Modeling multipolar gravitational-wave emission from small mass-ratio mergers

Using the effective-one-body (EOB) formalism and a time-domain Teukolsky code, we generate inspiral, merger, and ringdown waveforms in the small-mass-ratio limit. We use EOB inspiral and plunge trajectories to build the Teukolsky-equation source term, and compute full coalescence waveforms for a range of black-hole spins. By comparing EOB waveforms that were recently developed for comparable-mass binary black holes to these Teukolsky waveforms, we improve the EOB model for the (2, 2), (2, 1), (3, 3), and (4, 4) modes. Our results can be used to quickly and accurately extract useful information about merger waves for binaries with spin and should be useful for improving analytic models of such binaries.

Figure 1

Errors in amplitude (left panel) and phase (right panel) due to grid discretization for $a=0$ at mass ratio $\mu /M={10}^{-3}$. These errors are the residual we find following the Richardson extrapolation procedure described in the text.

Figure 2

Errors in amplitude (left panel) and phase (right panel) following extrapolation to infinity for the nonspinning case at mass ratio $\mu /M={10}^{-3}$.

Figure 3

Amplitude of the dominant modes during plunge, merger, and ringdown for $a=0$ (left panel) and $a/M=0.9$ (right panel). We also show orbital frequency, scaled to fit on the plot. As expected, the orbital frequency asymptotes to the horizon’s angular velocity at late times because the frame dragging locks the particle’s motion to that of the horizon. The vertical dashed line in the two panels marks the position of the peak of the orbital frequency.

Figure 4

The evolution of wave frequency for the (2, 2) mode for several spins. Notice that the strong late oscillations (during ringdown), which are particularly prominent as the spin decreases towards negative values; they are especially large for $a/M=-0.9$. These oscillations are physical and arise from superposition of the $(2,\pm 2)$ QNMs.

Figure 5

The EOB equatorial orbit used to produce the Teukolsky waveforms for $a/M=-0.9$, focusing on the transition between the inspiral and the plunge. As shown here, the Kerr black hole rotates clockwise. The red external circle is the ISCO; the blue internal one is the event horizon. The particle initially moves anticlockwise on quasicircular orbits but flips to clockwise motion during the plunge as its angular motion becomes locked to the horizon’s motion.

Figure 6

The amplitude $h$ (top panels) and gravitational-wave frequency $\omega$ (bottom panels) when the (2, 2) (left) and (4, 4) (right) modes reach their peak. Circles at $\nu \ge 0.12$ denote data points extracted from the numerical-relativity simulations; the left-most points at $\nu ={10}^{-3}$ are data extracted with the Teukolsky code. The solid lines are quadratic fits to the data points.

Figure 7

Comparison between (2, 2) and (3, 3) modes generated by the Teukolsky code for $a=0$, and the corresponding modes produced with the EOB model of Ref. 28. The EOB model was calibrated to numerical-relativity simulations at mass ratios 1, $1/2$, $1/3$, $1/4$, $1/6$ and extrapolated to $\nu ={10}^{-3}$. Upper panels show the real part of the modes; lower panels show the phase and fractional amplitude differences.

Figure 8

The same as Fig. 7 but for the (2, 1) and (4, 4) modes.

Figure 9

The same as Fig. 7 but for the Teukolsky-calibrated EOB model described in Sec. 4b.

Figure 10

The same as Fig. 8 but for the Teukolsky-calibrated EOB model described in Sec. 4b.

Figure 11

Comparison of Teukolsky-calibrated EOB and Teukolsky (2, 2) modes for $a/M=-0.5$ (left panel) or $a/M=0.5$ (right panel). Upper panels show the real part of the modes; lower panels show phase and fractional amplitude differences.

Figure 12

The same as Fig. 11 but for $a/M=-0.9$ (left panel) and $a/M=0.7$ (right panel).

Figure 13

Peak amplitude and corresponding frequency of Teukolsky modes as a function of spins. Top panel is the peak amplitude $|h_{\ell m,\mathrm{peak}}^{\mathrm{Teuk}}|$; bottom is the frequency ${\omega }_{\ell m,\mathrm{peak}}^{\mathrm{Teuk}}$ extracted when the amplitude reaches this peak. Data is for the (2, 2), (2, 1), (3, 3), and (4, 4) modes, for spins $a/M=-0.9$, $-0.5$, 0, 0.5, 0.7, and 0.9. Solid lines are quadratic fits to these; the fits are given in Eq. (56).

Figure 14

Comparison between Teukolsky-calibrated EOB and Teukolsky $h_{+}$ and $h_{\times }$ polarizations for $a/M=0$. The four dominant modes (2, 2), (2, 1), (3, 3), and (4, 4) are included.