Reverse phase transitions in binary neutron-star systems with exotic-matter cores

Multimessenger observations of binary neutron-star mergers provide a unique opportunity to constrain the dense-matter equation of state. Although it is known from quantum chromodynamics that hadronic matter will undergo a phase transition to exotic forms of matter—e.g., quark matter—the onset density of such a phase transition cannot be computed from first principles. Hence, it remains an open question if such phase transitions occur inside isolated neutron stars or during binary neutron-star mergers, or if they appear at even higher densities that are not realized in the cosmos. In this article, we perform numerical relativity simulations of neutron-star mergers and investigate scenarios in which the onset density of such a phase transition is exceeded in at least one inspiraling binary component. Our simulations reveal that shortly before the merger, it is possible that such stars undergo a “reverse phase transition”—i.e., densities decrease and the quark core inside the star disappears, leaving a purely hadronic star at merger. After the merger, when densities increase once more, the phase transition occurs again and leads, in the cases considered in this work, to the rapid formation of a black hole. We compute the gravitational-wave signal and the mass ejection for our simulations of such scenarios and find clear signatures that are related to the postmerger phase transition—e.g., smaller ejecta masses due to the softening of the equation of state through the quark core formation. Unfortunately, we do not find measurable imprints of the reverse phase transition.

Figure 1

EOSs represented by the pressure vs baryon number density curves (upper panel) and mass-radius curves (bottom panel). The marked points in the top panel identify the central rest-mass density of the stars in the models of Table 1: PT1 (black circle), noPT1 (black diamond), PT2 star A (blue $\times$), PT2 star B (blue circle), noPT2 star A (blue up triangle), noPT2 star B (blue down triangle). Masses and radii for the corresponding stars in isolation are marked accordingly in the bottom panel.

Figure 2

Snapshots of the baryon number density on the equatorial plane (upper panels) and the respective baryon number density–pressure phase diagram (lower panels), together with the zero-temperature EOS (black line) and color-coded specific internal energy $\epsilon$, are depicted for the noPT1 setup employing the highest resolution. Also, we represent the hadronic matter density $n_b$ in the upper-left color bar and the density $n_{b,u}$ of matter identified as ejecta in the upper-right color bar.

Figure 3

Snapshots of the baryon number density in the equatorial plane (upper panels) and the respective baryon number density–pressure phase diagram (lower panels) for the PT1 case employing the highest resolution. We represent the hadronic matter density in the upper-left color bar, the quark matter density in the upper-middle color bar, and the ejecta density in the upper-right color bar. The lower color bar refers to the specific internal energy. Upper panels, first column: in this instance, the inner core contains quarks at ${\log }_{10}(n_b[{\mathrm{fm}}^{-3}])\sim -0.1$. Upper panels, second column: the stars are tidally deformed, decreasing the core densities. The inner cores no longer contain quarks, since they reach maximum densities of ${\log }_{10}(n_b[{\mathrm{fm}}^{-3}])\sim -0.3$, below the threshold value at which quarks are present. Upper panels, third column: moment of merger, where we remark on the complete absence of quarks in the coalescing material. Upper panels, fourth column: in less than 1 ms after the merger, a sizeable quark-matter core is formed, which then quickly collapses to a BH. Lower panels: The zero-temperature PT1 EOS is depicted (black line) along with the color-coded specific internal energies for the respective snapshots; from this plot it is clear that at late stages of the inspiral, the quarks in the cores of both stars vanish, marking “the reverse phase transition”.

Figure 4

Maximum baryon number density evolution of all simulations for the highest resolution R4. The red-filled region corresponds to the density interval spanning the phase transition for the PT1 EOS. The blue-filled region spans the range of densities of the phase transition for the PT2 EOS. Here we note that all runs exhibit almost no density oscillations before the merger. The thick lines representing PT1 (black) and PT2 (blue) stay above their respective onset densities during the inspiral, which indicates the presence of quarks. Around 1 ms before the merger, the PT1 and PT2 maximum densities drop noticeably to values below the onset of the phase transition, evidencing the reverse phase transition. Comparatively, noPT1 and noPT2 (dashed lines) experience a smaller decrease of maximum density. Finally, all setups show the characteristic increase of maximum density within 1 ms after the merger, signaling the prompt BH formation.

Figure 5

Top panel: ($2,-2$) mode of the highest-resolution (R4) waveform for the PT1 (blue) and noPT1 (orange) scenarios. Bottom panel: The black line shows the phase difference between the Richardson extrapolated waveforms for PT1 and noPT1. The red-shaded area shows the error on the GW phase $\varphi$ for noPT1 due to the finite-resolution effects computed with Eq. (a1), while the gray-shaded area shows the error coming from the ID resolution for PT1, Eq. (a2). Finally, the blue-shaded area shows the total error on PT1 computed following Eq. (a3).

Figure 6

Convergence of the (2,2) mode for all the setups. We show the GW phase differences between consecutive resolutions and the rescaled phase differences assuming second-order convergence for the noPT cases and first-order convergence for the PT cases. The black vertical line marks the merger time.

Figure 7

Top panel: (2,2) mode of the highest-resolution waveform for PT2 (blue) and noPT2 (orange) scenarios. Bottom panel: The black line shows the GW phase difference between the Richardson extrapolated waveforms for PT2 and noPT2. Shaded areas show the error on the GW $\varphi$ for each system computed with Eq. (a1). In contrast to Fig. 5, we do not incorporate an additional phase difference based on simulations with different ID resolutions.

Figure 8

Snapshots of the baryon number density on the equatorial plane (upper panels) and the respective baryon number density-pressure phase diagram (lower panels), together with the zero-temperature EOS (black line) and color-coded specific internal energy, are depicted for the noPT2 setup employing the highest resolution. Hadronic matter is represented by the upper-left color bar, and matter is identified as ejecta by the upper-right color bar.

Figure 9

Snapshots of the baryon number density on the equatorial plane (upper panels) and the respective baryon number density–pressure phase diagram (lower panels) for the PT2 case are depicted employing the highest resolution. Similarly to Fig. 3, we represent hadronic matter in the upper-left color bar, the quark matter using the upper-middle colorbar, and the ejecta using the color bar on the upper right. The lower color bar refers to the specific internal energy. Upper panels, first column: in this instance, we notice that the inner core of one of the stars contains quarks at ${\log }_{10}(n_b[{\mathrm{fm}}^{-3}])\sim -0.1$. Upper panels, second column: the densest portions of the stars are found to be tidally deformed. The inner core containing quarks is smaller and less dense, with ${\log }_{10}(n_b[{\mathrm{fm}}^{-3}])<-0.1$. Upper panels, third column: the moment of merger, where we note the complete absence of quarks in the coalescing material. Upper panels, fourth column: in less than 1 ms after the merger, a sizeable quark-matter core is formed, which then quickly collapses to a BH. Lower panels: The zero-temperature PT2 EOS is depicted (black line) along with the color-coded specific internal energies for the respective snapshots; similarly to the PT1 case, we see the decrease of the quark’s content within the star towards the merger and its subsequent increase rapidly after the merger.