High precision ringdown modeling: Multimode fits and BMS frames

Quasinormal mode (QNM) modeling is an invaluable tool for characterizing remnant black holes, studying strong gravity, and testing general relativity. Only recently have QNM studies begun to focus on multimode fitting to numerical relativity strain waveforms. As gravitational wave observatories become even more sensitive they will be able to resolve higher-order modes. Consequently, multimode QNM fits will be critically important, and in turn require a more thorough treatment of the asymptotic frame at ${\mathcal{I}}^{+}$. The first main result of this work is a method for systematically fitting a QNM model containing many modes to a numerical waveform produced using Cauchy-characteristic extraction (CCE), a waveform extraction technique which is known to resolve memory effects. We choose the modes to model based on their power contribution to the residual between numerical and model waveforms. We show that the all-mode strain mismatch improves by a factor of $\sim {10}^5$ when using multimode fitting as opposed to only fitting the $(2,\pm 2,n)$ modes. Our most significant result addresses a critical point that has been overlooked in the QNM literature: the importance of matching the Bondi-van der Burg-Metzner-Sachs (BMS) frame of the numerical waveform to that of the QNM model. We show that by mapping the numerical waveforms—which exhibit the memory effect—to a BMS frame known as the super rest frame, there is an improvement of $\sim {10}^5$ in the all-mode strain mismatch compared to using a strain waveform whose BMS frame is not fixed. Furthermore, we find that by mapping CCE waveforms to the super rest frame, we can obtain all-mode mismatches that are, on average, a factor of $\sim 4$ better than using the publicly available extrapolated waveforms. We illustrate the effectiveness of these modeling enhancements by applying them to families of waveforms produced by numerical relativity and comparing our results to previous QNM studies.

Figure 1

Right-pointing triangles are prograde modes, left-pointing triangles are retrograde. Note that prograde and retrograde modes are present both in the left half-plane (so-called “mirror” modes) and the right half-plane (ordinary modes).

Figure 2

Fraction of unmodeled power that is obtained when comparing a QNM model built from the $(2,\pm 2,0-n)$ mode(s) to a CCE strain waveform as a function of the QNM model start time $u_0$. We compute the fraction of unmodeled power in the waveform using the news waveforms, i.e., by using Eq. (30). This includes the power that is unmodeled because of neglecting higher modes in the QNM model. BBH merger: SXS:BBH:0305.

Figure 3

Fraction of unmodeled power (solid) that is obtained when comparing QNM models built by our greedy algorithm with various number of modes to a CCE strain waveform. Again, the power, which is plotted and used in our algorithm to pick modes to model, is computed using the news waveforms, i.e., by using Eq. (30). The dashed curves are QNM fits using the fixed set of modes determined by the greedy algorithm at the time $u_0-u_{\mathrm{peak}}=20M$. On the other hand, the solid curves have their set of modes determined for each $u_0$ independently, which causes these curves to not be smooth. BBH merger: SXS:BBH:0305.

Figure 4

All-mode mismatches between the CCE waveform and a QNM model fitting $N$ number of modes. The solid curves correspond to QNM models with a varying number of modes that are modeled. The dashed curves are QNM amplitude fits using the modes from $u_0-u_{\mathrm{peak}}=20M$. The dash-dotted curves represent the all-mode mismatches from just the $(2,\pm 2)$ modes—one with the $n=0$ tone and another with the $n=0–7$ tones. Finally, the top of the black region illustrates the mismatch between the highest and the next-highest resolution waveforms to provide a reference for the numerical error that is present in our strain waveform. BBH merger: SXS:BBH:0305.

Figure 5

Various distributions of the epsilons [see Eq. (31)] that have been obtained by minimizing the residual between the strain waveforms from 86 SXS simulations used in the NRHyb3dq8 surrogate and a QNM model that has either been built with the (2, 2, 0) mode (blue), the $(2,2,0-3)$ modes (orange), the $(2,2,0-7)$ modes (green), or the 40 modes that are chosen by the multimode algorithm (red) (see Sec. 3a). On the $\epsilon$-axis, we also provide the median values of epsilon for each distribution. The starting time for the distributions created using the (2, 2) mode and various overtones is taken to be $u_0-u_{\mathrm{peak}}=0M$ while for the distribution created using 40 modes as chosen by the greedy algorithm it is $u_0-u_{\mathrm{peak}}=20M$.

Figure 6

Comparison between the real component of the (2, 0) mode of a CCE waveform and the QNM model built from the (2, 0) mode with $n=0$ and 7 overtones. The upper panels show both waveforms, while the lower panels show the residual between the two. In the plots on the left, we are using a NR waveform in the center-of-mass frame of the remnant BH, while in the plots on the right we have mapped the NR waveform to the super rest frame using the method outlined in Sec. 2 and Appendix A of [33]. In the bottom right plot, we also show a residual curve in red, whose NR waveform has been mapped to the center-of-mass frame of the remnant BH and changed by a constant so that it obtains a final value of zero. We include this curve to illustrate that by performing a supertranslation, rather than changing the strain by a constant, one can obtain much more accurate QNM fits due to the mode-mixing that is induced by supertranslations [36]. BBH merger: SXS:BBH:0305.

Figure 7

The same as Fig. 6, but for the (3, 2) mode. Note that when building this QNM model we have included not only the $(3,2,0-7)$ modes, but also the $(2,2,0-7)$ modes because these modes are needed to accurately represent the (3, 2) mode due to the spherical-spheroidal mixing that occurs when changing the basis of the QNM model.

Figure 8

The same as Fig. 6, but for the (2, 2) mode. We also include the purple curve, which illustrates the previous result obtained by [23, 25] when using an extrapolated waveform which has been changed by a constant so that its final value is zero.

Figure 9

Comparing the mismatch curves for the (2, 2) mode and its overtones obtained from the extrapolated waveform used in [18, 19] (dashed curves) as well as the corresponding CCE waveform, after it was mapped to the super rest frame (solid curves). This plot also serves the purpose of clarifying why Fig. 1 of [18] and Fig. 2 of [19] are different. In [18], they performed an ad hoc subtraction of their waveform, while in [19] no such change to the extrapolated waveform was performed. Note that for this plot we only include the prograde modes in our QNM model to remain consistent with the results of [18, 19]. BBH merger: SXS:BBH:0305.

Figure 10

Examining the mismatch between a NR waveform and a QNM model that is built from 100 modes as a function of the BMS frame that the numerical waveform is mapped to. The QNM model start time $u_0$ is taken to be the time at which the $L^2$ norm of the news takes on its maximum value. We show four bars that correspond to the extrapolated waveform (EXT) and the CCE waveform in three different BMS frames: the arbitrary BMS frame that the output of CCE is in, the remnant BH’s center-of-mass frame, and the super rest frame. In the top plot, we show the mismatch between the strain waveforms, while in the bottom plot we show the mismatch between the news waveforms. The parameters of the 14 binary black holes mergers that appear on the horizontal axis can be found in Table 1.

Figure 11

Examining the fraction of unmodeled power between a NR waveform and a QNM model that n is built from 168 modes as a function of mode for extrapolated (EXT) and CCE waveforms. The QNM model start time $u_0$ is taken to be $u_{\mathrm{peak}}$. We show three bars that correspond to the extrapolated waveform, the CCE waveform in the remnant BH’s center-of-mass frame, and the CCE waveform in super rest frame. In the top plot, we show the fraction of unmodeled power between the strain waveforms, while in the bottom plot we show the fraction of unmodeled power between the news waveforms. The modes are organized in terms of the largest relative difference in the fraction of unmodeled power in the strain domain between the center-of-mass and super rest frame CCE waveforms. BBH merger: SXS:BBH:0305.