Vacuum fluctuations in axion-dilaton cosmologies

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 $\mathrm{}(n=1\mathrm{})\mathrm{}$ spectra. We note that for $n<~1$ the metric is nonsingular in the conformal frame in which the axion is minimally coupled.