Inspiral, merger, and ringdown of unequal mass black hole binaries: A multipolar analysis

We study the inspiral, merger, and ringdown of unequal mass black hole binaries by analyzing a catalogue of numerical simulations for seven different values of the mass ratio (from $q=M_2/M_1=1$ to $q=4$). We compare numerical and post-Newtonian results by projecting the waveforms onto spin-weighted spherical harmonics, characterized by angular indices $(l,m)$. We find that the post-Newtonian equations predict remarkably well the relation between the wave amplitude and the orbital frequency for each $(l,m)$, and that the convergence of the post-Newtonian series to the numerical results is nonmonotonic. To leading order, the total energy emitted in the merger phase scales like ${\eta }^2$ and the spin of the final black hole scales like $\eta$, where $\eta =q/(1+q{)}^2$ is the symmetric mass ratio. We study the multipolar distribution of the radiation, finding that odd-$l$ multipoles are suppressed in the equal mass limit. Higher multipoles carry a larger fraction of the total energy as $q$ increases. We introduce and compare three different definitions for the ringdown starting time. Applying linear-estimation methods (the so-called Prony methods) to the ringdown phase, we find resolution-dependent time variations in the fitted parameters of the final black hole. By cross correlating information from different multipoles, we show that ringdown fits can be used to obtain precise estimates of the mass and spin of the final black hole, which are in remarkable agreement with energy and angular momentum balance calculations.

Figure 1

$|\mathrm{Re}(Mr{\psi }_{l,m})|$ for $q=1.0$ (left) and $q=2.0$ (right). For the equal mass ($q=1.0$) binary the $l=m=3$ component is strongly suppressed, and we do not show it.

Figure 2

$|Mr{\psi }_{l,m}|$ for different mass ratios. Each plot shows only some of the dominant components: $l=m=2$, 3, 4 and ($l=2$, $m=1$). The initial burst of radiation is induced by the initial data, and the wiggles at late times are due to numerical noise.

Figure 3

Convergence plots for $Mr{\psi }_{22}$ (left) and $Mr{\psi }_{44}$ (right). These plots show the differences between runs at different resolutions (as indicated in the inset), scaled to be consistent with second-order accuracy. They refer to run D7, mass ratio $q=4$, and $r_{\mathrm{ext}}=30M$.

Figure 4

Effect of changing the extraction radius on the memory effect when we do not apply corrections to the integration constants. These plots refer to the dominant multipole ($l=m=2$) of run D10 with $q=1.0$.

Figure 5

Effect of changing the extraction radius on the memory effect when we do not apply corrections to the integration constants. These plots refer to a small amplitude mode ($l=m=4$) of run D10 with $q=1.0$. The left panel shows the effect of changing the extraction radius at fixed resolution. In the right panel, we change the resolution at fixed extraction radius.

Figure 6

Ratio of radial and tangential velocities for D8 runs and for two values of the mass ratio ($q=2.0$ and $q=3.0$).

Figure 7

Orbital frequencies from the puncture motion ($\Omega ={\omega }_c$), from the gravitational wave frequency ($\Omega ={\omega }_{Dm}$), and from the PNQC approximation ($\Omega ={\omega }_{\mathrm{PNQC}}$). Here ${\omega }_{\mathrm{PNQC}}$ is computed by inverting Eqs. (3.7a, 3.7b, 3.7c), and keeping only the leading order in the PN expansions on the right-hand side. Times are measured starting from $t_{\mathrm{peak}}$, the peak of the $l=m=2$ mode amplitude for the given mass ratio (see Table ). Horizontal lines mark 2PN and 3PN estimates of the ISCO frequency (as listed in Tables ); the vertical dashed line marks (an estimate of) the CAH formation. The plots refer to runs D8 and two different mass ratios ($q=2.0$ on the left, and $q=3.0$ on the right).

Figure 8

Orbital frequencies from the puncture motion ($\Omega ={\omega }_c$), from the gravitational wave frequency ($\Omega ={\omega }_{Dm}$), and from the PNQC approximation ($\Omega ={\omega }_{\mathrm{PNQC}}$) in the region around the ISCO. Here ${\omega }_{\mathrm{PNQC}}$ is computed inverting Eqs. (3.7a, 3.7b, 3.7c), and keeping only the leading order in the PN expansions on the right-hand side.

Figure 9

Amplitudes obtained by substituting ${\omega }_{Dm}$ into the PNQC equations are compared with the numerical amplitude. All plots refer to runs D8. Figures on the left refer to $q=2.0$, those on the right to $q=3.0$. The top row shows amplitudes for $l=m=2$, the bottom row for $l=m=3$. Vertical lines mark 2PN and 3PN estimates of the ISCO.

Figure 10

Relative deviation between the orbital frequency obtained from the PNQC inspiral formulas and the true frequency of the signal ${\omega }_{D2}$. Plots refer to runs D8 with $q=2.0$ (left), $q=3.0$ (right).

Figure 11

The effect of changing extraction radius (top) and resolution (bottom) on the errors. These plots refer to mass ratio $q=1.0$, run D10. The extraction radii ($r_{\mathrm{ext}}=30$, 40, and 50) and resolutions (HR, MR, LR stands for high, medium, and low resolution, respectively) are indicated in the inset.

Figure 12

Left: total energy $E_{\mathrm{tot}}/M$ radiated in the merger (including the initial data burst) fitted by Eqs. (3.13, 3.14), respectively. Right: numerical data for the energy $E_l/M$ emitted into each multipole $l$ are compared with the fitting functions (3.15). Error bars are estimated by Richardson extrapolation. Notice the suppression of odd multipoles as $q\to 1$.

Figure 13

Angular momentum estimated from a QNM fit and from wave extraction, and corresponding fits. Error bars are estimated by Richardson extrapolation.

Figure 14

Total energy flux computed numerically (thick solid line), and substituting two different estimates of the orbital frequency (${\omega }_{Dm}$ and ${\omega }_c$) into the 2PN and 3.5PN energy fluxes, i.e., keeping different numbers of terms in Eq. (3.21). The 2PN and 3.5PN fluxes are so close they cannot be resolved “by eye” on the scale of this plot. The plots refer to runs D8. On the left we show results for $q=2.0$, on the right for $q=3.0$.

Figure 15

Dominant multipolar components of the energy flux computed numerically (thick solid lines), and substituting two different estimates of the orbital frequency (${\omega }_{Dm}$ and ${\omega }_c$) into the 2PN multipolar contribution to the energy flux, Eq. (3.24). The plots refer to runs D8. On the left we show results for $q=2.0$, on the right for $q=3.0$.

Figure 16

Relative contribution of different multipolar components to the integrated energy flux as a function of time. The plots refer to runs D8. On the left we show results for $q=2.0$, on the right for $q=3.0$.

Figure 17

Total angular momentum flux computed numerically (solid line) and substituting two different estimates of the orbital frequency, ${\omega }_{Dm}$ and ${\omega }_c$, into Eq. (3.25). We overplot the energy flux (multiplied by 50) for run D8 with $q=2.0$, to show that zero crossings of the angular momentum flux correspond (roughly) to local extrema of the energy flux.

Figure 18

Estimate of the angular momentum from a fit of the $l=m=2$ waveform using different methods. Top panels refer to a merger with $q=1.5$, bottom panels to a merger with $q=3.0$. Results on the left were obtained from low-resolution D7 runs, and those on the right from high-resolution D7 runs.

Figure 19

Richardson extrapolation of the estimated angular momentum for $l=m=2$ and $q=1.5$, assuming second-order (thick line) and fourth-order (thin line) convergence.

Figure 20

Angular momentum estimated applying the MP method (thick lines) and the LM method (thin lines) to the $l=m=2$ waveforms. Lines from top to bottom refer to different mass ratios: $q=1.0$, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0.

Figure 21

Consistency in the radiated angular momentum, as predicted by different multipoles, for a $q=1.5$, D7 run. Thick lines use the MP method, thin lines use the LM method. Solid lines are obtained by removing all points for which the absolute amplitude drops below ${10}^{-4}$, and dashed lines by removing all points for which the relative amplitude drops below ${10}^{-3}$ the peak value. The choice of fitting method and truncation criterion has negligible influence on the results. Arrows mark the last point in time when the fit can still be considered reliable, and different multipoles agree with linear perturbation theory of Kerr black holes.

Figure 22

Consistency in the energy radiated, as predicted by different multipoles, for a $q=1.5$, D7 run. Linestyles are the same as in Fig. 21.

Figure 23

Angular momentum estimated applying the method to $l=m=2$ waveforms (thick lines) and to $l=m=3$ ($l=m=4$ in the case $q=1.0$) waveforms (thin lines). Hollow circles mark the “perturbation theory time” for each mass ratio (see text). From top to bottom, different linestyles refer to $q=1.0$, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0, respectively.

Figure 24

Norm (4.3) as a function of the trial starting time for the dominant components $(l,m)=(2,2)$, (3, 3), (4, 4) in a merger with $q=2.0$ (we consider a D7 run here). Thick lines are obtained by fitting the frequency each time we change ${\tau }_0$. Thin lines use the following fixed values for the QNM frequency: $M{\omega }_R=0.51677$, $M{\omega }_I=0.08586$ for $l=m=2$, $M{\omega }_R=0.82210$, $M{\omega }_I=0.08571$ for $l=m=3$, $M{\omega }_R=1.12152$, $M{\omega }_I=0.08577$ for $l=m=4$.

Figure 25

EMOP for $l=2$ computed using run D7 (and high resolution). In the left panel we overplot ${ϵ}_{+}$ and the actual waveform, marking $t_{\mathrm{EMOP}}$ by a vertical dashed line. In the right panel we show that results are quite insensitive to $q$: each line corresponds to a different mass ratio, and linestyles are the same as in Fig. 23.

Figure 26

Energy and angular momentum radiated in the plunge using the 2PN and 3PN definitions of the ISCO are compared against the simple estimate by Blanchet et al. [20] (BQW in the legend). All estimates were computed using the D7 runs (except for the inverse triangles, which refer to D8 runs).

Figure 27

The kick velocity accumulated after plunge using the 2PN and 3PN definitions of the ISCO is compared against the corresponding estimates by Blanchet et al. [20] (BQW in the legend). In the two BQW estimates the integration is truncated at the horizon or at the light ring, respectively.

Figure 28

Top: $|\mathrm{Re}(Mr{\psi }_{22})|$ and $|\mathrm{Im}(Mr{\psi }_{22})|$ for $q=2.0$, D8 (top). Bottom: the degree of elliptic polarization $P_E$ for the $l=2$ waveform as viewed from the normal to the orbital plane.