A general proof of the equivalence between the \delta N and covariant formalisms

TL;DR

This paper proves the equivalence between the el N and covariant formalisms for curvature perturbations without assuming Einstein gravity, extending previous results to a more general gravity framework.

Contribution

It provides a general proof of the equivalence between the el N and covariant formalisms independent of slicing conditions or gravity theories.

Findings

Equivalence holds without assuming Einstein gravity.

The proof is valid for arbitrary slicing conditions.

The result generalizes previous superhorizon scale findings.

Abstract

Recently, the equivalence between the \delta N and covariant formalisms has been shown (Suyama et al. 2012), but they essentially assumed Einstein gravity in their proof. They showed that the evolution equation of the curvature covector in the covariant formalism on uniform energy density slicings coincides with that of the curvature perturbation in the \delta N formalism assuming the coincidence of uniform energy and uniform expansion (Hubble) slicings, which is the case on superhorizon scales in Einstein gravity. In this short note, we explicitly show the equivalence between the \delta N and covariant formalisms without specifying the slicing condition and the associated slicing coincidence, in other words, regardless of the gravity theory.