Local and Global Variations of The Fine Structure Constant

Using the BSBM varying-alpha theory, with dark matter dominated by magnetic energy, and the spherical collapse model for cosmological structure formation, we have studied the effects of the dark-energy equation of state and the coupling of alpha to the matter fields on the space and time evolution of alpha. We have compared its evolution inside virialised overdensities with that in the cosmological background, using the standard ($% \Lambda =0$) $CDM$ model of structure formation and the dark-energy modification, $wCDM$. We find that, independently of the model of structure formation one considers, there is always a difference between the value of alpha in an overdensity and in the background. In a $SCDM$ model, this difference is the same, independent of the virialisation redshift of the overdense region. In the case of a $wCDM$ model, especially at low redshifts, the difference depends on the time when virialisation occurs and the equation of state of the dark energy. At high redshifts, when the $wCDM$ model becomes asymptotically equivalent to the $SCDM$ one, the difference is constant. At low redshifts, when dark energy starts to dominate the cosmological expansion, the difference between alpha in a cluster and in the background grows. The inclusion of the effects of inhomogeneity leads naturally to no observable local time variations of alpha on Earth and in our Galaxy even though time variations can be significant on quasar scales. The inclusion of the effects of inhomogeneous cosmological evolution are necessary if terrestrial and solar-system bounds on the time variation of the fine structure 'constant' are to be correctly compared with extragalactic data.