Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features
Abstract
Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, P(cal N), of the stochastic number of e-folds, cal N, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine P(cal N). First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail, and a power-law behaviour, P(cal N) ∝ cal N−3/2, in the intermediate regime between the peak and the tail of the PDF. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.
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BibTeX
@misc{mishra2026eigenvalue,
title = {{Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features}},
author = {Swagat S. Mishra and Edmund J. Copeland and Anne M. Green},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.00176},
eprint = {2606.00176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.00176},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features
AU - Swagat S. Mishra
AU - Edmund J. Copeland
AU - Anne M. Green
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2606.00176
UR - https://arxiv.org/abs/2606.00176
AB - Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, P(cal N), of the stochastic number of e-folds, cal N, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine P(cal N). First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail, and a power-law behaviour, P(cal N) ∝ cal N^−3/2, in the intermediate regime between the peak and the tail of the PDF. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.
ER -