Parametrizing superfluid dark matter with rational approximations

We investigate how a spatially modulated real scalar background [Formula: see text] can modify phonon propagation in the context of Superfluid dark matter (SFDM). Using a simple toy model with quartic condensate and coupling [Formula: see text], we derive the local equation of state and the effective sound velocity [Formula: see text]. For [Formula: see text], modulation tends to increase the effective mass of the condensate and make the medium less rigid, suppressing [Formula: see text] up to a “dust-like” regime, [Formula: see text]. We implement this modulation for the background scalar field by imposing rational profiles, through Padé radial profiles, and show the corresponding variation of [Formula: see text] for different [Formula: see text], discussing implications for the structure of SFDM cores and the possible formation of inhomogeneous regions of dark matter.