Matter seen at many scales and the geometry of averaging in relativistic cosmology

TL;DR

This paper explores how averaging scalar functions on curved 3D spaces varies with scale in relativistic cosmology, linking it to Ricci flow and offering insights into the geometry of the universe at different scales.

Contribution

It introduces a novel perspective on scale-dependent averaging in cosmology, connecting Eulerian and Lagrangian viewpoints and rigorously deriving the Ricci deformation flow as a natural averaging process.

Findings

Establishes a rigorous connection between averaging procedures and Ricci flow.

Provides a new geometric framework for understanding scale dependence in cosmological models.

Suggests Ricci flow as a candidate for smoothing initial data in relativistic cosmology.

Abstract

We investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian scaling of variables as opposed to their Eulerian averaging on spatial domains. This program explains rigorously the origin of the Ricci deformation flow for the metric, a flow which, on heuristic grounds, has been already suggested as a possible candidate for averaging the initial data set for cosmological spacetimes.