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T-Dual Cosmological Solutions of Double Field Theory II

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T-Dual Cosmological Solutions of Double Field Theory II T-Dual Cosmological Solutions of Double Field Theory II Heliudson Bernardo,1, ∗ Robert Brandenberger,2, † and Guilherme Franzmann2, ‡ 1Instituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, S˜ao Paulo 01140-070, SP, Brazil, and Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada 2Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada (Dated: January 7, 2019) In this paper we present cosmological solutions of Double Field Theory in the supergravity frame and in the winding frame which are related via T-duality. In particular, we show that the solutions can be viewed without the need of complexifying the cosmological scale factor. I. INTRODUCTION trum of cosmological perturbations with a slight red tilt, like the spectrum which simple models of inflation pre- dict [6]. If the string scale corresponds to that of Grand Target space duality [1] is a key symmetry of su- Unification, then the observed amplitude of the spectrum perstring theory. Qualitatively speaking, it states that emerges naturally. String Gas Cosmology also predicts a physics on small compact spaces of radius R is equiva- roughly scale-invariant spectrum of gravitational waves, lent to physics on large compact spaces of radius 1/R (in but this time with a slight blue tilt [7], a prediction with string units). This duality is a symmetry of the mass which the scenario can be distinguished from simple in- spectrum of free strings: to each momentum mode of en- flationary models (see also [8] and [9] for other distinctive ergy n/R (where n is an integer) there is a winding mode predictions). of energy mR, where m is an integer. Hence, the spec- trum is unchanged under the symmetry transformation The phase transition at the end of the Hagedorn phase R → 1/R if the winding and momentum quantum num- allows exactly three spatial dimensions to expand, the bers m and n are interchanged. The energy of the string others being confined forever at the string scale by the oscillatory modes is independent of R. This symmetry is winding and momentum modes about the extra dimen- obeyed by string interactions, and it is also supposed to sion (see [10, 11] for detailed discussions of this point). hold at the non-perturbative level (see e.g. [2]). The dilaton can be stabilized by the addition of a gaug- ino condensation mechanism [12], without disrupting the The exponential tower of string oscillatory modes leads stabilization of the radii of the extra dimensions. Gaug- to a maximal temperature for a gas of strings in thermal ino condensation also leads to supersymmetry breaking equilibrium, the Hagedorn temperature [3]. Combining at a high scale [13]. The reader is referred to [14] for these thermodynamic considerations with the T-duality detailed reviews of the String Gas Cosmology scenario. symmetry lead to the proposal of String Gas Comology [4] (see also [5]), a nonsingular cosmological model in which However, an oustanding issue in String Gas Cosmol- the Universe loiters for a long time in a thermal state of ogy is to obtain a consistent description of the back- strings just below the Hagedorn temperature, a state in ground space-time. Einstein gravity is clearly not appli- which both momentum and winding modes are excited. cable since it is not consistent with the basic T-duality This is the ‘Hagedorn phase’. After a phase transition in symmetry of string theory. Dilaton gravity, as studied which the winding modes interact to decay into loops, the in Pre-Big Bang Cosmology [15] is a promising starting T-duality symmetry of the state is spontaneously broken, point, but it also does not take into account the fact, discussed in detail in [4], that to each spatial dimension arXiv:1901.01209v1 [hep-th] 4 Jan 2019 the equation of state of the matter gas changes to that of radiation, and the radiation phase of Standard Big Bang there are two position operators, the first one (x) dual to expansion can begin. momentum, the second one (˜x) dual to winding. Dou- ble Field Theory (DFT) (see [16, 17] for original works In addition to providing a nonsingular cosmology, and [18] for a detailed review) is a field theory model String Gas Cosmology leads to an alternative to cosmo- which is consistent both with the T-duality symmetry of logical inflation for the origin of structure [6]: Accord- string theory and the resulting doubling of the number of ing to this picture, thermal fluctuations of strings in the spatial coordinates (see also [19] for some early works). Hagedorn phase lead to the observed inhomogeneities in Hence, as a stepping stone towards understanding the dy- the distribution of matter at late times. Making use of namics of String Gas Cosmology it is of interest to study the holographic scaling of matter correlation functions in cosmological solutions of DFT. the Hagedorn phase, one obtains a scale-invariant spec- In an initial paper [20], point particle motion in dou- bled space was studied, and it was argued that, when interpreted in terms of physical clocks, geodesics can be ∗Electronic address: [email protected] completed arbitrarily far into the past and future. In †Electronic address: [email protected] a next paper [21], the cosmological equations of dilaton ‡Electronic address: [email protected] gravity were studied with a matter source which has the 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.
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