Cosmological moduli problem from thermal effects

We estimate the cosmological abundance of a modulus field that has dilatonic couplings to gauge fields, paying particular attention to thermal corrections on the modulus potential. We find that a certain amount of the modulus coherent oscillations is necessarily induced by a linear thermal effect. We argue that such an estimate provides the smallest possible modulus abundance for a given thermal history of the Universe. As an example we apply our results to a saxion, a bosonic supersymmetric partner of an axion, and derive a tight bound on the reheating temperature. We emphasize that the problem cannot be avoided by fine-tuning the initial deviation of the modulus field, since the minimal amount of the modulus is induced by the dynamics of the scalar potential.