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 test the results with numerical simulations. Our examples 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 cutoffs in the probability of large positive fluctuations, meaning that Gaussian distributions would vastly overproduce 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 strongly 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.

• Received 15 November 1996

DOI:https://doi.org/10.1103/PhysRevD.55.7423