O$(d,d)$ covariant String Cosmology to all orders in $\alpha^{\prime}$

Recently, all duality invariant $\alpha'$-corrections to the massless NS-NS sector of string theory on time-dependent backgrounds were classified and the form of their contribution to the action were calculated. In this paper we introduce matter sources in the resulting equations of motion in an O$(d,d)$ covariant way. We show that either starting with the corrected equations and sourcing them with matter or considering corrections to the matter sourced lowest order equations give the same set of equations that defines string cosmology to all orders in $\alpha'$. We also discuss perturbative and non-perturbative de Sitter solutions including matter.