A measure on the set of compact FriedmannLemaîtreRobertsonWalker models

Boudewijn F Roukema (1) , Vincent Blanloeil (2)

1 Torun Centre for Astronomy
2 IRMA - Institut de Recherche Mathématique Avancée

Compact, flat Friedmann-Lematre-Robertson-Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly-flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Ω as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The requirement that the local and global values of Ω are equal implies a range in Ω that consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrised by Ω do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by a spatial size parameter, the injectivity radius r inj, suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.

Mots clés: 98.80.Jk; 04.20.Gz; 02.40.-k

Boudewijn F Roukema, Vincent Blanloeil. A measure on the set of compact FriedmannLemaîtreRobertsonWalker models. Classical and Quantum Gravity, 2010, 27 (24), pp.245001. ⟨10.1088/0264-9381/27/24/245001⟩. ⟨hal-00654150⟩