Local and global variations of the fine-structure constant

Using the Bekenstein-Sandvik-Barrow-Magueijo (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 to the matter fields on the space and time evolution of. We have compared its evolution inside virialized overdensities with that in the cosmological background, using the standard (Lambda = 0) cold dark matter (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 in an overdensity and in the background. In a standard CDM (SCDM) model, this difference is the same, independent of the virialization redshift of the overdense region. In the case of a dark-energy CDM (wCDM) model, especially at low redshifts, the difference depends on the time when virialization 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 in a cluster and in the background grows. The inclusion of the effects of inhomogeneity leads naturally to no observable local time variations of 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.