  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.
