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 conformally flat three-metric. We compare the results of our simulations, in which the space-time dynamics is coupled to the evolution of the fluid, to previous results performed in the Cowling approximation in which the space-time dynamics was neglected. We show that the conformal flatness condition has only a small effect on the dynamics of pulsating relativistic stars and the obtained pulsation frequencies are very close to those expected in full general relativity. The pulsations are studied along various sequences of both uniformly and differentially rotating relativistic polytropes with index N = 1. For small pulsation amplitudes 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 shifts mode frequencies to smaller values, increasing the likelihood of detection by current gravitational wave interferometric detectors. We observe an extended avoided crossing between the l= 0 and l= 4 first overtones (previously known to exist from perturbative studies), which is important for correctly identifying mode frequencies in the case of detection. For uniformly rotating stars near the mass-shedding limit, we confirm the existence of the mass-shedding-induced damping of pulsations and argue that it is still relevant for secularly unstable modes, even though the effect is not as strong as was previously found in the Cowling approximation. We also investigate non-linear harmonics of the linear modes and note that rotation changes the pulsation frequencies in a way that would allow for various parametric instabilities between two or three modes to take place. Although this scenario has been explored before for slowly rotating collapse, it could become very interesting in the case of rapidly rotating collapse, where the quasi-radial mode could be in resonance with inertial modes. 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.