Impact of inhomogeneities on slowly rolling Quintessence: implications for the local variations of the fine-structure constant

Abstract

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.