An update on adiabatic modes in cosmology and δN formalism

Abstract

In this paper, we generalize the Weinberg's procedure to determine the comoving curvature perturbation ℛ to non-attractor inflationary regimes. We show that both modes of ℛ are related to a symmetry of the perturbative equations in the Newtonian gauge. As a byproduct, we clarify that adiabaticity does not generally imply constancy of ℛ, not even in the k ⟶ 0 limit. We then show that there exist non-equivalent definitions of δN that would reproduce ℛ or the uniform density curvature perturbation ζ at linear order. We have then shown that the perturbative δN definition in terms of difference between the number of e-foldings of different gauges, can be extended non-perturbatively at leading order in gradient expansion. Nevertheless, the computer friendly definition in terms of the difference of e-foldings obtained from the evolution of a local FRW Universe, respectively with perturbed and un-perturbed initial conditions, might only give information about the linear order curvature perturbations, contrary to what is stated in the literature.