Evolution of the f-mode instability in neutron stars and gravitational wave detectability

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 $M=1.98{M}_{\ensuremath{\bigodot}}$. 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 ${10}^{\ensuremath{-}6}{M}_{\ensuremath{\bigodot}}{c}^{2}$ 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 ${10}^{12}\text{ }\text{ }\mathrm{G}$, while the effects of an unstable $r$ mode become dominant when this mode reaches the maximum saturation value allowed by nonlinear 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.