Conserved non-linear quantities in cosmology

TL;DR

This paper presents a covariant, fully non-perturbative framework for non-linear cosmological perturbations based on gradients of the energy density and the integrated expansion, encapsulated in the central covector $\zeta_a$. It derives an exact conservation equation ${\cal L}_u \zeta_a = -\Theta\Gamma_a/[3(\rho+P)]$, with $\Gamma_a$ encoding non-adiabatic pressure, and recovers the linear curvature perturbation on uniform density hypersurfaces in the appropriate limit. The authors extend the analysis to second order, relate $\zeta_a$ to other conserved quantities such as $R_a$ and $C_a$, and show a general construction for conserved covectors from any conserved density $n$. The approach is non-perturbative and gauge-invariant in a practical sense, does not rely on Einstein's equations, and applies to any gravity theory with conserved energy-momentum, providing a unified, scalable toolkit for studying non-linear cosmological perturbations across all scales.

Abstract

We give a detailed and improved presentation of our recently proposed formalism for non-linear perturbations in cosmology, based on a covariant and fully non-perturbative approach. We work, in particular, with a covector combining the gradients of the energy density and of the local number of e-folds to obtain a non-linear generalization of the familiar linear uniform density perturbation. We show that this covector obeys a remarkably simple conservation equation which is exact, fully non-linear and valid at all scales. We relate explicitly our approach to the coordinate-based formalisms for linear perturbations and for second-order perturbations. We also consider other quantities, which are conserved on sufficiently large scales for adiabatic perturbations, and discuss the issue of gauge invariance.