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.Comment: LateX, IOPstyle, 48 pages, 11 figures; matches published version in
C.Q.
