Parametrizing superfluid dark matter with rational approximations

We investigate how a spatially modulated real scalar background $\phi(\vec{x})$ can modify phonon propagation in the context of Superfluid dark matter (SFDM). Using a simple toy model with quartic condensate and coupling $-g\phi^2|\Psi|^2$, we derive the local equation of state and the effective sound velocity $c_s(\vec{x})$. For $g>0$, modulation tends to increase the effective mass of the condensate and make the medium less rigid, suppressing $c_s^2\propto m_{\Psi,\mathrm{eff}}^{-4}$ up to a ``dust-like'' regime, $c_s^2\to 0$. We implement this modulation for the background scalar field by imposing rational profiles, through Pad\'e radial profiles, and show the corresponding variation of $c_s^2(r)$ for different $g$, discussing implications for the structure of SFDM cores and the possible formation of inhomogeneous regions of dark matter.