Merger of binary neutron stars to a black hole: Disk mass, short gamma-ray bursts, and quasinormal mode ringing

Three-dimensional simulations for the merger of binary neutron stars are performed in the framework of full general relativity. We pay particular attention to the black hole formation case and to the resulting mass of the surrounding disk for exploring the possibility for formation of the central engine of short-duration gamma-ray bursts (SGRBs). Hybrid equations of state are adopted mimicking realistic, stiff nuclear equations of state (EOSs), for which the maximum allowed gravitational mass of cold and spherical neutron stars, $M_{\mathrm{sph}}$, is larger than $2M_{\odot }$. Such stiff EOSs are adopted motivated by the recent possible discovery of a heavy neutron star of mass $\sim 2.1\pm 0.2M_{\odot }$. For the simulations, we focus on binary neutron stars of the ADM mass $M\gtrsim 2.6M_{\odot }$. For an ADM mass larger than the threshold mass $M_{\mathrm{thr}}$, the merger results in prompt formation of a black hole irrespective of the mass ratio $Q_M$ with $0.65\lesssim Q_M\le 1$. The value of $M_{\mathrm{thr}}$ depends on the EOSs and is approximately written as $1.3–1.35M_{\mathrm{sph}}$ for the chosen EOSs. For the black hole formation case, we evolve the space-time using a black hole excision technique and determine the mass of a quasistationary disk surrounding the black hole. The disk mass steeply increases with decreasing the value of $Q_M$ for given ADM mass and EOS. This suggests that a merger with small value of $Q_M$ is a candidate for producing central engine of SGRBs. For $M<M_{\mathrm{thr}}$, the outcome is a hypermassive neutron star of a large ellipticity. Because of the nonaxisymmetry, angular momentum is transported outward. If the hypermassive neutron star collapses to a black hole after the long-term angular momentum transport, the disk mass may be $\gtrsim 0.01M_{\odot }$ irrespective of $Q_M$. Gravitational waves are computed in terms of a gauge-invariant wave extraction technique. In the formation of the hypermassive neutron star, quasiperiodic gravitational waves of frequency between 3 and 3.5 kHz are emitted irrespective of EOSs. The effective amplitude of gravitational waves can be $\gtrsim 5\times {10}^{-21}$ at a distance of 50 Mpc, and hence, it may be detected by advanced laser-interferometers. For the black hole formation case, the black hole excision technique enables a long-term computation and extraction of ring-down gravitational waves associated with a black hole quasinormal mode. It is found that the frequency and amplitude are $\approx 6.5–7\mathrm{kHz}$ and $\sim {10}^{-22}$ at a distance of 50 Mpc for the binary of mass $M\approx 2.7–2.9M_{\odot }$.

Figure 1

Pressure and specific internal energy as a function of baryon rest-mass density $\rho$ for the APR EOS. The solid and dotted curves denote the results by fitting formulas and numerical data tabulated, respectively.

Figure 2

(a) ADM mass (solid curves) and total baryon rest-mass (dashed curves) as a function of central baryon rest-mass density ${\rho }_c$ and (b) relation between the circumferential radius and the ADM mass for cold and spherical neutron stars in equilibrium. “APR” and “SLy” denote the sequences for the APR and SLy EOSs, respectively.

Figure 3

Evolution of the central value of the lapse function ${\alpha }_c$ and the maximum values of the rest-mass density ${\rho }_{\max }$ for models APR1313 (solid curves), APR135135 (long-dashed curves), APR1414 (dashed curves), and APR1515 (dot-dashed curves). The dotted horizontal lines denote the central values of the lapse and rest-mass density for the marginally stable, spherical neutron star in equilibrium with the cold APR EOS.

Figure 4

Snapshots of the density contour curves for $\rho$ in the equatorial plane for model APR1313. The solid contour curves are drawn for $\rho =2\times {10}^{14}\times i\mathrm{g}/{\mathrm{cm}}^3(i=1,2,3,\cdots )$ and for $1\times {10}^{14-0.5i}\mathrm{g}/{\mathrm{cm}}^3(i=1\sim 6)$. The (blue) thick dotted and solid curves denote $1\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ and $1\times {10}^{12}\mathrm{g}/{\mathrm{cm}}^3$, respectively. The number in the upper left-hand side denotes the elapsed time from the beginning of the simulation in units of ms. Vectors indicate the local velocity field $(v^x,v^y)$, and the scale is shown in the upper right-hand corner.

Figure 5

The same as Fig. 4 but for model APR1414.

Figure 6

The same as Fig. 4 but for model APR1515. The outermost (green) dotted curves denote $1\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$. The thick circles around the origin in the last three panels denote the location of the apparent horizon.

Figure 7

The same as Fig. 6 but for model APR1416.

Figure 8

The same as Fig. 6 but for model APR1317.

Figure 9

The density contour curves for $\rho$ and the local velocity field $(v^x,v^z)$ in the $y=0$ plane (a) at $t=10.668\mathrm{ms}$ for model APR1313, (b) at $t=10.319\mathrm{ms}$ for APR1414, (c) at $t=2.523\mathrm{ms}$ for APR1515, (d) at $t=2.911\mathrm{ms}$ for APR1416, (e) at $t=2.959\mathrm{ms}$ for APR135165, and (f) at $t=3.038\mathrm{ms}$ for APR1317. The contour curves and the velocity vectors are drawn in the same way as in Fig. 6.

Figure 10

(a) The angular velocity $\Omega$ of hypermassive neutron stars along $x$ (solid curves) and $y$ axes (dashed curves) for model APR1313 (blue) at $t=10.668\mathrm{ms}$ and for APR1414 (black) at $t=10.319\mathrm{ms}$. (b) $\Omega$ of accretion disk around a central black hole for model APR135165 along $x$ (solid curve) and $y$ (dashed curve) axes at $t=2.959\mathrm{ms}$. The apparent horizon is located at $r\approx 3\mathrm{km}$. For both figures, the horizontal axis denote the coordinate radius.

Figure 11

Evolution of $M_{*}(j)$ for models APR1313 (left) and APR1414 (right). The dotted vertical lines labeled $q=0.9$ and 0.7 denote the values of $j/M$ at ISCOs around a Kerr black hole of the spin parameter $q$.

Figure 12

(a) Evolution of the baryon rest-mass of disks surrounding black holes and (b) the evolution of irreducible mass of black holes for models APR1515 (solid curve), APR145155 (dashed curve), APR1416 (long-dashed curve), APR135165 (dot-dashed curve), APR1317 (dot-long-dashed curve), and SLy125155 (dotted curve). $t_{\mathrm{AH}}$ denotes the time at the first formation of an apparent horizon.

Figure 13

Baryon rest-mass of disks around a black hole as a function of $Q_M=M_{*2}/M_{*1}$ for the black hole formation cases. The disk mass is evaluated at $t-t_{\mathrm{AH}}=0.5\mathrm{ms}$. The open circles and squares denote the results with (633, 633, 317) grid size in the APR and SLy EOSs, respectively. The filled circles and squares denote the results with (377, 377, 189) grid size in the APR and SLy EOSs. The dashed curves denote the fitting formulas (29). The results for the APR and SLy EOSs are derived for $M\approx 2.96M_{\odot }$ and $2.76M_{\odot }$, respectively.

Figure 14

Gravitational waveforms, $R_{+}$ and $R_{\times }$, (a) for model APR1313 at $r_{\mathrm{obs}}=36M_0$ and (b) for model APR1414 at $r_{\mathrm{obs}}=32M_0$.

Figure 15

Energy and angular momentum emission rates, $dE/dt$ and $dJ/dt$, of gravitational waves for models APR1313 (solid curves) and APR1414 (dashed curves). The units are $\mathrm{erg}/\mathrm{s}$ and $\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$, respectively.

Figure 16

The solid curves denote the evolution of angular momentum for models (a) APR1313 and (b) APR1414. The dashed curves denote $J_0-\Delta J$ where $J_0$ denotes the angular momentum at $t=0$.

Figure 17

Fourier power spectrum of gravitational waves $dE/df$ for models APR1313 (solid curve) and APR1414 (dashed curve). Since the simulations are started when the characteristic frequency of gravitational waves is $\sim 1\mathrm{kHz}$, the spectrum for $f<1\mathrm{kHz}$ cannot be presented. The dotted curve in the panel denotes the analytical result of $dE/df$ in the second post-Newtonian and point-particle approximation by which the real spectrum for $f\lesssim 1\mathrm{kHz}$ is approximated.

Figure 18

Nondimensional effective amplitude of gravitational waves from hypermassive neutron stars for models APR1313 (solid curve) and APR1414 (dashed curve). The assumed distance is 50 Mpc. The dotted line denotes the planned noise level of the advanced LIGO.

Figure 19

The same as Fig. 14, but for models APR135165 (left) at $r_{\mathrm{obs}}=27M_0$ and model SLy125155 at $r_{\mathrm{obs}}=29M_0$ (right). The lower panel is the enlargement of the upper panel for ring-down gravitational waves associated with a quasinormal mode of the formed black hole. The formation time of the apparent horizon $t_{\mathrm{AH}}$ is 2.38 ms for model APR135165 and 3.04 ms for model SLy125155, and hence, the quasinormal mode is excited for $t-r_{\mathrm{obs}}\gtrsim t_{\mathrm{AH}}$. The thick dotted (blue) curves denote the fitting formulas described by Eq. (38).

Figure 20

Evolution of the total radiated energy $\Delta E$ and angular momentum $\Delta J$ for models APR1416 (solid curves), APR135165 (dashed curves), and APR1317 (long-dashed curves)

Figure 21

Summary about the outcome after the merger. “HMNS,” “GW,” “B-field,” and “J-transport” are hypermassive neutron star, gravitational wave emission, magnetic field, and angular momentum transport, respectively. “Small disk,” “massive disk,” and “heavy disk” imply that the disk mass is $M_d\ll 0.01M_{\odot }$, $0.01M_{\odot }\lesssim M_d\lesssim 0.03M_{\odot }$, and $M_d\gtrsim 0.05M_{\odot }$, respectively. The solid arrow denotes the route found in this paper and the dashed arrow is the possible route based on the results of 57 and the speculation. There are three possible fates of the ellipsoidal hypermassive neutron star depending on the EOS and the time scale of dissipation and angular momentum transport processes. We note that a spheroid is formed probably only for the APR EOS. See Secs. 6a, 6b, 6c, 6d for details.