Bouncing universe and non-BPS branes
Philipp Höffer v. Loewenfeld, Jin U Kang, Nicolas Moeller, Ivo Sachs
aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION LMU-ASC 26/09Boun ing universe and non-BPS branesPhilipp Hö(cid:27)er v. Loewenfeld , ∗ , Jin U Kang , , † , Ni olas Moeller , ‡ , Ivo Sa hs , § Arnold Sommerfeld Center for Theoreti al Physi s (ASC),Ludwig-Maximilians-Universität Mün hen, Theresienstr. 37, 80333 Mün hen, Germany Department of Physi s, Kim Il Sung University, Pyongyang, DPR. KoreaAbstra t: We des ribe string frame boun ing universe s enarios involving the reationand annihilation of a non-BPS D9-brane in type IIA superstring theory. We (cid:28)nd several lasses of solutions, in whi h the boun e is driven by the ta hyon dynami s of the non-BPSbrane. The metri and the dilaton are onsistently des ribed in terms of the lowest ordere(cid:27)e tive a tion. The boun e solutions interpolate between ontra ting and expanding pre-big bang (or post-big bang) solutions. The singular behavior of our boun e solutions isthe same as that of the pre-big bang (or post-big bang) solution. Upon adding a simpledilaton potential the asymptoti urvature singularity is removed but the dilaton still growswithout bound. Su h a potential may result from α ′ orre tions in the open string se tor.Keywords: string theory and osmology, osmi singularity. ∗ von.Loewenfeldphysik.lmu.de † Jin.U.Kangphysik.lmu.de ‡ Ni olas.Moellerphysik.lmu.de § Ivo.Sa hsphysik.lmu.deontents1. Introdu tion 12. The model 23. Einstein frame and null energy ondition 54. Asymptoti analysis 75. Numeri al results 96. Nonsingular solutions (cid:21) Example 137. Con lusion 151. Introdu tionThe resolution of the big-bang singularity is an important open problem in standard os-mology (see e. g. [1, 2℄ for a review and referen es therein). At the same time it is a naturalplayground for string theory sin e quantum gravity orre tions are expe ted to be relevantin this regime. In the ekpyroti s enario [3, 4℄ and a re(cid:28)ned version, the y li universe[5℄, the hot big bang is the result of the ollision of two branes. Expli it y li modelshave been suggested as e(cid:27)e tive four-dimensional models inspired from heteroti M-theory.The boun e in the ekpyroti s enario o urs at a real urvature singularity and thus doesnot resolve the urvature singularity. The new ekpyroti s enario [6℄, realizes an expli itboun e dynami s by addition of a ghost ondensate but su(cid:27)ers from a va uum instabilityproblem (see [7℄). More generally, due to the ne essary violation of the null energy ondition(NEC) during a boun e, phenomenologi al models produ ing a boun e often su(cid:27)er fromthe problem of introdu ing matter with negative energy density, i. e. ghosts.There are other ways to address the big-bang singularity problem in string theory1. Thepre-big bang s enario (see [11℄ for a review) is a onsequen e of the fa t that the tree-levelequations of motion of string theory are not only symmetri under time re(cid:29)e tion t
7→ − t but also symmetri under the s ale-fa tor duality transformation a a with an appro-priate transformation of the dilaton. The `post-big bang' solution of standard osmologywith de elerated expansion de(cid:28)ned for positive times is by these dualities onne ted to anin(cid:29)ationary `pre-big bang' solution for negative times. In this way the osmi evolution is1Alternative suggestions avoiding the problem of introdu ing matter with negative energy density in ludeloop quantum osmology [8℄ and matrix models (e. g. [9, 10℄).(cid:21) 1 (cid:21)xtended to times prior to the big bang in a self-dual way but the solution is still singular.One an obtain regular self-dual solutions by tuning a suitable potential for the dilaton.But albeit a potential of this form might be the result of higher-loop quantum orre tions,its form has not been derived from string theory.In this paper we onsider a novel s enario in string theory where a boun e o ursin the string frame due to the reation of an unstable (non-BPS) brane as the universeboun es through a string size regime before expanding as the brane de ays. Our solutionse(cid:27)e tively interpolate either between a ontra ting and an expanding pre-big bang solutionor between a ontra ting and an expanding post-big bang solution. The future (past)singularity of the pre (post)-big bang solution is not resolved in this s enario. The ni efeature in our model is that the urvature as well as the dilaton and its derivative remainsmall (in string units) through the boun e so that referring to perturbative string theoryand the simplest low energy e(cid:27)e tive a tion for these (cid:28)elds is justi(cid:28)ed. In addition no (cid:28)netuning is required. On the other hand, we do not address issues like dilaton and modulistabilization whi h are of ourse important problems to embed this model into late timestandard osmology but are not relevant during the string s ale regime where the boun eo urs. We should stress that our string frame boun e solutions des ribe monotonously ontra ting or expanding geometries in the Einstein frame. A boun e in the Einstein framemay o ur upon stabilizing the dilaton asymptoti ally. However, this entails violating theNEC or, alternatively, allowing the gravitational oupling to hange sign in the string frame.We will dis uss a model where the latter e(cid:27)e t o urs.Our work is organized as follows. This paper onsists of seven se tions, of whi h thisintrodu tion is the (cid:28)rst. The model we employ is des ribed in Se tion 2. In this se tionwe present the e(cid:27)e tive a tions and the equations of motions for the metri , dilaton andta hyon from a non-BPS brane, whi h makes the boun e possible. In Se tion 3 we showthat our model has no ghost by onsidering the null energy ondition in the Einstein frame,and some features of string frame boun e s enarios are studied in relation to the Einsteinframe. In Se tion 4, the asymptoti behavior of the solutions is analyzed and its qualitativesimilarity to pre-big bang s enario is lari(cid:28)ed. The numeri al determination of the globalboun e solution is presented in Se tion 5. In Se tion 6 we present a simple model thatresolves the asymptoti urvature singularity in the string frame. Finally we on lude inthe last se tion.2. The modelWe onsider a non-BPS spa e-(cid:28)lling D9-brane in type IIA superstring theory. The detailsof the ompa ti(cid:28) ation will not play a role here. Con retely we onsider the lowest ordere(cid:27)e tive a tion for the metri and dilaton in the string frame as well as an e(cid:27)e tive a tionfor the open ta hyoni mode of the non-BPS D-brane. For now, the only assumption beingmade for the ta hyon a tion is that only the (cid:28)rst derivatives of the ta hyon appear in it. Theansatz for the gravitational a tion is justi(cid:28)ed provided the dilaton and metri are slowly(cid:21) 2 (cid:21)arying in string units. We write S = 12 κ Z d x √− g e − ( R + 4 ∂ µ Φ ∂ µ Φ) + S T (2.1)with S T = Z d x √− g e − Φ L ( T, ∂ µ T ∂ µ T ) , (2.2)where Φ is the dilaton, T is the ta hyon, and κ = 8 πG with G the ten-dimensionalNewton onstant. We use the signature ( − , + , . . . , +) for the metri . With these onven-tions, the matter energy-momentum tensor is given by T µν = 2 √− g δS T ( T, ∂ ρ T ∂ ρ T, Φ) δg νµ . (2.3)For a Lagrangian minimally oupled to gravity, the metri appears only in ( ∂T ) = ∂ ρ T ∂ ρ T ,and we an write the energy-momentum tensor as T µν = 2 e − Φ ∂L ( T, ( ∂T ) ) ∂ (( ∂T ) ) ∂ µ T ∂ ν T − δ µν e − Φ L (2.4)In a homogeneous isotropi universe that we will onsider, all (cid:28)elds are assumed to dependonly on time; the energy density ǫ and pressure p are given by ǫ = T , p = − T ii (no sum) = e − Φ L. (2.5)We are now ready to make an ansatz for the metri . We take a four-dimensionalspatially (cid:29)at FRW spa etime times a six-torus hara terized by a single modulus σ . Namely d s = − d t + a ( t ) δ ij d x i d x j + e σ ( t ) δ IJ d x I d x J , (2.6)where the lower- ase Latin indi es run over the three un ompa ti(cid:28)ed spa e oordinates,while the upper- ase Latin indi es label the six ompa ti(cid:28)ed dimensions. We further sim-plify the problem by restri ting to the ase where a ( t ) = e σ ( t ) . Although equality of thetwo s ale fa tors is phenomenologi ally not satisfying at late times it may be assumed nearthe osmologi al boun e whi h is the prime fo us of this paper. In parti ular, we will notaddress the important problem of moduli- and dilaton stabilization required to onne t tothe standard osmology at late times. With the ansatz (2.6) the Einstein equations are nowe(cid:27)e tively isotropi . And in parti ular the relations (2.5) an be used. From now on theindi es i, j in lude I, J .The equations of motion for g , g ii and Φ are then given by the following (cid:28)rst threeequations H − H ˙Φ + 4 ˙Φ − κ e ǫ = 0 , (2.7) − H + 16 H ˙Φ − − H − κ e p = 0 , (2.8) H ˙Φ − − H − H − κ p = 0 , (2.9) ˙ ǫ + 9 H ( ǫ + p ) − ˙Φ p = 0 , (2.10)(cid:21) 3 (cid:21)nd the last equation follows from the generalized onservation law ∇ µ T µν = ( ∂ ν Φ) e − Φ L .In the ase where ǫ = 0 and p = 0 , these equations allow exa t solutions, namely H = ± t , ˙Φ = − ± t . These are the spe ial ases of the pre -big bang ( t < ) and post-bigbang ( t > ) solutions, whi h respe t the time re(cid:29)e tion symmetry ( t
7→ − t ) and s ale-fa torduality symmetry ( a a − ) (see for example [12℄).Let us explore the possibility of the boun e. Subtra ting Eq. (2.9) from Eq. (2.8) we(cid:28)nd ˙ H + 9 H − H ˙Φ − κ p = 0 . (2.11)This is an important equation. It tells us that a ne essary ondition to have a boun e, H = 0 and ˙ H > , (2.12)is that the ta hyon pressure p must be positive. This is a tight onstraint for a s alar(cid:28)eld a tion; the Born-Infeld a tion for instan e, whi h is an often used ansatz as a higherderivative s alar (cid:28)eld a tion, gives a pressure that is always negative.Furthermore, assuming that H is negative during a ontra ting phase with growingdilaton and a negative pressure, (2.11) implies ˙ H < , i. e. a elerated ontra tion. Onthe other hand, for positive equation of state for the s alar (cid:28)eld, w > , the onservationequation in (2.10) gives a growing pressure p in the ontra ting phase so that a (cid:16)turn around(cid:17) ˙ H = 0 is ompatible with (2.11).Of ourse, a Born-Infeld a tion for the ta hyoni se tor of non-BPS branes is not in anyway suggested by string theory. On the other hand, within the restri tion to (cid:28)rst derivativea tions it is possible to derive an approximate e(cid:27)e tive a tion from string theory for theopen string ta hyon of an unstable brane. This a tion, onstru ted in [13℄ and furtherstudied in [14℄ is given by L = −√ τ e − T α ′ (cid:16) e − ( ∂T ) + p π ( ∂T ) erf (cid:16)p ( ∂T ) (cid:17)(cid:17) , (2.13)where τ is the tension of a BPS 9-brane, and therefore √ τ is the tension of a non-BPS9-brane [15℄. Let us shortly summarize how this a tion was onstru ted. First, setting ( ∂T ) to zero, we see that the potential is given by V ( T ) = √ τ e − T α ′ . (2.14)This is the exa t potential for the open string ta hyon potential found in boundary su-perstring (cid:28)eld theory [16, 17, 18℄. The lo ations of the minima of V ( T ) are at T = ±∞ .At these values the energy is degenerate with the losed string va uum whi h means thatthe non-BPS brane is absent. The onstru tion of the full a tion (2.13) is based on theobservation that the ta hyon kink T ( x ) = χ sin( x/ √ α ′ ) , where x is one of the spatialworld volume oordinates, is an exa tly marginal deformation of the underlying boundary onformal (cid:28)eld theory [19, 20℄ and thus should be a solution of the equations of motionobtained from (2.13). It turns out that this requirement determines uniquely the a tion(cid:21) 4 (cid:21)n e the potential has been hosen. Furthermore, it follows by analyti ontinuation thatSen's rolling ta hyon solution [21℄ T ( t ) = A sinh( t/ √ α ′ ) + B cosh( t/ √ α ′ ) is also a solution of the a tion (2.13) for all values of A and B . In this dynami al de ay(or reation) of the non-BPS brane the energy is onserved. The asymptoti state for largepositive (or negative) times has been argued to be given by "ta hyon matter" - essentially old dust made from very massive losed string states. - - PSfrag repla ements
Fw X Φ tTwH E H - - - - - PSfrag repla ements
F w X Φ tTwH E H Figure 1: X -dependen e of the pressure F ( X ) . Figure 2: Equation of state w of theta hyon as a fun tion of X .Let us now brie(cid:29)y explain how this a tion an allow a positive pressure, or equivalentlya positive Lagrangian [14℄. This follows from the fa t that it is real and ontinuous alsofor negative values of ( ∂T ) . Indeed if we write − ( ∂T ) = X (note that X = ˙ T in thehomogeneous ase), we an see that √− πX erf( √− X ) = − √ X Z √ X e s d s . (2.15)This is negative and grows in absolute value faster than the (cid:28)rst term e X in the Lagrangian;so for positive enough X (for negative enough ( ∂T ) ), the Lagrangian, and thus the pressure,is always positive. This an be seen from Fig. 1, where we show the X - dependen e ofthe pressure, F ( X ) ≡ − (cid:16) e X − √ X R √ X e s d s (cid:17) . In terms of F the Lagrangian an beexpressed as L = √ τ e − T α ′ F ( X ) . From the (cid:28)gure it is lear that d F/ d X > (this isalso lear from d F d X = √ X R √ X e s d s > ). From now on we work in the unit system with α ′ = 1 / . The energy density and pressure of the ta hyon are ǫ = √ τ e − Φ e − T + ˙ T , p = √ τ e − Φ e − T F ( ˙ T ) . (2.16)Note that the equation of state w = ǫ/p depends only on ˙ T as shown in Fig. 2. Inparti ular, w → as ˙ T → ∞ , while w → − as ˙ T → .3. Einstein frame and null energy onditionIn this se tion we show that our model has no ghost and satis(cid:28)es NEC in the Einsteinframe. This is important be ause there might be a pathology due to an instability oming(cid:21) 5 (cid:21)rom NEC violation. On the other hand, the onsideration on NEC will help us draw moregeneral on lusions on erning the boun e s enarios.The a tion (2.1)-(2.2) in the string frame with L given in (2.13) is expressed in theEinstein frame by means of a onformal transformation g µν = ˜ g µν e Φ2 (where ˜ g µν is themetri in the Einstein frame). This yields S = 12 κ Z d x p − ˜ g ˜ R − ( ˜ ∇ Φ) ! + S T (3.1)with S T = Z d x p − ˜ g e L ( T, ˜ X e − Φ2 ) , (3.2)where ˜ R and ˜ ∇ are the s alar urvature and the ovariant derivative asso iated with ˜ g , and ˜ X = − ( ˜ ∇ T ) = X e Φ2 .The relevant quantity for verifying lassi al stability and nonexisten e of ghost is thesign of the slope of the kineti term with respe t to the (cid:28)rst derivative of the (cid:28)elds, whi hshould be positive. If it is negative, it signals the existen e of a ghost (quantum me hani alva uum instability) and also that the squared speed of sound is negative ( lassi al instabil-ity) [7℄. Sin e the dilaton has the orre t sign for the kineti term in the Einstein frame,the only possible sour e of ghost is the ta hyon Lagrangian. Therefore, one should only he k the sign of d L d ˜ X . We have that d L d ˜ X = d L d X d X d ˜ X ∼ e Φ2 d F d X = e Φ2 √ X R √ X e s d s > . So thereis no ghost and therefore no violation of NEC in our model.Now we turn to the issue of the boun e. In the spatially (cid:29)at FRW spa etime we are onsidering, the Hubble parameter H E in the Einstein frame is related to that in the stringframe by H E = H − ˙Φ / . Sin e the null energy ondition is not violated in our model,the Hubble parameter in the Einstein frame monotonously de reases, so that the boun e annot arise in the Einstein frame. If the dilaton were frozen, it would be impossible tohave a boun e in the string frame as well be ause the two frames would be trivially related(in parti ular H E = H ). This is why the running dilaton is ru ial for the string frameboun e. Now we make some remarks on the boun e s enario in the string frame. Weassume that the boun e arises as an interpolation between two out of four di(cid:27)erent phases(i. e. ontra ting or expanding pre/post-big bang phases) in the pre-big bang s enario. Fourtransitions are then possible, namely from the ontra ting pre-big bang phase to expandingpre/post-big bang, or from ontra ting post-big bang to expanding post/pre-big bang. Butthe transition from pre-big bang to post-big bang annot happen in our model be ause theNEC in the orresponding Einstein frame is not violated. To show this, we note that in thepre-big bang s enario with S T = 0 the solutions are H = nt − t , ˙Φ = 9 n − t − t ) with n = ± . (3.3)Here, t < t orresponds to the pre-big bang phase, and t > t to the post-big bangphase. It then follows that H E = − n t − t ) is negative for pre-big bang solutions and positivefor post-big bang solutions, meaning that pre/post-big bang solutions orrespond to a ontra ting/expanding universe in the Einstein frame, regardless of whether the universe(cid:21) 6 (cid:21)s ontra ting or expanding in the string frame. This means that a transition from pre-bigbang to post-big bang in the string frame orresponds to a boun e in the Einstein frame,whi h is impossible unless the NEC is violated. Therefore, under the assumption mentionedabove, the only possible string frame boun e s enarios in our model are the interpolationseither between two pre-big bang phases or between two post-big bang phases.This argument an be extended to the ase where the asymptoti behavior of thesolutions in the string frame is qualitatively, but not exa tly, in agreement with the pre-big bang s enario, This is the ase in our model, as we will see in the next se tion. In on lusion, if one is given a model that does not violate the NEC and if one knows theasymptoti boundary onditions of the solution in the string frame, one an then predi tthe possible boun ing s enarios in the string frame by looking at the orresponding Einsteinframe. An example will be given in the next se tion.4. Asymptoti analysisIn this se tion we will try to obtain approximate analyti solutions. This will provide uswith the asymptoti boundary onditions for the numeri al solutions in the next se tion.We emphasize here that by (cid:16)asymptoti (cid:17) we mean t → −∞ , or t approa hing a pole t , atwhi h H diverges, as in the pre-big bang s enario. Similarly the present analysis applies to t → ∞ , or t approa hing a pole t , at whi h H diverges as in a post-big bang s enarioTo simplify our analysis, we will assume that in either of these limits, the ta hyonbehaves like dust, i. e. p = w ( t ) ǫ with w ( t ) → . This is equivalent to laiming that | ˙ T | → ∞ asymptoti ally be ause p ∝ ǫ/ ( ˙ T ) when | ˙ T | → ∞ (see [14℄ for details). To justifythis assumption, we look at the ta hyon equation of motion following from (2.13). We have ¨ T + (cid:16) H − ˙Φ (cid:17) D ( ˙ T ) − T = 0 , (4.1)where the fun tion D ( y ) = e − y R y e s d s is known as the Dawson integral. This fun tionis an odd fun tion, and thus vanishes at y = 0 . We will also use the fa t that D ( y ) = y + O ( y − ) for | y | → ∞ . For | t | → ∞ , we will see that (9 H − ˙Φ) tends to zero. We an thus ignore the se ond term in the equation of motion2 (4.1). Thus ¨ T = T and T ,as well as ˙ T , will grow exponentially; and the pressure will thus vanish for | t | → ∞ . Weemphasize that sin e the pressure vanishes exponentially fast, we an simply remove it fromthe asymptoti equations of motion be ause it will always be dominated by the other termsthat will vanish with a power law.When t approa hes a pole t the analysis is slightly di(cid:27)erent be ause there the term (9 H − ˙Φ) diverges (unless ˙Φ ∼ H , but we will see that this does not happen on ournumeri al solution). We will further assume that this term diverges at least as fast as t − t .Assuming that ˙ T is (cid:28)nite at t this then implies that either T or ¨ T will diverge like t − t orfaster, in ontradi tion with a (cid:28)nite ˙ T . We annot ex lude the ase where ˙ T ( t ) vanishesin su h a way that it pre isely an els the divergen e in (9 H − ˙Φ) . In that ase the se ond2Note that we ignore the possibility T → in the in(cid:28)nite past or future; we are only interested in the ases where the D-brane is absent in these limits. (cid:21) 7 (cid:21)erm of the equation of motion ould be regular at t , and thus T and ¨ T ould be regularthere as well. We will nevertheless ignore this possibility be ause ˙ T ( t ) orresponds tovery parti ular initial onditions. For generi initial onditions we will therefore have that | ˙ T | → ∞ when t → t . This then justi(cid:28)es our laim that we an ignore the pressure for theasymptoti analysis. An immediate onsequen e of Eq. (2.10) is then that ǫ ∼ a − .We now will pro eed by analyzing the system of equations (2.7-2.10) assuming thateither ˙Φ ∝ H , | ˙Φ | ≪ | H | or | ˙Φ | ≫ | H | asymptoti ally.i) Let us (cid:28)rst onsider the possibility ˙Φ ∝ H . In that ase (2.8) and (2.9) imply that either | ˙ H | ≫ H or ˙ H ∝ H . In the (cid:28)rst ase we get ˙Φ ≃ H and then (2.7) implies that − H = 2 κ e ǫ (4.2)i.e. negative energy. We thus ex lude that possibility. In the se ond ase from(2.8-2.9) we obtain the solutions of the pre-big bang s enario, i. e. Eq. (3.3). For onsisten y, we must verify that these solutions satisfy the onstraint (2.7). This isthe ase only when the energy density is subdominant ompared to the other termsin this equation. By using (3.3), one an see that e ǫ goes like | t − t | while the otherterms behave like t − t ) , so the energy is subdominant as t → t . Thus the solutions an be approximated to those of the pre-big bang s enario near the pole, t . At thesame time we see that this possibility is ex luded for | t | → ∞ .ii) For | ˙Φ | ≪ | H | Eqs. (2.8) and (2.9) imply H + 5 ˙ H = 0 . On the other hand, setting p = 0 in Eq. (2.11) gives us ˙ H + 9 H = 0 , a lear ontradi tion. Thus, | ˙Φ | ≪ | H | isex luded.iii) We are thus left with the sole possibility | ˙Φ | ≫ | H | . In that ase (2.7) implies = κ e ǫ, (4.3)and (2.8) together with (2.9) imply − ¨Φ + ˙Φ = 0 , whi h gives Φ = − log( | t − t | ) .With p = 0 , Eq. (2.11) then implies ˙ H − H ˙Φ = 0 , whi h gives H = h ( t − t ) , where h is some onstant. This is onsistent with Eq. (4.3) only for | t | → ∞ be ause e ǫ ∼ t as | t | → ∞ .To summarize, we (cid:28)nd that the only onsistent asymptoti solution for | t | → ∞ is givenby Φ ≃ − log( | t | ) , H ≃ ht . (4.4)and (3.3) for t → t . Note that H E ≃ − ˙Φ4 for (4.4).Now we are in the position to predi t the possible string frame boun e s enarios. Fol-lowing the same logi as in the last part of the previous se tion on erning the NEC in theEinstein frame, one an show that the only possible boun e s enario is the transition eitherfrom pre-big bang-like solution3 to pre-big bang solution or from post-big bang solution to3Note that what we refer to as pre/post-big bang-like solution is given in Eq. (4.4) with negative/positivetime (cid:21) 8 (cid:21)ost-big bang-like solution. This is be ause the other transitions orrespond to a boun e inEinstein frame, whi h is ex luded in our model whi h satis(cid:28)es the NEC. To explain this inmore detail, we onsider the ase where we start with ontra ting pre-big bang-like phase.For large negative times our boun e solution is in agreement with the pre-big bang-likesolution with a elerated ontra tion of the universe and growing dilaton. Then the uni-verse goes through a boun e and for t → t it approa hes a pre-big bang solution witha elerated expansion and growing dilaton. The Hubble parameter in the Einstein frameremains negative and keeps de reasing. As will be seen in the next se tion, the numeri alsolutions are in good agreement with this pi ture.5. Numeri al resultsIn this se tion, we numeri ally solve Eqs. (2.7-2.10) to obtain global solutions. In whatfollows we set √ κ τ = 1 (this an always be a hieved by adding a suitable onstant tothe dilaton), so that κ ǫ = e − Φ e − T + ˙ T (5.1) κ p = − e − Φ e − T e ˙ T − p ˙ T Z √ ˙ T e s ds . (5.2)With this setup, we performed the numeri al analysis and found a family of boun esolutions. For example, Figs. 3 show a boun e solution with the initial onditions, Φ(0) = − , ˙Φ(0) = 0 . , T (0) = 1000 and H (0) = − . . The graph in Fig. 3a shows theevolution of the Hubble parameter (solid line). The boun e takes pla e near t = 8 . Theboun ing solution an be seen as a transition from the ontra ting pre-big bang-like phase(short dashed line) to the expanding one (long dashed line). Both asymptoti solutions areobtained by setting p = 0 in the equations of motions sin e the pressure is negligible in thefar future and past (see Fig. 3b). The 'double bump' feature of the equation of state anbe understood by noting that as the non-BPS brane builds up | ˙ T | de reases and thus w in reases from zero as explained in se tion 2. Then as T rea hes the top of the potential | ˙ T | be omes small and onsequently w de reases again. Indeed for ˙ T = 0 the equation ofstate is that of a osmologi al onstant.We found that a broad range of the initial onditions are allowed for the boun e, sothere is no (cid:28)ne-tuning problem. For instan e, T (0) = ± and T (0) = ± (keepingthe same initial onditions as above for the other variables) gives boun e solutions withessentially the same behavior. Note that the large initial values for T do not represent a(cid:28)ne tuning. Rather it re(cid:29)e ts the ondition that the non-BPS brane is absent at very earlytimes. We see that the asymptoti behavior of this family of solutions is similar to theexpanding pre-big bang ase, in whi h the Hubble parameter blows up. This agrees withthe results of the previous se tion.If one hanges the sign of ˙Φ(0) , a very di(cid:27)erent kind of boun e is obtained. For instan e,Figs. 4 show a boun e solution with the initial onditions, Φ(0) = − , ˙Φ(0) = − . , T (0) = 5 and H (0) = − . . The graph in Fig. 4 shows the evolution of the Hubbleparameter H . The boun e takes pla e near t = 2 , and the universe smoothly evolves to the(cid:21) 9 (cid:21)
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FwX Φ tTwH E H (a) Hubble parameter H (solid line) PSfrag repla ements
FwX Φ tT wH E H (b) equation of state w - - - - - - - PSfrag repla ements
FwX Φ tTwH E H ( ) dilaton Φ PSfrag repla ements
FwX Φ tTwH E H (d) ta hyon T Figure 3: Boun ing numeri al solution with the initial onditions
Φ(0) = − , ˙Φ(0) = 0 . , T (0) = 1000 and H (0) = − . .standard osmologi al regime, where the Hubble parameter and the dilaton de reases withtime (Fig. 4). But going ba k in time further, we found a singularity where the Hubbleparameter blows up. This solution an be seen as a transition from the ontra ting post-big bang phase (short dashed line) to the expanding post-big bang-like phase (long dashedline). Both asymptoti solutions are obtained by setting p = 0 in the equations of motionsin e the pressure is negligible in the far future and past (see Fig. 4).In addition, we found os illatory solutions, in whi h a double boun e takes pla e (seeFigs. 5). The solution in Figs. 6 an be seen as a time re(cid:29)e ted one of the solution ofFigs. 5. Whi h solution is obtained depends on the sign of ˙Φ(0) . In both ases the asymp-toti behaviors towards the urvature singularity are analogous to the non-os illatory asesmentioned above. The evolutions of the equation of state are shown in Figs. 5 and 6. Notethat the negative equation of state implies small | ˙ T | (see Fig. 2), and this means that theos illatory solutions an arise if the speed of the ta hyon is small around the top of theta hyon potential (i. e. near T = 0 ). Alternatively this kind of solution an be obtainedwhen we arrange | ˙Φ | to be small enough around the top of the ta hyon potential (see Fig. 7).The variation of ˙Φ(0) with the other initial onditions (cid:28)xed (in this ase at Φ(0) = − , H (0) = 0 , and T (0) = 0 ) shows that de reasing ˙Φ(0) gives rise to a transition from singleboun e to y li boun e. (cid:21) 10 (cid:21) - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H (solid line) - PSfrag repla ements
FwX Φ tT wH E H (b) equation of state w - - - - - - PSfrag repla ements
FwX Φ tTwH E H ( ) dilaton Φ - - - PSfrag repla ements
FwX Φ tTwH E H (d) ta hyon T Figure 4: Boun ing, numeri al solution with non-singular future using initial onditions
Φ(0) = − , ˙Φ(0) = − . , T (0) = 5 and H (0) = − . . - - - - - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H - - - - - PSfrag repla ements
FwX Φ tT wH E H (b) equation of state w Figure 5: Os illatory, numeri al solution with initial onditions
Φ(0) = − , H (0) = 0 , ˙Φ(0) = 0 . and T (0) = 0 .In on lusion, we obtained boun e solutions, where the universe smoothly evolves fromthe ontra ting phase to the expansion. The boun e s enario that we here found an be lassi(cid:28)ed into the following two ases. Note that these are exa tly in agreement with ourpredi tions on the boun e s enarios in the previous se tion:(cid:21) 11 (cid:21) - - - - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H - - - - - PSfrag repla ements
FwX Φ tT wH E H (b) equation of state w Figure 6: Os illatory, numeri al solution with initial onditions
Φ(0) = − , H (0) = 0 , ˙Φ(0) = − . and T (0) = 0 . - - - PSfrag repla ements
FwX Φ tTwH E H Figure 7: Hubble parameter H for ˙Φ(0) = 0 . (solid line) and ˙Φ(0) = 0 . (dashed line).1. Transition from a elerating ontra tion (the ontra ting pre-big bang-like phase) toa elerating expansion (the pre-big bang in(cid:29)ation): In this ase the dilaton grows up,and if the speed of the ta hyon (or dilaton) is small enough near the maximum ofthe ta hyon potential, a double boun e an take pla e (Figs. 5) before the universeevolves to the pre-big bang phase.2. Transition from de elerating ontra tion (the ontra ting post-big bang phase) tode elerating expansion (post-big bang like phase): In this ase the dilaton de ays, anda double boun e an also take pla e (Figs. 6) under the same ondition mentionedabove.In all ases the ta hyon rolls over the top of the potential in the ourse of its evolution,and the boun e seems to happen when the ta hyon rea hes around the top of the potential.The pressure is important only around the boun e, and negligible (dust) asymptoti ally.Our string frame boun e solutions orrespond to monotonously ontra ting (i. e. H E = H − ˙Φ4 < ) or expanding geometries ( H E > ) in the Einstein frame (see Figs. 8-9),meaning that there is no boun e in the Einstein frame for our solutions. This is the fa tthat our model does not violate the NEC in the Einstein frame.(cid:21) 12 (cid:21)
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FwX Φ tTw H E H PSfrag repla ements
FwX Φ tTw H E H Figure 8: Hubble parameter H E in the Ein-stein frame orresponding to Fig. 3 (a). Figure 9: Hubble parameter H E in the Ein-stein frame orresponding to Fig. 4 (a).The numeri al results presented in this se tion verify the results on the asymptoti sin the previous se tion. Interestingly, as far as the dilaton Φ is on erned the asymptoti solution obtained here agrees qualitatively with the global numeri al solution. In otherwords the dynami s of the dilaton is not mu h a(cid:27)e ted by the presen e of the non-BPSbrane and is qualitatively similar to that of the pre-big bang s enario. Con erning H ( t ) things are di(cid:27)erent: See Fig. 3 for example. For large negative times H ( t ) is well des ribedby the asymptoti solution des ribed in Eq. (4.4). Then near the boun e whi h takes pla eat t ≃ H ( t ) signi(cid:28) antly. Then for t → t > (near thepole), where the pre big-bang singularity o urs, the Hubble onstant is well des ribed bythe solution in the pre-big bang s enario. This an be understood from the fa t that thebrane has already de ayed for t → t .6. Nonsingular solutions (cid:21) ExampleAs we have seen in the previous se tion, there is a singularity either in the future or in thepast, depending on the sign of ˙Φ , and we expe t that this singularity may be resolved inthe same way as in pre-big bang s enarios, e. g. relying on α ′ orre tions or quantum loop orre tions or alternatively using a dilaton potential (see [12℄ for some expli it examples).As a matter of fa t, resolving this kind of asymptoti singularities is less di(cid:30) ult than thebig-bang singularity. In this se tion we will give an example of resolution of this asymptoti singularity.Near the singularity the urvature and the dilaton blow up, and this suggests that apotential term of the form R e Φ in the Lagrangian may smooth out the singularity. Herethe oupling between the Ri i s alar and the dilaton is introdu ed be ause the singularityappears both in the urvature and dilaton. Su h a term is quite likely to appear as α ′ orre tion in the open string se tor, sin e it has the form of a tree level orre tion in theopen string oupling onstant and R is the natural invariant built from ba kground metri derivatives5.4In fa t one an always arrange the boun e to take pla e at t = 0 by shifting the time variable.5A similar potential has been motivated in the ontext of string gas osmology in [22℄ as a Casimir-typepotential. (cid:21) 13 (cid:21)hus as an example, in whi h su h an additional term may resolve the singularities,we study the dynami s of the system where the a tion (2.1) is supplemented by a potential κ Z d x √− g e − RV (Φ) . (6.1)The equations of motion then take the form H (1 + V (Φ)) + 4 ˙Φ − H ˙Φ (cid:18) V (Φ) − V ′ (Φ)2 (cid:19) − κ e ǫ = 0 (6.2) − + 18 H ˙Φ − (cid:16) H + 45 H (cid:17) (cid:18) V (Φ) − V ′ (Φ)2 (cid:19) − κ e p/ (6.3) ( ¨Φ + 8 H ˙Φ) (cid:0) V ′ (Φ) − V (Φ) − (cid:1) + ˙Φ (cid:0) V ′′ (Φ) − V ′ (Φ) + 4 V (Φ) + 2 (cid:1) +(1 + V (Φ)) (36 H + 8 ˙ H ) + κ e p = 0 (6.4) ˙ ǫ + 9 H ( ǫ + p ) − ˙Φ p = 0 . (6.5)We require that the additional term should not spoil the boun e, namely this term isimportant only when the urvature be omes very big. For on reteness we hoose V = − e Φ+5 / . Using the same setup as in the previous se tion, we perform the numeri alanalysis.First, let us onsider the ase in whi h the dilaton grows; in this ase we fa ed a futuresingularity. We found that the addition of a potential (6.1) an resolve the future urvaturesingularity. This is shown in Figs. 10, where the dashed (solid) urve orresponds to the ase without (with) the additional term. Here the initial onditions are hosen su h that theboun e takes pla e at t = 0 (in other words we impose the initial onditions at the boun eand extrapolate in both dire tions in time). In the ase without the additional term, thereis a singularity, while in the other ase the universe evolves to the standard osmologi alregime where the Hubble parameter de reases. As an be seen from the plot, the dynami sare almost the same in both ases before the Hubble parameter gets signi(cid:28) antly big, sothat the boun e is not spoiled. On e the Hubble parameter is large enough, the additionalterm smooths out the singularity.Now we turn to the ase in whi h the dilaton de reases. In the previous se tion wehave seen that in this ase there is a past singularity. With the help of the additional termmentioned above, we found that this singularity an be resolved as well. This is shown inFigs. 11. The me hanism of resolving the singularity is analogous to the ase of growingdilaton that we have seen above. What is interesting is that this solution orresponds tothe time re(cid:29)e ted version of the ase of growing dilaton sin e H
7→ − H and ˙Φ
7→ − ˙Φ underthe time re(cid:29)e tion, t
7→ − t .In both ases, the dilaton dynami s still has a singularity either in the future or in thepast. This is in ontrast to the ase where the ta hyon se tor is absent (i. e. ǫ = 0 and p = 0 ). (see Figs. 12, where the pre-big bang singularity is resolved not only in H , butalso in Φ ). Sin e the singular behavior of our solutions is the same as in the pre-big bangs enario, we expe t that in prin iple the dilaton singularity an be resolved as in Figs. 12,but this may require (cid:28)ne-tuning. (cid:21) 14 (cid:21) - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H - - - - - - PSfrag repla ements
FwX Φ tTwH E H (b) dilaton Φ Figure 10: Numeri al solution with growing dilaton (solid lines: with additional term, dashedlines: without additional term). - - - - - - - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H - - - - - - PSfrag repla ements
FwX Φ tTwH E H (b) dilaton Φ Figure 11: Numeri al solution with de reasing dilaton (solid lines: with additional term, dashedlines: without additional term).To sum up, we have shown an expli it example in whi h an additional term that mightarise from higher order orre tions an resolve the urvature singularity without a(cid:27)e tingthe boun e dynami s, though the singularity in dilaton has not been resolved.7. Con lusionWe suggested boun e s enarios, in whi h a non-BPS spa e-(cid:28)lling D9-brane in type IIAsuperstring theory drives a boun e of the s ale fa tor in the string frame. We employedthe lowest order e(cid:27)e tive a tion for the metri and dilaton in the string frame as well as ane(cid:27)e tive a tion for the open ta hyoni mode of the non-BPS D-brane. The positivity of thepressure of the ta hyon (cid:28)eld is responsible for the boun e, whi h is why the DBI a tion,for instan e, an not drive the boun e. The urvature as well as the time derivative of thedilaton remain small during the boun e. In other words, the gravitational se tor is entirely lassi al. (cid:21) 15 (cid:21) - - - PSfrag repla ements
FwX Φ tTwH E H (a) Hubble parameter H - -
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FwX Φ tTwH E H (b) dilaton Φ Figure 12: Numeri al solution with ta hyon se tor absent (solid lines: with the potential given in(6.1), dashed lines: pre-big bang solution).Asymptoti ally our boun e solutions look like pre-big bang or post-big bang solutions,with singular behavior of the urvature and the dilaton. The asymptoti string frame ur-vature singularity an be resolved by adding a phenomenologi al potential, ∝ R e − Φ , whi hmay or may not result from α ′ orre tions in the open string se tor. It would be desirable todetermine the sign and the pre ise numeri al value of the proportionality oe(cid:30) ient. Withour hoi e of the sign the gravitational oupling hanges sign in the string frame. This re-sults in a boun e in the Einstein frame at some time after the boun e has taken pla e in thestring frame without violating the null energy ondition. An interesting observation is thatwhile our phenomenologi al potential stabilizes the dilaton within the perturbative regimeit fails to do so on e the ta hyoni se tor is in luded. An obvious question is then whethera modi(cid:28)ed potential exists whi h stabilizes the dilaton in our model, and if so, whether it an be derived from string theory. We should also mention that throughout this paper weassumed the isotropy in 9-dimensional spa e (modulo ompa ti(cid:28) ation, the details of whi hdid not play a role here) during the era of the boun e. For phenomenologi al reasons itmay be preferable to onsider s enarios with a di(cid:27)erent dynami s for the s ale fa tor of theinternal dimensions. In parti ular, orbifold ompa ti(cid:28) ations are interesting sin e they area ompanied by a redu tion of supersymmetry. A preliminary analysis shows that for a T /Z2