$\mathrm{\Delta }N$ Formalism and Conserved Currents in Cosmology

The $\mathrm{\Delta }N$ 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\mathrm{.}Byusinganovelexactexpressionfor$\zeta_\mu$, 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 $\mathrm{\Delta }N$ formalism. Moreover, a similar analysis is done for the proposed non-perturbative generalization ${\mathcal{R}}_{\mu }$ of the comoving curvature perturbation $\mathcal{R}$.