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

TL;DR

This paper introduces a new formulation of the $\delta N$ formalism, called the $\delta n$ formalism, which uses the forward counting of e-folds to compute the probability density of curvature perturbations more straightforwardly.

Contribution

The paper re-formulates the $\delta N$ formalism using the forward e-folding number $n$, simplifying the calculation of the curvature perturbation's PDF.

Findings

Derived a simple formula for the curvature perturbation using the $\delta n$ formalism.

Established the PDF of initial conditions based on solutions of perturbation equations.

Showed the correlation between $\delta\pi_0$ and $\delta\phi_0$ after horizon exit.

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 initial conditions $\delta\phi_0$ and $\delta\pi_0$ 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 simple formula to calculate the curvature perturbation as well as its PDF by using the $\delta N$ method reformulated in terms of $n$, the $\delta n$ formalism.