Implications of stochastic effects for primordial black hole production
  in ultra-slow-roll inflation
  We study the impact of stochastic noise on the generation of primordial black
hole (PBH) seeds in ultra-slow-roll (USR) inflation with numerical simulations.
We consider the non-linearity of the system by consistently taking into account
the noise dependence on the inflaton perturbations, while evolving the
perturbations on the coarse-grained background affected by the noise. We
capture in this way the non-Markovian nature of the dynamics, and demonstrate
that non-Markovian effects are subleading. Using the $\Delta N$ formalism, we
find the probability distribution $P(\mathcal{R})$ of the comoving curvature
perturbation $\mathcal{R}$. We consider inflationary potentials that fit the
CMB and lead to PBH dark matter with $i)$ asteroid, $ii)$ solar, or $iii)$
Planck mass, as well as $iv)$ PBHs that form the seeds of supermassive black
holes. We find that stochastic effects enhance the PBH abundance by a factor of
$\mathcal{O}(10)-\mathcal{O}(10^8)$, depending on the PBH mass. We also show
that the usual approximation, where stochastic kicks depend only on the Hubble
rate, either underestimates or overestimates the abundance by orders of
magnitude, depending on the potential. We evaluate the gauge dependence of the
results, discuss the quantum-to-classical transition, and highlight open issues
of the application of the stochastic formalism to USR inflation.
