The problem of the reheating of the Universe after inflation is considered. Our approach is based on the use, as "first principles," of the renormalized version of (a) the evolution equation for the mean value of an interacting scalar field and (b) the semiclassical Einstein equations. We compute (for two different toy models) the characteristic time for the damping of the mean-value oscillations. We show that when the oscillations are damped the Universe becomes radiation dominated and inflation ends. The techniques used in this paper are those of quantum field theory in curved space-time and can be generalized to more realistic models.
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Mazzitelli et al. (1989) studied this question.