Quantum diffusion during inflation and primordial black holes. (arXiv:1707.00537v3 [hep-th] UPDATED)

<a href="http://arxiv.org/find/hep-th/1/au:+Pattison_C/0/1/0/all/0/1">Chris Pattison</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Vennin_V/0/1/0/all/0/1">Vincent Vennin</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Assadullahi_H/0/1/0/all/0/1">Hooshyar Assadullahi</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Wands_D/0/1/0/all/0/1">David Wands</a>

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-$delta N$ 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 $sim 1$

$e$-fold can be spent in regions of the potential dominated by quantum

diffusion. We explain how this requirement constrains inflationary potentials

with two examples.

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-$delta N$ 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 $sim 1$

$e$-fold can be spent in regions of the potential dominated by quantum

diffusion. We explain how this requirement constrains inflationary potentials

with two examples.

http://arxiv.org/icons/sfx.gif