We study nonsingular cosmological scenarios in a general D-dimensional string effective action of the dilaton-modulus-axion system in the presence of the matter source. In the standard dilatonic Brans-Dicke parameter (ω = -1) with radiation, we analytically obtain singularity-free bouncing solutions where the universe starts out in a state with a finite curvature and evolves towards the weakly coupled regime. We apply this analytic method to the string-gas cosmology including the massive state in addition to the leading massless state (radiation), with and without the axion. We numerically find bouncing solutions which asymptotically approach an almost radiation dominant universe with a decreasing curvature irrespective of the presence of the axion, implying that inclusion of the matter source is crucial for the existence of such solutions for ω = -1. In the theories with ω ≠ -1, it is possible to obtain complete regular bouncing solutions with a finite dilaton and curvature in both past and future asymptotics for the general dimension, D. We also discuss the case where dilatonic higher-order corrections are involved to the tree-level effective action and demonstrate that the presence of axion/modulus fields and the matter source does not significantly affect the dynamics of the dilaton-driven inflation and the subsequent graceful exit.
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