A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models
arXiv:0912.2300 · doi:10.1088/0264-9381/27/24/245001
Abstract
Compact, flat Friedmann-Lemaitre-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 Omega 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 inferred range in Omega 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 Omega do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by the injectivity radius r_inj ("size") 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.
19 pages, 4 figures; v2: minor language improvements; v3: generalisation: m, H functions of M
References in corpus (14)
- The Measure Problem in Cosmology
- The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps
- Extending the WMAP Bound on the Size of the Universe
- How flat can you get? A model comparison perspective on the curvature of the Universe
- Do we Live in a "Small Universe"?
- A new analysis of Poincaré dodecahedral space model
- Imprints of spherical non-trivial topologies on the CMB
- A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
- CMB Alignment in Multi-Connected Universes
- A spatial correlation analysis for a toroidal universe
- A weak acceleration effect due to residual gravity in a multiply connected universe
- The residual gravity acceleration effect in the Poincare dodecahedral space
- Poincare dodecahedral space parameter estimates
- Hot pixel contamination in the CMB correlation function?