Impact of inhomogeneities on slowly rolling Quintessence: implications
  for the local variations of the fine-structure constant
  We study how the evolution of a Dark Energy Quintessence fluid is modified by
the presence of a matter inhomogeneity. To do so, we study linear perturbations
of a flat FLRW background containing dust and a slowly rolling scalar field.
Under the assumptions of spherical symmetry and a static density contrast,
\textit{i.e.} $\dot{\delta}=0$, we obtain simple analytical solutions for
perturbations in the matter and dark energy-dominated epochs. As a consequence,
we show that perturbations of the scalar field, if a coupling \textit{\`a la}
Bekenstein is assumed, trigger a spatial dependence of the fine-structure
"constant" which then varies as $\Delta \alpha \propto 1/r$. We finally
highlight that such variations can be constrained with spectroscopic
observations of stars from within our galaxy, therefore offering a new probe of
the nature of Dark Energy.
