$g$-modes of neutron stars with hadron-to-quark crossover transitions

We perform the first study of the principal core $g$-mode oscillation in hybrid stars containing quark matter, utilizing a crossover model for the hadron-to-quark transition inspired by lattice QCD. The ensuing results are compared with our recent findings of $g$-mode frequencies in hybrid stars with a first-order phase transition using Gibbs constructions. We find that models using Gibbs construction yield $g$-mode amplitudes and the associated gravitational energy radiated that dominate over those of the chosen crossover model owing to the distinct behaviors of the equilibrium and adiabatic sound speeds in the various models. Based on our results, we conclude that were $g$-modes to be detected in upgraded LIGO and Virgo detectors it would indicate a first-order phase transition akin to a Gibbs construction.