  We explore the role of non-Gaussian fluctuations in primordial black hole
(PBH) formation and show that the standard Gaussian assumption, used in all PBH
formation papers to date, is not justified. Since large spikes in power are
usually associated with flat regions of the inflaton potential, quantum
fluctuations become more important in the field dynamics, leading to mode-mode
coupling and non-Gaussian statistics. Moreover, PBH production requires several
sigma (rare) fluctuations in order to prevent premature matter dominance of the
universe, so we are necessarily concerned with distribution tails, where any
intrinsic skewness will be especially important. We quantify this argument by
using the stochastic slow-roll equation and a relatively simple analytic method
to obtain the final distribution of fluctuations. We work out several examples
with toy models that produce PBH's, and show that the naive Gaussian assumption
can result in errors of many orders of magnitude. For models with spikes in
power, our calculations give sharp cut-offs in the probability of large
positive fluctuations, meaning that Gaussian distributions would vastly
over-produce PBH's. The standard results that link inflation-produced power
spectra and PBH number densities must then be reconsidered, since they rely
quite heavily on the Gaussian assumption. We point out that since the
probability distributions depend on the nature of the potential, it is
impossible to obtain results for general models. However, calculating the
distribution of fluctuations for any specific model seems to be relatively
straightforward, at least in the single inflaton case.
