Stringy corrections to a time-dependent background solution of string and M-Theory

We consider one of the simplest time-dependent backgrounds in M-theory, describing the shrinking away of the M-theory dimension with the other spatial dimensions static. As the M-theory dimension becomes small, the situation becomes well-described by string theory in a singular cosmological background where the string coupling tends to zero but the $\alpha'$-corrections become large, near the cosmic singularity. We compute these $\alpha'$-corrections, both for the background and for linearized perturbations, in heterotic string theory, and show they may be reproduced by a map from eleven-dimensional M-theory.