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Quantum diffusion during inflation and primordial black holes

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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.
Original languageEnglish
Article number046
JournalJournal of Cosmology and Astroparticle Physics
Volume2017
Issue number10
DOIs
Publication statusPublished - 26 Oct 2017

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  • 1707.00537-2

    Rights statement: The final published version of this article by Chris Pattison et al. JCAP10(2017)046 is available online at http://dx.doi.org/10.1088/1475-7516/2017/10/046. © 2017 IOP Publishing Ltd and Sissa Medialab.

    Accepted author manuscript (Post-print), 1.18 MB, PDF document

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