2017/07/31 by Chris Pattison, Vincent Vennin, Hooshyar Assadullahi +1 · 5 citations
Physics and Astronomy · #Astronomy and Astrophysical Research #Cosmology and Gravitation Theories #Diffusion #Function (biology) #Galaxies: Formation, Evolution, Phenomena #Inflation (cosmology) #Limit (mathematics) #Planck #Planck mass #Primordial black hole #Primordial fluctuations #Probability density function #Quantum #astro-ph.CO #gr-qc #hep-th
paper · pdf · doi:10.1088/1475-7516/2017/10/046
30 pages without appendices (total 42 pages), 9 figures, matches published version in JCAP where one typo in the equation given in the last line of page 23 has also been corrected
openalex created_date 2017/07/14 · openalex publication_date 2017/10/26 · arxiv created 2020/09/15 · arxiv updated 2020/09/16 · openalex updated_date 2026/08/05
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.