Regional averaging and scaling in relativistic cosmology

doi:10.1088/0264-9381/19/23/314

Regional averaging and scaling in relativistic cosmology

Thomas Buchert et al 2002 Class. Quantum Grav. 19 6109-6145 doi: 10.1088/0264-9381/19/23/314

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Thomas Buchert¹ and Mauro Carfora²
¹ Theoretische Physik, Ludwig-Maximilians-Universität, Theresienstr. 37, 80333 München, Germany
² Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy
E-mail: buchert@theorie.physik.uni-muenchen.de and mauro.carfora@pv.infn.it

Abstract. Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters such as 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 programme 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 dataset 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. In particular, we find that on the smoothed spatial domain $\overline{\cal B}$ evaluated cosmological parameters obey $\overline{\Omega}_{\overline{\cal B}}^{m} + \overline{\Omega}_{\overline{\cal B}}^{R} + \overline{\Omega}_{\overline{\cal B}}^{\Lambda} + \overline{\Omega}_{\overline{\cal B}}^{\cal Q}$ = 1, where $\overline{\Omega}_{\overline{\cal B}}^m,\overline{\Omega}_{\overline{\cal B}}^R$ and $\overline{\Omega}_{\overline{\cal B}}^{\Lambda}$ correspond to the standard Friedmannian parameters, while $\overline{\Omega}_{\overline{\cal B}}^{\cal Q}$ is a remnant of cosmic variance of expansion and shear fluctuations on the averaging domain. All these parameters are 'dressed' after smoothing out the geometrical fluctuations, and we give the relations of the 'dressed' to the 'bare' parameters. While the former provide the framework of interpreting observations with a 'Friedmannian bias', the latter determine the actual cosmological model.

PACS numbers: 0420, 9880, 0240K

Print publication: Issue 23 (7 December 2002)
Received 2 July 2002, in final form 8 October 2002
Published 11 November 2002

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Regional averaging and scaling in relativistic cosmology

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