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

In string theory the closed-string massless NS-NS sector forms a multiplet of $\mathbf{O}(D,D)$ symmetry. This suggests a specific modification to General Relativity in which the entire NS-NS sector is promoted to stringy graviton fields. Imposing off-shell $\mathbf{O}(D,D)$ symmetry fixes the correct couplings to other matter fields 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 framework. We derive the most general homogeneous and isotropic ansatzes for both stringy graviton fields and the $\mathbf{O}(D,D)$-covariant energy-momentum tensor. Crucially, the former admits space-filling magnetic $H$-flux. 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 solutions in this framework may be characterized by two equation-of-state parameters, $w$ and $\lambda$, where the latter characterizes the relative intensities of scalar and tensor forces. 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 $w$). We further point out that, in contrast to General Relativity, neither an $\mathbf{O}(D,D)$-symmetric cosmological constant nor a scalar field with positive energy density gives rise to a de Sitter solution.