Aspects of multimode Kerr ringdown fitting

A black hole that is ringing down to quiescence emits gravitational radiation of a very specific nature that can inform us of its mass and angular momentum, test the no-hair theorem for black holes, and perhaps even give us additional information about its progenitor system. This paper provides a detailed description of, and investigation into the behavior of, multimode fitting of the ringdown signal provided by numerical simulations. We find that there are at least three well-motivated multimode fitting schemes that can be used. These methods are tested against a specific numerical simulation to allow for comparison to prior work.

Figure 1

The unprimed $\{x,y,z\}$ coordinate frame corresponds to the numerical relativity frame in which the data for ${\psi }_{\mathrm{NR}}$ are extracted. The primed $\{x^{\prime },y^{\prime },z^{\prime }\}$ coordinate frame has its $z$-axis aligned with the angular momentum vector of the black hole. $\overline{\beta }$ represents the angle of the black hole’s angular momentum vector relative to the numerical-relativity $z$-axis.

Figure 2

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22\text{−}\mathcal{A}\}$ and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$. This case uses ${\mathcal{A}}_{\acute{\ell }\ell mn}={\delta }_{\acute{\ell }\ell }$ and can be compared directly with Fig. 1 of GIST.

Figure 3

Comparison of the waveform and fit using case $\{22\text{−}\mathcal{A}\}$. The upper panel shows $h_{+}$ from the numerical-relativity waveform and the $N=7$ version of the fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals. This case uses ${\mathcal{A}}_{\acute{\ell }\ell mn}={\delta }_{\acute{\ell }\ell }$ and can be compared directly with Fig. 2 of GIST.

Figure 4

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for ${\mathrm{\Psi }}_4$ using fitting case $\{22\text{−}\mathcal{A}\}$ and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 5

Comparison of the waveform and fit using case $\{22\text{−}\mathcal{A}\}$. The upper panel shows $\mathrm{Re}[{\mathrm{\Psi }}_4]=-{\ddot{\mathrm{h}}}_{+}$ from the numerical-relativity waveform and the $N=7$ version of the fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 6

This figure plots the same case as Fig. 2 but with SVD tolerance $\tau ={10}^{-16}$.

Figure 7

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 8

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for ${\mathrm{\Psi }}_4$ using fitting case $\{22\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 9

Same as Fig. 7 but with the mismatch recomputed using ${\mathbb{B}}_{\mathrm{ml}}$ as described in the text.

Figure 10

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22+3\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 11

Same as Fig. 10, but with the mismatch recomputed using ${\mathbb{B}}_{\mathrm{ml}}$ as described in the text.

Figure 12

Same as Fig. 10, but fit with the LS method using least-squares fitting based on Eq. (31) instead of using the EV method. As with the EV method, the mismatch is computed using $\mathbb{B}$.

Figure 13

Same as Fig. 12, but with the mismatch recomputed using ${\mathbb{B}}_{\mathrm{ml}}$ as described in the text.

Figure 14

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22+4\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 15

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22,32\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 16

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ using fitting case $\{22,32,42\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 17

Same as Fig. 16, but with the mismatch recomputed using ${\mathbb{B}}_{\mathrm{ml}}$ as described in the text.

Figure 18

Comparison of the $C_{22}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (2, 2) mode of $h_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 19

Comparison of the $C_{32}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (3, 2) mode of $h_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 20

Comparison of the $C_{42}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (4, 2) mode of $h_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 21

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for ${\mathrm{\Psi }}_4$ using fitting case $\{22,32,42\}$, the EV method, and SVD tolerance $\tau =0$. The number $N$ associated with each line denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$.

Figure 22

Comparison of the $C_{22}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (2, 2) mode of $\mathrm{Re}[{\mathrm{\Psi }}_4]=-{\ddot{\mathrm{h}}}_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 23

Comparison of the $C_{32}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (3, 2) mode of $\mathrm{Re}[{\mathrm{\Psi }}_4]=-{\ddot{\mathrm{h}}}_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 24

Comparison of the $C_{42}$ waveform and fit using case $\{22,32,42\}$. The upper panel shows the (4, 2) mode of $\mathrm{Re}[{\mathrm{\Psi }}_4]=-{\ddot{\mathrm{h}}}_{+}$ from the numerical-relativity waveform and the $N=7$ version of its fit with $t_i=t_{\mathrm{peak}}$. The lower panel shows the magnitude of the difference of the two complex signals.

Figure 25

Mismatches as a function of $\delta$ and ${\chi }_f$ at $t_i=t_{\mathrm{peak}}$ for fitting case $\{22,32,42\}$. Fitting was performed using the EV method with an SVD tolerance of $\tau =0$. The number $N$ associated with each panel denotes the maximum value of the overtone index $n$ used in the fitting mode set $\{\mathrm{QNM}\}$. The dotted lines indicate the location of the minimum of the overlap in each panel. The coordinates of the remnant black hole, as obtained directly from the simulation, are (0.692, 0.952).

Figure 26

The same as Fig. 25, but at $t_i-t_{\mathrm{peak}}=20M$.

Figure 27

The same as Fig. 26, but with $\tau ={10}^{-16}$.

Figure 28

The same as Fig. 25, but for fitting case $\{22\}$.

Figure 29

The same as Fig. 28, but with $\mathcal{M}$ computed using ${\mathbb{B}}_{\mathrm{ml}}$.

Figure 30

The same as Fig. 28, but fitting was performed using the mlEV method (same as GIST, or LS method with $\mathcal{M}$ computed using ${\mathbb{B}}_{\mathrm{ml}}$).

Figure 31

The fit value of the mismatch $\mathcal{M}$ as a function of $t_i-t_{\mathrm{peak}}$. Each panel compares the results when using fitting cases $\{22\}$ (dashed red line), $\{22,32\}$ (dotted blue line), and $\{22,32,42\}$ (solid black line). The number $N$ associated with each panel denotes the maximum value of the overtone index $n$ used in each fitting case. The overlaps are computed using the eigenvalue method with SVD tolerance $\tau =0$.

Figure 32

The fit value of the mass ratio $\delta$ as a function of $t_i-t_{\mathrm{peak}}$. Each panel compares the results when using fitting cases $\{22\}$ (dashed red line), $\{22,32\}$ (dotted blue line), and $\{22,32,42\}$ (solid black line). The mass of the remnant black hole obtained directly from the numerical simulation is shown as a horizontal line (solid orange). The overlaps are computed using the eigenvalue method with SVD tolerance $\tau =0$.

Figure 33

The fit value of the dimensionless angular momentum $\chi$ as a function of $t_i-t_{\mathrm{peak}}$. Each panel compares the results when using fitting cases $\{22\}$ (dashed red line), $\{22,32\}$ (dotted blue line), and $\{22,32,42\}$ (solid black line). The number $N$ associated with each panel denotes the maximum value of the overtone index $n$ used in each fitting case. The dimensionless angular momentum of the remnant black hole obtained directly from the numerical simulation is shown as a horizontal line (solid orange). The overlaps are computed using the eigenvalue method with SVD tolerance $\tau =0$.

Figure 34

Same as Fig. 31 but with SVD tolerance $\tau ={10}^{-16}$.

Figure 35

Same as Fig. 32 but with SVD tolerance $\tau ={10}^{-16}$.

Figure 36

Same as Fig. 33 but with SVD tolerance $\tau ={10}^{-16}$.

Figure 37

The deviation of the fit values of the mass ratio $\delta$ and ${\chi }_f$ from ${\mathcal{R}}_{\mathrm{NR}}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $N=7$. The upper two panels display $\mathrm{\Delta }\delta \equiv |0.952033-\delta |$ for the cases of $\tau =0$ and $\tau ={10}^{-16}$. The lower two panels display $\mathrm{\Delta }{\chi }_f\equiv |0.692085-{\chi }_f|$ for the cases of $\tau =0$ and $\tau ={10}^{-16}$. Each panel compares the results when using fitting cases $\{22\}$ (dashed red line), $\{22,32\}$ (dotted blue line), and $\{22,32,42\}$ (solid black line). The overlaps are computed using the eigenvalue method.

Figure 38

The fit of the mismatch $\mathcal{M}$ (top row), mass ratio $\delta$ (middle row), and dimensionless angular momentum $\chi$ (bottom row) as functions of $t_i-t_{\mathrm{peak}}$. Each panel compares the results when using fitting cases $\{22\}$ (dashed red line), $\{22,32\}$ (dotted blue line), and $\{22,32,42\}$ (solid black line). For brevity, we include only examples where all eight overtones ($N=7$) are used for each mode, and SVD tolerance $\tau =0$ in each case. The left column shows the results for the mismatch computed using the EV method, but with the overlap recomputed using Eq. (22) and ${\mathbb{B}}_{\mathrm{ml}}$. The right column shows the results for the mismatch computed using the mlEV method.

Figure 39

Absolute amplitude plots of $|C_{220}^{+}|$ obtained using fitting case $\{22,32,42\}$. Amplitudes displayed in the left column are from fits where the remnant parameters were fixed at $\mathcal{R}={\mathcal{R}}_{\mathrm{NR}}$, while those in the right column had $\mathcal{R}={\mathcal{P}}_m(t_i)$ chosen to minimize the mismatch $\mathcal{M}$ at each time $t_i$. Results in the top row are for $N=7$ but with the SVD tolerance varied. The black line represents $\tau =0$, the dotted blue line represents $\tau ={10}^{-16}$, the dashed red line represents $\tau ={10}^{-12}$, and the orange line represents $\tau ={10}^{-8}$. The lower two rows compare the amplitude as $N$ is varied, but the SVD tolerance if fixed. The middle row uses a linear scale with $\tau ={10}^{-12}$, while the bottom row uses a log scale with $\tau =0$. In these panels, the rightmost line(blue) is $N=0$ and the black line is $N=7$. Error bars from the fit at each $t_i$ are included in all curves but are only discernible on some curves in the middle-left panel.

Figure 40

Relative amplitude plots of a subset $\{{\mathrm{QNM}}_{\mathrm{rob}}\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau =0$ as they contribute to $C_{22}$. The solid lines are continuously fit, while the dashed lines are discretely fit with $t_i=t_{\mathrm{peak}}$ and the dotted lines are discretely fit with $t_i-t_{\mathrm{peak}}=20M$.

Figure 41

Relative amplitude plots of a subset $\{{\mathrm{QNM}}_{\mathrm{rob}}\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau =0$ as they contribute to $C_{32}$. See Fig. 40 for additional detail.

Figure 42

Relative amplitude plots of a subset $\{{\mathrm{QNM}}_{\mathrm{rob}}\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau =0$ as they contribute to $C_{42}$. See Fig. 40 for additional detail.

Figure 43

Relative amplitude plots of a subset $\{{\mathrm{QNM}}_{\mathrm{rob}}\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau ={10}^{-12}$ as they contribute to $C_{22}$. See Fig. 40 for additional detail.

Figure 44

Relative amplitude plots of a subset $\{223+,224+,225+,321+,421+\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau =0$ as they contribute to $C_{22}$. See Fig. 40 for additional detail.

Figure 45

Relative amplitude plots of a subset $\{223+,224+,225+,321+,421+\}$ of the QNMs for fitting case $\{22,32,42\}$ with $N=7$ and SVD tolerance $\tau ={10}^{-12}$ as they contribute to $C_{22}$. See Fig. 40 for additional detail.

Figure 46

Relative amplitude plots for fitting case $\{22,32,42*\}$ with SVD tolerance $\tau ={10}^{-12}$ as they contribute to $C_{22}$. See Fig. 40 for additional detail.

Figure 47

Mismatch $\mathcal{M}$ plotted as a function of $t_i-t_{\mathrm{peak}}$ for $h$ and SVD tolerance $\tau ={10}^{-12}$. The line labeled with an asterisk (${}^{*}$) corresponds to fitting case $\{22,32,42*\}$. The remain two lines correspond to fitting case $\{22,32,42\}$ with $N=7$. The line labeled $N=7!!$ was computed using all mode amplitudes. The line labeled $N=7\#$ was computed using only the five mode amplitudes from case $\{22,32,42*\}$.

Figure 48

Comparison of the $C_{22}$ waveform and various fits using SVD tolerance $\tau ={10}^{-12}$. The upper panel shows the (2, 2) mode of $h_{+}$ from the numerical-relativity waveform and three comparison fits. The fit line labeled $N=7\#$ uses only the five robust modes from the $\{22,32,42\}$ fitting case with $N=7$ at $t_i=t_{\mathrm{peak}}$. The other lines correspond to fitting case $\{22,32,42*\}$. The line labeled $*(0)$ is for the fit at $t_i=t_{\mathrm{peak}}$, while the line labeled $*(16)$ is for the fit at $t_i-t_{\mathrm{peak}}=16M$. The lower panel shows the magnitude of the difference of the complex signals.