Qualitative Analysis of Universes with Varying Alpha
John D. Barrow, David F. Mota

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, , we analyse the phase space of the system of equations that describes a time-varying fine structure 'constant', , in the Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We have classified all the possible behaviours of in ever-expanding universes with different and find new exact solutions for . We find the attractors points in the phase space for all . In general, will be a non-decreasing function of time that increases logarithmically in time during a period when the expansion is dust dominated (), but becomes constant when . This includes the case of negative-curvature domination (). also tends rapidly to a constant when the expansion scale factor increases exponentially. A general set of…
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