$δn$ formalism: A new formulation for the probability density of the curvature perturbation

Abstract

$δN$ formalism is a useful method to calculate the curvature perturbation. Contrary to what it is typically done in the literature, we re-formulate the $δN$ 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 $δϕ_0$ and its velocity $δπ_0$ are specified by the solutions of the perturbation equation on subhorizon scales. As $δπ_0$ is fully correlated with $δϕ_0$ after horizon exit, we find a novel $δn$ 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.