On Generalized Theories of Varying Fine Structure Constant

We work with a class of scalar extended theory of gravity that can drive the present cosmic acceleration as well as accommodate a mild cosmic variation of the fine structure constant $\alpha$. The motivation comes from a vintage theory developed by Bekenstein, Sandvik, Barrow and Magueijo. The $\alpha$ variation is introduced by a real scalar field interacting with charged matter. We execute a cosmological reconstruction based on a parametrization of the present matter density of the Universe. Observational consistency is ensured by comparing the theoretical estimates with JLA + OHD + BAO data sets, using a Markov chain Monte Carlo simulation. An analysis of molecular absorption lines from HIRES and UVES spectrographs is considered as a reference for the variation of $\alpha$ at different redshifts. Two examples are discussed. The first explores a field-dependent kinetic coupling of the scalar field interacting with charged matter. The second example is a generalized Brans-Dicke formalism where the varying $\alpha$ is fitted in as an effective matter field. This generates a simultaneous variation of the Newtonian constant $G$ and $\alpha$. The pattern of this variation can have a crucial role in cosmic expansion history.