Echoes of Stars in Tracking Rastall Gravity

We develop a model of a star with uniform density within the context of tracking Rastall gravity, where the density of the effective dark matter closely follows that of dark energy. To ascertain the stability of this star, we conduct an analysis of the axial gravitational perturbations. Our findings reveal that echo wavelets naturally emerge within the axial gravitational waves when dark matter substantially influences the star's construction. We argue that the fundamental cause of these echoes lies in the potential valley that resides near the star's surface. This phenomenon provides a promising avenue for detecting Rastall dark energy through the ringdown phase of gravitational waves emitted by binary stars that are enriched with dark matter.