Quasinormal-mode filters: A new approach to analyze the gravitational-wave ringdown of binary black-hole mergers

We propose two frequency-domain filters to analyze ringdown signals of binary black hole mergers. The first rational filter is constructed based on a set of (arbitrary) quasinormal modes (QNMs) of the remnant black holes, whereas the second full filter comes from the transmissivity of the remnant black holes. The two filters can remove corresponding QNMs from original time-domain ringdowns, while changing early inspiral signals in a trivial way—merely a time and phase shift. After filtering out dominant QNMs we can visualize the existence of various subdominant effects. For example, by applying our filters to a GW150914-like numerical relativity (NR) waveform, we find second-order effects in the $(l=4,m=4),(l=5,m=4)$, and $(l=5,m=5)$ harmonics; the spherical-spheroidal mixing mode in the $(l=2,m=2)$ harmonic; and a mixing mode in the $(l=2,m=1)$ harmonic due to a gravitational recoil. In another NR simulation where two component spins are antialigned with the orbital angular momentum we also find retrograde modes. The filters are sensitive to the remnant properties (i.e., mass and spin) and thus have a potential application to future data analyses and parameter estimations. We also investigate the stability of the full filter. Its connection to the instability of QNM spectra is discussed.

Figure 1

The pole of the original waveform $\tilde{h}(\omega )$ (in blue) and the filtered one ${\tilde{h}}^{\text{filter}}(\omega )$ (in orange). The contour is closed from the upper (lower) plane when $t<t_0$ ($t>t_0$).

Figure 2

The effect of the frequency-domain filter in Eq. (11) on a single QNM signal. The mode is chosen to be the fundamental $(l=2,m=2)$ QNM of a Kerr BH with dimensionless spin 0.69. The signal starts at $t=0$, and it is padded with 0 for $t<0$. After applying the filter, the original signal (its real part is shown as the black curve) is removed from the regime of interest ($t>0$), whereas an undesired “flipped ringdown” is introduced for $t<0$ (red curve). This “flipped ringdown” resembles the original signal, but decays backwards in time.

Figure 3

The impact of the filter ${\mathcal{F}}_{220}$ on the GW emitted by a single, perturbed Schwarzschild BH. In the upper panel, the real part of the filtered waveform (red curve) is compared with the original $h_{22}$ (black curve). Note that here we have undone the time shift induced by the filter by aligning two waveforms in the early regime. In the lower panel, the difference between the two waveforms corresponds to the combination of the “flipped ringdown” and the real ringdown (see the black and red curves in Fig. 2). Its peak (the vertical dashed line) represents the start time of the ringdown.

Figure 4

The physical meaning of ${\mathcal{F}}_{lm}^D$ based on the hybrid approach. The spacetime is split by a timelike world tube ${\mathrm{\Sigma }}_{\text{Shell}}$ (red line) into an inner PN regime II and an outer BHP regime I. During the spacetime reconstruction, we take a waveform from NR at null infinity ${\mathcal{I}}^{+}$, and evolve it backwards into the bulk using BHP theory as if ${\mathrm{\Sigma }}_{\text{Shell}}$ were not there. The result is proportional to the up-mode solution to the homogeneous Teukolsky equation. In particular, an image wave $\tilde{h}{D}_{lm}^{\mathrm{out}}$ needs to appear at the past horizon ${\mathcal{H}}^{-}$, and it is proportional to the filtered waveform. The image wave is spurious since the entire ${\mathcal{H}}^{-}$ lies inside the PN regime II, where the BHP theory does not apply. It exists there as a source to drive the wave in regime I. During the ringdown phase of $\tilde{h}$, the linear QNMs are free ringing of the remnant BH and hence can be annihilated by $D_{lm}^{\mathrm{out}}$, whereas nonlinear pieces are driven by some sources and thus cannot be removed.

Figure 5

The effect of the filter ${\mathcal{F}}_{lmn}$ on $h_{22}$ of SXS:BBH:0305. Here we have aligned the early inspiral portion between the original signal $h_{22}$ (black) and the filtered waveforms. After removing ${\omega }_{220}$ from the original waveform, the oscillation in the ringdown of the filtered waveform (red) is consistent with the QNM ${\omega }_{221}$ (blue). If we further remove ${\omega }_{22,n=1\dots 7}$, the residual shows the existence of the QNM ${\omega }_{320}$ (cyan), which is caused by the spherical-spheroidal mixing. For comparison, we evaluate the numerical error of this waveform (gray) by taking the difference between two adjacent numerical resolutions.

Figure 6

A comparison between the full filter ${\mathcal{F}}_{lm}^D$ [Eq. (22)] and the rational filter ${\mathcal{F}}_{\mathrm{tot}}$ [Eq. (17)] associated with ${\omega }_{22,n=0\dots 7}$. The latter one is more accurate to reveal the existence of the QNM ${\omega }_{320}$ in $h_{22}$ of SXS:BBH:0305. We attribute the inaccuracy of the full filter to the numerical noise when we interpolate the value of $D_{lm}^{\mathrm{out}}$ from the Black Hole Perturbation Toolkit.

Figure 7

Second-order modes in $h_{44}$ (top), $h_{54}$ (bottom left), $h_{55}$ (bottom right) of SXS:BBH:0305. After removing linear QNMs and relevant spherical-spheroidal mixing modes from original waveforms (black curves), filtered waveforms (red curves) contain oscillations that are consistent with the sum tone of $2{\omega }_{220}$ or ${\omega }_{220}+{\omega }_{330}$ (green dashed curves). As for the harmonics $h_{55}$ and $h_{54}$, the comparison is done in the superrest frame to avoid other mixing modes.

Figure 8

Leakage of the ${\omega }_{220}$ mode into the $h_{21}$ harmonic due to the gravitational recoil. After removing ${\omega }_{21,n=0\dots 2}$ and ${\omega }_{31,n=0,1}$ from the original $h_{21}$ waveform (black curve), the red curve exhibits the presence of the ${\omega }_{220}$ mode (yellow dashed curve). If we transform the waveform to the superrest frame (blue curve) and repeat our filtering process, the mixing mode ${\omega }_{220}$ will be completely removed (green curve).

Figure 9

Retrograde mode $-{\omega }_{2-20}^{*}$ in the ringdown of SXS:BBH:1936. Top panel: After removing the ${\omega }_{22,n=0\dots 3}$ modes and the spherical-spheroidal mixing mode ${\omega }_{320}$ from the original harmonic $h_{22}$ (black curve), we reveal the presence of $-{\omega }_{2-20}^{*}$ (green dashed curve) in the residual waveform (red curve). Bottom panel: The phase evolution of the original waveform (black curve) and the filtered waveform (the red curve). The phase of the original waveform decreases monotonically, indicating that the prograde modes are dominant. However, the phase of the filtered waveform starts to grow at the same time as the residual oscillations in the top panel appear, which demonstrates that the residual oscillations are retrograde modes.

Figure 10

Same as Fig. 9, the retrograde mode $-{\omega }_{2-20}^{*}$ in the $h_{22}$ of SXS:BBH:1107.

Figure 11

The up-mode solution of an ECO. We assume that a GW emerges from the horizon $(r_{*}=-\infty )$ and its amplitude is unity. It bounces back and forth within the cavity formed by the ECO surface and the BH potential. The GW seen by an observer at infinity consists of the main transmissive wave $1/D_{lm}^{\mathrm{out}}$ and a series of echoes.

Figure 12

The filter ${\mathcal{F}}_{lm}^{D\mathrm{ECO}}$ of a nonspinning ECO in the time domain. In the top panel, we set $b$ to $200M_f$ (blue) and $300M_f$ (red), while fixing the value of $\epsilon$ to ${10}^{-1}$. They are compared with that of a Schwarzschild BH (black). In the bottom panel, we choose $\epsilon ={10}^{-1},{10}^{-2},{10}^{-3}$ (blue, red, and yellow) and set $b$ to $200M_f$. In both cases, the original signal (around $t\sim 0$) remains unchanged. The perturbation appears as periodic echoes with the time interval $2b$. The amplitude of the $n$th echo is proportional to ${\epsilon }^n$.

Figure 13

Same as Fig. 12. The real part of ${\mathcal{F}}_{22}^D$ in the frequency domain.

Figure 14

The ringdown RSS of the filtered waveform as a function of ${\chi }_f$. The SXS:BBH:0305 waveform is used. The six panels correspond to different choices of the start time, i.e., $t_0$ in Eq. (30). In each panel, different colors indicate the results from removing different numbers of overtones. When $t_0$ is large $(\sim 50M_f)$, the true value of the spin ${\chi }_f^{\mathrm{true}}=0.692$ leads to the smallest RSS no matter how many overtones are removed. However, if we push $t_0$ to an early time, enough overtones need to be removed to obtain the true value. On the other hand, the RSS depends strongly on ${\chi }_f$; a 2% change in ${\chi }_f$ can result in around two orders of magnitude change in the RSS, when $t_0$ and $N$ are fixed to their true values.

Figure 15

Continuation of Fig. 14, except that the onset of the ringdown window $t_0$ is set to $-10M_f$.

Figure 16

Contours of RSS with varying $M_f$ and ${\chi }_f$. To avoid redundancy, we set $t_0$ to 0 and choose $N=2$ (left panels) and $N=7$ (right panels). In the top row we explore the parameter space near the true remnant properties, whereas in the bottom row we investigate a larger area. The true remnant mass and spin are marked with a cross. The effects of $M_f$ and ${\chi }_f$ are degenerate; their difference is more constrained than their sum. In addition, we find there is a second local minimum in Fig. 16.

Figure 17

An explanation for the second local minimum in Fig. 16. The blue dashed line corresponds to the original harmonic $h_{22}$ of SXS:BBH:0305. Using the true remnant properties, the corresponding QNMs are removed (red curve). However, it has a larger amplitude at around 0. This is because adjacent overtones contribute destructively to the original waveform. Fewer QNMs reduce this cancellation and lead to a larger amplitude. On the contrary, using the remnant properties at the second local minimum (black curve), the amplitude of the original waveform diminishes even though the corresponding QNMs are not filtered away. As a result, two systems lead to similar RSS.

Figure 18

Same as Fig. 16, except that the real and imaginary parts of the fundamental mode are used as two independent variables. The start time $t_0$ is set to $50M_f$. Similar to Fig. 16, there is a second local minimum.

Figure 19

Same as Fig. 12, but for a spinning ECO with ${\chi }_f=0.692$.

Figure 20

Same as Fig. 13, but for a spinning ECO with ${\chi }_f=0.692$.

Figure 21

Same as Fig. 14, but with varying $M_f$ and fixed ${\chi }_f$.