  Particle production in the background of an external classical oscillating
field is a key process describing the stage of preheating after inflation. For
sufficiently strong couplings between the inflaton and matter fields, this
process is known to proceed non-perturbatively. Parametric resonance plays
crucial role for bosonic fields in this case, and the evolution of the
occupation numbers for fermions is non-perturbative as well. In the Minkowski
space, parametric resonance for bosons and non-perturbative effects for
fermions would still persist even in the case of weak coupling. In particular,
the energy density of created bosons would grow exponentially with time.
However, the situation is quite different in the expanding universe. We give a
simple demonstration how the conditions of the expanding universe,
specifically, redshift of the field modes, lead to the usual perturbative
expressions for particle production by an oscillating inflaton in the case of
weak couplings. The results that we obtain are relevant and fully applicable to
the Starobinsky model of inflation.
