A Measure on a Subspace of FRW Solutions and ``The Flatness Problem" of Standard Cosmology

TL;DR

This paper constructs a measure on the space of FRW solutions using DeWitt's metric, revealing a singularity at =1 that influences the perceived distribution of the density parameter over cosmic time.

Contribution

It introduces a unique kinematic measure on FRW solutions and analyzes how its singularity affects interpretations of the flatness problem in cosmology.

Findings

The measure is singular at =1.

Distribution of  values is distorted over time.

Misinterpretation of =1 as a necessary condition for flatness.

Abstract

We use the metric on the space of gravity fields given by DeWitt to construct a unique kinematic measure on the space of FRW simple fluids and show that when the mass parameter $\Omega$ is used as a coordinate this measure is singular at $\Omega = 1$. This singularity, combined with the time evolution of $\Omega$, distorts distributions of $\Omega$ values to be concentrated in the neighborhood of 1 at early times. It is a distorted distribution of $\Omega$ values that sometimes misleads the casual observer to conclude that $\Omega$ must be exactly equal to 1.