A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models

TL;DR

This paper develops a probability measure on the set of compact FLRW models, showing that flat models are almost surely probable when parametrized by size, challenging the notion of their improbability.

Contribution

It introduces a physically motivated probability measure on compact FLRW models, using the injectivity radius to show flat models are almost surely likely.

Findings

Flat models have a non-zero probability under size parametrization.

Probability measure based on the injectivity radius favors flat models.

Discrete measure over 3-manifolds implies flat models are almost surely.

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.