Fundamental Physics with ESPRESSO, Constraining a simple parametrisation
  for varying $\alpha$
  The spectrograph ESPRESSO obtained updated limits on the variation of the
fine-structure constant, $\alpha$, through measurements along the line of sight
of a bright quasar with unprecedented precision and accuracy. These impose new
constraints on cosmological models with a varying $\alpha$. We assume such a
model where the electromagnetic sector is coupled to a scalar field dark energy
responsible for the current acceleration of the Universe. We parametrise the
variation of $\alpha$ with two extra parameters, one defining the cosmological
evolution of the quintessence component and the other fixing the coupling with
the electromagnetic field. The objective of this work is to constrain these
parameters with both astrophysical and local probes. We also make a comparative
analysis on how each data probe may constrain our parametrisation. We perform a
Bayesian analysis by comparing the predictions of the model with observations.
The astrophysical datasets are composed of quasars spectra measurements,
including the latest ESPRESSO data point, as well as Planck observations of the
cosmic microwave background. We combine them with local results from atomic
clocks and the MICROSCOPE experiment. The constraints placed on the
quintessence parameter are consistent with a null variation of the field, i.e.
compatible with a $\Lambda$CDM cosmology. The constraints on the coupling to
the electromagnetic sector are dominated by the E\"otv\"os parameter local
bound. The ESPRESSO data point is extremely important to study the cosmological
evolution of $\alpha$ as it probes an interval of redshift not accessible to
other observations. However, for this particular model, current available data
favours a null variation of $\alpha$ resulting mostly from the strong
MICROSCOPE limits.
