# Overtones, mirror modes, and mode-mixing in binary black hole mergers

Arnab Dhani<EMAIL_ADDRESS>Institute for Gravitation and the Cosmos,
Department of Physics, Pennsylvania State University, University Park, PA
16802, USA  B. S. Sathyaprakash  Institute for Gravitation and the Cosmos,
Department of Physics, Pennsylvania State University, University Park, PA
16802, USA  Department of Astronomy & Astrophysics, Pennsylvania State
University, University Park, PA 16802, USA  School of Physics and Astronomy,
Cardiff University, Cardiff, UK, CF24 3AA

###### Abstract

Gravitational waves emitted in the aftermath of a black hole binary
coalescence have characteristic complex frequencies called quasinormal modes
(QNMs). These can be used to test the nature of the merger remnant, e.g. a
test of the black hole ”no-hair” theorem. The relative excitation amplitudes
of the QNMs contain information about the progenitor system and can be
employed to test the consistency between the pre- and post-merger signals. The
post-merger signal from numerical relativity (NR) simulations of binary black
holes exhibit a rich structure that has to be understood to fully exploit the
information contained therein. In this study, we examine the importance of
_overtones_ and _mirror_ modes for a set of mass ratios of non-spinning black
hole binaries and a host of multipole modes corresponding to spherical
harmonic indices $l\leq 4$ and $m\leq 3.$ We find that the first overtone is
most important in the leading mode for a given $m$ and decreases for sub-
dominant modes. The contribution of the fundamental mirror mode is most
significant for $m=1$ and increases with mass ratio, although the model
systematics are not affected significantly by mirror modes. Mode-mixing is the
dominant effect for the sub-leading modes of a given $m$ and minimally affects
the leading mode.

Gravitational waves, black hole, perturbation theory, ringdown

##  I Introduction

Gravitational waves with characteristic complex frequencies, called
quasinormal mode (QNM) frequencies, are a feature of any dynamical horizon
that settles down to an isolated black hole Vishveshwara (1970). The QNMs,
often referred to as _ringdown_ signals, for an isolated Kerr black hole are
purely a function of its mass and spin and do not depend on the details of its
dynamical past. This property is known as the “no-hair” theorem Misner  _et
al._ (1973). The validity of this theorem is an important test for general
relativity (GR) and the nature of the final compact object formed in a black
hole binary merger Dreyer  _et al._ (2004); Berti  _et al._ (2007); Gossan
_et al._ (2012). Any deviation of the measured QNM spectra from its GR
prediction would be a tell-tale signature of modified gravity effects or the
existence of exotic compact objects Cardoso and Pani (2019); Carballo-Rubio
_et al._ (2018); Raposo  _et al._ (2019).

The detection of a single QNM mode together with an inspiral signal allows for
an inspiral-merger-ringdown consistency test of GR and the nature of the
remnant. In this test, the remnant parameters, namely the mass and spin of the
compact object, are independently estimated from the inspiral and the ringdown
signals and checked for consistency Abbott  _et al._ (2016); Ghosh  _et al._
(2018). Nevertheless, a direct test of the “no-hair” theorem involves the
detection of at least two ringdown modes Dreyer  _et al._ (2004); Berti  _et
al._ (2018); Isi and Farr (2021). Such experimental tests of the no-hair
theorem have been carried out using GW150914 Carullo  _et al._ (2019); Isi
_et al._ (2019) and GW190521 Capano  _et al._ (2021). Isi  _et al._ (2019)
report the detection of the first overtone $(l,m,n)=(221)$ in addition to the
dominant $(l,m,n)=(220)$ to constrain the fractional deviation of the overtone
frequency from its GR value to $-0.05\pm 0.2$ while Capano  _et al._ (2021)
uses their secondary detection of the $(l,m,n)=(330)$ mode to constrain the
frequency’s fractional deviation to $-0.01^{+0.07}_{-0.11}$.

Planned upgrades of current detectors LSC (2016); Kuns  _et al._ (2020);
Adhikari  _et al._ (2019) and addition of new ground- Punturo  _et al._
(2010); Abbott  _et al._ (2017); Ackley  _et al._ (2020) and space-based
detectors Amaro-Seoane  _et al._ (2017) would observe multiple higher-order
modes (both overtones and higher angular modes) in addition to the dominant
$(l,m,n)=(2,2,0)$ spherical harmonic mode, allowing us to test the “no-hair”
theorem to exquisite precision. It is, therefore, important to understand the
various features of the different modes in a ringdown waveform beyond the
dominant quadruple mode.

There has been a multitude of studies in the literature that have modeled the
higher spherical harmonic modes of a ringdown waveform Kamaretsos  _et al._
(2012); Baibhav  _et al._ (2018); London (2020); Ota and Chirenti (2020) using
the fundamental QNM. Other studies have added overtones and mirror modes
Giesler  _et al._ (2019); Ota and Chirenti (2020); Bhagwat  _et al._ (2020);
Jiménez Forteza  _et al._ (2020); Hughes  _et al._ (2019); Lim  _et al._
(2019); Dhani (2021); Forteza and Mourier (2021) to model the $l=2$ modes. In
the effective one-body formalism, the ringdown is modeled using a set of QNMs
and pseudo-modesBuonanno  _et al._ (2007); Pan  _et al._ (2014); Taracchini
_et al._ (2014); Babak  _et al._ (2017).

QNMs are the eigenmodes of a perturbed Kerr black hole as the angular part of
the perturbation equations separate out in terms of the _spheroidal_ harmonic
functions Teukolsky (1973); Berti  _et al._ (2006). This is similar in spirit
to how the radial wavefunction of a hydrogen atom is given by the Laguerre
polynomials as the angular part of the Schrodinger equation separates out in
terms of spherical harmonic functions. On the contrary, numerical relativity
(NR) simulations of a binary black hole merger are decomposed in the $-2$
spin-weighted _spherical_ harmonic basis. However, since the latter are not
the angular eigenfunctions of the problem, the radial solution is not given by
QNMs. By virtue of the fact that $-2$ spin-weighted spherical harmonics do
form a complete set of angular functions, the $-2$ spin-weighted spheroidal
harmonics can be decomposed in terms of the former and, hence, the radial
solution is a linear combination of QNMs. This is called mode-mixing Berti and
Klein (2014) because a given spherical harmonic mode is described by a set of
QNMs with different angular indices. Specifically, spheroidal harmonic modes
with different $l$ but same $m$ contribute to a spherical harmonic mode.

Kelly and Baker (2013) looked into the effect of mode-mixing in the $(3,2)$
mode of an unequal mass binary black hole merger. Mehta  _et al._ (2019)
include mode-mixing to construct phenomenological waveform models of non-
spinning binary black holes. In both of these studies, mode-mixing is
considered only for the higher $l$ modes for a given $m$ which are expected to
be affected the most in the case of non-precessing systems. When fitting a set
of NR modes with the same $m$ to a given set of QNMs that would account for
the effect of mode-mixing, the various NR modes should be fit simultaneously
to avoid multiple amplitude estimates for the same QNM. To our knowledge, only
London  _et al._ (2014) in their greedy-OLS algorithm and Cook (2020) in their
multimode modeling of $m=2$ and $l\leq 4$ modes take this into account.

In this work, we model the ringdown waveform of all spherical harmonic modes
up to $l=4$ and $m=3$ for non-spinning binaries of mass ratios
$q\in\\{1.25,5,10\\}$. The salient results of our study are as follows:

  1. 1.

In the leading (or dominant) mode for a given $m$ the correction due to the
first overtone is more important than that due to other modes but the effect
of the overtone decreases for sub-dominant (i.e., $l>m$) modes111Note that
except for $m=1,$ for which the leading mode is $l=2,$ for all other values of
$m$ the leading mode is essentially $l=m.$ We do not consider the $m=0$ modes
in this study..

  2. 2.

The contribution of the fundamental mirror mode is most significant for the
$m=1$ mode and this contribution increases for larger mass ratios becoming the
same order of magnitude as mode-mixing for the sub-leading modes of the set
$m=1$.

  3. 3.

The model systematics (e.g., biases in parameter estimation) are not affected
significantly by mirror modes.

  4. 4.

Most importantly, for a given $m$, mode-mixing is the dominant effect for the
sub-leading modes and minimally affects the leading mode corresponding that
$m$.

The outline of the paper is as follows. In Sec. II, we introduce our ringdown
waveform models and the various approximations in those models. In Sec. III,
we describe the fits of the various ringdown models to NR waveforms and
delineate the features arising from those fits. Sec. IV presents the
systematics of each model, specifically how the parameters of the system will
be affected due to inaccurate modeling. Finally, in Secs. V and VI we conclude
and lay out the directions for future work.

##  II Ringdown waveform model

A dynamical horizon settles down to an isolated Kerr black hole of mass
$M_{f}$ and dimensionless spin $a_{f}$ via the emission of gravitational
waves, which for an asymptotic observer at a distance $r$ takes the form222We
use the sign convention in SXS simulations.

$\begin{split}h(t)=h_{+}-ih_{\times}=&\frac{M_{f}}{r}\sum_{lmn}\Big{[}\mathcal{C}_{lmn}e^{-i\omega_{lmn}t}\prescript{}{-2}{S}_{lm}(\Omega,a_{f}\omega_{lmn})\\\
&+\mathcal{C}_{lmn}^{\prime}e^{-i\omega_{lmn}^{\prime}t}\prescript{}{-2}{S}_{lm}(\Omega,a_{f}\omega_{lmn}^{\prime})\Big{]}\\\
\end{split}$ (1)

where $\prescript{}{-2}{}S_{lm}(\Omega,a_{f}\omega_{lmn})$ is the $-2$ spin-
weighted spheroidal harmonic angular eigenfunction, $\Omega$ is the coordinate
on a sphere $(\theta,\phi)$, and $t$ is the retarded time at null infinity.
The QNMs $\omega_{lmn}$ and $\omega^{\prime}_{lmn}$ are the positive- and
negative-oscillation frequency solutions, respectively, of the radial
Teukolsky equation. The negative oscillation frequency solutions are often
called _mirror_ modes. The azimuthal symmetry of a Kerr spacetime imposes the
following relation between the two sets of QNMs:

$\omega^{\prime}_{lmn}=-\omega^{*}_{l-mn}.$ (2)

In Eq. (1), $\mathcal{C}_{lmn}=\mathcal{A}_{lmn}\,e^{i\phi_{lmn}}$ and
$\mathcal{C}^{\prime}_{lmn}=\mathcal{A}^{\prime}_{lmn}\,e^{i\phi^{\prime}_{lmn}}$
are the complex excitation amplitudes of the regular and mirror modes,
respectively. They depend on the nature of the initial perturbation, or when
the perturbation is produced by the merger of a pair of compact objects, the
parameters and initial configuration of the progenitor binary. The excitation
amplitudes cannot be analytically computed from the initial condition of a
black hole binary. We will, therefore, find fits for them using waveforms
obtained from NR simulations.

The symmetry relation in Eq. (2) can be used to simplify the waveform in Eq.
(1) to

$\begin{split}h(t)=h_{+}-ih_{\times}=&\frac{M_{f}}{r}\sum_{lmn}[\mathcal{C}_{lmn}e^{-i\omega_{lmn}t}\prescript{}{-2}{S}_{lm}(\Omega,a_{f}\omega_{lmn})\\\
&+\mathcal{C}_{lmn}^{\prime}e^{i\omega^{*}_{l-mn}t}\prescript{}{-2}{S}^{*}_{lm}(\Omega^{\prime},-a_{f}\omega_{l-mn}^{*})]\\\
\end{split}$ (3)

where $\Omega^{\prime}=(\pi-\theta,\phi)$.

The waveform, expressed in terms of spheroidal harmonic angular
eigenfunctions, cannot be fit to an NR waveform directly since the latter are
decomposed in terms of $-2$ spin-weighted spherical harmonics
$\prescript{}{-2}{}Y_{lm}(\Omega)$ and can be written as

$h(t)=h_{+}-ih_{\times}=\frac{M_{f}}{r}\sum_{lm}h_{lm}(t)\prescript{}{-2}{}Y_{lm}(\Omega).$
(4)

We will, therefore, expand the spheroidal harmonic angular eigenfunctions in
the orthonormal basis of spherical harmonics as

$\prescript{}{-2}{S}_{lm}(\Omega,a_{f}\omega_{lmn})=\sum_{l^{\prime}}\mu_{lm}^{l^{\prime}}(a_{f}\omega_{lmn})\prescript{}{-2}{}Y_{l^{\prime}m}(\Omega),$
(5)

where $\mu_{lm}^{l^{\prime}}(a_{f}\omega_{lmn})$ is called the spherical-
spheroidal mixing coefficient. An NR spherical harmonic mode of the form

$\begin{split}h_{l^{\prime}m}&=\sum_{ln}\left[\mathcal{C}_{lmn}\,\mu_{lm}^{l^{\prime}}\,(a_{f}\omega_{lmn})\,e^{-i\omega_{lmn}t}\right.\\\
&\left.+(-1)^{l+l^{\prime}}\mathcal{C}_{lmn}^{\prime}\,\mu_{l-m}^{*l^{\prime}}\,(a_{f}\omega_{l-mn})\,e^{i\omega_{l-mn}^{*}t}\right]\end{split}$
(6)

can then be arrived at by substituting Eq. (5) into Eq. (3) and using the
symmetry and orthogonality properties of $-2$ spin-weighted spherical
harmonics333For some of the intermediate steps, see Dhani (2021).. Given the
mass and spin of the final black hole, we can now fit for the complex
excitation amplitudes to NR waveform.

We have to now choose a set of QNMs that we want to fit to an NR mode
$(l^{\prime},m)$. In this study, we take that set to comprise of the
fundamental mode $(l,m,0)$, the fundamental mirror mode $(l,-m,0)$, and the
first overtone $(l,m,1)$. Furthermore, in order to determine the importance of
each of these QNMs to the different spherical harmonic modes and physical
systems, we define the following ringdown models: (i) Model ‘F’ is a ringdown
model that contains only the fundamental QNM, (ii) model ‘FM’ contains both
the fundamental mode and the fundamental mirror mode, (iii) model ‘FO’
contains the first overtone together with the fundamental mode, and (iv) model
‘FMO’ contains all the three QNMs. The various models are enumerated in Table.
1.

Model Name | Mode Content
---|---
F | Fundamental mode
FM | Fundamental mode + mirror mode
FO | Fundamental mode + overtone
FMO | Fundamental mode + mirror mode + overtone
Table 1: The different ringdown models defined in this study and their
symbols. We build four different models with varied mode content. Starting
from the simplest model with just the fundamental mode, FM, FO, and FMO
represent progressively greater mode content.

A spherical harmonic mode $(l^{\prime},m)$ has contributions not only from the
corresponding spheroidal harmonic quantum number $l=l^{\prime}$ but from all
$l$ for a given $m$, though the dominant contribution of mode-mixing is from
$l=l^{\prime}$ as can be seen from Berti and Klein (2014). Consequently, we do
a simultaneous fit of all the modes with the same $m$. For the modes
considered in this paper ($\forall l\leq 4,m\leq 3$), this implies a
simultaneous fit of the following three sets of NR modes:

  * •

$(2,2),(3,2),(4,2)$

  * •

$(2,1),(3,1),(4,1)$

  * •

$(3,3),(4,3)$.

The inclusion of mode-mixing entails that, for the ‘FMO’ model, the first two
sets that contain three modes are fit to $3\times 3=9$ QNMs and the last set
containing two modes is fit to $3\times 2=6$ QNMs. Additionally, we want to
ascertain the impact of mode-mixing to the different modes under
consideration. For that, we establish two additional classes of models, the
first of which would include the effect of mode-mixing in the above defined
models and the second class containing our four models would disregard mode-
mixing in the ringdown waveform. The latter involves setting
$\mu_{lmn}^{l^{\prime}}=\delta_{l}^{l^{\prime}}$ in Eq. (6). In this case, the
general ringdown model simplifies to

$h_{l^{\prime}m}=\sum_{n}\left[\mathcal{C}_{l^{\prime}mn}e^{-i\omega_{lmn}t}+\mathcal{C}_{lmn}^{\prime}e^{i\omega_{l-mn}^{*}t}\right].$
(7)

##  III Waveform fits

We consider three non-spinning black hole binaries from the publicly available
SXS444Simulating eXtreme SpacetimesSXS Collaboration  catalog of NR
simulations corresponding to three different mass ratios. The SXS IDs, mass
ratios, final masses, and dimensionless final spins of the simulations are
give in Table 2. We have chosen three simulations corresponding roughly to
mass ratios 10:8, 5:1, and 10:1. For the sake of convenience we call them
S10:8, S10:2, and S10:1, respectively. The chosen systems span the range of
mass ratios of interest to ground-based detectors. We set the complex QNM
frequencies to their GR values and fit the complex amplitudes in each of the
ringdown models introduced in the previous section to the NR waveform via a
linear least-squares method. To calculate the goodness of the fit, we define
the mismatch between the fit and the NR waveform as

$\mathcal{M}=1-\frac{\langle h_{\rm NR}|h_{\rm fit}\rangle}{\sqrt{\langle
h_{\rm NR}|h_{\rm NR}\rangle\langle h_{\rm fit}|h_{\rm fit}\rangle}}$ (8)

where the inner product is defined as

$\langle a(t)|b(t)\rangle=\int_{t_{0}}^{t}dt^{\prime}\oint d\Omega
a^{*}(t^{\prime},\Omega)b(t^{\prime},\Omega).$ (9)

Since we fit modes with the same $m$ simultaneously, the combined mismatch for
a given set of modes takes the form

$\mathcal{M}_{m}=1-\frac{\sum_{l}\langle h^{lm}_{\rm NR}|h^{lm}_{\rm
fit}\rangle}{\sqrt{\sum_{l}\langle h^{lm}_{\rm NR}|h^{lm}_{\rm
NR}\rangle\sum_{l}\langle h^{lm}_{\rm fit}|h^{lm}_{\rm fit}\rangle}}.$ (10)

Additionally, to measure the effect of the individual modes, we define

$\mathcal{M}_{lm}=1-\frac{\langle h^{lm}_{\rm NR}|h^{lm}_{\rm
fit}\rangle}{\sqrt{\langle h^{lm}_{\rm NR}|h^{lm}_{\rm NR}\rangle\langle
h^{lm}_{\rm fit}|h^{lm}_{\rm fit}\rangle}}$ (11)

for the individual modes of the same fits.

We vary the start time of the ringdown waveform between $t_{0}=t_{\rm peak}$
and $t_{0}=t_{\rm peak}+50M,$ where the reference time $t_{\rm peak}$ is
defined as the peak of the $h_{22}$ mode. The stop time is taken to be
$t=t_{\rm peak}+90M$.

SXS ID | Proxy | $q$ | $M_{f}$ | $a_{f}$
---|---|---|---|---
SXS:1143 | S10:8 | 1.250 | 0.9528 | 0.6795
SXS:0187 | S10:2 | 5.039 | 0.9825 | 0.4148
SXS:1107 | S10:1 | 10.00 | 0.9917 | 0.2605
Table 2: The SXS IDs, proxies, mass ratios, final masses, and dimensionless
spins of the binary black hole simulations considered in this study from the
publicly available catalog of SXS simulations.

In the rest of the section, we will discuss the different effects and QNMs
that affect the mismatch curves for the ringdown signal of an NR waveform.
Specifically, we will discuss the effects of mode-mixing, overtones, and
mirror modes to the various sets of modes considered. The mismatch curves as a
function of the ringdown start time $t_{0}$ for the four different models and
the two classes of models for the set of modes with $m=2,1,3,$ and the three
simulations considered, are given in Fig. 1, Fig.2 and Fig. 3, respectively.
In Appendix. A, we plot the mode amplitudes $|h_{lm}|$ and mode frequencies
$f_{lm}=-\Im(\dot{h}_{lm}/h_{lm})$ as a function of time $t-t_{\rm peak}$. We
additionally show our four best-fit ringdown models that include mode-mixing
in Fig. A1 for the mode amplitudes and Figs. A2, Fig. A3, and Fig. A4 for the
mode frequencies. The ringdown start time is taken to be $t_{0}-t_{\rm
peak}=20M$ for these plots.

Figure 1: Comparison of the mismatches $\mathcal{M}$ corresponding to the
$m=2$ mode, for the four models enumerated in Table 1 and three mass ratios
$q=1.25$ (S10:8, top row), $q=5$ (S10:2, middle row) and $q=10$ (S10:1, bottom
row), considered in this study as a function of the ringdown start time
$t_{0}$. The reference time for $t_{0}$ is taken to be the peak of the
$h_{22}$ mode. The green curves denote the class of models that include mode-
mixing while mode-mixing is ignored for the orange curves. The left most panel
in each row shows the mismatch $\mathcal{M}_{m}$ for the combined set of three
modes with the same $m$ given in Eq. (10), the other panels of the row show
the mismatches $\mathcal{M}_{lm}$ given in Eq. (11) and depict the
contribution of the individual modes to the total mismatch. In all cases the
fits were obtained simultaneously for all the modes. The red, dotted line
shows the mismatch between the two highest resolution NR simulations and is a
proxy for the NR noise floor. We note that the mirror modes do not improve the
mismatch and, hence, the curves for ‘F’ and ‘FM’ models essentially overlay on
top of each other as do ‘FO’ and ‘FMO’ models. On the contrary, the overtone
model essentially captures most of the features in the NR waveform and closely
follow the ‘FMO’ model. See text for other details.

###  III.1 Mode-mixing

We describe the mixing of various spheroidal to spherical modes described
above in this subsection.

The effect of mode-mixing in a given spherical harmonic mode is determined by
two independent factors: (a) the relative excitation amplitudes of the QNMs
with the same $m$ and (b) the magnitude of the mixing-coefficients between
them. For the physical systems considered (non-spinning binaries), the mode
with a smaller $l$ for a given $m$ is more strongly excited and, therefore,
the corresponding $l^{\prime}$ is affected less by mode-mixing from a larger
$l$. This is accentuated by the fact that the mixing coefficient is close to
unity for $l=l^{\prime}$ and at most a few percent otherwise Berti and Klein
(2014). Let’s consider the set $\\{(2,2),\,(3,2)\\}$ as an example and for
simplicity of argument consider the ringdown model ‘F’. Our fits show that
$\mathcal{C}_{220}$ is more strongly excited than $\mathcal{C}_{320}.$ This
can be inferred from the fact that for the $(3,2)$ mode (the bottom left sub-
figure for each mass ratio), the green curves (which include mode-mixing) show
considerably lower mismatches over orange curves (which don’t). On the
contrary, for the $(2,2)$ mode orange curves show similar mismatches to green
curves. Both of these excitation amplitudes contribute to the spherical
harmonic modes $(2,2)$ and $(3,2)$ with their relative contribution governed
by their mixing-coefficients as can be seen from Eq. (6).

The coefficients with $l^{\prime}=l$, $\mu^{2}_{220}$ and $\mu^{3}_{320}$, are
close to unity while the ones with $l^{\prime}\neq l$, $\mu^{2}_{320}$ and
$\mu^{3}_{220}$, are a few percent. Consequently, the $(2,2)$ mode doesn’t
have a considerable effect due to mode mixing but the effects of the same on
$(3,2)$ can be easily seen in the mismatch curves in Fig. 1, the mode
amplitudes $|h_{lm}|$ in Fig. A1, and the mode frequencies in Fig. A2. Similar
arguments follow for the mode $(4,2),$ which is affected by both of the
stronger modes, as can be seen in the two distinct modulations of the mode
amplitudes and mode frequencies in Fig. A1 and Fig. A2, respectively, for mass
ratio $q=1.25$. For the other mass ratios, the subdominant modulation due to
$(3,2)$ is not visible. On the other hand $(3,2)$ is modulated only by the
frequency of the sole stronger mode. In the absence of mode-mixing, the
$(3,2)$ and $(4,2)$ modes would be described by a single damped sinusoid for
the ‘F’ model and these are shown as a straight line whose slope corresponds
to the damping time and a horizontal line corresponding to the oscillation
frequency in the mode amplitude and mode frequency plots, respectively.

Figure 2: Same as Fig. 1 but for $m=1$ modes. The left most panels show that a
mere overtone (FO) or a mirror model (FM) is not enough to obtain the best fit
model. The second column of panels show that mode-mixing is unimportant for in
the leading mode (2,1), while the third and fourth columns show that sub-
leading modes (3,1) and (4,1) are significantly affected by mode-mixing.

The effect of mode-mixing is similar for all the sets of modes considered in
this study. The modes with the smallest $l$ for a given $m$ are least affected
by mode-mixing. The mode amplitudes for $m=1$ and $m=3$ are given in A1 and
the mode frequencies for the same modes are given in Figs. A3 and A4,
respectively.

We additionally observe from the mode amplitude and mode frequency plots that
the mass ratio least affected by mode-mixing is the largest mass ratio we
consider, $q=10$. As one goes to larger mass ratios, the final black hole
tends towards a Schwarzschild black hole, because we started with an initially
non-spinning binary, and, hence, the angular eigenfunctions tend towards $-2$
spin-weighted spherical harmonics making the NR modes the eigenmodes of the
system. Note that we stave off making a similar statement comparing the
smaller two mass ratios because even though the mixing coefficients are
monotonically decreasing between the final spins of the $q=1.25$ and $q=5$
binary, the relative excitation amplitudes are not and, therefore, the
contribution of mode-mixing is maximum for some mass ratio between them.

Figure 3: Same as Fig. 3 but for $m=3$. Since $l\geq m,$ there are only two
columns showing ${\cal M}_{lm},$ in addition to the left most column showing
${\cal M}_{m},$ for each mass ratio, in contrast to three columns for $m=2,1$
in Figs. 1 and 2. As in the case of $m=2,$ here too we see that the
contribution of the mirror modes is sub-dominant compared to the overtones.
This can be seen from the fact that the ‘FO’ (dot-dashed curves) and ‘FMO’
(solid curves) models overlay on top of each other as do ‘F’ (dotted curves)
and ‘FM’ models (dashed curves). The middle and right columns once again shows
that mode-mixing is less of an issue for the leading (3,3) mode but is
significant for the sub-leading (4,3) mode, respectively.

###  III.2 Overtones

We next turn to the effect of overtones (dot-dashed curves) and look at the
$m=2$ modes first. Some features of the mismatch curves are qualitatively
similar as a function of the mass ratio. We observe that adding the first
overtone improves the match of the fit waveform with the NR waveform for early
times. Indeed, the dot-dashed curves closely follow the solid curves
corresponding to the FMO model for all mass ratios for $m=2.$ Our results are
consistent with previous studies which either only considered the
$(l,m)=(2,2)$ mode Giesler  _et al._ (2019); Bhagwat  _et al._ (2020) or
included higher modes but only showed the combined mismatch $\mathcal{M}_{m}$
Cook (2020). Although the improvement is most prominent for the $(2,2)$ mode,
we report similar conclusions for the individual modes $(3,2)$ and $(4,2)$ of
this set for the mode-mixing class of models. We note, however, that the level
of improvement with the addition of the first overtone decreases for higher
angular modes.

If mode-mixing is ignored, we see that adding the first overtone doesn’t
improve the match much for the $(3,2)$ and $(4,2)$ modes, presumably because
mode-mixing is the dominant effect over overtones even at early times. We
further note that the higher modes of smaller mass ratio systems perform
poorer than larger mass ratios. This is most stark in the $m=1$ modes where
the first overtone of the $(4,1)$ mode barely improves the mismatch especially
for $q=1.25$. The match of the NR waveform with the leading $(2,1)$ mode of
this set sees a huge improvement with the addition of the first overtone
dominating the combined mismatch curves. The $(4,3)$ mode responds much better
to the first overtone with significant improvement over excluding the QNM but
still the leading mode of the set $(3,3)$ governs the combined mismatch
curves.

The performance of the overtone for the higher angular modes of a set lead us
to caution against their use for spectroscopic tests.

###  III.3 Mirror modes

For the modes of the sets $m=2$ and $m=3$, we see that adding the fundamental
mirror mode does not improve the mismatch across mass ratios $q=1$ to $q=10.$
This is seen in Figs. 1 and 3 where the mismatch curves for the ‘F’ and ‘FM’
models and that of ‘FO’ and ‘FMO’ models, respectively, essentially coincide
with each other and is again consistent with the results of Jiménez Forteza
_et al._ (2020) and Dhani (2021) but where too only the $(2,2)$ mode was
considered and the fundamental mirror mode didn’t improve the mismatch
significantly.

For the $m=2$ set of modes the effect of adding mirror modes to a ringdown
model can be clearly seen from Fig. A2 where we plot the mode frequencies as a
function of time $t-t_{\rm peak}$. In the left panels, the plot for the ‘F’
model is represented by a horizontal line corresponding to the oscillation
frequency of the fundamental QNM (this is true for the $(2,2)$ mode because
mode-mixing is negligible in this case) while the ‘FM’ model shows
characteristic modulations on top of the fundamental mode frequency
corresponding to the mirror mode oscillation frequency. For the $(2,2)$ mode,
across mass ratios, the mirror modes are not visible to the eye in the
numerical waveform and the corresponding features in the best-fit model are a
result of over-fitting. Particularly, for the $(2,2)$ mode, notice that the
mirror mode modulations vanish for the ‘FMO’ model though they are visible for
the ‘FM’ model. We believe this is because the fitting procedure tries to fit
some of the overtone features at early times using the mirror mode QNM. In
fact, in the case of $m=2$ the high-frequency mirror mode modulations first
become visible for mass ratio $q=10$ and the $(4,2)$ mode, less so for the
$(3,2)$ mode. However, these features are not picked up in the mismatch plot
in Fig. 1, which shows the same mismatch for the models ‘FO’ and ‘FMO’ for the
said mode(s).

For the $m=3$ modes, even though mirror modes do not seem to play an important
role, we cannot quantify this due to the presence of spurious oscillations in
the $(3,3)$ mode. These are easily visible in the mode frequency plot, Fig.
A4. The ‘FM’ model shows a beat pattern which is dissimilar to the typical
mirror mode modulations that we have seen. We are not able to assign any
physical reasoning to these oscillations. These features are though not
present in the mismatch curves for this set.

We find that mirror modes are most strongly excited for $m=1$ out of the three
sets we consider in this study. Furthermore, the excitation amplitudes of the
mirror modes increase with mass ratio $q$ as can be readily seen from the mode
frequency plot Fig. A3, where the high frequency modulations on top of the
fundamental QNM is due to the mirror modes while the low frequency ones are
due to mode mixing. We further see that the excitation amplitudes weighed by
the mixing coefficients, and therefore, the effective contribution of a QNM to
an NR mode, are the same order of magnitude for the mirror mode and the mode-
mixing mode for the $(3,1)$ and $(4,1)$ modes of the mass ratio $q=10$ case.

##  IV Final mass and spin: ringdown model systematics

Until now, we have fixed the final mass and spin of the black hole and fit for
the complex excitation amplitudes. In this section, we look at the systematic
uncertainties in each of our models arising from their estimation of the
remnant parameters. For this, we will vary the remnant mass an spin on a grid
and calculate the mismatch across the grid. We will then associate the mass-
spin pair that minimizes the mismatch on this grid as the estimated remnant
parameters for our model. We will then estimate the systematic uncertainty of
a model by the deviation of the best-fit remnant parameters from the values
determined by NR. We quantify this deviation as the Euclidean distance in the
remnant parameter space given by

$\epsilon=\sqrt{\left(\frac{\delta M_{f}}{M}\right)^{2}+\left(\delta
a_{f}\right)^{2}}$ (12)

where $\delta M_{f}$ and $\delta a_{f}$ are the differences between the best-
fit values and the NR values. Such a measure will help us in distinguishing
models with the same or a similar mismatch. For example, as we discussed in
the previous section, the ‘FM’ model for $m=2$ gives rise to spurious mirror
mode modulations that cannot be seen in the NR waveform. Even so, it produces
the same mismatch as the ‘F’ model. These spurious modulations have no
physical reason to produce the minimum mismatch for the true remnant values
and as such we expect the systematic uncertainties for such models to be
larger.

In the following, we will depict the mismatches on a grid of remnant
parameters for a ringdown start time of $t_{0}-t_{\rm peak}=20M$. We choose
this time because it is approximately the latest time when the mismatch curves
bottom out for the models with the best mismatches. We will later look at the
effect of the start time on the model systematics by calculating $\epsilon$
for different choices of start times.

###  IV.1 Modes: (2,2), (3,2), (4,2)

We will look at the $m=2$ modes first shown in Figs. 4, 5, and 6,
corresponding to mass ratios $q=1.25$, $q=5$, $q=10$, respectively. Each pair
of rows from the top depict the angular modes $l=2$, $l=3$, and $l=4$,
respectively; the top panels of each pair show the class of models that
include the effect of mode mixing while the bottom panels take no mode-mixing
into account.

For all mass ratios, the most important effect that stands out is mode-mixing.
In line with our earlier observation that mode-mixing does not improve the
mismatch for the $l=2$ mode of this set, we find that it does not notably
affect the remnant parameter estimates either across all mass ratios. This is
not the case for the higher $l$ modes of this set, which are significantly
affected by mode mixing. We additionally note that the class of models with
mode-mixing produces a deeper mismatch minima, which would assist in more
accurate parameter estimation.

The impact of the first overtone to model the systematics is detailed in Fig.
B1, where one can see that the best-fit remnant parameters for the ‘FO’ model
is closer to the true values determined by the NR waveform than both the ‘F’
and ‘FM’ models for $l=2,$ which shows the greatest improvement in mismatch.
The case for higher $l$ values is more nuanced, with clear improvement in the
remnant parameter estimate for the smallest mass ratio $q=1.25$ and for start
times before $t_{0}-t_{\rm peak}=20M$. For the two larger mass ratios, the
first overtone does not affect $\epsilon$ much. The ‘FMO’ model closely
follows the ‘FO’ model reinforcing that the fundamental mirror mode does not
play an important role for $m=2$.

(a)

(b)

(c)

Figure 4: Mismatch $\mathcal{M}_{lm}$ for the best-fit ringdown model on a
grid of $M_{f}$–$a_{f}$ for a ringdown start time of $t_{0}-t_{\rm peak}=20M$.
The white cross hairs show the value of the remnant determined by NR and the
orange crosses show the best-fit value. The different sub-figures correspond
to the angular modes of the set $m=2$, namely, $l=2$, $l=3$, and $l=4$,
respectively. The columns plot the mismatches on the remnant parameter space
for the four different models considered in this study (see Table. 1). The top
and bottom rows of each sub-figure plot the mode-mixing and the non-mode-
mixing classes of models, respectively. The scale for the color bars are the
same for each sub-figure. These set of plots correspond to the simulation
S10:8.

(a)

(b)

(c)

Figure 5: Same as Fig. 4 but for the simulation S10:2.

(a)

(b)

(c)

Figure 6: Same as Fig. 4 but for the simulation S10:1.

###  IV.2 Modes: (2,1), (3,1), (4,1)

(a)

(b)

(c)

Figure 7: This figure plots the mismatch $\mathcal{M}_{lm}$ in the remnant
parameter space for a ringdown start time of $t_{0}-t_{\rm peak}=20M$ as in
Fig. 4 but for $m=1$.

(a)

(b)

(c)

Figure 8: Same as Fig. 7 but for the simulation S10:2.

(a)

(b)

(c)

Figure 9: Same as Fig. 7 but for the simulation S10:1.

The mismatch plots in the remnant parameter space can be seen in Figs. 7, 8,
and 9 for mass ratios $q=1.25$, $q=5$, $q=10$, respectively. We report similar
conclusions for the $m=1$ modes as we did for the $m=2$ modes with regard to
mode-mixing. It is the most important effect for modes with $l>2$ and when
ignored gives wildly different remnant parameter estimates. The $l=2$ mode of
this set is again unaffected by mode-mixing as expected.

Overtones are important for certain cases of this set at early times as can be
seen from Fig. B1. The $\epsilon$ values for the modes of this set fluctuates
across start times and mass ratios and no clear pattern can be made out though
there is a general improvement in $\epsilon$ for the ‘FO’ model compared to
the ‘F’ or ‘FM’ models. It is not clear whether this is a feature of these
modes or if there are hidden numerical artifacts causing this behaviour.

A surprising feature of these modes is the effect of the mirror modes.
Although the addition of mirror modes improves the mismatch curves
significantly, as we saw in the previous section, and its effect can be
clearly seen in the mode frequency plots in Fig. A3, they do not seem to
improve the model systematic in the remnant parameter space. In fact, the
amplitude of mirror mode modulations in the modes $(3,1)$ and $(4,1)$ for the
$q=10$ case is of the same order of magnitude as the amplitude of mode-mixing
modulations. Yet, where mode-mixing improves the remnant parameter estimates
by an order of magnitude, the inclusion of mirror modes do not. This indicates
that there are other effects, either numerical or physical, that dominate the
deviation of the best-fit parameters from their true values. At this point, we
cannot rule out either. The mismatch curves for the $m=1$ modes too indicate
that we are missing some features since the mismatches of our best models fall
way short of the NR error floor, suggesting that either some physical
structures are being missed or there are systematic numerical artifacts across
resolutions.

###  IV.3 Modes: (3,3), (4,3)

(a)

(b)

Figure 10: This figure plots the mismatch $\mathcal{M}_{lm}$ in the remnant
parameter space for a ringdown start time of $t_{0}-t_{\rm peak}=20M$ as in
Fig. 4 but for $m=3$.

(a)

(b)

Figure 11: Same as Fig. 10 but for the simulation S10:2.

(a)

(b)

Figure 12: Same as Fig. 10 but for the simulation S10:1.

The best-fit mismatches on an $M_{f}$–$a_{f}$ grid for the set of modes with
$m=3$ are given in Figs. 10, 11, and 12, for the three mass ratios under
consideration. We report similar findings for mode-mixing, namely, the $(3,3)$
mode is unaffected though the $(4,3)$ mode gets significant contribution
across all mass ratios.

The ‘FO’ model that includes the first overtone performs significantly better
than the ‘F’ model for the $(3,3)$ mode for all mass ratios but the overtone
does not play much of a role for the $(4,3)$ mode as can be seen in Fig. B2.
The fundamental mirror mode is again unimportant as can be seen by the
overlapping $\epsilon$ curves for the ‘F’ and ‘FM’ models as well as the ‘FO’
and ‘FMO’ models.

##  V Discussion and conclusions

In this work, we examined the features of ringdown waveforms using NR
simulations for the modes $l\leq 4$ and $m\leq 3$. We constructed four
ringdown models by adding the fundamental mirror mode, the first overtone, or
both to a base model model consisting of only the fundamental QNM to assess
the importance of each of these QNMs in a ringdown waveform.

We found that overtones improve the mismatch to an NR waveform for all modes,
though the improvement is most prominent for the modes with the smallest $l$
for a given $m$. The refinement is insignificant for $l=4,$ which is the
largest $l$ considered in this study. We also found that adding the first
overtone improves the model systematic in the remnant parameter space for the
$(2,2)$ and $(3,3)$ modes across all mass ratios. The $(3,2)$, $(4,2)$, and
$(4,3)$ modes see refinement only for mass ratio $q=1.25$ and times between
$t_{0}-t_{\rm peak}=10M$ and $t_{0}-t_{\rm peak}=15M$. The other mass ratios
for these modes do not see any significant improvement with the inclusion of
the first overtone. The case for the set of modes with $m=1$ is not very clear
because the epsilon values for the various models fluctuate though a general
improvement can be seen with the addition of the overtone.

Moreover, we found that mirror modes are most significantly excited for the
$m=1$ modes with their relative strengths growing for higher $l$ modes and
larger mass ratios. Curiously, we observed that even though the mirror modes
improve the mismatches by orders of magnitude and are clearly visible in the
mode frequency plots of the waveforms, they do not improve the model
systematics in the remnant parameter space. This indicates that the departure
of the best-fit values of the remnant parameters from their true values is
dominated by other features, either physical or numerical, though we cannot
rule out either.

The angular part of a gravitational waveform in an NR simulation of a binary
black hole merger is decomposed in $-2$ spin-weighted spherical harmonics
whereas the Teukolsky equation, which describes the perturbations of a Kerr
black hole, separates out the angular part in terms of $-2$ spin-weighted
spheroidal harmonics. As a result, an NR spherical harmonic mode has
contributions from multiple spheroidal harmonic angular quantum numbers though
the azimuthal quantum numbers $m$ stays the same. This is known as mode-
mixing. We estimated the effect of mode-mixing for the various spherical
harmonic modes under consideration by further classifying the above four
models into a class that accounts for mode-mixing and one that does not. To
account for mode-mixing, we fit the set of spherical harmonic modes with the
same $m$ simultaneously to prevent multiple fit amplitudes from being
associated to the same QNM. We note that mode-mixing affects the dominant mode
of a set, or the mode with the smallest $l$ for a given $m$, the least. This
is true for both the mismatch curves and the model systematic in the remnant
parameter space. For the higher $l$ modes for a given $m$, we find that mode-
mixing plays a significant role, with the $\epsilon$ value that provides a
measure of the deviation of the best-fit remnant parameters from the true
values superior by an order of magnitude or better for the class of models
that account for mode-mixing.

##  VI Future work

A natural extension of this work would be to fit the amplitudes as a function
of the mass ratio for the best models of a given $m$ as determined in this
study. LISA is expected to observe binary black hole mergers with total mass
greater than $10^{8}\,M_{\odot}$ that would have little to no inspiral part.
Such an amplitude model can be used to infer the progenitor properties as has
already been done in Capano  _et al._ (2021) for GW190521. Additionally, such
amplitude fits can be used for testing the strong field dynamics as proposed
by Bhagwat and Pacilio (2021).

In this study, we focused on non-spinning binaries. An interesting
continuation would be to consider initially spinning binaries which are
expected to play a role for modes with odd $l+m$ and look at their effects on
these modes.

Finally, we plan on simulating real gravitational wave signals for the various
current, planned, and proposed gravitational wave detector networks to
determine the bias in parameter estimation due to ignoring mode-mixing for the
relevant modes considered in this study.

###### Acknowledgements.

This research was supported by National Science Foundation, grant number
PHYS-2012083, PHY-1836779 and AST-2006384.

##  Appendix A Waveform fits

In this section, we plot the mode amplitudes $|h_{lm}|$ and mode frequencies,
$f_{lm}=-\Im(\dot{h}_{lm}/h_{lm})$, as a function of time $t-t_{\rm peak}$ for
our four best-fit ringdown models. We only consider the models from the mode-
mixing class. We overlay the models on top of the corresponding NR mode to
depict the features of the NR mode that each QNM takes into account. We take
the ringdown start time to be $t_{0}-t_{\rm peak}=20M$ for these plots.

Let us now consider a simplified example to make apparent the various features
seen in the mode amplitude and mode frequency plots. Consider that we model a
given NR mode $h$ using two QNMs with complex frequencies
$\omega=\omega_{o}-i/\tau$ and
$\omega^{\prime}=\omega^{\prime}_{o}-i/\tau^{\prime}$ and their respective
mixing coefficients are $\mu$ and $\mu^{\prime}$. Let us further assume that
$\mu\mathcal{C}\gg\mu^{\prime}\mathcal{C}^{\prime}$ where $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ are the complex excitation amplitudes of the two QNMs,
respectively. The NR mode can then be written as

$h=\mathcal{C}\mu e^{-i\omega
t}+\mathcal{C}^{\prime}\mu^{\prime}e^{-i\omega^{\prime}t}$ (13)

The above formula is valid for any pair of QNMs and, therefore, the relevant
indices $(l,m)$ can be introduced as necessary.

When the above approximation for the effective excitation amplitudes holds,
the mode amplitude $|h|$ takes the form

$|h|\approx|\mathcal{C}\mu|e^{-t/\tau}\left[1+\Re\left(\frac{\mu^{\prime}}{\mu}\frac{\mathcal{C}^{\prime}}{\mathcal{C}}e^{i(\omega-\omega^{\prime})t}\right)\right]$
(14)

and the mode frequency $f$ is given by

$f\approx\omega\left[1-\Re\left(\left(1-\frac{\omega^{\prime}}{\omega}\right)\frac{\mu^{\prime}}{\mu}\frac{\mathcal{C}^{\prime}}{\mathcal{C}}e^{i(\omega-\omega^{\prime})t}\right)\right].$
(15)

###  A.1 Mode amplitudes

Figure A1: The mode amplitudes (black) for $m=2$, $m=1$, and $m=3$ (top to
bottom) are plotted as a function of time $t-t_{\rm peak}$ for the three
simulations in this study (left to right). The four different best-fit mode-
mixing models are overlaid on top for comparison. The slanted straight curves
are fits to their respective modes ignoring mode-mixing and using only the
fundamental mode. The start time for the fits are taken to be $t_{0}-t_{\rm
peak}=20M$. The effect of mode-mixing is negligible for $l=2$ but becomes more
important as $l$ increases.

In Fig. A1, the mode amplitudes for $m=2$, $m=1$, and $m=3$ are plotted. The
straight lines are the best-fits using the corresponding fundamental QNM
without the inclusion of mode-mixing. The foremost feature of these plots is
the presence of characteristic modulations about the straight line for the
higher modes of a set. More specifically, the second mode of a set sees a
single modulating frequency whereas the third mode, if present, sees an
interference between two modulating frequencies. The secondary modulations are
visible to the eye for the smaller mass ratios only.

Let us take the mode $(3,2)$ as a case study to describe the behavior of the
second mode of a set and use the above defined model. The two QNMs we use to
model are the fundamental modes $(3,2,0)$ ($\omega=\omega_{320}$) and
$(2,2,0)$ ($\omega^{\prime}=\omega_{220}$). The mixing coefficients are then
$\mu\equiv\mu^{3}_{320}\sim 1$ and $\mu^{\prime}\equiv\mu^{2}_{320}\sim
10^{-2}$, respectively, for the two modes. The relative excitation amplitudes
are $\mathcal{C}^{\prime}/\mathcal{C}\sim 10^{1}$ from our fits. Therefore the
amplitude of modulations are $\sim 10^{-1}$ with the modulating frequency
$\omega_{320}-\omega_{220}$.

If we carry out the same exercise for the $(3,2)$ mode using the fundamental
QNMs $(3,2,0)$ ($\omega=\omega_{320}$) and $(4,2,0)$
($\omega^{\prime}=\omega_{420}$), we find the mixing coefficients to be
$\mu\equiv\mu^{3}_{320}\sim 1$ and $\mu^{\prime}\equiv\mu^{4}_{320}\sim
10^{-2}$, respectively, with the relative excitation amplitudes being
$\mathcal{C}^{\prime}/\mathcal{C}\sim 10^{-1}$. Therefore, the amplitude of
modulations are $\sim 10^{-3}$ and, hence, not visible in the figures.

The above model can be easily extended to describe the third mode of a set, if
present. Note that till this point we are not describing the high frequency
modulations seen in the mode $(4,1)$ for the mass ratio $q=10$. We will
describe this in the next section.

###  A.2 Mode frequencies

Figure A2: The mode frequencies (black) for $m=2$ as a function of time
$t-t_{\rm peak}$ for the mass ratios examined in this study. The four
different best-fit mode-mixing models are overlaid on top for comparison. The
horizontal curves show the fundamental mode frequency for the respective
modes. The start time for the fits are taken to be $t_{0}-t_{\rm peak}=20M$.
Note that for modes $(3,2)$ and $(4,2)$ and mass ratio $q=1.25$, the
fundamental QNM grossly under estimates the true mode frequency. Note,
further, that for mode $(4,2)$ and mass ratio $q=10$, we start seeing the
modulations due to the fundamental mirror mode on top of mode-mixing for the
first time for $m=2$. Figure A3: Same as Fig. A2 but for $m=1$. Figure A4:
Same as Fig. A2 but for $m=3$.

The modulations due to mode-mixing can also be seen in the mode frequency
plots in Figs. A2, A3, and A4 for the sets of modes with $m=2$, $m=1$, and
$m=3$, respectively. The horizontal line in the plots for each mode
corresponds to the oscillation frequency of the corresponding fundamental QNM.

The foremost difference from the mode amplitude plots is the clear presence of
high frequency modulations on top of the mode-mixing modulations. These are
due to the fundamental mirror mode. We see from Eq. (15) that the modulating
frequency is $\Re(\omega-\omega^{\prime})$. For mode-mixing, this is the
difference in the oscillation frequencies of the fundamental QNM both of which
are positive whereas the oscillation frequency of mirror modes are negative
and, hence, the modulating frequency is typically higher. Further, note that
for the $(2,1)$ mode, even though the ‘FM’ model generally traces out the
mirror mode modulations, it is off in the amplitude and the phase whereas the
‘FMO’ model reproduces the mode frequency curves exactly indicating the role
played by the first overtone.

Finally, we point to the $(3,3)$ mode plots where the ‘FM’ model shows beat
patterns which are atypical of mirror mode modulations we have seen so far.
These are due to the spurious oscillatory features we see for this mode and
are not able to assign any physical reasoning for the presence of these
features.

##  Appendix B Effect of start time on model systematics

Throughout the main text of this paper, we showed the results for the model
systematics on the remnant parameter space taking the ringdown start time to
be $t_{0}-t_{\rm peak}=20M$ because that is approximately the time when the
mismatch curves for the model with the lowest mismatch bottom out for all the
modes considered in this study. In this section we look at the effect of
varying the start time of ringdown as this is a phenomenologically defined
quantity. Specifically, we look at its effect on $\epsilon$ (see Eq. (12))
which provides a measure of the deviation of the best-fit remnant parameters
from its true value as determined from NR simulations.

In Figs. B1 and B2, we plot $\epsilon$ for the sets $m=2$, $m=1$, and $m=3$,
respectively. The different panels in each figure correspond to the different
mass ratios considered. We find that, qualitatively, the conclusions we drew
from choosing the ringdown start time to be $t_{0}-t_{\rm peak}=20M$ remain
valid across the a range of start times. The first overtone improves
$\epsilon$ for the dominant mode of a set but not for the subdominant modes.
Mode-mixing is of prime importance for the subdominant modes of a set but not
the dominant mode, as expected. Ignoring mode-mixing gives wildly different
remnant values compared to the true value. In fact, the $\epsilon$ values for
the subdominant modes of the non-mode-mixing class of models are a
conservative estimate as the best-fit values rail against the boundaries of
our remnant space for most cases meaning that their correct estimates are much
worse than what is indicated here. The fundamental mirror mode do not seem to
improve the remnant estimates even though these model the modulations seen in
the waveform and improve the mismatches significantly indicating that there
are other features in the waveform dominating the error budget. These
conclusions are true across mass ratios.

(a)  (b)

Figure B1: The error in the estimation of the remnant parameter $\epsilon$ as
a function of the start time of ringdown $t_{0}-t_{\rm peak}$ for the four
individual models and the two classes of models including (green) and
excluding (orange) mode-mixing. This figure corresponds to the set $m=2$ (top)
and $m=1$ (bottom) for the three mass ratios (rows in a sub-figure) considered
in this study. Figure B2: Same as Fig. B1 but for $m=3$.

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