Probing hybrid stars with gravitational waves via interfacial modes

Shu Yan Lau and Kent Yagi

Phys. Rev. D 103, 063015 – Published 12 March 2021

Abstract

One of the uncertainties in nuclear physics is whether a phase transition between hadronic matter to quark matter exists in supranuclear equations of state. Such a feature can be probed via gravitational-wave signals from binary neutron star inspirals that contain information of the induced tides. The dynamical part of the tides is caused by the resonance of pulsation modes of stars, which causes a shift in the gravitational-wave phase. In this paper, we investigate the dynamical tides of the interfacial mode ( $i$ -mode) of spherical degree $l = 2$ , a nonradial mode caused by an interface associated with a quark-hadron phase transition inside a hybrid star. In particular, we focus on hybrid stars with a crystalline quark matter core and a fluid hadronic envelope. We employ a hybrid method which consists of a general relativistic calculation of the stellar structure, together with Newtonian formulations of tidal couplings and mode excitations. We find that the resonant frequency of such $i$ -modes typically ranges from 300 Hz to 1500 Hz, and the frequency increases as the shear modulus of the quark core increases. We next estimate the detectability of such a mode with existing and future gravitational-wave events from the inspiral waveform with a Fisher analysis. We find that GW170817 and GW190425 have the potential to detect the $i$ -mode if the quark-hadron phase transition occurs at a sufficiently low pressure and the shear modulus of the quark matter phase is large enough. We also find that the third-generation gravitational-wave detectors can further probe the $i$ -mode with intermediate transition pressure. Finally, we check our hybrid method against a fully-Newtonian analysis and find that the two results can be off by a factor of a few. Thus, the results presented here should be valid as an order-of-magnitude estimate and provide a new, interesting direction for probing the existence of quark core inside a neutron star using the $i$ -mode. A full general relativistic formalism, however, needs to be pursued for further analysis.

Received 24 December 2020
Accepted 19 February 2021

DOI:https://doi.org/10.1103/PhysRevD.103.063015

Physics Subject Headings (PhySH)

Gravitational waves Nuclear astrophysics QCD phase transitions

Gravitation, Cosmology & AstrophysicsNuclear Physics

Authors & Affiliations

Shu Yan Lau ^* and Kent Yagi ^†

Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

^*[email protected]
^†[email protected]

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Vol. 103, Iss. 6 — 15 March 2021

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Images

Figure 1
$M - R$ relations of the HS models (dashed lines) and hadronic matter models (solid lines) constructed with intermediate (left) and stiff (right) HEOSs.
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Figure 2
(Left) The weight-averaged $i$ -mode frequency of (1.4, 1.4) $M_{⊙}$ HS binary models against $Δ$ . A low $P_{t}$ EOS (MS1-QM; in black squares) and an intermediate $P_{t}$ EOS (Heb3-QM-3; in orange dots) are chosen to construct the models. (Right) Similar to the left panel but for the total overall phase shift. The phase shift of Heb3-QM-3 near $Δ = 10 MeV$ exceeds over 10 due to the avoided crossing between the $i$ -mode and the $f$ -mode (not shown in this figure). Near this region, the phase shift of the two modes comes close to each other and the resonant frequencies repel to avoid a degeneracy.
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Figure 3
The avoided crossing of the $i$ -mode and $f$ -mode as $Δ$ changes from 7 to 11 MeV of a $1.4 M_{⊙}$ HS model with the EOS Heb3-QM-3. The left panel shows the repulsion of the frequencies of the higher frequency mode and the lower frequency mode. The right panel shows the exchange in $| δ ϕ |$ between the two modes.
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Figure 4
(Left) The magnitude of the $i$ -mode’s total overall phase shift $| δ \bar{ϕ} |$ and the corresponding weight-averaged resonant frequency $\bar{f}$ for each HS EOS from Table 1, together with the detectability threshold with aLIGO (green solid) and CE (red dashed). If a point is above these curves, such an effect is detectable with the corresponding detector. Here we have assumed an equal-mass HS system with an individual mass of $1.4 M_{⊙}$ . We consider intermediate $P_{t}$ models (Heb3-QM-3, Heb3-QM-2, DD2-QM in blue) and low $P_{t}$ models (MS1-QM, Heb3-QM-1, NL3-QM, TM1-QM in black). The $i$ -mode becomes undetectable if the frequency $\bar{f}$ is higher than the inspiral cutoff frequency (shaded region) that we choose to be at ISCO ( $f_{ISCO}$ ). (Right) Similar to Fig. 4 but for individual masses of $1.8 M_{⊙}$ . We also present the high $P_{t}$ models (MPa1-QM, Heb2-QM, $DDH δ$ -QM) in magenta symbols. The SNR for the $1.4 M_{⊙}$ system is $20$ for aLIGO and 620 for CE, and that of the $1.8 M_{⊙}$ system is $24$ for aLIGO and 760 for CE respectively.
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Figure 5
Left: similar to Fig. 4 but for parameters consistent with GW170817. The detection threshold curve is computed with the noise curve of aLIGO O2 run. We present the intermediate $P_{t}$ models (Heb3-QM-3 in blue) and low $P_{t}$ models (MS1-QM, Heb3-QM-1 in black). Right: similar to the left panel but for parameters consistent with GW190425. The detection threshold curve is computed with the noise curve of the aLIGO O3 run. We present the high $P_{t}$ models (MPa1-QM in magenta), the intermediate $P_{t}$ models (Heb3-QM-3 in blue) and low $P_{t}$ models (MS1-QM, Heb3-QM-1 in black).
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Physical Review D

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