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(phi) by using the delta N formalism. We find a general formula for R(delta phi, delta pi), consisting of a sum of logarithmic functions of the field perturbation delta phi and the velocity perturbation delta pi at the point of interest, as well as of delta pi* at the boundaries of each quadratic piece, which are functions of (delta phi, delta pi) 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 phi. We also clarify the condition under which R(delta phi, delta pi) reduces to a single logarithm, which yields either the renowned "exponential tail" of the probability distribution function of R or a Gumbeldistribution-like tail.