Constraints on moduli cosmology from the production of dark matter and baryon isocurvature fluctuations

We set constraints on moduli cosmology from the production of dark matter -- radiation and baryon -- radiation isocurvature fluctuations through modulus decay, assuming the modulus remains light during inflation. We find that the moduli problem becomes worse at the perturbative level as a significant part of the parameter space m_\sigma (modulus mass) -- \sigma_{inf} (modulus vev at the end of inflation) is constrained by the non-observation of significant isocurvature fluctuations. We discuss in detail the evolution of the modulus vev and perturbations, in particular the consequences of Hubble scale corrections to the modulus potential and the stochastic motion of the modulus during inflation. We show, in particular, that a high modulus mass scale m_\sigma > 100 TeV, which allows the modulus to evade big-bang nucleosynthesis constraints is strongly constrained at the perturbative level. We find that generically, solving the moduli problem requires the inflationary scale to be much smaller than 10^{13} GeV.