Difference between quark stars and neutron stars in universal relations and their effect on gravitational waves

TL;DR

This work investigates whether universal relations involving tidal responses in compact stars extend from neutron stars to quark stars and how dynamical tides influence gravitational waves during binary inspirals. By solving full GR perturbations for QSs and comparing to NS results, the authors show that while the $A=Q_fR/M$–$\Lambda$ relation is universal, the $B=\omega R$–$\Lambda$ relation differs by about 20% due to QS radii, leading to a modest dynamical-tide effect on the GW phase. Using a PN framework, they find the dynamical-tide-induced dephasing between QSs and NSs is small, with static tides dominating, and conclude that current detectors (aLIGO) cannot distinguish QSs from NSs via inspiral signals; next-generation detectors (ET/CE) offer only marginal prospects for certain low-mass systems. Overall, the work highlights that universal relations help constrain EOSs but distinguishing QSs from NSs requires precise modelling of dynamical tides and favorable detector sensitivities. The results show a clear path to leveraging $f$-mode dynamics, but also underline the limited practical detectability with present GW observatories.

Abstract

We calculate the $f$-mode frequency and tidal overlap of quark stars using the full general relativity method. We verify the universal relations obtained from conventional neutron stars in the case of quark stars and explore the cases with different values of parameters of the quark star equation of state. Since quark stars have significantly smaller radii compared to neutron stars in the low mass range, the relation between the tidal defomability and $f$-mode frequency times radius is different for neutron stars and quark stars. This difference has an impact on dynamical tide, which is the lowest-order effect we know of that can distinguish quark stars and neutron stars from the gravitational wave during the inspiral phase. We calculate the tidal dephasing caused by this effect in the post-Newtonian method and find that it can not be detected even by the next-generation gravitational wave detectors.

Figures (8)

• Figure 1: Metric perturbations (solid lines), fluid displacement functions (dotted lines), and metric functions of the static background (dashed lines) as functions of radius inside a 1.4 $M_{\odot}$ QS computed with a MIT bag model with $B = 47.202~\mathrm{MeV\,fm}^{-3}$. $H_0$, $H_1$ and $K$ are in units of $\varepsilon_s=152.26$ MeV fm$^{-3}$, $X$ is in units of $\varepsilon_s^2$, and $W$, $V$, $\nu$ and $\lambda$ are dimensionless. The central pressure is $p_c=85$ MeV fm$^{-3}$, the radius is $R=11.72$ km, and the corresponding $f$-mode frequency is $\omega=(9.835\times 10^3+6.386~i)$ Hz, and the imaginary component of the frequency gives a damping time of 0.16 s.
• Figure 2: Top panel: universal relation between the $f$-mode frequency times $M_{1.4}$ and the tidal defomability. The star-shaped points represent our data for QSs, some of them are modeled in the MIT bag model with the different values of the effective bag parameter $B$, which reflect the surface density of QSs, as indicated in the legend. For other QSs, the EoS in Ref.2021PhRvD.103f3018Z is used, all possible strongly interacting phases of strange quark matter and ud quark matter is considered in this model. The parameter $\bar{\lambda}$ characterizing the strength of the related strong interaction. In the legend, the values of this parameter are showed by the number after "L". The circular points represent the results of NSs. The blue solid line show the result in Ref. PhysRevD.104.123002. Bottom panel: relative deviation between our results and the relation discussed in Ref. PhysRevD.104.123002 as a function of the tidal defomability.
• Figure 3: Universal relation between $\mathcal{A} = Q_fR/M$ and the tidal defomability. The point shapes have the same meaning with Fig. \ref{['fig:ol']}. The blue solid line shows our fitting result which is showed in Eq. (\ref{['eq:al']}).
• Figure 4: Universal relation between $\mathcal{B}(\Lambda) = \omega R$ and the tidal defomability. The point shapes have same the meaning with \ref{['fig:ol']}. The blue solid line shows our fitting result of NSs which is shown in \ref{['eq:bln']} and the orange solid line shows the fitting result of QSs which is shown in \ref{['eq:blq']}.
• Figure 5: Tidal dephasing of NSs and QSs as a function of GW frequency. Solid lines denote QSs and dashed lines denote NSs. The data are taken from \ref{['table:data']}, where QSs and NSs of the same mass have approximately equal tidal defomabilities.