Gauge-Invariant Perturbations of Varying-Alpha Cosmologies

Using a gauge-invariant formalism we derive and solve the perturbed cosmological equations for the BSBM theory of varying fine structure ‘constant’. We calculate the time evolution of inhomogeneous perturbations of the fine structure constant, $\frac{\delta \alpha }{\alpha }$ on small and large scales with respect to the Hubble radius. In a radiation-dominated universe small inhomogeneities in $\alpha$ will decrease on large scales but on scales smaller than the Hubble radius they will undergo stable oscillations. In a dust-dominated universe small inhomogeneous perturbations in $\alpha$ will become constant on large scales and on small scales they will increase as $t^{2\mathrm{/}3}$, and $\frac{\delta \alpha }{\alpha }$ will track $% \frac{\delta \rho {m}}{\rho{m}}$\mathrm{.}Iftheexpansionaccelerates,asinthecaseofa$\Lambda $orquintessence-dominatedphase,inhomogeneitiesin$\alpha $ will decrease on both large and small scales. The amplitude of perturbations in $\alpha$ will be much smaller than that of matter or radiation perturbations. We also present a numerical study of the non-linear evolution of spherical inhomogeneities in radiation and dust universes by means of a comparison between the evolution of flat and closed Friedmann models with time-varying $\alpha \mathrm{.}$Various limitations of these simple models are also discussed.