Accurate evolutions of inspiralling neutron-star binaries: Prompt and delayed collapse to a black hole

Binary neutron-star systems represent primary sources for the gravitational-wave detectors that are presently operating or are close to being operating at the target sensitivities. We present a systematic investigation in full general relativity of the dynamics and gravitational-wave emission from binary neutron stars which inspiral and merge, producing a black hole surrounded by a torus. Our results represent the state of the art from several points of view: (i) We use high-resolution shock-capturing methods for the solution of the hydrodynamics equations and high-order finite-differencing techniques for the solution of the Einstein equations; (ii) We employ adaptive mesh-refinement techniques with “moving boxes” that provide high-resolution around the orbiting stars; (iii) We use as initial data accurate solutions of the Einstein equations for a system of binary neutron stars in irrotational quasicircular orbits; (iv) We exploit the isolated-horizon formalism to measure the properties of the black holes produced in the merger; (v) Finally, we use two approaches, based either on gauge-invariant perturbations or on Weyl scalars, to calculate the gravitational waves emitted by the system. Within our idealized treatment of the matter, these techniques allow us to perform accurate evolutions on time scales never reported before (i.e. $\sim 30\mathrm{ms}$) and to provide the first complete description of the inspiral and merger of a neutron-star binary leading to the prompt or delayed formation of a black hole and to its ringdown. We consider either a polytropic equation of state or that of an ideal fluid and show that already with this idealized treatment a very interesting phenomenology can be described. In particular, we show that while higher-mass polytropic binaries lead to the prompt formation of a rapidly rotating black hole surrounded by a dense torus, lower-mass binaries give rise to a differentially rotating star, which undergoes large oscillations and emits large amounts of gravitational radiation. Eventually, also the hyper-massive neutron star collapses to a rotating black hole surrounded by a torus. Finally, we also show that the use of a nonisentropic equation of state leads to significantly different evolutions, giving rise to a delayed collapse also with high-mass binaries, as well as to a more intense emission of gravitational waves and to a geometrically thicker torus.

Figure 1

Isodensity contours on the $(x,y)$ (equatorial) plane for the evolution of the high-mass binary with the polytropic EOS (i.e. model 1.62-45-P in Table ). The time stamp in ms is shown on the top of each panel, the color-coding bar is shown on the right in units of $\mathrm{g}/{\mathrm{cm}}^3$, and the thick dashed line represents the AH.

Figure 2

Evolution of the maximum rest-mass density normalized to its initial value for the high-mass binary. Indicated with a dotted vertical line is the time at which the stars merge, while a vertical dashed line shows the time at which an AH is first found and which is a few ms only after the merger in this case. After this time, the maximum rest-mass density is computed in a region outside the AH and therefore it refers to the density of the oscillating torus. It is only a few orders of magnitude smaller. Note that the non-normalized value of the maximum rest-mass density at $t=0$ is $5.91\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ (see Table ). The binary has been evolved using the polytropic EOS.

Figure 3

Isodensity contours on the $(x,z)$ plane highlighting the formation of a torus surrounding the central BH, whose AH is indicated with a thick dashed line. The data refers to the high-mass binary evolved with the polytropic EOS.

Figure 4

Evolution of the proper separation (top part) and of the coordinate separation (bottom part) for binaries with initial coordinate separation of either 45 or 60 km (i.e. models 1.62-45-P and 1.62-60-P in Table ). Indicated with a dashed line is the proper separation for the binary starting at 45 km and suitably shifted in time.

Figure 5

Coordinate (dashed line) and proper (solid line) trajectory of one of the stars for the high-mass binary from a coordinate distance of 45 km (about 2.2 orbits).

Figure 6

Evolution of the maximum rest-mass density normalized to its initial value for high-mass binaries with initial coordinate separation of either 45 or 60 km. The vertical dashed lines denote the time at which an AH was found. The polytropic EOS was used for the evolutions.

Figure 7

Isodensity contours on the $(x,y)$ (equatorial) plane for the evolution of the low-mass binary with the polytropic EOS (i.e. model 1.46-45-P in Table ). The time stamp in ms is shown on the top of each panel, the color-coding bar is shown on the right in units of $\mathrm{g}/{\mathrm{cm}}^3$, and the thick dashed line represents the AH.

Figure 8

The same as in Fig. 2 but for the low-mass binary. Note that the merger time is essentially the same as for the high-mass binary but there is a long delay in the collapse and the onset of quasiharmonic oscillations in the HMNS. The binary has been evolved using the polytropic EOS. Note that the non-normalized value of the maximum rest-mass density at $t=0$ is $4.58\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ (see Table ).

Figure 9

Conservation of the total angular momentum for the high-mass binary (upper plot) and the low-mass one (lower plot). Indicated with different lines are the computed values of the volume-integrated angular momentum (solid line), of the angular momentum lost to gravitational waves (dotted line), and of their sum (dashed line). The dot-dashed line marks a 3% error.

Figure 10

Isodensity contours on the $(x,z)$ plane, highlighting the formation of a torus surrounding the central BH, whose AH is indicated with a thick dashed line. The data refers to the low-mass binary evolved with the polytropic EOS (cf. Fig. 3).

Figure 11

Isodensity contours on the $(x,y)$ (equatorial) plane for the evolution of the high-mass binary with the ideal-fluid EOS (i.e. model 1.62-45-IF in Table ). The time stamp in ms is shown on the top of each panel, the color-coding bar is shown on the right in units of $\mathrm{g}/{\mathrm{cm}}^3$, and the thick dashed line represents the AH.

Figure 12

Evolution of the maximum rest-mass density normalized to its initial value for the high-mass binary using the ideal-fluid EOS. Indicated with a dotted vertical line is the time at which the binary merges, while a vertical dashed line shows the time at which an AH is found. After this time, the maximum rest-mass density is computed in a region outside the AH. This figure should be compared with Fig. 2.

Figure 13

Evolution of the coordinate volume-integral of the thermal part of the internal energy density. Note the secular increase in the thermal energy after the merger (vertical dotted line). The shaded area refers to a window in time in which the calculation of ${ϵ}_{\mathrm{th}}$ via Eq. (25) becomes inaccurate as a result of the steep gradients in the hydrodynamical variables developing inside the AH.

Figure 14

Isodensity contours on the $(x,z)$ plane highlighting the formation of a torus surrounding the central BH, whose AH is indicated with a thick dashed line. The data refer to the high-mass binary evolved with the ideal-fluid EOS (cf. Figs. 3, 10 and the different vertical scales.)

Figure 15

Evolution of the maximum rest-mass density normalized to its initial value for the low-mass binary evolved using the ideal-fluid EOS. Indicated with a dotted vertical line is the time at which the binary merges. This figure should be compared with Figs. 2, 8, 12, of which maintains the same scale.

Figure 16

Left panel: Isodensity contours and velocity vector field (with the orbital component removed) on the $(x,y)$ (equatorial) plane at a selected time soon after the merger. Note the presence of localized vortices in the shear layer between the two stars. Right panel: contours of the weighted vorticity $\rho |\nabla \times \mathbf{v}{|}^z$ (that is the rest-mass density multiplied by the module of the $z$ component of the vorticity) for the same time shown in the left panel. This rendering highlights that in the shear layer the vorticity can be up to 3 orders of magnitude larger than in the bulk of the stars. Both panels refer to a high-mass binary evolved with the polytropic EOS.

Figure 17

Left panel: Maximum of the weighted vorticity $\rho |\nabla \times \mathbf{v}{|}^z$ on the equatorial plane normalized by the maximum of the rest-mass density ${\rho }_{\max }$ during the evolution of the high (solid line) and low (dashed line) mass binaries evolved with the polytropic EOS. Indicated with a dotted vertical line is the time at which the binaries merge. Both the curves are plotted until the formation of an AH. Right panel: The same as in the left panel but for the ideal-fluid EOS.

Figure 18

Left panel: Retarded-time evolution of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ as extracted from a 2-sphere at a coordinate radius $r=200M_{\odot }=295\mathrm{km}$ for the high-mass binary. Indicated in the inset is the final part of the signal corresponding to the BH quasinormal ringing. The merger takes place at $(t-r)\sim 5.3\mathrm{ms}$. Right panel: The same as in the left panel but shown in terms of the real part of the gauge-invariant quantity $Q_{22}^{+}$. In both cases the binaries have been evolved using the polytropic EOS.

Figure 19

Comparison of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ for the high-mass binary evolved with the polytropic EOS when extracted at different radii: $r=160M_{\odot }=236\mathrm{km}$ (solid line), $r=200M_{\odot }=295\mathrm{km}$ (dashed line), and $r=240M_{\odot }=354\mathrm{km}$ (dotted line).

Figure 20

Comparison of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ (upper panel) and of its amplitude (lower panel) for the high-mass binaries evolved with the polytropic EOS starting from an initial separation of 45 or 60 km. Indicated with a dashed line are the values after a time-shift.

Figure 21

Left panel: Retarded-time evolution of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ for the low-mass binary. Indicated in the inset is the final part of the signal corresponding to the BH quasinormal ringing. The merger takes place at $(t-r)\sim 5.3\mathrm{ms}$. Right panel: The same as in the left panel but shown in terms of the real part of the gauge-invariant quantity $Q_{22}^{+}$. In both cases the binaries have been evolved using the polytropic EOS.

Figure 22

Left panel: Retarded-time evolution of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ for the high-mass binary evolved with the ideal-fluid EOS. Indicated in the inset is the final part of the signal corresponding to the BH quasinormal ringing. Right panel: The same as in the left panel but for the low-mass binary. Here the inset does not refer to the BH quasinormal ringing; it highlights, instead, the periodic nature of the gravitational radiation after the merger. In both cases the merger takes place at $(t-r)\sim 5.8\mathrm{ms}$.

Figure 23

Left panel: Comparison in retarded-time evolution of the real part of the $\ell =m=2$ component of $r{\Psi }_4$ for the high-mass binary when evolved with the polytropic or with the ideal-fluid EOS. Right panel: The same as in the left panel but for the low-mass binary.

Figure 24

Left panel: Energy emitted in gravitational waves for the high-mass binary evolved with a polytropic EOS (solid line), for the low-mass binary evolved with a polytropic EOS (dotted line), for the high-mass binary evolved with the ideal-fluid EOS (dashed line), and for the low-mass binary evolved with the ideal-fluid EOS (dot-dashed line). Note that the largest amount of radiation comes from the low-mass binary whose emission has not been computed before. Indicated with a long-dashed line is the high-mass polytropic binary starting at 60 km. Right panel: The same as in the left panel but for the rate of the energy loss. Note that the largest burst of radiation is produced by the high-mass polytropic binary at the time of the prompt collapse to a BH.

Figure 25

Left panel: The same as in Fig. 24 but for the orbital angular momentum normalized to its initial value (cf. Table ). Right panel: The same as in Fig. 24 but for the rate of loss of orbital angular momentum.

Figure 26

Left panel: Comparison of retarded-time evolution of the amplitude the $\ell =m=2$ component of $h=(h_{+}^2+h_{\times }^2)$ for the high-mass binary when evolved with the polytropic (solid line) or with the ideal-fluid (dashed line) EOS; cf., Fig. 23, left panel. Right panel: The same as in the left panel but for the low-mass binary; cf., Fig. 23, right panel.

Figure 27

Left panel: Comparison of the PSD of the $\ell =m=2$ component of $h(f)f$ for the high-mass binary when evolved with the polytropic (solid line) or with the ideal-fluid (dashed line) EOS; cf., Fig. 23, left panel. Right panel: The same as in the left panel but for the low-mass binary; cf., Fig. 23, right panel. Indicated with a vertical long-dashed line is twice the initial orbital frequency.

Figure 28

Comparison of the rest-mass density normalized to its value at $t=0$ for evolutions performed with different numerical methods; the solid line refers to an evolution performed using the Marquina flux formula and a PPM reconstruction, the dotted line to HLLE-PPM, and the dashed line to HLLE-TVD (van Leer slope limiter). These data refer to an initial configuration not present in Table  (see text for details) and to an evolution with the ideal-fluid EOS.

Figure 29

Comparison of the time evolution of the coordinate separation (upper panel) and the proper separation (lower panel) between the stellar centers in case the initial Meudon shift is used (dashed line) and in case the initial shift is set to zero (continuous line).