonsingular ones. In plain wonis, one mayformulate dualities in terms of physieally eqllivalcnt vacua of

I

lhe lheory. Imagine one has lwo backgrounds A and B, whereevery physical quanlily of A may be wriuen in lerms of quan-lilies of B. Now, if A is singular bul B is nOl, we can say !hallhe strings don't care, and the singularity of A is just irrele-vant. This is lo say thal we were using the wrong degrees offreedoll1, oc wrong "tools" lo describe the low energy limit oflhe lheory.Anolher hinl !hal differenl degrces of frcedom mighl be

relevanl comes already from lhe following simple [10] ex-ample. Imagine there is a hidden extra dimension in the the-ory and lake lhe scalar field spatially nal FRW cosmology infour dimensions. It is of course singular at the "Bang". Now,imagine lhis sealar field is just a "trace" of an extra dimen-sion. The 5 dimensions Iifted cosmology has a nonsingularsealar curvaturc, This may hint mat the 4 dimensions singu.larily is jusI an artifacl of integrating lhe degrees of frccdomassociated with the extra dimensiono The extra dimensionseome naturally, oc 1 would ralhcr say are mUSl, if ene censid-ers S/M -lheory. So, lel USlry [he lwo lhings:

l. Use the extra dimensions.

2. Use lhe M-lheory symmelries, lo gel some info aboulsingularities.

Our starting poinl would be a 4-D cosmology wi!h a bunch ofma"less minimally coupled scalar ficlds (for more details seeReL II l. Imagine we have designed an algorilhm lo conSlroclsuch solUlions. (One can do il for quile a general e1ass ofmodels). We will concentrale on homogeneous scalar fields'P,(t) = q''Po(t). JuSI lhink of lhese fields as all having lhesame functional dependence, bUIdifferent ampliludes.The field equalions are, of course, invarianl under O(N)

rolation belween lhe fields. To be even more specific, lel uslook at !he inhomogeneous generalizalion of spatially openFRW cosmo!ogy wilh N scalar fields.

ds' = (s¡nh 2t) ~(3"-I) (cosh 4t - cosh 4z) l(i -") (-dt' + dz') + ~sinh 2t s¡nh 2z (tanh z dx' + cotanh z dy') .

Here </Jo = (../3/2) log tan(t), ami .\ is a sum of lhe squares of lhe ampliludes of lhe scalar fields and is an O(N) invarian!.The expression is wriuen for generie .\, bUl in particular case.\ = 1 lhe model is jusI an open FRW Solulion (lhough in unusualen-ordinales) coupled lo N scalar fields </J, ~ Pi log tan(t).

Lct us now "Iift" lhis solution to 4 + N dirncnsions ami sec whal happens.The 4 + N dimensional metric is:

N

dS~+N = 2 (sinh t) \- E!"., P' (cosh t)l+ E!"., P' (-dt' + dz' + sinh' z dx' + cosh' z dy') +L tanh'P' t (dwi)'.

i=l

The curvature invariant for the metrie:

nabcdnabcd ~ C(p,)t'(5-3) + 0[t'(5-')J

where we have wrillen S = L::\ Pi and C(Pi) is defined by

3 N N

C(Pi) = -(S - I)'(S' - 2S + 5) + LP;[3 +pi + (S - 3) (Pi + S)J + LP'Pj'lO . . .1=1 1<)

Rel~Mex. Fú. 49 SI COO::!)30-37

34 A. FEli'~"TEIN

The remarkable poinl aboul aH lhis is lhat lhe Krelehmanscalar RabcdRabcd is regular whcncvcr the condition ). = 1is satisficd. Thus, it may rcally be the case that the singular-ity in 4 dimensions is just an artifact of intcgrating the extradcgrees of frecdom cncodcd in highcr dirnensions.

Let lIS now movc forward and specify the numbcr of ex-1radimcnsions lo 7. In the 11 dimensional gcomctry wc canparametrizc the moduli mctric with the numbcrs Pi, whichmusl satisfy the following constraint:

,,7 , 2"I=L-,IPi +3L-,'#jP'Pj'

Now, wc want lo think of these solutions as rcprcscnt-ing vaeua of lhe M-lheory. Therefore we ean look now allhe U duality group of M eompaelificd on (SI f. It happensthai this group is gencrated by thc so.called ~ transfonnationwhieh permules the

(PI,'" ,P7)--+

(28 2s 28 8 s)

PI - 3,1"2 - 3,P3 - 3,P3 + 3"" ,P7+ 3

wilh 8 = PI + 1"2 +P3.Thcsc transfonnations map vacum solutions ¡rILO vac-

uurn solutions and ane can check thal singular solutions maybe trallsformed by lhis lransfonnation into no-singular ones.The U-dualily lrausformalions are eonjeetured lo be an exaelsymmelry of lhe M-lheory, and lhe fael lhal lhe lwo baek-grounds, olle singular. ami one regular are conncctcd via aU-duality transfonnation may indicate. as in the case of ho.mogeneity ami isotropy. lhat certain kinds of singularilies donot matter. The results presented hcre is not a rigorous study,of course, nevertheless one can gel an idea of what mighlhappen near the singularily if lhe ideas o slring eosmologyapply.

Finally, I would like lo presenl you wilh a seenario ofa PBB universe where one slarts wilh a small pcrturhationwhieh eollapses, inllales, gels rid of all iLSirregularities be-fore approaching the intennediate singularity and after exil-ing approaehes lhe slandard model.

4. A Universe from "almost nothing"

Il is more or less aeeepled lhal lo solve lhe fialness/horizonproblem in cosmo!ogy one must invoke lhe idea of acceler-aled expansiono There are IwO possibilities whieh provide usWilh aeeelerated behaviour of lhe seale faetor of lhe universe:

1. The standard inftation

2. Thc PB S scenario.

In lhe PBB pielllre one starLswilh lhe weakly eoupled regimeof lhe string lheory and ils dynamies is eontrolled by lhe lowenergy effective action. This is in conlrast with the standardinfiation where one start\) with physics on Plank scalc amithere is no way to extract a "decent" Lagrangean description

for lhe inflaton fíeld from a fundamental lheory. Moreover,lhe mere exiSlenee of lhe dilalon eoming from !he slring lhe-ory is quile "unhealthy" for standard pielure of inllation.

To address lhe problem of initial conditions in PBS sce-nario, Bounanno. Damollr, and Veneziano (BDV) [12J haveformulated lhe basic postulate of "asymptotic past triviality"idcntifying the initial state of lhe universe with lhe genericperturhative solution of lhe tree-Ievel low encrgy effccliveaClioTl. In this picture the initial state consists of a bath ofgravilational and dilatonic waves, some of which could haveeollapsed leadiug/modulo sollllion lo exit problem lo an FRWuniverse. This piclure generalizes ami makes more concreteprcvious sludies of inhomogeneous versions of PBS cosmol-ogy [13, 151.

However, wilhin lhis generie approaeh one may perfeeliywell formulale aud idenlify lhe problem, yel lhere are diffi-elllties in resolving it. Whal we proposed [16,171 is lo prohelhe model wilh redueed diffieullies, starling wilh slriellyplane waves. Recently this work was generalized to higherdimensions and extra fields in Ref. 19. In this picture onesolves lhe eollapselinllation problem analylieally relating lheKasner exponents near the singularity with the initial data forthe dilaton ami gravity waves. Once we do so, lhe path is freeto ask some phenomenological questions as lo the conditiollsfor successful inf1alion. or entropy generalion in these mod-els.

We will be imposing lhe two BDV conditions in lhe dis-tant past:

l. The slring lheory musl be weakly eoupled,exp('P/2« 1).

2. The eurvature is smal! in string unils.

lb simplify lhe lhings we slarl wilh lhe gravilon-dilalon sys-temo

Furtherrnore, we will assurne throughout thal the extra sixspatial dimensions are compactified in sorne intemal appro-priate manifold cOllsidercd lo be non-dynamical.

The gravily wave being linearly polarized is speeifiedhy a single funelion 1/J(r,8) and lhe sealar wave hy 4>(r, 8).where T ami s are nul! coordinates T = t - z and s = t + z.

Both waves satisfy ¡he same linear Euler Darboux cqua-lion in lhe inleraetion region (e.[ ReL 18).

11/J.r>+ 2(r + 8) (1/J,r + 1/J,,) = 0,

1rP,r> + 2(r + s) (rP,r + 4>.,) = O.

Thcse lwo equations. togcther with the initial data on thenul! boun<!aries of lhe inleraetiou region {1/J(r, 1), 1/J(I, s)}anel {4>(r, 1),4>(I,s)} pose a wel! defined initial value prob-lem. Bolh 1/J(r, s) and 4>(r,8) are el (and pieeewise e')functions.

Re\'. Mex. Fís. 49 SI (2002) 30-37

SOME ASPECfS OF PRE BlG BANG COSMOLOGY 35

4.1. Entropy generalion in a wave~collision-induced PUB

and a similar expression for lhe dilaton

Dne of lhe intcresting qucslions in lhis scenario is the waythe cntropy is generaled in lhe collision. First, lel me argue

that the enlropy in lhe plane wave region is O (both lhe mat-ter and gravitational). Imagine you wanl lo identify lhe statesof O gravitalional entropy. Firsl to mind comes Flal SpaeeTime. Then? Penrose suggesled FRW spaeetime due 10 OWeyl tensor. But if we takc the string lheories as a guide lheFRW doesn't fil that well. Plane waves on lhe eontrary rep-resent lhe most simple exael [20] string baekgrounds (1): ailthe curvature invariants vanish. So (2) if one would try to de-scribe the gravitational enlropy wilh a homogeneous functionvanishing aLlhe origin (not jusI lhe Weyl tensor) one wouldehoose the plane waves as O entropy.

The 3rd poinl is due to triviality of plane wave s-t with re-specl to vacuum polarization [21]: no particle creation! Dneusually relates the quanlum particle produetion (graviton,)10gravilational entropy-no such production t..:'1kesplace in lhevicinily of plane wave. stressing againlhe trivial entropy con-lenl.

WhaL happens then, when two sueh waves eollide? Theinteraclion region of lhe two incoming waves can be dividedinto three regions. Just aftcr the collision takes place al tithe ma1ter contenl of the un¡verse can be satisfaclorily de-scribed in lenns of a superposition of the lwo incoming non-interacting null fluids. In lhis regime the evolution is approx-imately adiabatie and almost no entropy is produeed. Thisalmost linear rcgirne comes 10 an end ao;;soon a"ilhe gravita-tional nonlinearities t.'1keover and lhe collision enters an in-termediale phase where lhe dynamies of lhe universe is dom-inated by both vclocities and spatial gradients. Now lhe evo-lution is no longer adiabatie and matter entropy is generated.In lhis regimc the ma1tcrcontenl of lhe universe cannol be de-seribed in terms of a perfeet fluid equation of stale [22] andsorne effective macroscopic dcscription of lhc fluid, such asan anisolropic fluid or sorne other phenomenological stress-energy tensor, should be invoked.

The produetion of entropy will stop al lhe momenL thevelocities begin to dominate over spatial gradienL"iami lheevolution becomes adiabatie again, the maller now being de-seribed by a perfeel fluid with stiff equation of state p = p.The lransition to an adiabatie phase will happen eilher be-fore or al t K when lhe universe cnters lhe Kmmer phase.From lhat moment on no further entropy is produeed up tolhe end of DDI. The relative duration of lhese three regimeserueially depends on the strenglh of lhe ineoming waves andon lhe iniLial dal,". lt is straightforward to show lhal for bothgravitational and scalar wavcs lhe ratio of spatial gradientsversus time derivatives dies off as 1/"¡ £1£2 and therefore itlakes the univcrsc a time of arder .J £1 £2 before lhe Kasner-like regime is reached. fixing the duralion of the nonadiabalicphase.

As diseussed abo ve, aH the entropy is generatcd in lhe in-terrnediale phase betwecn the adiabatic and Ka"inerregimes.On lhe other hund, in the Kasner regime lhe lOlal entropygenerated in lhis inlermediale region is carned adiabaticallyby the perfeet stiff fluid represenled by lhe dilaton field<p(t). If we consider a generie bariotropic cquation of state,

[ , ] (r+1)~(1 + r)' <p(r, 1) --,r r + z

[1 ] (r+1)~(1 + r)'1/I(r, 1) -

Ir r + z

1 11

+--- dr,,~ -,

+ __ 1__ r' dr,,~J_,

1 11[ , ] (s + 1) ~<p(z)= vT+Z ds (l+s)'<p(l,s) -

7r 1+ z Z ,8 S - Z

The € and 'P are lhe sourcc functionsexpresscd in tenns ofthe gravitational and scalar field respectively. So, for exam-pie, wc may ",k whieh are the models undergoing the PBBinflation among the all possible solutions pararnetrized by ,and 1", and the answer is "the piece" of cake in the plane ,and 1", meaning that the initial data leading to PBB is dense.

If lbe ineoming data is weak (, elose 10 O and 1" around-1) lbe PBB inflation takes place. The faet that Lheinflation-ary data is dense one may say that PBB is stable a feature ofthis eollision.

How weak should Lhewaves be lO solve horizonlf1atnessproblem?

One usually defines lbe Z faeLor whieh is a ratio of co-rnoving Hubblc radius at lhe time when lhe Dilalon DrivcnInflation slart, 10 the Hubble radius al the time the pieLurebreaksdown either due to the strong coupling or to the strongcurvature. Dile may work lhal the slrong curvature regimc,and therefore lhe loop eorreetions are reached before lhe eor-rections in c/ become impOrlanl. Expressing the conditionsof successful inflatioll in tcrms of the focusing Icnglhs of lhewaves one finds:

Now, you may analytically intcgrate the solutions intolhe interaction region, ami work oul lhe Kasner exponell1sin tenns of the incorning inilial dala.

The line elernenl near the singularily is

ds2 = ~a(,) (_~2 + dz2) + ~1+,(') dx2 +e-,(,)dy2,where a(z) ;: }[,2(Z) + <p2(Z) - 1], with

,(z)= ~r'dS [(l+s)~1/I(l,s)] (S+l)~7r l+zJz ,8 s-z

{3 = 2<p(z)- a(z) + 2

Herc £1 and £2 are lhe focallengths of the waves. Thereforethe waves must be extremely weak 10 solve lhe flatness amihorizon problems. But lhis is nice, since lhis is whal one cx-pecto;;anyway, if one hao;;in mind a piclure of lhesc waves assmall perlurbations on an olherwise flat background.

Rev. Mex. Fú'. 49 SI C(X)2) 30-37

36 A. FEI~STEIN

p = h- l)p, the first law of thermodynamics implies thatthe energy density p and entropy density s == S/V evolvewith !he temperature T as

-"-p = O"T"I-1,

,s = "(aT"'I-1 1

where the tcmperature is given by the usual relation T-I(8s/op)v. Here a is a dimensionful eonstant whieh in theparticular case of radiation ('y = 4/3) is just !he Stefan-Boltzmann conslant. Using these equations we easily fiudlhat for a stiff perfect nuid ('y = 2) the entropy density scan be expressed in terms of the energy density as

8 = 2,¡ap.

It is important 10 stress that (J is a pararneter that givcs the en-tropy comem of, and thus lhe number of degrees of freedomthat we a,"ociale wilh, the effective perfect nuid. In principieil could be computed provided a microscopical description ofthe nuid is available. However in the case at hand, the stiffperlect nuid is jusI an effective description of lhe e1assicaldilatan field in the Kasner regimc. Since the cvolution in theDDI phase is adiabatic, a can be seeu as a phenomenologi-cal parameter that measures !he amoum of enlropy generatedduring the intermediate region where the dilaton field shouldbe described by an imperfect effective fluid. This effectivedescriplion of the dilalon condensate, and lhe entropy gener-ated, would !hen depend on a number of phenomeuologicalpararncters.

Wc can now give an exprcssion foc the total cntropy in-side a Hubble volume at the beginning of DDI in terms of theinitial data. The energy density carried by the dilaton field inthe Kasner epoch is given by

~2 (32p(t) = 4&. = 4(2 t2'

PI Pl

Thus, the lotal entropy inside a Hubble volume al the be-ginning of the Kasncr regimc can be written in teems of theKasner exponeuts and lhe source funclion for lhe dilalon as

1. C.W.Misner, Phys. Rev. LeIl. 22 (1969) 107l.

2. A. Gulh, Phys. Rev. D 23 (1981) 347.

3. R. Pemase, "Singularities and lime-asymmctry", in GeneralRelativilY, an EinSleÚ¡ Cenrenary Survey', edited by S.W.Hawking and W. Israel, (Cambridge Univcrsity Press, Cam-bridge, UK, 1979).

4. G. Vcneziano, Phys. Leu. B 265 (1991) 287; M. Gaspcrini andG. Veneziano, ASlropart. Phy.'i. 1 (1993) 317; G. Vencziano,

Now we can wrile tK = r¡y'L,L2 wilh 0< r¡.:s 1, so finallywc arrive al

SI/(tK) = [r¡2Jat2PI 1(31 ] L~L2 == ",L,L2.0'10'20'3 {Pi (lPI

11 is intcrcsting 10 natice that this scaling [oc the en.lropy in lhe Hubble volume al lhe beginning of DDI can bealso retrieved by considering a particular example when, asa result of the wavc collision, a spacc-time locally ¡somet-ric to lhal of a Schwarzschild black hole is produced in theinteraction region. In that case lhe focal lengths of the in-coming waves are related wilh the ma" of thc black holeby [23J M = y'L,L,j (¡'l' Now we can write lhe Bekenstein-Hawking entropy ofthe black hole S = 47f(j"M2 in terms ofthcfocallengths ofthe incoming waves as S = 47fL1 L,j (~,.Thls analogy IS further supported by the fact that lhe temper-ature of the created quanlum partides in both the black holeami the colliding wave space-time scales Iike [24J T ~ l/Mand T ~ 1/ y'L, L2 respectively, as well as by the similar-ities between the thermodynamics of black holes and stifffluids [25J. It is importanl lo nOlice tbal this scaling of thelemperature of the created partieles with lhe focallenglhs im-plies that, whenevercnough innation oeeurs, the contributionof these partieles lo the total entropy is negligible.

One may have thought, in principIe, that it is possible loavoid tbe entropy prot1uction bclore lhe DDI by jusI lakinga soIution "inthe intcraction region for which the evolution isglobally adiabatic. The simplesl possibilily for such a solu-tion would be a Bianchi 1metric for wbich the Kasner regimeexlends all the way back lo lhe null boundaries. This, how-ever, should be immediately discarded due lo !he constrainlsposed by the boundary condilions in the colliding wave prob-lem. Or, put in terms of the null data, it is not possible tochoose the initial data on thc nuH houndaries in su eh a waythat lhe metric is globally of Bianchi 1 type in the whole in-teraetion region and at the same time el and pieeewise e2

across lhe boundary.

Acknowledgments

1am graleful to Nora Brelon of!he organizing commiuee forhospilality at Huatulco, and to Kerstin Kunze and lo MiguelVazquez-Mozo for many hours of fruitful collaboration anddiseussions.

"Infiating, warming up, and probing the prc-bangian universc",G. Vcncziano, "String cosmology: The prc-big bang scenari-o", in; 1he Primordial Universe, proceedings lo thc 1999 LesHouchcs SurnrncrSchool, edition by P. Binetruy, R. Schacffer,J. Silk, and F.David, (Springcr-Verlag, 2001) hcp-th!OOO2094;E.J. Copcland, A. Lahiri, and D. Wands, Phys. Re\~D 50 (1994)4868; J.E. Lidsey, D. Wands, and EJ. CopeJand, Phys. Rep. 337(2000) 343 hep-lhI9909061, for an updatcd collcction ol'paperson string cosmology, sce http://www.to.infn.itTgaspcrin.

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