  We study the class of two-field inflationary Universe models \lambda\phi^4/4
+ g^2\chi^2\phi^2/2, in which parametric resonance during the initial stages of
reheating can lead to an exponential amplification of the amplitude of
cosmological fluctuations. Employing both analytical arguments and numerical
simulations, we determine the time at which backreaction of fluctuations on the
background fields shuts off the exponential growth, making use of the Hartree
approximation, and including scalar metric perturbations. For the case
g^2/\lambda=2, we find that the amplitude of fluctuations after preheating will
exceed the observational upper bound independent of the value of \lambda,
unless the duration of inflation is very long. Cosmological fluctuations are
acceptably small for g^2/\lambda>=8. We also find that the addition of
\chi-field self-interaction can limit the growth of fluctuations, and in the
negative-coupling case the system can become effectively single-field, removing
the resonance.
