  Searching for variations of nature's fundamental constants is a crucial step
in our quest to go beyond our current standard model of fundamental physics. If
they exist, such variations will be very likely driven by the existence of a
new fundamental field. The Bekenstein model and its extensions introduce such a
scalar field in a purely phenomenological way, inducing a variation of the
fine-structure constant on cosmological scales. This theoretical framework is
as simple and general as possible while still preserving all the symmetries of
standard quantum electrodynamics. When allowing for couplings to the other
sectors of the Universe, such as baryons, dark matter, and the cosmological
constant, the Bekenstein model is expected to reproduce the low energy limits
of several grand unification, quantum gravity, and higher dimensional theories.
In this work, we constrain different versions of the Bekenstein model by
confronting the full cosmological evolution of the field with an extensive set
of astrophysical, cosmological, and local measurements. We show that couplings
of the order of parts per million (ppm) are excluded for all the cases
considered, imposing strong restrictions on theoretical frameworks aiming to
deal with variations of the fine-structure constant.
