Numerical stochastic inflation constrained by frozen noise

Abstract

Stochastic inflation can resolve strong inflationary perturbations, which seed primordial black holes. I present a fast and accurate way to compute these perturbations in typical black hole producing single-field models, treating the short-wavelength Fourier modes beyond the de Sitter approximation. The squeezing and freezing of the modes reduces the problem to one dimension, and the resulting new form of the stochastic equations, dubbed `constrained stochastic inflation,' can be solved efficiently with semi-analytical techniques and numerical importance sampling. In an example case, the perturbation distribution is resolved in seconds deep into its non-Gaussian tail, a speed-up of factor 109 compared to a previous study. Along the way, I comment on the role of the momentum constraint in stochastic inflation.

Primordial black holes and stochastic inflation beyond slow roll. Part I. Noise matrix elements

“…Numerical simulations of the stochastic dynamics carried out in refs. [71,113] also indicate that the mass fraction of PBHs do not depend upon the particular choice for the value of σ, as long as it is not arbitrarily small.…”

“…In our analysis, we have considered σ = 0.01 in order to account for substantial non-linearity in the evolution by including as many modes into the long-wavelength regime as possible without violating the stochastic nature of the noise terms (see refs. [71,113]). Nevertheless, our results given in eqs.…”

“…However, numerical simulations demonstrate that such non-Markovian corrections due to the backreaction effects of the stochastic IR background are negligible for single field inflationary potentials with a large class of PBH forming features, such as a flat segment, an inflection point, or a bump/dip and hence can be safely ignored (see refs. [71,113]).…”

“…This property of maximal correlation between the noise terms is a direct consequence of the fact that for σ 1, the super-Hubble UV mode functions, φ k , are frozen by the time they join the coarse-graining scale k = σaH (see refs. [71,113] for a detailed discussion on the freezing behaviour of the UV modes). Consequently, we conclude that quantum diffusion can be assumed to be sourced by a single random noise term.…”

Monomial warm inflation revisited

“…[20]. This procedure differs from the one in section 6 and other recent works [46,[50][51][52] in specific points that we discuss below. The results that follow reach the same conclusion as in section 6: the…”