A general proof of the conservation of the curvature perturbation

TL;DR

This work presents a non-linear generalisation of the curvature perturbation, $\zeta$, and proves its conservation on super-Hubble scales whenever the local pressure is a unique function of the local energy density (adiabatic). Using a 3+1 ADM decomposition and a gradient expansion, the authors show that locally the universe behaves like FLRW on large scales, yielding a local Hubble rate and a local Friedmann equation, while the curvature perturbation remains constant on suitable slicings. They derive a non-linear $\Delta N$ formula, expressing the difference between curvature perturbations on different slicings in terms of $e$-foldings, and identify a gauge-invariant conserved quantity $-\zeta$ valid for $P=P(\rho)$. The results reproduce and extend known second-order perturbation theory findings, providing a transparent, non-perturbative framework for studying non-Gaussianity and large-scale cosmological perturbations with broad theoretical relevance.

Abstract

Without invoking a perturbative expansion, we define the cosmological curvature perturbation, and consider its behaviour assuming that the universe is smooth over a sufficiently large comoving scale. The equations are simple, resembling closely the first-order equations, and they lead to results which generalise those already proven in linear perturbation theory and (in part) in second-order perturbation theory. In particular, the curvature perturbation is conserved provided that the pressure is a unique function of the energy density.