Asteroseismology using quadrupolar f -modes revisited: Breaking of universal relationships in the slow hadron-quark conversion scenario

In this work, we consider polar perturbations and we calculate the frequency and damping time of the quadrupolar fundamental f -mode of compact objects, constructed using a wide range of model-independent hybrid equations of state that include quark matter. We give special attention to the impact of the hadron-quark conversion speed that, in the slow case, gives rise to a branch of slow stable hybrid stars. Moreover, we study the validity of universal relationships proposed in the literature and find out that none of them remains valid when slow stable hybrid stars are taken into account. This fact could constrain the applicability of asteroseismology methods with fundamental modes designed to estimate the properties of pulsating compact objects. We hope that this result could be tested with the start up of the third-generation gravitational wave observatories, which might shed some light on the f -mode emission from compact objects.