New Mathematical Analysis of Stochastic Inflation Near Hilltop Potentials
A new preprint on arXiv presents an improved analytical treatment of stochastic constant-roll inflation using the spectral method applied to a quadratic hilltop potential. The work finds that rare trajectories crossing the hilltop and becoming temporarily trapped near a reflecting boundary dominate the mean first-passage time, undermining the mean as a reliable statistical descriptor of the inflationary background. The results have implications for models of primordial black hole formation, where similar statistical subtleties are expected to arise.
Researcher Eemeli Tomberg has submitted a 40-page preprint to arXiv detailing a spectral-method solution to the Fokker-Planck equation governing large inflationary perturbations in a quadratic hilltop potential, a setting corresponding to constant-roll inflation. Unlike prior studies, the analysis explicitly accounts for trajectories that cross the hilltop and become transiently trapped near a reflecting boundary on the far side, escaping only slowly via quantum tunneling governed by the lowest eigensolution. Although such trajectories are rare, they are shown to dominate the mean first-passage time, making the mean a misleading summary statistic for the inflationary background. Tomberg argues the median is a more appropriate descriptor and uses it to compute the coarse-grained ΔN distribution, finding that its well-known exponential tail first flattens and then develops a peak near a maximum ΔN value. The paper suggests that analogous complications affect primordial black hole models, where the statistics of large perturbations are central to abundance predictions.
What's missing
As a preprint, this work has not yet undergone formal peer review. The broader implications for primordial black hole abundance estimates are noted qualitatively but not worked out in detail.
What different sources said
- arXiv astro-phCenter
A simple mechanism for the enhancement of the inflationary power spectrum
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