  We study the effective action for gravity obtained after the integration of
scalar matter fields, using the local momentum representation based on the
Riemann normal coordinates expansion. By considering this expansion around
different space-time points, we also compute the non-local terms together with
the more usual divergent ones. We discuss the applicability of our results to
the calculation of particle production rates in cosmological backgrounds and
compare this method with the traditional Bogolyubov transformations.
