Universal relations for neutron star f -mode and g -mode oscillations

Abstract

Among the various oscillation modes of neutron stars, $f$- and $g$- modes are the most likely to be ultimately observed in binary neutron star mergers due to their relatively large coupling and shared frequencies with tidal excitations. The $f$-mode frequency and damping time are known to correlate in normal neutron stars with their compactness, and previous fits to hadronic stars are extended and shown to be valid for an extremely broad sampling of equations of state using a piecewise polytropic parametrization scheme for hadrons and a constant sound-speed parametrization for quark matter. Separate fits applicable to quark (self-bound) stars are improved. Much more significant correlations exist with tidal deformability, and therefore with moment of inertia and quadrupole moment. It is conclusively demonstrated that these correlations are the same for all types of stars, whether hadronic, hybrid, or pure quark, and its accuracy is quantified. A novel 1-node branch of the $f$-mode that occurs in low-mass hybrid stars in a narrow mass range just beyond the critical mass necessary for a phase transition to appear is identified. This 1-node branch shows the largest, but still small, deviations from the universal correlation for any configuration. It is characterized by a nonmonotonic relation between neutron star mass and $f$-mode frequency, in contrast to the behavior otherwise observed in normal, quark and hybrid stars. The $g$-mode only exists in matter with a nonbarotropic equation of state involving temperature, chemical potential or composition (such as being out of beta equilibrium), or a phase transition in barotropic matter. The $g$-mode therefore could serve as a probe for studying phase transitions in hybrid stars. In contrast with the $f$-mode, $g$-mode frequencies do not correlate well with tidal deformability, but depend strongly on properties of the transition (the density and the magnitude of the discontinuity) at the transition. Imposing causality and maximum mass constraints, a fit involving neutron star and phase transition properties is found and the $g$-mode frequency is determined to have an upper bound of about 1.25 kHz. However, if the sound speed ${c}_{s}$ in the inner core at densities above the phase transition density is restricted to ${c}_{s}^{2}\ensuremath{\le}1/3$, $g$-mode frequencies can only reach about 0.8 kHz, which are significantly lower than $f$-mode frequencies (1.3--2.8 kHz). $g$-mode gravitational wave damping times are found to be extremely long, $>{10}^{4}\text{ }\text{ }\mathrm{s}$ (${10}^{2}\text{ }\text{ }\mathrm{s}$) in the inner core with ${c}_{s}^{2}\ensuremath{\le}1/3$ (1), in comparison with $f$-mode damping times (0.1--1 s).