The exponential tail of inflationary fluctuations: consequences for
  primordial black holes
  The curvature perturbations produced during an early era of inflation are
known to have quasi-Gaussian distribution functions close to their maximum,
where they are well constrained by measurements of the cosmic microwave
background anisotropies and of the large-scale structures. In contrast, the
tails of these distributions are poorly known, although this part is the
relevant one for rare, extreme objects such as primordial black holes. We show
that these tails are highly non-Gaussian, and cannot be described with standard
non-Gaussian expansions, that are designed to approximate the distributions
close to their maximum only. Using the stochastic-$\delta N$ formalism, we
develop a generic framework to compute the tails, which are found to have an
exponential, rather than Gaussian, decay. These exponential tails are
inevitable, and do not require any non-minimal feature as they simply result
from the quantum diffusion of the inflaton field along its potential. We apply
our formalism to a few relevant single-field, slow-roll inflationary
potentials, where our analytical treatment is confirmed by comparison with
numerical results. We discuss the implications for the expected abundance of
primordial black holes in these models, and highlight that it can differ from
standard results by several orders of magnitude. In particular, we find that
potentials with an inflection point overproduce primordial black holes, unless
slow roll is violated.
