Gravitational waves from black hole-neutron star binaries: Classification of waveforms

Using our new numerical-relativity code SACRA, long-term simulations for inspiral and merger of black hole (BH)-neutron star (NS) binaries are performed, focusing particularly on gravitational waveforms. As the initial conditions, BH-NS binaries in a quasiequilibrium state are prepared in a modified version of the moving-puncture approach. The BH is modeled by a nonspinning moving puncture and, for the NS, a polytropic equation of state with $\Gamma =2$ and the irrotational velocity field are employed. The mass ratio of the BH to the NS, $Q=M_{\mathrm{BH}}/M_{\mathrm{NS}}$, is chosen in the range between 1.5 and 5. The compactness of the NS, defined by $\mathcal{C}=G{M}_{\mathrm{NS}}/c^2{R}_{\mathrm{NS}}$, is chosen to be between 0.145 and 0.178. For a large value of $Q$ for which the NS is not tidally disrupted and is simply swallowed by the BH, gravitational waves are characterized by inspiral, merger and ringdown waveforms. In this case, the waveforms are qualitatively the same as that from BH-BH binaries. For a sufficiently small value of $Q\lesssim 2$, the NS may be tidally disrupted before it is swallowed by the BH. In this case, the amplitude of the merger and ringdown waveforms is very low, and thus, gravitational waves are characterized by the inspiral waveform and subsequent quick damping. The difference in the merger and ringdown waveforms is clearly reflected in the spectrum shape and in the “cutoff” frequency above which the spectrum amplitude steeply decreases. When a NS is not tidally disrupted (e.g., for $Q=5$), kick velocity, induced by asymmetric gravitational-wave emission, agrees approximately with that derived for the merger of BH-BH binaries, whereas for the case when the tidal disruption occurs, the kick velocity is significantly suppressed.

Figure 1

Evolution of orbital separation (a) for model M20.145 and (b) for model M50.145. The solid and dotted curves denote the results with the initial conditions obtained in the 3PN-J and ${\beta }^{\phi }$ conditions, respectively. The dashed curve in panel (a) denotes the result for model M20.145N. To align the curves at the onset of the merger, the time is appropriately shifted for the results of M20.145N, M20.145b, and M50.145b. Note that the merger sets in when the orbital separation becomes $\sim 5M_0$. The binaries for models M20.145 and M50.145 are in the inspiral phase for $\sim 5$ and 7.5 orbits, respectively (see Sec. 4), whereas those for models M20.145b and M50.145b are only for $\sim 4$ and 5.25 orbits, respectively.

Figure 2

Initial conditions in the parameter space of $(\mathcal{C},Q)$ are plotted by the solid circles. The meaning of the dashed curve is as follows [7]: For the binaries located above the dashed curve, mass shedding (MS) does not occur until the binaries reach the ISCO, whereas for the binaries below the dashed curve, the mass shedding will occur for the NS due to the tidal force of the BH before the ISCO is reached.

Figure 3

Coordinate separation between the BH and NS (a) for models M20.145, (b) M40.145, and (c) M50.145. Here, $r_{\mathrm{sep}}=|x_{\mathrm{NS}}^i-x_{\mathrm{BH}}^i|$; see Eq. (21).

Figure 4

Snapshots of the density contour curves and density contrasts as well as the location of the BH in the merger and ringdown phases for model M20.145. The contour curves are plotted for $\rho ={10}^{-i}$ where $i=2,3,4,5$ in the first four panels, whereas in the last two, $\rho ={10}^{-i}$ where $i=3$, 4, and 5. The first panel denotes the state just at the onset of the merger. The filled circles show the region inside the apparent horizon. Note that the maximum value of $\rho$ for the NS in the inspiral phase is $\approx 0.126$.

Figure 5

The same as Fig. 4 but for model M40.145.

Figure 6

The same as Fig. 4 but for model M50.145.

Figure 7

Evolution of the rest mass of the material located outside the apparent horizon (a) for models M15.145, M20.145, M30.145, M40.145, and M50.145 with $N=36$; (b) for model M20.145, M20.145N, and M20.145b with $N=36$; (c) for models M20.145, M20.160, M20.178, M30.145, M30.160, and M30.178 with $N=36$; and (d) for models M15.145, M20.145, M20.160, and M20.178 with $N=36$ (solid curves) and $N=30$ (dotted curves). For a hypothetical value of $M_{\mathrm{NS}}=1.35M_{\odot }$, $100M_0\approx 1.66$, 2.00, 2.66, 3.33, and 3.99 ms for $Q=1.5$, 2, 3, 4, and 5, respectively.

Figure 8

Evolution of $C_p/C_e$, $C_e/4\pi M_0$, and $M_{\mathrm{irr}}/M_0$ as functions of time for models (a) M20.145 and (b) M40.145.

Figure 9

Gravitational waveforms observed along the $z$ axis (solid curve) for models (a) M20.145, (b) M30.145, (c) M40.145, (d) M50.145, (e) M20.160, and (f) M30.178. $t_{\mathrm{ret}}$ denotes the retarded time [see Eq. (31) for definition] and $m_0$ is the total mass defined by $M_{\mathrm{BH}}+M_{\mathrm{NS}}$. The amplitude at a hypothetical distance $D$ can be found from Eq. (32). The dot-dotted curves denote the waveforms derived by the Taylor-T4 formula.

Figure 10

Orbital angular velocity computed from ${\Psi }_4$ as a function of the retarded time for models (a) M20.145, (b) M30.145, (c) M40.145, and (d) M50.145. $t_{\mathrm{ret}}$ denotes the retarded time and $m_0$ is the total mass. For all the panels, the results for two different grid resolutions with $N=30$ (dashed curve) and 36 (solid curve) and the results for M20.145b–M50.145b (dot-dashed curves) are shown together. The dot-dotted curve denotes the result derived by the Taylor-T4 formula.

Figure 11

Gravitational waves (the real part of ${\Psi }_4$) emitted in the merger and ringdown phases for models (a) M30.145, (b) M40.145, (c) M50.145, and (d) M30.145 (dashed curve), M30.160 (dotted curve), and M30.178 (solid curve). For (a)–(c), a fitting waveform given by Eq. (34) is plotted together by the dot-dotted curve.

Figure 12

(a)–(d) The spectrum of gravitational waves $fh(f)D/m_0$ (solid curves) for models (a) M20.145 and M20.160, (b) M30.145, (c) M40.145, and (d) M50.145, respectively. The dashed and long-dashed curves denote the relation according to Eq. (40) and spectra of gravitational waves computed in the Taylor-T4 formula. The dotted curves denote the results of fitting. The upper horizontal and right vertical axes show the value in a hypothetical value of $M_{\mathrm{NS}}=1.35M_{\odot }$ and $D=100\mathrm{Mpc}$. The arrow indicates the frequency of the fundamental quasinormal mode calculated by Eq. (36). (e) The same as (a) but for gathering the spectra for models M20.145–M50.145 (long-dashed, dotted, dashed, and solid curves). To clarify the qualitative feature of the spectra, a smoothing procedure is applied for the numerical data. The three lines in the upper left side denote the predicted frequency at which mass shedding of the NS occurs due to tidal field of the companion BH for models M20.145–M40.145 ($Q$ denotes the mass ratio). (f) The same as (b) but for models M30.145 (dot-dotted curve), M30.160 (dotted curve), and M30.178 (solid curve). A smoothing procedure is also applied for the numerical data.

Figure 13

$f_{\mathrm{cut}}$ as a function of $\mathcal{C}$ for various values of $Q$. The dashed and dotted lines denote the value of $f_{\mathrm{QNM}}$ for $Q=2$ and 3, respectively. For $Q=4$ and 5, $f_{\mathrm{QNM}}$ is slightly smaller than that for $Q=3$, and $\approx 0.078/m_0$ and $0.075/m_0$, respectively.

Figure 14

Kick velocity as a function of $\nu \equiv Q/(1+Q{)}^2$ for different compactness of the NSs. The dot-dotted curve denotes the fitting formula derived in Ref. 40 for the merger of nonspinning BH-BH binaries.