  We study the matter-coupled equations of motion for cosmological NS massless
fields including all $\alpha'$ corrections in an O$(d,d)$ duality invariant
approach, with emphasis on the Kalb-Ramond two-form field $B_{(2)}$ and its
source. Solutions for the vacuum and matter case are found and the
corresponding Einstein frame cosmologies are discussed. We also show that the
ansatz for $B_{(2)}$ required by the duality invariant framework implies that
the two-form is non-isotropic.
