Logarithmic Duality of the Curvature Perturbation

We study the comoving curvature perturbation R in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential V () by using the formalism. We find a general formula for R (, ), consisting of a sum of logarithmic functions of the field perturbation and the velocity perturbation at the point of interest, as well as of * at the boundaries of each quadratic piece, which are functions of (, ) 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. We also clarify the condition under which R (, ) reduces to a single logarithm, which yields either the renowned ``exponential tail'' of the probability distribution function of R or a Gumbel-distribution-like tail.