The $g$-mode frequencies in cold neutron stars and nuclear saturation parameters

Oscillation frequencies excited in neutron stars are crucial for extracting their interior properties. In addition to the fundamental and pressure modes, the gravity ($g$-) modes can be excited even in zero-temperature stellar models due to the composition gradient. In this study, we systematically study the $g$-mode frequencies, focusing on the nucleonic equation of state. Then, we can derive an empirical relation for the 1st $g$-mode frequencies as a function of the stellar compactness and the combination of the nuclear saturation parameters, $\eta_0 \equiv L/K_0$, where $K_0$ and $L$ denote the incompressibility of symmetric nuclear matter and the density dependence of the nuclear symmetry energy, respectively. If an observed 1st $g$-mode frequency significantly deviates from our empirical relation, it may indicate the emergence of additional degrees of freedom or new compositions inside the star.