Highly Unstable Fundamental Stringsin

We characterize a regime of extreme (Jeans-like) instability, for strings evolving in cosmological backgrounds, in which the string's proper size grows asymptotically like the scale factor of the expanding Universe.We develop a new approximation scheme, based on the asymptotic proportionality of world-sheet and conformal times, for the systematic, quantitative description of such non-linear regime. We nd that only innationary geometries (accelerated expansion) are compatible with this instability, and we derive an equation of state for a perfect uid of unstable strings. The eeective pressure is negative, but not large enough to sustain by itself a phase of accelerated expansion.