On spherical dust fluctuations: the exact vs. the perturbative approach

TL;DR

This paper explores the relationship between exact LTB dust models and standard cosmological perturbation theories, showing their equivalence in the linear regime and clarifying the concept of density contrast in a covariant framework.

Contribution

It establishes a covariant, gauge-invariant formalism linking exact LTB fluctuations with conventional perturbation approaches, enhancing understanding of dust dynamics in GR.

Findings

Exact LTB fluctuations are equivalent to CPT and GIC perturbations in the linear regime.

Non-local density fluctuations provide a covariant definition of density contrast.

Spatial curvature perturbation remains time-invariant in the studied models.

Abstract

We examine the relation between the dynamics of Lema\^{\i}tre-Tolman-Bondi (LTB) dust models (with and without $\Lambda$) and the dynamics of dust perturbations in two of the more familiar formalisms used in cosmology: the metric based Cosmological Perturbation Theory (CPT) and the Covariant Gauge Invariant (GIC) perturbations. For this purpose we recast the evolution of LTB models in terms of a covariant and gauge invariant formalism of local and non-local "exact fluctuations " on a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background defined by suitable averages of covariant scalars. We examine the properties of these fluctuations, which can be defined for a confined comoving domain or for an asymptotic domain extending to whole time slices. In particular, the non-local density fluctuation provides a covariant and precise definition for the notion of the "density contrast ". We show that in their linear regime these LTB exact fluctuations (local and non-local) are fully equivalent to the conventional cosmological perturbations in the synchronous-comoving gauge of CPT and to GIC perturbations. As an immediate consequence, we show the time-invariance of the spatial curvature perturbation in a simple form. The present work may provide important theoretical connections between the exact and perturbative (linear or no-linear) approach to the dynamics of dust sources in General Relativity.