Effects of inner crusts on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>g</mml:mi></mml:math>-mode oscillations in neutron stars

In this work, we investigate the influence of neutron stars' crusts on the nonradial $g$-mode oscillations and examine their correlations with nuclear matter properties fixed by adopting ten different relativistic density functionals. At subsaturation densities, neutron star matter takes nonuniform structures and form the crusts. We find that the Brunt-V"ais"al"a (BV) frequency increases significantly at densities slightly above the neutron drip density (i.e., neutron stars' inner crusts), which leads to crust $g$-mode oscillations with their frequencies insensitive to the adopted density functional. At larger densities, BV frequency increases as well due to the core-crust transitions and emergence of muons, which lead to core $g$-mode oscillations. It is found that the obtained core $g$-mode frequencies generally increase with the slope of nuclear symmetry energy $L$, which eventually intersect with that of the crust $g$-modes adopting large enough $L$. This leads to the avoid-crossing phenomenon for the global $g$-modes that encompass contributions from both the crust and core. For neutron stars with fixed masses, the frequency of the global ${g}_{1}$-mode generally increases linearly with $L$, where the Pearson correlation coefficient ranges from 0.7 to 0.9. The corresponding linear functions are then fixed by fitting to the ${g}_{1}$-mode frequencies predicted by the ten relativistic density functionals, which enables the possible measurements of $L$ based on gravitational wave observations. In our future study, the effects of the discontinuities in density or shear modulus should be considered, while the temperature, rotation, magnetic field, and superfluid neutron gas in neutron stars could also play important roles.