Quantum diffusion during inflation and primordial black holes

Abstract

Abstract. We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-δN formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than ∼ 1 e-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

The exponential tail of inflationary fluctuations: consequences for primordial black holes

“…This expression is always well approximated by its first term, and one can see that for PBHs not to be over produced, one needs to impose µ ζ c , (4.6) in agreement with the conclusions of Ref. [61]. This places an upper bound on the width, or a lower bound on the height, of flat sections in the potential.…”

“…it contains first and second-order derivatives with respect to the field coordinates Φ i ). The PDF of the number of e-folds realised from Φ until the end of inflation, N , can then be shown to obey the adjoint Fokker-Planck equation [53,61]…”

“…More precisely, in Refs. [53,61], it is shown that, in general, when the condition v v 2 v 2 1 (3.37) holds, the potential can be assumed to be exactly flat (hence dominated by quantum diffusion), while when the opposite condition applies, the dynamics of the inflaton is essentially classical. The condition (3.37) is saturated at a value of x such that |x| = [2v 0 /(3β)] 1/3 ≡ ∆φ well /(2M Pl ), which defines the width…”

Quantum loop effects to the power spectrum of primordial perturbations during ultra slow-roll inflation

“…Hence, we can use the solution of the Hankel function of order ν in Eq. (19) to compute the power spectrum which is explicitly given by…”

“…A natural framework to involve quantum kicks is through the stochastic inflation [18]. Several works [19][20][21][22] have been devoted to the study of this quantum noise effects and found a significant boost of PBH production whereas the work in Ref. [23] has claimed that quantum diffusion effects in the USR inflation are insignificant by keeping the formalism in its regime of validity.…”