$\mathbf{O}(D,D)$ completion of the Friedmann equations

String theory suggests a unique and unambiguous modification to General Relativity: the symmetry of $\mathbf{O}(D,D)$ T-duality promotes the entire closed-string massless NS-NS sector to stringy graviton fields. The symmetry fixes the couplings to other matter fields unambiguously and the Einstein field equations are enriched to comprise $D^{2}+1$ components, dubbed recently as the Einstein Double Field Equations. Here we explore the cosmological implications of this `Stringy Gravity'. We derive the most general homogeneous and isotropic ansatzes for both stringy graviton fields and the stringy energy-momentum tensor. Substituting them into the Einstein Double Field Equations, we obtain the $\mathbf{O}(D,D)$ completion of the Friedmann equations along with a generalized continuity equation. We discuss how this gives an enriched and novel framework beyond typical string cosmology, with solutions that may be characterized by two equation-of-state parameters, $w$ (conventional) and $\lambda$ (new). When $\lambda+3w=1$, the dilaton remains constant throughout the cosmological evolution, and one recovers the standard Friedmann equations for generic matter content (i.e. for any $w$), an improvement over conventional string cosmology where this occurs only for a radiation equation of state ($w=1/3$). We further point out that, in contrast to General Relativity, in Stringy Gravity there is no de Sitter solution arising from either an $\mathbf{O}(D,D)$-symmetric cosmological constant or scalar field with positive energy density.


Introduction
Despite its many successes, General Relativity (GR) faces several well-known shortcomings when applied to cosmology. In order to explain the large-scale dynamics of the universe, one needs to introduce dark matter and dark energy. Furthermore, solving the horizon and flatness problems requires new dynamics, such as inflation or bouncing cosmologies, involving additional degrees of freedom which may come into play near the strong-coupling regime at which GR breaks down.
In order to make quantitative predictions about the very early universe, we need to invoke a consistent theory of quantum gravity. String theory is currently the best-developed candidate, and its effects on cosmology may be studied in the low-energy effective supergravity (SUGRA) limit at weak coupling [1]. However, string theory does not predict GR exactly. In GR the spacetime metric, g µν , is the only gravitational field.
On the other hand, string theory predicts its own gravity, or Stringy Gravity, of which the fundamental fields With the EDFEs as our starting point, in this work we derive generalized Friedmann equations which are O(D, D) symmetric. Hereafter, we refer to them as the O(D, D)-complete Friedmann Equations (OFEs).
As will be shown, imposing the O(D, D) symmetry results in modifications to the conventional SUGRA equations for Riemannian backgrounds. Perhaps surprisingly, one finds that whenever the dilaton is kept constant, whether dynamically or through some appropriate coupling to the matter sector, the standard Friedmann equations are recovered in the presence of any matter sources, not just for a radiation-dominated universe as is the case in the usual string cosmology [31].
The organization of the present paper is as follows. In section 2 we review briefly the Einstein Double Field Equations [21,22]. In section 3 we derive our main result, the O(D, D) complete Friedman Equations with D = 4. In particular, we introduce two equation-of-state parameters, w (usual) and λ (new). In the following subsections we apply some DFT results from [21] to cosmology: in section 3.1 we summarize the stringy energy-momentum tensors of various types of matter; and in section 3.2 we discuss energy conditions. Appendix A contains a derivation of the most general cosmological, i.e. homogeneous and isotropic, form of the stringy energy-momentum tensor within the framework of Double Field Theory. Section 4 discusses various solutions, such as a (generalized) perfect fluid, scalar field, and radiation.
In standard cosmology there are several key scenarios in which the evolution of the universe approaches a de Sitter spacetime. These include the late-time Λ-dominated expansion and the hypothesized period of inflation in the early universe. However, in recent years the question of whether or not de Sitter solutions can be realized in a consistent theory of quantum gravity, such as string theory, has been widely debated (see, for example, [32][33][34][35][36][37][38][39][40][41][42][43][44][45]). Hence in section 5 we investigate in detail the possibility of realizing de Sitter solutions in O(D, D)-symmetric cosmology, and find that it appears to require solutions with negative energy density, a violation of the weak energy condition. We conclude with comments in section 6.

The O(D, D) paradigm: review of the Einstein Double Field Equations
In General Relativity (GR) the metric, which sets the local geometry, is the only field responsible for gravitational phenomena. All other fields are classified as additional matter and couple unambiguously to the geometry via a minimal coupling, i.e. promoting ordinary derivatives to covariant ones and generalizing volume elements. This procedure ensures covariance under both diffeomorphisms and, with the vielbein formalism, local Lorentz symmetry.
The situation is more involved in supergravity, in which the gravitational sector includes three different fields: the metric g µν , the B-field B µν , and the dilaton φ. Together with the symmetries from GR, SUGRA is also invariant under gauge transformations of the B-field. Although both setups are very similar, in SUGRA not all of the fields responsible for gravity are also geometric: that role is exclusively reserved for the metric.
Only since the introduction of DFT [6][7][8][9][10][11] has it been possible to consider all gravitational fields on the same footing, as being responsible for defining geometry. Note that the DFT geometry is still not fully understood in the mathematical literature, and although it recovers Riemannian backgrounds under certain conditions, it is certainly much more general, see e.g. [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61]. The geometrical framework of DFT allows us to define a generalization of the scalar and Ricci curvatures, S (0) and P A CP B D S CD [62] (c.f. [63,64]), respectively, both of which may reduce to the usual definitions with a trivial dilaton and B-field. The natural next step is to include matter content into such a framework: this was accomplished in [14] and the resulting interactions between gravity and matter are described by the Einstein Double Field Equations (EDFEs) [21,22], where the above indices are charged under the O(D, D) symmetry group. The left-hand side corresponds to the string-theoretic extension of the Einstein tensor which is conserved off-shell [65], while the right-hand side is the corresponding generalization of the energy-momentum tensor, which is conserved on-shell [21], These objects respect all symmetries of DFT.
Our starting point for this paper is to consider purely Riemannian backgrounds in the above equation (c.f. non-Riemannian ones [58]). In such cases, (2.1) reduces to where R and R µν stand for the usual Ricci scalar and tensor in string frame, respectively, while H µνρ is the field strength of the B-field. The skew-symmetric K [µν] and the symmetric K (µν) can be understood (up to equations of motion) as the matter sources for the B-field and the traceless part of the metric, respectively, while T (0) corresponds to the matter sourcing the trace of the metric and the dilaton. The terms on the right-hand side of the latter two equations are absent in SUGRA, and their inclusion above characterizes a modification, or conceptual generalization, of standard SUGRA cosmology in terms of how the matter sources are coupled to the stringy gravity sector. The inclusion of these terms puts all gravitational fields on an equal footing, and crucially is a direct result of imposing thoroughly the O(D, D) symmetry on the DFT action coupled to matter.
Together with the above equations, there is also an on-shell conservation law for the stringy energymomentum tensor, arising from 'doubled' general covariance. This is also consistent with the fact that the stringy Einstein curvature tensor is, by construction, covariantly conserved off-shell [65]. On Riemannian backgrounds this conservation law reduces to the equations The above equations may be derived from the spacetime action This differs from the usual SUGRA cosmology action by the fact that the O(D, D)-invariant measure, e −2d ≡ √ −ge −2φ , couples to the entire matter Lagrangian as √ −ge −2φ L matter ≡ L matter (where L matter is the Lagrangian density of (2.3)) and hence matter is coupled not only to the metric but also to the dilaton and B-field (through the covariant derivatives). Also note that the metric g µν above is the string (Jordan)frame metric rather than the Einstein-frame metric.

O(D, D) completion of the Friedmann equations
In this section we obtain the most general ansatz for a D = 4 homogeneous and isotropic cosmological background. This allows us to write down the resulting O(D, D) completion of the Friedmann equations.
The DFT-Killing equations for Riemannian backgrounds are given by [21] (see Appendix A) where ζ a are ordinary GR Killing vectors, while theζ a are corresponding one-forms required to complete the parametrization of DFT isometries. In order to study cosmology we should consider homogeneous and isotropic backgrounds, in which case these DFT-Killing vectors will correspond to spatial rotations and translations. In such cases the most general solution is given by where dΩ 2 = dϑ 2 + sin 2 ϑdϕ 2 , B (2) = 1 2 B µν dx µ ∧ dx ν , k corresponds to the spatial curvature, and h is a constant corresponding to the magnetic H-flux We emphasize that non-trivial H-flux is compatible with the cosmological principle only for D = 4. Moreover, we must impose the same symmetry conditions on the matter sector, resulting in the Killing equations 2 The latter implies that T (0) (t) is a time-dependent function, while the former implies that K µν is diagonal and spatially homogeneous, where K t t (t) and K r r (t) are time-dependent functions, and K 1 Note that the equations of motion (2.4) imply that for homogeneous and isotropic solutions (3.2), the antisymmetric part of Kµν must vanish. 3 Since the above ansatz has been written for D = 4, we are not considering critical strings. However, it can easily be generalized to arbitrary dimensions.
Applying the ansatz (3.2) and (3.5) to (2.4), we obtain the O(D, D)-complete Friedmann equations, where H ≡ a /(N a) and the prime denotes differentiation with respect to the time coordinate of (3.2).
Furthermore, applying the ansatz to (2.5) yields one non-trivial conservation equation, In order to make contact with known physics we must rewrite these equations in terms of standard physical quantities such as energy density and pressure. Our basic assumption is that standard FLRW cosmology should be recovered from (3.6) and (3.7) in the case where the dilaton is constant, φ = φ = 0, and the H-flux vanishes, h = 0. One might also be tempted to set T (0) = 0, however this cannot in general be the case, as seen, for example, from the first equation of (3.6): instead we find T (0) = K µ µ in this limit. Thus we should find a definition of energy density and pressure in terms of K µ ν and T (0) . It turns out that the appropriate definitions are given by (3.8) As we will see shortly, this definition reduces (3.6) to the standard Friedmann equations in the limit of constant dilaton and vanishing H-flux. It is further justified from writing the Hamiltonian of (2.6) in a cosmological background, from which one can easily see that the corresponding energy density should be given by the above formula. Note that the e −2φ factors arise as a direct consequence of the same overall factor appearing in the O(D, D)-invariant matter action (2.6). As shown below, it is through this identification that the SUGRA cosmological equations can also be recovered.
It also is worthwhile to rearrange (3.9) and (3.10) to obtain Then all the terms appearing on the left-hand sides of (3.9), (3.10), (3.11), (3.15), and (3.16) correspond precisely to the quantities appearing in various energy conditions, as will be discussed further in section 3.2.
We also emphasize that this set of equations differs from the conventional SUGRA cosmological equations crucially because the matter sector couples minimally to the dilaton as well as the string-frame metric, which is again a consequence of extending the O(D, D) symmetry of the DFT gravitational sector to the matter sector.
It is known that the SUGRA limit for which the dilaton is stabilized (in the absence of spatial curvature and with trivial B-field) recovers a radiation-dominated universe, with a linear barotropic equation of state given by w = 1/3 in D = 4 [31]. In the present case, for vanishing H-flux and constant dilaton, for which we choose φ = 0 without loss of generality, the OFEs reduce to where we have chosen cosmic gauge (N = 1), with the dot '˙' denoting differentiation with respect to cosmic time. The first two equations are the standard Friedmann equations for generic matter content, while the final equation fixes T (0) in terms of density and pressure. Therefore, the low-energy DFT limit with a stabilized dilaton is consistent with introducing any type of matter, as opposed to only radiation. This is surprising and hints towards the fact that the current framework might be the natural extension of GR to higher energies.
Looking to the last equation above, it is also clear that in the limit of vanishing T (0) one recovers standard SUGRA cosmology, for which the equation of state corresponds to radiation if the dilaton is stabilized. With that in mind, we may define the ratios: where they are constant, 4 w is the conventional parameter corresponding to the effective pressure of the generalized fluid, while the new parameter, λ, measures the density rate at which the matter is coupled to the dilaton in comparison to the coupling with the metric. Since the dilaton is now also part of the gravity sector, naturally one can understand dilaton-induced interactions as an additional component of the gravitational interactions of matter. It is only in the limit of vanishing λ that we recover standard SUGRA in our equations above. We can also see that this limit is obtained not only when the matter decouples from the dilaton, but also when the energy density is much larger than T (0) , or when the dilaton field becomes very large (provided ρ does not simultaneously decrease too rapidly). Note that for a general barotropic fluid, the quantity λ should be provided a priori, so one can think of it as providing a generalized equation The appearance of T (0) is the main feature that distinguishes the OFEs from the standard SUGRA cosmological backgrounds studied in the literature (see [1] for a review). Thus in the next subsection we summarize how the stringy energy-momentum tensor, in particular T (0) , is evaluated for different types of matter content. Note that in all cases where the dilaton is minimally coupled (i.e. via the DFT volume element) to a dilaton-independent spacetime Lagrangian L matter , we find T (0) = −2L matter . Hence in such cases, vanishing T (0) simply corresponds to the Lagrangian vanishing on-shell.

Examples of stringy energy-momentum tensors in cosmology
In [21] many different examples of matter content were considered, with the stringy energy-momentum tensor components K µν and T (0) computed for each. Here we collect and summarize these results, and comment on their respective cosmological implications.
• Cosmological constant: The DFT cosmological constant simply couples minimally to the DFT volume element, which crucially includes the dilaton [62]. Varying such a term in the action gives a non-vanishing contribution only for T (0) , such that K µν = 0 and T (0) = 1 4πG Λ DFT .
• Scalar field: The canonical Lagrangian for a scalar field Φ in DFT also couples minimally to the dilaton, and further couples linearly to the (inverse) string-frame metric, giving K µν = ∂ µ Φ∂ ν Φ and In particular, we will see later in (4.46) that this implies λ = −2w. The presence of non-zero T (0) in the OFEs should have intriguing consequences for the general dynamics of a scalar field in cosmological O(D, D)-symmetric backgrounds. In particular, inflationary models in the context of supergravity should be revisited in future work.
• Fermionic fields: The fermionic Lagrangian is proportional to its equation of motion, thus minimal dilaton coupling implies that T (0) vanishes on-shell. Furthermore, it turns out that in this case we may change variables to a spinor density whose Lagrangian decouples completely from the dilaton, giving • Gauge fields: (Heterotic) Yang-Mills fields also couple minimally to the dilaton in DFT [14,15,[66][67][68] and consequently have a non-zero dilaton charge, given by T (0) = −2L YM [21], similarly to the scalar case. This implies that whenever the dilaton is dynamical, the fine structure constant may vary, leading to significant observational constraints [69,70]. We leave a detailed analysis of these constraints to future work, and for now content ourselves to look for solutions where the dilaton is either constant or slowly varying at late times.
• Ramond-Ramond sector: With an O(D, D)-symmetric unifying formulation [17], • Point Particle sources: It has been shown that T (0) vanishes for this case. The point particle follows the geodesics defined with respect to not the Einstein-frame metric but the string-frame metric [3].
• String sources: To zeroth order in α , strings do not couple to the dilaton, so their T (0) also vanishes.
In particular, in order to study the effects of non-trivial T (0) , we will derive cosmological solutions for the cosmological constant, study the scalar case thoroughly, and consider a generalized perfect fluid. The gauge fields alone shall be considered in future works.

Energy conditions
In section 4.3 of [21], various energy conditions were considered in the context of static, spherically symmetric solutions in Stringy Gravity, which are conjectured to constrain which solutions are allowed physically.
Here we discuss how they are expressed and extended in homogeneous and isotropic backgrounds.
• The strong energy condition (SEC) is defined such that it includes magnetic H-flux as well as matter contributions. On cosmological backgrounds it reduces to the usual strong energy condition in GR plus a flux term, Note that the flux contribution is always positive, so SEC violations in GR do not necessarily imply violations here.
• The positive mass condition 5 depends in general on electric H-flux and the stringy energy-momentum Since electric H-flux is forbidden on cosmological backgrounds due to the requirement of homogeneity, this constraint simply becomes (3.20) In [21] it was noted that local violations of this condition may give rise to a regime where gravity becomes repulsive. 6 • The weak energy condition (WEC) can be defined, in consistent analogy with GR, as For the spherical solution considered in [21], 7 this implies that the Noether charge associated with time translation invariance should be non-negative.
• The pressure condition depends only on magnetic H-flux and the spatial components of the stringy energy-momentum tensor. In a cosmological context it becomes In [21] this was labelled as the "weak energy condition", in heuristic analogy with GR, since it similarly pertains to only the tt-component, −Kt t . However, for consistency with our definitions (3.8), we hereby consider it as another independent condition.
Furthermore, the inequality ρ + p + h 2 e −2φ 16πGa 6 ≥ 0 has been included in (3.19) and (3.21) (here a strict inequality) in order to be fully consistent with the standard definitions of the energy conditions in GR. Genuine DFT justification remains to be found, but this is beyond the scope of the present work. 6 Technically this required a corresponding density condition, which was defined without the spatial integral present in the positive mass condition. However on homogeneous backgrounds the integral simply yields a constant factor, so these two conditions become identical. 7 See (4.78) therein.

Solutions
Having obtained the generalization of the Friedmann equations in Stringy Gravity, we turn to the important matter of finding solutions. We first give an exposition of the general framework, before investigating examples of analytic solutions for various types of matter.

Generalized perfect fluid
First of all, let us consider a "generalized perfect fluid" in which w and λ are constant, corresponding to a linear equation of state. In cosmic gauge, the conservation equation (3.12) iṡ For constant w and λ, this can be integrated to give where, in keeping with usual conventions, we have defined a 0 ≡ 1 and φ 0 ≡ 0. Note that the terms in the OFEs which depend on the spatial curvature k and magnetic H-flux h can be interpreted as a particular case of (4.2): specifically, (w, λ) h = (1, 2) and (w, λ) k = (−1/3, 2). Therefore in this subsection we absorb them into ρ and consider a single contribution to the energy density of the form (4.2), which is equivalent to setting the parameters h = k = 0. We will reintroduce them in the following (sub)sections when we examine specific solutions in detail.
It is instructive to consider a power law ansatz such that Comparing this ansatz with the OFEs, we see that solutions with non-trivial ρ are possible only if implying the constraint The OFEs reduce to (1 + 3w)ρ 0 = −n 2 + n − ns − 2 3 s 2 + s , (4.8) where we define a dimensionless quantityρ Note that (4.7)-(4.9) are valid for anyρ 0 , while (4.6) holds only for non-vanishingρ 0 .
Solving for n and s with generic w and λ, we find and the (not necessarily non-vanishing) energy density is proportional tô To obtain these we have made use of the fact that, from rearranging (4.7)-(4.9), In order to make contact with conventional cosmology, we would like to understand the circumstances in which the dilaton φ may become constant, i.e. s = 0. From (4.11) we see that this is only possible on the critical line, which is also the case for more general solutions (not just perfect fluids), and which from (4.11) and (4.12) corresponds to We will see in the next subsection that this corresponds to massless scalar field solutions.
We observe from (4.4) and (4.11) that for the power-law ansatz (4.3), lying on the critical line 8 (4.14) is a necessary and sufficient condition for the dilaton to be constant. However, we emphasize that the sufficient condition applies specifically to power-law behaviour, whereas more general solutions on the critical line may have a non-trivial dilaton profile, as will be seen explicitly in some examples below.
As a final remark, note that (4.12) is smooth asρ 0 → 0: in this limit it coincides with the DFT vacuum solution [73] (i.e. ρ = 0; see (4.34) and (4.35)) with h = k = 0. This is a power law with corresponding to the boundaries of the gray regions in Figure 1,

Analytic solutions
In order to investigate specific cases in detail, we introduce a useful gauge choice which we dub 'Einsteinconformal' gauge, which is defined by the parametrization While the first equality fixes the time reparametrization symmetry, the second simply defines a new timedependent function b, which corresponds to the Einstein-frame scale factor. The OFEs are given in this gauge by where w and λ were defined in (3.18), and we have taken suitable linear combinations of (3.9)-(3.11) for later convenience. In addition, the conservation equation becomes (4.25) Various types of matter will in general occupy different positions on the (w, λ)-plane, see Figure 1. However, note that in general w and λ need not be constant, for example if there are multiple competing contributions to the total energy density.
One interesting scenario emerges on the critical line, 3w + λ = 1 (4.14). Here (4.23) can be integrated to give where h o is a real constant. Plugging this back into (4.21) yields forces the H-flux and variation of the dilaton to vanish, and hence our framework recovers the standard Friedmann equations in this limit. More generally, we can solve for φ by casting (4.26) in the form .

Pure DFT vacuum
In Stringy Gravity, the metric g µν is supplemented by the additional fields B µν and φ. Therefore it is worth considering purely stringy gravitational solutions, since these may be non-trivial due to possible interactions within the extended gravitational sector. This simple scenario, which corresponds to ρ = p = T (0) = 0, can thus yield some initial insight into the nature of cosmological evolution in Stringy Gravity, and provide a foundation for more general solutions featuring additional matter.
In this scenario, equation (4.22) can be recast as which has a general solution given by where C 1 and η 0 are integration constants, and we have defined Consistency with (4.21) requires that the integration constants are related by Using the fact that dτ dη = 1 + kτ 2 , (4.33) and using (4.32), the general solution for the dilaton can be expressed as [73] e 2φ = τ τ * where τ * is an integration constant. Note that this has a minimum at e 2φ min = |h/( factor of the original string-frame metric is thus given by [73]

O(D, D)-symmetric DFT cosmological constant
The action for a DFT cosmological constant in a Riemannian background is simply given by [62] S which implies Note that this corresponds to a scalar field (discussed below) in the limit Φ = 0 and V = V 0 ≡ Λ/8πG.
Choosing cosmic gauge (N = 1), there is a solution for a static universe (H = 0) with the dilaton evolving as In this case, non-trivial H-flux implies positive spatial curvature, and physical solutions require Λa 2 ≥ 2k (note that a must be constant). For h = k = 0 and Λ > 0 this recovers the static solution in flat Minkowskian spacetime with a linear dilaton, initially derived in [74]: where m ≡ Λ/2 > 0.
More generally, there are expanding solutions for h = k = 0 and Λ > 0 with positive m = Λ/2 [75] given by which is real and positive for t > t 0 , and which is similarly defined for t < t 0 and can be obtained from (4.40) by a combined spatial T-duality (Table 1)  converges to the positive-sign case as t → −∞, with C φ = 1 2 e 2φ 0 . For further discussion of linear dilaton solutions, see [74]. Scalar field, e.g. massless limit The action for a spatially homogeneous, canonical scalar field in a Riemannian DFT background is In Einstein-conformal gauge, this yields the equation of motion For the energy-momentum tensor components, we have and thus the density and pressure of Φ are given by We see that the equation of state is confined to the range −1 ≤ w ≤ 1 along the line In the limit of vanishing potential, V (Φ) = 0, from (4.45) we find that ρ = p and thus w = 1. In such cases (4.22) once again reduces to yielding again the solution (4.30). Furthermore, from (4.46) we have λ = −2, which lies on the critical line; see Figure 1. Hence we can solve for the dilaton by integrating (4.28), giving This has a minimum at e 2φ min = |h/h o |. Combining this with (4.30), the scale factor associated with the string-frame metric is given by Note that for consistency with (4.21) we now require .
Finally, note that for h = k = 0 we have a power-law solution. This corresponds to the family of solutions satisfying (4.18), with as can be verified explicitly by converting to cosmic time. which can be solved to give

Radiation solution: with H-flux and dynamically frozen dilaton
where C 1 is an integration constant, E 0 ≡ 8πGρ 0 /3, and τ is as defined in (4.31).
Since we are on the critical line we can integrate (4.23), giving (4.26). Applying this to (4.21) and using the solution (4.54) for the conformal scale factor, one can verify explicitly that as in the vacuum solution. Integrating (4.26) using the new scale factor (4.54) yields where C r is another integration constant. Note that for E 0 = 0 we should recover the vacuum solution, which implies that τ * = C 1 C ∓1/ √ 3 r . Thus we can express the resulting scale factor as As a bonus, we can generalize this solution to the case of radiation plus a scalar with vanishing potential (ignoring interactions). The inhomogeneous piece of equation (4.22) depends only on the energy density in radiation, since the equivalent contribution from the scalar field vanishes, as w Φ = 1. Therefore b 2 must take the form (4.54). Furthermore, being on the critical line, both solutions satisfy (4.26) such that the OFE (4.21) will split into a linear sum of terms for the massless scalar and radiation, respectively. Similarly, in the non-interacting limit, the scalar field and radiation must independently satisfy the conservation equation (4.24) (which is guaranteed by their respective equations of motion). In all, the combined exact solution amounts to a relaxation of (4.55), giving Here we observe that setting E 0 = 0 recovers the solution for a scalar field given in (4.48) and (4.49), while gives the radiation solution of (4.56) and (4.57). Taking both conditions simultaneously, we recover (4.34) and (4.35), the pure vacuum solution. The scalar field evolution is again determined by (4.50), however the solution to this equation now takes the form Note that in the absence of radiation, E 0 = 0, this reduces to (4.51) as expected.
Consider a flat or hyperbolic universe, k ∈ {0, −1}, in which we may study the asymptotic behaviour at large η. In the presence of radiation, we can see from both (4.56) and (4.58) that the dilaton tends towards a constant value and is thus dynamically frozen. Hence at late times this scenario recovers the standard Friedmann equations with a radiation equation of state. Similarly, the scalar field (4.60) also becomes frozen at late times.

de Sitter solutions?
In the absence of external guidance, there lies an inevitable ambiguity in how the string dilaton, φ, and the B-field should couple to matter -point particles, Maxwell fields, spinor fields, any scalar fields, etc.
-in the conventional Riemannian framework. For example, from the stringent experimental constraints supporting the Equivalence Principle and against any "fifth force", it is required that a particle (or a planet) should follow a pure geodesic, However, in the conventional Riemannian or GR framework, it is unclear which metric, string-frame or Einstein-frame, should constitute the Christoffel connection, γ λ µν . With non-trivial string dilaton, the different choices are physically inequivalent: although one may freely perform a field redefinition to switch between frames, this will generically introduce a non-trivial fifth force.
The O(D, D) symmetry principle fixes all the couplings of the closed-string massless sector to any matter. In particular, the O(D, D)-symmetric doubled formulation of a point particle action dictates that the particle follows a geodesic with respect to the string-frame metric [3]. On the other hand, the O(D, D)symmetric action of a canonical scalar field χ reads, with the string-frame metric, which can be rewritten in terms of the Einstein-frame metric, g E µν = g µν exp (−2φ), as Thus, in particular for a massless field, V (χ) = 0, the dilaton completely drops out of the massless scalar action in the Einstein frame. In accordance with [76], in order to generate almost scale-invariant cosmological perturbations of χ, it might be necessary for the Einstein-frame metric g E µν to follow either an inflationary or bouncing-type evolution. However, this is somewhat in contrast to the case of Maxwell fields, where, due to the presence of a (classical) Weyl symmetry, the change of frames does not remove the dilaton.
Within conventional Riemannian cosmology where the dilaton is neglected, there is strong evidence (and arguments) that the expansion of our universe is currently accelerating. However, the above analyses show subtleties regarding the dilaton and may implore us to revisit the "evidence": it implies that, pro- In this section, rather than attempting to resolve the subtle issue of frame dependence, we simply test whether the de Sitter solution is natural in O(D, D)-symmetric cosmology, both in string and Einstein frames separately. In order to make contact with concrete examples, here we will focus in particular on DFT coupled to a (spatially homogeneous) scalar field with arbitrary potential, which also includes the limiting case of a DFT cosmological constant. From (4.44), these solutions satisfy We will show that de Sitter solutions for such models would require exotic matter with negative energy density, violating the weak energy condition (3.21).

String frame
First of all we wish to investigate whether de Sitter solutions are allowed for the string-frame metric. To this end, we set k = 0 and consider the ansatz Here t is cosmic time in string frame, which is defined by setting N = 1. Imposing (5.5) and applying (5.6), solving the OFEs (3.9)-(3.11) forφ givesφ Inserting this into (3.9) yields a general expression for the energy density, Further, plugging (5.7) and (5.8) into either (3.10) or (3.11) gives a similar expression for the pressure, From these we can see that as t → ∞ the energy density and pressure become negative, and hence the weak Since K t t corresponds to minus the kinetic energy for any known type of matter, positive kinetic energy implies ρ < 0, so here also we expect the WEC to be violated.

Einstein frame
To construct the Einstein frame metric, g E µν = g µν exp(−2φ), it is sufficient to consider the general ansatz (3.2) in the gauge We also define the energy density and pressure in Einstein frame as such that the Hamiltonian density is frame-independent: equations then take the form where we have set k = 0 and defined the Hubble parameter in Einstein frame, We now impose the de Sitter ansatz in Einstein frame, Taking the difference of (5.13) and (5.14) yields for which the right-hand side is positive-definite, implying that ρ E + p E = ρ E (1 + w) ≤ 0, violating the strong and weak energy conditions, (3.19) and (3.21), respectively. This is the case either for ρ E < 0 and w ≥ −1, suggesting negative-energy-density solutions as before, or ρ E ≥ 0 and w ≤ −1. Here w = −1 is obtained only forφ = h = 0, which in turn implies, from (5.13) and (5.15), that Finally, it is crucial to investigate whether O(D, D) is truly a symmetry of our universe at early times, and whether or not it is broken at late times. Only further exploration will reveal the answer.
In this parametrization, the section condition can be expressed simply as∂ ν ≡ 0, where the partial derivative is understood to act on or contract with all DFT fields. 9 The gravitational sector consists of a DFT dilaton, d, and a dynamical DFT metric, H AB , which can be decomposed into a pair of projectors, P AB = 1 2 (J +H) AB andP AB = 1 2 (J −H) AB . Furthermore the corresponding local frame has symmetry group Spin(1, D−1)×Spin(D−1, 1), under which the projectors can be decomposed into vielbeins {V Ap ,V Ap } as P AB = V Ap V B p andP AB =V ApVB p , where the local Lorentz indices are raised and lowered using the metrics η pq = diag(− + + · · · +) andηpq = diag(+ − − · · · −), respectively.
Isometries in DFT are best studied using a further-generalized Lie derivative L, which acts on O(D, D) vector indices as well as local Spin(1, D − 1) × Spin(D − 1, 1) indices, as defined in [21]. For isometries parametrized by some set of N DFT vectors, {ζ a }, a = 1, . . . , N , the further-generalized Lie derivatives of 9 Note that for Yang-Mills fields we must generalize this to a covariant derivative, (∂A − iAA), such that when we write AA = (Ã ν , Aµ), the section condition also impliesÃ ν = 0.
each gravitational field with respect to these DFT-Killing vectors should vanish, This in turn implies that the DFT-Killing equations, which read should be satisfied.
In order to couple Stringy Gravity to matter, we must introduce a non-trivial energy-momentum tensor T AB , which appears on the right-hand side of the Einstein Double Field Equations (2.1) and can be derived case-by-case from the gravitational variation of an appropriate O(D, D)-invariant matter Lagrangian [21].
Combining these results and solving, we find that the most general form of K µν in a homogeneous and isotropic universe is diagonal, where K t t (t) and K r r (t) are time-dependent functions. Note in particular that the antisymmetric part K [µν] = 0, which in three spatial dimensions is consistent with homogeneous and isotropic H-flux (A. 12) under the second Einstein Double Field Equation (2.4).