Non-linear axisymmetric pulsations of rotating relativistic stars in the conformal flatness approximation

We study non-linear axisymmetric pulsations of rotating relativistic stars using a general relativistic hydrodynamics code under the assumption of a conformal flatness. We compare our results to previous simulations where the spacetime dynamics was neglected. The pulsations are studied along various sequences of both uniformly and differentially rotating relativistic polytropes with index N = 1. We identify several modes, including the lowest-order l = 0, 2, and 4 axisymmetric modes, as well as several axisymmetric inertial modes. Differential rotation significantly lowers mode frequencies, increasing prospects for detection by current gravitational wave interferometers. We observe an extended avoided crossing between the l = 0 and l = 4 first overtones, which is important for correctly identifying mode frequencies in case of detection. For uniformly rotating stars near the mass-shedding limit, we confirm the existence of the mass-shedding-induced damping of pulsations, though the effect is not as strong as in the Cowling approximation. We also investigate non-linear harmonics of the linear modes and notice that rotation changes the pulsation frequencies in a way that would allow for various parametric instabilities between two or three modes to take place. We assess the detectability of each obtained mode by current gravitational wave detectors and outline how the empirical relations that have been constructed for gravitational wave asteroseismology could be extended to include the effects of rotation.