  Moduli fields generically produce strong dark matter -- radiation and baryon
-- radiation isocurvature perturbations through their decay if they remain
light during inflation. We show that existing upper bounds on the magnitude of
such fluctuations can thus be translated into stringent constraints on the
moduli parameter space m_\sigma (modulus mass) -- \sigma_{inf} (modulus vacuum
expectation value at the end of inflation). These constraints are complementary
to previously existing bounds so that the moduli problem becomes worse at the
perturbative level. In particular, if the inflationary scale H_{inf}~10^{13}
GeV, particle physics scenarios which predict high moduli masses m_\sigma >
10-100 TeV are plagued by the perturbative moduli problem, even though they
evade big-bang nucleosynthesis constraints.
