Universal Relations for Neutron Star f-mode and g-mode Oscillations*

Among the various asteroseismology oscillation modes, f- and g- modes are most likely to be ultimately observed in BNS mergers, due to the relatively large coupling and shared frequencies with tidal excitations. However, the f-mode correlates strongly with static NS properties (known as I-love-Q-f relations) even for hybrid NS and QS. Nevertheless, observations of the g-mode could give us information not only about the composition but also a potential phase transition. The discontinuity g-mode frequency depends strongly on the transition density and the magnitude of the density discontinuity. Due to causality and maximum mass constraints, the discontinuity g-mode frequency has an upper bound of about 1.5 kHz. However, if the squared sound speed ratio in the inner core is restricted to 1/3, the discontinuity g-mode can only reach about 0.8 kHz, which is significantly lower than the f-mode frequency, 1.3-2.8 kHz. Also, the g-mode gravitational wave damping time is usually extremely long >10^4 s (10^2 s) for inner core with the squared sound speed ratio less than 1/3 (1), which is large compared with the f-mode damping time, 0.1-1 s. The compositional g-mode of realistic hadonic NS depends mostly on the lepton fraction at the center of the NS, with an upper bound of about 0.7 kHz.