Dynamical Tides in Neutron Stars with First-Order Phase Transitions: The Role of the Discontinuity Mode

During the late stages of a binary neutron star inspiral, dynamical tides induced in each star by its companion become significant and should be included in complete gravitational-wave (GW) modeling. We investigate the coupling between the tidal field and quasinormal modes in hybrid stars and show that the discontinuity mode (g mode) intrinsically associated with first-order phase transitions and buoyancy contributes non-negligibly compared with the fundamental f mode. We find that the g-mode overlap integral can reach up to ∼10% of the f-mode value for hybrid star masses in the range 1.4-2.0M_{⊙}, with the largest values generally associated with larger density jumps. This leads to a GW phase shift due to the g mode of Δϕ_{g}≲0.1-1  rad (i.e., up to ∼5%-10% of Δϕ_{f}), with the largest shifts occurring for masses near the phase transition. At higher masses, the shifts remain smaller and nearly constant, with Δϕ_{g}≲0.1  rad (roughly ∼1% of Δϕ_{f}). These GW shifts may be relevant even at the design sensitivity of current second-generation GW detectors in the most optimistic cases. Moreover, if a g mode is present and lies near the f-mode frequency, neglecting it in the GW modeling can lead to systematic biases in neutron star parameter estimation, resulting in radius errors of up to 1%-2%. These results show the importance of dynamical tides to probe neutron stars' equation of state, and to test the existence of dense-matter phase transitions.