The Universe after inflation: the wide resonance case
  We study numerically the decay of massive and massless inflatons into massive excitations, via a $\phi^2 X^2$ coupling, in the expanding Universe. We find that a wide enough resonance can survive the Universe expansion, though account for the expansion is very important for determining precisely how wide it should be. For a massive inflaton, the effective production of particles with mass ten times that of the inflaton requires very large values of the resonance parameter $q$, $q\gsim 10^8$. For these large $q$, the maximal size of produced fluctuations is significantly suppressed by the back reaction, but at least within the Hartree approximation they are still not negligible. For the massless inflaton with a $\lambda\phi^4/4$ potential, the Universe expansion completely prevents a resonance production of particles with masses larger than $\sqrt{\lambda}\phi(0)$ for $q$ up to $q=10^6$. 
S. Yu. Khlebnikov and I. I. Tkachev