Logarithmic Duality of the Curvature Perturbation

We study the comoving curvature perturbation $\mathcal{R}$ in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential $V(\varphi)$ by using the $\delta N$ formalism. We find a general formula for $\mathcal{R}(\delta\varphi, \delta\pi)$, consisting of a sum of logarithmic functions of the field perturbation $\delta\varphi$ 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 ($\delta\varphi, \delta\pi$) 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 $\varphi$. We also clarify the condition under which $\mathcal{R}(\delta\varphi, \delta\pi)$ 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.