Binary black hole merger in the extreme-mass-ratio limit: A multipolar analysis

Building up on previous work, we present a new calculation of the gravitational wave emission generated during the transition from quasicircular inspiral to plunge, merger, and ringdown by a binary system of nonspinning black holes, of masses $m_1$ and $m_2$, in the extreme mass ratio limit, $m_1{m}_2\ll (m_1+m_2{)}^2$. The relative dynamics of the system is computed without making any adiabatic approximation by using an effective one body (EOB) description, namely, by representing the binary by an effective particle of mass $\mu =m_1{m}_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M=m_1+m_2$ and submitted to an $\mathcal{O}(\nu )$ 5PN-resummed analytical radiation reaction force, with $\nu =\mu /M$. The gravitational wave emission is calculated via a multipolar Regge-Wheeler-Zerilli-type perturbative approach (valid in the limit $\nu \ll 1$). We consider three mass ratios, $\nu =\{{10}^{-2},{10}^{-3},{10}^{-4}\}$, and we compute the multipolar waveform up to $\ell =8$. We estimate energy and angular momentum losses during the quasiuniversal and quasigeodesic part of the plunge phase and we analyze the structure of the ringdown. We calculate the gravitational recoil, or “kick,” imparted to the merger remnant by the gravitational wave emission and we emphasize the importance of higher multipoles to get a final value of the recoil $v/(c{\nu }^2)=0.0446$. We finally show that there is an excellent fractional agreement ($\sim {10}^{-3}$) (even during the plunge) between the 5PN EOB analytically resummed radiation reaction flux and the numerically computed gravitational wave angular momentum flux. This is a further confirmation of the aptitude of the EOB formalism to accurately model extreme-mass-ratio inspirals, as needed for the future space-based LISA gravitational wave detector.

Figure 1

Transition from quasicircular inspiral orbit to plunge. Initial position is $r_0=7M$ and $\nu ={10}^{-3}$.

Figure 2

Complete $\ell =m=2$ gravitational (Zerilli) waveform corresponding to the dynamics depicted in Fig. 1. The waveform is extracted at $r_{*}^{\mathrm{obs}}/M=1000$.

Figure 3

Multipolar structure of the waveform. The left panels exhibit the moduli; the right panels the instantaneous gravitational wave frequencies for some representative multipoles. Note the oscillation pattern during ringdown (especially in the $\ell =2$, $m=1$ modulus and frequency) due to the interference between positive and negative frequency QNMs. The waveform refer to the $\nu ={10}^{-3}$ mass ratio.

Figure 4

“Convergence” of the waveform when $\nu \to 0$. Retarded times have been shifted so that the zero coincides with the maximum of the waveform modulus $|{\Psi }_{22}^{(\mathrm{e})}|$ for each value of $\nu$. The horizontal dashed line indicates the adiabatic LSO frequency. The vertical dashed line conventionally identifies the beginning of an approximately quasiuniversal and quasigeodesic plunge phase.

Figure 5

Parametric plot of $v_x$ versus $v_y$ (for $\nu ={10}^{-3}$) obtained from Eq. (22) with $v_0/{\nu }^2=(1.549-\mathrm{i}1.0644)\times {10}^{-3}$. The analogous plot with $v_0=0$ is shown in the inset.

Figure 6

Time-evolution of the magnitude of the recoil velocity. The figure shows the monotonic “multipolar” convergence to the final result. The plot refers to mass ratio $\nu ={10}^{-3}$.

Figure 7

Results of the fit of NR data of Refs. 12, 82 using Eq. (23). Note the good agreement between the NR-extrapolation and our test-mass result. The bottom panel contains the relative difference with the data. This plot corresponds to the second row of Table , without the test-mass point. See text for details.

Figure 8

Comparison between two angular momentum losses: the GW flux (solid line) computed à la Regge-Wheeler-Zerilli including up to $\ell =8$ radiation multipoles, and the mechanical angular momentum loss $-{\mathcal{F}}_{\phi }$ (dash line). The two vertical lines correspond (from left to right) to the particle crossing, respectively, the adiabatic LSO location ($r=6M$) and the light-ring location ($r=3M$). The plot refers to $\nu ={10}^{-3}$.

Figure 9

Difference between mechanical angular momentum loss and GW energy flux shown versus twice the orbital frequency $2\Omega$. The vertical line locates the adiabatic LSO frequency. The main panel focuses on the inspiral phase, while the inset shows the full range until the $2{\Omega }_{\max }$, where ${\Omega }_{\max }$ indicates the maximum of orbital frequency.

Figure 10

Self-convergence factor computed from $p$-norms of ($\ell =8$, $m=8$) inspiral plunge waveforms. Resolutions used are: $\Delta r_{*}=\{0.05,0.025,0.0125\}$. The particle is initially at $R(0)=6.5M$ and $\nu =0.01$.

Figure 11

Comparison between the old and the new code. The plot shows the amplitude of the (2, 2) mode of the Regge-Wheeler-Zerilli as computed with the old code (at resolutions $\Delta r_{*}=0.5$ and 0.25) based on Lax-Wendroff scheme and with the new code (at resolution $\Delta r_{*}=0.5$). Other parameters of the runs are $\mu =0.001$, $R(0)=6.5$, and $c_{\mathrm{cfl}}=0.9$.

Figure 12

Waveform emitted by a particle plunging radially into the black hole (along the $z$-axis) from $r=10M$. The waveform is extracted at $r_{*}=1000M$.