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Cosmology Gas String the of [ reviews scale breaking detailed high supersymmetry Gaug- a to dimensions. leads at extra also the condensation of ino radii the of stabilization dimen- the extra by the [ scale (see about string sion modes the the momentum at expand, and forever winding to confined dimensions being spatial others three exactly allows in- simple predictions). [ from also distinguished (see be models flationary can scenario the which pre- inflation of models simple [ tilt, which dict red spectrum slight a the with like perturbations cosmological of trum n [ and l il Theory Field ble na nta ae [ paper initial an In Cosmol- Gas String in issue oustanding an However, phase Hagedorn the of end the at transition phase The r-i agCosmology Bang Pre-Big 2, 6 18 .I h tigsaecrepnst hto Grand of that to corresponds scale string the If ]. † o ealdrve)i edter model theory field a is review) detailed a for ] n ulem Franzmann Guilherme and l hoyi h uegaiyframe supergravity the in Theory eld riua,w hwta h solutions the that show we articular, clsaefactor. scale ical 10 , 11 21 o ealddsusoso hspoint). this of discussions detailed for ] ,tecsooia qain fdilaton of equations cosmological the ], 13 DT se[ (see (DFT) .Terae srfre o[ to referred is reader The ]. T,Canada 2T8, A lPaulista, al T,Canada 2T8, A 20 4 ,ta oec pta dimension spatial each to that ], ,pitpril oini dou- in motion particle point ], 8 and n [ and ] [ 15 12 x 19 16 ult winding. to dual ) sapoiigstarting promising a is ] ,wtotdsutn the disrupting without ], o oeeryworks). early some for ] , 9 17 o te distinctive other for ] 2, 7 o rgnlworks original for ] ,apeito with prediction a ], ‡ x ulto dual ) 14 Dou- for ] 2 equation of state of a gas of closed strings. Again, it was fields can be defined as shown that the cosmological dynamics is non-singular. 1 The full DFT equations of motion in the case of homoge- a(t) → (1) neous and isotropic cosmology were then studied in [22]. a(t) The consistency of DFT with the underlying string the- d(t) → d(t) , (2) ory leads to a constraint. In DFT, in general a stronger where d(t) is the shifted dilaton version of this constraint is used, namely the assumption D − 1 that the fields only depend on one subset of the doubled d(t) = φ(t) − ln a(t) (3) coordinates. There are various possible frames which re- 2 alize this (see the discussion in the following section). which is invariant under a T-duality transformation. In In the supergravity frame it is assumed that the fields DFT this definition can be generalized to be do not depend on the “doubled” coordinatesx ˜, while in the winding frame it is assumed that the fields only de- 1 a(t, t˜) → (4) pend onx ˜ and not on the x coordinates. It was shown a(t,t˜ ) that for solutions with constant dilaton in the supergrav- ˜ → ˜ ity frame, the consistency of the equations demands that d(t, t) d(t,t) . (5) the equation of state of matter is that of relativistic ra- This implies that dilaton transforms as diation, while constant dilaton in the winding frame de- mands that the equation of state of matter is that of a gas φ(t, t˜) → φ(t,t˜ ) − (D − 1) ln a(t,t˜ ) . (6) of winding modes. These two solutions, however, are not T-dual. In this paper we will look for solutions which are An important assumption of DFT is the need to im- T-dual. We expand on the analysis of [22] and present pose a section condition, a condition which states that improvements in the solutions. the fields only depend on a D-dimensional subset of the In the following section we discuss different frames space-time variables. The different choices of this sec- which can be used. They can be obtained from each other tion condition are called frames, and different frames are by T-duality transformations. We also discuss the T- related via T-duality transformations. The supergravity duality transformation of fields. In Section 3 we present frame is the frame in which the fields only depend on the equations of DFT for a homogeneous and isotropic the (t, x). The second frame which we will consider is cosmology. In Section 4 we introduce a T-duality pre- the winding frame in which the fields only depend on the serving ansatz for the solutions, before finding solutions (t,˜ x˜) coordinates. of these equations in Section 5. We conclude with a dis- In this paper we are interested in finding supergravity cussion of our results. frame solutions

(φ(t),a(t), d(t)) (7) II. T-DUAL FRAMES VS. T-DUAL VARIABLES and winding frame solutions We consider an underlying D-dimensional space-time. The fields of DFT then live in a 2D dimensional space (φ(t˜),a(t˜), d(t˜)) (8) with coordinates (t, x) and dual coordinates (t,˜ x˜), where t is time and x denote the D − 1 spatial coordinates. which are T-dual to each other, i.e. generalized metric In general, the of DFT is made up of ˜ ˜ the D−dimensional space-time metric, the dilaton and d(t) = d(t(t)) (9) 1 an antisymmetric tensor field, all being functions of the a(t˜) = , (10) 2D coordinates. a(t(t˜)) In this section (like in the rest of this paper) we con- sider only homogeneous and isotropic space-times and where t(t˜)= t˜. transformations which preserve the symmetries. In this case, the basic fields reduce to the cosmological scale fac- tor a(t, t˜) and the dilaton φ(t, t˜). It is self-consistent to III. EQUATIONS neglect the antisymmetric tensor field. These are the same fields which also appear in dilaton gravity. Our starting point is the equations for DFT under a In supergravity, the T-duality transformation of the cosmological ansatz [23] (Eqs. (8) in [22]): 3

4d′′ − 4(d′)2 − (D − 1)H˜ 2 +4d¨− 4d˙2 − (D − 1)H2 =0 (D − 1)H˜ 2 − 2d′′ − (D − 1)H2 +2d¨=0 H˜ ′ − 2Hd˜ ′ + H˙ − 2Hd˙ =0 , (11) where the prime denotes the derivative with respect to t˜, and the overdot the derivative with respect to t. In addition,

a˙ a′ H = , H˜ = . (12) a a These equations are invariant under T-duality, since d(t, t˜) is a scalar and H ↔−H˜ under this transformation. Then, we couple these equations with matter in the following way [22]

4d′′ − 4(d′)2 − (D − 1)H˜ 2 +4d¨− 4d˙2 − (D − 1)H2 =0 1 (D − 1)H˜ 2 − 2d′′ − (D − 1)H2 +2d¨= e2dE 2 1 H˜ ′ − 2Hd˜ ′ + H˙ − 2Hd˙ = e2dP. (13) 2 Now, these new equations are invariant under T-duality provided E →−E and P →−P . But this is exactly the case since, as explained in [22], the T-dual of the energy and pressure are given by

˜ − δF →− − 2 ˜ δF − ˜ E(t, t)= 2 2 gtt(t,t) = E(t,t), δgtt(t, t˜)  δgtt(t,t˜ ) ˜ − 2 δF ˜ − δF →− δF − ˜ P (t, t)= − gij (t, t)= = P (t, t), (14) D 1 δgij (t, t˜) δ ln a(t, t˜) δ ln(1/a(t,t˜ ))

where we used gtt = 1 for our case and assumed that the dilaton in the way it was considered in [22]. However, matter action in double space F is O(D,D) invariant. since the SuGra and winding frame solutions are not T- The invariance of Eqs. (13) under T-duality is a strong dual to each other, the comparison of these solutions used support for the correctness of the coupling with matter. to motivate the correspondence t˜→ t−1 is tenuous. Solutions to Eqs. (13) may be found after imposing the In this work, we look for equations and solutions that strong condition of DFT. One may impose that all func- respect T-duality, and specifically with constant dilaton tions are t˜-independent or t-independent, corresponding only in the SuGra frame or in the winding frame. We to the supergravity (SuGra) or winding frames, respec- also solve an apparent inconsistency with positive energy tively. In [22], solutions based on either the SuGra or density in the winding frame, found in [22]. winding frames were found for the case of constant dila- ton φ(t, t˜)= φ0. But notice that by (6) the dilaton trans- forms non-trivially under T-duality. Hence, the solutions IV. T-DUALITY PRESERVING ANSATZ AND found in [22] in the SuGra and winding frames, respec- EQUATIONS FOR EACH FRAME tively, are not T-dual to each other. The fact that two solutions both with constant dilaton in the respective Starting from the supergravity frame, let us look for frames are not related by T-duality (or O(D,D, ) more solutions with constant dilaton. In this case generally) can be confirmed by noting that equations (12) in [22] obtained from (13) after assuming constant dilaton 2d(t) = 2φ0 − (D − 1) ln a(t) are not T-dual invariant. These equations were obtained ⇒ ˙ − − by imposing = 2d = (D 1)H. (16)

2d(t, t˜) = 2φ0 − (D − 1) ln a(t, t˜) (15) We now seek solutions in the winding frame which are T-dual. By the invariance of d, d(t)= d(t˜(t)), we have =⇒ 2d˙ = −(D − 1)H, 2d′ = −(D − 1)H,˜

′ D − 1 D − 1 which is not compatible with T-duality, since 2d does φ0 − ln a = φ(t˜) − ln a(t˜) (17) not transform to 2d˙ as it should. 2 2 From the point of view of a field theory with doubled D − 1 a(t(t˜)) =⇒ φ(t˜)= φ0 − ln . coordinates, there is no problem in considering constant 2  a(t˜)  4

Now by the scale-factor duality which comes from the Equations (20) and (21) are ansaetze compatible with transformation of the generalized metric, a(t(t˜)) = T-duality between the SuGra and winding frames. 1/a(t˜), and so To find the equations in each frame under these as- φ(t˜) = φ0 + (D − 1) ln a(t˜) , (18) sumptions, let us consider and hence ˙ ′ ˜ D − 1 2d(t) = α(D − 1)H, 2d (t˜)=α ˜(D − 1)H, (22) d(t˜) = φ + ln a(t˜) 0 2 =⇒ 2d′(t˜) = (D − 1)H.˜ (19) which takes both cases into account: for (α, α˜) = (−1, 1) Thus, the ansatz for the rescaled dilaton d(t, t˜) in the we have a constant dilaton in the SuGra frame and winding frame will be such that non-constant dilaton in the winding frame; for (α, α˜) = (1, −1), we have constant dilaton in the winding frame 2d˙(t) = −(D − 1)H, 2d′(t˜) = (D − 1)H,˜ (20) and non-constant dilaton in the SuGra frame. The case (α, α˜) = (−1, −1) corresponds to having the dilaton con- which is related to the supergravity frame dilaton by T- stant in both frames and was considered in [22]. But, duality. Similarly, for a constant dilaton in the winding as already argued, this breaks the T-duality between the frame we have frames. Here, we are looking for solutions in each frame that are T-dual to each other, so we will not consider the ˙ − ′ ˜ − − ˜ 2d(t) = (D 1)H, 2d (t) = (D 1)H. (21) case (α, α˜)=(1, 1).

Applying the section conditions, we get equations for SuGra and winding frame,

4d¨− 4d˙2 − (D − 1)H2 =0 4d′′ − 4(d′)2 − (D − 1)H˜ 2 =0 1 1 −(D − 1)H2 +2d¨= e2dE(t) (D − 1)H˜ 2 − 2d′′ = e2dE(t˜) 2 2 (23) 1 1 H˙ − 2Hd˙ = e2dP (t) H˜ ′ − 2Hd˜ ′ = e2dP (t˜) 2 2

Before solving them, notice that the energy and pressure in the winding frame is different from E(t˜). in the winding frame are given by Using (22) in SuGra frame, we have 2 2 δF δF 2αH˙ − H (α (D − 1)+1)=0, E˜(t˜)= −2 = −2 −g2 (t˜) = −E(t˜), ˜ tt ˜ 1 δgt˜t˜(t)  δgtt(t)  αH˙ − H2 = e2dE, (24) 2(D − 1) 2 δF δF ˙ − − 2 1 2d P˜(t˜)= − g (t˜)= −2 a−2(t˜) H α(D 1)H = e P, (26) − ˜ ˜i˜j −2 ˜ 2 D 1 δg˜i˜j (t) δ(a (t)) which implies δF = = −P (t˜). (25) 2φ (α+1)(D−1) ˜ e 0 a δ ln a(t) H2 = ρ, (D − 1)(α2(D − 1) − 1) Thus, under T-duality, E(t) → E˜(t˜) and P (t) → P˜(t˜). 1 1 w = − , (27) This observation allows to reinterpret the minus sign ap- α D − 1 ˜ 2 pearing in the equation for H in [22]. In contrast to ρ˙ + (D − 1)H(ρ + p)=0 . what happens in the SuGra frame, the energy measured in the winding frame is not simply the function E(t, t˜) Notice that φ0 is the value of the dilaton in the frame projected to E(t˜) upon applying the section condition, where it is constant. but actually the negative of it. The difference appears In winding frame we obtain ˜ because the definition of E(t, t) selects the SuGra frame 2˜αH˜ ′ − H˜ 2(˜α2(D − 1)+1)=0, (28) as a preferred frame, since g does not enter in this def- t˜t˜ 1 inition. As explained in [22], to work only with E(t, t˜) −H˜ 2 +α ˜H˜ ′ = e2dE,˜ 2(D − 1) was a choice since the variations with respect to gtt can 1 be written as gt˜t˜ variations. But this choice selects t as −H˜ ′ +α ˜(D − 1)H˜ 2 = e2dP,˜ a preferred variable and so it is natural that the energy 2 5 which are equivalent to For constant dilaton in the winding frame, we find

e2φ0 a(˜α+1)(D−1) ρ(t) ∝ a−(D−2)(t), ρ˜(t˜) ∝ a−D(t˜), (37) H˜ 2 = ρ,˜ − 2 − − (D 1)(˜α (D 1) 1) a(t) ∝ t−2/D, a(t˜) ∝ t˜2/D, (38) 1 1 which shows that a fluid with winding equation of state w = − , (29) α˜ D 1 has time dependence of the scale factor like radiation in (D − 1) − 1/α˜ ρ˜′ + (D − 1) H˜ (˜ρ +˜p)=0 , the winding frame. (D − 1)+1/α˜  As we can check from the above results, we found solu- tions in the SuGra and winding frame which are T-dual to where w is the equation of state parameter each other. Also, the solutions exhibit a symmetry con- p nected with T-duality: if we change t to t˜ in the SuGra w = , (30) frame solution with constant dilaton in that frame, we ρ get the winding frame solution with constant dilaton in p and ρ being pressure and energy density, respectively. the winding frame, and vice-versa. From these equations, we conclude that the equation of state is the same in both frames regardless in which frame the dilaton is taken to be constant. For constant VI. DISCUSSION dilaton in the SuGra frame we obtain the equation of state of radiation, for constant dilaton in the winding In this paper we have constructed supergravity and frame, on the other hand, the equation of state is that of winding frame solutions of the cosmological equations of a gas of winding modes. Double Field Theory which are T-dual to each other. When the correct transformation of the energy and pres- sure is taken into account, there is no need for complex- V. SOLUTIONS ification of the scale factor. Since Double Field Theory is based on the same T- Solving the equations of the previous section in the duality symmetry which is key to superstring theory, one SuGra frame, we obtain could hope that Double Field Theory could provide a con- sistent background for superstring cosmology, and pro- − − ρ(t) ∝ a (D 1)+1/α(t), (31) vide a good background for String Gas Cosmology. Let 2 − − − us consider the background space to be toroidal. In this α 1 α(D 1) 1/α 2 a(t) ∝ (D − 1) − t −α(D−1)−1/α , case, as argued in [4], for large values of the radius R of  2 2α the torus (in string units), the light degrees of freedom (32) correspond to the momenta, and the supergravity frame is hence the one in which observers made up of light de- while in the winding frame we get grees of freedom measure physical quantities. In contrast, ρ˜(t˜) ∝ a−(D−1)+1/α˜ (t˜), (33) for small values of R, it is the winding modes which are light, and hence the winding frame is the frame in which 2 − − − −α˜ 1 α˜(D 1) 1/α˜ 2 observers describe the physics. In the transition region a(t˜) ∝ (D − 1) − t˜−α˜(D−1)−1/α˜ .  2 2˜α (the Hagedorn phase) the full nature of double space will (34) be important. It is possible that the section condition becomes dynamical [25]. It would be interesting in this In particular, for constant dilaton in the SuGra frame, context to explore the connection with the recent ideas we have in [24].

ρ(t) ∝ a−D(t), ρ˜(t˜) ∝ a−(D−2)(t˜), (35) Acknowledgement a(t) ∝ t2/D, a(t˜) ∝ t˜−2/D. (36)

We see that given a radiation equation of state in both The research at McGill is supported in part by funds frames, the energy density in the winding frame has the from NSERC, from the Canada Research Chair program, same a dependence as a fluid with winding equation of from a John Templeton Foundation grant to the Univer- state. The reason for this is that in the winding frame sity of Western Ontario and by the IRC - South Africa the dilaton is not constant, and hence the relationship - Canada Research Chairs Mobility Initiative Grant No. between equation of state and scale factor dependence of 109684. HB would like to thank CAPES for supporting the energy density which we are used to from Einstein his work and McGill University for hospitality during an gravity changes. exchange period as a Graduate Research Trainee. 6

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