Small mass plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown waveforms

We numerically solve the Teukolsky equation in the time domain to obtain the gravitational-wave emission of a small mass inspiraling and plunging into the equatorial plane of a Kerr black hole. We account for the dissipation of orbital energy using the Teukolsky frequency-domain gravitational-wave fluxes for circular, equatorial orbits, down to the light-ring. We consider Kerr spins $-0.99\le q\le 0.99$, and compute the inspiral-merger-ringdown (2,2), (2,1), (3,3), (3,2), (4,4), and (5,5) modes. We study the large-spin regime, and find a great simplicity in the merger waveforms, thanks to the extremely circular character of the plunging orbits. We also quantitatively examine the mixing of quasinormal modes during the ringdown, which induces complicated amplitude and frequency modulations in the waveforms. Finally, we explain how the study of small mass-ratio black-hole binaries helps extending effective-one-body models for comparable-mass, spinning black-hole binaries to any mass ratio and spin magnitude.

Figure 1

Numerical discretization errors in the phase (left panel) and amplitude (right panel) of the Teukolsky waveforms for the (2,2), (2,1), (3,3), (3,2), (4,4), and (5,5) modes. The plots are for spin $q=0.9$. A vertical line marks the position of the peak of the orbital frequency, at time $t_{\text{peak}}^{\mathrm{\Omega }}$, which occurs close to merger.

Figure 2

Late inspiral, plunge, merger and ringdown of the Teukolsky $h_{22}^{\text{Teuk}}$ waveform (upper panel), its GW frequency ${\omega }_{22}^{\text{Teuk}}$ and orbital frequency $\mathrm{\Omega }$ of the underlying dynamics (lower panel) for spin $q=0.99$. We note the simplicity of the amplitude during the last phase of the evolution. The plot spans a radial range from $r=2.21M$ to the horizon, located at $r=1.14M$; here, $r_{\mathrm{ISCO}}=1.45M$ and $r_{\mathrm{LR}}=1.17M$. Vertical dashed lines mark the position of the ISCO and the light-ring. A vertical green line marks the position $t_{\text{match}}^{22}=t_{\text{peak}}^{\mathrm{\Omega }}+\mathrm{\Delta }{t}_{\text{peak}}^{22}$ of the ringdown matching as prescribed in the EOB model of Ref. [29] (see discussion in Sec. 6). $R$ is the distance to the source.

Figure 3

Flattening of the peak amplitude of the Teukolsky (2,2) mode as the spin grows towards 1. The curves are normalized by the values of the amplitude at the peak. We align the waveforms in time at $t_{\text{peak}}^{22}$, and plot them as functions of the number of GW cycles from the peak.

Figure 4

Ratio between the radiation-reaction timescale $T_{\text{rad}}$ and the orbital period $T_{\text{orb}}$ as a function of the radial separation for large positive spins. The curves extend up to the peak of the orbital frequency. The solid lines are computed from the numerical integration of the equations of motion. The dashed lines are the analytical predictions for the quasicircular regime, using only quadrupolar emission [Eq. (4)]. Vertical lines mark the position of the respective ISCOs.

Figure 5

(2,2) component of the ingoing $+$ outgoing Teukolsky GW flux. The curves extend down to $r_{\min }=r_{\mathrm{LR}}+0.01M$.

Figure 6

Radial velocity $\dot{r}$ during the plunge. The curves cover the range $r_{+}<r<r_{\mathrm{ISCO}}$.

Figure 7

GW frequencies of Teukolsky (3,2) ringdown waveforms with positive spin. For the common $x$ axis, we use the time elapsed from the orbital frequency peak in units of $2\pi /{\omega }_{320}$. Solid horizontal lines indicate the value $M{\omega }_{320}$, while dashed horizontal lines indicate the value $M{\omega }_{220}$.

Figure 8

Teukolsky (2,2) mode waveforms for spin $q=-0.7$ (left panel) and $-0.99$ (right panel), displaying mode mixing during the ringdown phase. For $q=-0.7$ we plot a ringdown waveform which contains the mode $(2,-2,0)$ besides the usual $(2,2,n)$ ($n=0,1,\dots$). The vertical dashed lines mark $t=t_{\text{match}}^{22}$. Note that the amplitudes have been rescaled by a factor of 5. $R$ is the distance to the source.

Figure 9

Teukolsky (2,1) mode for spin $q=-0.8$ (left panel) and (3,2) mode for spin $q=0.9$ (right panel). For the $q=-0.8$ waveform, the modulations come from the interference of (2,1,0) and $(2,-1,0)$; note how the amplitude peak is affected by the mixing, starting at a time where $\mathrm{\Omega }=0$, i.e., the turning point of the azimuthal motion. For the $q=0.9$ waveform, instead, the ringdown contains the modes $(3,2,n)$’s and (2,2,0).The vertical dashed lines mark $t=t_{\text{match}}^{\ell m}$. $R$ is the distance to the source.

Figure 10

Teukolsky (3,2) waveforms for spin $q=-0.5$ (left panel) and $-0.95$ (right panel). For $q=-0.5$, the ringdown waveform contains the modes $(3,-2,0)$ and (2,2,0) besides the usual $(3,2,n)$’s. For the $q=-0.95$ waveform, the GW frequency modulations in the late ringdown are not centered neither about $-{\omega }_{3-20}$ nor about $-{\omega }_{2-20}$, and we cannot apply the simple model of Eq. (5). The vertical dashed lines mark $t=t_{\text{match}}^{32}$. $R$ is the distance to the source.

Figure 11

For spin 0.5, comparison between Teukolsky (2,2) mode waveform (solid blue lines) and the EOB model of Ref. [22] evaluated in the test-particle limit (dashed red lines). The Teukolsky waveform is evaluated along the EOB trajectory. The waveforms are aligned at their amplitude peak, which corresponds to 0 retarded time; 50 GW cycles before the peak are shown. $R$ is the distance to the source.

Figure 12

Same as Figure 11, but for spin 0.8.

Figure 13

Time delay between the orbital frequency peak and the Teukolsky amplitude peak, defined as $\mathrm{\Delta }{t}_{\text{peak}}^{\ell m}\equiv t_{\text{peak}}^{\ell m}-t_{\text{peak}}^{\mathrm{\Omega }}$. The value of $\mathrm{\Delta }{t}_{\text{peak}}^{22}$ for spin 0.95 is $-103M$, and exceeds the range of the plot.

Figure 14

Amplitude and curvature of the Teukolsky waveforms at their amplitude peak. $R$ is the distance to the source.

Figure 15

Frequency and derivative of the frequency of the Teukolsky waveforms at their amplitude peak.