Physics

Exponential function

Non-Gaussianity

Probability density function

Cosmic microwave background

Slow roll

Mathematics

Statistical physics

Statistics

Mathematical analysis

Inflaton

Inflation (cosmology)

Langevin equation

Quantum mechanics

Anisotropy

Classical mechanics

Quantum

Journal of Cosmology and Astroparticle Physics

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-δ𝒩 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.

Ultra-slow-roll inflation with quantum diffusion