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We study axion-dilaton cosmologies derived from the low-energy string effective action. We present the classical homogeneous Friedmann-Robertson-Walker solutions and derive the semiclassical perturbation spectra in the dilaton, axion, and moduli fields in the pre-big-bang scenario. By constructing the unique S-duality-invariant field perturbations for the axion and dilaton fields we derive S-duality-invariant solutions, valid when the axion field is time dependent as well as in a dilaton-vacuum cosmology. Whereas the dilaton and moduli fields have steep blue perturbation spectra (with spectral index n=4) we find that the axion spectrum depends upon the expansion rate of the internal dimensions (0. 54<n<~4) which allows scale-invariant (n=1) spectra. We note that for n<~1 the metric is nonsingular in the conformal frame in which the axion is minimally coupled.
Copeland et al. (Tue,) studied this question.