S-Duality in String Gas Cosmology

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arXiv:hep-th/0611193v2 27 Nov 2006 ‡ † ∗ ue sas aal ffiigteflxadteshape the and sional flux with the gas fixing string of A [12]. capable moduli [9–11]. also universe is the fluxes en- of the density on naturally ergy bounds phenomenological can spe- the one the exceeding radius, avoid at compactification is massless self-dual cosmology become cial the which states that mo- by Assuming positive dominated result and natu- [8]. a winding a pressures negative as mentum offers between dimensions SGC balance extra a times, stabilize of late to In mechanism expanding ral an needed). un- in is complete thermodynamics universe more string a some of and are [4–7] derstanding and there argument seems that three it in observed might (however caveats dimensions the [1] directions of between internal section number hierarchy cross the the on annihilation explain strings level winding tree naive of the the Moreover, of cosmology. dependence big-bang is initial standard invariance the of resolve T-duality singularity may which stringy symmetry key The important the an play exist role. directions, which compact modes, cosmological the- of winding presence string the that the of out in SGC freedom turns of in it degrees and excitations, ory new theoretic 3]). considers field [2, also usual see one review to (for addition cosmology In to theory string plying ntesetu hntefis oooycasvanishes. class exist homology first not the do Indeed, when modes spectrum non- string the topologically winding in stabilize since to cycles needed accord- toroidal are times dimen- branes early higher sional [1], in mechanism annihilate Brandenberger-Vafa to to ing expected are these [email protected] [email protected] [email protected] G a eeoe n[31]t nld ihrdimen- higher include to [13–15] in developed was SGC ap- of way natural a is [1] (SGC) cosmology gas String p bae hteiti tigMter.Although, theory. string/M in exist that -branes p bae r aal ffiigntol h vol- the only not fixing of capable are -branes oe f( of modes ne -ult rvddthat provided S-duality under aho edrligdw oisgon tt,w lopoint also we const state, and ground radius its Hubble decreasing to with down phase moduli rolling cosmological of field mechanism tachyon new a a offers which invariance, S-duality ecnie o omlgclmdli tigter involv theory string in model cosmological toy a consider We .INTRODUCTION I. ,n m, tig,ie on ttsof states bound i.e. strings, ) aa Arapoglu Savas -ult nSrn a Cosmology Gas String in S-duality mkMh o6,Ç engelköy, No:68, Mah. Emek 2 og z¸ iUiest,Dprmn fPhysics, of Department Bo˜gazi¸ci University, m 44,Bebek, 34342, and 1 1 n eaGusyInstitute, Gürsey Feza Dtd uy2 2018) 2, July (Dated: , ∗ r necagd h iao sntrlysaiie u t due stabilized naturally is dilaton The interchanged. are t Karakci Ata m udmna and fundamental sabl Turkey Istanbul, ˙ acdsmer on a oc h ais(n thus (and point radius S-dual the self the force en- towards can dilaton) the point in symmetry production that hanced membrane a [25] cosmological on in the supergravity shown circle dimensional was eleven it the Indeed di- compactifying dilaton. extra may T- fixing of dilaton, in As on stabilization assist acts the basically superpotentials). for which S-duality, by role mensions, key solution a a plays for duality however [23] ra- (see and 23] and [24] [22, dilaton problematic is of simultaneously) stabilization cosmology,dion gas (or string/brane stabilization in dilaton natural is dimensions tra gener- they of stress modulus the shape by [21]. the wrapped ate they also torus but (radion) internal [18–20]) manifold an also flat (see Ricci 17] internal [16, an of modulus ume ope aeonpaesneprubtv edequa- field perturbative since strongly phase the of Hagedorn description theory coupled field difficult effective an seems give it to supersymmetry), (like renormalization [29]. should zero dilaton phase thus to Hagedorn and equal state coupled be S-dual strongly self a the to criticism that Hubble corresponds this assume of frame out to Einstein way is possible and A for string be coincide. [30], the should parameters dilaton in that the criticized so out, As work constant really to out. mechanism freeze this the and exit radius Hage- perturbations During called Hubble cosmological radius. so phase Hubble the Hagedorn decreasing by the has Hubble followed which infinite be phase with should dorn phase it should meta-stable and spacetime this radius [26–29], in to static Hagedorn According be coupled strongly [26–29]. a phase in mechanism seeded fluctuations perturbations stringy thermal cosmological by free proposed scale recently generate to the from comes 1 ( ebaecmatfiainsuidi [25]. in studied compactification membrane ept h atta h aetm tblzto fex- of stabilization time late the that fact the Despite nteasneo pca ymtyta mle non- implies that symmetry special a of absence the In SGC in S-duality consider to motivation Another ,n m, 2 , † sabl Turkey Istanbul, ˙ tig htw osdri hsppras rs nthe in arise also paper this in consider we that strings ) n l Kaya Ali and n dilaton. ant xn nsrn a omlg.Using cosmology. gas string in fixing n u osbewyo elzn a realizing of way possible a out n h idn n momentum and winding the ing -tig.Temdli invariant is model The D-strings. 1 , 2 ‡ 1 . o 2 tions cannot be trusted. However, it should be possible The field equations following from this action in the to model the Hagedorn phase even if one enters into the proper time A = 0 (and in string units ls = 1) read strong coupling regime above the zero dilaton, since in that case one can apply an S-duality map to yield a weak ϕ˙ 2 B˙ 2 = eϕE, − i coupling description. i X Alternatively, one can also consider the possibility of 1 B¨ ϕ˙B˙ = eϕP , (7) having constant dilaton also in the Hagedorn phase. In i − i 2 i that case, the equations of dilaton gravity reduce to that 2¨ϕ ϕ˙ 2 B˙ 2 = eϕP , of usual Einstein’s equations. Although it is not di- − − i − ϕ i rectly related to the main interest of this paper, namely X S-duality, we also point out that using a tachyon field where rolling down to its ground state it is possible to obtain ∂F ∂F ∂F a cosmological evolution with decreasing Hubble radius E = F + β , P = , P = . (8) ∂β i ∂B ϕ ∂ϕ which can be matched to the static strongly coupled − i − Hagedorn phase. As a direct consequence of field equations, the coupled sources should obey the energy-momentum conservation equation II. S-DUALITY

E˙ + B˙ iPi +ϕP ˙ ϕ =0. (9) Let us start with the low energy eﬀective action in 10- i X dimensions Here, we see that the shifted dilaton looks like an ex-

1 2φ 2 tra dimension, which is not surprising due to the 11- S = √ ge− R + 4( φ) . (1) 10 l8 − ∇ dimensional origin of dilaton. s Z The conservation equation (9) implies that the entropy It is well known that (1) is invariant under S-duality ∂F transformation S β2 (10) ≡ ∂β φ φ , →− φ is a constant of motion, hence the evolution is adiabatic. g e− g . (2) µν → µν The temperature will adjust itself to yield constant entropy and one can write β = β(B , ϕ). Therefore in the Since the string coupling constant g is given by g = eφ, i s s adiabatic approximation energy E can be viewed as a this map corresponds to a strong-weak coupling duality. function of B and ϕ. In that case the pressures can be Assuming a metric of the form, i calculated as2

2 2A 2 2Bi 2 ds = e dt + e dθi , (3) ∂E ∂E − Pi = , Pϕ = , (11) i ∂B ∂ϕ X − i − where the functions A, Bi depend only on time t, the which means that the total action becomes equivalent to action (1) reduces to S = S dt eA E(B , ϕ). (12) 10 − i ϕ A 2 2 S = l dt e− − B˙ ϕ˙ , (4) Z 10 s i − " i # The action will be T-duality invariant provided Z X where the shifted dilaton is defined as E(Bi, ϕ)= E( Bi, ϕ). (13) − ϕ =2φ B . (5) On the other hand, from (2), S-duality invariance re- − i i quires X φ/2 Here we take the dimensionful coordinates θ to have E(Bi, φ)= e E(Bi φ/2, φ). (14) i − − length one in string units. Note that in this equation we treat E as a function of φ, To add stringy matter to (1), the corresponding free not the shifted dilaton ϕ. energy F should be coupled minimally to the string metric in the form of an effective Lagrangian [31]. Differing from [31], we assume that the free energy also depends on the string coupling constant, hence dilaton (or shifted 2 To verify this note that when β = β(Bi,ϕ) and S = S0, E = dilaton), and the total action becomes ∂E Fˆ + S0/β where Fˆ(Bi,ϕ) = F (Bi,ϕ,β(Bi,ϕ)). Then = ∂Bi ∂Fˆ ∂β S0 ∂F 2 = = Pi. The same manipulations for ϕ S = S dt eA F (B ,ϕ,βeA). (6) ∂Bi − ∂Bi β ∂Bi − 10 − i gives (11). Z 3

III. STABILIZATION OF DILATON V

In type IIB string theory, an S-duality transformation (2) converts a fundamental string into a D-string. Actu- ally, the action of S-duality in type IIB theory is more general than the mere strong-weak coupling map (2). It can be represented as an SL(2,Z) transformation and by acting on the fundamental string one can obtain an SL(2,Z) multiplet of strings characterized by two inte- gers m and n [32]. An (m,n) string can be viewed as ϕ the bound state of m fundamental and n D-strings [33]. The transformation (2) simply interchanges m and n. In string frame the tension of the (m,n) string is given by [32] FIG. 1: The dilaton potential for m > n.

2 2 n T = Ts m + , (15) 2 2 φ 2 φ A+Bi A Bi s gs dt Ew m e + n e− e + Em e − . − i Z where gs is the string coupling constant and Ts is the X p tension of the fundamental (1, 0) string determining the S-duality transformation in the Einstein frame is simply string length ls. In seeking how S-duality acts in the given by context of string gas cosmology, it is clear that one should study a model involving (m,n) strings. φ φ, (18) →− The energy of an (m,n) string winding a compact direction is proportional to its tension (15) times the radius and the invariance of the action SE is obvious provided 3 of the circle. On the other hand, the energy of a momen- m n. Note that S-duality leads to a potential for ↔ √ 2 φ 2 φ tum mode (either corresponding to a small vibration of dilaton V (φ) = Ew m e + n e− which has a global a winding string or to an unwound string circling around minimum (see figure). that direction) is inversely proportional to the compact- In the proper time, the dilaton equation that follows ification radius. Note that the energy of the momentum from the above action reads (we set ls = 1) mode should not depend on the tension (15). It should 2 φ 2 φ 1 Bi k m e n e− be measured in units of the corresponding angular coor- φ¨ + k˙ φ˙ = Ew e − − , (19) −2 √ 2 φ 2 φ dinate parametrizing the circle and thus from (1) and (4) i m e + n e− X in terms of the string length ls. As a result, the string frame matter action for the where winding and momentum modes of (m,n) strings should take the form k = Bi. (20) i X 2 2 2φ A+Bi A Bi S = dt E m + n e e + E e − , − w − m From (19) one sees that the constant dilaton i X Z p where the sum should go over compact directions, and Ew φ = ln(n/m) (21) and Em are constants characterizing winding and momentum energies, respectively. It is easy to see that the is a solution. Moreover, (21) is also consistent with other matter action is S-duality invariant with m n. field equations. Note that it is possible to stay in the ↔ weak coupling by choosing m n. The S-duality transformation (2) is more manifest in ≫ Einstein frame defined by To see that this is a stable point for dilaton and hence ˙ (E) φ/2 it is fixed, we first point out that k never vanishes, which gµν = e− gµν . (16) follows from the variation of the action with respect to ˙ Writing the Einstein frame metric as (note that we are A. To preserve adiabatic approximation we impose k> 0 ˙ ˙ using the same letters for the metric components in Ein- and thus kφ term in (19) acts like a frictional force that stein frame, which should not be confused with (3)) dumps the motion. Moreover, from the right hand side we see that when φ > ln(n/m) there is a negative and ds2 = e2Adt2 + e2Bi dθ2, (17) when φ < ln(n/m) there is a positive acceleration that E − i i X the total action becomes

3 Actually V (φ) is not an honest potential since the action is not of A+ Bj 1 j ˙ 2 ˙ 2 ˙2 SE = ls dt e− Bi ( Bj ) + φ the form V √ g. However, it still implies a non-trivial dilaton  − 2  − i j dependence of non-kinetic energy. Z P X X   R 4 force the equilibrium value. Indeed, looking for a small We see that δφ and δC are nicely decoupled. The corre- perturbation δφ around (21) we ﬁnd that it obeys sponding linearly independent solutions for them can be found as

mn Bi k δφ¨ + k˙ δφ˙ = Ew e − δφ, (22) t√4λ 1 t√4λ 1 − 2 cos ln 2 − sin ln 2 − r i ! , , (30) X h √t i h √t i which is similar to a dumped oscilator with time depen- dent friction and frequency. where λ =8/d for δφ and λ = 4(d 5)/(dp) for δC, and To quantify these arguments and to see whether it in each case λ> 1/4. Thus both δφ−and δC fall for large t is possible to stabilize both dilaton and radion simul- as 1/√t, which proves that they are stabilized. Similarly, tanously let us consider (1 + d + p) dimensional split of it can be seen from (29) that the perturbation δB also the spacetime with the following metric goes like 1/√t as t . One may think that→ ∞ a cosmology dominated by (m,n) ds2 = dt2 + e2Bds2 + e2Cds2. (23) E − d p strings of one type is not realistic. Indeed, (m,n) strings have a larger tension compared to, say, (1, 0) or (0, 1) Here, ds2 and ds2 are flat metrics on Rd and T p, playing d p strings, which become lightest strings in Einstein frame the roles of the observed and internal dimensions. From at weak and strong couplings, respectively. However, as- the Einstein frame action it is possible to obtain the fol- suming that the numbers of fundamental and D-strings lowing field equations do not change and, for the moment, they are equal to 2 φ 2 φ each other, then it is energetically more favorable to form ¨ ˙ ˙ p C k m e n e− φ + kφ = Ewe − − , (1, 1) strings, since the tension of a (1, 1) string is less −2 √m2eφ + n2e φ − than the sum of the tensions of (1, 0) and (0, 1) strings. (d 5) C¨ + C˙ k˙ = Fm − Fw, (24) Similarly, when the ratio of the numbers is m/n, the cos- − 4 mology will be dominated by (m,n) strings. p B¨ + B˙ k˙ = Fw, The restriction d > 5 on the number of observed di- 4 mensions in the above toy model can easily be circum- together with the initial constraint vented by including higher dimensional S-duality related branes. For instance, in type IIB string theory one can 2 2 2 1 2 k˙ = dB˙ + pC˙ + φ˙ +2pFw +2pFm, (25) consider a bound state of m Dirichlet and n solitonic five- 2 branes which has the following tension in the Einstein frame [33] where the functions Fw and Fm are given by

Ew 2 φ 2 φ k+C 2 Fw = m e + n e e− , 2 n 2 − T = T5 m gs + . (31) s gs Emp k C Fm = e− − . (26) 2 As for strings, the duality map (18) interchanges m and Note that k = dB + pC and d + p = 9. n. One can consider (1+3+5+1) dimensional split of the For d > 5, there is a special solution with constant spacetime where (m,n) fivebranes and strings are wrap- dilaton and radion given by ping over the five and the one dimensional subspaces of the six dimensional toroidal internal space, respectively. 2 It can be verified that in this S-duality invariant model φ = ln(n/m), B = ln(αt), C = C0, (27) d both radion and dilaton are fixed in a three dimensional where observed space, where again φ = ln(n/m). The above results indicate that S-duality can play a 8E e2C0 = m , crucial role in SGC, especially in the late time stabiliza- (d 1)E √2mn tion of dilaton. For early time applications, it seems a − w √ detailed information about the free energy of the string 2 dpEw 2mn (1 p)C0 α = e − . (28) gas, particularly its dependence on the string coupling 16 constant, is needed. To verify stabilization let us consider small perturbations around this background which obey IV. DISCUSSION ¨ 2 ˙ 8 δφ δφ + δφ + 2 =0, t d t As a final comment we would like to point out a pos- 2 4(d 5) δC δC¨ + δC˙ + − =0, (29) sible way of getting a decreasing Hubble radius in string t dp t2 theory. This should mimic the cosmological evolution 4 2 2p δC˙ (2p 2) δC in the Hagedorn phase [29], following the static strongly δB¨ + δB˙ + δB + + − =0. t t2 d t d t2 coupled Hagedorn phase. We assume that the dilaton 5 and internal dimensions are already stabilized leaving a t four dimensional Einstein theory. A decreasing Hubble radius necessarily implies accelerated expansion, but the opposite is not correct. Therefore, it seems harder to obtain former compared to the acceleration. Indeed, if the spatial sections of the universe are flat, one can only get a decreasing Hubble radius by violating the null energy condition, i.e. ρ + P < 0. One method of obtaining acceleration in string theory is via a tachyon field T rolling down to its ground state [34]. The tachyon potential V (T ) has a positive R H maximum V0 at T = 0 and a minimum at T = T0 with V (T0) = 0. The Lagrangian is given by FIG. 2: The time evolution of the Hubble radius in the = V (T ) 1 T˙ 2. (32) L − − tachyon cosmology with a spherical space. p Considering the evolution of tachyon field from T = 0 (with a small initial velocity) towards the minimum at Note that, although the curvature of the sphere (σ-term) ˙ T = T0, one sees that T increases in time. From the contributes negatively to the expansion speed B˙ , it in- corresponding energy momentum tensor it follows that creases B¨, which is precisely what we were looking for. 2 for T˙ < 2/3, ρ + 3P < 0 which implies acceleration Since we would like to match the background to the [34]. However, in this interval ρ + P > 0 which gives an static strongly coupled Hagedorn phase, we assume that increasing Hubble radius. Thus the tachyon field alone initially at t = 0, B˙ (0) = 0. For the tachyon field we cannot yield a decreasing Hubble radius. impose T (0) = 0 and take a small positive initial velocity, Actually, there is another problem here which is also i.e 0 < T˙ (0) 1 (assuming T˙ (0) = 0 gives the Einstein crucial for our discussion. From field equations one can static universe≪ which is unstable). Then from (35) and see that during this accelerating phase the expansion (36) one sees that B¨(0) > 0. Recalling that the Hubble speed can never vanish. Therefore, it is not possible to radius R is given by R 1/B˙ , it decreases initially. match this evolution to the static strongly coupled Hage- H H To understand the succeeding∼ evolution we note that dorn phase. To cure this difficulty one should essentially since B¨ + B˙ 2 (1 3T˙ 2/2) the acceleration stops when try to slow down the expansion speed. To that effect, the ˙ 2 ∼ −¨ easiest madification is to assume that the space is posi- T = 2/3. Thus, B should turn to negative at some tively curved rather than being flat. Taking the space- earlier time and subsequently the Hubble radius increases time metric as (see figure). Here, one should also check that in this period B˙ 2 given 2 2A 2 2B 2 in (35) is always positive. To see this is indeed the case, ds = e dt + e dΩ3, (33) − 2B 2 let us define f = σe− and g = V/(6 1 T˙ ). Then, where dΩ2 is the unit round metric on the three sphere, initially f(0) = g(0). From (37) we find that− 3 p and coupling the tachyon field with the canonical potential V (T ), one obtains the action d ln g d ln f 3T˙ 2 = . (38) dt dt 2 ! A+3B 2 A+B S = l dt e− ( 6B˙ )+6σe 4 p − Z Since initially B˙ (0) = 0 and B¨(0) > 0, there is an interval dt eA+3B V (T ) 1 e 2AT˙ 2, (34) during which B˙ > 0 and consequently d(ln f)/dt < 0. − − − Z Also from (37), we have d(ln g)/dt < 0. Then (38) implies p ˙ where σ = 1 and that term is related to the curvature d(ln g)/dt > d(ln f)/dt since initially T 1. Therefore, ≪ ˙ 2 of the sphere. Varying with respect to A, B and T one g>f during this period and from (35) one sees B > 0. ˙ 2 ˙ 2 finds (we set lp = 1) Iterating, it is guaranteed that B > 0 until T =2/3. As a result, we see that it is possible to obtain a cosmological phase with decreasing Hubble radius which can be ˙ 2 2B V B = σe− + , (35) matched to the static strongly coupled Hagedorn phase, − 6 1 T˙ 2 − provided that the flat space is replaced by a sphere, the ˙ 2 2B VpT tachyon field dominates the cosmic evolution and the ini- B¨ = σe− , (36) − 4 1 T˙ 2 tial tachyon potential energy is fine tuned with the curva- − ture of the three sphere. Of course, it should be checked d V p V T˙ = 3B˙ . (37) that these assumptions do not invalidate the results of dt 1 T˙ 2 ! − 1 T˙ 2 [26–29]. Especially, since the curvature of the sphere in- − − p p 6 troduces a new length scale, it may alter the scale inde- Acknowledgments pendence of the perturbation spectrum. Moreover, the existence of the tachyon field should be explained. In any case, we find it interesting that a slight modification The work of Ali Kaya is partially supported by Turk- of the tachyon cosmology yields a decreasing Hubble ra- ish Academy of Sciences via Young Investigator Award dius. Program (TUBA-GEB¨ IP).˙

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