  We provide a simple but robust argument that primordial black hole (PBH)
production generically does {\em not} exceed astrophysical bounds during the
resonant preheating phase after inflation. This conclusion is supported by
fully nonlinear lattice simulations of various models in two and three
dimensions which include rescattering but neglect metric perturbations. We
examine the degree to which preheating amplifies density perturbations at the
Hubble scale and show that at the end of the parametric resonance, power
spectra are universal, with no memory of the power spectrum at the end of
inflation. In addition we show how the probability distribution of density
perturbations changes from exponential on very small scales to Gaussian when
smoothed over the Hubble scale -- the crucial length for studies of primordial
black hole formation -- hence justifying the standard assumption of
Gaussianity.
