Massive string modes and non-singular pre-big-bang cosmology

Perturbative $\alpha '$ corrections to the low energy string effective action have been recently found to have a potentially regularizing effect on the singularity of the lowest order pre-big-bang solutions. Whether they actually regularize it, however, cannot be determined working at any finite order in a perturbative expansion in powers of the string constant $\alpha '$, because of scheme dependence ambiguities. Physically, these corrections are dominated by the integration over the first few massive string states. Very massive string modes, instead, always have a regularizing effect which is non-perturbative in field with Hubble constant $H$ they are produced with an effective Hawking temperature $T=H/(2\pi)$, and an infinite production rate would occur if this temperature exceeded the Hagedorn temperature. Depending on which regularizing mechanism takes place first, different cosmological scenarios are possible in the string phase of the model, in which the Hubble parameter either approaches asymptotically a constant value or gets close to a maximum value and then bounces back.