Averaging from a global point of view

TL;DR

This paper investigates the averaging problem in spherically symmetric dust solutions, demonstrating that the FLRW solution is a valid average and analyzing deviations through second variation of spatial volume.

Contribution

It provides a mathematical analysis showing the FLRW solution as an averaged solution in a specific class of dust models and examines deviations via second variation.

Findings

Variation of volume vanishes at FLRW solution.

Supports FLRW as an averaged solution.

Analyzes leading deviations from FLRW.

Abstract

We study the averaging problem from a point of view of variation of spatial volume $V$. We show that in the space of spherically symmetric dust solutions which are regular on the spatial manifold $S^3$ the variation $\delta V$ vanishes at the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution in an appropriate sense, which supports the validity of the FLRW solution as the averaged solution. We also present the second variation $\delta^2 V$, giving the leading effect of the deviation from the FLRW solution.