$\delta n$ formalism: A new formulation for the probability density of the curvature perturbation

Abstract

$\delta N$ formalism is a useful method to calculate the curvature perturbation. Contrary to what it is typically done in the literature, we re-formulate the $\delta N$ formalism by using the $e$-folding number $n$ counted forward in time. For a fixed initial time $\bar{n}_0$, the probability density function (PDF) of the field perturbation $\delta\phi_0$ and its velocity $\delta\pi_0$ are specified by the solutions of the perturbation equation on subhorizon scales. As $\delta\pi_0$ is fully correlated with $\delta\phi_0$ after horizon exit, we find a novel $\delta n$ formalism to calculate the curvature perturbation as well as its PDF. It can naturally incorporate those trajectories for which inflation never ends, which are lost when counting $N$ backward in time.