  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 ${\cal R}$.
