# $\Delta N$ Formalism and Conserved Currents in Cosmology

## TL;DR

This paper critically examines the $
abla N$ formalism and conserved currents in cosmology, revealing issues with their equivalence and proposing a novel exact expression for the conserved current $
abla $, challenging previous assumptions.

## Contribution

It introduces a new exact expression for the conserved current $
abla $ in barotropic fluids and analyzes its relation to standard curvature perturbations, questioning prior equivalences.

## Key findings

- The new expression for $
abla $ differs from the standard $$ in general.
- The claimed equivalence between $
abla N$ formalism and conserved currents is not generally valid.
- Analysis of the non-perturbative generalization ${\u03c1}_$ of the comoving curvature perturbation.

## Abstract

The $\Delta N$ formalism, based on the counting of the number of e-folds during inflation in different local patches of the Universe, has been introduced several years ago as a simple and physically intuitive approach to calculate (non-linear) curvature perturbations from inflation on large sales, without resorting to the full machinery of (higher-order) perturbation theory. Later on, it was claimed the equivalence with the results found by introducing a conserved fully non-linear current $\zeta_\mu$, thereby allowing to directly connect perturbations during inflation to late-Universe observables. We discus some issues arising from the choice of the initial hyper-surface in the $\Delta N$ formalism. By using a novel exact expression for $\zeta_\mu$, valid for any barotropic fluid, we find that it is not in general related to the standard uniform density curvature perturbation $\zeta$; such a result conflicts with the claimed equivalence with $\Delta N$ formalism. Moreover, a similar analysis is done for the proposed non-perturbative generalization ${\cal R}_\mu$ of the comoving curvature perturbation ${\cal R}$.