Weighed scalar averaging in LTB dust models, part II: a formalism of exact perturbations

TL;DR

This paper develops a formalism for exact, non-linear perturbations in LTB dust models using q-averages, linking local and non-local perturbations to FLRW backgrounds and deriving their evolution equations.

Contribution

It introduces a new formalism for exact perturbations in LTB models based on q-averages, connecting local and non-local perturbations to FLRW backgrounds without requiring actual matching.

Findings

Derived evolution equations for q-scalars and perturbations.

Established a formal link between local/non-local perturbations and FLRW backgrounds.

Illustrated differences with an example of a cosmic density void.

Abstract

We examine the exact perturbations that arise from the q-average formalism that was applied in the preceding article (part I) to Lemaitre-Tolman-Bondi (LTB) models. By introducing an initial value parametrization, we show that all LTB scalars that take a FLRW "look alike" form (frequently used in the literature dealing with LTB models) follow as q-averages of covariant scalars that are common to FLRW models. These q--scalars determine for every averaging domain a unique FLRW background state through Darmois matching conditions at the domain boundary, though the definition of this background does not require an actual matching with a FLRW region (Swiss cheese type models). Local perturbations describe the deviation from the FLRW background state through the local gradients of covariant scalars at the boundary of every comoving domain, while non-local perturbations do so in terms of the intuitive notion of a "contrast" of local scalars with respect to FLRW reference values that emerge from q-averages assigned to the whole domain or the whole time slice in the asymptotic limit. We derive fluid flow evolution equations that completely determine the dynamics of the models in terms of the q-scalars and both types of perturbations. A rigorous formalism of exact spherical non-linear perturbations is defined over the FLRW background state associated to the q-scalars, recovering the standard results of linear perturbation theory in the appropriate limit. We examine the notion of the amplitude and illustrate the differences between local vs non-local perturbations by qualitative diagrams and through an example of a cosmic density void that follows from the numeric solution of the evolution equations.