The fine tuning problem in pre-big-bang inflation

We examine the effect of spatial curvature in the pre-big-bang inflationary model suggested by string theory. We study $O(\alpha ')$ corrections and we show that, independently of the initial curvature, they lead to a phase of exponential inflation. The amount of inflation in this phase is long enough to solve the horizon and flatness problems if the evolution starts deeply into the weak coupling regime. There is a region of the parameter space of the model where such a long inflationary phase at the string scale is consistent with COBE anisotropies, millisecond pulsar timing and nucleosynthesis constraints. We discuss implications for the spectrum of relic gravitational waves at the frequencies of LIGO and Virgo.