| Summary: | Inflationary stage is followed by particle production in the background of an oscillating inflaton field, which process is called preheating. For sufficiently strong couplings between the inflaton and matter fields, it is known to proceed non-perturbatively, with parametric resonance playing crucial role for bosonic fields. 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 inflationary model.
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