emphasize the higher harmonics of the signal, and a luminosity distance of
$700\leavevmode\nobreak\ \text{Mpc}$ to give a network SNR of $\sim 16.6$.
These and the selected injected values for the phase and the sky-location
parameters are listed in the left column of Table 3.

We employ the Bilby parameter-estimation code [174], with version $2.0.0$ and
the nested sampler dynesty [197] using the acceptance-walk method, which is
well-suited for executing on a multicore machine, in particular, we run on 1
node of 64 CPUs. For the sampler settings for the recovery, we employ a number
of accepted jumps during each MCMC chain $\mathrm{naccept}=20$ and a total
number of live points $\mathrm{nlive}=1000$. We employ the sky
parameterization option H1L1, which enables us to sample the sky position in
azimuth and zenith, converted in post-processing to right ascension and
declination, since this typically improves the convergence of the sampler, and
we enable distance marginalization, to further improve convergence. We leave
the rest of the sampler parameters with their default values.

The prior distributions are uniform for most of the parameters, except for the
individual dimensionless spin values, which follow a distribution implied by
the isotropic spin prior commonly employed in GW parameter estimation. Though
a non-uniform prior could shift the posterior from the true values for
moderate SNR in a zero-noise setup, we decide to employ this spin prior as it
is commonly employed in actual analyses [198, 2, 4, 5, 6].

Figure 21: 2D and 1D posterior distributions for some relevant parameters
measured from the synthetic BBH signal with mass ratio $q=15$, total source-
frame mass of $162M_{\odot}$, dimensionless spins $\chi_{1z}=0.5$ and
$\chi_{2z}=0.0$. The inclination with respect to the line of sight of the
binary is $\iota=\pi/3$ rad. The other parameters are specified in the text
and in Table 3. The injected signal is the SXS NR waveform SXS:BBH:2464. In
the 2D posteriors the solid contours represent the $90\%$ credible intervals
and black dots show the values of the parameters of the injected signal. In
the 1D posteriors they are represented by dashed and solid vertical lines,
respectively. The parameter estimation is performed with the SEOBNRv5HM model
(green) and the IMRPhenomXHM model (orange).

We perform two parameter estimation runs on this injected signal, one with the
SEOBNRv5HM model presented in this paper, and a run with the state-of-the-art
waveform model IMRPhenomXHM from the 4th generation of Fourier-domain
phenomenological waveform models, to crosscheck the results. For SEOBNRv5HM,
we employ the conditioning routine implemented in pySEOBNR, which closely
mimics the procedure of LALSimulation [189].

The median recovered values for both models, and the $90\%$ confidence
intervals, are listed in Table 3, and some relevant 2D contours are
highlighted in Fig. 21. The results show that the SEOBNRv5HM model is able to
recover better the synthetic signal, especially for the intrinsic parameters,
with the injected value of all the parameters inside the $90\%$ confidence
intervals and very small deviations between the median values of the posterior
distributions and the actual injected values (the main deviation is in the
reference phase parameter, whose recovered distribution is prior-dominated).
On the contrary, the results inferred by the IMRPhenomXHM model contain
important biases in most of the intrinsic parameters, with the injected values
outside the $90\%$ confidence intervals for the component masses, the total
mass, the mass-ratio, and the effective-spin parameter $\chi_{\mathrm{eff}}$.
For the extrinsic parameters, both models recover the injected values within
the $90\%$ confidence intervals, with small but similar deviations in the
median values for the distance and the inclination. The improved accuracy of
SEOBNRv5HM in this challenging region of parameter space (high asymmetric
masses and spinning primary black hole) is also reflected in the recovered
matched-filter SNR in the three detectors and the Bayes factor of the
inference run, which are consistently higher than the corresponding values for
IMRPhenomXHM. These results are consistent with the fact that SEOBNRv5HM has
lower unfaithfulness than IMRPhenomXHM against this NR simulation, $0.5\%$ and
$6.7\%$ respectively, for the injected value of the total mass.

###  VI.2 Inference of real gravitational-wave events

We then perform parameter estimation on three real GW events: GW150914 [1],
the first detection which has become a benchmark for testing new waveform
models, GW170729 [199], an interesting event from O2, which has been analysed
with multimode waveform models, and GW190412 [200], the first confident mass-
asymmetric binary reported during O2. For each event, we employ the strain
data, detector calibration uncertainties and PSD provided by the Gravitational
Wave Open Science Center (GWOSC) [201]. We perform the runs using Bilby [174]
version $2.0.0$ with the nested sampler dynesty [197], and we employ the same
settings as in the previous section, except for the number of accepted jumps
during each MCMC chain that we set to $\mathrm{naccept}=60$. For each GW
signal, we perform a run with SEOBNRv5HM, employing the PA approximation, and
a crosschecking run with the IMRPhenomXHM waveform model.

In Fig. 22 we show some relevant 2D posterior distribution for the parameters,
and observe good agreement between waveform models. These results are also
consistent with the published results for the events, taking into account that
LVK catalog results employ precessing-spin waveform models and therefore minor
differences are expected. The good agreement between the SEOBNRv5HM and
IMRPhenomXHM posteriors is consistent with the fact that, for the events
considered here, the recovered parameters are within the NR calibration region
of both models. As in the case of the NR-injected signal, we observe a slight
improvement in matched-filter SNR and Bayes factor for SEOBNRv5HM with respect
to IMRPhenomXHM, more pronounced for the two more massive events, as seen in
Fig 23. Although the improvement is not drastic, these results suggest that
SEOBNRv5HM describes the data more accurately, which is consistent with the
unfaithfulness results discussed in Sec. V.1.

Run/event | GW150914 | GW170729 | GW190412
---|---|---|---
SEOBNRv5HM Bilby (64 cores) | 23h | 20h | 1d 18h
Table 4: Evaluation time for the different parameter estimation runs on real
GW events with the SEOBNRv5HM model. The time reported is actual real-time,
while the total computational cost in CPU hours can be obtained by multiplying
this time by the reported number of CPU cores employed.

In Table 4 we report the real-time spent on the inference for the parameters
of these events for the waveform model SEOBNRv5HM. Employing Bilby on a single
computing node (of 64 cores) requires less than a day for GW150914 and
GW170729, and less than two days for GW190412, with a moderately low chirp
mass. Therefore, the model is sufficiently efficient to be employed with the
preferred parameter estimation pipeline by the LVK Collaboration.

(a) GW150914.

(b) GW170729.

(c) GW190412.

Figure 22: 1D and 2D posterior distributions for several parameters for the GW
events GW150914, GW170729 and GW190412.

Figure 23: Network matched-filter SNR recovered by IMRPhenomXHM and SEOBNRv5HM
for the three analyzed GW events.

##  VII Frequency domain reduced order model

The requirement of integrating a system of ODEs to solve for the dynamics of
the binary in EOB models increases the time needed for generating a waveform.
Surrogate or reduced order modeling (ROM) techniques [91, 92, 93, 21, 23, 22,
95, 202, 203, 24, 25, 96, 97, 100] have been applied in several contexts to
accelerate slow waveform computation, in both EOB and NR models. These
techniques involve decomposing the waveforms from a training set in
orthonormal bases on sparse grids in time or frequency and then interpolating
or fitting the resulting waveform data pieces over the binary parameter space.
The result is a highly accurate, yet fast, method for generating waveforms for
data analysis applications, which can reduce computational time by orders of
magnitude compared to ODE-based waveform models.

A frequency domain (FD) ROM of SEOBNRv4HM was built in Ref. [96], with
modeling error introduced in building the ROM below the unfaithfulness of
SEOBNRv4HM against NR simulations used to calibrate the model, and waveform
evaluation times reduced by two orders of magnitude. In this section we show
the performance of SEOBNRv5_ROM, a FD ROM of SEOBNRv5, built following the
same techniques of SEOBNRv4HM_ROM [96, 93, 92]. These mostly involve modeling
in FD the phase of a carrier signal, based on the time-domain orbital phase,
and the “coorbital modes” obtained after extracting the carrier phasing from
each FD mode. The coorbital modes have an almost constant phase in the
inspiral, and allow us to avoid zero-crossings in the subdominant harmonics
which would complicate the interpolation of the training data. As for
SEOBNRv4HM_ROM, the SEOBNRv5_ROM model combines a higher resolution high-
frequency ROM, starting from $20\leavevmode\nobreak\ \text{Hz}$ for binaries
with total mass of $50M_{\odot}$, and a lower resolution low-frequency ROM,
starting from $20\leavevmode\nobreak\ \text{Hz}$ for binaries with total mass
of $5M_{\odot}$, and can be extended to arbitrarily low frequencies by
hybridizing it with multipolar PN waveforms. SEOBNRv5_ROM can be generated for
mass-ratios between $1$ and $100$, dimensionless spins between
$[-0.998,0.998]$, and includes only the dominant $(\ell,|m|)=(2,2)$ mode. A
multipolar reduced order model of SEOBNRv5HM (SEOBNRv5HM_ROM), including the
$(\ell,|m|)=(2,2),(3,3),(2,1),(4,4),(5,5),(3,2),(4,3)$ modes, is also under
development, and will be presented in near future work. Despite the speed of
SEOBNRv5HM being sufficient for many GW data analysis applications, using a
ROM can still lead to a significant increase in efficiency. Additionally,
there are several applications for which it is desirable to be able to
generate clean FD waveforms of any length.

In Fig. 24 we show a histogram of the unfaithfulness between SEOBNRv5_ROM and
SEOBNRv5, for different values of the total mass, for $10^{5}$ configurations
with mass-ratios between $1$ and $100$ and dimensionless spins between
$[-0.998,0.998]$. We observe an excellent agreement, with median values
$\lesssim 10^{-5}$. The unfaithfulness increases with the total mass of the
system, as in previous ROM models [96, 75], because the ROM modes are
generated up to a maximum frequency that scales with the inverse of the total
mass. In particular, the mismatch is larger for cases with high mass ratio and
negative spins, as the maximum frequency of each mode is proportional to its
least damped QNM frequency [96], which decreases in this region of the
parameter space. Nonetheless, the modeling error introduced in the
construction of the ROM is negligible compared to the inaccuracy of the
SEOBNRv5 waveforms with respect to the NR simulations.

Figure 25 highlights the speedup of the ROM with respect to SEOBNRv5, by
comparing walltimes of the two models for generating a FD waveform with the
same parameters ($q=1,\leavevmode\nobreak\ \chi_{1}=0.8,\leavevmode\nobreak\
\chi_{2}=0.3$), as a function of the total mass $M$. For SEOBNRv5 we also
employ the PA approximation. As for Fig. 20, we use a starting frequency
$f_{\rm{start}}=10\leavevmode\nobreak\ \rm{Hz}$ and we choose the sampling
rate for the time domain model such that the Nyquist criterion is satisfied
for the $\ell=2$ multipoles. The FD SEOBNRv5_ROM model is instead generated up
to a maximum frequency equal to the corresponding Nyquist frequency. Notably,
we obtain an improvement from a factor $\sim 7$ for low total-mass binaries,
to more than $\sim 20$ for $M\sim 100M_{\odot}$. Overall, we can appreciate
that SEOBNRv5_ROM can be generated in less than $10\leavevmode\nobreak\
\text{ms}$ for $M\gtrsim 20M_{\odot}$.

Figure 24: Mismatch of SEOBNRv5_ROM against SEOBNRv5 for different values of
the total mass, for $10^{5}$ random configurations. The dashed vertical lines
show the medians. Figure 25: Walltime comparison between SEOBNRv5 and
SEOBNRv5_ROM, using a strarting frequency
$f_{\rm{start}}=10\leavevmode\nobreak\ \rm{Hz}$, as a function of the total
mass $M$. For the time domain SEOBNRv5 model this also includes the conversion
in Fourier domain.

##  VIII Conclusions

In this paper we have presented SEOBNRv5HM, a new EOBNR waveform model for
quasi-circular, spinning, nonprecessing BBHs, which improves the previous
generation, SEOBNRv4HM [76], on both speed and accuracy against NR
simulations. The waveform model includes the modes
$(\ell,|m|)=(2,2),(3,3),(2,1),(4,4),(5,5),(3,2),(4,3)$, and models the mode-
mixing in the merger-ringdown for the modes $(3,2),(4,3)$, which were not
included in SEOBNRv4HM.

Sections II and III have outlined the building blocks of the waveform model.
The aligned-spin SEOBNRv5 Hamiltonian is based on a deformation of the
equatorial Kerr Hamiltonian, and includes most of the known 5PN nonspinning
and full 4PN information for spinning binaries, with improved resummations
[117]. The factorized waveform modes and RR force of SEOBNRv4HM have been
enhanced by additional PN information (as well as corrections to some of the
terms) from Ref. [118], and have been calibrated to 2GSF fluxes in the
nonspinning limit in Ref. [116]. To improve the accuracy of the model in the
inspiral, we have refined the calibration pipeline employed by Ref. [75], and
largely upgraded its efficiency, in order to be able to tune the model to a
large catalog of 442 NR simulations as shown in Sec. IV. We have also improved
the modeling of the merger and ringdown using the full NR dataset at our
disposal, as well as 13 waveforms from BH perturbation theory.

In Sec. V we have compared SEOBNRv5HM and other state-of-the-art waveform
approximants to NR simulations, using mismatch calculations. Results showed
that the dominant (2,2) mode of SEOBNRv5 is, on average, more accurate than
SEOBNRv4 [75] by an order of magnitude, it is more accurate than the other
state-of-the-art EOB model, TEOBResumS-GIOTTO [105, 104, 62, 81], by more than
a factor 2, and is overall comparable to the Fourier-domain phenomenological
model IMRPhenomXAS [42]. By computing mismatches of the full polarizations at
inclination $\iota=\pi/3$, we noted that all models become less accurate,
nonetheless SEOBNRv5HM outperforms both SEOBNRv4HM [76] and TEOBResumS-GIOTTO,
as well as the phenomenological model IMRPhenomXHM [44], both considering
average values and number of cases above $0.01$. We have validated the model
against the NR surrogate models NRHybSur3dq8 [24] and NRHybSur2dq15 [30] and
found results consistent with the NR comparison, demonstrating the robustness
of our calibration procedure. Further tests of the calibration pipeline are
described in Appendix E. In particular, we show that the accuracy of the model
against the entire NR dataset of 442 waveforms does not change appreciably
when using only 137 to calibrate the model. We also show that our calibration
pipeline is robust with respect to changes in the shape of the PSD used, as
the unfaithfulness of the model against NR remains very similar when using a
white noise curve, the Einsten Telescope [204] and Cosmic Explorer [16] PSDs.
Nonetheless, such a comparison is limited by the length of the available NR
simulations, which do not cover the entire bandwidth of next-generation GW
detectors, and more detailed studies will be needed to assess the accuracy
SEOBNRv5HM in such a context. Comparing SEOBNRv5HM and IMRPhenomXHM against
each other in a larger parameter space, we have seen instead that the
mismatches can become very large in the region where both models are not
calibrated to NR waveforms, in particular for both high mass ratio, say $\geq
5$, and high positive spin, say $\geq 0.8$, configurations. Thus, producing
new NR simulations for these parameters would be critical to reduce modeling
systematics. Comparing the angular-momentum flux and binding energy of
SEOBNRv5 and SEOBNRv4 against NR, we have highlighted a similar improvement,
also thanks to the 2GSF information [116], despite the fact that these
quantities do not enter directly the calibration. This is a powerful check of
the physical robustness of the model, and provides confidence in its
reliability when extrapolating outside of the NR calibration region. We have
implemented SEOBNRv5HM in a flexible, high-performance, Python package
pySEOBNR, and we have shown that the model is fast enough for typical GW data-
analysis applications: it is more than 10 times faster than SEOBNRv4HM without
using the PA approximation, up to two times faster than SEOBNRv4HM_PA when
using it, and overall close to TEOBResumS-GIOTTO.

In Sec. VI we have demonstrated that the model can be reliably used for GW
parameter estimation, by performing a recovery on a NR injection, and by
analyzing 3 events observed by LIGO and Virgo, GW150914, GW170729, GW190412.
For the 3 events, we have found consistent results when comparing the
parameters recovered by SEOBNRv5HM and by IMRPhenomXHM, while still observing
a slight improvement in matched-filter SNR and Bayes factor for SEOBNRv5HM
with respect to IMRPhenomXHM. On the other hand, the NR injection in a
challenging region of parameter space (high asymmetric masses and spinning
primary black hole) shows more significant differences. SEOBNRv5HM accurately
recovers all the binary parameters, while the results inferred by the
IMRPhenomXHM model contain biases in most of the intrinsic parameters due to
larger modeling errors.

In Sec. VII we have finally shown the performance of a FD ROM model
(SEOBNRv5_ROM) developed following the techniques used in Ref. [96], which
allows for a significant speedup in evaluation time, while retaining identical
accuracy against NR.

The pySEOBNR code infrastructure [135] is a framework developed with the goal
of facilitating the development of future SEOBNR waveform models, and upcoming
work would naturally revolve around adding more physical effects to the
SEOBNRv5 family, as well as improving its efficiency and accuracy by including
ever more information from both NR simulations and different analytical
frameworks. The first extension of SEOBNRv5HM, as far as additional physical
effects are concerned, involves modeling spin precession, and such a model
(SEOBNRv5PHM) has been developed in parallel to this work in Ref. [136].

An upcoming extension would also involve eccentric and hyperbolic orbits
(SEOBNRv5EHM), following similar strategies adopted in the SEOBNRv4EHM model
of Ref. [89]. The more efficient, flexible and parallelized calibration
pipeline described in this work would allow having a more accurate eccentric
model, with a consistent treatment of eccentric corrections in the waveform
modes and RR force, after appropriately recalibrating the quasi-circular limit
of the model. Moreover, the efficiency of SEOBNRv5HM compared to SEOBNRv4HM
without PA approximation, which is slow to use when adding eccentricity, means
that one could expect a significant speed-up of SEOBNRv5EHM compared to
SEOBNRv4EHM. Further ongoing developments in the SEOBNRv5 family involve the
modeling of tidal effects, already incorporated in SEOBNRv4 models [205, 206,
95, 207], and the addition of parametrized GR deviations to perform theory
agnostic tests of GR [208, 209, 210].

Another direction for improvements revolves around pushing the accuracy of the
model against NR even further. Exploring different ways to incorporate and
resum information from the PN, PM and 2GSF approximations, while still
retaining flexibility in the calibration, would be crucial, and an efficient
calibration code (pySEOBNR [135]) is essential to understand the impact of
different modeling and resummation choices. At the same time, more and better
NR simulations, especially in currently unexplored regions of the binary
parameter space, are also critical to reach the accuracy requirements of next-
generation detectors [171, 211]. A limitation of all state-of-the-art
approximants is the modeling of the higher-modes, and a straightforward
improvement to be done in future work would be to add all terms through 3.5PN
to the waveform modes and RR force from Refs. [118, 154]. Further work should
also go into improving the modeling of the HMs through merger and ringdown, as
well as including additional modes. Finally, a calibration pipeline similar to
the one developed here could be used to calibrate SEOBNRv5PHM [136] to spin-
precessing NR simulations.

## Acknowledgments

It is our pleasure to thank Geraint Pratten, Stanislav Babak, Alice Bonino,
Eleanor Hamilton, N.V. Krishnendu, Piero Rettegno, Riccardo Sturani and
Jooheon Yoo for performing the LIGO-Virgo-KAGRA review of the SEOBNRv5 models.

Part of M.K.’s work on this paper is supported by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is supported in part by
the Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Colleges and Universities. M.v.d.M. is supported by VILLUM FONDEN (grant no.
37766), and the Danish Research Foundation. R.C. is supported by NSF Grants
No. AST-2006538, PHY-2207502, PHY-090003 and PHY-20043, and NASA Grants No.
19-ATP19-0051, 20-LPS20-0011 and 21-ATP21-0010. H.R. is supported by the
Fundação para a Ciência e Tecnologia (FCT) within the projects UID/04564/2021,
UIDB/04564/2020, UIDP/04564/2020 and EXPL/FIS-AST/0735/2021. This work was
supported in part by the Sherman Fairchild Foundation, by NSF Grants No.
PHY-2011961, No. PHY-2011968, and No. OAC-1931266 at Caltech, by NSF Grants
No. PHY-1912081, , No. PHY-2207342, No. OAC-1931280 and No. OAC-2209655 at
Cornell, by NSF awards No. AST-1559694 and No. PHY-1654359, by Nicholas and
Lee Begovich, and by the Dan Black Family Trust.

The computational work for this manuscript was carried out on the Hypatia
computer cluster at the Max Planck Institute for Gravitational Physics in
Potsdam.

SEOBNRv5HM is publicly available through the python package pySEOBNR
git.ligo.org/waveforms/software/pyseobnr. Stable versions of pySEOBNR are
published through the Python Package Index (PyPI), and can be installed via
pip install pyseobnr.

This research has made use of data or software obtained from the Gravitational
Wave Open Science Center (gwosc.org), a service of LIGO Laboratory, the LIGO
Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory
and Advanced LIGO are funded by the United States National Science Foundation
(NSF) as well as the Science and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO and
construction and operation of the GEO600 detector. Additional support for
Advanced LIGO was provided by the Australian Research Council. Virgo is
funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education,
Culture, Sports, Science and Technology (MEXT), Japan Society for the
Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and
Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National
Science and Technology Council (NSTC) in Taiwan.

##  Appendix A Hamiltonian coefficients

We summarize here results that were derived in Ref. [117]. In the nonspinning
limit, the Hamiltonian is given by Eq. (6). The 5PN-expanded
$\bar{D}_{\text{noS}}$, which enters the Hamiltonian through Eq. (8), is given
by [140, 141]

$\displaystyle\bar{D}_{\text{noS}}^{\text{Tay}}(u)$ $\displaystyle=1+6\nu
u^{2}+\left(52\nu-6\nu^{2}\right)u^{3}+\bigg{[}\nu\left(-\frac{533}{45}-\frac{23761\pi^{2}}{1536}+\frac{1184\gamma_{E}}{15}-\frac{6496\ln
2}{15}+\frac{2916\ln 3}{5}\right)$
$\displaystyle\qquad+\left(\frac{123\pi^{2}}{16}-260\right)\nu^{2}+\frac{592\nu}{15}\ln
u\bigg{]}u^{4}+\left(-\frac{3392\nu^{2}}{15}-\frac{1420\nu}{7}\right)u^{5}\ln
u$
$\displaystyle\quad+\bigg{[}\nu\left(\frac{294464}{175}-\frac{2840\gamma_{E}}{7}-\frac{63707\pi^{2}}{512}+\frac{120648\ln
2}{35}-\frac{19683\ln
3}{7}\right)+\left(\frac{1069}{3}-\frac{205\pi^{2}}{16}\right)\nu^{3}$
$\displaystyle\qquad+\left(d_{5}^{\nu^{2}}-\frac{6784\gamma_{E}}{15}+\frac{67736}{105}+\frac{58320\ln
3}{7}-\frac{326656\ln 2}{21}\right)\nu^{2}\bigg{]}u^{5},$ (99)

where we set the remaining unknown coefficient $d_{5}^{\nu^{2}}$ to zero. To
improve agreement with NR, we perform a (2,3) Padé resummation of
$\bar{D}_{\text{noS}}^{\text{Tay}}(u)$.

For $Q_{\text{noS}}$, we use the full 5.5PN expansion derived in Refs. [141,
212], which is expanded in eccentricity to
$\operatorname{\mathcal{O}}(p_{r}^{8})$. Instead of using $p_{r}$, we write
$Q_{\text{noS}}$ in terms of $p_{r_{*}}$ using Eq. (8), then PN expand to
5.5PN order, leading to

$\displaystyle Q_{\text{noS}}$
$\displaystyle=\frac{p_{r_{*}}^{4}}{\mu^{2}}\Bigg{\\{}2(4-3\nu)\nu
u^{2}+u^{3}\left[10\nu^{3}-131\nu^{2}+\nu\left(-\frac{4348}{15}+\frac{496256\ln
2}{45}-\frac{33048\ln
3}{5}\right)\right]+u^{4}\Bigg{[}\left(792-\frac{615\pi^{2}}{32}\right)\nu^{3}$
$\displaystyle\qquad+\nu^{2}\left(-\frac{592\ln
u}{5}+\frac{31633\pi^{2}}{512}-\frac{1184\gamma_{E}}{5}+\frac{45683}{105}+\frac{33693536\ln
2}{105}-\frac{6396489\ln 3}{70}-\frac{9765625\ln 5}{126}\right)$
$\displaystyle\qquad+\nu\left(\frac{5428\ln
u}{105}+\frac{1249177}{1050}-\frac{93031\pi^{2}}{1536}+\frac{10856\gamma_{E}}{105}-\frac{4396376}{105}\ln
2+\frac{9765625\ln 5}{504}-\frac{601911\ln
3}{280}\right)\Bigg{]}+\frac{88703\pi\nu u^{9/2}}{1890}\Bigg{\\}}$
$\displaystyle\quad+\frac{p_{r_{*}}^{6}}{\mu^{4}}\Bigg{\\{}u^{2}\left[6\nu^{3}-\frac{27\nu^{2}}{5}+\nu\left(-\frac{827}{3}-\frac{1}{25}2358912\ln
2+\frac{1399437\ln 3}{50}+\frac{390625\ln 5}{18}\right)\right]$
$\displaystyle\qquad+u^{3}\Bigg{[}-14\nu^{4}+188\nu^{3}+\nu^{2}\left(\frac{154229}{75}-\frac{4998308864\ln
2}{1575}+\frac{26171875\ln 5}{18}-\frac{45409167\ln 3}{350}\right)$
$\displaystyle\quad\qquad+\nu\left(-\frac{860317}{1050}+\frac{305146624\ln
2}{945}+\frac{35643726\ln 3}{175}-\frac{52468750\ln
5}{189}\right)\Bigg{]}-\frac{2723471\pi\nu u^{7/2}}{756000}\Bigg{\\}}$
$\displaystyle\quad+\frac{p_{r_{*}}^{8}}{\mu^{6}}\Bigg{\\{}u\nu\left[-\frac{35772}{175}+\frac{21668992\ln
2}{45}+\frac{6591861\ln 3}{350}-\frac{27734375\ln
5}{126}\right]+\frac{5994461\pi\nu
u^{5/2}}{12700800}+u^{2}\Bigg{[}-6\nu^{4}+\frac{24\nu^{3}}{7}$
$\displaystyle\quad\qquad+\nu^{2}\left(\frac{870976}{525}+\frac{703189497728\ln
2}{33075}+\frac{332067403089\ln 3}{39200}-\frac{13841287201\ln
7}{4320}-\frac{468490234375\ln 5}{42336}\right)$
$\displaystyle\quad\qquad+\nu\left(-\frac{2222547}{2450}-\frac{1347019456}{525}\ln
2+\frac{278690984375\ln 5}{169344}+\frac{13841287201\ln
7}{17280}-\frac{346536085761\ln 3}{156800}\right)\Bigg{]}\Bigg{\\}}.$ (100)

In the aligned-spin Hamiltonian, the 3.5PN SO gyro-gravitomagnetic factors in
Eq. (II.2) are given by

$\displaystyle g_{a_{+}}^{\text{3.5PN}}$
$\displaystyle=\frac{7}{4}+\left[\tilde{L}^{2}u^{2}\left(-\frac{45\nu}{32}-\frac{15}{32}\right)+u\left(\frac{23\nu}{32}-\frac{3}{32}\right)\right]$
$\displaystyle\quad+\Bigg{[}\tilde{L}^{4}u^{4}\left(\frac{345\nu^{2}}{256}+\frac{75\nu}{128}+\frac{105}{256}\right)+\tilde{L}^{2}u^{3}\left(-\frac{1591\nu^{2}}{768}-\frac{267\nu}{128}+\frac{59}{256}\right)+u^{2}\left(\frac{109\nu^{2}}{192}-\frac{177\nu}{32}-\frac{5}{64}\right)\Bigg{]},$
(101a) $\displaystyle g_{a_{-}}^{\text{3.5PN}}$
$\displaystyle=\frac{1}{4}+\left[\tilde{L}^{2}u^{2}\left(\frac{15}{32}-\frac{9\nu}{32}\right)+u\left(\frac{11\nu}{32}+\frac{3}{32}\right)\right]$
$\displaystyle\quad+\Bigg{[}\tilde{L}^{4}u^{4}\left(\frac{75\nu^{2}}{256}-\frac{45\nu}{128}-\frac{105}{256}\right)+\tilde{L}^{2}u^{3}\left(-\frac{613\nu^{2}}{768}-\frac{35\nu}{128}-\frac{59}{256}\right)+u^{2}\left(\frac{103\nu^{2}}{192}-\frac{\nu}{32}+\frac{5}{64}\right)\Bigg{]},$
(101b)

where the square brackets collect different PN orders, and we defined
$\tilde{L}\equiv L/(M\mu)\equiv p_{\phi}/(M\mu)$. The cubic-in-spin term
$G_{a^{3}}$ reads

$G_{a^{3}}^{\text{align}}=\frac{Mp_{\phi}}{4r^{2}}\left(\delta
a_{-}a_{+}^{2}-a_{+}^{3}\right).$ (102)

The potentials in the even-in-spin part of the effective Hamiltonian in Eq.
(II.2) include the 4PN SS information, and are given by

$\displaystyle A^{\text{align}}$
$\displaystyle=\frac{a_{+}^{2}/r^{2}+A_{\text{noS}}+A_{\text{SS}}^{\text{align}}}{1+(1+2M/r)a_{+}^{2}/r^{2}},$
(103a) $\displaystyle B_{np}^{\text{align}}$
$\displaystyle=-1+\frac{a_{+}^{2}}{r^{2}}+A_{\text{noS}}\bar{D}_{\text{noS}}+B_{np,\text{SS}}^{\text{align}},$
(103b) $\displaystyle B_{npa}^{\text{Kerr\,eq}}$
$\displaystyle=-\frac{1+2M/r}{r^{2}+a_{+}^{2}(1+2M/r)},$ (103c) $\displaystyle
Q^{\text{align}}$ $\displaystyle=Q_{\text{noS}}+Q_{\text{SS}}^{\text{align}},$
(103d)

where

$\displaystyle A_{\text{SS}}^{\text{align}}$
$\displaystyle=\frac{M^{2}}{r^{4}}\left[\frac{9a_{+}^{2}}{8}-\frac{5}{4}\delta
a_{-}a_{+}+a_{-}^{2}\left(\frac{\nu}{2}+\frac{1}{8}\right)\right]$
$\displaystyle\quad+\frac{M^{3}}{r^{5}}\bigg{[}a_{+}^{2}\left(-\frac{175\nu}{64}-\frac{225}{64}\right)+\delta
a_{-}a_{+}\left(\frac{117}{32}-\frac{39\nu}{16}\right)$
$\displaystyle\quad\qquad+a_{-}^{2}\left(\frac{21\nu^{2}}{16}-\frac{81\nu}{64}-\frac{9}{64}\right)\bigg{]},$
(104a) $\displaystyle B_{np,\text{SS}}^{\text{align}}$
$\displaystyle=\frac{M}{r^{3}}\left[a_{+}^{2}\left(3\nu+\frac{45}{16}\right)-\frac{21}{8}\delta
a_{-}a_{+}+a_{-}^{2}\left(\frac{3\nu}{4}-\frac{3}{16}\right)\right]$
$\displaystyle\quad+\frac{M^{2}}{r^{4}}\bigg{[}a_{+}^{2}\left(-\frac{1171\nu}{64}-\frac{861}{64}\right)+\delta
a_{-}a_{+}\left(\frac{13\nu}{16}+\frac{449}{32}\right)$
$\displaystyle\quad\qquad+a_{-}^{2}\left(\frac{\nu^{2}}{16}+\frac{115\nu}{64}-\frac{37}{64}\right)\bigg{]},$
(104b) $\displaystyle Q_{\text{SS}}^{\text{align}}$
$\displaystyle=\frac{Mp_{r}^{4}}{\mu^{2}r^{3}}\bigg{[}a_{+}^{2}\left(\frac{25}{32}-5\nu^{2}+\frac{165\nu}{32}\right)+\delta
a_{-}a_{+}\left(\frac{45\nu}{8}-\frac{5}{16}\right)$
$\displaystyle\quad\qquad+a_{-}^{2}\left(-\frac{15\nu^{2}}{8}+\frac{75\nu}{32}-\frac{15}{32}\right)\bigg{]}.$
(104c)

##  Appendix B Expressions for the factorized waveform modes

In this Appendix, we list the expressions for $\rho_{\ell m}$, $f_{\ell m}$
and $\delta_{\ell m}$, which are used in the factorized modes (see Eqs. (25)
and (37)).

In the $(2,2)$ mode, $\rho_{22}$ and $\delta_{22}$ are given by

$\displaystyle\rho_{22}$
$\displaystyle=1+v_{\Omega}^{2}\left(\tfrac{55}{84}\nu-\tfrac{43}{42}\right)+v_{\Omega}^{3}\left[\left(\tfrac{2}{3}\nu-\tfrac{2}{3}\right)\chi_{S}-\tfrac{2}{3}\delta\chi_{A}\right]+v_{\Omega}^{4}\Big{[}\tfrac{19583}{42336}\nu^{2}-\tfrac{33025}{21168}\nu-\tfrac{20555}{10584}+\left(\tfrac{1}{2}-2\nu\right)\chi_{A}^{2}+\delta\chi_{A}\chi_{S}+\tfrac{1}{2}\chi_{S}^{2}\Big{]}$
$\displaystyle\quad+v_{\Omega}^{5}\left[\delta\left(-{\color[rgb]{0,0,0}\tfrac{19}{42}}\nu-\tfrac{34}{21}\right)\chi_{A}+\left(\tfrac{209}{126}\nu^{2}+\tfrac{49}{18}\nu-\tfrac{34}{21}\right)\chi_{S}\right]$
$\displaystyle\quad+v_{\Omega}^{6}\Big{[}\tfrac{10620745\nu^{3}}{39118464}-\tfrac{6292061\nu^{2}}{3259872}+\tfrac{41\pi^{2}\nu}{192}-\tfrac{48993925\nu}{9779616}-\tfrac{428}{105}\text{eulerlog}(2,v_{\Omega})+\tfrac{1556919113}{122245200}$
$\displaystyle\qquad{\color[rgb]{0,0,0}+\delta\left(\tfrac{89}{126}-\tfrac{781}{252}\nu\right)\chi_{A}\chi_{S}+\left(-\tfrac{27}{14}\nu^{2}-\tfrac{457}{504}\nu+\tfrac{89}{252}\right)\chi_{A}^{2}+\left(\tfrac{10}{9}\nu^{2}-\tfrac{1817}{504}\nu+\tfrac{89}{252}\right)\chi_{S}^{2}}\Big{]}$
$\displaystyle\quad+v_{\Omega}^{7}\Big{[}\delta\left(\tfrac{97865}{63504}\nu^{2}+\tfrac{50140}{3969}\nu+\tfrac{18733}{15876}\right)\chi_{A}+\left(\tfrac{50803}{63504}\nu^{3}-\tfrac{245717}{63504}\nu^{2}+\tfrac{74749}{5292}\nu+\tfrac{18733}{15876}\right)\chi_{S}{\color[rgb]{0,0,0}+\delta\chi_{A}^{3}\left(\tfrac{1}{3}-\tfrac{4}{3}\nu\right)+\delta(2\nu+1)\chi_{A}\chi_{S}^{2}}$
$\displaystyle\qquad{\color[rgb]{0,0,0}+\left(-4\nu^{2}-3\nu+1\right)\chi_{A}^{2}\chi_{S}+\left(\nu+\tfrac{1}{3}\right)\chi_{S}^{3}}\Big{]}+v_{\Omega}^{8}\left[\tfrac{9202}{2205}\text{eulerlog}(2,v_{\Omega})-\tfrac{387216563023}{160190110080}\right]$
$\displaystyle\quad+v_{\Omega}^{10}\left[\tfrac{439877}{55566}\text{eulerlog}(2,v_{\Omega})-\tfrac{16094530514677}{533967033600}\right],$
(105a) $\displaystyle\delta_{22}$ $\displaystyle=\tfrac{7}{3}\Omega
H_{\text{EOB}}+\big{(}\Omega
H_{\text{EOB}}\big{)}^{2}\left[\left(\tfrac{8}{3}\nu-\tfrac{4}{3}\right)\chi_{S}-\tfrac{4}{3}\delta\chi_{A}+\tfrac{428}{105}\pi\right]+\big{(}\Omega
H_{\text{EOB}}\big{)}^{3}\left[\tfrac{1712}{315}\pi^{2}-\tfrac{2203}{81}\right]-24\nu
v_{\Omega}^{5},$ (105b)

where $\text{eulerlog}(m,v_{\Omega})$ is defined by Eq. (50). The coefficient
$19/42$ of $\mathcal{O}(v_{\Omega}^{5}\delta\chi_{A}\nu)$ in $\rho_{22}$
corrects a typo in the SEOBNRv4 code, and we added in $\rho_{22}$ the NLO
spin-squared and LO spin-cubed contributions, which are given by Eq. (4.11a)
of Ref. [118].

The $(2,1)$ mode reads

$\displaystyle\rho_{21}^{\text{NS}}$
$\displaystyle=1+v_{\Omega}^{2}\left(\tfrac{23}{84}\nu-\tfrac{59}{56}\right)+v_{\Omega}^{4}\left(\tfrac{617}{4704}\nu^{2}-\tfrac{10993}{14112}\nu-\tfrac{47009}{56448}\right)+v_{\Omega}^{6}\left[\tfrac{7613184941}{2607897600}-\tfrac{107}{105}\text{
eulerlog }\left(1,v_{\Omega}\right)\right]$
$\displaystyle\quad+v_{\Omega}^{8}\left[-\tfrac{1168617463883}{911303737344}+\tfrac{6313}{5880}\text{
eulerlog
}\left(1,v_{\Omega}\right)\right]+v_{\Omega}^{10}\left[-\tfrac{63735873771463}{16569158860800}+\tfrac{5029963}{5927040}\text{
eulerlog }\left(1,v_{\Omega}\right)\right],$ (106a) $\displaystyle
f_{21}^{\text{S}}$
$\displaystyle=-\tfrac{3}{2}v_{\Omega}\left(\frac{\chi_{A}}{\delta}+\chi_{S}\right)+v_{\Omega}^{3}\Big{[}\left(\tfrac{131}{84}\nu+\tfrac{61}{12}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{79}{84}\nu+\tfrac{61}{12}\right)\chi_{S}\Big{]}+v_{\Omega}^{4}\left[(-2\nu-3)\chi_{A}^{2}+\left(\tfrac{21}{2}\nu-6\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{1}{2}\nu-3\right)\chi_{S}^{2}\right]$
$\displaystyle\quad+v_{\Omega}^{5}\Big{[}\left(-\tfrac{703}{112}\nu^{2}+\tfrac{8797}{1008}\nu-\tfrac{81}{16}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{613}{1008}\nu^{2}+\tfrac{1709}{1008}\nu-\tfrac{81}{16}\right)\chi_{S}+\left(\tfrac{3}{4}-3\nu\right)\frac{\chi_{A}^{3}}{\delta}+\left(\tfrac{9}{4}-6\nu\right)\frac{\chi_{A}\chi_{S}^{2}}{\delta}+\left(\tfrac{9}{4}-3\nu\right)\chi_{A}^{2}\chi_{S}+\tfrac{3}{4}\chi_{S}^{3}\Big{]}$
$\displaystyle\quad+{\color[rgb]{0,0,0}v_{\Omega}^{6}\Big{[}\left(\tfrac{5}{7}\nu^{2}-\tfrac{9287}{1008}\nu+\tfrac{4163}{252}\right)\chi_{A}^{2}+\left(\tfrac{139}{72}\nu^{2}-\tfrac{2633}{1008}\nu+\tfrac{4163}{252}\right)\chi_{S}^{2}+\left(\tfrac{9487}{504}\nu^{2}-\tfrac{1636}{21}\nu+\tfrac{4163}{126}\right)\frac{\chi_{A}\chi_{S}}{\delta}\Big{]}},$
(106b) $\displaystyle\delta_{21}$ $\displaystyle=\tfrac{2}{3}\Omega
H_{\text{EOB}}+\tfrac{107}{105}\pi\big{(}\Omega
H_{\text{EOB}}\big{)}^{2}+\left(\tfrac{214}{315}\pi^{2}-\tfrac{272}{81}\right)\big{(}\Omega
H_{\text{EOB}}\big{)}^{3}-{\color[rgb]{0,0,0}\tfrac{25}{2}}\nu
v_{\Omega}^{5},$ (106c)

where the $\mathcal{O}(v_{\Omega}^{6}\chi^{2}\nu^{2})$ terms in
$f_{21}^{\text{S}}$ correct those used in the SEOBNRv4HM model [76]. We also
fixed the coefficient $-25/2$ of $\mathcal{O}(\nu v_{\Omega}^{5})$ in
$\delta_{21}$, which was the result of an error in Ref. [153], which was later
corrected in an erratum, as noted in Ref. [118].

The $(3,3)$ mode is given by

$\displaystyle\rho_{33}^{\text{NS}}$
$\displaystyle=1+v_{\Omega}^{2}\left(\tfrac{2}{3}\nu-\tfrac{7}{6}\right)+v_{\Omega}^{4}\left(-\tfrac{6719}{3960}-\tfrac{1861}{990}\nu+\tfrac{149}{330}\nu
v_{\Omega}^{2}\right)$
$\displaystyle\quad+v_{\Omega}^{6}\left[\tfrac{3203101567}{227026800}+\left(-\tfrac{129509}{25740}+\tfrac{41\pi^{2}}{192}\right)\nu-\tfrac{274621}{154440}\nu^{2}+\tfrac{12011}{46332}\nu^{3}-\tfrac{26}{7}\text{eulerlog}\left(3,v_{\Omega}\right)\right]$
$\displaystyle\quad+v_{\Omega}^{8}\left[-\tfrac{57566572157}{8562153600}+\tfrac{13}{3}\text{
eulerlog
}\left(3,v_{\Omega}\right)\right]+v_{\Omega}^{10}\left[-\tfrac{903823148417327}{30566888352000}+\tfrac{87347}{13860}\text{eulerlog}\left(3,v_{\Omega}\right)\right],$
(107a) $\displaystyle f_{33}^{\text{S}}$
$\displaystyle=v_{\Omega}^{3}\left[\left(\tfrac{19}{2}\nu-2\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{5}{2}\nu-2\right)\chi_{S}\right]+v_{\Omega}^{4}\bigg{[}\left(\tfrac{3}{2}-6\nu\right)\chi_{A}^{2}+(3-12\nu)\frac{\chi_{A}\chi_{S}}{\delta}+\tfrac{3}{2}\chi_{S}^{2}\bigg{]}$
$\displaystyle\quad+v_{\Omega}^{5}\left[\left(\tfrac{407}{30}\nu^{2}-\tfrac{593}{60}\nu+\tfrac{2}{3}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{241}{30}\nu^{2}+\tfrac{11}{20}\nu+\tfrac{2}{3}\right)\chi_{S}\right]$
$\displaystyle\quad+v_{\Omega}^{6}\bigg{[}\left(-12\nu^{2}+\tfrac{11}{2}\nu-\tfrac{7}{4}\right)\chi_{A}^{2}+\left(44\nu^{2}-\nu-\tfrac{7}{2}\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(6\nu^{2}-\tfrac{27}{2}\nu-\tfrac{7}{4}\right)\chi_{S}^{2}\bigg{]}$
$\displaystyle\quad+i\big{(}\Omega
H_{\rm{EOB}}\big{)}^{2}\left[\left(\tfrac{7339}{540}\nu-\tfrac{81}{20}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{593}{108}\nu-\tfrac{81}{20}\right)\chi_{S}\right],$
(107b) $\displaystyle\delta_{33}$
$\displaystyle=\tfrac{13}{10}\left(H_{\mathrm{EOB}}\Omega\right)+\tfrac{39\pi}{7}\left(H_{\mathrm{EOB}}\Omega\right)^{2}+\left(-\tfrac{227827}{3000}+\tfrac{78\pi^{2}}{7}\right)\left(H_{\mathrm{EOB}}\Omega\right)^{3}-\tfrac{80897}{2430}\nu
v_{\Omega}^{5},$ (107c)

where the imaginary part of $f_{33}^{\text{S}}$ is included in $\delta_{33}$
in Ref. [118], but we moved it to $f_{33}^{\text{S}}$ to facilitate the
implementation in the equal-mass limit, for which we pull the factor $\delta$
from the leading order $h_{33}^{\text{N}}$ into $f_{\ell m}$ to cancel the
divergent $1/\delta$.

For the $(4,4)$ mode, we use

$\displaystyle\rho_{44}$
$\displaystyle=1+v_{\Omega}^{2}\left[\tfrac{1614-5870\nu+2625\nu^{2}}{1320(-1+3\nu)}\right]+v_{\Omega}^{3}\left[\left(\tfrac{2}{3}-\tfrac{41\nu}{15}+\tfrac{14\nu^{2}}{5}\right)\tfrac{1}{(-1+3\nu)}\chi_{S}+\delta\left(\tfrac{2}{3}-\tfrac{13\nu}{5}\right)\tfrac{1}{(-1+3\nu)}\chi_{A}\right]$
$\displaystyle\quad+v_{\Omega}^{4}\left[-\tfrac{14210377}{8808800(1-3\nu)^{2}}+\tfrac{32485357\nu}{4404400(1-3\nu)^{2}}-\tfrac{1401149\nu^{2}}{1415700(1-3\nu)^{2}}-\tfrac{801565\nu^{3}}{37752(1-3\nu)^{2}}+\tfrac{3976393\nu^{4}}{1006720(1-3\nu)^{2}}+\tfrac{1}{2}\chi_{A}^{2}-2\nu\chi_{A}^{2}+\delta\chi_{A}\chi_{S}+\tfrac{1}{2}\chi_{S}^{2}\right]$
$\displaystyle\quad+v_{\Omega}^{5}\Bigg{[}\left(-\tfrac{69}{55}+\tfrac{16571\nu}{1650}-\tfrac{2673\nu^{2}}{100}+\tfrac{8539\nu^{3}}{440}+\tfrac{591\nu^{4}}{44}\right)\tfrac{1}{(1-3\nu)^{2}}\chi_{S}+\delta\left(-\tfrac{69}{55}+\tfrac{10679\nu}{1650}-\tfrac{1933\nu^{2}}{220}+\tfrac{597\nu^{3}}{440}\right)\tfrac{1}{(1-3\nu)^{2}}\chi_{A}\Bigg{]}$
$\displaystyle\quad+v_{\Omega}^{6}\left[\tfrac{16600939332793}{1098809712000}-\tfrac{12568}{3465}\text{eulerlog}\left(4,v_{\Omega}\right)\right]+v_{\Omega}^{8}\left[-\tfrac{172066910136202271}{19426955708160000}+\tfrac{845198}{190575}\text{eulerlog}\left(4,v_{\Omega}\right)\right]$
$\displaystyle\quad+v_{\Omega}^{10}\left[-\tfrac{17154485653213713419357}{568432724020761600000}+\tfrac{22324502267}{3815311500}\text{eulerlog}\left(4,v_{\Omega}\right)\right]\,,$
(108a) $\displaystyle\delta_{44}$
$\displaystyle=\tfrac{(112+219\nu)}{120(1-3\nu)}(\Omega
H_{\mathrm{EOB}})+\tfrac{25136\pi}{3465}(\Omega
H_{\mathrm{EOB}})^{2}+\left(\tfrac{201088}{10395}\pi^{2}-\tfrac{55144}{375}\right)(\Omega
H_{\mathrm{EOB}})^{3}\,,$ (108b)

and for the $(5,5)$ mode

$\displaystyle\rho_{55}^{\mathrm{NS}}=$ $\displaystyle
1+v_{\Omega}^{2}\left[\tfrac{487}{390(-1+2\nu)}-\tfrac{649\nu}{195(-1+2\nu)}+\tfrac{256\nu^{2}}{195(-1+2\nu)}\right]-\tfrac{3353747}{2129400}v_{\Omega}^{4}$
$\displaystyle+v_{\Omega}^{6}\left[\tfrac{190606537999247}{11957879934000}-\tfrac{1546}{429}\text{eulerlog}\left(5,v_{\Omega}\right)\right]+v_{\Omega}^{8}\left[-\tfrac{1213641959949291437}{118143853747920000}+\tfrac{376451}{83655}\text{eulerlog}\left(5,v_{\Omega}\right)\right]$
$\displaystyle+v_{\Omega}^{10}\left[-\tfrac{150082616449726042201261}{4837990810977324000000}+\tfrac{2592446431}{456756300}\text{eulerlog}\left(5,v_{\Omega}\right)\right]\,,$
(109a) $\displaystyle f_{55}^{\mathrm{S}}=$
$\displaystyle\left[\left(-\tfrac{70\nu}{3(-1+2\nu)}+\tfrac{110\nu^{2}}{3(-1+2\nu)}+\tfrac{10}{3(-1+2\nu)}\right)\frac{\chi_{A}}{\delta}+\left(\tfrac{10}{3(-1+2\nu)}-\tfrac{10\nu}{-1+2\nu}+\tfrac{10\nu^{2}}{-1+2\nu}\right)\chi_{S}\right]v_{\Omega}^{3}+\left[\tfrac{5}{2}\delta^{2}\chi_{A}^{2}+5\delta\chi_{A}\chi_{S}+\tfrac{5}{2}\chi_{S}^{2}\right]v_{\Omega}^{4},$
(109b) $\displaystyle\delta_{55}=$
$\displaystyle\tfrac{(96875+857528\nu)}{131250(1-2\nu)}(\Omega
H_{\mathrm{EOB}})+\tfrac{3865\pi}{429}(\Omega
H_{\mathrm{EOB}})^{2}+\tfrac{-7686949127+954500400\pi^{2}}{31783752}(\Omega
H_{\mathrm{EOB}})^{3}\,,$ (109c)

which are both the same as in SEOBNRv4HM [76].

The (3,2) mode is given by

$\displaystyle\rho_{32}$
$\displaystyle=1+v_{\Omega}\frac{4\nu\chi_{S}}{3(1-3\nu)}+v_{\Omega}^{2}\left[\frac{-\tfrac{32}{27}\nu^{2}+\tfrac{223}{54}\nu-\tfrac{164}{135}}{1-3\nu}{\color[rgb]{0,0,0}-\frac{16\nu^{2}\chi_{S}^{2}}{9(1-3\nu)^{2}}}\right]$
$\displaystyle\quad+v_{\Omega}^{3}\left[{\color[rgb]{0,0,0}\left(\tfrac{13}{9}\nu+\tfrac{2}{9}\right)\frac{\delta\chi_{A}}{1-3\nu}+\left(\tfrac{607}{81}\nu^{3}+\tfrac{503}{81}\nu^{2}-\tfrac{1478}{405}\nu+\tfrac{2}{9}\right)\frac{\chi_{S}}{(1-3\nu)^{2}}}{\color[rgb]{0,0,0}+\frac{320\nu^{3}\chi_{S}^{3}}{81(1-3\nu)^{3}}}\right]$
$\displaystyle\quad+v_{\Omega}^{4}\Bigg{[}\frac{\tfrac{77141}{40095}\nu^{4}-\tfrac{508474}{40095}\nu^{3}-\tfrac{945121}{320760}\nu^{2}+\tfrac{1610009\nu}{320760}-\tfrac{180566}{200475}}{(1-3\nu)^{2}}+{\color[rgb]{0,0,0}\left(4\nu^{2}-3\nu+\tfrac{1}{3}\right)\frac{\chi_{A}^{2}}{1-3\nu}+\left(-\tfrac{50}{27}\nu^{2}-\tfrac{88}{27}\nu+\tfrac{2}{3}\right)\frac{\delta\chi_{A}\chi_{S}}{(1-3\nu)^{2}}}$
$\displaystyle\qquad{\color[rgb]{0,0,0}+\left(-\tfrac{2452}{243}\nu^{4}-\tfrac{1997}{243}\nu^{3}+\tfrac{1435}{243}\nu^{2}-\tfrac{43}{27}\nu+\tfrac{1}{3}\right)\frac{\chi_{S}^{2}}{(1-3\nu)^{3}}}\Bigg{]}$
$\displaystyle\quad+{\color[rgb]{0,0,0}v_{\Omega}^{5}\Bigg{[}\left(-\tfrac{1184225}{96228}\nu^{5}-\tfrac{40204523}{962280}\nu^{4}+\tfrac{101706029}{962280}\nu^{3}-\tfrac{14103833}{192456}\nu^{2}+\tfrac{20471053}{962280}\nu-\tfrac{2788}{1215}\right)\frac{\chi_{S}}{(1-3\nu)^{3}}+\left(\tfrac{608}{81}\nu^{3}+\tfrac{736}{81}\nu^{2}-\tfrac{16}{9}\nu\right)\frac{\delta\chi_{A}\chi_{S}^{2}}{(1-3\nu)^{3}}}$
$\displaystyle\qquad{\color[rgb]{0,0,0}+\left(\tfrac{889673}{106920}\nu^{3}-\tfrac{75737}{5346}\nu^{2}+\tfrac{376177}{35640}\nu-\tfrac{2788}{1215}\right)\frac{\delta\chi_{A}}{(1-3\nu)^{2}}+\left(\tfrac{96176}{2187}\nu^{5}+\tfrac{43528}{2187}\nu^{4}-\tfrac{40232}{2187}\nu^{3}+\tfrac{376}{81}\nu^{2}-\tfrac{8\nu}{9}\right)\frac{\chi_{S}^{3}}{(1-3\nu)^{4}}}$
$\displaystyle\qquad{\color[rgb]{0,0,0}+\left(-\tfrac{32}{3}\nu^{3}+8\nu^{2}-\tfrac{8}{9}\nu\right)\frac{\chi_{A}^{2}\chi_{S}}{(1-3\nu)^{2}}\Bigg{]}}+v_{\Omega}^{6}\left[\tfrac{5849948554}{940355325}-\tfrac{104\text{
eulerlog}(2,v_{\Omega})}{63}\right]+v_{\Omega}^{8}\left[\tfrac{17056\text{
eulerlog}(2,v_{\Omega})}{8505}-\tfrac{10607269449358}{3072140846775}\right]$
$\displaystyle\qquad+{\color[rgb]{0,0,0}v_{\Omega}^{10}\left[-\tfrac{1312549797426453052}{176264081083715625}+\tfrac{18778864\text{
eulerlog}(2,v_{\Omega})}{12629925}\right]},$ (110a) $\displaystyle\delta_{32}$
$\displaystyle=\left(\tfrac{11}{5}\nu+\tfrac{2}{3}\right)\frac{\Omega
H_{\text{EOB}}}{1-3\nu}+\tfrac{52}{21}\pi(\Omega
H_{\mathrm{EOB}})^{2}+\left(\tfrac{208}{63}\pi^{2}-\tfrac{9112}{405}\right)(\Omega
H_{\mathrm{EOB}})^{2},$ (110b)

where we added all spin contributions beyond the LO spin-orbit in $\rho_{32}$,
as well as the test-mass limit terms given in Eq. (49).

The (4,3) mode is given by

$\displaystyle\rho_{43}^{\text{NS}}$
$\displaystyle=1+\frac{v_{\Omega}^{2}}{1-2\nu}\left(-\tfrac{10}{11}\nu^{2}+\tfrac{547}{176}\nu-\tfrac{111}{88}\right)-\tfrac{6894273}{7047040}v_{\Omega}^{4}+v_{\Omega}^{6}\left[\tfrac{1664224207351}{195343948800}-\tfrac{1571}{770}\text{
eulerlog}(3,v_{\Omega})\right]$
$\displaystyle\quad+{\color[rgb]{0,0,0}v}_{\Omega}^{8}\left[-\tfrac{2465107182496333}{460490801971200}+\tfrac{174381}{67760}\text{
eulerlog}(3,v_{\Omega})\right]$ (111a) $\displaystyle f_{43}^{\text{S}}$
$\displaystyle=\frac{v_{\Omega}}{1-2\nu}\left(\tfrac{5}{2}\nu\chi_{S}-\tfrac{5}{2}\nu\frac{\chi_{A}}{\delta}\right){\color[rgb]{0,0,0}+\frac{v_{\Omega}^{3}}{1-2\nu}\left[\left(\tfrac{887}{44}\nu-\tfrac{3143}{132}\nu^{2}\right)\frac{\chi_{A}}{\delta}+\left(-\tfrac{529}{132}\nu^{2}-\tfrac{667}{44}\nu\right)\chi_{S}\right]}$
$\displaystyle\quad+{\color[rgb]{0,0,0}\frac{v_{\Omega}^{4}}{1-2\nu}\Big{[}\left(12\nu^{2}-\tfrac{37}{3}\nu+\tfrac{3}{2}\right)\chi_{A}^{2}+\left(\tfrac{137}{6}\nu^{2}-18\nu+3\right)\frac{\chi_{A}\chi_{S}}{\delta}+\left(\tfrac{35}{6}\nu^{2}+\tfrac{1}{3}\nu+\tfrac{3}{2}\right)\chi_{S}^{2}\Big{]}},$
(111b) $\displaystyle\delta_{43}$
$\displaystyle=\left(\tfrac{4961}{810}\nu+\tfrac{3}{5}\right)\frac{\Omega
H_{\text{EOB}}}{1-2\nu}+\tfrac{1571}{385}\pi(\Omega H_{\mathrm{EOB}})^{2},$
(111c)

where we added all spin contributions beyond the LO spin-orbit in
$f_{43}^{\text{S}}$, and the test-mass limit terms of Eq. (49) in
$\rho_{43}^{\text{NS}}$.

All other modes, which are used in the RR force in Eq. (16), are the same as
in SEOBNRv4HM. They are written in Refs. [73, 149], but we also list them here
for completeness:

$\displaystyle\rho_{31}^{\text{NS}}$
$\displaystyle=1-v_{\Omega}^{2}\left[\frac{2}{9}\nu+\frac{13}{18}\right]+v_{\Omega}^{4}\left[-\frac{829}{1782}\nu^{2}-\frac{1685}{1782}\nu+\frac{101}{7128}\right]+v_{\Omega}^{6}\left[\frac{11706720301}{6129723600}-\frac{26}{63}\text{eulerlog}(1,v_{\Omega}^{2})\right]$
$\displaystyle\quad+v_{\Omega}^{8}\left[\frac{169}{567}\text{eulerlog}(1,v_{\Omega}^{2})+\frac{2606097992581}{4854741091200}\right],$
(112) $\displaystyle f_{31}^{\text{S}}$
$\displaystyle=v_{\Omega}^{3}\left[\left(\frac{11}{2}\nu-2\right)\frac{\chi_{A}}{\delta}+\left(\frac{13}{2}\nu-2\right)\chi_{S}\right],$
(113) $\displaystyle\delta_{31}$ $\displaystyle=\frac{13}{30}\Omega
H_{\mathrm{EOB}}+\frac{13}{21}\pi(\Omega
H_{\mathrm{EOB}})^{2}+\left(\frac{26}{63}\pi^{2}-\frac{227827}{81000}\right)(\Omega
H_{\mathrm{EOB}})^{3},$ (114) $\displaystyle\rho_{42}$
$\displaystyle=1+\frac{285\,\nu^{2}-3530\,\nu+1146}{1320\,(3\,\nu-1)}v_{\Omega}^{2}-\frac{v_{\Omega}^{3}}{15(1-3\nu)}\left[(78\nu^{2}-59\nu+10)\chi_{S}+(10-21\nu)\delta\,\chi_{A}\right]$
$\displaystyle\quad+\frac{-379526805\,\nu^{4}-3047981160\,\nu^{3}+1204388696\,\nu^{2}+295834536\,\nu-114859044}{317116800\,(1-3\,\nu)^{2}}v_{\Omega}^{4}$
$\displaystyle\quad+\left[\frac{848238724511}{219761942400}-\frac{3142}{3465}\text{eulerlog}(2,v_{\Omega}^{2})\right]v_{\Omega}^{6}\,,$
(115) $\displaystyle\delta_{42}$
$\displaystyle=\left(\frac{7}{15}+\frac{14}{5}\nu\right)\frac{\Omega
H_{\mathrm{EOB}}}{1-3\nu}+\frac{6284}{3465}\pi(\Omega H_{\mathrm{EOB}})^{2},$
(116) $\displaystyle\rho_{41}^{\text{NS}}$
$\displaystyle=1+\frac{288\,\nu^{2}-1385\,\nu+602}{528\,(2\,\nu-1)}v_{\Omega}^{2}-\frac{7775491}{21141120}v_{\Omega}^{4}+\left[\frac{1227423222031}{1758095539200}-\frac{1571}{6930}\text{eulerlog}(1,v_{\Omega}^{2})\right]\,v_{\Omega}^{6}\,,$
(117) $\displaystyle f_{41}^{\text{S}}$
$\displaystyle=\frac{5}{2}\nu\frac{v_{\Omega}}{1-2\nu}\left(\chi_{S}-\frac{\chi_{A}}{\delta}\right),$
(118) $\displaystyle\delta_{41}$
$\displaystyle=\left(\frac{1}{5}+\frac{507}{10}\nu\right)\frac{\Omega
H_{\mathrm{EOB}}}{1-2\nu}+\frac{1571}{3465}\pi(\Omega H_{\mathrm{EOB}})^{2},$
(119) $\displaystyle\rho_{54}$
$\displaystyle=1+\frac{33320\,\nu^{3}-127610\,\nu^{2}+96019\,\nu-17448}{13650\,(5\,\nu^{2}-5\,\nu+1)}v_{\Omega}^{2}-\frac{16213384}{15526875}v_{\Omega}^{4}\,,$
(120) $\displaystyle\delta_{54}$ $\displaystyle=\frac{8}{15}\Omega
H_{\text{EOB}},$ (121) $\displaystyle\rho_{53}$
$\displaystyle=1+\frac{176\,\nu^{2}-850\,\nu+375}{390\,(2\,\nu-1)}v_{\Omega}^{2}-\frac{410833}{709800}v_{\Omega}^{4}\,,$
(122) $\displaystyle\delta_{53}$ $\displaystyle=\frac{31}{70}\Omega
H_{\text{EOB}},$ (123) $\displaystyle\rho_{52}$
$\displaystyle=1+\frac{21980\,\nu^{3}-104930\,\nu^{2}+84679\,\nu-15828}{13650\,(5\,\nu^{2}-5\,\nu+1)}v_{\Omega}^{2}-\frac{7187914}{15526875}v_{\Omega}^{4}\,,$
(124) $\displaystyle\delta_{52}$ $\displaystyle=\frac{4}{15}\Omega
H_{\text{EOB}},$ (125) $\displaystyle\rho_{51}$
$\displaystyle=1+\frac{8\,\nu^{2}-626\,\nu+319}{390\,(2\,\nu-1)}v_{\Omega}^{2}-\frac{31877}{304200}v_{\Omega}^{4}\,,$
(126) $\displaystyle\delta_{51}$ $\displaystyle=\frac{31}{210}\Omega
H_{\text{EOB}},$ (127) $\displaystyle\rho_{66}$
$\displaystyle=1+\frac{273\,\nu^{3}-861\,\nu^{2}+602\,\nu-106}{84\,(5\,\nu^{2}-5\,\nu+1)}v_{\Omega}^{2}-\frac{1025435}{659736}v_{\Omega}^{4}\,,$
(128) $\displaystyle\delta_{66}$ $\displaystyle=\frac{43}{70}\Omega
H_{\text{EOB}},$ (129) $\displaystyle\rho_{65}$
$\displaystyle=1+\frac{220\,\nu^{3}-910\,\nu^{2}+838\,\nu-185}{144\,(3\,\nu^{2}-4\,\nu+1)}v_{\Omega}^{2}\,,$
(130) $\displaystyle\delta_{65}$ $\displaystyle=\frac{10}{21}\Omega
H_{\text{EOB}},$ (131) $\displaystyle\rho_{64}$
$\displaystyle=1+\frac{133\,\nu^{3}-581\,\nu^{2}+462\,\nu-86}{84\,(5\,\nu^{2}-5\,\nu+1)}v_{\Omega}^{2}-\frac{476887}{659736}v_{\Omega}^{4}\,,$
(132) $\displaystyle\delta_{64}$ $\displaystyle=\frac{43}{105}\Omega
H_{\text{EOB}},$ (133) $\displaystyle\rho_{63}$
$\displaystyle=1+\frac{156\,\nu^{3}-750\,\nu^{2}+742\,\nu-169}{144\,(3\,\nu^{2}-4\,\nu+1)}v_{\Omega}^{2}\,,$
(134) $\displaystyle\delta_{63}$ $\displaystyle=\frac{2}{7}\Omega
H_{\text{EOB}},$ (135) $\displaystyle\rho_{62}$
$\displaystyle=1+\frac{49\,\nu^{3}-413\,\nu^{2}+378\,\nu-74}{84\,(5\,\nu^{2}-5\,\nu+1)}v_{\Omega}^{2}-\frac{817991}{3298680}v_{\Omega}^{4}\,,$
(136) $\displaystyle\delta_{62}$ $\displaystyle=\frac{43}{210}\Omega
H_{\text{EOB}},$ (137) $\displaystyle\rho_{61}$
$\displaystyle=1+\frac{124\,\nu^{3}-670\,\nu^{2}+694\,\nu-161}{144\,(3\,\nu^{2}-4\,\nu+1)}v_{\Omega}^{2}\,,$
(138) $\displaystyle\delta_{61}$ $\displaystyle=\frac{2}{21}\Omega
H_{\text{EOB}},$ (139) $\displaystyle\rho_{77}$
$\displaystyle=1+\frac{1380\nu^{3}-4963\nu^{2}+4246\nu-906}{714\left(3\nu^{2}-4\nu+1\right)}v_{\Omega}^{2}\,,$
(140) $\displaystyle\delta_{77}$ $\displaystyle=\frac{19}{36}\Omega
H_{\text{EOB}},$ (141) $\displaystyle\rho_{76}$
$\displaystyle=1+\frac{6104\nu^{4}-29351\nu^{3}+37828\nu^{2}-16185\nu+2144}{1666\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)}v_{\Omega}^{2}\,,$
(142) $\displaystyle\rho_{75}$
$\displaystyle=1+\frac{804\nu^{3}-3523\nu^{2}+3382\nu-762}{714\left(3\nu^{2}-4\nu+1\right)}v_{\Omega}^{2}\,,$
(143) $\displaystyle\delta_{75}$ $\displaystyle=\frac{95}{252}\Omega
H_{\text{EOB}},$ (144) $\displaystyle\rho_{74}$
$\displaystyle=1+\frac{41076\nu^{4}-217959\nu^{3}+298872\nu^{2}-131805\nu+17756}{14994\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)}v_{\Omega}^{2}\,,$
(145) $\displaystyle\rho_{73}$
$\displaystyle=1+\frac{420\nu^{3}-2563\nu^{2}+2806\nu-666}{714\left(3\nu^{2}-4\nu+1\right)}v_{\Omega}^{2}\,,$
(146) $\displaystyle\delta_{73}$ $\displaystyle=\frac{19}{84}\Omega
H_{\text{EOB}},$ (147) $\displaystyle\rho_{72}$
$\displaystyle=1+\frac{32760\nu^{4}-190239\nu^{3}+273924\nu^{2}-123489\nu+16832}{14994\left(7\nu^{3}-14\nu^{2}+7\nu-1\right)}v_{\Omega}^{2}\,,$
(148) $\displaystyle\rho_{71}$
$\displaystyle=1+\frac{228\nu^{3}-2083\nu^{2}+2518\nu-618}{714\left(3\nu^{2}-4\nu+1\right)}v_{\Omega}^{2}\,,$
(149) $\displaystyle\delta_{71}$ $\displaystyle=\frac{19}{252}\Omega
H_{\text{EOB}},$ (150) $\displaystyle\rho_{88}$
$\displaystyle=1+\frac{3482-26778\nu+64659\nu^{2}-53445\nu^{3}+12243\nu^{4}}{2736(-1+7\nu-14\nu^{2}+7\nu^{3})}v_{\Omega}^{2}\,,$
(151) $\displaystyle\rho_{87}$
$\displaystyle=1+\frac{23478-154099\nu+309498\nu^{2}-207550\nu^{3}+38920\nu^{4}}{18240(-1+6\nu-10\nu^{2}+4\nu^{3})}v_{\Omega}^{2}\,,$
(152) $\displaystyle\rho_{86}$
$\displaystyle=1+\frac{1002-7498\nu+17269\nu^{2}-13055\nu^{3}+2653\nu^{4}}{912(-1+7\nu-14\nu^{2}+7\nu^{3})}v_{\Omega}^{2}\,,$
(153) $\displaystyle\rho_{85}$
$\displaystyle=1+\frac{4350-28055\nu+54642\nu^{2}-34598\nu^{3}+6056\nu^{4}}{3648(-1+6\nu-10\nu^{2}+4\nu^{3})}v_{\Omega}^{2}\,,$
(154) $\displaystyle\rho_{84}$
$\displaystyle=1+\frac{2666-19434\nu+42627\nu^{2}-28965\nu^{3}+4899\nu^{4}}{2736(-1+7\nu-14\nu^{2}+7\nu^{3})}v_{\Omega}^{2}\,,$
(155) $\displaystyle\rho_{83}$
$\displaystyle=1+\frac{20598-131059\nu+249018\nu^{2}-149950\nu^{3}+24520\nu^{4}}{18240(-1+6\nu-10\nu^{2}+4\nu^{3})}v_{\Omega}^{2}\,,$
(156) $\displaystyle\rho_{82}$
$\displaystyle=1+\frac{2462-17598\nu+37119\nu^{2}-22845\nu^{3}+3063\nu^{4}}{2736(-1+7\nu-14\nu^{2}+7\nu^{3})}v_{\Omega}^{2}\,,$
(157) $\displaystyle\rho_{81}$
$\displaystyle=1+\frac{20022-126451\nu+236922\nu^{2}-138430\nu^{3}+21640\nu^{4}}{18240(-1+6\nu-10\nu^{2}+4\nu^{3})}v_{\Omega}^{2}\,.$
(158)

##  Appendix C Fits of nonquasicircular input values

In this appendix we provide fits for the nonquasicircular (NQC) input values,
$\left|h_{\ell m}\left(t_{\text{match }}^{\ell m}\right)\right|$,
$\partial_{t}\left|h_{\ell m}\left(t_{\text{match }}^{\ell m}\right)\right|$,
$\partial_{t}^{2}\left|h_{\ell m}\left(t_{\text{match }}^{\ell
m}\right)\right|$, $\omega_{\ell m}\left(t_{\text{match }}^{\ell m}\right)$,
$\partial_{t}\omega_{\ell m}\left(t_{\text{match }}^{\ell m}\right)$. To
produce the fits we used NR simulations with the highest level of resolution
available and extrapolation order $N=2$. Depending on the mode, we excluded a
different number of NR waveforms from the fits, where numerical errors
prevented us from fitting them accurately. As in Ref. [76] we define the
following combinations of $m_{1}$, $m_{2}$, $\chi_{1}$, $\chi_{2}$ to be used
in the fits.

$\displaystyle\delta$ $\displaystyle=\dfrac{(m_{1}-m_{2})}{(m_{1}+m_{2}}),$
(159) $\displaystyle\chi_{33}$ $\displaystyle=\chi_{S}\delta+\chi_{A}$ (160)
$\displaystyle\chi_{21A}$
$\displaystyle=\frac{\chi_{S}}{1-1.3\nu}\delta+\chi_{A}$ (161)
$\displaystyle\chi_{44A}$ $\displaystyle=(1-5\nu)\chi_{S}+\chi_{A}\delta$
(162) $\displaystyle\chi_{21D}$
$\displaystyle=\frac{\chi_{S}}{1-2\nu}\delta+\chi_{A}$ (163)
$\displaystyle\chi_{44D}$ $\displaystyle=(1-7\nu)\chi_{S}+\chi_{A}\delta$
(164) $\displaystyle\chi$
$\displaystyle=\chi_{S}+\chi_{A}\frac{\delta}{1-2\nu}$ (165)

The variables $\chi_{33}$, $\chi_{21A}$, $\chi_{21D}$ vanish by construction
for equal-mass equal-spin configurations, and are used to enforce that the
odd-$m$ modes also vanish in the same limit as required by symmetry.

###  C.1 Amplitude’s fits

$\displaystyle\dfrac{|h_{22}^{\rm{NR}}(t_{22}^{\rm{match}})|}{\nu}=$
$\displaystyle|0.430147\chi^{3}\nu-0.084939\chi^{3}+0.619889\chi^{2}\nu^{2}-0.020826\chi^{2}-13.357614\chi\nu^{3}$
$\displaystyle+7.194264\chi\nu^{2}-1.743135\chi\nu+0.18694\chi+71.979698\nu^{4}-46.87586\nu^{3}$
$\displaystyle+12.440405\nu^{2}-0.868289\nu+1.467097|$ (166)
$\displaystyle\dfrac{|h_{33}^{\rm{NR}}(t_{33}^{\rm{match}})|}{\nu}=$
$\displaystyle|-0.088371\chi_{33}^{2}\delta\nu+0.036258\chi_{33}^{2}\delta+1.057731\chi_{33}\nu^{2}-0.466709\chi_{33}\nu$
$\displaystyle+0.099543\chi_{33}+1.96267\delta\nu^{2}+0.027833\delta\nu+0.558808\delta|$
(167) $\displaystyle\dfrac{|h_{21}^{\rm{NR}}(t_{21}^{\rm{match}})|}{\nu}=$
$\displaystyle|-0.033175\chi_{21A}^{3}\delta+0.086356\chi_{21A}^{2}\delta\nu-0.049897\chi_{21A}^{2}\delta+0.012706\chi_{21A}\delta$
$\displaystyle+0.168668\chi_{21A}\nu-0.285597\chi_{21A}+1.067921\delta\nu^{2}-0.189346\delta\nu+0.431426\delta|$
(168) $\displaystyle\dfrac{|h_{44}^{\rm{NR}}(t_{44}^{\rm{match}})|}{\nu}=$
$\displaystyle|0.031483\chi_{44A}^{2}-0.180165\chi_{44A}\nu+0.063931\chi_{44A}+6.239418\nu^{3}-1.947473\nu^{2}$
$\displaystyle-0.615307\nu+0.262533|$ (169)
$\displaystyle\dfrac{|h_{55}^{\rm{NR}}(t_{55}^{\rm{match}})|}{\nu}=$
$\displaystyle|-7.402839\chi_{33}\nu^{3}+3.965852\chi_{33}\nu^{2}-0.762776\chi_{33}\nu+0.062757\chi_{33}$
$\displaystyle+1.093812\delta\nu^{2}-0.462142\delta\nu+0.125468\delta|$ (170)
$\displaystyle\dfrac{|h_{32}^{\rm{NR}}(t_{32}^{\rm{match}})|}{\nu}=$
$\displaystyle|0.022598\chi^{2}+0.307803\chi\nu-0.020771\chi+8.917771\nu^{3}-2.194506\nu^{2}$
$\displaystyle-0.387911\nu+0.155446|$ (171)
$\displaystyle\dfrac{|h_{43}^{\rm{NR}}(t_{43}^{\rm{match}})|}{\nu}=$
$\displaystyle|-0.071554\chi_{33}^{2}\delta\nu+0.021932\chi_{33}^{2}\delta-1.738079\chi_{33}\nu^{2}+0.436576\chi_{33}\nu$
$\displaystyle-0.020081\chi_{33}+0.809615\delta\nu^{2}-0.273364\delta\nu+0.07442\delta|$
(172)

###  C.2 Amplitude-first-derivative’s fits

$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{22}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{22}}\equiv$
$\displaystyle 0$ (173)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{33}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{33}}=$
$\displaystyle\chi_{33}^{2}\delta\left(0.004941\nu-0.002094\right)$
$\displaystyle+0.001781\left|{\chi_{33}^{2}+\chi_{33}\delta\left(39.247538\nu-2.986889\right)+\delta^{2}\left(85.173306\nu+4.637906\right)}\right|^{1/2}$
(174)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{21}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{21}}=$
$\displaystyle\chi_{21D}\delta\left(0.023534\nu-0.008064\right)+\delta\left(0.006743-0.0297\nu\right)$
$\displaystyle+0.008256\left|{\chi_{21D}-\delta\left(5.471011\nu^{2}+1.235589\nu+0.815482\right)}\right|$
(175)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{44}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{44}}=$
$\displaystyle-0.001251\chi_{44D}^{3}+0.006387\chi_{44D}^{2}\nu-0.001223\chi_{44D}^{2}-0.034308\chi_{44D}\nu^{2}$
$\displaystyle+0.014373\chi_{44D}\nu-0.000681\chi_{44D}+1.134679\nu^{3}-0.417056\nu^{2}$
$\displaystyle+0.024004\nu+0.003498$ (176)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{55}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{55}}=$
$\displaystyle\chi_{33}^{2}\delta\left(0.008568\nu-0.00155\right)+\chi_{33}\delta\left(0.002705\nu-0.001015\right)$
$\displaystyle+\delta\left(0.002563-0.010891\nu\right)+0.000284\left|{\chi_{33}+\delta\left(32.459725\nu+0.165336\right)}\right|$
(177)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{32}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{32}}=$
$\displaystyle-0.000806\chi^{3}-0.011027\chi^{2}\nu+0.002999\chi^{2}-0.14087\chi\nu^{2}+0.063211\chi\nu$
$\displaystyle-0.006783\chi+1.693423\nu^{3}-0.510999\nu^{2}+0.020607\nu+0.003674$
(178)
$\displaystyle\dfrac{1}{\nu}\dfrac{d|h_{43}^{\rm{NR}}(t)|}{dt}\bigg{|}_{t=t^{\rm{match}}_{43}}=$
$\displaystyle\chi_{33}^{2}\delta\left(0.001773-0.012159\nu\right)+\chi_{33}\delta\left(0.022249\nu-0.004295\right)$
$\displaystyle+\delta\left(0.012043\nu-0.001067\right)+0.00082\left|{\chi_{33}+\delta\left(3.880171-20.015436\nu\right)}\right|$
(179)

###  C.3 Amplitude-second-derivative’s fits

$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{22}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{22}}=$
$\displaystyle
0.000386\chi^{2}+0.003589\chi\nu+0.001326\chi-0.003353\nu^{2}-0.005615\nu-0.002457$
(180)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{33}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{33}}=$
$\displaystyle\chi_{33}\delta\left(0.000552\nu+0.001029\right)-0.000218$
$\displaystyle\cdot\left|{\chi_{33}+\delta\left(-2188.340923\nu^{4}+1331.981345\nu^{3}-289.772357\nu^{2}+32.212775\nu+3.396168\right)}\right|$
(181)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{21}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{21}}=$
$\displaystyle
0.00015\delta-\left|0.000316\chi_{21D}^{3}-\chi_{21D}^{2}\delta\left(-0.043291\nu^{2}+0.005682\nu+0.000502\right)\right.$
$\displaystyle\left.+0.000372\chi_{21D}\delta-\delta\left(0.003643\nu+2.8\cdot
10^{-5}\right)\right|$ (182)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{44}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{44}}=$
$\displaystyle-0.000591\chi^{2}\nu+0.000174\chi^{2}-0.000501\chi\nu+0.000318\chi+0.138496\nu^{3}$
$\displaystyle-0.047008\nu^{2}+0.003899\nu-0.000451$ (183)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{55}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{55}}=$
$\displaystyle\chi_{33}^{2}\cdot\left(0.000278\nu-5.6\cdot
10^{-5}\right)+\chi_{33}\delta\left(0.000246\nu-6.8\cdot
10^{-5}\right)+\delta\left(0.000118-5.9\cdot 10^{-5}\nu\right)$ (184)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{32}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{32}}=$
$\displaystyle-0.002882\chi^{2}\nu+0.000707\chi^{2}-0.027461\chi\nu^{2}+0.008481\chi\nu-0.000691\chi$
$\displaystyle+0.20836\nu^{3}-0.053191\nu^{2}+0.001604\nu-5.6\cdot 10^{-5}$
(185)
$\displaystyle\dfrac{1}{\nu}\dfrac{d^{2}|h_{43}^{\rm{NR}}(t)|}{dt^{2}}\bigg{|}_{t=t^{\rm{match}}_{43}}=$
$\displaystyle\chi_{33}\delta\left(0.00291\nu-0.000348\right)-5.0\cdot
10^{-6}$
$\displaystyle\cdot\left|{\chi_{33}+\delta\left(-25646.358742\nu^{4}+12647.805787\nu^{3}+291.751053\nu^{2}-531.965263\nu+23.849357\right)}\right|$
(186)

###  C.4 Frequency and frequency-derivative fits

$\displaystyle\omega_{22}^{\rm{NR}}(t_{22}^{\rm{match}})=$
$\displaystyle-0.015259\chi^{4}+0.241948\chi^{3}\nu-0.066927\chi^{3}-0.971409\chi^{2}\nu^{2}+0.518014\chi^{2}\nu$
$\displaystyle-0.087152\chi^{2}+3.751456\chi\nu^{3}-1.697343\chi\nu^{2}+0.250965\chi\nu-0.091339\chi$
$\displaystyle+5.893523\nu^{4}-3.349305\nu^{3}+0.285392\nu^{2}-0.317096\nu-0.268541$
(187) $\displaystyle\omega_{33}^{\rm{NR}}(t_{33}^{\rm{match}})=$
$\displaystyle-0.045141\chi^{3}+0.346675\chi^{2}\nu-0.119419\chi^{2}-0.745924\chi\nu^{2}+0.478915\chi\nu$
$\displaystyle-0.17467\chi+8.887163\nu^{3}-4.226831\nu^{2}-0.427167$ (188)
$\displaystyle\omega_{21}^{\rm{NR}}(t_{21}^{\rm{match}})=$
$\displaystyle-0.01009\chi^{3}+0.077343\chi^{2}\nu-0.02411\chi^{2}-0.168854\chi\nu^{2}+0.159382\chi\nu$
$\displaystyle-0.047635\chi-1.965157\nu^{3}+0.53085\nu^{2}-0.237904\nu-0.176526$
(189) $\displaystyle\omega_{44}^{\rm{NR}}(t_{44}^{\rm{match}})=$
$\displaystyle-0.042529\chi^{3}+0.415864\chi^{2}\nu-0.155222\chi^{2}-0.768712\chi\nu^{2}+0.592568\chi\nu$
$\displaystyle-0.244508\chi+13.651335\nu^{3}-5.490329\nu^{2}-0.574041$ (190)
$\displaystyle\omega_{55}^{\rm{NR}}(t_{55}^{\rm{match}})=$
$\displaystyle-0.091629\chi^{3}+0.802759\chi^{2}\nu-0.246646\chi^{2}-3.04576\chi\nu^{2}+1.43471\chi\nu$
$\displaystyle-0.329591\chi+13.81386\nu^{3}-6.61611\nu^{2}+0.472474\nu-0.589341$
(191) $\displaystyle\omega_{32}^{\rm{NR}}(t_{32}^{\rm{match}})=$
$\displaystyle-0.045647\chi^{2}-2.758635\chi\nu^{2}+0.811353\chi\nu-0.112477\chi-2.346024\nu^{3}$
$\displaystyle+1.57986\nu^{2}-0.317756\nu-0.331141$ (192)
$\displaystyle\omega_{43}^{\rm{NR}}(t_{43}^{\rm{match}})=$
$\displaystyle-0.037919\chi^{3}+0.226903\chi^{2}\nu-0.087288\chi^{2}-0.905919\chi\nu^{2}+0.291092\chi\nu$
$\displaystyle-0.1198\chi-55.534105\nu^{3}+23.913277\nu^{2}-3.487986\nu-0.34306$
(193) $\displaystyle\dot{\omega}_{22}^{\rm{NR}}(t_{22}^{\rm{match}})=$
$\displaystyle
0.000614\chi^{3}-0.008393\chi^{2}\nu+0.001948\chi^{2}+0.07799\chi\nu^{2}-0.028772\chi\nu$
$\displaystyle+0.001705\chi-0.237126\nu^{3}+0.092215\nu^{2}-0.03104\nu-0.005484$
(194) $\displaystyle\dot{\omega}_{33}^{\rm{NR}}(t_{33}^{\rm{match}})=$
$\displaystyle
0.001697\chi^{3}-0.016231\chi^{2}\nu+0.003985\chi^{2}+0.154378\chi\nu^{2}-0.050618\chi\nu$
$\displaystyle+0.002721\chi+0.255402\nu^{3}-0.08663\nu^{2}-0.027405\nu-0.009736$
(195) $\displaystyle\dot{\omega}_{21}^{\rm{NR}}(t_{21}^{\rm{match}})=$
$\displaystyle
0.00149\chi^{3}-0.008965\chi^{2}\nu+0.002739\chi^{2}+0.033831\chi\nu^{2}-0.005752\chi\nu$
$\displaystyle+0.002003\chi-0.204368\nu^{3}+0.120705\nu^{2}-0.035144\nu-0.006579$
(196) $\displaystyle\dot{\omega}_{44}^{\rm{NR}}(t_{44}^{\rm{match}})=$
$\displaystyle
0.001812\chi^{3}-0.024687\chi^{2}\nu+0.00568\chi^{2}+0.162693\chi\nu^{2}-0.061205\chi\nu$
$\displaystyle+0.003623\chi+0.536664\nu^{3}-0.094797\nu^{2}-0.045406\nu-0.013038$
(197) $\displaystyle\dot{\omega}_{55}^{\rm{NR}}(t_{55}^{\rm{match}})=$
$\displaystyle
0.001509\chi^{3}-0.01547\chi^{2}\nu+0.002802\chi^{2}+0.164011\chi\nu^{2}-0.056516\chi\nu$
$\displaystyle+0.002072\chi+0.043963\nu^{3}+0.048045\nu^{2}-0.045197\nu-0.008688$
(198) $\displaystyle\dot{\omega}_{32}^{\rm{NR}}(t_{32}^{\rm{match}})=$
$\displaystyle-0.036711\chi^{2}\nu+0.005532\chi^{2}+0.09192\chi\nu^{2}-0.030713\chi\nu+0.005927\chi$
$\displaystyle-2.494788\nu^{3}+0.995116\nu^{2}-0.10163\nu-0.010763$ (199)
$\displaystyle\dot{\omega}_{43}^{\rm{NR}}(t_{43}^{\rm{match}})=$
$\displaystyle
0.000537\chi^{3}-0.009876\chi^{2}\nu+0.003279\chi^{2}+0.13296\chi\nu^{2}-0.060884\chi\nu$
$\displaystyle+0.008513\chi-5.160613\nu^{3}+2.180781\nu^{2}-0.292607\nu-0.005308$
(200)

##  Appendix D Fits for amplitude and phase of merger-ringdown model

In this appendix we provide fits across parameter space for the free
coefficients in the merger-ringdown ansatz given by Eqs. (52) and (53). To
produce the fits we use NR simulations with the highest level of resolution
available and extrapolation order $N=2$. They read:

$\displaystyle c_{1,f}^{22}=$
$\displaystyle-0.001777\chi^{4}+0.062842\chi^{3}\nu-0.018908\chi^{3}+0.013161\chi^{2}\nu^{2}+0.049388\chi^{2}\nu$
$\displaystyle-0.019314\chi^{2}+1.867978\chi\nu^{3}-0.702488\chi\nu^{2}+0.033885\chi\nu-0.011612\chi$
$\displaystyle-4.238246\nu^{4}+2.043712\nu^{3}-0.406992\nu^{2}+0.053589\nu+0.086254$
(201) $\displaystyle c_{2,f}^{22}=$ $\displaystyle
1.021875\chi^{3}\nu-0.20348\chi^{3}-3.556173\chi^{2}\nu^{2}+1.970082\chi^{2}\nu-0.264297\chi^{2}$
$\displaystyle+2.002947\chi\nu^{3}-5.585851\chi\nu^{2}+1.837724\chi\nu-0.27076\chi-63.286459\nu^{4}$
$\displaystyle+44.331389\nu^{3}-9.529573\nu^{2}+1.155695\nu-0.528763$ (202)
$\displaystyle d_{1,f}^{22}=$
$\displaystyle-0.013321\chi^{4}+0.047305\chi^{3}\nu-0.024203\chi^{3}+1.033352\chi^{2}\nu^{2}-0.254351\chi^{2}\nu$
$\displaystyle-0.007847\chi^{2}+4.113463\chi\nu^{3}-1.652924\chi\nu^{2}+0.090834\chi\nu-28.423701\nu^{4}$
$\displaystyle+20.719874\nu^{3}-6.075679\nu^{2}+0.780093\nu+0.135758$ (203)
$\displaystyle d_{2,f}^{22}=$
$\displaystyle\exp(-0.163113\chi^{4}-3.398858\chi^{3}\nu+0.728816\chi^{3}+23.975132\chi^{2}\nu^{2}-10.064954\chi^{2}\nu$
$\displaystyle+1.2115\chi^{2}+9.057306\chi\nu^{3}-5.268296\chi\nu^{2}+0.464553\chi\nu+0.56269\chi$
$\displaystyle-352.249383\nu^{4}+275.843499\nu^{3}-81.483314\nu^{2}+11.184576\nu+0.03571)$
(204) $\displaystyle c_{1,f}^{33}=$
$\displaystyle-0.00956\chi^{3}+0.029459\chi^{2}\nu-0.020264\chi^{2}-0.494524\chi\nu^{2}+0.169463\chi\nu$
$\displaystyle-0.026285\chi-5.847417\nu^{3}+1.957462\nu^{2}-0.171682\nu+0.093539$
(205) $\displaystyle c_{2,f}^{33}=$
$\displaystyle-0.057346\chi^{3}+0.237107\chi^{2}\nu-0.094285\chi^{2}-4.250609\chi\nu^{2}+1.763105\chi\nu$
$\displaystyle-0.315826\chi+14.801916\nu^{3}-7.060581\nu^{2}+1.158627\nu-0.646888$
(206) $\displaystyle d_{1,f}^{33}=$
$\displaystyle-0.016524\chi^{3}+0.221466\chi^{2}\nu-0.066323\chi^{2}+0.678442\chi\nu^{2}-0.261264\chi\nu$
$\displaystyle+0.006664\chi+2.316434\nu^{3}-2.192227\nu^{2}+0.424582\nu+0.161577$
(207) $\displaystyle d_{2,f}^{33}=$
$\displaystyle\exp(0.275999\chi^{3}-1.830695\chi^{2}\nu+0.512734\chi^{2}+29.072515\chi\nu^{2}-10.581319\chi\nu$
$\displaystyle+1.310643\chi+324.310223\nu^{3}-124.681881\nu^{2}+13.200426\nu+0.410855)$
(208) $\displaystyle c_{1,f}^{21}=$ $\displaystyle
0.173462\chi^{2}\nu-0.028873\chi^{2}+0.197467\chi\nu^{2}-0.026139\chi-2.934735\nu^{3}$
$\displaystyle+1.009106\nu^{2}-0.112721\nu+0.099889$ (209) $\displaystyle
c_{2,f}^{21}=$ $\displaystyle
0.183489\chi^{3}+0.10573\chi^{2}-20.792825\chi\nu^{2}+6.867746\chi\nu-0.484948\chi$
$\displaystyle-54.917585\nu^{3}+16.466312\nu^{2}+0.426316\nu-0.92208$ (210)
$\displaystyle d_{1,f}^{21}=$ $\displaystyle
0.018467\chi^{4}+0.398621\chi^{3}\nu-0.050499\chi^{3}-0.877201\chi^{2}\nu^{2}+0.414553\chi^{2}\nu$
$\displaystyle-0.068277\chi^{2}-10.648526\chi\nu^{3}+4.104918\chi\nu^{2}-0.723576\chi\nu+0.039227\chi$
$\displaystyle+42.715534\nu^{4}-18.280603\nu^{3}+2.236592\nu^{2}-0.048094\nu+0.16335$
(211) $\displaystyle d_{2,f}^{21}=$
$\displaystyle\exp(0.814085\chi^{3}-1.197363\chi^{2}\nu+0.560622\chi^{2}+6.44667\chi\nu^{2}-5.630563\chi\nu$
$\displaystyle+0.949586\chi+91.269183\nu^{3}-27.329751\nu^{2}+1.101262\nu+1.040761)$
(212) $\displaystyle c_{1,f}^{44}=$ $\displaystyle
4.519504\chi\nu^{2}-1.489036\chi\nu+0.068403\chi-1656.065439\nu^{4}+817.835726\nu^{3}$
$\displaystyle-127.055379\nu^{2}+6.921968\nu+0.009386$ (213) $\displaystyle
c_{2,f}^{44}=$ $\displaystyle
0.964861\chi^{3}\nu-0.185226\chi^{3}-12.647814\chi^{2}\nu^{2}+5.264969\chi^{2}\nu-0.539721\chi^{2}$
$\displaystyle-254.719552\chi\nu^{3}+105.698791\chi\nu^{2}-12.107281\chi\nu+0.2244\chi-393.727702\nu^{4}$
$\displaystyle+145.32788\nu^{3}-15.556222\nu^{2}+1.592449\nu-0.677664$ (214)
$\displaystyle d_{1,f}^{44}=$
$\displaystyle-0.020644\chi^{3}+0.494221\chi^{2}\nu-0.127074\chi^{2}+4.297985\chi\nu^{2}-1.284386\chi\nu$
$\displaystyle+0.062684\chi-44.280815\nu^{3}+11.021482\nu^{2}-0.162943\nu+0.166018$
(215) $\displaystyle d_{2,f}^{44}=$
$\displaystyle\exp(37.735116\chi\nu^{2}-12.516669\chi\nu+1.309868\chi-528.368915\nu^{3}+155.115196\nu^{2}$
$\displaystyle-6.612448\nu+0.787726)$ (216) $\displaystyle c_{1,f}^{55}=$
$\displaystyle-0.009957\chi^{3}+0.059748\chi^{2}\nu-0.02146\chi^{2}-0.206811\chi\nu^{2}+0.055078\chi\nu$
$\displaystyle-0.014528\chi-5.966891\nu^{3}+1.76928\nu^{2}-0.055272\nu+0.080368$
(217) $\displaystyle c_{2,f}^{55}=$ $\displaystyle
0.119703\chi^{4}+1.638345\chi^{2}\nu^{2}-0.064725\chi^{2}-28.499278\chi\nu^{3}+3.73034\chi\nu^{2}$
$\displaystyle+1.853723\chi\nu-0.225283\chi-1887.591102\nu^{4}+794.134711\nu^{3}-107.010824\nu^{2}$
$\displaystyle+6.32117\nu-1.507483$ (218) $\displaystyle d_{1,f}^{55}=$
$\displaystyle-0.021537\chi^{3}+0.168071\chi^{2}\nu-0.050263\chi^{2}+0.871799\chi\nu^{2}-0.230057\chi\nu$
$\displaystyle+9.018546\nu^{3}-5.009488\nu^{2}+0.606313\nu+0.150622$ (219)
$\displaystyle d_{2,f}^{55}=$
$\displaystyle\exp(28.839035\chi\nu^{2}-9.726025\chi\nu+0.901423\chi+143.745208\nu^{3}-64.478227\nu^{2}$
$\displaystyle+6.223833\nu+2.058139)$ (220) $\displaystyle c_{1,f}^{32}=$
$\displaystyle-0.133035\chi^{3}+0.641681\chi^{2}\nu-0.111865\chi^{2}+8.987763\chi\nu^{2}-1.582259\chi\nu$
$\displaystyle+0.095604\chi-26.991806\nu^{3}+13.716801\nu^{2}-1.63083\nu+0.157543$
(221) $\displaystyle c_{2,f}^{32}=$ $\displaystyle
0.121608\chi^{3}-1.590623\chi^{2}\nu+0.167231\chi^{2}-25.544931\chi\nu^{2}+10.127968\chi\nu$
$\displaystyle-0.999062\chi-51.469773\nu^{3}+46.209833\nu^{2}-6.484571\nu-0.716883$
(222) $\displaystyle d_{1,f}^{32}=$
$\displaystyle\exp(-0.764015\chi^{3}-8.684722\chi^{2}\nu+0.691946\chi^{2}-0.518291\chi\nu^{2}-1.407934\chi\nu$
$\displaystyle+0.236427\chi+81.222175\nu^{3}-18.040529\nu^{2}+2.216406\nu-1.879455)$
(223) $\displaystyle d_{2,f}^{32}=$
$\displaystyle\exp(-1.819822\chi^{3}-24.501503\chi^{2}\nu+3.287882\chi^{2}-39.324579\chi\nu^{2}+14.379901\chi\nu$
$\displaystyle-215.372399\nu^{3}+136.20936\nu^{2}-16.842816\nu+1.463485)$
(224) $\displaystyle c_{1,f}^{43}=$ $\displaystyle
0.041585\chi^{3}+4.188908\chi\nu^{2}-1.365732\chi\nu+0.058908\chi+44.311948\nu^{3}$
$\displaystyle-22.114177\nu^{2}+3.386082\nu-0.035315$ (225) $\displaystyle
c_{2,f}^{43}=$ $\displaystyle
0.125764\chi^{3}+0.337235\chi^{2}\nu+0.146202\chi^{2}-9.803187\chi\nu^{2}+3.995199\chi\nu$
$\displaystyle-0.240976\chi-57.968821\nu^{3}+7.820929\nu^{2}+3.364741\nu-1.121716$
(226) $\displaystyle d_{1,f}^{43}=$
$\displaystyle\exp(-0.888286\chi^{3}+3.97869\chi^{2}\nu-1.047181\chi^{2}-14.823391\chi\nu^{2}+6.940856\chi\nu$
$\displaystyle-0.367801\chi+366.645645\nu^{3}-161.732513\nu^{2}+19.564699\nu-2.29578)$
(227) $\displaystyle d_{2,f}^{43}=$
$\displaystyle\exp(-0.950676\chi^{3}-0.31428\chi^{2}+39.21796\chi\nu^{2}-10.651167\chi\nu+1.339732\chi$
$\displaystyle+730.42296\nu^{3}-312.960598\nu^{2}+37.402567\nu-0.061894)$
(228)

##  Appendix E Robustness of the calibration pipeline

In this appendix we demonstrate that the calibration pipeline described in
Sec. IV is robust with regard to the number of NR waveforms and the PSD used
in the calibration likelihood.

For the first point, we repeat the procedure for the aligned-spin calibration
parameters $\bm{\theta}_{\rm{S}}\equiv\\{d_{\rm{SO}},\Delta
t^{22}_{\rm{ISCO,S}}\\}$, by using a representative subset of 119 aligned-spin
NR simulations, selected with a greedy algorithm following Ref. [91]. We do
not change the nonspinning fits for
$\bm{\theta}_{\rm{noS}}\equiv\\{a_{6},\Delta t^{22}_{\rm{ISCO,noS}}\\}$ (which
already used a subset of 18 simulations out of 39), and the fits for the
merger-ringdown and NQC corrections. This brings the total number of waveforms
used to 137, which is comparable to the 141 used in the calibration of
SEOBNRv4. As in Sec. V, we compute the (2,2)-mode mismatch of this model
against the entire set of 442 cases summarized in Sec. IV.

Figure 26: Histogram of the maximum (2,2)-mode mismatch over a range of total
masses between 10 and 300 $M_{\odot}$, between the 442 NR simulations used in
this work and SEOBNRv4 (blue), SEOBNRv5 calibrated to 442 NR simulations
(green) and SEOBNRv5 calibrated to 137 NR simulations (orange). The vertical
dashed lines show the medians.

Figure 26 shows the maximum mismatch across a range of total masses between
$[10,300]M_{\odot}$ for SEOBNRv5 calibrated to 442 simulations, SEOBNRv5
calibrated to 137 simulations, and SEOBNRv4. The median (dashed vertical line)
goes only from $1.99\times 10^{-4}$ for SEOBNRv5 calibrated to 442
simulations, to $2.74\times 10^{-4}$ for SEOBNRv5 calibrated to 137
simulations, which is more than $5$ times smaller than SEOBNRv4 ($1.44\times
10^{-3}$). Moreover for SEOBNRv5 calibrated to 137 simulations there is only a
single case with mismatch just above $0.003$ (BFI:q8-7d:0080, with parameters
$(q,\chi_{1},\chi_{2})=(8.0,0.0,-0.8)$). As already pointed out in Sec. V,
even the default SEOBNRv5 model can have mismatches slightly above $10^{-3}$
against similar cases, with high mass ratio, small $a_{+}$, but large
individual spins, as the calibration term $\propto a_{+}d_{\rm{SO}}$ does not
tend to be very effective. This shows that the calibration pipeline does not
rely on using the entire NR dataset described in Sec. IV. Although we added
some critical new NR simulations, especially for high mass ratios, the
improvement with respect to SEOBNRv4 is also largely due to the improved
analytical prescriptions, for the waveform modes, RR force and Hamiltonian,
and the more effective calibration procedure.

The calibration likelihood of Eq. (73) also depends on the Advanced LIGO [169]
PSD. To show that our calibration pipeline is robust with respect to changes
in the shape of the PSD used, we compute again (2,2)-mode mismatches against
NR simulations (as this is the metric used in the likelihood) using a white
noise curve, the Einsten Telescope [204] and Cosmic Explorer [16] PSDs. For
this purpose we use the original fits of $\\{d_{\rm{SO}},\Delta
t^{22}_{\rm{ISCO,S}}\\}$ given in Eqs. (IV.3) and (IV.3).

Figure 27: Histogram of the maximum (2,2)-mode mismatch over a range of total
masses between 10 and 300 $M_{\odot}$, between the 442 NR simulations used in
this work and SEOBNRv5, using a white noise curve and the PSDs of Advanced
LIGO, Einstein Telescope and Cosmic Explorer. The vertical dashed lines show
the medians. The (2,2)-mode of SEOBNRv5 is calibrated using the Advanced LIGO
PSD, but performs as well using different noise curves.

Figure 27 shows again the maximum mismatch, across a range of total masses
between $[10,300]M_{\odot}$, of SEOBNRv5 against the 442 NR simulations used
in this work, using a white noise curve and the PSDs of Advanced LIGO,
Einstein Telescope and Cosmic Explorer. We see that the result is very similar
for all the cases and, despite SEOBNRv5 being calibrated using the Advanced
LIGO PSD, it performs equally well using different noise curves.

##  Appendix F Impact of NQC corrections in the radiation-reaction force

In Sec. III.1 we pointed out that we do not include the NQC corrections in the
SEOBNRv5 RR force. Recently, Refs. [105, 184] implemented a fast prescription
in TEOBResumS to include fits of NQC corrections in both the waveform and RR
force, without requiring an iterative procedure. A similar prescription could
also be used in SEOBNRv5HM. Reference [116] tested it in the nonspinning
limit, finding that the inclusion of NQCs corrections has a smaller effect
than the calibration to 2GSF data in bringing the angular-momentum flux closer
to the NR’s one, except in the last fraction of a GW cycle before merger.
Moreover, Ref. [116] found that the inclusion of the nonspinning NQC
corrections has a negligible effect on the waveform after recalibrating the
conservative dynamics to NR, as the degeneracy between changes in the flux and
in the Hamiltonian reabsorbs any imperfection in the flux with the
calibration. The nonspinning limit of the model is however already very close
to the NR error, and the effect on more challenging aligned-spin cases could
be larger.

While for the current work we did not do a systematic study in the entire
aligned-spin parameter space to include the NQC corrections in the RR force,
in this Appendix we try to understand the potential improvement for one binary
configuration and compare the results with SXS:BBH:1432, having parameters
$(q,\chi_{1},\chi_{2})=(5.839,0.658,0.793)$. Thus, we iteratively include the
NQCs in the RR force and repeat the nested sampling procedure detailed in
Sec.IV.2 to find new values for the aligned-spin calibration parameters
$\\{\Delta t^{22}_{\rm{ISCO,S}},d_{\rm{SO}}\\}$, which are going to slightly
change compared to the default ones, given the different dissipative dynamics.
To do a comparison without performing fits of the NQCs and of the calibration
parameters across parameter space, we directly compare the maximum likelihood
points of the calibration posteriors, corresponding to the values of the
parameters giving the best unfaithfulness and time to merger.

Figure 28: $(2,2)$-mode mode mismatch against the NR simulation SXS:BBH:1432
for the maximum likelihood points of the SEOBNRv5 calibration posteriors, with
and without NQCs in the RR force. The unfaithfulness is very similar, as the
calibration of the Hamiltonian reabsorbs any difference in the dissipative
dynamics. We also show an estimate of the NR error obtained by comparing
simulations of different resolutions. Figure 29: Comparison of the Newtonian
normalized angular-momentum flux between SEOBNRv5 with and without NQCs in the
radiation-reaction, and the NR simulation SXS:BBH:1432. The triangle, square
and diamond correspond respectively to 3, 1 and 0 cycles before merger, which
for each model is taken as the peak of the (2,2)-mode amplitude. SEOBNRv5 with
NQCs matches NR at merger at expected, but does not agree with NR as well as
SEOBNRv5 without NQCs at low frequencies.

Figure 28 shows the $(2,2)$-mode mismatch against NR for the maximum
likelihood points of the SEOBNRv5 calibration posteriors with and without NQCs
in the RR force, together with an estimate of the NR error obtained by
comparing different resolutions. The unfaithfulness is very similar, as the
calibration of the Hamiltonian reabsorbs any difference in the dissipative
dynamics. Moreover, in Fig. 29 we compare to NR the Newtonian normalized
angular-momentum flux between SEOBNRv5 with and without NQCs in the RR force.
The triangle, square and diamond correspond respectively to 3, 1 and 0 cycles
before merger, which for each model is taken as the peak of the $(2,2)$-mode
amplitude. SEOBNRv5 with NQCs matches NR at merger at expected, but does not
agree with NR as well as SEOBNRv5 without NQCs at low frequencies, showing
that the addition of the NQCs does not necessarily improve the flux.

This behavior could be potentially improved by doing more iterations, finding
new values for the NQCs given the corrected calibration parameters, and
repeating the calibration, but would be time-consuming and not necessarily
bring a significant improvement in the waveforms. Nonetheless, a consistent
treatment of the NQCs both in the waveform and the RR force would most likely
provide more faithful representations of the angular-momentum and energy
fluxes, and a more systematic study across parameter space will be done in a
future update of the model, together with a recalibration of the conservative
dynamics. Alternative ways to improve the waveform close to merger, and reduce
the impact of the NQC corrections, should be also investigated, especially in
light of the upcoming eccentric generalization of this model SEOBNRv5EHM, as
past experience in developing SEOBNRv4EHM have demonstarted [89].

##  Appendix G Comparison against time-domain nonprecessing phenomenological
models

In this appendix we compare the performance of SEOBNRv5HM and the Fourier
domain IMRPhenomXHM [42] against the time-domain nonprecessing
phenomenological model IMRPhenomTHM [45, 46], which includes the
$(\ell,|m|)=(2,2),(3,3),(2,1),(4,4),(5,5)$ modes. In particular, we repeat the
mismatch calculation against the NR catalog detailed in Sec. V, both for the
dominant mode only (Fig. 30 and Fig. 31) and for the full polarizations, at
inclination $\iota=\pi/3$ (Fig. 32 and Fig. 33).

Considering the dominant mode mismatches, we see that IMRPhenomT performs
slightly worse than SEOBNRv5, considering both the median mismatch and the
fraction of cases below $10^{-4}$, while IMRPhenomXAS achieves on average
slightly lower values of the maximum unfaithfulness, as already noted in Sec.
V. More quantitatively, IMRPhenomT features $91\%$ of cases with maximum
unfaithfulness below $10^{-3}$, $5\%$ of cases below $10^{-4}$, and a median
of $2.31\times 10^{-4}$ (see also Table 1).

Considering the mismatches of the full polarizations at inclination
$\iota=\pi/3$ we note instead that IMRPhenomTHM is slightly more accurate than
SEOBNRv5HM, and both models are appreciably more accurate than IMRPhenomXHM.
More specifically, IMRPhenomTHM shows maximum unfaithfulness below $10^{-2}$
for $99\%$ of cases, below $10^{-3}$ for $57\%$ of cases, and a median of
$7.49\times 10^{-4}$ (see also Table 2). The NR simulation
BFI:ExtremeAligned:0003, with $q=10.0,\leavevmode\nobreak\
\chi_{1}=0.8,\leavevmode\nobreak\ \chi_{2}=0.5$, is the only outlier reaching
$3\%$ mismatch at high total mass, for both IMRPhenomTHM and SEOBNRv5HM, while
being at the level of $0.1\%$ when considering only the $(2,2)$ mode. This
suggests common limitations in the modeling of the higher modes for such
extreme configurations. In Fig. 34 we show the unfaithfulness between
SEOBNRv5HM, IMRPhenomTHM and BFI:ExtremeAligned:0003, using multipoles up to
$\ell_{\rm{max}}=5$ and $\ell_{\rm{max}}=4$, in both the models and the NR
waveform. We note that the mismatch for high total masses reduces
significantly for both models when removing $\ell=5$ multipoles, indicating
that part of the disagreement is due to mismodelling of the $(5,5)$ mode, as
well as missing contributions in the models from the $(\ell=5,m\neq 5)$
multipoles, which can have a non-negligible impact in the ringdown of such
high mass-ratio binaries. From the behavior of the unfaithfulness as a
function of the total mass in Fig. 32 we appreciate that SEOBNRv5HM is overall
closer to IMRPhenomTHM than to IMRPhenomXHM, as expected given the similar
merger-ringdown model, and the comparable NR calibration coverage.

Figure 30: (2,2)-mode mismatch over a range of total masses between 10 and 300
$M_{\odot}$, between SEOBNRv5, IMRPhenomXAS, IMRPhenomT and the 442 NR
simulations used in this work. The colored lines highlight cases with the
worst maximum mismatch for each model. Figure 31: Histogram of the maximum
(2,2)-mode mismatch over a range of total masses between 10 and 300
$M_{\odot}$, between SEOBNRv5, IMRPhenomXAS, IMRPhenomT and the 442 NR
simulations used in this work. The NR error is estimated by computing the
mismatch between NR simulations with the highest and second-highest
resolutions. The vertical dashed lines show the medians.  Figure 32: The sky-
and-polarization averaged, SNR-weighted mismatch, for inclination
$\iota=\pi/3$, over a range of total masses between 20 and 300 $M_{\odot}$
between SEOBNRv5HM, IMRPhenomXHM, IMRPhenomTHM and the 441 SXS NR simulations
used in this work. The colored lines highlight cases with the worst maximum
mismatch for each model. This comparison highlights the similarity in
performance of the time-domain models SEOBNRv5HM and IMRPhenomTHM. Figure 33:
Histogram of the maximum sky-and-polarization averaged, SNR-weighted mismatch,
for inclination $\iota=\pi/3$, over a range of total masses between 20 and 300
$M_{\odot}$, between SEOBNRv5HM, IMRPhenomXMH, IMRPhenomTHM and the 441 SXS NR
simulations used in this work. The NR error is estimated by computing the
mismatch between NR simulations with the highest and second-highest
resolutions. The vertical dashed lines show the medians. Figure 34: Sky-and-
polarization averaged, SNR-weighted mismatch, for inclination $\iota=\pi/3$ as
a function of the total mass between SEOBNRv5HM, IMRPhenomTHM and the NR
simulation BFI:ExtremeAligned:0003, with $q=10.0,\leavevmode\nobreak\
\chi_{1}=0.8,\leavevmode\nobreak\ \chi_{2}=0.5$, using $\ell_{\rm{max}}=5$ and
$\ell_{\rm{max}}=4$, in both the models and the NR waveform.

## References

  * Abbott  _et al._ [2016a] B. P. Abbott _et al._ (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016a), arXiv:1602.03837 [gr-qc] .
  * Abbott  _et al._ [2019] B. P. Abbott _et al._ (LIGO Scientific, Virgo), GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE] .
  * Venumadhav  _et al._ [2020] T. Venumadhav, B. Zackay, J. Roulet, L. Dai, and M. Zaldarriaga, New binary black hole mergers in the second observing run of Advanced LIGO and Advanced Virgo, Phys. Rev. D 101, 083030 (2020), arXiv:1904.07214 [astro-ph.HE] .
  * Abbott  _et al._ [2021a] R. Abbott _et al._ (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11, 021053 (2021a), arXiv:2010.14527 [gr-qc] .
  * Abbott  _et al._ [2021b] R. Abbott _et al._ (LIGO Scientific, VIRGO), GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, (2021b), arXiv:2108.01045 [gr-qc] .
  * Abbott  _et al._ [2021c] R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run, (2021c), arXiv:2111.03606 [gr-qc] .
  * Nitz  _et al._ [2021] A. H. Nitz, C. D. Capano, S. Kumar, Y.-F. Wang, S. Kastha, M. Schäfer, R. Dhurkunde, and M. Cabero, 3-OGC: Catalog of Gravitational Waves from Compact-binary Mergers, Astrophys. J. 922, 76 (2021), arXiv:2105.09151 [astro-ph.HE] .
  * Olsen  _et al._ [2022] S. Olsen, T. Venumadhav, J. Mushkin, J. Roulet, B. Zackay, and M. Zaldarriaga, New binary black hole mergers in the LIGO-Virgo O3a data, Phys. Rev. D 106, 043009 (2022), arXiv:2201.02252 [astro-ph.HE] .
  * Aasi  _et al._ [2015] J. Aasi _et al._ (LIGO Scientific), Advanced LIGO, Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * Acernese  _et al._ [2015] F. Acernese _et al._ (VIRGO), Advanced Virgo: a second-generation interferometric gravitational wave detector, Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc] .
  * Buikema  _et al._ [2020] A. Buikema _et al._ (aLIGO), Sensitivity and performance of the Advanced LIGO detectors in the third observing run, Phys. Rev. D 102, 062003 (2020), arXiv:2008.01301 [astro-ph.IM] .
  * Tse  _et al._ [2019] M. Tse _et al._ , Quantum-Enhanced Advanced LIGO Detectors in the Era of Gravitational-Wave Astronomy, Phys. Rev. Lett. 123, 231107 (2019).
  * Acernese  _et al._ [2019] F. Acernese _et al._ (Virgo), Increasing the Astrophysical Reach of the Advanced Virgo Detector via the Application of Squeezed Vacuum States of Light, Phys. Rev. Lett. 123, 231108 (2019).
  * Punturo  _et al._ [2010] M. Punturo _et al._ , The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27, 194002 (2010).
  * Reitze  _et al._ [2019] D. Reitze _et al._ , Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833 [astro-ph.IM] .
  * Evans  _et al._ [2021] M. Evans _et al._ , A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, (2021), arXiv:2109.09882 [astro-ph.IM] .
  * Amaro-Seoane  _et al._ [2017] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi, _et al._ , Laser interferometer space antenna, arXiv preprint arXiv:1702.00786 (2017).
  * Pretorius [2005] F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95, 121101 (2005), arXiv:gr-qc/0507014 .
  * Campanelli  _et al._ [2006] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96, 111101 (2006), arXiv:gr-qc/0511048 .
  * Baker  _et al._ [2006] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96, 111102 (2006), arXiv:gr-qc/0511103 .
  * Blackman  _et al._ [2015] J. Blackman, S. E. Field, C. R. Galley, B. Szilágyi, M. A. Scheel, M. Tiglio, and D. A. Hemberger, Fast and Accurate Prediction of Numerical Relativity Waveforms from Binary Black Hole Coalescences Using Surrogate Models, Phys. Rev. Lett. 115, 121102 (2015), arXiv:1502.07758 [gr-qc] .
  * Blackman  _et al._ [2017a] J. Blackman, S. E. Field, M. A. Scheel, C. R. Galley, D. A. Hemberger, P. Schmidt, and R. Smith, A Surrogate Model of Gravitational Waveforms from Numerical Relativity Simulations of Precessing Binary Black Hole Mergers, Phys. Rev. D 95, 104023 (2017a), arXiv:1701.00550 [gr-qc] .
  * Blackman  _et al._ [2017b] J. Blackman, S. E. Field, M. A. Scheel, C. R. Galley, C. D. Ott, M. Boyle, L. E. Kidder, H. P. Pfeiffer, and B. Szilágyi, Numerical relativity waveform surrogate model for generically precessing binary black hole mergers, Phys. Rev. D 96, 024058 (2017b), arXiv:1705.07089 [gr-qc] .
  * Varma  _et al._ [2019a] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, L. E. Kidder, and H. P. Pfeiffer, Surrogate model of hybridized numerical relativity binary black hole waveforms, Phys. Rev. D 99, 064045 (2019a), arXiv:1812.07865 [gr-qc] .
  * Varma  _et al._ [2019b] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses, Phys. Rev. Research. 1, 033015 (2019b), arXiv:1905.09300 [gr-qc] .
  * Williams  _et al._ [2020] D. Williams, I. S. Heng, J. Gair, J. A. Clark, and B. Khamesra, Precessing numerical relativity waveform surrogate model for binary black holes: A Gaussian process regression approach, Phys. Rev. D 101, 063011 (2020), arXiv:1903.09204 [gr-qc] .
  * Rifat  _et al._ [2020] N. E. M. Rifat, S. E. Field, G. Khanna, and V. Varma, Surrogate model for gravitational wave signals from comparable and large-mass-ratio black hole binaries, Phys. Rev. D 101, 081502 (2020), arXiv:1910.10473 [gr-qc] .
  * Islam  _et al._ [2021] T. Islam, V. Varma, J. Lodman, S. E. Field, G. Khanna, M. A. Scheel, H. P. Pfeiffer, D. Gerosa, and L. E. Kidder, Eccentric binary black hole surrogate models for the gravitational waveform and remnant properties: comparable mass, nonspinning case, Phys. Rev. D 103, 064022 (2021), arXiv:2101.11798 [gr-qc] .
  * Islam  _et al._ [2022] T. Islam, S. E. Field, S. A. Hughes, G. Khanna, V. Varma, M. Giesler, M. A. Scheel, L. E. Kidder, and H. P. Pfeiffer, Surrogate model for gravitational wave signals from nonspinning, comparable-to large-mass-ratio black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity, Phys. Rev. D 106, 104025 (2022), arXiv:2204.01972 [gr-qc] .
  * Yoo  _et al._ [2022] J. Yoo, V. Varma, M. Giesler, M. A. Scheel, C.-J. Haster, H. P. Pfeiffer, L. E. Kidder, and M. Boyle, Targeted large mass ratio numerical relativity surrogate waveform model for GW190814, Phys. Rev. D 106, 044001 (2022), arXiv:2203.10109 [gr-qc] .
  * Pan  _et al._ [2008] Y. Pan, A. Buonanno, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, F. Pretorius, and J. R. van Meter, A Data-analysis driven comparison of analytic and numerical coalescing binary waveforms: Nonspinning case, Phys. Rev. D 77, 024014 (2008), arXiv:0704.1964 [gr-qc] .
  * Ajith  _et al._ [2007] P. Ajith _et al._ , Phenomenological template family for black-hole coalescence waveforms, Class. Quant. Grav. 24, S689 (2007), arXiv:0704.3764 [gr-qc] .
  * Ajith  _et al._ [2011] P. Ajith _et al._ , Inspiral-merger-ringdown waveforms for black-hole binaries with non-precessing spins, Phys. Rev. Lett. 106, 241101 (2011), arXiv:0909.2867 [gr-qc] .
  * Santamaria  _et al._ [2010] L. Santamaria _et al._ , Matching post-Newtonian and numerical relativity waveforms: systematic errors and a new phenomenological model for non-precessing black hole binaries, Phys. Rev. D 82, 064016 (2010), arXiv:1005.3306 [gr-qc] .
  * Hannam  _et al._ [2014] M. Hannam, P. Schmidt, A. Bohé, L. Haegel, S. Husa, F. Ohme, G. Pratten, and M. Pürrer, Simple Model of Complete Precessing Black-Hole-Binary Gravitational Waveforms, Phys. Rev. Lett. 113, 151101 (2014), arXiv:1308.3271 [gr-qc] .
  * Husa  _et al._ [2016] S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. Jiménez Forteza, and A. Bohé, Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal, Phys. Rev. D 93, 044006 (2016), arXiv:1508.07250 [gr-qc] .
  * Khan  _et al._ [2016] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. Jiménez Forteza, and A. Bohé, Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era, Phys. Rev. D 93, 044007 (2016), arXiv:1508.07253 [gr-qc] .
  * London  _et al._ [2018] L. London, S. Khan, E. Fauchon-Jones, C. García, M. Hannam, S. Husa, X. Jiménez-Forteza, C. Kalaghatgi, F. Ohme, and F. Pannarale, First higher-multipole model of gravitational waves from spinning and coalescing black-hole binaries, Phys. Rev. Lett. 120, 161102 (2018), arXiv:1708.00404 [gr-qc] .
  * Khan  _et al._ [2019] S. Khan, K. Chatziioannou, M. Hannam, and F. Ohme, Phenomenological model for the gravitational-wave signal from precessing binary black holes with two-spin effects, Phys. Rev. D 100, 024059 (2019), arXiv:1809.10113 [gr-qc] .
  * Khan  _et al._ [2020] S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam, Including higher order multipoles in gravitational-wave models for precessing binary black holes, Phys. Rev. D 101, 024056 (2020), arXiv:1911.06050 [gr-qc] .
  * Dietrich  _et al._ [2019] T. Dietrich, A. Samajdar, S. Khan, N. K. Johnson-McDaniel, R. Dudi, and W. Tichy, Improving the NRTidal model for binary neutron star systems, Phys. Rev. D 100, 044003 (2019), arXiv:1905.06011 [gr-qc] .
  * Pratten  _et al._ [2020a] G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. Ramos-Buades, H. Estelles, and R. Jaume, Setting the cornerstone for a family of models for gravitational waves from compact binaries: The dominant harmonic for nonprecessing quasicircular black holes, Phys. Rev. D 102, 064001 (2020a), arXiv:2001.11412 [gr-qc] .
  * Pratten  _et al._ [2021] G. Pratten _et al._ , Computationally efficient models for the dominant and subdominant harmonic modes of precessing binary black holes, Phys. Rev. D 103, 104056 (2021), arXiv:2004.06503 [gr-qc] .
  * García-Quirós  _et al._ [2020] C. García-Quirós, M. Colleoni, S. Husa, H. Estellés, G. Pratten, A. Ramos-Buades, M. Mateu-Lucena, and R. Jaume, Multimode frequency-domain model for the gravitational wave signal from nonprecessing black-hole binaries, Phys. Rev. D 102, 064002 (2020), arXiv:2001.10914 [gr-qc] .
  * Estellés  _et al._ [2021] H. Estellés, A. Ramos-Buades, S. Husa, C. García-Quirós, M. Colleoni, L. Haegel, and R. Jaume, Phenomenological time domain model for dominant quadrupole gravitational wave signal of coalescing binary black holes, Phys. Rev. D 103, 124060 (2021), arXiv:2004.08302 [gr-qc] .
  * Estellés  _et al._ [2022a] H. Estellés, S. Husa, M. Colleoni, D. Keitel, M. Mateu-Lucena, C. García-Quirós, A. Ramos-Buades, and A. Borchers, Time-domain phenomenological model of gravitational-wave subdominant harmonics for quasicircular nonprecessing binary black hole coalescences, Phys. Rev. D 105, 084039 (2022a), arXiv:2012.11923 [gr-qc] .
  * Estellés  _et al._ [2022b] H. Estellés, M. Colleoni, C. García-Quirós, S. Husa, D. Keitel, M. Mateu-Lucena, M. d. L. Planas, and A. Ramos-Buades, New twists in compact binary waveform modeling: A fast time-domain model for precession, Phys. Rev. D 105, 084040 (2022b), arXiv:2105.05872 [gr-qc] .
  * Hamilton  _et al._ [2021] E. Hamilton, L. London, J. E. Thompson, E. Fauchon-Jones, M. Hannam, C. Kalaghatgi, S. Khan, F. Pannarale, and A. Vano-Vinuales, Model of gravitational waves from precessing black-hole binaries through merger and ringdown, Phys. Rev. D 104, 124027 (2021), arXiv:2107.08876 [gr-qc] .
  * Buonanno and Damour [1999] A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59, 084006 (1999), arXiv:gr-qc/9811091 .
  * Buonanno and Damour [2000] A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Phys. Rev. D 62, 064015 (2000), arXiv:gr-qc/0001013 .
  * Damour  _et al._ [2000] T. Damour, P. Jaranowski, and G. Schaefer, On the determination of the last stable orbit for circular general relativistic binaries at the third postNewtonian approximation, Phys. Rev. D 62, 084011 (2000), arXiv:gr-qc/0005034 .
  * Damour [2001] T. Damour, Coalescence of two spinning black holes: an effective one-body approach, Phys. Rev. D 64, 124013 (2001), arXiv:gr-qc/0103018 .
  * Buonanno  _et al._ [2006] A. Buonanno, Y. Chen, and T. Damour, Transition from inspiral to plunge in precessing binaries of spinning black holes, Phys. Rev. D 74, 104005 (2006), arXiv:gr-qc/0508067 .
  * Buonanno  _et al._ [2007a] A. Buonanno, G. B. Cook, and F. Pretorius, Inspiral, merger and ring-down of equal-mass black-hole binaries, Phys. Rev. D 75, 124018 (2007a), arXiv:gr-qc/0610122 .
  * Buonanno  _et al._ [2007b] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, and J. R. van Meter, Toward faithful templates for non-spinning binary black holes using the effective-one-body approach, Phys. Rev. D 76, 104049 (2007b), arXiv:0706.3732 [gr-qc] .
  * Damour and Nagar [2008] T. Damour and A. Nagar, Comparing Effective-One-Body gravitational waveforms to accurate numerical data, Phys. Rev. D 77, 024043 (2008), arXiv:0711.2628 [gr-qc] .
  * Damour  _et al._ [2009] T. Damour, B. R. Iyer, and A. Nagar, Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries, Phys. Rev. D 79, 064004 (2009), arXiv:0811.2069 [gr-qc] .
  * Buonanno  _et al._ [2009] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, and L. E. Kidder, Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of non-spinning, equal-mass black holes, Phys. Rev. D 79, 124028 (2009), arXiv:0902.0790 [gr-qc] .
  * Pan  _et al._ [2011a] Y. Pan, A. Buonanno, M. Boyle, L. T. Buchman, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism, Phys. Rev. D 84, 124052 (2011a), arXiv:1106.1021 [gr-qc] .
  * Damour  _et al._ [2013] T. Damour, A. Nagar, and S. Bernuzzi, Improved effective-one-body description of coalescing nonspinning black-hole binaries and its numerical-relativity completion, Phys. Rev. D 87, 084035 (2013), arXiv:1212.4357 [gr-qc] .
  * Damour  _et al._ [2015] T. Damour, P. Jaranowski, and G. Schäfer, Fourth post-Newtonian effective one-body dynamics, Phys. Rev. D 91, 084024 (2015), arXiv:1502.07245 [gr-qc] .
  * Nagar  _et al._ [2020a] A. Nagar, G. Pratten, G. Riemenschneider, and R. Gamba, Multipolar effective one body model for nonspinning black hole binaries, Phys. Rev. D 101, 024041 (2020a), arXiv:1904.09550 [gr-qc] .
  * Damour  _et al._ [2008a] T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, Faithful Effective-One-Body waveforms of equal-mass coalescing black-hole binaries, Phys. Rev. D 77, 084017 (2008a), arXiv:0712.3003 [gr-qc] .
  * Damour  _et al._ [2008b] T. Damour, P. Jaranowski, and G. Schaefer, Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling, Phys. Rev. D 78, 024009 (2008b), arXiv:0803.0915 [gr-qc] .
  * Pan  _et al._ [2010] Y. Pan, A. Buonanno, L. T. Buchman, T. Chu, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-precessing, spinning, equal-mass black holes, Phys. Rev. D 81, 084041 (2010), arXiv:0912.3466 [gr-qc] .
  * Damour  _et al._ [2008c] T. Damour, A. Nagar, M. Hannam, S. Husa, and B. Bruegmann, Accurate Effective-One-Body waveforms of inspiralling and coalescing black-hole binaries, Phys. Rev. D 78, 044039 (2008c), arXiv:0803.3162 [gr-qc] .
  * Barausse and Buonanno [2010] E. Barausse and A. Buonanno, An Improved effective-one-body Hamiltonian for spinning black-hole binaries, Phys. Rev. D 81, 084024 (2010), arXiv:0912.3517 [gr-qc] .
  * Barausse and Buonanno [2011] E. Barausse and A. Buonanno, Extending the effective-one-body Hamiltonian of black-hole binaries to include next-to-next-to-leading spin-orbit couplings, Phys. Rev. D 84, 104027 (2011), arXiv:1107.2904 [gr-qc] .
  * Nagar [2011] A. Nagar, Effective one body Hamiltonian of two spinning black-holes with next-to-next-to-leading order spin-orbit coupling, Phys. Rev. D 84, 084028 (2011), [Erratum: Phys.Rev.D 88, 089901 (2013)], arXiv:1106.4349 [gr-qc] .
  * Damour and Nagar [2014] T. Damour and A. Nagar, New effective-one-body description of coalescing nonprecessing spinning black-hole binaries, Phys. Rev. D 90, 044018 (2014), arXiv:1406.6913 [gr-qc] .
  * Balmelli and Damour [2015] S. Balmelli and T. Damour, New effective-one-body Hamiltonian with next-to-leading order spin-spin coupling, Phys. Rev. D 92, 124022 (2015), arXiv:1509.08135 [gr-qc] .
  * Khalil  _et al._ [2020] M. Khalil, J. Steinhoff, J. Vines, and A. Buonanno, Fourth post-Newtonian effective-one-body Hamiltonians with generic spins, Phys. Rev. D 101, 104034 (2020), arXiv:2003.04469 [gr-qc] .
  * Taracchini  _et al._ [2012] A. Taracchini, Y. Pan, A. Buonanno, E. Barausse, M. Boyle, T. Chu, G. Lovelace, H. P. Pfeiffer, and M. A. Scheel, Prototype effective-one-body model for nonprecessing spinning inspiral-merger-ringdown waveforms, Phys. Rev. D 86, 024011 (2012), arXiv:1202.0790 [gr-qc] .
  * Taracchini  _et al._ [2014a] A. Taracchini _et al._ , Effective-one-body model for black-hole binaries with generic mass ratios and spins, Phys. Rev. D 89, 061502 (2014a), arXiv:1311.2544 [gr-qc] .
  * Bohé  _et al._ [2017] A. Bohé _et al._ , Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors, Phys. Rev. D 95, 044028 (2017), arXiv:1611.03703 [gr-qc] .
  * Cotesta  _et al._ [2018] R. Cotesta, A. Buonanno, A. Bohé, A. Taracchini, I. Hinder, and S. Ossokine, Enriching the Symphony of Gravitational Waves from Binary Black Holes by Tuning Higher Harmonics, Phys. Rev. D 98, 084028 (2018), arXiv:1803.10701 [gr-qc] .
  * Pan  _et al._ [2014] Y. Pan, A. Buonanno, A. Taracchini, L. E. Kidder, A. H. Mroué, H. P. Pfeiffer, M. A. Scheel, and B. Szilágyi, Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism, Phys. Rev. D 89, 084006 (2014), arXiv:1307.6232 [gr-qc] .
  * Babak  _et al._ [2017] S. Babak, A. Taracchini, and A. Buonanno, Validating the effective-one-body model of spinning, precessing binary black holes against numerical relativity, Phys. Rev. D 95, 024010 (2017), arXiv:1607.05661 [gr-qc] .
  * Ossokine  _et al._ [2020] S. Ossokine _et al._ , Multipolar Effective-One-Body Waveforms for Precessing Binary Black Holes: Construction and Validation, Phys. Rev. D 102, 044055 (2020), arXiv:2004.09442 [gr-qc] .
  * Nagar  _et al._ [2019] A. Nagar, F. Messina, P. Rettegno, D. Bini, T. Damour, A. Geralico, S. Akcay, and S. Bernuzzi, Nonlinear-in-spin effects in effective-one-body waveform models of spin-aligned, inspiralling, neutron star binaries, Phys. Rev. D 99, 044007 (2019), arXiv:1812.07923 [gr-qc] .
  * Nagar  _et al._ [2018] A. Nagar _et al._ , Time-domain effective-one-body gravitational waveforms for coalescing compact binaries with nonprecessing spins, tides and self-spin effects, Phys. Rev. D 98, 104052 (2018), arXiv:1806.01772 [gr-qc] .
  * Akcay  _et al._ [2021] S. Akcay, R. Gamba, and S. Bernuzzi, Hybrid post-Newtonian effective-one-body scheme for spin-precessing compact-binary waveforms up to merger, Phys. Rev. D 103, 024014 (2021), arXiv:2005.05338 [gr-qc] .
  * Gamba  _et al._ [2022] R. Gamba, S. Akçay, S. Bernuzzi, and J. Williams, Effective-one-body waveforms for precessing coalescing compact binaries with post-Newtonian twist, Phys. Rev. D 106, 024020 (2022), arXiv:2111.03675 [gr-qc] .
  * Bini and Damour [2012] D. Bini and T. Damour, Gravitational radiation reaction along general orbits in the effective one-body formalism, Phys. Rev. D 86, 124012 (2012), arXiv:1210.2834 [gr-qc] .
  * Hinderer and Babak [2017] T. Hinderer and S. Babak, Foundations of an effective-one-body model for coalescing binaries on eccentric orbits, Phys. Rev. D 96, 104048 (2017), arXiv:1707.08426 [gr-qc] .
  * Chiaramello and Nagar [2020] D. Chiaramello and A. Nagar, Faithful analytical effective-one-body waveform model for spin-aligned, moderately eccentric, coalescing black hole binaries, Phys. Rev. D 101, 101501 (2020), arXiv:2001.11736 [gr-qc] .
  * Nagar  _et al._ [2021] A. Nagar, A. Bonino, and P. Rettegno, Effective one-body multipolar waveform model for spin-aligned, quasicircular, eccentric, hyperbolic black hole binaries, Phys. Rev. D 103, 104021 (2021), arXiv:2101.08624 [gr-qc] .
  * Khalil  _et al._ [2021] M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Radiation-reaction force and multipolar waveforms for eccentric, spin-aligned binaries in the effective-one-body formalism, Phys. Rev. D 104, 024046 (2021), arXiv:2104.11705 [gr-qc] .
  * Ramos-Buades  _et al._ [2022] A. Ramos-Buades, A. Buonanno, M. Khalil, and S. Ossokine, Effective-one-body multipolar waveforms for eccentric binary black holes with nonprecessing spins, Phys. Rev. D 105, 044035 (2022), arXiv:2112.06952 [gr-qc] .
  * Albanesi  _et al._ [2022] S. Albanesi, A. Placidi, A. Nagar, M. Orselli, and S. Bernuzzi, New avenue for accurate analytical waveforms and fluxes for eccentric compact binaries, Phys. Rev. D 105, L121503 (2022), arXiv:2203.16286 [gr-qc] .
  * Field  _et al._ [2014] S. E. Field, C. R. Galley, J. S. Hesthaven, J. Kaye, and M. Tiglio, Fast prediction and evaluation of gravitational waveforms using surrogate models, Phys. Rev. X 4, 031006 (2014), arXiv:1308.3565 [gr-qc] .
  * Pürrer [2014] M. Pürrer, Frequency domain reduced order models for gravitational waves from aligned-spin compact binaries, Class. Quant. Grav. 31, 195010 (2014), arXiv:1402.4146 [gr-qc] .
  * Pürrer [2016] M. Pürrer, Frequency domain reduced order model of aligned-spin effective-one-body waveforms with generic mass-ratios and spins, Phys. Rev. D 93, 064041 (2016), arXiv:1512.02248 [gr-qc] .
  * Lackey  _et al._ [2017] B. D. Lackey, S. Bernuzzi, C. R. Galley, J. Meidam, and C. Van Den Broeck, Effective-one-body waveforms for binary neutron stars using surrogate models, Phys. Rev. D 95, 104036 (2017), arXiv:1610.04742 [gr-qc] .
  * Lackey  _et al._ [2019] B. D. Lackey, M. Pürrer, A. Taracchini, and S. Marsat, Surrogate model for an aligned-spin effective one body waveform model of binary neutron star inspirals using Gaussian process regression, Phys. Rev. D 100, 024002 (2019), arXiv:1812.08643 [gr-qc] .
  * Cotesta  _et al._ [2020] R. Cotesta, S. Marsat, and M. Pürrer, Frequency domain reduced order model of aligned-spin effective-one-body waveforms with higher-order modes, Phys. Rev. D 101, 124040 (2020), arXiv:2003.12079 [gr-qc] .
  * Gadre  _et al._ [2022] B. Gadre, M. Pürrer, S. E. Field, S. Ossokine, and V. Varma, A fully precessing higher-mode surrogate model of effective-one-body waveforms, (2022), arXiv:2203.00381 [gr-qc] .
  * Tissino  _et al._ [2022] J. Tissino, G. Carullo, M. Breschi, R. Gamba, S. Schmidt, and S. Bernuzzi, Combining effective-one-body accuracy and reduced-order-quadrature speed for binary neutron star merger parameter estimation with machine learning, (2022), arXiv:2210.15684 [gr-qc] .
  * Khan and Green [2021] S. Khan and R. Green, Gravitational-wave surrogate models powered by artificial neural networks, Phys. Rev. D 103, 064015 (2021), arXiv:2008.12932 [gr-qc] .
  * Thomas  _et al._ [2022] L. M. Thomas, G. Pratten, and P. Schmidt, Accelerating multimodal gravitational waveforms from precessing compact binaries with artificial neural networks, Phys. Rev. D 106, 104029 (2022), arXiv:2205.14066 [gr-qc] .
  * Dax  _et al._ [2021] M. Dax, S. R. Green, J. Gair, J. H. Macke, A. Buonanno, and B. Schölkopf, Real-Time Gravitational Wave Science with Neural Posterior Estimation, Phys. Rev. Lett. 127, 241103 (2021), arXiv:2106.12594 [gr-qc] .
  * Dax  _et al._ [2022] M. Dax, S. R. Green, J. Gair, M. Pürrer, J. Wildberger, J. H. Macke, A. Buonanno, and B. Schölkopf, Neural Importance Sampling for Rapid and Reliable Gravitational-Wave Inference, (2022), arXiv:2210.05686 [gr-qc] .
  * Mihaylov  _et al._ [2021] D. P. Mihaylov, S. Ossokine, A. Buonanno, and A. Ghosh, Fast post-adiabatic waveforms in the time domain: Applications to compact binary coalescences in LIGO and Virgo, Phys. Rev. D 104, 124087 (2021), arXiv:2105.06983 [gr-qc] .
  * Nagar  _et al._ [2020b] A. Nagar, G. Riemenschneider, G. Pratten, P. Rettegno, and F. Messina, Multipolar effective one body waveform model for spin-aligned black hole binaries, Phys. Rev. D 102, 024077 (2020b), arXiv:2001.09082 [gr-qc] .
  * Riemenschneider  _et al._ [2021] G. Riemenschneider, P. Rettegno, M. Breschi, A. Albertini, R. Gamba, S. Bernuzzi, and A. Nagar, Assessment of consistent next-to-quasicircular corrections and postadiabatic approximation in effective-one-body multipolar waveforms for binary black hole coalescences, Phys. Rev. D 104, 104045 (2021), arXiv:2104.07533 [gr-qc] .
  * [106] https://observing.docs.ligo.org/plan/, accessed: 2023-02-25.
  * Akutsu  _et al._ [2021] T. Akutsu _et al._ (KAGRA), Overview of KAGRA: Detector design and construction history, PTEP 2021, 05A101 (2021), arXiv:2005.05574 [physics.ins-det] .
  * Pratten  _et al._ [2020b] G. Pratten, P. Schmidt, R. Buscicchio, and L. M. Thomas, Measuring precession in asymmetric compact binaries, Phys. Rev. Res. 2, 043096 (2020b), arXiv:2006.16153 [gr-qc] .
  * Colleoni  _et al._ [2021] M. Colleoni, M. Mateu-Lucena, H. Estellés, C. García-Quirós, D. Keitel, G. Pratten, A. Ramos-Buades, and S. Husa, Towards the routine use of subdominant harmonics in gravitational-wave inference: Reanalysis of GW190412 with generation X waveform models, Phys. Rev. D 103, 024029 (2021), arXiv:2010.05830 [gr-qc] .
  * Damour [2016] T. Damour, Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory, Phys. Rev. D 94, 104015 (2016), arXiv:1609.00354 [gr-qc] .
  * Damour [2018] T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97, 044038 (2018), arXiv:1710.10599 [gr-qc] .
  * Antonelli  _et al._ [2019] A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent, and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99, 104004 (2019), arXiv:1901.07102 [gr-qc] .
  * Damgaard and Vanhove [2021] P. H. Damgaard and P. Vanhove, Remodeling the effective one-body formalism in post-Minkowskian gravity, Phys. Rev. D 104, 104029 (2021), arXiv:2108.11248 [hep-th] .
  * Khalil  _et al._ [2022] M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Energetics and scattering of gravitational two-body systems at fourth post-Minkowskian order, Phys. Rev. D 106, 024042 (2022), arXiv:2204.05047 [gr-qc] .
  * Damour and Rettegno [2022] T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical Relativity meets Post-Minkowskian gravity, (2022), arXiv:2211.01399 [gr-qc] .
  * van de Meent  _et al._ [2023] M. van de Meent, A. Buonanno, D. Mihaylov, S. Ossokine, L. Pompili, N. Warburton, A. Pound, B. Wardell, L. Durkan, and J. Miller, Enhancing the SEOBNRv5 effective-one-body waveform model with second-order gravitational self-force fluxes, (2023).
  * Khalil  _et al._ [2023] M. Khalil, A. Buonanno, H. Estellés, D. Mihaylov, S. Ossokine, L. Pompili, and A. Ramos-Buades, Theoretical groundwork supporting the precessing-spin two-body dynamics of the effective-one-body waveform models SEOBNRv5, (2023).
  * Henry  _et al._ [2022] Q. Henry, S. Marsat, and M. Khalil, Spin contributions to the gravitational-waveform modes for spin-aligned binaries at the 3.5PN order, Phys. Rev. D 106, 124018 (2022), arXiv:2209.00374 [gr-qc] .
  * Warburton  _et al._ [2021] N. Warburton, A. Pound, B. Wardell, J. Miller, and L. Durkan, Gravitational-Wave Energy Flux for Compact Binaries through Second Order in the Mass Ratio, Phys. Rev. Lett. 127, 151102 (2021), arXiv:2107.01298 [gr-qc] .
  * Wardell  _et al._ [2021] B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Gravitational waveforms for compact binaries from second-order self-force theory, (2021), arXiv:2112.12265 [gr-qc] .
  * [121] http://www.black-holes.org/waveforms.
  * Boyle  _et al._ [2019] M. Boyle _et al._ , The SXS Collaboration catalog of binary black hole simulations, Class. Quant. Grav. 36, 195006 (2019), arXiv:1904.04831 [gr-qc] .
  * Chu  _et al._ [2016] T. Chu, H. Fong, P. Kumar, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E. Kidder, M. A. Scheel, and B. Szilagyi, On the accuracy and precision of numerical waveforms: Effect of waveform extraction methodology, Class. Quant. Grav. 33, 165001 (2016), arXiv:1512.06800 [gr-qc] .
  * Hemberger  _et al._ [2013] D. A. Hemberger, G. Lovelace, T. J. Loredo, L. E. Kidder, M. A. Scheel, B. Szilagyi, N. W. Taylor, and S. A. Teukolsky, Final spin and radiated energy in numerical simulations of binary black holes with equal masses and equal, aligned or anti-aligned spins, Phys. Rev. D88, 064014 (2013), arXiv:1305.5991 [gr-qc] .
  * Scheel  _et al._ [2015] M. A. Scheel, M. Giesler, D. A. Hemberger, G. Lovelace, K. Kuper, M. Boyle, B. Szilagyi, and L. E. Kidder, Improved methods for simulating nearly extremal binary black holes, Class. Quant. Grav. 32, 105009 (2015), arXiv:1412.1803 [gr-qc] .
  * Lovelace  _et al._ [2015] G. Lovelace _et al._ , Nearly extremal apparent horizons in simulations of merging black holes, Class. Quant. Grav. 32, 065007 (2015), arXiv:1411.7297 [gr-qc] .
  * Abbott  _et al._ [2016b] B. P. Abbott _et al._ (Virgo, LIGO Scientific), Directly comparing GW150914 with numerical solutions of Einsteins equations for binary black hole coalescence, Phys. Rev. D94, 064035 (2016b), arXiv:1606.01262 [gr-qc] .
  * Lovelace  _et al._ [2016] G. Lovelace _et al._ , Modeling the source of GW150914 with targeted numerical-relativity simulations, Class. Quant. Grav. 33, 244002 (2016), arXiv:1607.05377 [gr-qc] .
  * Abbott  _et al._ [2016c] B. P. Abbott _et al._ (Virgo, LIGO Scientific), GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116, 241103 (2016c), arXiv:1606.04855 [gr-qc] .
  * Kumar  _et al._ [2015] P. Kumar, K. Barkett, S. Bhagwat, N. Afshari, D. A. Brown, G. Lovelace, M. A. Scheel, and B. Szilagyi, Accuracy and precision of gravitational-wave models of inspiraling neutron star-black hole binaries with spin: Comparison with matter-free numerical relativity in the low-frequency regime, Phys. Rev. D92, 102001 (2015), arXiv:1507.00103 [gr-qc] .
  * Mroue  _et al._ [2013] A. H. Mroue _et al._ , Catalog of 174 Binary Black Hole Simulations for Gravitational Wave Astronomy, Phys. Rev. Lett. 111, 241104 (2013), arXiv:1304.6077 [gr-qc] .
  * Haas  _et al._ [2022] R. Haas, C.-H. Cheng, P. Diener, Z. Etienne, G. Ficarra, T. Ikeda, H. Kalyanaraman, N. Kuo, L. Leung, C. Tian, B.-J. J. Tsao, A. Wen, M. Alcubierre, D. Alic, G. Allen, M. Ansorg, F. G. L. Armengol, M. Babiuc-Hamilton, L. Baiotti, W. Benger, E. Bentivegna, S. Bernuzzi, T. Bode, G. Bozzola, S. R. Brandt, B. Brendal, B. Bruegmann, M. Campanelli, F. Cipolletta, G. Corvino, S. Cupp, R. D. Pietri, H. Dimmelmeier, R. Dooley, N. Dorband, M. Elley, Y. E. Khamra, J. Faber, T. Font, J. Frieben, B. Giacomazzo, T. Goodale, C. Gundlach, I. Hawke, S. Hawley, I. Hinder, E. A. Huerta, S. Husa, S. Iyer, L. Ji, D. Johnson, A. V. Joshi, W. Kastaun, T. Kellermann, A. Knapp, M. Koppitz, P. Laguna, G. Lanferman, F. Löffler, H. Macpherson, J. Masso, L. Menger, A. Merzky, J. M. Miller, M. Miller, P. Moesta, P. Montero, B. Mundim, P. Nelson, A. Nerozzi, S. C. Noble, C. Ott, R. Paruchuri, D. Pollney, D. Radice, T. Radke, C. Reisswig, L. Rezzolla, D. Rideout, M. Ripeanu, L. Sala, J. A. Schewtschenko, E. Schnetter, B. Schutz, E. Seidel, E. Seidel, J. Shalf, K. Sible, U. Sperhake, N. Stergioulas, W.-M. Suen, B. Szilagyi, R. Takahashi, M. Thomas, J. Thornburg, M. Tobias, A. Tonita, P. Walker, M.-B. Wan, B. Wardell, L. Werneck, H. Witek, M. Zilhão, B. Zink, and Y. Zlochower, The einstein toolkit (2022), to find out more, visit http://einsteintoolkit.org.
  * Barausse  _et al._ [2012] E. Barausse, A. Buonanno, S. A. Hughes, G. Khanna, S. O’Sullivan, and Y. Pan, Modeling multipolar gravitational-wave emission from small mass-ratio mergers, Phys. Rev. D 85, 024046 (2012), arXiv:1110.3081 [gr-qc] .
  * Taracchini  _et al._ [2014b] A. Taracchini, A. Buonanno, G. Khanna, and S. A. Hughes, Small mass plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown waveforms, Phys. Rev. D 90, 084025 (2014b), arXiv:1404.1819 [gr-qc] .
  * Mihaylov  _et al._ [2023] D. Mihaylov, S. Ossokine, A. Buonanno, H. Estellés, L. Pompili, M. Pürrer, and A. Ramos-Buades, pySEOBNR: a software package for the next generation of effective-one-body multipolar waveform models, (2023).
  * Ramos-Buades  _et al._ [2023] A. Ramos-Buades, A. Buonanno, S. Ossokine, H. Estellés, M. Khalil, L. Pompili, and M. Shiferaw, SEOBNRv5PHM: Next generation of accurate and efficient multipolar precessing-spin effective-one-body waveforms for binary black holes, (2023).
  * Barausse  _et al._ [2009] E. Barausse, E. Racine, and A. Buonanno, Hamiltonian of a spinning test-particle in curved spacetime, Phys. Rev. D 80, 104025 (2009), [Erratum: Phys.Rev.D 85, 069904 (2012)], arXiv:0907.4745 [gr-qc] .
  * Vines  _et al._ [2016] J. Vines, D. Kunst, J. Steinhoff, and T. Hinderer, Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin, Phys. Rev. D 93, 103008 (2016), [Erratum: Phys.Rev.D 104, 029902 (2021)], arXiv:1601.07529 [gr-qc] .
  * Damour and Nagar [2007] T. Damour and A. Nagar, Faithful effective-one-body waveforms of small-mass-ratio coalescing black-hole binaries, Phys. Rev. D 76, 064028 (2007), arXiv:0705.2519 [gr-qc] .
  * Bini  _et al._ [2019] D. Bini, T. Damour, and A. Geralico, Novel approach to binary dynamics: application to the fifth post-Newtonian level, Phys. Rev. Lett. 123, 231104 (2019), arXiv:1909.02375 [gr-qc] .
  * Bini  _et al._ [2020a] D. Bini, T. Damour, and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders, Phys. Rev. D 102, 024062 (2020a), arXiv:2003.11891 [gr-qc] .
  * Barack and Sago [2010] L. Barack and N. Sago, Gravitational self-force on a particle in eccentric orbit around a Schwarzschild black hole, Phys. Rev. D 81, 084021 (2010), arXiv:1002.2386 [gr-qc] .
  * Isoyama  _et al._ [2014] S. Isoyama, L. Barack, S. R. Dolan, A. Le Tiec, H. Nakano, A. G. Shah, T. Tanaka, and N. Warburton, Gravitational Self-Force Correction to the Innermost Stable Circular Equatorial Orbit of a Kerr Black Hole, Phys. Rev. Lett. 113, 161101 (2014), arXiv:1404.6133 [gr-qc] .
  * Antonelli  _et al._ [2020a] A. Antonelli, C. Kavanagh, M. Khalil, J. Steinhoff, and J. Vines, Gravitational spin-orbit coupling through third-subleading post-Newtonian order: from first-order self-force to arbitrary mass ratios, Phys. Rev. Lett. 125, 011103 (2020a), arXiv:2003.11391 [gr-qc] .
  * Antonelli  _et al._ [2020b] A. Antonelli, C. Kavanagh, M. Khalil, J. Steinhoff, and J. Vines, Gravitational spin-orbit and aligned spin1-spin2 couplings through third-subleading post-Newtonian orders, Phys. Rev. D 102, 124024 (2020b), arXiv:2010.02018 [gr-qc] .
  * Mandal  _et al._ [2022a] M. K. Mandal, P. Mastrolia, R. Patil, and J. Steinhoff, Gravitational Spin-Orbit Hamiltonian at NNNLO in the post-Newtonian framework, (2022a), arXiv:2209.00611 [hep-th] .
  * Kim  _et al._ [2022a] J.-W. Kim, M. Levi, and Z. Yin, N3LO Spin-Orbit Interaction via the EFT of Spinning Gravitating Objects, (2022a), arXiv:2208.14949 [hep-th] .
  * Vines and Steinhoff [2018] J. Vines and J. Steinhoff, Spin-multipole effects in binary black holes and the test-body limit, Phys. Rev. D 97, 064010 (2018), arXiv:1606.08832 [gr-qc] .
  * Pan  _et al._ [2011b] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi, Post-Newtonian factorized multipolar waveforms for spinning, non-precessing black-hole binaries, Phys. Rev. D 83, 064003 (2011b), [Erratum: Phys.Rev.D 87, 109901 (2013)], arXiv:1006.0431 [gr-qc] .
  * Nagar and Rettegno [2019] A. Nagar and P. Rettegno, Efficient effective one body time-domain gravitational waveforms, Phys. Rev. D 99, 021501 (2019), arXiv:1805.03891 [gr-qc] .
  * Rettegno  _et al._ [2020] P. Rettegno, F. Martinetti, A. Nagar, D. Bini, G. Riemenschneider, and T. Damour, Comparing Effective One Body Hamiltonians for spin-aligned coalescing binaries, Phys. Rev. D 101, 104027 (2020), arXiv:1911.10818 [gr-qc] .
  * Blanchet [1998] L. Blanchet, Gravitational wave tails of tails, Class. Quant. Grav. 15, 113 (1998), [Erratum: Class.Quant.Grav. 22, 3381 (2005)], arXiv:gr-qc/9710038 .
  * Blanchet  _et al._ [2008] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, The Third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits, Class. Quant. Grav. 25, 165003 (2008), [Erratum: Class.Quant.Grav. 29, 239501 (2012)], arXiv:0802.1249 [gr-qc] .
  * Henry [2022] Q. Henry, Complete gravitational-waveform amplitude modes for quasi-circular compact binaries to the 3.5PN order, (2022), arXiv:2210.15602 [gr-qc] .
  * Bardeen  _et al._ [1972] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation, Astrophys. J. 178, 347 (1972).
  * Jiménez-Forteza  _et al._ [2017] X. Jiménez-Forteza, D. Keitel, S. Husa, M. Hannam, S. Khan, and M. Pürrer, Hierarchical data-driven approach to fitting numerical relativity data for nonprecessing binary black holes with an application to final spin and radiated energy, Phys. Rev. D 95, 064024 (2017), arXiv:1611.00332 [gr-qc] .
  * Hofmann  _et al._ [2016] F. Hofmann, E. Barausse, and L. Rezzolla, The final spin from binary black holes in quasi-circular orbits, Astrophys. J. Lett. 825, L19 (2016), arXiv:1605.01938 [gr-qc] .
  * Price and Khanna [2016] R. H. Price and G. Khanna, Arrival Times of Gravitational Radiation Peaks for Binary Inspiral, Phys. Rev. D 94, 104026 (2016), arXiv:1607.04226 [gr-qc] .
  * Stein [2019] L. C. Stein, qnm: A Python package for calculating Kerr quasinormal modes, separation constants, and spherical-spheroidal mixing coefficients, J. Open Source Softw. 4, 1683 (2019), arXiv:1908.10377 [gr-qc] .
  * Guyon  _et al._ [2002] I. Guyon, J. Weston, S. Barnhill, and V. Vapnik, Gene Selection for Cancer Classification Using Support Vector Machines, Machine Learning 46, 389 (2002).
  * Kelly and Baker [2013] B. J. Kelly and J. G. Baker, Decoding mode mixing in black-hole merger ringdown, Phys. Rev. D 87, 084004 (2013), arXiv:1212.5553 [gr-qc] .
  * Berti  _et al._ [2006] E. Berti, V. Cardoso, and M. Casals, Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions, Phys. Rev. D 73, 024013 (2006), [Erratum: Phys.Rev.D 73, 109902 (2006)], arXiv:gr-qc/0511111 .
  * Berti and Klein [2014] E. Berti and A. Klein, Mixing of spherical and spheroidal modes in perturbed Kerr black holes, Phys. Rev. D 90, 064012 (2014), arXiv:1408.1860 [gr-qc] .
  * London and Fauchon-Jones [2019] L. London and E. Fauchon-Jones, On modeling for Kerr black holes: Basis learning, QNM frequencies, and spherical-spheroidal mixing coefficients, Class. Quant. Grav. 36, 235015 (2019), arXiv:1810.03550 [gr-qc] .
  * Kumar Mehta  _et al._ [2019] A. Kumar Mehta, P. Tiwari, N. K. Johnson-McDaniel, C. K. Mishra, V. Varma, and P. Ajith, Including mode mixing in a higher-multipole model for gravitational waveforms from nonspinning black-hole binaries, Phys. Rev. D 100, 024032 (2019), arXiv:1902.02731 [gr-qc] .
  * Nagar and Rettegno [2021] A. Nagar and P. Rettegno, Next generation: Impact of high-order analytical information on effective one body waveform models for noncircularized, spin-aligned black hole binaries, Phys. Rev. D 104, 104004 (2021), arXiv:2108.02043 [gr-qc] .
  * Finn and Chernoff [1993] L. S. Finn and D. F. Chernoff, Observing binary inspiral in gravitational radiation: One interferometer, Phys. Rev. D 47, 2198 (1993), arXiv:gr-qc/9301003 .
  * Sathyaprakash and Dhurandhar [1991] B. S. Sathyaprakash and S. V. Dhurandhar, Choice of filters for the detection of gravitational waves from coalescing binaries, Phys. Rev. D 44, 3819 (1991).
  * Barsotti  _et al._ [2018] L. Barsotti, P. Fritschel, M. Evans, and S. Gras (LIGO Collaboration), Updated advanced ligo sensitivity design curve (2018), LIGO Document T1800044-v5.
  * McKechan  _et al._ [2010] D. J. A. McKechan, C. Robinson, and B. S. Sathyaprakash, A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries, Class. Quant. Grav. 27, 084020 (2010), arXiv:1003.2939 [gr-qc] .
  * Pürrer and Haster [2020] M. Pürrer and C.-J. Haster, Gravitational waveform accuracy requirements for future ground-based detectors, Phys. Rev. Res. 2, 023151 (2020), arXiv:1912.10055 [gr-qc] .