  We consider the production of gravitinos and moduli fields from quantum
vacuum fluctuations induced by the presence of scalar metric perturbations at
the end of inflation. We obtain the corresponding occupation numbers, up to
first order in perturbation theory, in terms of the power spectrum of the
metric perturbations. We compute the limits imposed by nucleosynthesis on the
spectral index $n_s$ for different models with constant $n_s$. The results show
that, in certain cases, such limits can be as strong as $n_s<1.12$, which is
more stringent than those coming from primordial black hole production.
