Logarithmic Duality of the Curvature Perturbation

Abstract

We study the comoving curvature perturbation $\mathcal{R}$ in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential $V(φ)$ by using the $δN$ formalism. We find a general formula for $\mathcal{R}(δφ, δπ)$, consisting of a sum of logarithmic functions of the field perturbation $δφ$ and the velocity perturbation $δπ$ at the point of interest, as well as of $δπ_*$ at the boundaries of each quadratic piece, which are functions of ($δφ, δπ$) through the equation of motion. Each logarithmic expression has an equivalent dual expression, due to the second-order nature of the equation of motion for $φ$. We also clarify the condition under which $\mathcal{R}(δφ, δπ)$ reduces to a single logarithm, which yields either the renowned ``exponential tail'' of the probability distribution function of $\mathcal{R}$ or a Gumbel-distribution-like tail.