Cosmology at the top of the $\alpha'$ tower
  The cosmology of the fully $\alpha'$-corrected duality-invariant action for
the Neveu-Schwarz sector of string theory is revisited, with special emphasis
on its coupling to matter sources. The role of the duality covariant pressure
and dilatonic charge of the matter sector is explored in various contexts, from
the low-curvature regime to non-perturbative solutions in $\alpha'$. We comment
on how an infinite tower of $\alpha'$ corrections allows for fixed-dilaton de
Sitter solutions, even in vacuum. We further investigate the necessary
conditions for accelerated expansion in the Einstein frame, as well as for
non-singular bounces that could resolve the big bang singularity. In
particular, explicit examples are constructed, which show that the tower of
$\alpha'$ corrections may support an Einstein-frame non-singular cosmological
bouncing background, even when the matter sector respects the null energy
condition.
