Binary black hole coalescence in the large-mass-ratio limit: The hyperboloidal layer method and waveforms at null infinity

We compute and analyze the gravitational waveform emitted to future null infinity by a system of two black holes in the large-mass-ratio limit. We consider the transition from the quasiadiabatic inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic-radiation reaction. To compute the waveforms, we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foliation, which coincides with the standard Schwarzschild foliation in the region including the motion of the small black hole, and is globally hyperboloidal, allowing us to include future null infinity in the computational domain by compactification. This method is called the hyperboloidal layer method, and is discussed here for the first time in a study of the gravitational radiation emitted by black hole binaries. We consider binaries characterized by five mass ratios, $\nu ={10}^{-2,-3,-4,-5,-6}$, that are primary targets of space-based or third-generation gravitational wave detectors. We show significative phase differences between finite-radius and null-infinity waveforms. We test, in our context, the reliability of the extrapolation procedure routinely applied to numerical relativity waveforms. We present an updated calculation of the final and maximum gravitational recoil imparted to the merger remnant by the gravitational wave emission, $v_{\mathrm{kick}}^{\mathrm{end}}/(c{\nu }^2)=0.04474\pm 0.00007$ and $v_{\mathrm{kick}}^{\max }/(c{\nu }^2)=0.05248\pm 0.00008$. As a self-consistency test of the method, we show an excellent fractional agreement (even during the plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the gravitational wave angular momentum flux computed at null infinity. New results concerning the radiation emitted from unstable circular orbits are also presented. The high accuracy waveforms computed here could be considered for the construction of template banks or for calibrating analytic models such as the effective-one-body model.

Figure 1

Level sets of the hyperboloidal time $\tau$ as defined by Eqs. (8, 15, 29) with respect to standard Schwarzschild coordinates $\{t,r_{*}\}$. The dashed line at $r_{*}=R_{*}=50$ depicts the location of the interface between the inner domain and the hyperboloidal layer.

Figure 2

Penrose diagram of Schwarzschild spacetime depicting the causal properties of the foliation plotted partially in Fig. 1. The dashed line indicates the interface to the hyperboloidal layer. The time surfaces agree with standard Schwarzschild time surfaces to the left of the interface. The diagram also shows that the foliation stays spacelike everywhere, including the asymptotic domain near null infinity.

Figure 3

Characteristic speeds on the numerical grid as given in Eq. 3. The dashed line denotes the location of the interface between the inner domain and the hyperboloidal layer. The outgoing speed has the same value in the inner domain as in the layer, whereas the incoming speed smoothy approaches zero in the layer.

Figure 4

The structure of the characteristics on the numerical grid. Compare Figs. 1, 2, 3.

Figure 5

Stable circular orbits: fractional difference between the RWZ total energy flux computed with our code and extracted at ${\mathcal{I}}^{+}$ (up to $\ell =8$) and the corresponding semianalytic data computed by Fujita et al. [113].

Figure 6

Newton-normalized total gravitational wave energy flux summed up to $\ell =8$. The analytical (5PN-accurate, EOB-resummed) flux is compared with the numerical points, that include also unstable circular orbits. The vertical dashed line indicates the LSO location at $x=1/6$.

Figure 7

The exact functions ${\rho }_{\ell m}$ extracted from the numerical fluxes for $1/7.9456\le x\le 1/3.1$. The vertical dashed line indicates the LSO location, $x=1/6$.

Figure 8

The $\mathcal{R}{h}_{+}/(M\nu )$ polarization (from Eq. (b1)) of the gravitational waveform for $\nu ={10}^{-3}$. The top panel shows the complete wave train ($\sim 40$ orbits up to merger). The bottom panel focuses around the merger time and illustrates the impact of subdominant multipoles. The vertical dashed line indicates the light-ring crossing time by the point-particle.

Figure 9

Phase difference (left panels) and relative amplitude difference (right panels) between multipoles extracted at ${\mathcal{I}}^{+}$ and at finite radii. Extraction radii are $r_{*}^{\mathrm{extr}}/M=(250,500,1000)$. Data refer to the $\nu ={10}^{-3}$ binary.

Figure 10

Phase difference between the $\mathcal{R}{h}_{+}/(M\nu )$ total gravitational wave polarization at ${\mathcal{I}}^{+}$ and at finite radii. Data refer to the $\nu ={10}^{-3}$ binary.

Figure 11

Fractional amplitude difference (top panels) and dephasing (bottom panel) between ${\mathcal{I}}^{+}$ and extrapolated waveforms. Note that we plot the ${log}_{10}$. Multipoles are $\ell =m=2$ (left panels) and $\ell =2$, $m=1$ (right panels). Extraction radii are $r_{*}/M\simeq (250,500,750,1000)$. Different lines refer to different polynomial order in the extrapolation i.e. $K$ in Eq. (35). The plot refers to the $\nu ={10}^{-3}$ binary.

Figure 12

Residual of amplitude (top) and phase (bottom) between the ${\mathcal{I}}^{+}$ and the extrapolated $\ell =m=2$ waveform. Note that we plot the ${log}_{10}$. Extraction radii are $r_{*}^{\mathrm{extr}}/M=(250,500,750,1000,2000,4000)$. Different lines refer to different polynomial order in the extrapolation i.e. different $K$ in Eq. (35). Data refer to $\nu ={10}^{-3}$ binary.

Figure 13

Late-time comparison between two angular momentum losses for the binary with $\nu ={10}^{-4}$. The GW flux (${\dot{J}}_{\mathrm{GW}}/{\nu }^2$, solid line) computed from the RWZ waveform and extracted at ${\mathcal{I}}^{+}$ (including up to ${\ell }_{\max }=8$ radiation multipoles) is contrasted with the EOB-resummed, analytical mechanical angular momentum loss $-\widehat{\mathcal{F}}/\nu$ (dashed line). The two vertical lines correspond (from left to right) to the particle crossing, respectively, the adiabatic LSO location ($r=6$, $t_{\mathrm{LSO}}=39974.40$), and the light-ring location ($r=3$, $t_{\mathrm{LR}}=40388$).

Figure 14

(Color online). Relative difference between the mechanical angular momentum loss and the GW energy flux for the five mass ratios considered. The figure highlights how a very small fractional difference is maintained also after the LSO crossing.

Figure 15

Convergence of real (top) and imaginary (bottom) part of the $\ell =m=2$ waveform generated by a particle on a circular orbit at $r_0=7.9456$. The plots show the differences between low and medium resolution data and the difference between medium and high resolution data scaled for 4th order convergence.

Figure 16

(Color online) Multipolar waveforms generated by the quasicircular inspiral, plunge, merger and ringdown of the $\nu ={10}^{-2}$ binary initially at separation $r_0=7$. Main panel: the ringdown phase for the $\ell =m=2$ and $\ell =6$, $m=1$ modes. Top-right inset: the complete $\ell =m=2$ waveform. Bottom-left inset: the time-evolution of the $r_{*}$ coordinates of the point particle. The vertical dashed line on the main panel marks the hyperboloidal time ${\tau }_{\mathrm{end}}\approx 762$ corresponding to the dynamical time $t$ where the particle reaches the internal boundary of the numerical grid, ${\rho }_{\min }=-50$.

Figure 17

Same as Fig. 15 but for a particle in a inspiral to plunge orbit.

Figure 18

Convergence of the phase of the $\ell =m=2$ waveform for inspiral to plunge orbit. The plot shows the differences between low and medium resolution data and the difference between medium and high resolution data scaled for 4th order convergence.