In the present paper we study the oscillations of fast rotating neutron stars with realistic equations of state (EoS) within the Cowling approximation. We derive improved empirical relations for gravitational wave asteroseismology with f-modes, and for the first time we consider not only quadrupolar oscillations but also modes with higher spherical order (l ¼ jmj ¼ 3, 4). After performing a systematic comparison with polytropic EoS, we show that the empirical relations found in this case approximately also hold for realistic EoS. In addition, we show that these relations will not change significantly even if the Cowling approximation is dropped and the full general relativistic case is considered, although the normalization used here (frequencies and damping times in the nonrotating limit) could differ considerably. We also address the inverse problem; i.e., we investigate in detail what kind of observational data are required in order to determine characteristic neutron star parameters. It is shown that masses, radii and rotation rates can be estimated quite accurately using the derived asteroseismology relations. We also compute the instability window for certain models, i.e., the limiting curve in a T À plane where the secular Chandrasekhar-Friedman-Schutz instability overcomes dissipative effects, and we show that some of the modern realistic EoS will lead to a larger instability window compared to all of the polytropic ones presented so far in the literature. Additionally, we calculate the r-mode instability window and compare it with the f-mode case. The overall results for the instability window suggest that it is vital to take into account oscillations with l ¼ 3, 4 when considering gravitational wave asteroseismology using the f-mode in rapidly rotating neutron stars, as these modes can become unstable for a much larger range of parameters than pure quadrupolar oscillations.

We investigate damping and growth times of the quadrupolar f -mode for rapidly rotating stars and a variety of different polytropic equations of state in the Cowling approximation. This is the first study of the damping/growth time of this type of oscillations for fast rotating neutron stars in a relativistic treatment where the spacetime degrees of freedom of the perturbations are neglected. We use these frequencies and damping/growth times to create robust empirical formulae which can be used for gravitational wave asteroseismology. The estimation of the damping/growth time is based on the quadrupole formula and our results agree very well with Newtonian ones in the appropriate limit.

Nonaxisymmetric oscillations of rapidly rotating relativistic stars are studied using the Cowling approximation. The oscillation spectra have been estimated by Fourier transforming the evolution equations describing the perturbations. This is the first study of its kind and provides information on the effect of fast rotation on the oscillation spectra while it offers the possibility of studying the complete problem by including space-time perturbations. Our study includes both axisymmetric and nonaxisymmetric perturbations and provides limits for the onset of the secular bar mode rotational instability. We also present approximate formulas for the dependence of the oscillation spectrum from rotation. The results suggest that it is possible to extract the relativistic star's parameters from the observed gravitational wave spectrum

We study the dynamical evolution of the gravitational-wave driven instability of the f -mode in rapidly rotating relativistic stars. With an approach based on linear perturbation theory we describe the evolution of the mode amplitude and follow the trajectory of a newborn neutron star through its instability window. The influence on the f -mode instability of the magnetic field and the presence of an unstable r-mode is also considered. Two different configurations are studied in more detail; an N = 1 polytrope with a typical mass and radius and a more massive polytropic N = 0.62 model with gravitational mass M = 1.98M . We study several evolutions with different initial rotation rates and temperature and determine the gravitational waves radiated during the instability. In more massive models, an unstable f -mode with a saturation energy of about 10 −6 M c 2 may generate a gravitational-wave signal which can be detected by the Advanced LIGO/Virgo detector from the Virgo cluster. The magnetic field affects the evolution and then the detectability of the gravitational radiation when its strength is higher than 10 12 G, while the effects of an unstable rmode become dominant when this mode reaches the maximum saturation value allowed by non-linear mode couplings. However, the relative saturation amplitude of the f -and r-modes must be known more accurately in order to provide a definitive answer to this issue. From the thermal evolution we find also that the heat generated by shear viscosity during the saturation phase completely balances the neutrinos' cooling and prevents the star from entering the regime of mutual friction. The evolution time of the instability is therefore longer and the star loses significantly larger amounts of angular momentum via gravitational waves.