Regional averaging and scaling in relativistic cosmology

TL;DR

This paper explores scale-dependent averaging methods in relativistic cosmology, proposing a Ricci flow-based smoothing technique to better understand inhomogeneities and their impact on cosmological parameters.

Contribution

It introduces a Lagrangian smoothing approach using Ricci flow to address scale dependence in averaging inhomogeneous cosmologies, contrasting it with Newtonian models.

Findings

Ricci flow provides a rigorous smoothing of the spatial geometry.

Smoothing affects the effective cosmological parameters derived from inhomogeneous models.

The approach offers a new perspective on scale-dependent averaging in relativistic cosmology.

Abstract

Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters like the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with observations. For this reason the relevant parameters are intrinsically scale-dependent and one wishes to control this dependence without restricting the cosmological model by unphysical assumptions. In the latter respect we contrast our way to approach the averaging problem in relativistic cosmology with shortcomings of averaged Newtonian models. Explicitly, we investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian smoothing of (tensorial) variables as opposed to their Eulerian averaging on spatial domains. This program is realized with the help of a global Ricci deformation flow for the metric. We explain rigorously the origin of the Ricci flow which, on heuristic grounds, has already been suggested as a possible candidate for smoothing the initial data set for cosmological spacetimes. The smoothing of geometry implies a renormalization of averaged spatial variables. We discuss the results in terms of effective cosmological parameters that would be assigned to the smoothed cosmological spacetime.