Cosmological Phases of the String Thermal Effective Potential

In a superstring framework, the free energy density, F, can be determined unambiguously at the full string level once supersymmetry is spontaneously broken via geometrical fluxes. We show explicitly that only the moduli associated to the supersymmetry breaking may give relevant contributions. All other spectator moduli \mu_I give exponentially suppressed contributions for relatively small (as compared to the string scale) temperature, T, and supersymmetry breaking scale, M. More concisely, for \mu_I > T and M, F takes the form F(T,M; \mu_I)=F(T,M)+O[exp(- {\mu_I\over T}), exp(- {\mu_I\over M})] We study the cosmological regime where T and M are below the Hagedorn temperature scale T_H. In this regime, F remains finite for any values of the spectator moduli \mu_I. We investigate extensively the case of one spectator modulus \mu_d corresponding to R_d, the radius-modulus field of an internal compactified dimension. We show that its thermal effective potential admits five phases, each of which can be described by a distinct but different effective field theory. For late cosmological times, the Universe is attracted to a "Radiation-like evolution" with M(t) ~ T(t)~ 1/a(t)~ t^{-2/d}. The spectator modulus \mu(t) is stabilized either to the stringy enhanced symmetry point where R_d=1, or fixed at an arbitrary constant \mu_0>T,M. For arbitrary boundary conditions at some initial time, t_E, \mu(t) may pass through more than one effective field theory phase before its final attraction.