Ultra-slow-roll inflation with quantum diffusion

Abstract

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.

Baryogenesis from ultra-slow-roll inflation

“…To suppress the non-Gaussian tail in the high-sigma limit of ζ led by quantum diffusion in the exact USR limit (δ 2 → −3) [51,[89][90][91][92], we put an upper bound to δ 2 as discussed in appendix A. Note that for δ ≤ −3, the curvature perturbation ζ is led by the late-time scaling dimension (or conformal weight) ∆ ≡ 3/2 − 9/4 + 3δ + δ 2 = 3/2 − |3/2 + δ|, and thus the dimensionless power spectrum…”

“…The statistics of PBH abundance in the exact USR inflation with δ 2 = −3 is very sensitive to the effect of quantum diffusion [89,91]. For more physical cases with δ 2 < −3, the inflaton mass (A.1) led by the constant-rate condition δ2 = 0 in fact has fixed solutions of the inflaton mode functions (with the de Sitter condition H → 0), see [24].…”

“…The exact USR limit with δ 2 = −3 corresponds to a region of completely flat potential in which the inflaton field φ is effectively massless. In this region stochastic effect driven by modes well inside the horizon can play an important role in the VEV of φ, leading to non-linear translation of the curvature perturbation ζ from φ beyond the standard Gaussian approximation [51,[89][90][91][92]. In this section, we drive an upper bound for δ 2 to prevent the breakdown of the Gaussian assumption for ζ.…”

Constant-rate inflation: primordial black holes from conformal weight transitions

“…The analytic structure of the power spectrum generated by a constant-rate inflation has been investigated in [11,12,18,19,41], showing an essential departure from the standard slow-roll predictions [9] due to the significant violation of slow-roll conditions in the background dynamics [8]. Quantum diffusion induced by the off-attractor nature of the constant-rate rolling can introduce non-negligible corrections on the large-scale spectrum via the stochastic effect [13,[42][43][44][45][46].…”

Rare Events Are Nonperturbative: Primordial Black Holes From Heavy-Tailed Distributions

“…However, the Gaussian approximation is clearly unjustified away from the peak of the PDF and a reliable calculation of ρ ζ at its tail is necessarily of nonperturbative nature. There have been a number of nonperturbative investigations taking into account various effects that give rise to an enhancement in the PBH abundance, sometimes by several orders of magnitude [14][15][16][17][18][19][20][21][22][23]. This enhancement is due to lifting the Gaussian fall-off of ρ ζ to a slower type of decay; the slowest single-field case being an exponential fall-off e −Λζ .…”