# Quantum diffusion during inflation and primordial black holes

## 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-δ 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 ∼ 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.

- Authors:

- Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX (United Kingdom)
- Laboratoire Astroparticule et Cosmologie, Université Denis Diderot Paris 7, 75013 Paris (France)

- Publication Date:

- OSTI Identifier:
- 22667621

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 10; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; BLACK HOLES; COMPUTERIZED SIMULATION; CORRECTIONS; DENSITY; DIFFERENTIAL EQUATIONS; DIFFUSION; DISTURBANCES; EXPANSION; INFLATIONARY UNIVERSE; MASS; PERTURBATION THEORY; PROBABILITY; PROBABILITY DENSITY FUNCTIONS; STOCHASTIC PROCESSES

### Citation Formats

```
Pattison, Chris, Assadullahi, Hooshyar, Wands, David, and Vennin, Vincent, E-mail: hooshyar.assadullahi@port.ac.uk, E-mail: christopher.pattison@port.ac.uk, E-mail: vincent.vennin@port.ac.uk, E-mail: david.wands@port.ac.uk.
```*Quantum diffusion during inflation and primordial black holes*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/10/046.

```
Pattison, Chris, Assadullahi, Hooshyar, Wands, David, & Vennin, Vincent, E-mail: hooshyar.assadullahi@port.ac.uk, E-mail: christopher.pattison@port.ac.uk, E-mail: vincent.vennin@port.ac.uk, E-mail: david.wands@port.ac.uk.
```*Quantum diffusion during inflation and primordial black holes*. United States. doi:10.1088/1475-7516/2017/10/046.

```
Pattison, Chris, Assadullahi, Hooshyar, Wands, David, and Vennin, Vincent, E-mail: hooshyar.assadullahi@port.ac.uk, E-mail: christopher.pattison@port.ac.uk, E-mail: vincent.vennin@port.ac.uk, E-mail: david.wands@port.ac.uk. Sun .
"Quantum diffusion during inflation and primordial black holes". United States.
doi:10.1088/1475-7516/2017/10/046.
```

```
@article{osti_22667621,
```

title = {Quantum diffusion during inflation and primordial black holes},

author = {Pattison, Chris and Assadullahi, Hooshyar and Wands, David and Vennin, Vincent, E-mail: hooshyar.assadullahi@port.ac.uk, E-mail: christopher.pattison@port.ac.uk, E-mail: vincent.vennin@port.ac.uk, E-mail: david.wands@port.ac.uk},

abstractNote = {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-δ 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 ∼ 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.},

doi = {10.1088/1475-7516/2017/10/046},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 10,

volume = 2017,

place = {United States},

year = {Sun Oct 01 00:00:00 EDT 2017},

month = {Sun Oct 01 00:00:00 EDT 2017}

}