Time variation of the fine structure constant in the early universe and the Bekenstein model

We calculate bounds on the variation of the fine structure constant at the time of primordial nucleosynthesis and at the time of neutral hydrogen formation. We use these bounds and other bounds from the late universe to test Bekenstein model. We modify the Kawano code, CAMB and CosmoMC in order to include the possible variation of the fine structure constant. We use observational primordial abundances of $\De$, $\He$ and $\Li$, recent data from the Cosmic Microwave Background and the 2dFGRS power spectrum, to obtain bounds on the variation of $\alpha$. We calculate a piecewise solution to the scalar field equation of Bekenstein model in two different regimes; i) matter and radiation, ii) matter and cosmological constant. We match both solutions with appropriate boundary conditions. We perform a statistical analysis using the bounds obtained from the early universe and other bounds from the late universe to constrain the free parameters of the model. Results are consistent with no variation of $\alpha$ in the early universe. Limits on $\alpha$ are inconsistent with the scale length of the theory $l$ being larger than Planck scale. In order to fit all observational and experimental data, the assumption $l>L_p$ implied in Bekenstein's model has to be relaxed.