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.
