Modeling ringdown: Beyond the fundamental quasinormal modes

While black hole perturbation theory predicts a rich quasinormal mode structure, technical challenges have limited the numerical study of excitations to the fundamental, lowest order modes caused by the coalescence of black holes. Here, we present a robust method to identify quasinormal mode excitations beyond the fundamentals within currently available numerical relativity waveforms. In applying this method to waveforms of 68 initially nonspinning black hole binaries, of mass ratios $1:1$ to $15:1$, we find not only the fundamental quasinormal mode amplitudes, but also overtones, and evidence for second order quasinormal modes. We find that the mass-ratio dependence of quasinormal mode excitation is very well modeled by a post-Newtonian-like sum in symmetric mass ratio. Concurrently, we find that the mass-ratio dependence of some quasinormal mode excitations is qualitatively different from their post-Newtonian inspired counterparts, suggesting that the imprints of nonlinear merger are more evident in some modes than in others. We present new fitting formulas for the related quasinormal mode excitations, as well as for remnant black hole spin and mass. We also discuss the relevance of our results in terms of gravitational wave detection and characterization.

Figure 1

Ringdown for a $2:1$ mass-ratio, initially nonspinning BH binary calculated via the GaTech MAYA code [18, 19, 20, 21, 22]. The solid gray lines show the time domain envelope of NR ringdown for two different lines of sight. Here $\theta$ and $\varphi$ are polar and azimuthal angles relative to the remnant BH’s spin vector. The dashed black lines show the corresponding model ringdowns (QNM sums) calculated using the results of this paper: estimation of spheroidal QNM excitations from NR, including and beyond the fundamental overtones.

Figure 2

Remnant BH spin for initially nonspinning systems of varying mass ratio. The black dots are final spin values calculated using the isolated horizon formalism [49]. The trend is monotonic and well fitted with a fourth order polynomial (Appendix pp3).

Figure 3

As demonstrated by this set of $2:1$ mass-ratio nonspinning waveforms, fitting a single decaying sinusoid to ${\psi }_{lm}^{\text{NR}}$ incurs systematic residuals. Top panel: The time-domain envelopes for (2,2),(3,3),(3,2),(4,4) spherical multipoles and related fits, starting $10M$ after the peak luminosity of ${\psi }_{22}^{\text{NR}}$. Bottom panel: The frequency-domain envelopes, $|{\tilde{\psi }}_{l,m}^{\text{NR}}|$. All fits correspond to the lowest, $n=0$, QNMs. While the (2,2) and (3,3) multipole waveforms are best described by a single QNM fit, all fits display visible deviations from the raw data.

Figure 4

Here we see the fundamental QNM excitations estimated by single-mode fitting. Left: The black dots are the excitation amplitudes estimated from fitting. For reference, the dashed grey lines are phenomenological fits from Kamaretsos [36], and the solid red lines are phenomenological fits from the more recent study by Meidam et al. [53]. The error bars were calculated as described in Sec. 3a. The right set of panels shows the related fractional residual errors calculated via Eq. (18).

Figure 5

Time domain comparison of different fitting methods for artificial multimode data.

Figure 6

Top panels: Multimode fitting results for ${\psi }_{22}^{\text{NR}}$. Bottom panels: Multimode fitting results for ${\psi }_{44}^{\text{NR}}$. Left: QNMs recovered, plotted in central frequency and decay time. Each point is labeled with its QNM index in $(\ell ,m,n)$ format. Right: Frequency domain envelopes of component QNMs (color), NR data (grey), and total fit (black). Within each right panel, the shaded region denotes the frequency cutoff. Points in the left panels correspond to curves in the right panels of the same color and QNM label. For reference, we have overlayed the results of the modified Prony method in Fig. 6’s lower left panel.

Figure 7

Fundamentals. The error bars were calculated as described in Sec. 3a.

Figure 8

Examples of phases relative to $m{\varphi }_{22}/2$.

Figure 9

Estimated overtone (top) and second order (bottom) excitation amplitudes via multimode fitting. The error bars were calculated as described in Sec. 3a.

Figure 10

Ratio of inner-products between spherical and spheroidal harmonics estimated via multimode fitting and direct calculation. The error bars were calculated as described in Sec. 3a.

Figure 11

Difference between phase of $(l,m,n)=(2,2,0)$ QNM excitation ($10M$ after the peak luminosity in ${\psi }_{22}^{\text{NR}}$) and the scaled kick direction, $m{\varphi }_{\text{kick}}$ (Sec. 4).

Figure 12

Comparison of the PN strain amplitudes with QNM amplitudes. Top: Amplitude of dimensionless post-Newtonian strain for a selection of $(\ell ,m)$ spherical multipoles. Values were calculated at $M\omega =0.18$ using Ref. [66]. Bottom: Amplitude only fits for fundamental QNM excitations.

Figure 13

Frequency domain envelopes of strain and fitted QNM amplitudes for a $2:1$ mass-ratio system ($\eta =0.22$) of 350 $M_{\odot }$, at a distance of 100 Mpc. Left: Signal for line of sight along final spin direction [e.g. $(\theta ,\varphi )=(0,0)$]. Right: Line of sight $\pi /3\mathrm{rad}$ with respect to final spin direction, $(\theta ,\varphi )=(\pi /3,0)$. Noise curves for the Einstein Telescope and Adv. LIGO are shown for reference. For each panel, the color of each quasinormal mode curve, along with its relative position, labels the mode’s contribution to total signal to noise ratio. In each case, the $(l,m,n)=(2,2,0)$ mode is the most dominant.

Figure 14

The $n=0,1$ and 2 overtones of the $l=m=2$ QNM excitation recovered from NR ringdown if initially nonspinning unequal mass-ratio BH binaries. The error bars were calculated as described in Sec. 3a.

Figure 15

Mean fractional root-mean-square error [Eq. (18)] for the $l=m=2$ multipole with respect to fitting region start time, $T_0$. Here the greedy-OLS (Sec. 3a) algorithm was used to used to perform a multimode fit for each fitting region.