Dynamical tides in neutron stars with first-order phase transitions: the role of the discontinuity mode

TL;DR

This work analyzes dynamical tides in neutron stars with a first-order phase transition, focusing on the discontinuity $g$-mode that arises at the hadron–quark interface. Using full general-relativistic nonradial perturbation theory for $l=2$ and a family of hybrid EOSs with tunable density jumps $n_q/n_h$, the authors compute $f$- and $g$-mode overlaps and the resulting GW phase shifts, showing that the $g$-mode can contribute up to $\sim$10% of the $f$-mode effect and produce phase shifts $\Delta\phi_g$ up to $\sim$1 rad near the transition mass. The largest effects occur near the phase-transition mass and for strong transitions, while heavier stars exhibit smaller, nearly constant shifts; neglecting the $g$-mode can bias neutron-star radius inferences by about $1$–$2\%$. These findings imply that dynamical tides, including the $g$-mode, are important for constraining the EOS and testing dense-matter phase transitions with current and next-generation GW and EM observations.

Abstract

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 quasi-normal 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 $\sim 10\%$ of the $f$-mode value for hybrid star masses in the range 1.4-2.0$M_{\odot}$, with the largest values generally associated with larger density jumps. This leads to a GW phase shift due to the $g$-mode of $Δφ_g \lesssim 0.1$-$1$ rad (i.e., up to $\sim5\%-10\%$ 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 $Δφ_g \lesssim 0.1$ rad (roughly $\sim 1\%$ 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.

Figures (20)

• Figure 1: Mass-radius relations for hybrid stars for various EOSs characterized by $n_q/n_h$ values ranging from 1.1 to 1.9, spanning scenarios from weak to strong phase transitions. To facilitate identification of the properties of the stellar models analyzed, dashed lines have been plotted at $1.4\,M_{\odot}$, $1.8\,M_{\odot}$, and $2.0\,M_{\odot}$.
• Figure 2: Overlap integrals for the $f$- and $g$-modes for $1.4\,M_{\odot}$, $1.8\,M_{\odot}$ and $2.0\,M_{\odot}$ hybrid stars. As expected, $Q_f$ is dominant (due to the mode's influence on the entire star and its lack of nodes), but $Q_g$ is only approximately a factor of 10 smaller. The variations for $1.4\,M_{\odot}$ case are a consequence of the small quark phase and how it nonlinearly influences the eigenfrequencies, even allowing them to change sign in the hadronic phase (see the Supplemental Material).
• Figure 3: GW phase shift for the $f$- and $g$-modes for a stars with $1.4\,M_{\odot}$, $1.8\,M_{\odot}$ and $2.0\,M_{\odot}$ and different number baryon density jumps. We set the NS companion mass to $1.4\,M_{\odot}$. We adopted a denser sampling in $n_q/n_h-1$ (step $0.025$) for the $1.4M_\odot$ sequence over the range $n_q/n_h - 1 \leq 0.4$ because this mass lies closest to the phase-transition threshold in our models.
• Figure 4: Damping times for the $f$- and $g$-modes of hybrid stars with masses of $1.4\,M_{\odot}$, $1.8\,M_{\odot}$, and $2.0\,M_{\odot}$. The damping times of $g$-modes display nonlinear behavior with $n_q/n_h$, owing to their dependence on buoyancy. These damping times span roughly seven orders of magnitude and can reach values as large as $\sim (10–100)s$ for hybrid stars with significant density jumps.
• Figure 5: Frequencies for the $f$-mode and $g$-mode for stars with $1.4\,M_{\odot}$, $1.8\,M_{\odot}$ and $2.0\,M_{\odot}$ and different number baryon density jumps.