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 ΔN
formalism, we find the probability distribution P(ℛ) of
the comoving curvature perturbation 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
𝒪(10)–𝒪(108), 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.
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