Cosmological effects in the local static frame

¹ Institut Girard Desargues, CNRS UMR 5028, Bâtiment Jean Braconnier, Département de mathématiques, Université Lyon I, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France e-mail: Michel.Mizony@univ-lyon1.fr
² Service d'Astrophysique, CE Saclay, 91191 Gif-sur-Yvette Cedex, France CNRS FRE K25910 “Plasmas, Gravitation et Cosmologie” e-mail: marclr@cea.fr

Received: 18 October 2004
Accepted: 13 December 2004

Abstract

What is the influence of cosmology (the expansion law and its acceleration, the cosmological constant...) on the dynamics and optics of a local system like the solar system, a galaxy, a cluster or a supercluster? The answer requires the solution of the Einstein equation with the local source, which tends towards the cosmological model at large distances. There is, in general, no analytic expression for the corresponding metric, but we calculate here an expansion in a small parameter, which allows us to answer this question. First, we derive a static expression for the pure cosmological (Friedmann-Lemaître) metric, whose validity, although local, extends in a very large region around the observer. This expression appears as the metric of an osculating de Sitter model. 
We then propose an expansion of the cosmological metric with a local source, which is valid in the very wide neighborhood of the local system. This allows us to calculate exactly the (tiny) influence of cosmology on the dynamics of the solar system: it results that, contrary to some claims, cosmological effects fail to account for the unexplained acceleration of the Pioneer probe by several orders of magnitudes. Our expression provides estimations of the cosmological influence in the calculations of rotation or dispersion velocity curves in galaxies, clusters, and any type of cosmic structure, necessary for precise evaluation of dark matter and/or cosmic flows. The same metric can also be used to estimate the influence of cosmology on gravitational optics in the vicinity of such systems.