Qualitative Analysis of Universes with Varying Alpha

TL;DR

This paper analyzes how the fine structure constant $mbda$ evolves over time in different expanding universe models, classifying behaviors, finding exact solutions, and establishing conditions for $mbda$ to stabilize.

Contribution

It provides a comprehensive phase space analysis of a varying $mbda$ in power-law expanding universes, including new exact solutions and attractor points.

Findings

$mbda$ generally increases logarithmically during dust domination

$mbda$ becomes constant for $n>2/3$

$mbda$ rapidly stabilizes in exponential expansion

Abstract

Assuming a Friedmann universe which evolves with a power-law scale factor, $a=t^{n}$, we analyse the phase space of the system of equations that describes a time-varying fine structure 'constant', $\alpha$, in the Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We have classified all the possible behaviours of $\alpha (t)$ in ever-expanding universes with different $n$ and find new exact solutions for $\alpha (t)$. We find the attractors points in the phase space for all $n$. In general, $\alpha $ will be a non-decreasing function of time that increases logarithmically in time during a period when the expansion is dust dominated ($n=2/3$), but becomes constant when $n>2/3$. This includes the case of negative-curvature domination ($n=1$). $\alpha $ also tends rapidly to a constant when the expansion scale factor increases exponentially. A general set of conditions is established for $\alpha $ to become asymptotically constant at late times in an expanding universe.