Inflationary cosmologies from compactification?

We consider the compactification of $\mathrm{}(d+n)\mathrm{}$-dimensional pure gravity and of superstring or M-theory on an n-dimensional internal space to a d-dimensional Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, with a spatial curvature $k=0,\pm 1,$ in the Einstein conformal frame. The internal space is taken to be a product of Einstein spaces, each of which is allowed to have arbitrary curvature and a time-dependent volume. By investigating the effective d-dimensional scalar potential, which is a sum of exponentials, it is shown that such compactifications, in the $k=0,+1\mathrm{}$ cases, do not lead to large amounts of accelerating expansion of the scale factor of the resulting FLRW universe, and, in particular, do not lead to inflation. The case $k=-1$ admits solutions with eternal accelerating expansion for which the acceleration, however, tends to zero at late times.