Frozen and $\beta$-equilibrated $f$ and $p$ modes of cold neutron stars: nuclear metamodel predictions

When the chemical re-equilibration timescale is sufficiently long, the normal and quasi-normal mode frequencies of neutron stars should be calculated in the idealized limit that the internal composition of each fluid element is fixed over the oscillation period. However, many studies rely on a barotropic equation of state, which implicitly overlooks potential out-of-$\beta$-equilibrium effects. To investigate possible biases arising from this assumption, we calculate the non-radial fundamental ($f$) and first pressure ($p_1$) modes for a wide range of neutron star structures, each governed by different nucleonic equations of state. This ensemble is generated using the metamodel technique, a phenomenological framework that incorporates constraints from experimental nuclear physics and chiral effective field theory. The metamodel also provides the internal composition of $\beta$-equilibrated $npe\mu$ matter, allowing us to calculate oscillation modes beyond those supported by a purely barotropic fluid. Thus, we systematically assess the impact of assuming a barotropic equation of state across various equations of state and provide a distribution of expected $f$ and $p_1$ mode frequencies that may be detectable by next-generation gravitational wave interferometers.