Logarithmic Duality of the Curvature Perturbation

Logarithmic Duality of the Curvature Perturbation by Shi Pi et al. on Sunday 27 November We study the comoving curvature perturbation $\mathcal{R}$ in general 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 $\mathcal{R}(\delta\varphi)$, which consists of a sum of logarithmic functions of the field perturbation $\delta\varphi$ at the point of interest, as well as of its field velocity perturbations $\delta\pi_*$ at the boundaries of each quadratic piece, which are functions of $\delta\varphi$ through the equations of motion. In some simple cases, $\mathcal{R}(\delta\varphi)$ reduces to a single logarithm, which yields either the renowned ``exponential tail'' of the probability distribution function of $\mathcal{R}$ or the Gumbel distribution. arXiv: http://arxiv.org/abs/http://arxiv.org/abs/2211.13932v1