We study the comoving curvature perturbation $\mathcal{R}$ in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential $V(\phi )$ by using the $\delta N$ formalism. We find a general formula for $\mathcal{R}(\delta \phi ,\delta \pi )$, 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 $\mathcal{R}(\delta \phi ,\delta \pi )$ 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.

• Received 7 December 2022
• Revised 17 April 2023
• Accepted 2 June 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.011002