Systematics of prompt black-hole formation in neutron star mergers

This study addresses the collapse behavior of neutron star (NS) mergers expressed through the binary threshold mass $M_{\text{thres}}$ for prompt black-hole (BH) formation, which we determine by relativistic hydrodynamical simulations for a set of 40 equation of state (EOS) models of NS matter. $M_{\text{thres}}$ can be well described by various fit formulas involving stellar parameters of nonrotating NSs, which are employed to characterize the EOS models. Using these relations we compute which constraints on NS radii and the tidal deformability are set by current and future merger detections that reveal information about the merger product. We systematically investigate the impact of the binary mass ratio $q=M_1/M_2$ and assemble various fits, which make different assumptions about a-priori knowledge. This includes fit formulas for $M_{\text{thres}}$ for a fixed mass ratio or a range of $q$ if this parameter is not known very well. Also, we construct relations describing the threshold to prompt collapse for different classes of candidate EOSs, which for instance do or do not include models with a phase transition to quark matter. In particular, we find fit formulas for $M_{\text{thres}}$ including an explicit $q$ dependence, which are valid in a broad range of $0.7\le q\le 1$ and which are nearly as tight as relations for fixed mass ratios. For most EOS models except for some extreme cases the threshold mass of asymmetric mergers is equal or smaller than the one of equal-mass binaries. Generally, the impact of the binary mass asymmetry on $M_{\text{thres}}$ becomes stronger with more extreme mass ratios, while $M_{\text{thres}}$ is approximately constant for small deviations from $q=1$, i.e., for $0.85\le q\le 1$. The magnitude of the reduction of $M_{\text{thres}}$ with the binary mass asymmetry follows a systematic EOS dependence. We also describe in more detail that a phase transition to deconfined quark matter can leave a characteristic imprint on the collapse behavior of NS mergers. The occurrence of quark matter can reduce the stability of the remnant and thus the threshold mass relative to a purely hadronic reference model. Comparing specifically the threshold mass and the combined tidal deformability ${\tilde{\mathrm{\Lambda }}}_{\text{thres}}$ of a system with $M_{\text{thres}}$ can yield peculiar combinations of those two quantities, where $M_{\text{thres}}$ is particularly small in relation to ${\tilde{\mathrm{\Lambda }}}_{\text{thres}}$. Since no purely hadronic EOS can yield such a combination of $M_{\text{thres}}$ and ${\tilde{\mathrm{\Lambda }}}_{\text{thres}}$, a combined measurement or a constraint on both quantities can indicate the onset of quark deconfinement. Finally, we point out new univariate relations between $M_{\text{thres}}$ and stellar properties of high-mass NSs, which can be employed for direct EOS constraints or consistency checks in combination with other measurements.

Figure 1

Upper panels: Threshold binary mass $M_{\text{thres}}$ for prompt BH formation as function of the maximum mass $M_{\max }$ of nonrotating NSs and the radius of a $1.6M_{\odot }$ NS for $q=M_1/M_2=1$ (left) and $q=0.7$ (right) with the base EOS sample. Blue plane is a bilinear fit (see Table 1) to the data (blue points). Short black lines visualize deviations between fit and data. Lower panels: Same as the upper panels but with the tidal deformability ${\mathrm{\Lambda }}_{1.4}$ of a $1.4M_{\odot }$ NS instead of $R_{1.6}$ ($q=1$ on the left, $q=0.7$ on the right; see Table 2).

Figure 2

Bounds on NS radius $R_{1.6}$ implied by GW detections with total binary mass $M_{\mathrm{tot}}$. Black lines show a lower limit on $R_{1.6}$ if an observation provides evidence for no direct BH formation. Red line displays an upper limit on $R_{1.6}$ inferred from events with indications for a prompt collapse of the merger remnant. The shaded areas indicate the maximum scatter of the underlying empirical relations for $M_{\text{thres}}$; the dashed black line employs a very conservative assumption about the maximum stiffness of NS matter at higher densities (see text). The blue vertical line provides the total binary mass of GW170817 with the error bar indicated by the blue shaded area.

Figure 3

Difference $\mathrm{\Delta }{M}_{\text{thres}}$ between the threshold mass of equal-mass mergers and the threshold mass of asymmetric binaries with mass ratio $q=0.7$ as function of $M_{\max }$ and $R_{1.6}$ (blue dots). The blue plane displays a bilinear fit (see Table 5). Deviations between the fit and the underlying data are illustrated by black lines. This figure is identical to Fig. 3 in the Supplemental Material of [49].

Figure 4

Left panel: Threshold mass $M_{\text{thres}}$ for prompt BH formation as function of mass ratio $q$ for the DD2F EOS. Black error bars indicate finite precision to which $M_{\text{thres}}$ is given from merger simulations for fixed $q$. Blue dashed curve displays a fit $M_{\text{thres}}(q)=\alpha (1-q{)}^{2.5}+\gamma$ to the data. Red curves are directly computed from specific data points assuming a function $M_{\text{thres}}(q)=\alpha (1-q{)}^{2.5}+\gamma$ (see main text). Right panel: Threshold chirp mass ${\mathcal{M}}_{\text{thres}}$ for prompt BH formation as function of the mass ratio $q$ for the DD2F EOS. Black error bars indicate finite precision to which ${\mathcal{M}}_{\text{thres}}$ is given from merger simulations. Red curves are directly computed from specific data points assuming a function ${\mathcal{M}}_{\text{thres}}(q)=\alpha (1-q{)}^{2.5}+\gamma$ (see main text).

Figure 5

Threshold binary mass for prompt collapse as function of binary mass ratio $q$ for different EOS properties based on Eq. (10). Solid curves assume a fixed maximum mass of $M_{\max }=2.0M_{\odot }$ but different NS radii. Blue curves show $M_{\text{thres}}(q)$ for a fixed radius $R_{1.6}=13\mathrm{km}$ but with $M_{\max }$ being $2.0M_{\odot }$ (solid), $2.1M_{\odot }$ (dashed) and $2.2M_{\odot }$ (dotted).

Figure 6

Angular momentum of a binary with a total mass of $3M_{\odot }$ as function of the mass ratio $q$ within a Newtonian point-particle approximation at an orbital distance of 24 km. Within this model the total angular momentum decreases with binary mass asymmetry (black). The contribution from the lighter binary component grows (green curve), whereas the angular momentum of the more massive binary component decreases even stronger (blue curve).

Figure 7

Toy model estimate of the difference in the remnant’s angular momentum between symmetric ($q=1$) and asymmetric mergers ($q=0.7$) as function of the radius $R_{1.6}$ of a $1.6M_{\odot }$ NS for different EOSs of the base sample. See text for more explanations.

Figure 8

Threshold mass $M_{\text{thres}}$ and combined tidal deformability ${\tilde{\mathrm{\Lambda }}}_{\text{thres}}$ of binary systems with $M_{\mathrm{tot}}=M_{\text{thres}}$ for different EOSs and $q=1$. EOS models are classified as in Sec. 2: large black dots show the purely hadronic base sample, small black dots display the excluded hadronic sample, i.e., models which are incompatible with GW170817, and green symbols indicate hybrid models. (Two excluded models with $M_{\text{thres}}>3.4M_{\odot }$ lie outside the plot limits.) Overplotted crosses mark hyperonic EOSs. The red plus sign corresponds to the ALF2 EOS, where quark matter resembles properties of hadronic matter [84, 87]. The black dashed line separates an area where only hybrid models occur, from a “mixed” regime with both hybrid and purely hadronic EOS. Blue lines display curves of constant $M_{\max }$, which are given by fit 43 in Table 2 for the hadronic base sample. Since viable hadronic models should have $M_{\max }\ge 1.97M_{\odot }$, no hadronic EOS (large black dots) occurs significantly to the left the blue solid line. Note that all except for one hybrid model (green dots) fulfill $M_{\max }\ge 1.97M_{\odot }$, although some models occur on the left of the blue solid line. The red curve marks configurations which have approximately a NS radius of 13.5 km (with a mass of $M_{\text{thres}}/2$) as an approximate upper bound on NS radii from GW170817. See main text for more explanations. A very similar figure can be found in [49].

Figure 9

Same as Fig. 8 but for a binary mass ratio $q=0.85$ (left panel) and $q=0.7$ (right panel). For orientation the gray dashed line in the left panel is identical to the black dashed line in Fig. 8.

Figure 10

Left panel: same as Fig. 8 with only purely hadronic models. Red arrows link EOS models with comparable radii at $M_{\text{thres}}/2$, but which differ with respect to the presence or absence of hyperons above a certain central density and yield different $M_{\max }$. Blue arrows link EOS models with comparable $M_{\max }$. See main text and caption of Fig. 8 for more explanations; see Table I of Supplemental Material to [49] for data of the individual EOS models. Right panel: mass-radius relations of nonrotating NSs for the EOS models connected with arrows in the left panel.

Figure 11

Radius of nonrotating NSs with a mass of $2.0M_{\odot }$ as function of $M_{\text{thres}}$ for equal-mass mergers. Black symbols refer to models of the hadronic base sample, green dots show hybrid EOSs. Dashed line is a linear fit to the data (see Table 8). The bluish band displays the total binary mass of GW170817 [26].

Figure 12

Tidal deformability for NSs with a mass of $2.0M_{\odot }$ as function of $M_{\text{thres}}$ for equal-mass mergers. Black symbols refer to models of the hadronic base sample, green dots show hybrid EOSs. Dashed line is a linear fit to the data (see Table 8). The bluish band displays the total binary mass of GW170817 [26].

Figure 13

Difference $\mathrm{\Delta }{M}_{\text{thres}}$ between the threshold mass of equal-mass mergers and the threshold mass of asymmetric binaries with mass ratio $q=0.85$ (left panel) or $q=0.7$ (right panel) as function of $M_{\max }$ and $R_{1.6}$ (black dots). Results for the complete EOS sample are shown. The blue planes display bilinear fits as in Sec. 4 with parameters $a=-0.136$, $b=2.495\times {10}^{-2}$ and $c=0.001$ (left panel) and $a=-0.304$, $b=5.198\times {10}^{-2}$ and $c=0.083$ (right panel, see Table 5). Deviations between the fit and the underlying data are illustrated by black lines: maximum and average deviations are $0.053M_{\odot }$ and $0.018M_{\odot }$ for $q=0.85$ in the left panel and $0.051M_{\odot }$ and $0.016M_{\odot }$ for $q=0.7$ in the right panel. The red dots show cases where $\mathrm{\Delta }{M}_{\text{thres}}$ is negative, i.e., the threshold mass of the asymmetric merger is slightly higher than that of the corresponding equal-mass case.