S-duality in String Gas Cosmology 1 2 1,2 Savas Arapoglu ,∗ Ata Karakci ,† and Ali Kaya ‡ 1Feza G¨ursey Institute, Emek Mah. No:68, C¸engelk¨oy, Istanbul,˙ Turkey 2Bo˜gazi¸ci University, Department of Physics, 34342, Bebek, Istanbul,˙ Turkey (Dated: July 2, 2018) We consider a toy cosmological model in string theory involving the winding and momentum modes of (m, n) strings, i.e. bound states of m fundamental and n D-strings. The model is invariant under S-duality provided that m and n are interchanged. The dilaton is naturally stabilized due to S-duality invariance, which offers a new mechanism of moduli fixing in string gas cosmology. Using a tachyon field rolling down to its ground state, we also point out a possible way of realizing a cosmological phase with decreasing Hubble radius and constant dilaton. I. INTRODUCTION ume modulus of an internal Ricci flat manifold (radion) [16, 17] (see also [18–20]) but also the shape modulus of String gas cosmology (SGC) [1] is a natural way of ap- an internal torus they wrapped by the stress they gener- plying string theory to cosmology (for review see [2, 3]). ate [21]. In addition to usual field theoretic excitations, in SGC Despite the fact that the late time stabilization of ex- one also considers new degrees of freedom of string the- tra dimensions is natural in string/brane gas cosmology, ory and it turns out that the winding modes, which exist dilaton stabilization (or stabilization of dilaton and ra- in the presence of compact directions, play an important dion simultaneously) is problematic [22, 23] (see however cosmological role. The stringy T-duality invariance is [24] and [23] for a solution by superpotentials). As T- the key symmetry which may resolve the initial big-bang duality plays a key role for the stabilization of extra di- singularity of standard cosmology. Moreover, the naive mensions, S-duality, which basically acts on dilaton, may dependence of the tree level annihilation cross section assist in fixing dilaton. Indeed it was shown in [25] that of winding strings on the number of dimensions might compactifying the eleven dimensional supergravity on a explain the hierarchy between the observed three and circle the cosmological membrane production in the en- internal directions [1] (however it seems there are some hanced symmetry point can force the radius (and thus 1 caveats in that argument [4–7] and a more complete un- dilaton) towards the self S-dual point . derstanding of string thermodynamics in an expanding Another motivation to consider S-duality in SGC universe is needed). In late times, SGC offers a natu- comes from the recently proposed stringy mechanism ral mechanism to stabilize extra dimensions as a result to generate scale free cosmological perturbations seeded of a balance between negative winding and positive mo- by thermal fluctuations in a strongly coupled Hagedorn mentum pressures [8]. Assuming that the cosmology is phase [26–29]. According to [26–29], spacetime should dominated by states which become massless at the spe- be static in this meta-stable phase with infinite Hubble arXiv:hep-th/0611193v2 27 Nov 2006 cial self-dual compactification radius, one can naturally radius and it should be followed by the so called Hage- avoid exceeding the phenomenological bounds on the en- dorn phase which has decreasing Hubble radius. During ergy density of the universe [9–11]. A string gas with the Hagedorn phase cosmological perturbations exit the fluxes is also capable of fixing the flux and the shape Hubble radius and freeze out. As criticized in [30], for moduli [12]. this mechanism to really work out, the dilaton should be SGC was developed in [13–15] to include higher dimen- constant so that the string and Einstein frame Hubble sional p-branes that exist in string/M theory. Although, parameters coincide. A possible way out of this criticism these are expected to annihilate in early times accord- is to assume that the strongly coupled Hagedorn phase ing to Brandenberger-Vafa mechanism [1], higher dimen- corresponds to a self S-dual state and thus dilaton should sional branes are needed to stabilize topologically non- be equal to zero [29]. toroidal cycles since winding string modes do not exist In the absence of a special symmetry that implies non- in the spectrum when the first homology class vanishes. renormalization (like supersymmetry), it seems difficult Indeed, p-branes are capable of fixing not only the vol- to give an effective field theory description of the strongly coupled Hagedorn phase since perturbative field equa- ∗[email protected] †[email protected] 1 (m, n) strings that we consider in this paper also arise in the ‡[email protected] membrane compactification studied in [25]. 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 en- tropy 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 met- ric 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 − .