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.
