Constant roll and non-Gaussian tail in light of logarithmic duality

TL;DR

This paper explores how constant-roll inflation models exhibit non-Gaussian tails in curvature perturbations, linked to a logarithmic duality, impacting primordial black hole formation estimates.

Contribution

It reveals the role of logarithmic duality in constant-roll inflation and characterizes the non-Gaussian tail behavior of curvature perturbations.

Findings

Critical value $eta=-3/2$ determines stability of the CR condition.

Curvature perturbations follow a logarithmic mapping of Gaussian fields.

Tail behaviors include exponential and Gumbel-like distributions.

Abstract

The curvature perturbation in a model of constant-roll (CR) inflation is interpreted in view of the logarithmic duality discovered in Ref. [1] according to the $\delta N$ formalism. We confirm that the critical value $\beta:=\ddot{\varphi}/(H\dot{\varphi})=-3/2$ determining whether the CR condition is stable or not is understood as the point at which the dual solutions, i.e., the attractor and non-attractor solutions of the field equation, are interchanged. For the attractor-solution domination, the curvature perturbation in the CR model is given by a simple logarithmic mapping of a Gaussian random field, which can realise both the exponential tail (i.e., the single exponential decay) and the Gumbel-distribution-like tail (i.e., the double exponential decay) of the probability density function, depending on the value of $\beta$. Such a tail behaviour is important for, e.g., the estimation of the primordial black hole abundance.