Qualitative analysis of universes with varying alpha

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-BarrowMagueijo generalization 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 attractor 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). a also tends rapidly to a constant when the expansion scale factor increases exponentially. A general set of conditions is established for a to become asymptotically constant at late times in an expanding universe.