Δ N formalism and conserved currents in cosmology

The ∆N formalism, based on the counting of the number of e-folds during in- flation 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) pertur- bation theory. Later on, it was claimed the equivalence with the results found by introducing a conserved fully non-linear current ζμ, 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 ∆N formalism. By using a novel exact expression for ζμ, valid for any barotropic fluid, we find that it is not in general related to the standard uni- form density curvature perturbation ζ; such a result conflicts with the claimed equivalence with ∆N formalism. Moreover, a similar analysis is done for the proposed non-perturbative generalization Rμ of the comoving curvature perturbation R.