An update on adiabatic modes in cosmology and $\delta$N formalism

TL;DR

This paper extends Weinberg's method to non-attractor inflation, clarifies the relationship between adiabaticity and curvature perturbations, and discusses different definitions of the $ abla$ formalism, highlighting limitations of some approaches.

Contribution

It generalizes the $ abla$ formalism to non-attractor regimes and clarifies the non-constancy of $ abla$ in adiabatic conditions, also analyzing $ abla N$ definitions.

Findings

Both modes of $ abla$ relate to symmetry in perturbative equations.

Adiabaticity does not necessarily imply constant $ abla$.

Non-perturbative $ abla N$ can be extended at leading order in gradient expansion.

Abstract

In this paper, we generalize the Weinberg's procedure to determine the comoving curvature perturbation $\cal R$ to non-attractor inflationary regimes. We show that both modes of $\cal R$ 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 $\cal R$, not even in the $k\rightarrow 0$ limit. We then show that there exist non-equivalent definitions of $\delta N$ that would reproduce $\mathcal{R}$ or the uniform density curvature perturbation $\zeta$ at linear order. We have then shown that the perturbative $\delta N$ 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 stated in the literature.