Qualitative Analysis of Universes with Varying Alpha

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Assuming a Friedmann universe which evolves with a power-law scale factor, a=tⁿ, we analyse the phase space of the system of equations that describes a time-varying fine structure 'constant', α, in the Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We have classified all the possible behaviours of α(t) in ever-expanding universes with different n and find new exact solutions for α(t). We find the attractors points in the phase space for all n. In general, α 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). α also tends rapidly to a constant when the expansion scale factor increases exponentially. A general set of conditions is established for α to become asymptotically constant at late times in an expanding universe.