  We study the problem of scalar particle production after inflation by a
rapidly oscillating inflaton field. We use the framework of the chaotic
inflation scenario with quartic and quadratic inflaton potentials. Particular
attention is paid to parametric resonance phenomena which take place in the
presence of the quickly oscillating inflaton field. We have found that in the
region of applicability of perturbation theory the effects of parametric
resonance are crucial, and estimates based on first order Born approximation
often underestimate the particle production. In the case of the quartic
inflaton potential $V(\varphi) = \lambda \varphi^4$, the particle production
process is very efficient even for small values of coupling constants. The
reheating temperature of the universe in this case is $\left[\lambda\, \log\,
(1/\lambda) \right]^{- 1}$ times larger than the corresponding estimates based
on first order Born approximation. In the case of the quadratic inflaton
potential the reheating process depends crucially on the type of coupling
between the inflaton and the other scalar field and on the magnitudes of the
coupling constants. If the inflaton coupling to fermions and its linear (in
inflaton field) coupling to scalar fields are suppressed, then, as previously
discussed by Kofman, Linde and Starobinsky (see e.g. Ref. 13), the inflaton
field will eventually decouple from the rest of the matter, and the residual
inflaton oscillations may provide the (cold) dark matter of the universe. In
the case of the quadratic inflaton potential we obtain the lowest and the
highest possible bounds on the effective energy density of the inflaton field
when it freezes out.
