  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.
