Ultra-slow-roll inflation with quantum diffusion

Christopher David Pattison*, Vincent Vennin, David Wands, Hooshyar Assadullahi

*Corresponding author for this work

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We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-δΝ formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inflation. To complement these analytical techniques, we present numerical results found both by very large numbers of simulations of the Langevin equations, and through a new, more efficient approach based on iterative Volterra integrals. We illustrate these techniques with two examples of potentials that exhibit an ultra-slow-roll phase leading to the possible production of primordial black holes.
Original languageEnglish
Article number080
Pages (from-to)1-40
Number of pages40
JournalJournal of Cosmology and Astroparticle Physics
Issue number04
Publication statusPublished - 30 Apr 2021


  • UKRI
  • STFC
  • ST/S000550/1
  • inflation
  • primordial black holes
  • TC_highlight


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