Covariant generalization of cosmological perturbation theory

TL;DR

This paper introduces a covariant framework for cosmological perturbation theory, defining conserved scalar quantities and gauge-invariant perturbations at all orders, enhancing understanding of inhomogeneities in the universe.

Contribution

It develops a covariant perturbative approach with exact scalar quantities and gauge-invariant methods applicable at arbitrary perturbation order, improving analysis of cosmological inhomogeneities.

Findings

Defines an exact scalar for inhomogeneities in e-folds conserved on all scales.

Derives a compact, physical conservation equation valid at all perturbation orders.

Provides gauge-invariant expressions for curvature perturbations at any order.

Abstract

We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the n-th order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order.