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, i.e. δ = 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 a la Bekenstein is assumed, trigger a spatial dependence of the fine-structure 'constant' which then varies as Δα ⋊ 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 renergy.