Constraining the onset density of the hadron-quark phase transition with gravitational-wave observations

We study the possible occurrence of the hadron-quark phase transition (PT) during the merging of neutron star binaries by hydrodynamical simulations employing a set of temperature-dependent hybrid equations of state (EOSs). Following previous work, we describe an unambiguous and measurable signature of deconfined quark matter in the gravitational-wave (GW) signal of neutron star binary mergers including equal-mass and unequal-mass systems of different total binary mass. The softening of the EOS by the PT at higher densities, i.e., after merging, leads to a characteristic increase of the dominant postmerger GW frequency $f_{\mathrm{peak}}$ relative to the tidal deformability $\mathrm{\Lambda }$ inferred during the premerger inspiral phase. Hence, measuring such an increase of the postmerger frequency provides evidence for the presence of a strong PT. If the postmerger frequency and the tidal deformability are compatible with results from purely baryonic EOS models yielding very tight relations between $f_{\mathrm{peak}}$ and $\mathrm{\Lambda }$, a strong PT can be excluded up to a certain density. We find tight correlations of $f_{\mathrm{peak}}$ and $\mathrm{\Lambda }$ with the maximum density during the early postmerger remnant evolution. These GW observables thus inform about the density regime which is probed by the remnant and its GW emission. Exploiting such relations, we devise a directly applicable, concrete procedure to constrain the onset density of the QCD PT from future GW measurements. We point out two interesting scenarios: if no indications for a PT are inferred from a GW detection, our procedure yields a lower limit on the onset density of the hadron-quark PT. On the contrary, if a merger event reveals evidence for the occurrence of deconfined quark matter, the inferred GW parameters set an upper limit on the PT onset density. Both scenarios would thus have strong implications for high-density matter physics, e.g., determining the range of validity of nuclear physics and constraining the properties for quark deconfinement. These prospects demonstrate the importance of simultaneously measuring pre- and postmerger GW signals to exploit the complementarity of the information encoded in both phases. Hence, our work stresses the value added by dedicated high-frequency GW instruments.

Figure 1

Rest-mass density (color-coded) in the equatorial plane of a merger simulation of two $1.35{\mathrm{M}}_{\odot }$ NSs described by the DD2F-SF-6 EOS. The dashed, black line marks the region corresponding to the onset of the hadron-quark PT. The solid, blue line encloses regions with pure quark matter.

Figure 2

Rest mass distribution of matter in the baryon chemical potential–temperature plane for the merger simulation from Fig. 1 normalized to the total mass of the system. Note the logarithmic mass scale. The black line is the temperature-dependent phase boundary between the hadronic phase at low chemical potential and the deconfined quark phase at high chemical potential. The total amount of matter at temperatures below 5 MeV in the graphs (a)–(d) are about 92%, 92%, 42%, and 19%, respectively.

Figure 3

Left panel: maximum rest-mass density as a function of time in binary NS merger simulations of two $1.35{\mathrm{M}}_{\odot }$ NSs with different hybrid DD2F-SF EOSs (colored curves) in comparison to the hadronic DD2F reference model (black curve). The merging time is shown by the vertical dashed line. Red dots show the value of ${\rho }_{\max }^{\max }$ (the highest maximum density within the first 5 ms after merging) for every simulation. Right panel: total fraction of mass being present as deconfined quark matter as a function of time for the same binary systems as in the left panel.

Figure 4

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ as a function of the tidal deformability $\mathrm{\Lambda }$ for $1.2–1.2{\mathrm{M}}_{\odot }$ [graph (a)], $1.35–1.35{\mathrm{M}}_{\odot }$ [graph (b)], $1.4–1.4{\mathrm{M}}_{\odot }$ [graph (c)], and $1.5–1.5{\mathrm{M}}_{\odot }$ [graph (d)] mergers with different microphysical EOSs. Black crosses display results with purely hadronic EOSs, while green plus signs depict results with the hybrid DD2F-SF models. The solid curves are least squares fits to data points from purely hadronic EOSs. The gray shaded area illustrates the largest deviation of the data of purely hadronic models from the least squares fit. For $1.35–1.35{\mathrm{M}}_{\odot }$ and $1.4–1.4{\mathrm{M}}_{\odot }$ binaries, the results from the hybrid models appear as clear outliers at higher GW frequencies.

Figure 5

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$ of the respective binary system. Red symbols refer do data from $1.3–1.4{\mathrm{M}}_{\odot }$ binaries, while black symbols refer do data from equal-mass binaries with the same chirp mass of a $1.3–1.4{\mathrm{M}}_{\odot }$ binary. Crosses represent data from purely hadronic EOSs, while plus signs display data obtained with hybrid DD2F-SF models. The solid black line shows a least squares fit with a second-order polynomial to the data (excluding the hybrid models). The gray shaded areas illustrate the maximum deviation of the data of hadronic models from the fit.

Figure 6

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$. Different colors refer to data from different total binary masses. Crosses refer to data from purely hadronic models, while plus signs represent data with hybrid DD2F-SF models. Solid black line is a least squares fit with a second-order polynomial to the data (excluding the DD2F-SF models). The gray shaded area illustrates the maximum deviation of the data of hadronic models from the fit.

Figure 7

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after merging as a function of the dominant postmerger GW frequency $f_{\mathrm{peak}}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ [graph (a)], $1.35–1.35{\mathrm{M}}_{\odot }$ [graph (b)], $1.4–1.4{\mathrm{M}}_{\odot }$ [graph (c)], and $1.5–1.5{\mathrm{M}}_{\odot }$ [graph (d)] mergers with different microphysical EOSs. Black crosses show results with purely hadronic EOSs, while green plus signs depict results with hybrid DD2F-SF models. Solid curves display least squares fits to results for purely hadronic EOSs.

Figure 8

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger as a function of the dominant postmerger GW frequency $f_{\mathrm{peak}}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ (black crosses), $1.35–1.35{\mathrm{M}}_{\odot }$ (red crosses), $1.4–1.4{\mathrm{M}}_{\odot }$ (blue crosses), and $1.5–1.5{\mathrm{M}}_{\odot }$ (green crosses) mergers with different microphysical EOSs. This plot contains the entire data of Figs. 7 excluding the results from the DD2F-SF EOSs. Solid black line shows a least squares fit to all shown data points [Eq. (5)].

Figure 9

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ (black), $1.35–1.35{\mathrm{M}}_{\odot }$ (red), $1.4–1.4{\mathrm{M}}_{\odot }$ (blue), and $1.5–1.5{\mathrm{M}}_{\odot }$ (green) mergers with different purely hadronic microphysical EOSs. The solid black line shows a least squares fit to all shown data points [Eq. (6)]. The gray shaded area illustrates the maximum deviation of the data from the fit.

Figure 10

Constraints on ${\rho }_{\text{onset}}$ as a function of $\mathrm{\Lambda }$ for a hypothetical $2.65{\mathrm{M}}_{\odot }$ binary. The black solid line shows the empirical ${\rho }_{\max }^{\max }(\mathrm{\Lambda },M_{\mathrm{tot}})$ relation [Eq. (13)] with the parameters from Eq. (14). The dashed lines display the uncertainty of the ${\rho }_{\max }^{\max }(\mathrm{\Lambda },M_{\mathrm{tot}})$ relation. Depending on the consistency of $f_{\mathrm{peak}}$ with Eq. (11), these curves illustrate upper or lower limits on ${\rho }_{\text{onset}}$. The red lines show more conservative constraints introduced in Sec. 5b. See text for more details.

Figure 11

Rest mass distribution (color-coded) of matter in the density-temperature plane for the merger simulation from Fig. 1 normalized to the total mass of the system. Note the logarithmic mass scale. The dashed, black line displays the temperature-dependent onset density of the hadron-quark phase transition. The solid black line is the boundary of the pure quark matter phase. The total amounts of matter at temperatures lower than 5 MeV in the graphs (a)–(d) are about 92%, 92%, 42%, and 19%, respectively.

Figure 12

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$. Different colors refer to data from different total binary masses. Crosses refer to data from purely hadronic models, while plus signs represent data with hybrid DD2F-SF models. The solid black line is a least squares fit with a second-order polynomial to the data (excluding the DD2F-SF models). The gray shaded area illustrates the maximum deviation of the data from hadronic models from the fit. Compared to Fig. 6, this plot also contains data from $1.3-1.4{\mathrm{M}}_{\odot }$ binaries.

Figure 13

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger as a function of the dominant postmerger GW frequency $f_{\mathrm{peak}}$. Red symbols refer to data from $1.3-1.4{\mathrm{M}}_{\odot }$ binaries, while black symbols refer to data from equal-mass binaries with the chirp mass of a $1.3–1.4{\mathrm{M}}_{\odot }$ binary. Crosses represent data from purely hadronic models, while plus signs illustrate data with hybrid DD2F-SF EOSs. The solid black line is a least squares fit with a second-order polynomial to the data (excluding the hybrid models).

Figure 14

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger as a function of the dominant postmerger GW frequency $f_{\mathrm{peak}}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ (black crosses), $1.35–1.35{\mathrm{M}}_{\odot }$ (red crosses), $1.4–1.4{\mathrm{M}}_{\odot }$ (blue crosses) $1.5–1.5{\mathrm{M}}_{\odot }$ (green crosses), and $1.3–1.4{\mathrm{M}}_{\odot }$ (cyan crosses) mergers with different microphysical EOSs. This plot contains the entire data of Fig. 8 together with data from asymmetric mergers and additional results from grid-based calculations (orange circles). The solid black line is a least squares fit to all shown data points [Eq. (5)] excluding results from grid-based calculations.

Figure 15

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ (black), $1.35–1.35{\mathrm{M}}_{\odot }$ (red), $1.4–1.4{\mathrm{M}}_{\odot }$ (blue), $1.5–1.5{\mathrm{M}}_{\odot }$ (green), and $1.3–1.4{\mathrm{M}}_{\odot }$ (cyan) mergers with different microphysical EOSs. This plot contains the entire data of Fig. 9 together with results from asymmetric binaries. The solid black line is a least squares fit to all shown data points [Eq. (6)]. The gray shaded area illustrates the maximum deviation of the data from the fit.

Figure 16

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$. Plotted are results from $1.2-1.2{\mathrm{M}}_{\odot }$ and $1.35-1.35{\mathrm{M}}_{\odot }$ binaries [graph (a)], from $1.35-1.35{\mathrm{M}}_{\odot }$ and $1.4-1.4{\mathrm{M}}_{\odot }$ binaries [graph (b)], and from $1.4-1.4{\mathrm{M}}_{\odot }$ and $1.5-1.5{\mathrm{M}}_{\odot }$ binaries [graph (c)]. Different colors refer to data from different total binary masses. Crosses refer to data from purely hadronic models, while plus signs represent data with hybrid DD2F-SF models. The solid black lines are least squares fits with a second-order polynomial to the data (excluding the DD2F-SF models) in the respective plot. The gray shaded areas illustrate the maximum deviation of the data of hadronic models from each fit. At binary masses of $1.35–1.35{\mathrm{M}}_{\odot }$ and $1.4–1.4{\mathrm{M}}_{\odot }$, DD2F-SF models appear as outliers.

Figure 17

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger as a function of the dominant postmerger GW frequency $f_{\mathrm{peak}}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ and $1.35–1.35{\mathrm{M}}_{\odot }$ binaries [graph (a)], $1.35–1.35{\mathrm{M}}_{\odot }$ and $1.4–1.4{\mathrm{M}}_{\odot }$ binaries [graph (b)], and for $1.4–1.4{\mathrm{M}}_{\odot }$ and $1.5–1.5{\mathrm{M}}_{\odot }$ binaries [graph (c)] with purely hadronic EOSs. Different colored crosses refer to data from different total binary masses. The solid black lines are least squares fits with a second-order polynomial to the data.

Figure 18

Maximum rest-mass density ${\rho }_{\max }^{\max }$ in the remnant during the first 5 ms after the merger scaled by the total binary mass $M_{\mathrm{tot}}$ as a function of the combined tidal deformability $\tilde{\mathrm{\Lambda }}$ for $1.2–1.2{\mathrm{M}}_{\odot }$ and $1.35–1.35{\mathrm{M}}_{\odot }$ binaries [graph (a)], $1.35–1.35{\mathrm{M}}_{\odot }$ and $1.4–1.4{\mathrm{M}}_{\odot }$ binaries [graph (b)], and for $1.4–1.4{\mathrm{M}}_{\odot }$ and $1.5–1.5{\mathrm{M}}_{\odot }$ binaries [graph (c)] with purely hadronic EOSs. Different colored crosses refer to data from different total binary masses. The solid black lines are least squares fits with Eq. (6) to the data. The gray shaded areas illustrate the maximum deviation of the data from the respective fit.

Figure 19

The slope $\frac{d\mathrm{\Lambda }}{d{M}_{\mathrm{tot}}}$ of the tidal deformability $\mathrm{\Lambda }$ with respect to the total binary mass $M_{\mathrm{tot}}$ as a function of $\frac{\mathrm{\Lambda }}{M_{\mathrm{tot}}}$ for all hadronic EOSs used in this paper. The red line shows a least squares fit $\frac{d\mathrm{\Lambda }}{d{M}_{\mathrm{tot}}}=z\frac{\mathrm{\Lambda }}{M_{\mathrm{tot}}}$.

Figure 20

Maximum rest-mass density within the first 5 ms after the merger ${\rho }_{\max }^{\max }$ as a function of the total binary mass $M_{\mathrm{tot}}$ for the purely hadronic DD2F EOS. Crosses show actual simulation data between these points linear interpolation are used. The cyan (orange) point indicates the $M_{\mathrm{tot}}$ value for which ${\rho }_{\max }^{\max }$ is equal to the transition density of the hybrid DD2F-SF-6 (DD2F-SF-7) EOS.

Figure 21

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ as a function of the total binary mass $M_{\mathrm{tot}}$ for the purely hadronic DD2F EOS (black) and the hybrid DD2F-SF-6 EOS (cyan). Crosses show simulation data, which we connect with a linear interpolation. The dots highlight data points with a binary mass $M_{\mathrm{tot},\mathrm{onset}}$ where ${\rho }_{\max }^{\max }$ in a simulation with the purely hadronic DD2F model is expected to reach the transition density of the DD2F-SF-6 EOS (see Fig. 20).

Figure 22

Dominant postmerger GW frequency $f_{\mathrm{peak}}$ as a function of the total binary mass $M_{\mathrm{tot}}$ for the purely hadronic DD2F EOS (black) and the hybrid DD2F-SF-7 EOS (orange). Crosses show data points between these points linear interpolation are used. Dots highlight the data points with a total binary mass $M_{\mathrm{tot},\mathrm{onset}}$ where ${\rho }_{\max }^{\max }$ in a simulation with the purely hadronic DD2F model is expected to reach the transition density of the DD2F-SF-7 EOS (see Fig. 20).

Figure 23

Average mass fraction of quark matter in the merger remnant during the first 5 ms after the merger as a function of the shift in $f_{\mathrm{peak}}$ with respect to the corresponding purely hadronic merger for binaries with different masses and different hybrid EOSs. Crosses represent actual simulation results; each color refers to a specific hybrid model. The black dashed line marks a shift of $f_{\mathrm{peak}}$ by 200 Hz caused by the appearance of quark matter; any shift larger than this would be a clear indication of the presence of a PT.

Figure 24

Constraints on the onset density of deconfinement ${\rho }_{\text{onset}}$ as a function of $\mathrm{\Lambda }$ for hypothetical binaries of different total binary masses. The black solid lines show the empirical ${\rho }_{\max }^{\max }(\mathrm{\Lambda },M_{\mathrm{tot}})$ relation [Eq. (13)] with the parameters from Eq. (14) for each mass. The dashed lines display the uncertainty of the ${\rho }_{\max }^{\max }(\mathrm{\Lambda },M_{\mathrm{tot}})$ relation. Depending on the consistency of $f_{\mathrm{peak}}$ with Eq. (11), these represent upper or lower limits on ${\rho }_{\text{onset}}$. The red lines show more conservative constraints introduced in Sec. 5b. See text for more details.