  We use Fisher Matrix analysis techniques to forecast the cosmological impact
of astrophysical tests of the stability of the fine-structure constant to be
carried out by the forthcoming ESPRESSO spectrograph at the VLT (due for
commissioning in late 2017), as well by the planned high-resolution
spectrograph (currently in Phase A) for the European Extremely Large Telescope.
Assuming a fiducial model without $\alpha$ variations, we show that ESPRESSO
can improve current bounds on the E\"{o}tv\"{o}s parameter---which quantifies
Weak Equivalence Principle violations---by up to two orders of magnitude,
leading to stronger bounds than those expected from the ongoing tests with the
MICROSCOPE satellite, while constraints from the E-ELT should be competitive
with those of the proposed STEP satellite. Should an $\alpha$ variation be
detected, these measurements will further constrain cosmological parameters,
being particularly sensitive to the dynamics of dark energy.
