http://arxiv.org/abs/2211.13932
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.
S. Pi and M. Sasaki
Mon, 28 Nov 22
25/93
Comments: 7 pages, 3 figures