Redshift Dependence of CMB Temperature in BSBM $\alpha$-Varying Theories

We have investigated some cosmological consequences of variation of the fine structure constant, $\alpha$, in the context of the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) model. In this context, such a variation is attributed to electric charge by letting the latter take on the value of a real scalar field, $\phi$, which varies generally in space and time. We consider the dynamics of $\phi$ in the case that it is allowed to have an exponential potential. We have found solutions for evolutions of $\phi$ and $\alpha$ in the radiation and the matter dominated eras. By employing observational bounds on the temporal variation of $\alpha$, we constrain the model parameters. We then investigate the impact of $\alpha$-variation on redshift dependence of CMB temperature. It is shown that there is a deviation from the standard evolution of the temperature which contains two different contributions, one is important at high redshift eras and the other is mostly effective at low redshift eras.