Rare Events Are Nonperturbative: Primordial Black Holes From Heavy-Tailed Distributions

In recent years it has been noted that the perturbative treatment of the statistics of fluctuations may fail to make correct predictions for the abundance of primordial black holes (PBHs). Moreover, it has been shown in some explicit single-field examples that the nonperturbative effects may lead to an exponential tail for the probability distribution function (PDF) of fluctuations responsible for PBH formation -- in contrast to the PDF being Gaussian, as suggested by perturbation theory. In this paper, we advocate that the so-called $\delta N$ formalism can be considered as a simple, yet effective, tool for the nonperturbative estimate of the tail of the PDF. We discuss the criteria a model needs to satisfy so that the results of the classical $\delta N$ formalism can be trusted and most possible complications due to the quantum nature of fluctuations can be avoided. As a proof of concept, we then apply this method to a simple example and show that the tail of the PDF can be even {\it heavier} than exponential, leading to a significant enhancement of the PBH formation probability, compared with the predictions of the perturbation theory. Our results, along with other related findings, motivate the invention of new, nonperturbative methods for the problem and open up new ideas on generating PBHs with notable abundance.