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

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