On average properties of inhomogeneous fluids in general relativity III: general fluid cosmologies

TL;DR

This paper develops generalized averaging formalisms for inhomogeneous cosmologies with arbitrary fluids, including vorticity and tilt, extending previous models by considering more general foliations and flow conditions.

Contribution

It introduces new averaging schemes for inhomogeneous cosmologies with arbitrary fluids, broadening the scope of previous irrotational models and addressing limitations like rest mass conservation.

Findings

Two different averaging systems are formulated and compared.

The second scheme simplifies equations by focusing on fluid flow and 1+3 threading.

Application to constant fluid proper time slices yields particularly transparent results.

Abstract

We investigate effective equations governing the volume expansion of spatially averaged portions of inhomogeneous cosmologies in spacetimes filled with an arbitrary fluid. This work is a follow-up to previous studies focused on irrotational dust models (Paper I) and irrotational perfect fluids (Paper II) in flow-orthogonal foliations of spacetime. It complements them by considering arbitrary foliations, arbitrary lapse and shift, and by allowing for a tilted fluid flow with vorticity. As for the first studies, the propagation of the spatial averaging domain is chosen to follow the congruence of the fluid, which avoids unphysical dependencies in the averaged system that is obtained. We present two different averaging schemes and corresponding systems of averaged evolution equations providing generalizations of Papers I and II. The first one retains the averaging operator used in several other generalizations found in the literature. We extensively discuss relations to these formalisms and pinpoint limitations, in particular regarding rest mass conservation on the averaging domain. The alternative averaging scheme that we subsequently introduce follows the spirit of Papers I and II and focuses on the fluid flow and the associated 1+3 threading congruence, used jointly with the 3+1 foliation that builds the surfaces of averaging. This results in compact averaged equations with a minimal number of cosmological backreaction terms. We highlight that this system becomes especially transparent when applied to a natural class of foliations which have constant fluid proper time slices.