Variation of the fine structure constant in QSO spectra from coherent dark matter oscillations

We consider the problem of the evolution of the fine structure coefficient alpha under the assumption that the scalar field coupling to the Maxwell term satisfies the condition mt>>1 for coherent dark matter oscillations. In this case we find that the coupling scale f in the leading order coupling -(phi/4f)F^2 affects the cosmological evolution of alpha according to ln(alpha/alpha_0) xi(m_{Pl}/f)ln(tanh(t/2 tau)/tanh(t_0/2 tau)). A fit to the QSO observations by Murphy et al. yields f/xi= 2.12^{+0.58}_{-0.37} 10^5m_{Pl}. Here m_{Pl} is the reduced Planck mass, and xi^2=rho_phi/rho_m parametrizes the contribution of phi to the matter density in the universe.