# Fundamental Tone and Overtones of Quasinormal Modes in Ringdown Gravitational Waves: A Detailed Study in Black Hole Perturbation

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## Abstract

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## 1. Introduction

## 2. Post-Newtonian Initial Conditions in the Close Limit: Head-On Collisions

#### Model Problem: Head-On Collisions of Two Non-Spinning Black Holes

## 3. Numerical Method

#### 3.1. Initial Data and Gravitational Waveform in Terms of Master Functions

#### 3.2. Time Domain Integration of the Zerilli Equation

#### 3.3. Simulation Parameters

- Case A: an equal mass collision with the initial separation of ${r}_{12}=3.5M$.
- Case B: an asymmetric mass collision with the mass ratio of $\nu =0.2$ and the initial separation of ${r}_{12}=4.4M$.

#### 3.4. Code Validation

## 4. Modeling of Ringdown Waveforms

#### 4.1. Standard Quasinormal-Mode Fitting Formula

#### 4.2. Modified Quasinormal-Mode Fitting Formula with an Orthonormal Set of Mode Functions

## 5. Results

#### 5.1. Quasinormal-Mode Fitting and Residuals

#### 5.2. Convergence of the Fitting Coefficients

## 6. Summary and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analysis of Some Numerical-Relativity Waveforms in Time Domain

**Table A1.**Masses and spins of remnant BHs from NR simulations of binary BHs examined in this Appendix. ${M}_{\mathrm{rem}}/M$ and ${\chi}_{\mathrm{rem}}$ are the nondimensional mass ans spin parameters, respectively. The ‘SXS’ data is presented in the Simulating eXtreme Spacetimes (SXS) catalogue [66], while the ‘RIT’ data is imported from the CCRG@RIT Catalog of Numerical Simulations [67]. Each reference in the table is cited from the corresponding metadata file. SXS:BBH:0305 is used in Section 6, and the remaining NR data are studied in Appendix A and Appendix B.

ID | Mass (${\mathit{M}}_{\mathbf{rem}}/\mathit{M}$) | Spin (${\mathit{\chi}}_{\mathbf{rem}}$) | Reference |
---|---|---|---|

SXS:BBH:0305 | 0.952032939704 | 0.6920851868180025 | [66,124,125] |

SXS:BBH:1936 | 0.9851822160611967 | 0.021659378750190413 | [66,125,126] |

SXS:BBH:0260 | 0.9810057011067479 | 0.12447236057508855 | [66,125,127] |

SXS:BBH:1501 | 0.93633431069 | 0.8085731624240002 | [66,125,128] |

SXS:BBH:1477 | 0.911077401717 | 0.907542632208 | [66,125,128] |

SXS:BBH:0178 | 0.8866898235070239 | 0.9499311295284206 | [66,125,129] |

SXS:BBH:1124 | 0.8827804590335694 | 0.9506671398803149 | [66,125] |

RIT:BBH:0062 | 0.9520211506 | 0.6919694604 | [130,131,132] |

RIT:BBH:0604 | 0.9361520656 | 0.8101416903 | [130,131,132] |

RIT:BBH:0558 | 0.9108618514 | 0.9077062488 | [130,131,132] |

RIT:BBH:0767 | 0.9057246958 | 0.9462438132 | [130,131,132] |

**Figure A2.**Real part of the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=2)$ spheroidal harmonic mode $\Re ({\tilde{h}}^{22})$ of RIT NR waveforms listed in Table A1 and their QNM fit residuals.

## Appendix B. Analysis of the Fitting Coefficients of Some Numerical-Relativity Data

**Figure A3.**The waveforms and fitting coefficients of samples of SXS data listed in Table A1. The plots in the (

**left column**) show the real part of $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=2)$ spheroidal harmonic mode (the blue solid line) and the QNM fit to the model with the orthonormal set given in Equation (37) (the orange dashed line). The plots in the (

**center column**) present the absolute values of the fitting coefficients for ${\tilde{C}}_{k}^{\ell m}$ (the blue filled circles) and ${C}_{k}^{\ell m}$ (the orange filled triangles). The plots in the (

**right column**) show ${\Delta}_{n}$ in Equation (41).

**Figure A4.**The waveforms and fitting coefficients of samples of RIT data listed in Table A1. The plots in the (

**left column**) show the real part of the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=2)$ spheroidal harmonic mode (the blue solid line) and the QNM fit to the model with the orthonormal set given in Equation (37) (the orange dashed line). The plots in the (

**center column**) present the absolute values of the fitting coefficients for ${\tilde{C}}_{k}^{\ell m}$ (the blue filled circles) and ${C}_{k}^{\ell m}$ (the orange filled triangles). The plots in the (

**right column**) show ${\Delta}_{n}$ in Equation (41).

## Appendix C. Frequencies of Kerr Quasinormal Modes

**Figure A5.**Fundamental tone and overtones of Kerr QNMs for $m=\ell $ modes. Here, ${\omega}_{R}$ and ${\omega}_{I}$ mean the real and imaginary parts of the QNM frequency, respectively. We plot the values for $2\le \ell \le 7$ in the range of the nondimensional spin of $0\le \chi \le 0.999$. The filled circles show the QNM values at $\chi =0.95$. For $\chi \le 0.95$, the $n=0$ mode always has the highest real frequency and the slowest damping time. There is an interesting feature in the fifth overtone ($n=5$) of the $m=\ell =2$ mode. To make these plots, we use the QNM data provided in Ref. [103].

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**Figure 1.**(

**Left**) Schematic picture of the numerical domain in the double null coordinates, $(u=t-{r}^{*},\phantom{\rule{0.166667em}{0ex}}v=t+{r}^{*})$. We impose the close-limit, PN initial condition in Equations (13) and (14) on the surface with $t={t}_{i}$ (the bold dashed line) and extract the waveform on the surface with $v={v}_{f}$ (the bold solid line). (

**Right**) A cell in the numerical domain with the size of $h\times h$, where h is the resolution, and C denotes the center of the cell.

**Figure 2.**Absolute differences between the results with different resolutions, $h/M=(0.08,\phantom{\rule{0.166667em}{0ex}}0.04,$$0.02,\phantom{\rule{0.166667em}{0ex}}0.01,\phantom{\rule{0.166667em}{0ex}}0.005)$. For example, ${\Psi}_{h=0.04}-{\Psi}_{h=0.08}$ (the blue solid curve) denotes the absolute differences between $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=0)$ mode of the Zerilli-Moncrief function for case A in the $h/M=0.08$ and $0.04$ simulations. The plot demonstrates the second order convergence of our time domain code. The numerical accuracy with the highest resolution is roughly estimated as ${10}^{-7}$ at the time of the peak amplitude ${u}_{\mathrm{peak}}$, and ${10}^{-13}$ in the late time of $(u-{u}_{\mathrm{peak}})/M\gtrsim 200$.

**Figure 3.**TD data of the Zerilli-Moncrief functions ${\Psi}^{\ell m}$ with the 2PN close-limit initial data for the head-on collision (the blue solid lines) and QNM fitting with $N=7$ in Equation (37) (the orange dashed lines). The TD data is extracted as a function of the retarded time, u, on the $v={v}_{f}=3000M$ surface. The (

**top**panel) shows the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=0)$ mode computed for case A. The (

**bottom**two panel) shows the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=0)$ mode (left) and the $(\ell =3,\phantom{\rule{0.166667em}{0ex}}m=1)$ mode (right) computed for case B.

**Figure 4.**Fit residuals (the blue solid lines) between the Zerilli-Moncrief TD data ${\Psi}_{\mathrm{num}}^{\ell m}$ and the QNM fit including overtones with $N=7$. The grouping of panels are the same as for Figure 3. For reference, we plot $\Delta \Psi $ in Equation (22) with ${h}_{1}=0.01M$ and ${h}_{2}=0.005M$ (the green dotted lines) to roughly estimates the numerical error in the data. We also show the lines proportional to ${(u-{u}_{\mathrm{peak}})}^{-3}$ in the (

**top**) and (

**bottom-left**panel) and to ${(u-{u}_{\mathrm{peak}})}^{-4}$ in the (

**bottom-right**panel) in order to see the late-time behavior (the orange dashed lines).

**Figure 5.**Analysis of mock data of the tail component in Equation (38) (the orange dashed lines). We plot the $(\ell =2)$ and $(\ell =3)$ modes in the (

**left**) and (

**right**) panels, respectively. The green dotted lines show the QNM fit including overtones with $N=7$, and the blue solid lines are the fit residuals. Here, we choose ${D}_{l}=1$, ${u}_{1}={u}_{\mathrm{peak}}+10M$ and ${u}_{2}={u}_{\mathrm{peak}}+30M$ in Equation (38) with Equation (39).

**Figure 6.**(

**Left**) the absolute values of the fitting coefficients, ${\tilde{C}}_{k}^{\ell m}$ (the blue filled circles) and ${C}_{k}^{\ell m}$ (the orange filled triangles) for the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=0)$ mode in case A, as a function of mode k. ${\tilde{C}}_{k}^{\ell m}$ is calculated by Equation (36), while ${C}_{k}^{\ell m}$ is obtained by the least square fit of the TD data to the model in Equation (25). (

**Right**) the estimator ${\Delta}_{n}$ in Equation (41) as a function of overtones n.

**Figure 7.**The same figure as for Figure 6, but using the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=0)$ mode (

**top**) and $(\ell =3,\phantom{\rule{0.166667em}{0ex}}m=1)$ mode (

**bottom**) in case B.

**Figure 8.**(

**Left**) the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=2)$ spheroidal harmonic mode of the NR waveform SXS:BBH:0305 (the blue solid line) and the QNM fit including overtone with orthonormalized mode functions introduced in Equation (37) (the orange dashed line). (

**Right**) the NR data (the blue solid line) and the fit residuals (the orange dashed line).

**Figure 9.**Fitting coefficients of the $(\ell =2,\phantom{\rule{0.166667em}{0ex}}m=2)$ spheroidal harmonic mode of SXS:BBH:0305. The (

**left**panel) displays the absolute values of the fitting coefficients for ${\tilde{C}}_{k}^{\ell m}$ (the blue filled circles) and ${C}_{k}^{\ell m}$ (the orange filled triangles). The plots in the (

**right**column) show ${\Delta}_{n}$ in Equation (41).

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**MDPI and ACS Style**

Sago, N.; Isoyama, S.; Nakano, H.
Fundamental Tone and Overtones of Quasinormal Modes in Ringdown Gravitational Waves: A Detailed Study in Black Hole Perturbation. *Universe* **2021**, *7*, 357.
https://doi.org/10.3390/universe7100357

**AMA Style**

Sago N, Isoyama S, Nakano H.
Fundamental Tone and Overtones of Quasinormal Modes in Ringdown Gravitational Waves: A Detailed Study in Black Hole Perturbation. *Universe*. 2021; 7(10):357.
https://doi.org/10.3390/universe7100357

**Chicago/Turabian Style**

Sago, Norichika, Soichiro Isoyama, and Hiroyuki Nakano.
2021. "Fundamental Tone and Overtones of Quasinormal Modes in Ringdown Gravitational Waves: A Detailed Study in Black Hole Perturbation" *Universe* 7, no. 10: 357.
https://doi.org/10.3390/universe7100357