Quantum diffusion beyond slow-roll: implications for primordial
  black-hole production
  Primordial black-holes (PBH) can be produced in single-field models of
inflation with a quasi-inflection point in the potential. In these models, a
large production of PBHs requires a deviation from the slow-roll (SR)
trajectory. In turn, this SR violation can produce an exponential growth of
quantum fluctuations. We study the back-reaction of these quantum modes on the
inflationary dynamics using stochastic inflation in the Hamilton-Jacobi
formalism. We develop a methodology to solve quantum diffusion beyond SR in
terms of the statistical moments of the probability distribution. We apply
these techniques to a toy model potential with a quasi-inflection point. We
find that there is an enhancement of the power spectrum due to the dominance of
the stochastic noise in the phase beyond SR. Moreover, non-Gaussian corrections
become as well relevant with a large positive kurtosis. Altogether, this
produces a significant boost of PBH production. We discuss how our results
extend to other single-field models with similar dynamics. We conclude that the
abundance of PBHs in this class of models should be revisited including quantum
diffusion.
