A novel non-perturbative approach to String Cosmology
  We develop an exact functional method applied to the bosonic string on a shperical world sheet, in graviton and dilaton backgrounds, consistent with conformal invariance. In this method, quantum fluctuations are controled by the amplitude of the kinetic term of the corresponding stringy sigma-model, and we exhibit a novel non-perturbative non-critical string configuration which appears as a fixed point of our evolution equation. We argue that this string configuration is an exact solution, valid to all orders in alpha', which is consistent with string scattering amplitudes. The dilaton configuration is logarithmic in terms of the string coordinate X^0, and the amplitude of the corresponding quantum fluctuations is independent of the target space dimension D; for D=4, the corresponding Universe, in the Einstein frame, is static and flat. A linearization around this fixed point leads to a slowly expanding, decelerating Universe, reaching asymptotically (in Einstein time) the Minkowski Universe. Moreover, the well-known linear (in terms of X^0) dilaton background, which is a trivial fixed point of our evolution equation, is recovered by our non trivial fixed point for early times. This feature explains the time evolution from a linearly expanding Universe to a Minkowski Universe. 
Jean Alexandre and Nick E. Mavromatos