Avoiding unrealistic priors: the case of dark energy constraints from
  the time variation of the fine-structure constant
  We critically assess recent claims suggesting that upper limits on the time
variation of the fine-structure constant tightly constrain the coupling of a
dark energy scalar field to the electromagnetic sector, and, indirectly, the
violation of the weak equivalence principle. We show that such constraints
depend crucially on the assumed priors, even if the dark energy was described
by a dynamical scalar field with a constant equation of state parameter $w$
linearly coupled to the electromagnetic sector through a dimensionless coupling
$\zeta$. We find that, although local atomic clock tests, as well as other
terrestrial, astrophysical and cosmological data, put stringent bounds on
$|\zeta| {\sqrt {|w+1|}}$, the time variation of the fine-structure constant
cannot be used to set or to improve upper limits on $|\zeta|$ or $|w+1|$
without specifying priors, consistent but not favoured by current data, which
strongly disfavour low values of $|w+1|$ or $|\zeta|$, respectively. We briefly
discuss how this might change with a new generation of high-resolution
ultra-stable spectrographs, such as ESPRESSO and ELT-HIRES, in combination with
forthcoming missions to map the geometry of the Universe, such as Euclid, or to
test the equivalence principle, such as MICROSCOPE or STEP.
