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

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 δN formalism. We confirm that the critical value β := φ̈ /( Hφ̇ )= -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 β . Such a tail behaviour is important for, e.g., the estimation of the primordial black hole (PBH) abundance.