  We consider the coupled evolution of density, (scalar) metric and dilaton
perturbations in the transition from a ``stringy" phase of growing curvature
and gravitational coupling to the standard radiation-dominated cosmology. We
show that dilaton production, with a spectrum tilted towards large frequencies,
emerges as a general property of this scenario. We discuss the
frame-independence of the dilaton spectrum and of the inflationary properties
of the metric background by using, as model of source, a pressureless gas of
weakly interacting strings, which is shown to provide an approximate but
consistent solution to the full system of background equations and string
equations of motion. We combine various cosmological bounds on a growing
dilaton spectrum with the bound on the dilaton mass obtained from tests of the
equivalence principle, and we find allowed windows compatible with a universe
presently dominated by a relic background of dilatonic dark matter.
