and the co-precessing NR waveform. We considered a system of total mass 100M⊙
and performed the match over the frequency range for which the NR data was
available; from $\left(f_{\text{ref}}+5\right)$Hz to 244Hz. The value of the
reference frequency $f_{\text{ref}}$ for each simulation is given in Table 1.
The co-precessing-frame models we consider are PhenomPv3, PhenomXP and
PhenomDCP.

As can be seen from Fig. 17, the assumptions that go into producing the
aligned-spin mapping used in the production of modified PhenomD and modified
PhenomXAS become less accurate as both mass ratio and spin are increased.
PhenomDCP performs better than both modified PhenomD and modified PhenomXAS
for almost all cases, with the most noticeable improvement for the higher mass
ratio, high-spin cases where we are in greatest need of a tuned co-precessing
model. In the cases where PhenomDCP has a similar or slightly worse
performance than either of the other two models the match is generally already
comparable to the accuracy level of our input NR waveforms.

###  XI.4 Matches: Accuracy of the angle model

In order to test the accuracy of the angle model we constructed a set of
precessing waveforms by calculating the symmetrised frequency-domain co-
precessing NR waveform containing only the $\ell=2$ multipoles and “twisting”
this waveform up with the modelled precession angles. We constructed two sets
of precessing waveforms in this fashion; one using the model for the angles
presented in this paper, and the other using the MSA angles, in order to
quantify the effect of modelling the merger-ringdown behaviour of the angles.
We then calculated the SNR-weighted match between these waveforms and the
symmetrised NR waveforms in the $\mathbf{J}$-aligned frame comprising only the
$\ell=2$ multipoles. As with the co-precessing matches described above, these
matches were calculated at a fixed total mass $M=100$M⊙ and performed over a
frequency range from $\left(f_{\text{ref}}+5\right)$Hz to 244Hz (the frequency
range for which the NR data was available).

In Fig. 18, we have shown the inclination average of the full precessing match
for ease of presentation. We can see that the matches using the improved angle
model are above 0.99 across the majority of the parameter space. The only
cases for which this is not true are in the most extreme corner of the
parameter space we modelled; cases with $q=8$, $\chi=0.6$ and
$\theta_{\mathrm{LS}}\geq 90^{\circ}$. In these cases we find the PN
expressions used for $\alpha$ during inspiral deviate from those calculated
from the NR waveform at reasonably low frequencies. In the case of
$(q,\chi,\theta_{\mathrm{LS}})=(8,0.8,120^{\circ})$ this is before the start
of the NR waveform, as shown in Fig. 19. Improving the model for these cases
would require a model for the intermediate region between where the PN
expression ceases to be accurate and where the current model begins, which may
require longer NR waveforms to be produced. Additionally, we expect that
modelling this intermediate region will improve matches for several other
cases as well, where the PN expressions for the angles deviate from what we
see in the NR data at lower frequencies than are covered by our current
merger-ringdown model for the angles. Nonetheless, in all cases we see
significant improvement over the previous model.

Figure 19: Comparison MSA $\alpha$ (blue dashed line) with the value
calculated from the NR waveform (black solid line) for the
$(q,\chi,\theta_{\rm LS})=(8,0.8,120^{\circ})$ configuration. In order to see
a region over which the two values agree well we would need a longer NR
waveform; see text for more details.  Figure 20: Mismatch as a function of the
inclination of the binary, quantified by the angle between the line of sight
and the total angular momentum $\theta_{\text{JN}}$, for four cases, at
100$M_{\odot}$. These mismatches consider the co-precessing NR waveform
twisted up with the angle model used by PhenomPv3 (steel blue) and PhenomPNR
(purple). The solid markers show the SNR-weighted average mismatch while the
shaded regions show the variation with respect to signal polaristation and
phase.

The best matches are seen in the least extreme part of parameter space; namely
for low mass ratio systems. This is the region of parameter space where
existing models for the angles already perform reasonably well. The biggest
improvement in the matches as a result of the improved model for the angles is
seen at higher mass ratios, particularly for larger $\theta_{\mathrm{LS}}$.

For a selection of these cases we show the mismatch as a function of
$\theta_{\rm JN}$ in Fig. 20. The figure shows both the SNR-weighted average,
and the range of mismatches with respect to signal polarisation and phase. We
see that the mismatches against symmetrised NR waveforms are approximately
symmetric about $\theta_{\text{JN}}=\pi/2$. The MSA angles generally give the
worst SNR-weighted average mismatch for systems with
$\theta_{\text{JN}}=0,\pi$, although this is not always the case, and the
variations with respect to different choices of polarisation and phase are
often larger than those with respect to inclination. This mismatch then
typically improves as it approaches $\theta_{\text{JN}}=\pi/2$ systems, with a
slight increase for systems at exactly $\pi/2$ in most cases. In contrast, the
SNR-weighted average mismatches involving the new angle model show one of two
main behaviours with respect to inclination: the first gives the lowest
mismatches for systems with $\theta_{\text{JN}}=0,\pi$ with a marked
degradation towards $\theta_{\text{JN}}=\pi/2$, while the second shows
approximately constant values for the mismatch with respect to inclination,
with a possible slight improvement for systems with
$\theta_{\text{JN}}=\pi/2$. However, we do not observe any clear pattern in
how these two trends manifest themselves across the parameter space. The most
important result to note is that in comparing the new angle model with the MSA
angles in this figure, for the new angle model the lowest mismatch is always
better (the lower edge of the envelopes), the highest mistmatch is always
better (the upper edge of the envelopes), and the SNR-weighted mismatch is
always better.

In general we might expect errors in the angle models to lead to worse
mismatches for edge-on configurations, since at these orientations the
contributions of the subdominant $\ell=2$ multipoles are largest, and the
strength of those multipoles in our model is directly related to the
precession angles, in particular $\beta$. However, $\theta_{\rm JN}=\pi/2$
doesn’t necessarily correspond to the binary being edge-on to the detector,
unless $\beta$ is close to zero; in general, a system viewed from $\theta_{\rm
JN}=\pi/2$ is _never_ edge on. Because of this, and because of the large
variation in matches across the cases shown in Fig. 20, we revisit this
question in the full-model mismatches in the next section, where we specify
the binary orientation at the beginning of the waveform (so
$\theta_{\text{LN}}=\pi/2$ corresponds to edge-on at least at one point in the
inspiral), and consider an exhaustive set of masses, orientations and
polarisations for every NR waveform.

###  XI.5 Matches: Accuracy of PhenomPNR

In this section we compare the accuracy of the complete PhenomPNR model to
existing precessing waveform models by computing SNR-weighted mismatches
between these approximants and the various NR waveforms detailed in Sec. XI.2.
Each SNR-weighted mismatch is computed over a range of total masses
$M_{\text{total}}\in[100,120,140,160,180,200,220,240]$M⊙ and at four
inclination values, $\theta_{\text{LN}}\in[0,\pi/6,\pi/3,\pi/2]$, specified at
the reference frequencies given in the waveform tables. The choice to sample
in $\theta_{\text{LN}}$, rather than $\theta_{\text{JN}}$ as done above, was
motivated partially by the frame convention of LALsuite LIGO Scientific
Collaboration (2018), which specifies that the LAL inertial frame Schmidt  _et
al._ (2017) in which the waveforms are generated be instantaneously
$\hat{\mathbf{L}}$-aligned at the given reference frequency. This choice
allows for comparisons with match results already present in the literature.
As was also noted in the previous section, the conventional wisdom gleaned
from non-precessing signals regarding model performance and the importance of
higher multipoles for configurations with $\theta_{\text{LN}}\sim\pi/2$ also
holds for precessing cases where $\beta$ remains small throughout most of the
inspiral, as is the case with most of the NR waveforms we consider. The
matches were performed starting at a frequency of 20Hz or
$f_{\text{ref}}+5$Hz, whichever was higher, with $f_{\text{ref}}$ listed for
each NR waveform in Tables 1 and 2. The nominal starting frequency of 20 Hz
was chosen to match the approximate low-frequency cut-off of typical signal
analysis, and dictated our choice of 100 $M_{\odot}$ as the lowest total mass
we consider.

####  XI.5.1 Match variation with inclination

Figure 21: Histograms of the SNR-weighted mismatches between the NR waveforms
listed in Tables 1-2 and the waveform models PhenomPNR and PhenomPv3. Each
subplot contains the SNR-weighted mismatches for all total masses separated by
inclination descending as
$\theta_{\text{LN}}\in[0^{\circ},30^{\circ},60^{\circ},90^{\circ}]$. The
mismatches for all total mass values listed in Sec. XI.5 are included at each
inclination. The results for PhenomPNR are present in solid black while the
results for PhenomPv3 are given in dashed red.

As discussed in the previous section, we would expect improvement in PhenomPNR
to be most apparent when trying to replicate highly-precessing signals at high
inclination, $\theta_{\text{LN}}\sim\pi/2$, where the modulations in the
signal due to precession grow stronger as more power is distributed across the
$\ell=2$ multipoles. We therefore compare the performance of PhenomPNR with
the earlier precessing model PhenomPv3, plotting the SNR-weighted mismatches
between these two models and the NR waveforms for each inclination value used.
We choose PhenomPv3, since it was the base model that we modified to produce
PhenomPNR, and this comparison provides the most direct measure of the level
of improvement achieved by including NR-tuned precession effects in both the
co-precessing-frame and angle models. The overall distribution of SNR-weighted
mismatches is shown in Fig. 21. For PhenomPv3, which uses the uncalibrated MSA
angles, the performance noticably degrades as the signal inclination
increases, whereas the mismatches for PhenomPNR remain relatively unchanged
with respect to changes in inclination. This is largely consistent with Fig.
20, where the average match shows little variation with respect to inclination
for three out of the four configurations shown, and suggests that the
behaviour of the model for the $(q,\chi,\theta_{\rm{LS}})=(8,0.8,90^{\circ})$
case is atypical.

####  XI.5.2 General match results

Figure 22: Histograms of the SNR-weighted mismatches between various models in
comparison and the NR waveforms listed in Tables 1-2. The mismatches for all
inclination and total mass values listed in Sec. XI.5 are included. In all
three subplots, the results for PhenomPNR (“PNR”) are presented with a solid
black outline, with the other model results given with dashed outlines from
left to right as PhenomXP (“XP”) in red, SEOBNRv4P (“EOB”) in blue, and
NRSur7dq4 (“SUR”) in green. For the comparison plot between PhenomPNR and
NRSur7dq4 we only include results of NR waveforms for which both models are
run.

Figure 23: SNR-weighted mismatches averaged over total mass and inclination
between the precessing waveform models PhenomPNR (“PNR”), PhenomXP (“XP”), and
SEOBNRv4P (“EOB”), and the NR waveforms listed in Tables 1-2, shown in order
of the table listings. For the BAM cases, the solid vertical lines separate
cases by mass ratio, and the dashed vertical lines separate spin magnitude.
For the SXS and Maya cases, the solid vertical line splits by NR catalogue,
and the dashed vertical line indicates a transition from single-spin to two-
spin cases.

We compare the performance or PhenomPNR against the precessing waveform
approximants PhenomXP, SEOBNRv4P, and NRSur7dq4. The full results are shown
for all inclinations and total masses in Fig. 22, and the mass- and
inclination-averaged SNR-weighted mismatches are shown per waveform in Fig.
23. The model NRSur7dq4 was calibrated only up to $q=4$, and while its
implementation in LALsuite allows for extrapolation beyond this, we choose to
limit the comparison with this model to the subset of the available NR
waveforms with $q\leq 4$ to ensure accuracy is maintained.

Overall we see an improvement in the mismatches between PhenomPNR and the NR
waveforms compared to PhenomXP and SEOBNRv4P. The mismatch results show
comparable performance between PhenomPNR and NRSur7dq4, but we caution a
reminder that NRSur7dq4 is a model that does not make the simplifying
assumptions outlined in Sec. III, and while the effects of these additional
physical features are generally small, we expect that their presence in
NRSur7dq4 compared to the NR data used for this comparison would bias the
results toward slightly higher mismatches. Nonetheless, it is encouraging to
observe that the PhenomPNR model, while tuned to a comparatively small number
of waveforms over a large configuration parameter space, and using a simple
set of model ansätze, and several simplifying assumptions, in general has
comparable mismatches to the NRSur7dq4 model.

From Fig. 23 it is apparent that the overall mismatch increases with mass-
ratio, and for each mass ratio the mismatch generally worsens with increasing
spin magnitude. Such a trend is also visible in Figs. 17-18. A simple
explanation for this observation arises from the PN scaling of the opening
angle $\iota$ with symmetric mass ratio and spin magnitude in quasi-circular
binaries with simple precession Apostolatos  _et al._ (1994),

$\sin\iota=\frac{S_{\perp}}{\sqrt{(\eta\sqrt{MR}+S_{\parallel})^{2}+S_{\perp}^{2}}},$
(77)

where $M$ is the system’s total mass and $R$ its orbital separation. A larger
opening angle increases the impact of precession modulations on the signal,
and these are where model inaccuracies will be most apparent. One would
similarly expect to see worsening mismatches as $S_{\perp}$ is maximised,
i.e., $\theta_{\mathrm{LS}}=90^{\circ}$ for single-spin cases; however this
trend is not as apparent in the results. The results in Fig. 23 show that
PhenomPNR is an improvement over PhenomXP in the most extreme region of
parameter space for $q=8,\chi\in[0.6,0.8]$, while SEOBNRv4P yields better
results in this region when $\theta_{\mathrm{LS}}>90^{\circ}$.

Regarding the performance of PhenomPNR for the two-spin NR cases listed in
Table 2, specifically cases 8-19 and cases 23-27, we observe that PhenomPNR
and PhenomXP perform surprisingly similarly for these cases, both for the SXS
and MAYA cases, whereas PhenomPNR provides a general improvement over
SEOBNRv4P for the two-spin cases. These results provide a reassuring
validation of the single-spin mapping detailed in Sec. IV.

Finally we remark on the impact of the fixed-$\hat{\mathbf{J}}$ assumption
used in the modeling of PhenomPNR and outlined in Sec. I.1. We computed the
SNR-weighted mismatches between the raw NR signals in an initially
$\hat{\mathbf{J}}$-aligned frame and those in the fixed-$\hat{\mathbf{J}}$
frame and find that the resulting mismatches are more than an order of
magnitude lower than the mismatches between PhenomPNR and the
fixed-$\hat{\mathbf{J}}$ frame NR signals presented in this section, and in
all cases lower than $5.1\times 10^{-4}$ at $100M_{\odot}$. The full
comparison is displayed in Fig. 24, and shows that the
fixed-$\hat{\mathbf{J}}$ approximation remains valid over a broad range of
parameter space where $\theta_{\mathrm{LS}}<90^{\circ}$ but begins to break
down for systems with higher mass ratio and opening angle, implying that
future modelling efforts should take care to re-evaluate the validity of this
approximation in more extreme regions of parameter space.

Figure 24: SNR-weighted mismatches computed at 100M⊙ and averaged over
inclination between the precessing waveform model PhenomPNR (“PNR”) and the NR
waveforms listed in Table 1, shown in order of the table listings. Alongside
these results are plotted the SNR-weighted mismatches computed between the NR
waveforms in the initially $\hat{\mathbf{J}}$-aligned frame and the
fixed-$\hat{\mathbf{J}}$ frame (“NR”). The solid vertical lines separate cases
by mass ratio, and the dashed vertical lines separate spin magnitude.

##  XII Conclusion

We have presented a new model of the GW signal from the inspiral, merger and
ringdown of precessing non-eccentric black-hole binaries, PhenomPNR. This is
the first model to explicitly calibrate precession effects through merger and
ringdown to NR simulations, and to use higher-order PN amplitude terms to
consistently define a signal-based co-precessing frame (the “quadrupole
aligned” (QA), or “optimal emission direction”) throughout the model.

The model is calibrated to 40 NR simulations of binaries where only the larger
black hole is spinning; the simulations cover four mass ratios ($q=1,2,4,8$),
two spin magnitudes ($\chi=0.4,0.8$), and five values of the spin misalignment
angle ($\theta_{\rm
LS}=30^{\circ},60^{\circ},90^{\circ},120^{\circ},150^{\circ}$). In the
frequency domain we separately model the co-precesing-frame signal,
$h_{2,2}^{\rm CP}(f)$, and the precession angles, $(\alpha,\beta,\gamma)$. We
model only the dominant $(\ell=2,|m|=2)$ multipoles in the co-precessing
frame, and neglect $\pm m$ asymmetries in the multipoles.

The co-precessing-frame model, PhenomDCP, is an extension of the earlier
aligned-spin model PhenomD, which was calibrated to 19 NR simulations of
either single-spin or equal-spin binaries, up to $q=18$ and spins of
$|\chi|\leq 0.85$. Our extension captures the effect of in-plane spin on the
amplitude and phase of the co-precessing-frame signal in the late inspiral and
merger-ringdown. We note for the first time that the final black hole’s
ringdown frequency is shifted to an _effective ringdown frequency_ in the co-
precessing frame. For this reason we explicitly model the effective ringdown
frequency across the single-spin parameter space, and do not make use of
estimates of the final black hole’s mass and spin.

The angle model, PhenomAngles, uses during inspiral the MSA PN angles used in
previous Phenom models. These angles describe the dynamics of the orbital
plane of the binary, which is only approximately equal to the QA direction
that we require to correctly model the signal. We find that this approximation
holds well throughout the inspiral for the angles $\alpha$ and $\gamma$, but
is not sufficiently accurate for $\beta$. However, it is possible to use
higher-order PN amplitude expressions, and reduction to a single-spin
subspace, to transform the MSA binary inclination into a good approximation of
the QA $\beta$. This is discussed in Sec. VI.2. Note that current EOBNR models
also use the orbital precession dynamics as an approximation to the signal
precession dynamics, and so an approach like the one used here is likely to
also improve the accuracy of those models.

The PhenomAngles precession angles through merger and ringdown are where our
model differs most significantly from previous Phenom and EOBNR models. We
observe and model a “dip” in $\alpha(f)$ (and therefore $\gamma(f)$) around
the effective ringdown frequency, similar to that found in the phase
derivative when constructing PhenomD Husa  _et al._ (2016); Khan  _et al._
(2016). Most importantly, we also model the steep collapse of $\beta$ through
merger. As we discuss in Sec. IX, this feature is quite distinct from the
asymptotic ringdown behaviour of $\beta$, and results in a shift in the
frequency location of the peak amplitude for each of the $\ell=2$ multipoles;
see Fig. 13.

Our precession model is tuned to single-spin NR simulations, but we make use
of a non-bijective mapping between the six spin components required to
describe a two-spin system, and the two components required in our single-spin
fits to NR data; see Sec. IV. In some parts of our model, this is equivalent
to using the $\chi_{\rm p}$ parameter from earlier Phenom models, but we also
introduce modifications to produce a mapping with greater physical fidelity
near $q=1$, and in the precession dynamics we taper away two-spin oscillations
as the system approaches merger. The result is an approximate IMR model for
two-spin systems.

In Sec. XI we demonstrate the accuracy of PhenomDCP, PhenomAngles, and the
complete IMR model PhenomPNR, by calculating matches against NR waveforms. The
matches are calculated against not only the 40 calibration waveforms, but an
additional 36 BAM verification waveforms from across the same single-spin
parameter space, plus 27 SXS and Maya waveforms, which include two-spin
systems. We find that our model in general improves significantly over
previous Phenom and EOBNR models, as illustrated in Fig. 23.

There are several immediate directions for future work. PhenomPNR does not
model subdominant multipoles in the co-precessing frame, but these will be
essential for measuring the properties of observations at larger mass ratios,
which is the very region of parameter space where PhenomPNR shows the greatest
improvement over previous models. This could be achieved through directly
modelling each of the multipoles, and including mode-mixing effects as in Ref.
García-Quirós  _et al._ (2020). Alternatively, one could estimate the
subdominant multipoles through the approximation used in Ref. London  _et al._
(2018).

Beyond this, the model needs to be extended to include explicit NR calibration
to two-spin systems, and to model $\pm m$ multipole asymmetries. Our results
also suggest that the angle modelling needs to be improved at lower
frequencies for cases with large mass ratios, large spins, and large values of
$\theta_{\rm LS}$; it is possible that this will require longer NR
simulations.

PhenomPNR models the signal in a frame where the direction of the total
angular momentum is constant, by first transforming the calibration NR
waveforms to a frame that tracks the evolution of $\hat{\mathbf{J}}(t)$. The
error incurred by this approximation is evaluated in Fig. 24, and we note that
this error is in general well below the other sources of modelling error.
However, in future, if we wish to construct models with mismatch errors below
$10^{-4}$, this approximation will need to be removed.

Finally, although most GW observations to date have been of systems with
comparable masses, there has been one observation (GW190814 Abbott  _et al._
(2020c)) where the mass ratio is likely outside the calibration region of this
model, and so it is necessary that the calibration region be extended to
higher mass ratios. All of these areas are the subject of ongoing work.

##  XIII Acknowledgements

We would like to thank other members of the Cardiff Gravity Exploration
Institute who performed simulations that were used in this project: Shrobana
Ghosh, Charlie Hoy, Panagiota Kolitsidou and David Yeeles.

The authors were supported in part by Science and Technology Facilities
Council (STFC) grant ST/V00154X/1 and European Research Council (ERC)
Consolidator Grant 647839. E. Hamilton was supported in part through the COST
Action CA18108, supported by COST (European Cooperation in Science and
Technology), by Swiss National Science Foundation (SNSF) grant IZCOZ0-189876.
L. London was supported at Massachusetts Institute of Technology (MIT) by
National Science Foundation Grant No. PHY-1707549 as well as support from
MIT’s School of Science and Department of Physics. A. Vano-Vinuales also
thanks the PhD researcher Decree-Law no. 57/2016 of August 29 (Portugal) and
Project No. UIDB/00099/2020 for support.

M. Hannam thanks Università di Roma “Sapienza” for hospitality while part of
this work was completed.

Simulations used in this work were performed on the DiRAC@Durham facility,
managed by the Institute for Computational Cosmology on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
funding via STFC capital grants ST/P002293/1 and ST/R002371/1, Durham
University and STFC operations grant ST/R000832/1. In addition, several of the
simulations used in this work were performed as part of an allocation
graciously provided by Oracle to explore the use of our code on the Oracle
Cloud Infrastructure.

This research also used the supercomputing facilities at Cardiff University
operated by Advanced Research Computing at Cardiff (ARCCA) on behalf of the
Cardiff Supercomputing Facility and the HPC Wales and Supercomputing Wales
(SCW) projects. We acknowledge the support of the latter, which is part-funded
by the European Regional Development Fund (ERDF) via the Welsh Government. In
part the computational resources at Cardiff University were also supported by
STFC grant ST/I006285/1.

Various plots and analyses in this paper were made using the Python software
packages LALSuite LIGO Scientific Collaboration (2018), Matplotlib Hunter
(2007), Numpy Harris  _et al._ (2020), PyCBC Nitz  _et al._ (2021), and Scipy
Virtanen  _et al._ (2020).

##  Appendix A Calculation of Precession Angles

Here we outline the calculation of the coprecessing frame quantified by
$\alpha(f)$, $\beta(f)$ and $\gamma(f)$; see Fig. 3. To calculate each angle
we use the rotationally invariant eigenvalue method O’Shaughnessy  _et al._
(2011); Boyle  _et al._ (2011). As shown in Ref. Boyle  _et al._ (2011), when
multipole moments are limited to cases where $\ell=2$ this is equivalent to
the original QA method Schmidt  _et al._ (2011). The result is independent of
the initial inertial frame when the minimum rotation condition is imposed
Boyle  _et al._ (2011).

The eigenvalue method and minimal rotation convention are described in Refs.
O’Shaughnessy  _et al._ (2011); Pekowsky  _et al._ (2013) and Boyle  _et al._
(2011). Ref. O’Shaughnessy  _et al._ (2011) introduces the eigenvalue method.
Ref. Pekowsky  _et al._ (2013) details the practical structure of this method
in its Appendix A, and Ref. Boyle  _et al._ (2011) adds the minimal rotation
convention which defines the optimal emission direction in a frame invariant
way. Here we provide a self-contained description of the algorithm to
calculate $\alpha(f)$, $\beta(f)$ and $\gamma(f)$.

Starting with the discrete Fourier transform of $\Psi_{4}$ decomposed into
spin weight $-2$ spherical harmonics, $\tilde{\psi}_{\ell m}$, we compute the
effect of all pair-wise angular momentum generators averaged about the
binary’s centre of mass. This is
$\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$, where

$\displaystyle\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle\;=$
$\displaystyle\;\frac{1}{2}\langle\mathcal{L}_{a}\mathcal{L}_{b}+\mathcal{L}_{b}\mathcal{L}_{a}\rangle$
(78) $\displaystyle\;=$
$\displaystyle\;\frac{\int_{\Omega}\tilde{\Psi}_{4}^{*}(f)\mathcal{L}_{(a}\mathcal{L}_{b)}\tilde{\Psi}_{4}(f){d}\Omega}{\int_{\Omega}|\tilde{\Psi}_{4}(f)|^{2}{d}\Omega}\;,$

with $a$ and $b$ over $\\{x,y,z\\}$, and where

$\displaystyle\mathcal{L}_{x}$
$\displaystyle=\frac{1}{2}(\mathcal{L}_{+}+\mathcal{L}_{-})\text{,
}\mathcal{L}_{y}=-i\frac{1}{2}(\mathcal{L}_{+}-\mathcal{L}_{-})\;,$ (79)
$\displaystyle\mathcal{L}_{\pm}$ $\displaystyle=e^{\pm i\varphi}\left[\pm
i\partial_{\theta}-\cot\theta\partial_{\varphi}-is\csc\theta\right]\;,$

and

$\displaystyle\mathcal{L}_{z}=\partial_{\varphi}\;.$ (80)

In Eq. (79), $s$ is the spin weight of the object being acted upon Newman and
Penrose (1966); Shah and Whiting (2016). As we are only interested in outgoing
gravitational wave radiation, $s=-2$.

In practice, evaluation of Eq. (78) need not require direct integration when
$\tilde{\Psi}_{4}$ is written in terms of its multipole moments,
$\tilde{\psi}_{\ell m}$; see Eq. (1). That is, as the operation of
$\mathcal{L}_{\pm}$ and $\mathcal{L}_{z}$ on ${{}_{-2}}Y_{\ell m}$ are known
Ruiz  _et al._ (2008); Newman and Penrose (1966), one finds that

$\displaystyle\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$ $\displaystyle=$
$\displaystyle\small{\frac{1}{\sum_{\ell,m}|\tilde{\psi}_{\ell
m}|^{2}}\begin{bmatrix}I_{0}+\text{Re}(I_{2})&\text{Im}I_{2}&\text{Re}I_{1}\\\
&I_{0}-\text{Re}(I_{2})&\text{Im}I_{1}\\\ &&I_{zz}\end{bmatrix}}$ (81a) where
$\displaystyle I_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\,(\tilde{\Psi},\mathcal{L}_{+}\mathcal{L}_{+}\tilde{\Psi})$
(81b) $\displaystyle=$ $\displaystyle\frac{1}{2}\,\sum_{\ell,m}c_{\ell
m}c_{\ell m+1}\tilde{\psi}_{\ell m+2}^{*}\tilde{\psi}_{\ell m}$ $\displaystyle
I_{1}$ $\displaystyle\equiv$
$\displaystyle(\tilde{\Psi},\mathcal{L}_{+}(\mathcal{L}_{z}+1/2)\tilde{\Psi})$
(81c) $\displaystyle=$ $\displaystyle\sum_{\ell m}c_{\ell
m}(m+1/2)\tilde{\psi}_{\ell m+1}^{*}\tilde{\psi}_{\ell m}$ $\displaystyle
I_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(\tilde{\Psi}|\ell(\ell+1)-\mathcal{L}_{z}^{2}|\tilde{\Psi}\right)$
(81d) $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\ell
m}[\ell(\ell+1)-m^{2}]|\tilde{\psi}_{\ell m}|^{2}$ $\displaystyle I_{zz}$
$\displaystyle\equiv$
$\displaystyle(\tilde{\Psi},\mathcal{L}_{z}\mathcal{L}_{z}\tilde{\Psi})=\sum_{\ell
m}m^{2}|\tilde{\psi}_{\ell m}|^{2}$ (81e) and where $c_{\ell
m}=\sqrt{\ell(\ell+1)-m(m+1)}$.

The resulting tensor, $\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$, is
analogous to the Cauchy stress tensor in continuum mechanics, and describes
infinitesimal changes in momenta (linear and angular) associated with
$\tilde{\Psi}_{4}(f)$ averaged about the source.

From the discussion in Sec. III, we see that
$\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$ is unchanged when considering
$\tilde{h}(f)$ rather than $\tilde{\Psi}_{4}(f)$, as the factor of $1/2\pi f$
amounts to a simple overall rescaling that does not affect normalized
eigenvectors. From these points, it follows that the eigenvector of
$\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$ with the largest eigenvalue
describes the direction about the source that experiences the largest strain
(or curvature) and strain-rate (or curvature-rate). This is the coprecessing
frame.

If we label $\langle\mathcal{L}_{(a}\mathcal{L}_{b)}\rangle$’s dominant
normalised eigenvector as $\hat{V}=(v_{x},v_{y},v_{z})$, then the angles
associated with the coprecessing frame are given by

$\displaystyle\alpha(f)$
$\displaystyle=\arctan{\left(\frac{v_{y}(f)}{v_{x}(f)}\right)},$ (82)
$\displaystyle\beta(f)$ $\displaystyle=\arccos{\left(v_{z}(f)\right)},$ (83)
$\displaystyle\gamma(f)$
$\displaystyle=-\int^{f}(\partial_{f^{\prime}}\alpha(f^{\prime}))\cos\beta(f^{\prime})\;\text{d}f^{\prime}\;.$
(84)

Eqs. (82)-(83) follow from the use of a source centered spherical polar
coordinate system in the asymptotically flat decomposition frame.
Equivalently, this is related to the frame of a distant observer. Eq. (84) is
the minimum rotation condition presented in Ref. Boyle  _et al._ (2011), which
removes secular changes in phase due to the evolution of $\alpha$ and $\beta$.

##  Appendix B Waveforms used in analysis

The NR waveforms used in the analysis of the model are listed in Tables 1 and
2. Table 1 contains the 80 waveforms which comprise the BAM catalogue Fauchon-
Jones  _et al._ (2021) of single spin precessing systems up to mass ratio
$q=8$ and single-spin magnitude $\chi=0.8$. A subset of 40 of these waveforms
were also used in tuning the model. Table 2 lists the additional waveforms
taken from the SXS Mroue  _et al._ (2013); Boyle  _et al._ (2019) and Maya
catalogues Jani  _et al._ (2016) used in assessing the accuracy of the model
and ensuring it was not over-fitted. This selection of waveforms includes two-
spin cases.

Simulation ID |  $\frac{100M_{\odot}}{M}f_{\text{ref}}$ (Hz) | $q$ | $\chi$ | $\theta_{\text{LS}}(^{\circ})$
---|---|---|---|---
CF21-1 | 14.8 | 1 | 0.2 | 30
CF21-2 | 14.8 | 1 | 0.2 | 60
CF21-3 | – | – | – | –
CF21-4 | – | – | – | –
CF21-5 | 14.7 | 1 | 0.2 | 150
CF21-6 | 14.8 | 1 | 0.4 | 30
CF21-7 | 14.8 | 1 | 0.4 | 60
CF21-8 | 14.9 | 1 | 0.4 | 90
CF21-9 | 14.8 | 1 | 0.4 | 120
CF21-10 | 14.8 | 1 | 0.4 | 150
CF21-11 | 18.2 | 1 | 0.6 | 30
CF21-12 | 14.8 | 1 | 0.6 | 60
CF21-13 | 14.9 | 1 | 0.6 | 90
CF21-14 | 14.8 | 1 | 0.6 | 120
CF21-15 | 14.8 | 1 | 0.6 | 150
CF21-16 | 14.8 | 1 | 0.8 | 30
CF21-17 | 14.7 | 1 | 0.8 | 60
CF21-18 | 14.9 | 1 | 0.8 | 90
CF21-19 | 14.9 | 1 | 0.8 | 120
CF21-20 | 15.2 | 1 | 0.8 | 150
CF21-21 | 14.7 | 2 | 0.2 | 30
CF21-22 | 14.7 | 2 | 0.2 | 60
CF21-23 | 14.8 | 2 | 0.2 | 90
CF21-24 | 15.2 | 2 | 0.2 | 120
CF21-25 | 14.8 | 2 | 0.2 | 150
CF21-26 | 14.8 | 2 | 0.4 | 30
CF21-27 | 14.6 | 2 | 0.4 | 60
CF21-28 | 14.7 | 2 | 0.4 | 90
CF21-29 | 14.8 | 2 | 0.4 | 120
CF21-30 | 14.8 | 2 | 0.4 | 150
CF21-31 | 14.7 | 2 | 0.6 | 30
CF21-32 | 14.9 | 2 | 0.6 | 60
CF21-33 | 14.5 | 2 | 0.6 | 90
CF21-34 | 14.9 | 2 | 0.6 | 120
CF21-35 | 14.4 | 2 | 0.6 | 150
CF21-36 | 14.9 | 2 | 0.8 | 30
CF21-37 | 14.9 | 2 | 0.8 | 60
CF21-38 | 14.7 | 2 | 0.8 | 90
CF21-39 | 15.0 | 2 | 0.8 | 120
CF21-40 | 15.0 | 2 | 0.8 | 150

Simulation ID |  $\frac{100M_{\odot}}{M}f_{\text{ref}}$ (Hz) | $q$ | $\chi$ | $\theta_{\text{LS}}(^{\circ})$
---|---|---|---|---
CF21-41 | – | – | – | –
CF21-42 | 16.0 | 4 | 0.2 | 60
CF21-43 | 16.8 | 4 | 0.2 | 90
CF21-44 | 15.3 | 4 | 0.2 | 120
CF21-45 | 15.2 | 4 | 0.2 | 150
CF21-46 | 16.6 | 4 | 0.4 | 30
CF21-47 | 16.3 | 4 | 0.4 | 60
CF21-48 | 14.7 | 4 | 0.4 | 90
CF21-49 | 14.8 | 4 | 0.4 | 120
CF21-50 | 15.0 | 4 | 0.4 | 150
CF21-51 | 17.0 | 4 | 0.6 | 30
CF21-52 | 16.2 | 4 | 0.6 | 60
CF21-53 | 15.8 | 4 | 0.6 | 90
CF21-54 | 15.1 | 4 | 0.6 | 120
CF21-55 | 14.0 | 4 | 0.6 | 150
CF21-56 | 17.5 | 4 | 0.8 | 30
CF21-57 | 16.8 | 4 | 0.8 | 60
CF21-58 | 14.9 | 4 | 0.8 | 90
CF21-59 | 14.8 | 4 | 0.8 | 120
CF21-60 | 14.9 | 4 | 0.8 | 150
CF21-61 | 18.4 | 8 | 0.2 | 30
CF21-62 | 18.1 | 8 | 0.2 | 60
CF21-63 | 17.8 | 8 | 0.2 | 90
CF21-64 | 17.3 | 8 | 0.2 | 120
CF21-65 | 17.2 | 8 | 0.2 | 150
CF21-66 | 19.0 | 8 | 0.4 | 30
CF21-67 | 18.6 | 8 | 0.4 | 60
CF21-68 | 17.8 | 8 | 0.4 | 90
CF21-69 | 17.0 | 8 | 0.4 | 120
CF21-70 | 16.5 | 8 | 0.4 | 150
CF21-71 | 19.7 | 8 | 0.6 | 30
CF21-72 | – | – | – | –
CF21-73 | 17.9 | 8 | 0.6 | 90
CF21-74 | 16.7 | 8 | 0.6 | 120
CF21-75 | 17.0 | 8 | 0.6 | 150
CF21-76 | 20.5 | 8 | 0.8 | 30
CF21-77 | 19.5 | 8 | 0.8 | 60
CF21-78 | 18.0 | 8 | 0.8 | 90
CF21-79 | 16.0 | 8 | 0.8 | 120
CF21-80 | 15.2 | 8 | 0.8 | 150

Table 1: BAM single-spin configurations used in tuning the co-precessing and angle models as well as in the assessment of the accuracy of the model. Simulation ID |  $\frac{100M_{\odot}}{M}f_{\text{ref}}$ (Hz) | $q$ | $\chi$ | $\theta_{\text{LS}}(^{\circ})$ | $\chi_{1}$ | $\chi_{2}$
---|---|---|---|---|---|---
SXS0097 | 9.2 | 1.5 | 0.5 | 90 | $\left(-0.493,0,0.083\right)$ | $\left(0,0,0\right)$
SXS0018 | 7.9 | 1.5 | 0.5 | 90 | $\left(-0.494,0,0.078\right)$ | $\left(0,0,0\right)$
SXS0092 | 9.4 | 1.5 | 0.5 | 150 | $\left(-0.29,0,-0.407\right)$ | $\left(0,0,0\right)$
SXS0033 | 11.2 | 3.0 | 0.5 | 30 | $\left(-0.19,0,0.463\right)$ | $\left(0,0,0\right)$
SXS0035 | 8.5 | 3.0 | 0.5 | 90 | $\left(-0.476,0,0.154\right)$ | $\left(0,0,0\right)$
SXS1109 | 10.2 | 5.0 | 0.5 | 90 | $\left(-0.435,0,0.246\right)$ | $\left(0,0,0\right)$
SXS0062 | 14.1 | 5.0 | 0.5 | 116 | $\left(-0.492,0,0.088\right)$ | $\left(0,0,0\right)$
SXS0161 | 9.2 | 1.0 | 1.199 | 120 | $\left(-0.579,0,-0.158\right)$ | $\left(-0.579,0,-0.158\right)$
SXS0115 | 10.2 | 1.07 | 0.246 | 74 | $\left(-0.027,-0.016,-0.203\right)$ | $\left(-0.236,0.018,0.304\right)$
SXS0116 | 10.2 | 1.08 | 0.167 | 40 | $\left(0.022,-0.099,0.032\right)$ | $\left(-0.143,0.115,0.106\right)$
SXS0124 | 10.2 | 1.26 | 0.412 | 44 | $\left(-0.247,-0.041,0.091\right)$ | $\left(-0.079,0.065,0.294\right)$
SXS0102 | 9.3 | 1.5 | 0.5 | 90 | $\left(-0.486,0,0.116\right)$ | $\left(-0.486,0,0.116\right)$
SXS1397 | 5.1 | 1.56 | 0.299 | 111 | $\left(-0.242,0.037,-0.172\right)$ | $\left(0.458,-0.089,0.102\right)$
SXS0135 | 10.3 | 1.64 | 0.186 | 128 | $\left(-0.059,0.095,0.025\right)$ | $\left(0.003,-0.257,-0.228\right)$
SXS0143 | 10.2 | 1.92 | 0.441 | 28 | $\left(-0.072,0.041,0.443\right)$ | $\left(-0.413,-0.15,-0.056\right)$
SXS0144 | 10.2 | 1.94 | 0.214 | 146 | $\left(-0.135,0.011,-0.281\right)$ | $\left(0.05,-0.04,0.213\right)$
SXS0049 | 11.2 | 3.0 | 0.527 | 72 | $\left(-0.474,0,0.159\right)$ | $\left(0.159,0,0.474\right)$
SXS1160 | 11.1 | 3.0 | 0.658 | 63 | $\left(-0.455,-0.024,0.531\right)$ | $\left(0.41,0.212,-0.384\right)$
SXS0165 | 18.1 | 6.0 | 0.93 | 125 | $\left(-0.648,0.003,0.639\right)$ | $\left(-0.186,-0.094,-0.216\right)$
GT0745 | 22.3 | 6.0 | 0.6 | 91 | $\left(-0.437,0,0.411\right)$ | $\left(0,0,0\right)$
GT0742 | 21.9 | 7.0 | 0.6 | 91 | $\left(-0.404,0,0.444\right)$ | $\left(0,0,0\right)$
GT0834 | 20.9 | 7.0 | 0.8 | 168 | $\left(-0.375,0,0.706\right)$ | $\left(0,0,0\right)$
GT0880 | 22.0 | 4.5 | 0.537 | 52 | $\left(-0.269,0,0.536\right)$ | $\left(0.269,0,-0.536\right)$
GT0887 | 21.8 | 5.0 | 0.543 | 51 | $\left(-0.256,0,0.542\right)$ | $\left(0.256,0,-0.542\right)$
GT0889 | 21.6 | 6.0 | 0.552 | 50 | $\left(-0.234,0,0.552\right)$ | $\left(0.234,0,-0.552\right)$
GT0888 | 21.4 | 7.0 | 0.559 | 49 | $\left(-0.216,0,0.56\right)$ | $\left(0.216,0,-0.56\right)$
GT0886 | 21.4 | 8.0 | 0.564 | 49 | $\left(-0.2,0,0.566\right)$ | $\left(0.2,0,-0.566\right)$
Table 2: Additional configurations from the SXS and MAYA catalogues used in
the assessment of the accuracy of the model.

##  Appendix C Parameter-space fits

Here we show how each of the co-efficients that appear in PhenomPNR vary
across the parameter space. Figs. 25 and 26 show the variation of the co-
efficients which appear in PhenomDCP, as described in Sec. V. Fig. 27 shows
the co-efficients in the ansatz for $\alpha$ and Fig. 28 shows those in the
ansatz for $\beta$, which are presented in Sec. VII. As can be seen from these
figures, the co-efficients vary smoothly across the parameter space.

|
---|---
Figure 25: Amplitude parameters for tuned co-precessing waveform model, PhenomDCP. The fits are shown as two-dimensional surfaces covering the parameter space described by $\eta$ and $\cos\theta_{\mathrm{LS}}$. On the left in blue are the fits for the simulations with $\chi=0.4$ and on the right in red are the fits for $\chi=0.8$. Above each of these surfaces are shown the residuals. |
---|---
|
Figure 26: Phase parameters for tuned co-precessing waveform model, PhenomDCP.
The fits are shown as two-dimensional surfaces covering the parameter space
described by $\eta$ and $\cos\theta_{\mathrm{LS}}$. On the left in blue are
the fits for the simulations with $\chi=0.4$ and on the right in red are the
fits for $\chi=0.8$. Above each of these surfaces are shown the residuals.
Figure 27: Comparison of the fits for each of the co-efficients for the ansatz
for $\alpha$ given in equation 48 with the co-efficients found from the data
as described in Sec. VII. The fits are shown as two-dimensional surfaces
covering the parameter space described by $\eta$ and
$\cos\theta_{\mathrm{LS}}$. On the left in blue are the fits for the
simulations with $\chi=0.4$ and on the right in red are the fits for
$\chi=0.8$. Above each of these surfaces are shown the residuals.  Figure 28:
Comparison of the fits for each of the co-efficients for the ansatz for
$\beta$ given in equation 49 with the co-efficients found from the data as
described in Sec. VII. The fits are shown as two-dimensional surfaces covering
the parameter space described by $\eta$ and $\cos\theta_{\mathrm{LS}}$. On the
left in blue are the fits for the simulations with $\chi=0.4$ and on the right
in red are the fits for $\chi=0.8$. Above each of these surfaces are shown the
residuals.

## References

  * Aasi  _et al._ (2015) J. Aasi _et al._ (LIGO Scientific), Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * Acernese  _et al._ (2015) F. Acernese _et al._ (VIRGO), Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc] .
  * Abbott  _et al._ (2019) B. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE] .
  * Nitz  _et al._ (2020) A. H. Nitz, T. Dent, G. S. Davies, S. Kumar, C. D. Capano, I. hARRY, S. Mozzon, L. Nuttall, A. Lundgren, and M. Tápai, The Astrophysical Journal 891, 123 (2020).
  * Zackay  _et al._ (2019) B. Zackay, L. Dai, T. Venumadhav, J. Roulet, and M. Zaldarriaga, (2019), arXiv:1910.09528 [astro-ph.HE] .
  * Venumadhav  _et al._ (2020) T. Venumadhav, B. Zackay, J. Roulet, L. Dai, and M. Zaldarriaga, Phys. Rev. D 101, 083030 (2020), arXiv:1904.07214 [astro-ph.HE] .
  * Abbott  _et al._ (2020a) R. Abbott _et al._ (LIGO Scientific, Virgo), (2020a), arXiv:2010.14527 [gr-qc] .
  * Husa  _et al._ (2016) S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. Jiménez Forteza, and A. Bohé, 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, Phys. Rev. D 93, 044007 (2016), arXiv:1508.07253 [gr-qc] .
  * Hannam  _et al._ (2014) M. Hannam, P. Schmidt, A. Bohé, L. Haegel, S. Husa, F. Ohme, G. Pratten, and M. Pürrer, Phys. Rev. Lett. 113, 151101 (2014), arXiv:1308.3271 [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, Phys. Rev. Lett. 120, 161102 (2018), arXiv:1708.00404 [gr-qc] .
  * Khan  _et al._ (2019) S. Khan, K. Chatziioannou, M. Hannam, and F. Ohme, Phys. Rev. D100, 024059 (2019), arXiv:1809.10113 [gr-qc] .
  * Khan  _et al._ (2020) S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam, Phys. Rev. D 101, 024056 (2020), arXiv:1911.06050 [gr-qc] .
  * Pratten  _et al._ (2020a) G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. Ramos-Buades, H. Estelles, and R. Jaume, (2020a), arXiv:2001.11412 [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, (2020), arXiv:2001.10914 [gr-qc] .
  * Pratten  _et al._ (2020b) G. Pratten _et al._ , (2020b), arXiv:2004.06503 [gr-qc] .
  * Thompson  _et al._ (2020) J. E. Thompson, E. Fauchon-Jones, S. Khan, E. Nitoglia, F. Pannarale, T. Dietrich, and M. Hannam, Phys. Rev. D 101, 124059 (2020), arXiv:2002.08383 [gr-qc] .
  * Estellés  _et al._ (2020a) H. Estellés, A. Ramos-Buades, S. Husa, C. García-Quirós, M. Colleoni, L. Haegel, and R. Jaume, (2020a), arXiv:2004.08302 [gr-qc] .
  * Estellés  _et al._ (2020b) H. Estellés, S. Husa, M. Colleoni, D. Keitel, M. Mateu-Lucena, C. García-Quirós, A. Ramos-Buades, and A. Borchers, (2020b), arXiv:2012.11923 [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, Phys. Rev. D 86, 024011 (2012), arXiv:1202.0790 [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, Phys. Rev. D89, 084006 (2014), arXiv:1307.6232 [gr-qc] .
  * Taracchini  _et al._ (2014) A. Taracchini _et al._ , Phys. Rev. D89, 061502 (2014), arXiv:1311.2544 [gr-qc] .
  * Bohé  _et al._ (2017) A. Bohé _et al._ , 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, Phys. Rev. D 98, 084028 (2018), arXiv:1803.10701 [gr-qc] .
  * Ossokine  _et al._ (2020) S. Ossokine _et al._ , Phys. Rev. D 102, 044055 (2020), arXiv:2004.09442 [gr-qc] .
  * Matas  _et al._ (2020) A. Matas _et al._ , Phys. Rev. D 102, 043023 (2020), arXiv:2004.10001 [gr-qc] .
  * Ramos-Buades  _et al._ (2020) A. Ramos-Buades, P. Schmidt, G. Pratten, and S. Husa, Phys. Rev. D 101, 103014 (2020), arXiv:2001.10936 [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, Phys. Rev. D 95, 104023 (2017a), arXiv:1701.00550 [gr-qc] .
  * Varma  _et al._ (2019a) V. Varma, S. E. Field, M. A. Scheel, J. Blackman, L. E. Kidder, and H. P. Pfeiffer, Phys. Rev. D 99, 064045 (2019a), arXiv:1812.07865 [gr-qc] .
  * Apostolatos  _et al._ (1994) T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Phys. Rev. D 49, 6274 (1994).
  * Kidder (1995) L. E. Kidder, Phys. Rev. D 52, 821 (1995), arXiv:gr-qc/9506022 .
  * Schmidt  _et al._ (2011) P. Schmidt, M. Hannam, S. Husa, and P. Ajith, Phys. Rev. D84, 024046 (2011), arXiv:1012.2879 [gr-qc] .
  * Schmidt  _et al._ (2012) P. Schmidt, M. Hannam, and S. Husa, Phys. Rev. D 86, 104063 (2012), arXiv:1207.3088 [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, Phys. Rev. D 96, 024058 (2017b), arXiv:1705.07089 [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, Phys. Rev. Research. 1, 033015 (2019b), arXiv:1905.09300 [gr-qc] .
  * Ajith  _et al._ (2007) P. Ajith _et al._ , Class. Quant. Grav. 24, S689 (2007), arXiv:0704.3764 [gr-qc] .
  * Ajith  _et al._ (2008) P. Ajith _et al._ , Phys. Rev. D 77, 104017 (2008), [Erratum: Phys.Rev.D 79, 129901 (2009)], arXiv:0710.2335 [gr-qc] .
  * Ajith  _et al._ (2011) P. Ajith _et al._ , Phys. Rev. Lett. 106, 241101 (2011), arXiv:0909.2867 [gr-qc] .
  * Santamaria  _et al._ (2010) L. Santamaria _et al._ , Phys. Rev. D82, 064016 (2010), arXiv:1005.3306 [gr-qc] .
  * Pürrer  _et al._ (2016) M. Pürrer, M. Hannam, and F. Ohme, Phys. Rev. D 93, 084042 (2016), arXiv:1512.04955 [gr-qc] .
  * Abbott  _et al._ (2020b) B. Abbott _et al._ (KAGRA, LIGO Scientific, Virgo), Living Rev. Rel. 23, 3 (2020b).
  * Fairhurst  _et al._ (2020) S. Fairhurst, R. Green, M. Hannam, and C. Hoy, Phys. Rev. D 102, 041302 (2020), arXiv:1908.00555 [gr-qc] .
  * Estellés  _et al._ (2021) H. Estellés, M. Colleoni, C. García-Quirós, S. Husa, D. Keitel, M. Mateu-Lucena, M. d. L. Planas, and A. Ramos-Buades, (2021), arXiv:2105.05872 [gr-qc] .
  * Fauchon-Jones  _et al._ (2021) E. Fauchon-Jones, E. Hamilton, C. Kalaghatgi, S. Ghosh, M. Hannam, C. Hoy, S. Khan, P. Kolitsidou, L. London, J. Thompson, A. Vañó-Viñuales, and D. Yeeles, (2021), in preparation.
  * Pekowsky  _et al._ (2013) L. Pekowsky, R. O’Shaughnessy, J. Healy, and D. Shoemaker, Phys. Rev. D 88, 024040 (2013), arXiv:1304.3176 [gr-qc] .
  * Hofmann  _et al._ (2016) F. Hofmann, E. Barausse, and L. Rezzolla, Astrophys. J. Lett. 825, L19 (2016), arXiv:1605.01938 [gr-qc] .
  * O’Shaughnessy  _et al._ (2011) R. O’Shaughnessy, B. Vaishnav, J. Healy, Z. Meeks, and D. Shoemaker, Phys. Rev. D84, 124002 (2011), arXiv:1109.5224 [gr-qc] .
  * Boyle  _et al._ (2011) M. Boyle, R. Owen, and H. P. Pfeiffer, Phys. Rev. D84, 124011 (2011), arXiv:1110.2965 [gr-qc] .
  * Ochsner and O’Shaughnessy (2012) E. Ochsner and R. O’Shaughnessy, Phys. Rev. D86, 104037 (2012), arXiv:1205.2287 [gr-qc] .
  * Boyle  _et al._ (2014) M. Boyle, L. E. Kidder, S. Ossokine, and H. P. Pfeiffer, (2014), arXiv:1409.4431 [gr-qc] .
  * Hamilton and Hannam (2018) E. Hamilton and M. Hannam, Phys. Rev. D98, 084018 (2018), arXiv:1807.06331 [gr-qc] .
  * Chatziioannou  _et al._ (2017) K. Chatziioannou, A. Klein, N. Cornish, and N. Yunes, Phys. Rev. Lett. 118, 051101 (2017), arXiv:1606.03117 [gr-qc] .
  * O’Shaughnessy  _et al._ (2013) R. O’Shaughnessy, L. London, J. Healy, and D. Shoemaker, Phys. Rev. D 87, 044038 (2013), arXiv:1209.3712 [gr-qc] .
  * Marsat and Baker (2018) S. Marsat and J. G. Baker, (2018), arXiv:1806.10734 [gr-qc] .
  * Bruegmann  _et al._ (2008a) B. Bruegmann, J. A. Gonzalez, M. Hannam, S. Husa, and U. Sperhake, Phys. Rev. D77, 124047 (2008a), arXiv:0707.0135 [gr-qc] .
  * Kalaghatgi and Hannam (2020) C. Kalaghatgi and M. Hannam, (2020), arXiv:2008.09957 [gr-qc] .
  * LIGO Scientific Collaboration (2018) LIGO Scientific Collaboration, “LIGO Algorithm Library - LALSuite,” free software (GPL) (2018).
  * Buonanno  _et al._ (2004) A. Buonanno, Y.-b. Chen, Y. Pan, and M. Vallisneri, Phys. Rev. D 70, 104003 (2004), [Erratum: Phys.Rev.D 74, 029902 (2006)], arXiv:gr-qc/0405090 .
  * Schmidt  _et al._ (2015) P. Schmidt, F. Ohme, and M. Hannam, Phys. Rev. D 91, 024043 (2015), arXiv:1408.1810 [gr-qc] .
  * Abbott  _et al._ (2020c) R. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 896, L44 (2020c), arXiv:2006.12611 [astro-ph.HE] .
  * Bruegmann  _et al._ (2008b) B. Bruegmann, J. A. Gonzalez, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, Phys. Rev. D 77, 024027 (2008b), arXiv:gr-qc/0610128 .
  * Boyle  _et al._ (2019) M. Boyle _et al._ , Class. Quant. Grav. 36, 195006 (2019), arXiv:1904.04831 [gr-qc] .
  * (63) http://www.black-holes.org.
  * Jani  _et al._ (2016) K. Jani, J. Healy, J. A. Clark, L. London, P. Laguna, and D. Shoemaker, Class. Quant. Grav. 33, 204001 (2016), arXiv:1605.03204 [gr-qc] .
  * (65) http://www.einstein.gatech.edu/catalog/.
  * Campanelli  _et al._ (2006) M. Campanelli, C. Lousto, and Y. Zlochower, Phys. Rev. D 74, 041501 (2006), arXiv:gr-qc/0604012 .
  * Cook and York (1990) G. B. Cook and J. York, James W., Phys. Rev. D 41, 1077 (1990).
  * Oppenheim  _et al._ (1999) A. V. Oppenheim, R. W. Schafer, and J. R. Buck,  _Discrete-Time Signal Processing (2nd Ed.)_ (Prentice-Hall, Inc., USA, 1999).
  * Wigner (1959) E. P. Wigner,  _Group theory and its application to the quantum mechanics of atomic spectra_ (Academic Press, New York, 1959).
  * Bowen and York (1980) J. M. Bowen and J. W. York, Jr., Phys. Rev. D 21, 2047 (1980).
  * Ruiz  _et al._ (2008) M. Ruiz, M. Alcubierre, D. Núñez, and R. Takahashi, General Relativity and Gravitation 40, 1705 (2008).
  * Campanelli  _et al._ (2007) M. Campanelli, C. O. Lousto, Y. Zlochower, B. Krishnan, and D. Merritt, Phys. Rev. D 75, 064030 (2007), arXiv:gr-qc/0612076 .
  * Varma  _et al._ (2020) V. Varma, M. Isi, and S. Biscoveanu, Phys. Rev. Lett. 124, 101104 (2020), arXiv:2002.00296 [gr-qc] .
  * Cutler and Flanagan (1994) C. Cutler and E. E. Flanagan, Phys. Rev. D49, 2658 (1994), arXiv:gr-qc/9402014 [gr-qc] .
  * Poisson and Will (1995) E. Poisson and C. M. Will, Phys. Rev. D 52, 848 (1995), arXiv:gr-qc/9502040 .
  * Ajith (2011) P. Ajith, Phys. Rev. D 84, 084037 (2011), arXiv:1107.1267 [gr-qc] .
  * Kumar  _et al._ (2016) P. Kumar, T. Chu, H. Fong, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E. Kidder, M. A. Scheel, and B. Szilagyi, Phys. Rev. D 93, 104050 (2016), arXiv:1601.05396 [gr-qc] .
  * Chatziioannou  _et al._ (2017) K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish, Phys. Rev. D95, 104004 (2017), arXiv:1703.03967 [gr-qc] .
  * Racine (2008) E. Racine, Phys. Rev. D 78, 044021 (2008), arXiv:0803.1820 [gr-qc] .
  * Gerosa  _et al._ (2020) D. Gerosa, M. Mould, D. Gangardt, P. Schmidt, G. Pratten, and L. M. Thomas, (2020), arXiv:2011.11948 [gr-qc] .
  * Thomas  _et al._ (2020) L. M. Thomas, P. Schmidt, and G. Pratten, (2020), arXiv:2012.02209 [gr-qc] .
  * Arun  _et al._ (2009) K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D79, 104023 (2009), [Erratum: Phys. Rev.D84,049901(2011)], arXiv:0810.5336 [gr-qc] .
  * London and Fauchon-Jones (2019) L. London and E. Fauchon-Jones, Class. Quant. Grav. 36, 235015 (2019), arXiv:1810.03550 [gr-qc] .
  * London  _et al._ (2020) L. London, E. Fauchon, and EZHamilton, “llondon6/positive: map,”  (2020).
  * Kesden  _et al._ (2015) M. Kesden, D. Gerosa, R. O’Shaughnessy, E. Berti, and U. Sperhake, Phys. Rev. Lett. 114, 081103 (2015), arXiv:1411.0674 [gr-qc] .
  * Klein  _et al._ (2013) A. Klein, N. Cornish, and N. Yunes, Phys. Rev. D 88, 124015 (2013), arXiv:1305.1932 [gr-qc] .
  * Chatziioannou  _et al._ (2013) K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish, Phys. Rev. D 88, 063011 (2013), arXiv:1307.4418 [gr-qc] .
  * Virtanen  _et al._ (2020) P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, Nature Methods 17, 261 (2020).
  * Leaver (1985) E. Leaver, Proc. Roy. Soc. Lond. A A402, 285 (1985).
  * Kamaretsos  _et al._ (2012a) I. Kamaretsos, M. Hannam, and B. Sathyaprakash, Phys. Rev. Lett. 109, 141102 (2012a), arXiv:1207.0399 [gr-qc] .
  * Kamaretsos  _et al._ (2012b) I. Kamaretsos, M. Hannam, S. Husa, and B. S. Sathyaprakash, Phys. Rev. D 85, 024018 (2012b), arXiv:1107.0854 [gr-qc] .
  * Giesler  _et al._ (2019) M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky, Phys. Rev. X 9, 041060 (2019), arXiv:1903.08284 [gr-qc] .
  * (93) LSC, “LIGO Document T1800044-v5,” https://dcc.ligo.org/LIGO-T1800044/public.
  * Harry  _et al._ (2016) I. Harry, S. Privitera, A. Bohé, and A. Buonanno, Phys. Rev. D 94, 024012 (2016), arXiv:1603.02444 [gr-qc] .
  * Mroue  _et al._ (2013) A. H. Mroue _et al._ , Phys. Rev. Lett. 111, 241104 (2013), arXiv:1304.6077 [gr-qc] .
  * Schmidt  _et al._ (2017) P. Schmidt, I. W. Harry, and H. P. Pfeiffer, (2017), arXiv:1703.01076 [gr-qc] .
  * Hunter (2007) J. D. Hunter, Computing in Science & Engineering 9, 90 (2007).
  * Harris  _et al._ (2020) C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Nature 585, 357 (2020).
  * Nitz  _et al._ (2021) A. Nitz, I. Harry, D. Brown, C. M. Biwer, J. Willis, T. D. Canton, C. Capano, T. Dent, L. Pekowsky, A. R. Williamson, G. S. C. Davies, S. De, M. Cabero, B. Machenschalk, P. Kumar, D. Macleod, S. Reyes, dfinstad, F. Pannarale, T. Massinger, S. Kumar, M. Tápai, L. Singer, S. Khan, S. Fairhurst, A. Nielsen, S. Singh, shasvath, B. U. V. Gadre, and I. Dorrington, “gwastro/pycbc: Pycbc release 1.18.1,”  (2021).
  * Newman and Penrose (1966) E. T. Newman and R. Penrose, Journal of Mathematical Physics 7, 863 (1966), https://doi.org/10.1063/1.1931221 .
  * Shah and Whiting (2016) A. G. Shah and B. F. Whiting, Gen. Rel. Grav. 48, 78 (2016), arXiv:1503.02618 [gr-qc] .