Full higher-dimensional analysis of moduli oscillation and radiation in expanding universe

We investigate effects of the radiation and the moduli oscillation around the stabilized values on the evolution of a 6-dimensional spacetime compactified on $S^2$. In order to see the transition from the 5-dimensional space to the 3-dimensional one, we develop a procedure to pursue the spacetime evolution with appropriate approximations, which is valid until the spacetime behaves like 4-dimensional. In the case that the moduli stabilization process cannot be described in the context of the 4-dimensional effective theory, it takes quite a long time for the moduli oscillation to dominate the total energy density, in contrast to the conventional result obtained by the 4-dimensional effective theory approach. We also found that even if the moduli are set at the stabilized values, they start to oscillate due to the pressure in the extra space $S^2$ in some cases.