$\vartheta_{\text{LN},0}=\pi/2$ is due to a more faithful $h_{21}$ multipole
satisfying the relation $|h_{21}|\gtrsim|h_{22}|$ as we have previously
explored in detail. As we have only four cases with $Q\gtrsim 5$, we move on
to comparisons of longer waveforms.

Figure 17:  Trace plots of the mismatches between
$\\{{\texttt{SEOB}},\texttt{TEOB},\texttt{TPHM},\texttt{XPHM}\\}$ and
numerical relativity waveforms from 23 long SXS simulations as a function of
the total binary mass. We computed the mismatches at the values of
$M=\\{37.5,75,112.5,150,187.5,225\\}M_{\odot}$ (marked by the faint vertical
lines) and linearly connected the data points. The top (bottom) two rows show
${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$ (${\bar{\cal M}}_{\text{opt}}$)
with odd (even) rows corresponding to $\vartheta_{\text{LN},0}=0$
($\vartheta_{\text{LN},0}=\pi/2$) results. The four columns represent, from
left to right, the results for SEOB, TEOB, TPHM, XPHM.

###  V.2 Long SXS Waveforms

For this assessment, we employ 23 long SXS waveforms, namely simulations
1389-1411 Boyle  _et al._ (2019). The total number of GW cycles in these
simulations varies approximately between $128$ and $305$, and the relevant
parameters have the following ranges: $Q\leq 4,\chi_{\text{p}}\lesssim
0.49,\chi_{\rm eff}\in[-0.2,0.3]$ though only 1409-1411 have $Q=4$ with the
rest having $Q<2$. With only 23 cases as compared with $\mathcal{O}(10^{3})$
in Sec. IV and $\mathcal{O}(10^{2})$ in Sec. V.1, we could afford the
computational time to expand our total mass sample to the following values:
$M=\\{37.5,75,112.5,150,187.5,225\\}M_{\odot}$, with $f_{i}=f_{0}+3\,$Hz for
all match integrals. Note that because the ASD drops steeply from
$\mathcal{O}(10)$ Hz to 20 Hz (see Fig. 1), only 1 to 2 inspiral cycles
contribute to the match integrals for the $M=225M_{\odot}$ cases even though
$f_{i}\in[4,5]\,$Hz for these. Similarly, we record 2.5 to 4 inspiral cycles
for $M=187.5M_{\odot}$ and 4.5 to 7 cycles for $M=150M_{\odot}$. Accordingly,
we observe that $\text{(MR-only SNR)}\gtrsim\text{(inspiral-only SNR)}$ for
$M\gtrsim 175M_{\odot}$.

We present the resulting mismatches as trace plots in Fig. 17. In the top two
rows of the figure, we display ${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$ at
$\vartheta_{\text{LN},0}=0$ and $\pi/2$ then repeat this for ${\bar{\cal
M}}_{\text{opt}}$ in the bottom two rows with the four columns corresponding
to the four approximants in the usual order. First, we note that with the
exception of one TPHM $\vartheta_{\text{LN},0}=\pi/2$ case (corresponding to
SXS:1395), all mismatches are less than 0.035. Moreover, we also observe that
${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}(\vartheta_{\text{LN},0}=0)<0.01$
holds nearly universally and that this inequality still holds at
$\vartheta_{\text{LN},0}=\pi/2$ for the EOB models. Our finding of ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}\preceq{\bar{\cal M}}_{\text{opt}}$ from the
previous sections persists here as well, which can be gathered by comparing
row one with row three, and row two with row four. The dependence of the
mismatches on inclination also seems to be consistent with our previous
results as the relation ${\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)\preceq{\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=\pi/2)$ is retained as seen by
comparing row one with row two, and row three with row four of the same
figure.

As for the dependence of the mismatches on the total mass, we observe mostly a
flattening of each mismatch curve as $M$ exceeds a certain threshold, more
strongly so for the $\vartheta_{\text{LN},0}=0$ cases. This flattening is
consistent with the results of, e.g., Refs. Pratten  _et al._ (2020); Estellés
_et al._ (2022b); Gamba  _et al._ (2022); Ramos-Buades  _et al._ (2023).
Interestingly we observe it for both ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ and ${\bar{\cal M}}_{\text{opt}}$. For the
flattening of the former, our reasoning is as follows: as the binary gets
heavier, the signal shifts toward lower frequencies: e.g.,
$f_{\text{peak}}\lesssim 50\,$Hz for $M=150M_{\odot}$ and
$f_{\text{peak}}\lesssim 30\,$Hz for $M=225M_{\odot}$. In this regime, the
detectors are less sensitive (see Fig. 1), therefore the mismatch expression
is also less sensitive to the differences between waveforms. So the dependence
on the total mass becomes irrelevant for ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ beyond a certain threshold.

The flattening of the ${\bar{\cal M}}_{\text{opt}}$ curves are partly due to
this reason combined with the fact that for the cases with only a few inspiral
cycles, i.e., $M\gtrsim 150M_{\odot}$, the MR part of the signal becomes the
dominant contribution to the mismatch. In other words, the contribution to the
mismatch from $\lesssim 5$ inspiral cycles does not change the overall
${\bar{\cal M}}_{\text{opt}}$ significantly enough. However, the MR part of
some of the heavier cases falls in the $\mathcal{O}(100)\,$Hz region, where
the detectors are more sensitive, as is, accordingly, our full-band mismatch
${\bar{\cal M}}_{\text{opt}}$. This is why we observe flatter curves for
${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}(\vartheta_{\text{LN},0}=0)$ than
for ${\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)$ in Fig. 17. Also
note that some curves do not flatten out at all. We are unable to provide a
single universal, “one size fits all” explanation for these as the shape and
mass dependence of each curve varies from one model to another.

One final prominent feature of the figure is the change in XPHM’s
faithfulness, ${\bar{\cal M}}_{\text{opt}}$, in going from
$\vartheta_{\text{LN},0}=0$ to $\pi/2$, for which we see that the values of
the mismatches shift from mostly being below 0.01 to mostly being above it,
while still remaining below 0.035. This behavior might be indicative of more
severe MR-related issues for XPHM’s precessing $(2,\pm 1)$ multipoles than for
other models. We re-computed ${\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=\pi/2)$ using the updated SpinTaylor
version of XPHM Colleoni  _et al._ (2024) which showed improvement: the range
of ${\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=\pi/2)$ dropped from
$[0.004,0.03]$ to $[0.003,0.02]$. More noticeably, the $M\gtrsim 100M_{\odot}$
values of ${\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=\pi/2)$
obtained with the SpinTaylor version are all $\lesssim 0.02$ in contrast to
many cases of $0.02<{\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=\pi/2)\lesssim 0.035$ seen in the
lower right panel of Fig. 17.

##  VI Injection/Recovery Study

As the final part of our survey, we investigate the performance of the
precessing approximants in an injection-recovery PE study. Specifically, we
inject two different SXS ($\ell=2$)-only strains into LIGO noise and recover
the parameters of the injected waveforms with SEOB, TPHM and XPHM. We are
greatful to Charlie Hoy and Lorenzo Pompili for assisting us with the SEOB
runs. Our omission of the TEOB results is due to:  (i) the precessing model
can only be used for PE with the bajes library Breschi  _et al._ (2021) which
is external to the computational infrastructure that we employed for the PE
runs with $\\{{{\texttt{SEOB}}},\texttt{TPHM},\texttt{XPHM}\\}$;  (ii)
bajesruns crashed with waveform errors when the upper bounds of the
$\chi_{1,2}$ priors were above $0.8$ despite TEOB generating reasonable
waveforms in that regime. This error first arose when attempting to analyse
GW200129_065458 with TEOB and persisted when injection/recovery runs were
carried out. We hope to collaborate with bajes developers in the future so
that these issues can be resolved.

Returning to our study here. We performed “zero-noise” injections of the NR
waveforms SXS:0050 (henceforth 0050) and SXS:0628 (henceforth 0628) while
using an estimated O4a PSD aLI (2022) for our likelihood computations. These
simulations have key parameters
$\\{Q,\chi_{\mathrm{eff}},\text{max}(\chi_{\text{p}})\\}\approx\\{3,0.001,0.5\\},\\{2,-0.174,0.84\\}$
respectively. The former has a higher component mass asymmetry and moderate
spins, only in the orbital plane, whereas the latter has a moderate mass
ratio, but significant precession. We perform each injection twice: for a low
mass and a high mass BBH given in the detector frame, specifically,
$M^{\text{det}}=56.5M_{\odot},150M_{\odot}$ for 0050 and
$M^{\text{det}}=67M_{\odot},150M_{\odot}$ for 0628. The values for the lighter
masses are chosen so that the SXS reference frequency corresponds to
$f^{\text{det}}_{0}\approx 20\;\mathrm{Hz}$ for both simulations. The values
for the heavier masses are chosen to put more emphasis on the plunge-merger-
ringdown portions of the waveforms, consistent with the rest of this article.
The source-frame masses are smaller by a factor of $(1+z)$ where $z$ is the
redshift of the sources at the injected luminosity distance which we set to
$\\{\\{308,649\\},\\{555,1020\\}\\}$Mpc, respectively for light/heavy 0050 and
light/heavy 0628.

Figure 18:  Recovery performances of SEOB, TPHM and XPHM for a synthetic
signal generated by a “zero-noise” injection of the SXS simulation 0050 into
the O4a LIGO Hanford-Livingston network. In each panel, we present the results
from both the light and the heavy mass runs as the darker and lighter halves
of each violin, respectively. Shown are the recovered posteriors for $\\{{\cal
M}_{\text{c}}^{\text{det}},M^{\text{det}},q,\chi_{\mathrm{eff}},d_{\text{L}},\chi_{1},\theta_{1},\chi_{2},\theta_{2},{\vartheta_{\text{LN},0}}\\}$
with the dashed horizontal lines marking the injected values. For $\\{{\cal
M}_{\text{c}}^{\text{det}},M^{\text{det}},d_{\text{L}}\\}$, we present
posteriors for the relative difference between the injected and the recovered
values, i.e.,
$\Delta_{\text{rel}}X:=(X_{\text{r}}-X_{\text{inj}})/X_{\text{r}}$. The first
panel, i.e., $\Delta_{\text{rel}}{\cal M}^{\text{det}}_{\text{c}}$ is somewhat
hard to read because the models produced very narrow posteriors for the light
mass run, but much wider ones for the heavy one.

The reason why the value of $d_{\text{L}}$ for the $150M_{\odot}$ 0050 run
differs from that of 0628 is due to our choice of extrinsic parameters.
Specifically, we fine-tuned
$\\{\vartheta_{\text{LN},0},\theta_{s},\phi_{s},\psi_{s},d_{\text{L}}\\}$ such
that the total network SNR, $\rho_{\text{tot}}$, accumulated by the H1-L1
network is 40 for each case resulting in a precession SNR of
$\rho_{\text{p}}=\\{\\{11,6.4\\},\\{15,10\\}\\}$, respectively for light/heavy
0050 and light/heavy 0628. This quantity was first defined in Ref. Fairhurst
_et al._ (2019) and represents the contribution to the SNR coming from the
second most significant precessing harmonic. It is proportional to the total
SNR, thus becomes larger for louder events, and has to be greater than 2.1 for
the effects of precession on the signal to not be attributed to noise alone
Fairhurst  _et al._ (2019). Let us add that we opted here for
$\rho_{\text{tot}}=40$ as our trial runs with $\rho_{\text{tot}}=20$ resulted
in the posteriors of precession-related quantities, such as $\chi_{\text{p}}$,
being prior dominated, hence uninformative. The specific values for all our
injected parameters can be found in the config files uploaded to our git
repository.

We employ the bilby Ashton  _et al._ (2019); Romero-Shaw  _et al._ (2020b) and
bilby_pipe bil (2020) modules with the dynesty sampler Skilling (2004, 2006);
Speagle (2020); Koposov  _et al._ (2023). We set the number of live points,
`nlive`, to 1000, and `naccept` to 60. Other dynesty attributes are set to
their default values. When employing the IMRPhenom models, we run two parallel
chains and combine the results at the end. For SEOB, we run only one chain per
injection.

The priors for the parameters are as follows. For the lighter mass runs, we
use the bilby function `UniformInComponentsChirpMass` with the range ${\cal
M}_{\text{c}}\in[10,35]M_{\odot}$, the `UniformInComponentsMassRatio` function
with range $q\in[0.083,1]$, and constraints of $m_{i}\in[1,1000]M_{\odot}$.
The luminosity distance prior range is given by
$d_{\text{L}}\in[100,5000]\,$Mpc via the function `UniformSourceFrame` using
$\Lambda$CDM cosmology parameterized by Planck 2015 data Ade  _et al._ (2016).
The spin magnitudes are uniform in the range $\chi_{i}\in[0,0.99]$ while the
spin tilt angles are uniform in their sines: $\sin\theta_{i}\in[0,1]$, and the
$\mathbf{J}$ frame inclination angle is uniform in cosine:
$\cos\vartheta_{\text{JN,0}}\in[-1,1]$, where $\vartheta_{\text{JN,0}}$ is the
angle between $\mathbf{J}$ and $\hat{\mathbf{N}}$ at $f=f_{0}$. Other angles
are all uniform in their respective ranges. For the heavier
$M^{\text{det}}=150M_{\odot}$ runs, we set the chirp mass prior range to
${\cal M}_{\text{c}}\in[45,85]M_{\odot}\,([40,90]M_{\odot})$ for 0050(0628)
with the other priors kept the same. We find that such an injection/recovery
PE “job” consisting of two chains of IMRPhenomTPHM _and_ two chains of
IMRPhenomXPHM takes roughly one day using $100$ processors. With the same
resources, one SEOBNRv5PHM chain can be obtained in $\sim 1.5$ days.

###  VI.1 Main Results

Overall, we have 12 sets of posterior parameters to present resulting from
$\\{{\texttt{SEOB}},\texttt{TPHM},\texttt{XPHM}\\}\times\\{\texttt{0050},\texttt{0628}\\}\times\\{$light,
heavy$\\}$ runs. For the sake of brevity, we show only a subset of posteriors
for important intrinsic and extrinsic parameters, namely the set $\\{{\cal
M}_{\text{c}}^{\text{det}},M^{\text{det}},q,\chi_{\mathrm{eff}},d_{\text{L}},\chi_{1},\theta_{1},\chi_{2},\theta_{2},{\vartheta_{\text{LN},0}}\\}$
in Figs. 18 and 19, where, for $\\{{\cal
M}^{\text{det}}_{\text{c}},M^{\text{det}},d_{\text{L}}\\}$ we present the
relative differences between the injected and the recovered values. When
relevant, we also discuss the posteriors for other parameters besides these.
Table 4 presents the 5th, 50th and 95th percentiles of the recovered
posteriors for the relevant parameters. We discuss the recovery of precession-
related quantities further below in Sec. VI.2. For reference, we display in
Table 3, the mismatches between the injected waveforms and those generated by
SEOB, TPHM and XPHM using the injected parameters.

Model |  0050 light |  0050 heavy |  0628 light |  0628 heavy
---|---|---|---|---
SEOB | $1.28\times 10^{-2}$ | $1.39\times 10^{-2}$ | $1.05\times 10^{-2}$ | $7.94\times 10^{-3}$
TPHM | $2.13\times 10^{-2}$ | $1.00\times 10^{-2}$ | $1.75\times 10^{-2}$ | $9.81\times 10^{-3}$
XPHM | $2.43\times 10^{-2}$ | $1.73\times 10^{-2}$ | $1.67\times 10^{-2}$ | $1.63\times 10^{-2}$
Table 3: Mismatches calculated between the injected SXS waveforms and the
recovery models evaluated at the same parameters as the injected waveforms.
The values presented here are calculated using Eq. (13).

We start with Fig. 18, where we present both the light and heavy mass
posteriors from the 0050 recovery runs as violin plots. A common feature seems
to be that SEOB and TPHM posteriors are more similar to each other than XPHM.
In general, one model recovers a particular set of parameters better than the
other two. For example, for the light mass case, SEOB recovers $\\{{\cal
M}^{\text{det}}_{\text{c}},q,\chi_{\mathrm{eff}}\\}$ better, whereas XPHM
recovers $\\{\chi_{1},\theta_{1},\theta_{2},d_{\text{L}}\\}$ better. TPHM
recovers $\vartheta_{\text{LN},0}$ better than the other two models, and is
sometimes better than SEOB or XPHM for other parameters, but not both.
$\chi_{2}$ is recovered at roughly the same confidence by the models though
the XPHM posterior is skewed oppositely. Looking at the median values and the
90% confidence intervals (CIs) for $m_{1},m_{2}$, i.e., the source-frame
component masses, in Table 4, we see SEOB/TPHM slightly overestimating $m_{1}$
while underestimating $m_{2}$ with XPHM behaving in the opposite manner.
Additional corner plots of the posteriors can be found in our git repository.

SEOB’s and TPHM’s performances are even more similar for the heavy mass case.
SEOB/TPHM recover $\\{{\cal M}^{\text{det}}_{\text{c}},q,\chi_{1}\\}$ better,
whereas XPHM does so for
$\\{\theta_{1},\theta_{2},d_{\text{L}},\vartheta_{\text{LN},0}\\}$. $\chi_{2}$
posteriors for SEOB/TPHM look nearly flat indicating the dominance of the
uniform prior in the posterior, whereas XPHM’s posterior rails against the
$\chi=1$ Kerr bound. $\\{M^{\text{det}},\chi_{\mathrm{eff}}\\}$, are recovered
equally marginally by SEOB and XPHM. The recovery posteriors for ${\cal
M}^{\text{det}}_{\text{c}}$ are rather wide with the 90% CIs ranging roughly
from $54M_{\odot}$ to $68M_{\odot}$ for SEOB, $54M_{\odot}$ to $70M_{\odot}$
for TPHM, and from $58M_{\odot}$ to $70M_{\odot}$ for XPHM, given an injected
value of $55M_{\odot}$.

Figure 19:  Same as Fig. 18, but for the SXS simulation 0628.

The recovery of 0628’s parameters reveals yet more differences in model
performance, but with the similarity in SEOB and TPHM’s posteriors persisting.
Let us recall that 0628 has stronger precession than 0050, but non-zero
parallel spin projections. For the light-mass run, SEOB recovers $\\{{\cal
M}^{\text{det}}_{\text{c}},\chi_{\mathrm{eff}}\\}$ better while TPHM does so
for $\\{M^{\text{det}},q,\theta_{1}\\}$ with XPHM recovering $\theta_{2}$
better. SEOB, TPHM recover $\vartheta_{\text{LN},0}$ very similarly, and SEOB,
XPHM do so for $d_{\text{L}}$. SEOB/XPHM both rail against the $\chi_{1}=1$
bound, but not TPHM. On the other hand, SEOB/TPHM rail against the
$\chi_{2}=0$ bound, with XPHM somewhat railing against $\chi_{2}=1$. Though
XPHM recovers ${\cal M}^{\text{det}}_{\text{c}}$ in the 90% CI, its recovery
of the other mass parameters is rather poor. For example, its posteriors for
both the detector and the source frame (total, primary, secondary) masses do
not contain the injected parameters within their respective 90% CIs, whereas
SEOB’s and TPHM’s posteriors do, with the injected values recovered near the
peaks of the distributions. This can also be partly gathered by looking at the
recovered $\\{m_{1},m_{2}\\}$ posteriors in Table 4 under the 0628 Light rows.
The fact XPHM under(over)estimates $m_{1}(m_{2})$ is why it recovers ${\cal
M}_{\text{c}}$ within its 90% CI.

As for the heavy-mass 0628, SEOB/TPHM recover both the detector and source
frame masses robustly. XPHM underestimates the injected values for
$m_{1},m^{\text{det}}_{1}$ by three standard deviations or more. Despite this,
its $q$ posteriors are comparable to that of SEOB’s. Overall, SEOB recovers
$\\{M^{\text{det}},\chi_{\mathrm{eff}},d_{\text{L}}\\}$ better, while TPHM
does so for $\\{q,\chi_{1},\theta_{1}\\}$, and with XPHM doing so for
$\\{{\cal M}^{\text{det}}_{\text{c}},\theta_{2}\\}$, but cannot recover
$\theta_{1}$ with any confidence. The various railings of the $\chi_{1,2}$
posteriors against the $\chi_{1,2}=0,1$ bounds, seen for the light-mass
injection, are exacerbated. As was the case with 0050, the recovered
posteriors for ${\cal M}^{\text{det}}_{\text{c}}$ are wider for the heavy-mass
run with the 90% CIs ranging roughly from $58M_{\odot}$ to $67M_{\odot}$ for
SEOB, $59M_{\odot}$ to $69M_{\odot}$ for TPHM, and from $53M_{\odot}$ to
$64M_{\odot}$ for XPHM for an injected value of $60.5M_{\odot}$. Another
prominent feature to note is the bimodality in the $\vartheta_{\text{LN},0}$
posteriors which we did not observe for 0050. Given that we only inject and
recover the $\ell=2$ strain, some bimodality in the inclination is expected
due to the distance-inclination degeneracy. However, when there is precession,
the $(2,\pm 1)$ multipoles may contain enough power to break this, and more so
for larger $Q$. Indeed, we find that the ratio
$|{}_{-2}Y_{21}h_{21}|/|{}_{-2}Y_{22}h_{22}|$ is larger for 0050 than for
0628, thus explaining why we observe no bimodality in the inclination
posteriors for 0050.

An interesting result that warrants further discussion is the consistent
overestimation of $\chi_{\mathrm{eff}}$ for the 0050 run where the injection
has $\chi_{\mathrm{eff}}=0.001$. This can be seen in the upper right panel of
Fig. 18 (also upper left panels of Figs. 20, 21) and in Table 4. This is
consistent with the broad findings of Ref. Ng  _et al._ (2018). Given that the
total SNRs are fixed to 40 with precession SNRs all exceeding 6, this is
somewhat suprising. Perhaps, this may be due to the fact that the injected
value is small. Interestingly, App. D of Ref. Ng  _et al._ (2018) contradicts
us, mostly finding that the posteriors slightly underestimate the injected
values of $\chi_{\mathrm{eff}}=0$. We believe the disagreement is mostly due
to the facts that (i) they set $\theta_{1,2}=10^{\circ},30^{\circ}$ thus have
to fix $\chi_{1,2}=0$ to obtain $\chi_{\mathrm{eff}}=0$, which is different
than our situation which has $\theta_{1,2}\simeq 90^{\circ}$; (ii) they employ
the IMRPhenomPv2 model which is no longer considered the state of the art. It
might be case that the recovery of small injected values of
$\chi_{\mathrm{eff}}$ with large $\chi_{\text{p}}$ is in general more
challenging. Otherwise, we can expect $\chi_{\mathrm{eff}}$ to be well
recovered by the current models as seen in, e.g., Table I of Ref. Estellés
_et al._ (2022b) for TPHM and Table II of Ref. Pratten  _et al._ (2021) for
XPHM. A systematic injection/recovery campaign would certainly find a trend
(if there is any). Let us reiterate that $\chi_{\text{eff}}$ is an effective
measure of the impact of spin _in the inspiral_ , which may not necessarily
hold for high-mass systems, i.e., few inspiral cycles. Indeed, we observe that
the overestimation is more severe for our heavy mass injection. For the 0628
run, SEOB recovers $\chi_{\mathrm{eff}}$ very well, but TPHM and XPHM
overestimate it once again.

|  $m_{1}(M_{\odot})$ |  $m_{2}(M_{\odot})$ |  $\chi_{1}$ |  $\chi_{2}$ |  $\chi_{\mathrm{eff}}$ |  $\chi_{\text{p}}$ |  $\theta_{1}$ |  $\theta_{2}$ |  $d_{\text{L}}$(Mpc) |  $\vartheta_{\text{LN},0}$ |
---|---|---|---|---|---|---|---|---|---|---|---
0050 Light |  |  |  |  |  |  |  |  |  |  |
inj. | $39.74$ | $13.25$ | $0.50$ | $0.50$ | $0.001$ | $0.50$ | $1.57$ | $1.57$ | $308$ |  $1.05$ |
SEOB | $41.56^{+3.79}_{-3.21}$ | $12.75^{+1.08}_{-0.93}$ | $0.66^{+0.16}_{-0.18}$ | $0.32^{+0.48}_{-0.29}$ | $0.023^{+0.056}_{-0.059}$ | $0.66^{+0.16}_{-0.18}$ | $1.49^{+0.14}_{-0.15}$ | $1.78^{+0.92}_{-1.13}$ | $328^{+35}_{-36}$ |  $0.86^{+0.17}_{-0.12}$ |
TPHM | $42.57^{+4.08}_{-3.33}$ | $12.63^{+1.06}_{-0.97}$ | $0.58^{+0.12}_{-0.12}$ | $0.31^{+0.48}_{-0.28}$ | $0.077^{+0.068}_{-0.057}$ | $0.57^{+0.12}_{-0.13}$ | $1.38^{+0.16}_{-0.16}$ | $1.67^{+1.01}_{-1.15}$ | $322^{+36}_{-39}$ |  $0.95^{+0.20}_{-0.16}$ |
XPHM | $37.84^{+3.00}_{-2.85}$ | $14.17^{+1.09}_{-1.00}$ | $0.39^{+0.21}_{-0.15}$ | $0.64^{+0.29}_{-0.45}$ | $0.034^{+0.065}_{-0.067}$ | $0.38^{+0.22}_{-0.15}$ | $1.51^{+0.45}_{-0.44}$ | $1.44^{+0.89}_{-0.88}$ | $311^{+61}_{-62}$ |  $0.88^{+0.23}_{-0.19}$ |
0050 Heavy |  |  |  |  |  |  |  |  |  |  |
inj. | $99.24$ | $33.08$ | $0.50$ | $0.50$ | $0.001$ | $0.50$ | $1.57$ | $1.57$ | $649$ |  $1.05$ |
SEOB | $98.85^{+10.75}_{-9.01}$ | $39.01^{+11.26}_{-8.04}$ | $0.64^{+0.24}_{-0.29}$ | $0.49^{+0.43}_{-0.44}$ | $0.085^{+0.121}_{-0.111}$ | $0.62^{+0.24}_{-0.29}$ | $1.46^{+0.25}_{-0.36}$ | $1.36^{+1.12}_{-0.94}$ | $694^{+186}_{-174}$ |  $0.97^{+0.25}_{-0.17}$ |
TPHM | $100.63^{+15.24}_{-12.94}$ | $38.68^{+15.13}_{-8.97}$ | $0.57^{+0.21}_{-0.25}$ | $0.43^{+0.48}_{-0.38}$ | $0.122^{+0.105}_{-0.110}$ | $0.54^{+0.20}_{-0.24}$ | $1.32^{+0.31}_{-0.38}$ | $1.43^{+1.12}_{-1.01}$ | $705^{+198}_{-164}$ |  $1.01^{+0.23}_{-0.19}$ |
XPHM | $90.36^{+7.01}_{-8.13}$ | $46.31^{+13.09}_{-7.91}$ | $0.30^{+0.29}_{-0.20}$ | $0.80^{+0.17}_{-0.39}$ | $0.139^{+0.106}_{-0.123}$ | $0.36^{+0.22}_{-0.14}$ | $1.38^{+0.62}_{-0.67}$ | $1.16^{+0.70}_{-0.61}$ | $679^{+202}_{-168}$ |  $1.08^{+0.31}_{-0.24}$ |
0628 Light |  |  |  |  |  |  |  |  |  |  |
inj. | $40.44$ | $19.61$ | $0.85$ | $0.51$ | $-0.174$ | $0.84$ | $1.70$ | $2.23$ | $555$ |  $0.52$ |
SEOB | $38.65^{+3.54}_{-3.57}$ | $20.36^{+2.25}_{-1.88}$ | $0.87^{+0.10}_{-0.18}$ | $0.33^{+0.44}_{-0.29}$ | $-0.167^{+0.065}_{-0.066}$ | $0.84^{+0.10}_{-0.17}$ | $1.81^{+0.17}_{-0.19}$ | $1.89^{+0.82}_{-1.12}$ | $584^{+114}_{-114}$ |  $0.52^{+2.05}_{-0.22}$ |
TPHM | $40.44^{+4.66}_{-4.06}$ | $19.62^{+2.27}_{-1.96}$ | $0.78^{+0.14}_{-0.15}$ | $0.33^{+0.48}_{-0.28}$ | $-0.083^{+0.071}_{-0.065}$ | $0.77^{+0.13}_{-0.15}$ | $1.75^{+0.20}_{-0.16}$ | $1.46^{+1.00}_{-0.87}$ | $612^{+108}_{-110}$ |  $0.55^{+2.11}_{-0.28}$ |
XPHM | $34.46^{+2.85}_{-2.57}$ | $23.26^{+2.22}_{-2.51}$ | $0.83^{+0.14}_{-0.23}$ | $0.72^{+0.22}_{-0.50}$ | $-0.102^{+0.091}_{-0.131}$ | $0.63^{+0.21}_{-0.20}$ | $2.24^{+0.29}_{-0.30}$ | $0.79^{+0.56}_{-0.47}$ | $585^{+145}_{-134}$ |  $2.01^{+0.57}_{-1.53}$ |
0628 Heavy |  |  |  |  |  |  |  |  |  |  |
inj. | $84.07$ | $40.77$ | $0.85$ | $0.51$ | $-0.174$ | $0.84$ | $1.70$ | $2.23$ | $1020$ |  $0.52$ |
SEOB | $76.22^{+8.92}_{-8.92}$ | $46.75^{+12.28}_{-8.17}$ | $0.84^{+0.13}_{-0.24}$ | $0.32^{+0.52}_{-0.28}$ | $-0.179^{+0.094}_{-0.093}$ | $0.79^{+0.15}_{-0.24}$ | $1.89^{+0.27}_{-0.30}$ | $1.73^{+0.91}_{-1.13}$ | $1029^{+259}_{-255}$ |  $0.95^{+1.74}_{-0.56}$ |
TPHM | $78.37^{+13.73}_{-10.79}$ | $46.87^{+11.36}_{-8.94}$ | $0.78^{+0.17}_{-0.25}$ | $0.29^{+0.50}_{-0.26}$ | $-0.100^{+0.102}_{-0.096}$ | $0.75^{+0.19}_{-0.26}$ | $1.78^{+0.33}_{-0.31}$ | $1.62^{+0.98}_{-1.03}$ | $1130^{+323}_{-285}$ |  $0.85^{+1.77}_{-0.44}$ |
XPHM | $73.16^{+6.51}_{-5.59}$ | $45.84^{+8.28}_{-9.45}$ | $0.82^{+0.15}_{-0.30}$ | $0.81^{+0.16}_{-0.46}$ | $-0.139^{+0.163}_{-0.269}$ | $0.53^{+0.20}_{-0.17}$ | $2.42^{+0.31}_{-0.33}$ | $0.78^{+0.71}_{-0.50}$ | $909^{+315}_{-298}$ |  $1.13^{+1.27}_{-0.53}$ |
Table 4:  The results of our injection/recovery parameter estimation runs. For
a selected subset of parameters given in the first row, we present the medians
and $\\{$5th, 95th$\\}$ percentile error bars for the posteriors recovered by
SEOB, TPHM and XPHM for four injections of SXS numerical relativity waveforms
into LIGO Hanford-Livingston O4a sensitivity. The labels 0050, 0628 in the
first column are the SXS simulation numbers. Light and Heavy refer to the
total mass of the binary black hole system. The row label “inj.” denotes the
injected value. The $\\{$5th, 95th$\\}$ error bars are simply the following
percentile differences: $\\{$5th-50th, 95th-50th$\\}$.

Overall, the least informative recovery is for $\chi_{2}$ as can be gathered
from Figs. 18, 19 and Table 4. Though the models capture the injected values
within their 90% CIs, these are two to three times wider than the ones
corresponding to $\chi_{1}$, thus hardly informative. The only clear result is
the opposite behavior between the SEOB/TPHM and XPHM posteriors with the
former favoring low values and the latter high values. We should also remark
that the heavy 0050 SEOB/TPHM posteriors are mostly prior dominated. To better
quantify this, we compute the Jensen-Shannon (JS) divergences,
$D_{\text{JS}}$, between the posteriors and the priors. Given two
distributions $p_{1}(x),p_{2}(x)$, the JS divergence is a measure of how
similar they are, given by Lin (1991)

$D_{\text{JS}}=\frac{1}{2}\left[\sum_{x}p_{1}(x)\ln\left(\frac{p_{1}(x)}{\bar{p}(x)}\right)+p_{2}(x)\ln\left(\frac{p_{2}(x)}{\bar{p}(x)}\right)\right],$
(26)

where $\bar{p}(x)=(p_{1}(x)+p_{2}(x))/2$. $D_{\text{JS}}=0$ means identical
distributions. We find that for the heavy-mass 0050 run, both SEOB and TPHM
$\chi_{2}$ posteriors yield $D_{\text{JS}}<0.01$ indicating high resemblance
to the flat $\chi_{2}$ priors as can be seen from Fig. 18.

Similarly, for $\theta_{2}$, the SEOB/TPHM posteriors are mostly prior
dominated. For three TPHM runs, the JS divergence between the $\theta_{2}$
posteriors and the prior are all $\lesssim 0.01$, indicative of very similar
distributions. Even for the light-mass 0628 run, TPHM yields
$D_{\text{JS}}\approx 0.019$. SEOB posteriors give $D_{\text{JS}}\approx 0.03$
for the same run, and $D_{\text{JS}}<0.015$ for the other three runs. On the
other hand, the JS divergences between XPHM’s posteriors and the priors are
all greater than 0.1 except for the light 0050 run which still yields
$D_{\text{JS}}>0.02$, which is usually taken to be the threshold for
similarity between two distributions.

###  VI.2 Inferring Precession from the Injections

In the previous section, we deliberately omitted the discussion on the
recovery of $\chi_{\text{p}}$ as we want to present it along with the other
precession scalars, namely $\chi_{\text{p}}^{\text{Gen}}$ and
$\chi_{\perp,\text{J}}$. Recall that our injected signals have total SNRs of
40 each, part of which is due to the precession SNR, specifically,
$\rho_{\text{p}}=\\{\\{11,6.4\\},\\{15,10\\}\\}$, respectively for light/heavy
0050 and light/heavy 0628.

We present the recovered posteriors for
$\\{\chi_{\text{p}},\chi_{\text{p}}^{\text{Gen}},\chi_{\perp,\text{J}}\\}$
(along with $\chi_{\mathrm{eff}}$) from the light/heavy 0050 PE runs in Figs.
20 and 21. Similar results for the light/heavy 0628 recovery are shown in
Figs. 22 and 23. In all the figures, we observe posteriors not dominated by
the priors (dashed histograms) consistent with our having injected
$\rho_{\text{p}}\geq 6$ into all runs. We also see the tallest and narrowest
posteriors for the $\rho_{\text{p}}=15$ run, i.e., the light 0628. As was the
case in the previous section, the posteriors coming from the heavy mass
recovery runs are in general wider than their light-mass counterparts. They
are additionally not as centered on the injected values and less symmetric.

Starting with the $\chi_{\text{p}}$ posteriors, we note that TPHM consistently
recovers the injected value very close to its median with SEOB performing
similarly with some disagreement between the two models’ posteriors for the
light-mass injections. XPHM consistently underestimates $\chi_{\text{p}}$, but
still recovers it within its 90% CI except for heavy 0628 (see Table 4). The
underestimation by XPHM bias was not observed in Ref. Pratten  _et al._
(2021), where the recovery of a
$\\{Q=6,\chi_{\mathrm{eff}}<0,\chi_{\text{p}}\gtrsim 0.75\\}$ injection
(SXS:0165) was performed, albeit at an SNR of 26 with no information as to
what $\rho_{\text{p}}$ is. This might explain why a bias was not observed
there though we should be cautious about drawing conclusions regarding
systematic biases from a few PE runs Estellés  _et al._ (2022b).

Turning our attention to the $\chi_{\text{p}}^{\text{Gen}}$ posteriors, we
find that all models recover the injected values within their 90% CIs as can
be discerned from Figs. 20, 21, 22 and 23. The consistent underestimation of
$\chi_{\text{p}}$ by XPHM is not exhibited here for
$\chi_{\text{p}}^{\text{Gen}}$. SEOB/TPHM recover the injected values closer
to the medians than XPHM except for the heavy 0628 run. In any case, we do not
see the bias observed in the lower panels of Fig. 7 of Ref. De Renzis  _et
al._ (2022), where injections of NRSur7dq4 were carried out. The reason for
this most likely is that their injected SNRs were greater than 75.

As for the $\chi_{\perp,\text{J}}$ recovery, we find that both SEOB and TPHM
also recover this quantity rather well, whereby the injected values almost
line up with the peaks of nearly symmetric posterior distributions. TPHM’s
recovery is especially outstanding for the light mass runs. XPHM
underestimates the injected value every time and either barely recovers within
its 90% CI or just outside it. As far as we are aware, ours are the first PE
results for this quantity so we can not compare our findings with the
literature.

Overall, both SEOB and TPHM seem to recover any one of the three
$\chi_{\perp}$’s well at these relatively high SNRs. The models especially
recover $\chi_{\text{p}}^{\text{Gen}},\chi_{\perp,\text{J}}$ posteriors well
though this is likely just a coincidence, nonetheless warrants a more
systematic investigation. XPHM seems to best recover
$\chi_{\text{p}}^{\text{Gen}}$ of the three $\chi_{\perp}$’s. Focusing only on
the $\chi_{\text{p}}$ recovery, we see that SEOB, TPHM perform reliably, with
perhaps the latter being most reliable in terms of consistent recovery closest
to its median.

Figure 20:  Posterior distributions recovered by TPHM and XPHM for
$\\{\chi_{\mathrm{eff}},\chi_{\text{p}},\chi_{\text{p}}^{\text{Gen}},\chi_{\perp,\text{J}}\\}$
for the injection of the SXS simulation 0050 with
$M^{\text{det}}=56.5M_{\odot}$. The black dashed distribution is the prior for
each quantity. The orange line marks the injected value. The color coding of
each model is consistent with the entire article. Figure 21:  Same as Fig. 20,
but for the heavy 0050 PE run with an injected value of
$M^{\text{det}}=150M_{\odot}$. Note that the vertical scale is different from
Fig. 20, which is why the histograms for the priors may give the illusion of
looking different, when they are in fact identical.

Looking at the $\\{t_{c},\varphi_{c}\\}$-maximized mismatches of Table 3, we
can see some support for why we expect SEOB, TPHM to perform better: for the
heavy injections, SEOB/TPHM waveforms with injected parameters yield
mismatches that are nearly half of the corresponding XPHM waveforms. As for
the light cases, SEOB mismatches are lower, while the TPHM mismatch is lower
for 0050, and roughly the same for 0628. We can compare these mismatches with
what the indistinguishability criterion of Ref. Baird  _et al._ (2013) gives
us. Namely, for $k$ degrees of freedom and an SNR of $\rho$, the required
mismatch below which two waveforms will be indistinguishable at $100p\%$
confidence is given by Hannam  _et al._ (2022)

${\bar{\cal M}}\leq\frac{\chi_{k}^{2}(1-p)}{2\rho^{2}}.$ (27)

A precessing BBH system has eight degrees of freedom. This reduces to four if
we minimize the mismatches over the planar spin components
$\phi_{1},\phi_{2}$. To quantify waveform indistinguishability at 90%
confidence, we use $\chi_{8}^{2}(0.1)\simeq 13.362$ and
$\chi_{4}^{2}(0.1)\simeq 7.79$. Accordingly, for an SNR of 40, two waveforms
will be indistinguishable if their mismatch is less than $4.2\times 10^{-3}$
assuming $k=8$ and ${\bar{\cal M}}\leq 2.4\times 10^{-3}$ assuming $k=4$. The
mismatches we have in Table 3 are larger than these bounds, but SEOB’s are
only twice to thrice the $k=8$ bound, with TPHM’s slightly larger. Thus, we
expect the results of our PE recovery to be somewhat biased, but less so for
SEOB and TPHM. Perhaps, this is manifest mostly in the precession-related
posteriors. We can also solve Eq. 27 for an ‘indistinguishability SNR’ given a
mismatch. For the fiducial value of ${\bar{\cal M}}=0.01$, we obtain
$\rho\approx 26$. Thus, any injection with less than this SNR would not pick
up modeling related systematics in PE.

##  VII Summary

We have conducted a long survey of four precessing waveforms models:
SEOBNRv5PHM, TEOBResumS, IMRPhenomTPHM and IMRPhenomXPHM. Our survey has a
large part pertaining to assessing model faithfulness to the surrogate model
NRSur7dq4 and to numerical relativity waveforms from the SXS simulation
catalog. We quantify model faithfulness via the optimized mismatch,
${\bar{\cal M}}_{\text{opt}}$ [Eqs. (20, 21)], and the merger-ringdown
truncated (inspiral-only) mismatch, ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ [Eq. (23)]. The second shorter part of our
survey involves checking model performance in several injection/recovery
parameter estimation runs for which we could not employ TEOBResumS.

Figure 22:  Same as Fig. 20, but for the light 0628 run with an injected value
of $M^{\text{det}}=67M_{\odot}$.

The assessment of model faithfulness to NRSur7dq4 has two separate parts
distinguished by the chosen set of intrinsic parameters. In Secs. IV.1, IV.2,
we employ a discrete grid with emphasis on having large planar components for
the spin vectors $\mathbf{S}_{1},\mathbf{S}_{2}$ to “stress-test” the
precessing models. In Sec. IV.3, we use a random-uniformly filled intrinsic
parameter space. The latter is commonly encountered in reviews of waveform
models as well as in parameter estimation runs, where the parameter space is
randomly sampled. As can be seen from Fig. 2, the two parameter sets cover
different regions. For the discrete grid, we consider both a light and a
heavy-mass binary with total source frame masses of $37.5M_{\odot}$ and
$150M_{\odot}$. The cases with the latter mass contain fewer inspiral cycles,
thus the plunge-merger-ringdown portions contribute more to the signal than
for the light-mass cases. We further decompose each mass set into four subsets
of mass ratio, $Q=\\{1.\bar{1},2,4,6\\}$, the last of which is in the so-
called extrapolation region of NRSur7dq4.

Figure 23:  Same as Fig. 20, but for the heavy 0628 run with an injected value
of $M^{\text{det}}=150M_{\odot}$.

In Sec. V, we extend the above survey to comparisons with numerical relativity
waveforms from the SXS catalogs. We separate our investigation into a part
involving “short” waveforms, i.e., less than 70 GW cycles (Sec. V.1) and into
a part with longer waveforms ($\gtrsim 125$ GW cycles) in Sec. V.2. For the
former, we once again set the binary mass to $37.5M_{\odot}$ and
$150M_{\odot}$. For the latter, we consider a total of six values for the
total mass ranging from $37.5M_{\odot}$ to $225M_{\odot}$ in steps of
$37.5M_{\odot}$. For the short SXS set, we picked 317 simulations out of the
larger ensemble of approximately 1600 in order to have roughly 100 precessing
waveforms per mass ratio subsets: $\\{Q\lesssim 1.3,Q\approx 2,Q\approx 4\\}$.
This way, we were better able to draw parallels with the assessment conducted
using NRSur7dq4 in Secs. IV.1, IV.2.

We compute all mismatches at inclinations of $\vartheta_{\text{LN},0}=0$ and
$\vartheta_{\text{LN},0}=\pi/2$. For each case, we also compute the merger-
ringdown truncated, i.e., inspiral only, mismatch denoted by the noMR
superscript. Our findings are as follows:

  1. (i)

For each case in our parameter sets, ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}<{\bar{\cal M}}_{\text{opt}}$. The corresponding
distributions can be seen in Figs. 4, 10, 13 and 17. The ratio
$\cal{R}={\bar{\cal M}}_{\text{opt}}/{\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ can be $\mathcal{O}(10)$ or larger, but
remains $\lesssim 7$ for SEOB.

  2. (ii)

The offset between the ${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$
distributions and the ${\bar{\cal M}}_{\text{opt}}$ distributions is larger
for the heavier-mass binaries as can be seen by comparing the left and right
panels of Figs. 4, 13.

  3. (iii)

Model faithfulness deteriorates with increasing mass asymmetry as shown in
Figs. 5, 9 and 15.

  4. (iv)

For short waveforms, model faithfulness, specifically ${\bar{\cal
M}}_{\text{opt}}$, does not significantly degrade at higher inclinations as
exhibited in Figs. 8, 10, 13 and 15. However, for long waveforms, i.e.,
$\gtrsim 125$ cycles, faithfulness decreases as inclination goes from 0 to
$\pi/2$ as shown in Fig. 17. The models’ faithfulness under changing
inclination also varies when considering the inspiral only (${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$) and different mass ratios.

  5. (v)

Models exhibit higher unfaithfulness to NRSur7dq4/SXS in regions with strong
precession.

  6. (vi)

When large enough, the unfaithfulness of the co-precessing (aligned-spin)
$(2,\pm 2)$ multipoles to the corresponding NRSur7dq4/SXS multipoles can
become the dominant systematic in the precessing waveform unfaithfulness. This
is especially the case for TEOBResumS and IMRPhenomTPHM as detailed in Secs.
IV.1.3, V.1.4, and is manifested as a strong correlation shown in Figs. 7, 16.

  7. (vii)

Sufficiently faithful precessing $(2,\pm 1)$ multipoles can make high-
inclination waveforms more faithful than their low inclination counterparts as
explained in Secs. IV.2 and V.1.3.

Finally, in Sec. VI, we conduct a parameter estimation study by performing
zero-noise injections of two precessing SXS simulations (0050 and 0628) and
recovering the injected parameters with the models SEOBNRv5PHM, IMRPhenomTPHM
and IMRPhenomXPHM. Our work in this regard complements those of Refs. Ramos-
Buades  _et al._ (2023); Pratten  _et al._ (2021); Estellés  _et al._ (2022b),
where different SXS simulations were injected. We perform each injection
twice, at a low and high total binary mass. We set the extrinsic parameters so
that in each case the total network SNR equals 40 with 15% to 40% coming from
precession depending on the case. The recovered posterior distributions for a
subset of key parameters are shown in Figs. 18 and 19 for SXS:0050 and
SXS:0628, respectively. These are supplemented by Table 4 where we present the
median values as well as the $\\{5,95\\}$ percentile error bars of the
posteriors for a complementary subset of parameters with some overlap.

Overall, we observe mixed model performance where one model may recover
certain parameters better than the other. For example, SEOBNRv5PHM
consistently recovers $\chi_{\mathrm{eff}}$ better while IMRPhenomTPHM could
be argued to better recover $\chi_{1}$ (no railing), whereas IMRPhenomXPHM
better recovers $\theta_{2}$. However, when it comes to the recovery of the
effective precession scalars, $\chi_{\text{p}},\chi_{\text{p}}^{\text{Gen}}$
and the perpendicular spin component $\chi_{\perp,\text{J}}$, SEOBNRv5PHM and
IMRPhenomTPHM outperform IMRPhenomXPHM, which we have presented in Figs. 20,
21 for SXS:0050, and in Figs. 22, 23 for SXS:0628.
$\chi_{\text{p}}^{\text{Gen}}$ and $\chi_{\perp,\text{J}}$ are especially
rather well recovered by SEOBNRv5PHM and IMRPhenomTPHM as can be garnered from
the bottom panels of these figures. We must nonetheless repeat our cautionary
remark from Sec. VI that results from a few parameter estimation runs are
insufficient for drawing general, reliable conclusions regarding model
performance. Therefore, many more such studies are needed with special
attention paid to the precession SNR.

##  VIII Discussion

Our findings indicate that at present, the MR portion of the waveforms along
with the precessing $(2,\pm 1)$ multipoles need to be further improved to
attain an overall waveform faithfulness comparable to that of the inspiral-
only $(2,\pm 2)$ multipoles for all inclinations. As already mentioned, part
of the increased unfaithfulness when considering the full waveform is simply
due to changing signal morphology of the MR phase. One can, in principle,
determine how much of the increase is due to the actual mismodelling of the MR
regime by smoothly stitching the inspiral part of, e.g., an SEOBNRv5PHM
waveform with the MR part of an NRSur7dq4/SXS waveform and computing the
mismatch ${\bar{\cal M}}_{\text{opt}}$ between this hybrid waveform and the
full SEOBNRv5PHM waveform. A comparison of this mismatch with the already
computed ${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$ values with respect to
NRSur7dq4/SXS then roughly tells us how much of the mismatch comes from
waveform systematics in the MR regime. This comparison might provide an
estimation for an achievable waveform model unfaithfulness that can be
targeted.

As for the modelling of the precessing $(2,\pm 1)$ multipoles, we have seen
already that the main contribution to these comes from the AS $(2,\pm 2)$
multipoles which differ most from NRSur7dq4/SXS’s co-precessing $(2,\pm 2)$
multipoles for cases with $60^{\circ}\leq\theta_{1,2}\leq 120^{\circ}$ and
with increasing mass asymmetry. This region in $\theta_{1,2}$ space is also
where we believe the assumed $m\leftrightarrow-m$ symmetry of the AS
multipoles is most unfaithful. The obvious remedy is the abandonment of this
symmetry, at least for the AS $(2,\pm 2)$ multipoles which has been done for
the new model IMRPhenomXO4a Thompson  _et al._ (2023).

Though the AS $(2,\pm 1)$ multipoles do not matter nearly as much as their
$(2,\pm 2)$ counterparts, we have seen that their omission can cause
mismatches as large as 0.01 for $\vartheta_{\text{LN},0}=0$ and 0.03 for
$\vartheta_{\text{LN},0}=\pi/2$. As detector sensitivity steadily improves
such margins will become crucial to overcome. More work needs to be undertaken
to fully determine how much the unfaithfulness of the AS $(2,\pm 1)$
multipoles matters, while it is clear that this is subdominant to the
unfaithfulness of the AS $(2,\pm 2)$ multipoles. However, the ratio $|h_{2,\pm
1}^{\text{AS}}|/|h_{2,\pm 2}^{\text{AS}}|$ increases with increasing mass
asymmetry implying that unfaithfulness of the AS $(2,\pm 1)$ multipoles
becomes non-negligible beyond a certain mass ratio.

The repeatedly observed fact that faithfulness, in general, deterioriates with
increasing mass asymmetry ($Q\gtrsim 4$) should be addressed more
systematically. Though it is the case that these waveforms have more GW cycles
than the ones with $Q\leq 2$, we had argued that waveform length alone does
not explain the increased unfaithfulness. Part of the degradation is due to
the AS multipoles as we saw in Sec. IV.1.3: they become less faithful to
NRSur7dq4’s co-precessing multipoles as $Q$ increases. On the other hand, some
deterioration could be due to the mismodelling of the AS $(2,\pm 1)$
multipoles which have increasing contributions as discussed above. Another
part could be due to the mismodelling of the Euler angles which brings us to
the second part of this discussion.

Our survey is by no means a complete one, so let us expand on possible ways to
extend it here. The twist prescription has two essential ingredients: the AS
multipoles and the Euler angles used to rotate the multipoles from the
$\mathbf{L}_{\text{N}}(t)$ frame (co-precessing) to an inertial (e.g.,
$\mathbf{L}_{\text{N},0}$) frame. Though we investigated how the
unfaithfulness of the AS multipoles might affect the strain faithfulnes, we
did not conduct a study on how the systematics in the Euler angles affect the
faithfulness. We propose the following hybridized approach for such a study:
Euler rotation of NRSur7dq4’s co-precessing multipoles for which the angles
are to be input from various models under investigation. Such a study was
performed in Ref. Hamilton  _et al._ (2021) (Fig. 18) for assessing the MSA
angles and the angles from their new prescription.

In our case, we could, for example, use TEOBResumS Euler angles to twist the
co-precessing NRSur7dq4 multipoles then compute the mismatches between the
resulting hybrid waveforms and NRSur7dq4 waveforms. This investigation can
also be extended to the choice of co-precessing frame, i.e., since
$\mathbf{L}\neq\mathbf{L}_{\text{N}}$, the Euler angles for the
$\mathbf{L}(t)\to\mathbf{L}_{0}$ rotation will differ from those of the
$\mathbf{L}_{\text{N}}(t)\to\mathbf{L}_{\text{N},0}$ rotation, albeit
slightly. Moreover, neither direction coincides with the direction of maximum
GW emission which, in the time domain, differs depending on whether one uses
the GW strain, the Bondi news function, or the Weyl scalar $\psi_{4}$ Schmidt
_et al._ (2011); Ochsner and O’Shaughnessy (2012); Boyle  _et al._ (2014);
Hamilton and Hannam (2018); Hamilton  _et al._ (2021).

It would be useful to extend our total mass coverage in Secs. IV, V.1 to more
values as we did for our assessment using the long SXS waveforms. A range of
$M\approx 10M_{\odot}$ to $\approx 250M_{\odot}$ in steps of $10M_{\odot}$ is
common. Additionally, for random-uniformly filled parameter spaces, 5000 to
$\mathcal{O}(10^{4})$ cases are customary. However, given that we have to
compute ${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$ and ${\bar{\cal
M}}_{\text{opt}}$ separately at two different inclinations for each single
point in the BBH parameter space, we have four times the computational burden.
This is a considerable increase as each optimized mismatch requires up to
$\mathcal{O}(10^{5})$ waveform generations which we must then repeat for 42
points in our $\\{\varphi_{\text{ref}},\kappa\\}$ grid [Eq. (20)].

Another obvious way to extend our survey is by including higher multipoles in
the strain. Ref. Ramos-Buades  _et al._ (2020b) already conducted such a study
with the multipoles $(3,\pm 3),(3\pm 2),(4,\pm 4),(4,\pm 3)$ included. We can
extend our AS $(2\pm 2),(2\pm 1)$ multipole assessment of Sec. IV.1.3 to these
multipoles. We can also re-apply all the analyses that we performed on the
$\ell=2$ strain to the $\ell>2$ strain.

Yet another extension is to employ as many precessing numerical relativity
waveforms as feasible as was done in the LVK study using $\gtrsim 1500$ NR
waveforms that we mentioned in Sec. I. A shorter version of this at a single
inclination of $\pi/3$ was presented in Ref. Ramos-Buades  _et al._ (2023). We
can supplement these results with our ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ and ${\bar{\cal M}}_{\text{opt}}$ computed at
inclinations of 0 and $\pi/2$ and possibly more. All of this data could then
be presented in terms of mass ratio as in our Figs. 5, 9 and 15. We should
also add that the latest trend in the literature is to present model
unfaithfulness in terms of an SNR-weighted match Ossokine  _et al._ (2020);
Hamilton  _et al._ (2021). This quantity is different than the sky-optimized
match ${\cal M}_{\text{opt}}$ of Eq. (19). Therefore, a comparison of these
two faithfulness measures may also be informative.

A large scale injection/recovery parameter estimation campaign would make
certain vague trends either more concrete or obsolete. However, such studies
are computationally expensive which is why most model reviews suffice with one
or two of these. Nevertheless, we have found that using the parallel_bilby
library, we can complete two chains of IMRPhenomTPHM _and_ two chains of
IMRPhenomXPHM injection/recovery runs per day using $\mathcal{O}(100)$
processors. With the same resources, we can alternatively obtain one
SEOBNRv5PHM chain per $\sim 1.5$ days. So, $\gtrsim 30$ injection/recovery PE
runs can be completed in approximately one month with the IMRPhenom models.

The conclusions of injection/recovery studies are dependent on the choice of
injected SNRs. In our case, we opted for a total SNR of 40 which may seem
unrealistic given that 95% of the O1, O2, O3 events had $\text{SNR}\lesssim
23$777We extracted this value from the list of events provided in Ref.
Wikipedia contributors (2023).. However, there will be a few events with large
enough SNRs for which accurate modelling of precession will be crucial.
GW200129_065458, at an SNR of 26, was such an event where a re-analysis of the
data with the state of the art precessing model NRSur7dq4 discovered strong
precession Hannam  _et al._ (2022), consistent with an earlier analysis from
an IMRPhenomXPHM run, but not with the one from SEOBNRv4PHM Abbott  _et al._
(2021a). The precession SNR $\rho_{\text{p}}$ should also be chosen carefully
in such studies. Even if an SNR=40 system is very strongly precessing, if the
extrinsic parameters conspire to minimize $\rho_{\text{p}}$, the imprint of
precession on the received GWs will be weak.

Finally, let us add that a survey such as ours risks becoming outdated by the
time it may be complete. Indeed, during the writing of this article, the model
SEOBNR got upgraded from v4 to v5. Additionally, the MSA prescription of BBH
precession dynamics in IMRPhenomXPHM got replaced by one based on SpinTaylorT4
Colleoni  _et al._ (2024). Finally, an upgraded IMRPhenomX (O4a) model was
released Thompson  _et al._ (2023). The inclusion of any new models into a
similar survey inevitably delays the completion of the survey itself. However,
once a new model is incorporated into LALSimulation it should, in principle,
be straightforward to assess its faithfulness within the framework that we
have built.

###### Acknowledgements.

We thank Rossella Gamba for sharing codes and help with TEOBResumS, Matteo
Breschi for help with bajes, Marta Colleoni for answering questions regarding
IMRPhenomXP. We are grateful to Charlie Hoy for helping us with the
troubleshooting of the SEOBNRv5PHM PE runs and sharing the results of his
validation runs using the Sciama High Performance Compute (HPC) cluster which
is supported by the ICG, SEPNet and the University of Portsmouth. We are also
grateful to Lorenzo Pompili for pointing out the final fix to our SEOBNRv5PHM
PE runs and sharing, in particular, the results of the heavy mass PE
injection/recovery with SEOBNRv5PHM. This work makes use of the Black Hole
Perturbation Toolkit BHP  and the “pesummary“ package Hoy and Raymond (2021).
S.A. and J.MU. acknowledge support from the University College Dublin Ad Astra
Fellowship. J.T. acknowledges support from the NASA LISA Preparatory Science
grant 20-LPS20-0005.

Model | $M(M_{\odot})$ | $\ x\ $ |  $(n_{x},n_{y},n_{z})$ |  # params. | $\bar{R}^{2}$ | $\ \ \Delta_{\text{rel}}^{\text{av}}\ \ $ |  BIC |  AICc
---|---|---|---|---|---|---|---|---
SEOB | 37.5 | $\chi_{\text{p}}^{\text{Gen}}$ | (4,2,1) | 21 | 0.777 |  3.33$\times 10^{-3}$ | -696 | -793
TEOB | 37.5 | $\chi_{\text{p}}^{\text{Gen}}$ | (7,5,2) | 81 | 0.959 |  2.49$\times 10^{-3}$ | -1512 | -1858
TPHM | 37.5 | $\chi_{\text{p}}^{\text{Gen}}$ | (6,4,2) | 60 | 0.874 |  1.54$\times 10^{-2}$ | -304 | -565
XPHM | 150 | $\chi_{\text{p}}^{\text{Gen}}$ | (6,5,2) | 63 | 0.895 |  4.97$\times 10^{-3}$ | -481 | -754
SEOB | 150 | $\chi_{\text{p}}^{\text{Gen}}$ | (6,4,2) | 60 | 0.813 |  5.10$\times 10^{-3}$ | +76 | -182
TEOB | 150 | $\chi_{\text{p}}^{\text{Gen}}$ | (6,5,2) | 63 | 0.893 |  5.96$\times 10^{-3}$ | -479 | -751
TPHM | 150 | $\chi_{\text{p}}^{\text{Gen}}$ | (6,4,2) | 60 | 0.869 |  6.89$\times 10^{-3}$ | -319 | -579
XPHM | 150 | $\chi_{\text{p}}^{\text{Gen}}$ | (7,4,2) | 75 | 0.914 |  6.79$\times 10^{-3}$ | -248 | -570
SEOB | 37.5 | $\chi_{\perp,\text{J}}$ | (7,5,2) | 81 | 0.884 |  3.17$\times 10^{-3}$ | -823 | -1171
TEOB | 37.5 | $\chi_{\perp,\text{J}}$ | (7,3,2) | 66 | 0.965 |  1.72$\times 10^{-3}$ | -1646 | -1929
TPHM | 37.5 | $\chi_{\perp,\text{J}}$ | (6,5,3) | 73 | 0.898 |  5.63$\times 10^{-3}$ | -353 | -666
XPHM | 37.5 | $\chi_{\perp,\text{J}}$ | (7,4,1) | 55 | 0.904 |  3.96$\times 10^{-3}$ | -606 | -846
SEOB | 150 | $\chi_{\perp,\text{J}}$ | (6,5,2) | 63 | 0.884 |  5.10$\times 10^{-3}$ | -203 | -475
TEOB | 150 | $\chi_{\perp,\text{J}}$ | (6,5,2) | 63 | 0.918 |  4.85$\times 10^{-3}$ | -680 | -954
TPHM | 150 | $\chi_{\perp,\text{J}}$ | (6,4,2) | 60 | 0.908 |  5.79$\times 10^{-3}$ | -572 | -834
XPHM | 150 | $\chi_{\perp,\text{J}}$ | (7,3,2) | 66 | 0.933 |  5.35$\times 10^{-3}$ | -457 | -742
Table 5: List of the three dimensional fits to the mismatches between a given
waveform model and NRSur7dq4. Specifically, we have fitted to
$\log_{10}[{\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)]$ values
resulting from the discrete parameter set of Sec. IV.1. From left to right,
the columns are: the model name, the total mass, the $x$ variable used in the
fits [see Eq. (28)], the three dimensional polynomial order of the fits
(ibid.), the total number of parameters used, the adjusted R-squared value,
the average relative difference between the fit and the validation data [see
Eq. (29)], the Bayesian information criterion value, and the value of the
finite-sample size corrected Akaike information criterion.

We thank Niels Warburton for providing access to the ‘chirp2’ computer, which
is funded by Royal Society - Science Foundation Ireland University Research
Fellowship grant no. RGF\R1\180022. This research has made use of data,
software and/or web tools obtained from the Gravitational Wave Open Science
Center Abbott  _et al._ (2021f) (https://www.gw-openscience.org), a service of
LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo
Collaboration. We have additionally employed the computational resources of
LIGO Laboratory (CIT cluster) supported by the U.S. National Science
Foundation Grants PHY-0757058 and PHY-0823459. LIGO Laboratory and Advanced
LIGO are funded by the United States 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.

Software: The following software have been used for this work. The numpy
Harris  _et al._ (2020), matplotlib Hunter (2007), pandas The pandas
development team (2020); Wes McKinney (2010) and seaborn Waskom (2021)
libraries of python. LALSuite (7.10) LIGO Scientific Collaboration  _et al._
(2018), LALInference Veitch  _et al._ (2015), PyCBC (2.0.6) Biwer  _et al._
(2019) BILBY (2.1.2) Ashton  _et al._ (2019); Romero-Shaw  _et al._ (2020b),
the sxs package (v2022.5.2) Boyle and Scheel (2023), the Scri package
(2022.8.8) Boyle  _et al._ (2020), and TEOBResumSv4.1.5-GIOTTO 84b8f10 (6 July
2023). These results were found to be in good agreement with the more recent
TEOBResumS bd3452e (7 September 2023). Our notebooks and data are available at
our git repository.

##  Appendix A Multidimensional Fits of Mismatches for Bayesian Model
Selection

As briefly mentioned in Sec. I, we can use the results of our faithfulness
survey to inform Bayesian model selection. For example, one can employ the
waveform mismatch, ${\bar{\cal M}}_{\text{opt}}$ in our case, as a type of
weight to generate a weighted categorical prior. This then can act as a custom
prior in a joint Bayesian analysis Hoy (2022). Therefore, by constructing fits
to ${\bar{\cal M}}_{\text{opt}}$ over the intrinsic parameter space, we can
effectively provide a parameter space dependent prior for Bayesian model
selection. We take a first step toward building this categorical prior here by
constructing multidimensional fits to ${\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)$.

Though we have at our disposal several sets of intrinsic parameters, namely
the discrete grid of Sec. IV.1, the uniformly filled space of Sec. IV.3, and
the SXS sets of Sec. V, we choose the discrete grid, as this set seems to
cover the high-spin regions of the parameter space better (see Fig. 2). We
should keep in mind that we fixed $\chi_{1}=\chi_{2}=0.8$ for this set.

In more detail, we construct multidimensional polynomial fits to
$\log_{10}[{\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)]$ as
functions of $\eta,\chi_{\mathrm{eff}}$ and either
$\chi_{\text{p}}^{\text{Gen}}$ or $\chi_{\perp,\text{J}}$. We choose these
perpendicular projections as opposed to $\chi_{\text{p}}$ since these have
better coverage of the parameter space as shown in Fig. 2. Using the symmetric
mass ratio $\eta$ allows us to work with a compact fit domain since
$0<\eta\leq 0.25$. We do not include the $Q=6\,(\eta=6/49)$ subset for our
fits here though this can be included straightforwardly. For our specific
fits, we use the values for
$\\{\chi_{\mathrm{eff}},\chi_{\text{p}}^{\text{Gen}},\chi_{\perp,\text{J}}\\}$
at $f=f_{0}$, but have dropped the subscript 0 to reduce clutter. The total
mass, $M$, is the fourth parameter in our fits, but since we have only two
points in mass space, i.e., $M=37.5M_{\odot},150M_{\odot}$, we construct
separate 3D fits for each mass. Future work in this direction will have
additional data coming from other values of the total mass, thus necessitating
4D fits.

Figure 24: Contour plots of the relative difference between the validation
data (red points) and the third TEOB fit of Table 5, which is obtained from
Eq. (28) with $\\{n_{x},n_{y},n_{z}\\}=\\{7,3,2\\}$. This particular fit is a
function of $\chi_{\perp,\text{J}},\chi_{\mathrm{eff}}$ and $\eta$ and
represents the $M=37.5M_{\odot}$, $\log_{10}[{\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)]$ data, where ${\bar{\cal
M}}_{\text{opt}}$ is the sky-optimized mismatch between NRSur7dq4 and TEOB.
The three panels from left to right show contour plots of the relative
differences separated by mass ratios of $Q=1.\bar{1},2,4$. The black dots are
the data points used to construct the fit and are not covered by the contour
plot shown here which only applies to the red dots.

For fitting, we employ simple polynomials of the form

$\mathcal{P}_{\mathcal{M}}(x,y,z)=\sum_{j=0}^{n_{y}}\sum_{k=0}^{n_{z}}\sum_{i=0}^{n^{kj}_{\text{max}}}a_{ijk}\,x^{i}y^{j}z^{k},$
(28)

where $a_{ijk}$ are the fitting coefficients,
$\\{y,z\\}=\\{\chi_{\mathrm{eff}},\eta\\}$, $x=\chi_{\text{p}}^{\text{Gen}}$
or $\chi_{\perp,\text{J}}$, and
$n^{kj}_{\text{max}}=\max\\{n_{x},n_{y},n_{z}\\}-k-j$. We expect the highest
variation in mismatches to be along the planar spin direction and the least
variation in the mass ratio space. Therefore, we impose $n_{x}>n_{y}>n_{z}$
with $n_{x}\leq 7,n_{y}\leq 5,n_{z}\leq 4$. For our fits, we use a routine
that picks the best values for $n_{x},n_{y},n_{z}$ that minimize  (i)
$1-\bar{R}^{2}$, where $\bar{R}^{2}$ is the adjusted R-squared value,  (ii)
the Bayesian information criterion (BIC),  (iii) the Akaike information
criterion with the finite sample correction (AICc), and  (iv) the average
relative difference between the fit $\mathcal{P}_{\mathcal{M}}$ and the
validation data $d_{i}=(x_{i},y_{i},z_{i},w_{i})$ which we define as

$\Delta_{\text{rel}}^{\text{av}}:=\frac{1}{N}\left(\sum_{i=1}^{N}\left[1-\frac{w_{i}}{\mathcal{P}_{\mathcal{M}}(x_{i},y_{i},z_{i})}\right]^{2}\right)^{1/2},$
(29)

where $w_{i}$ represent $\log_{10}[{\bar{\cal
M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)]$. We employ two thirds of the
data set for fitting and the remaining one third for validation, where the
assignment fit/validation is made randomly. There are likely more informative
ways of choosing the fit/validation subsets, but we leave this for future
work. There is also not necessarily one set of unique values of
$\\{n_{x},n_{y},n_{z}\\}$ that satisfy all four criteria listed above. If more
than one set of values satisfy the criteria, we pick the one with smaller
values of $\\{n_{x},n_{y},n_{z}\\}$. On the other hand, if the routine can not
meet all four criteria, it looks for $\\{n_{x},n_{y},n_{z}\\}$ that minimize
three out of the four.

Using this routine and the fitting function (28), we construct 3D fits of
$\log_{10}[{\bar{\cal M}}_{\text{opt}}(\vartheta_{\text{LN},0}=0)]$ for all
four approximants for $M=37.5M_{\odot},150M_{\odot}$. We find that in most
cases, the best fit functions have $n_{x}=6$ or 7, $n_{y}=4$ or 5 and, not
surprisingly, $n_{z}\leq 2$ (since there are only three $\eta$ values) , thus
resulting in roughly $\mathcal{O}(60)$ parameters as shown in Table 5 where we
also report various goodness-of-fit metrics. As can be seen from the table,
the mismatches for TEOB and XPHM can be fit rather well for both light and
heavy masses with $\bar{R}^{2}$ values as high as 0.965 and an average
relative disagreement as low as $1.7\times 10^{-3}$. SEOB mismatches can still
be fit, but not as well, with $\bar{R}^{2}$ below 0.9. TPHM mismatches prove
to be hardest to fit with the corresponding $\Delta_{\text{rel}}^{\text{av}}$
values being some of the highest.

Table 5 also hints that fits constructed using $\chi_{\perp,\text{J}}$ yield
slightly higher (lower) values for
$\bar{R}^{2}(\Delta_{\text{rel}}^{\text{av}})$ than those using
$\chi_{\text{p}}^{\text{Gen}}$ though the difference is rather marginal and
requires a more thorough investigation to be conclusive. We show one of the
fits of Table 5 in Fig. 24 in terms of 2D contourplots of the relative
difference between the fit and the validation data for the $M=37.5M_{\odot}$
TEOB mismatches. This particular fit uses $\chi_{\perp,\text{J}}$ so
corresponds to the third TEOB row of Table 5 with $\bar{R}^{2}=0.965$. As the
figure shows, for most of the plot regions, the relative difference between
the validation data points (red) and the fit is less than 0.05. In fact,
according to Table 5, the average relative difference is $\approx 1.7\times
10^{-3}$. Fits of this level of faithfulness may prove quite useful for
Bayesian model selection.

We additionally briefly explored whether or not a more finely sampled set of
parameters improves the goodness of the fits. In particular, we re-computed
the $M=150M_{\odot}$ NRSur7dq4-XPHM mismatches for $\mathcal{O}(3000)$ BBHs,
i.e., nearly three times as many data points in the spin space. For both the
$\chi_{\text{p}}^{\text{Gen}}$ and $\chi_{\perp,\text{J}}$ fits, we observed
slight improvement with $\bar{R}^{2}$ almost equalling 0.95 and
$\Delta_{\text{rel}}^{\text{av}}$ dropping to $2\times 10^{-3}$ which can be
compared with the second and fourth XPHM rows of Table 5. We should add that
this improvement comes at the expense of increasing the computational burden
by a factor of three.

In summary, we have illustrated that, using certain projections of the spins,
it is possible to construct reasonably good three-dimensional fits of the
mismatches, effectively capturing most of the behavior of model faithfulness
over the seven-dimensional intrinsic parameter space. The fits can be improved
by including a denser coverage of the parameter space, but also need to be
extended along the fourth dimension (total mass) to be of actual use in
Bayesian model selection. There are further additional improvements that can
be made, but we leave it all to future work. Our goal here was to show that
faithful fits can be constructed in principle by using either
$\chi_{\text{p}}^{\text{Gen}}$ or $\chi_{\perp,\text{J}}$ and the goodness-of-
fit metrics in Table 5 seem to affirm this.

##  Appendix B A brief Comparison of SEOBNRv5PHM with SEOBNRv4PHM

As already mentioned, while in the middle of this project, the upgraded model
SEOBNRv5PHM Ramos-Buades  _et al._ (2023) was released along with a simple-to-
use python package pySEONBR Mihaylov  _et al._ (2023) for it. As the newer
model is considerably faster than its v4 predecessor, we were easily compelled
to use it. We had however already generated a considerable data set of
SEOBNRv4PHM mismatches by then. Therefore, we show a brief comparison between
the newest (v5) and the previous versions in Fig. 25 where we plot ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$ and ${\bar{\cal M}}_{\text{opt}}$ once again
at inclinations of $\vartheta_{\text{LN},0}=0$ and $\pi/2$ for the randomly-
uniformly filled 1000-case parameter set. We see that the upgraded model is
more faithful to NRSur7dq4 than its predecessor.

Figure 25:  Comparison of the faithfulness of SEOBNRv5PHM with that of
SEOBNRv4PHM for the 1000-case random-uniformly filled parameter space
described in Sec. IV.3. The baseline model used to measure faithfulness is
NRSur7dq4.

##  Appendix C Another Comparison: IMRPhenomXO4a vs. MSA and SpinTaylor
Implementations of IMRPhenomXPHM

Here, we provide a brief comparison between the original IMRPhenomXPHM (using
MSA angles) Pratten  _et al._ (2021) with the upgraded version that uses
SpinTaylorT4 dynamics to obtain the Euler angles Colleoni  _et al._ (2024). We
also include the new model IMRPhenomXO4a Thompson  _et al._ (2023) in the
comparison. We had briefly discussed the unfaithfulness performances of the
MSA and the SpinTaylorT4 versions at the end of Sec. V.2. We supplement it
here with Fig. 26 where we plot ${\bar{\cal M}}_{\text{opt}}^{\text{noMR}}$
and ${\bar{\cal M}}_{\text{opt}}$ yet again at inclinations of
$\vartheta_{\text{LN},0}=0$ and $\pi/2$ for the random-uniform parameter set.
We observe very similar performances between the MSA and SpinTaylor
implementations, and XO4a. The performance of the MSA version is only slightly
inferior at higher inclinations with the newer IMRPhenomX models slightly
outperforming their predecessor. All three models yield very similar
distributions for the inspiral-only mismatches ${\bar{\cal
M}}_{\text{opt}}^{\text{noMR}}$, indicating that the differences exhibited may
be more related to merger-ringdown modelling.

A more striking difference in the faithfulness of the MSA version vs. the
other two is exposed when considering the discrete parameter set of Secs.
IV.1, IV.2 where we had a small number of cases with parameters chosen such
that $|\mathbf{J}_{\text{N},0}|\approx 0.025M^{2}$ which caused the MSA
prescription to break down. This was exhibited as small secondary modes
located at ${\bar{\cal M}}_{\text{opt}}\approx 0.1\ (0.3)$ for
$M=37.5M_{\odot}\ (150M_{\odot})$ in the mismatch distributions plotted in
Figs. 3, 8, and re-plotted in Fig. 27, where we plot the histograms for
${\bar{\cal M}}_{\text{opt}}$ at $\vartheta_{\text{LN},0}=0,\pi/2$ for
$M=37.5M_{\odot},150M_{\odot}$. We can see from the figure that the newer
SpinTaylor version of XPHM and XO4a do not run into this issue.

Figure 26: Comparison of the faithfulness of the MSA implementation of
IMRPhenomXPHM with that of its more recent SpinTaylor implementation (ST in
the figure) and the new model IMRPhenomXO4a. The parameters used to generate
the waveforms are those of the random-uniformly filled set of Sec. IV.3. The
baseline model used to measure faithfulness is NRSur7dq4. Figure 27: Same
comparison as Fig. 26, but for the discrete parameter set of Sec. IV.1. Note
the disappearance of a secondary mode containing high mismatch values for the
SpinTaylor version of XPHM and IMRPhenomXO4a.

##  Appendix D Timing Benchmarks

Though detailed timing benchmarks can be found in, e.g., Refs. Pratten  _et
al._ (2021); Estellés  _et al._ (2022b); Ramos-Buades  _et al._ (2023), we
nonetheless present our own results here. We choose three different $Q=2$
configurations for our timing: (i) $\theta_{1,2}=30^{\circ}$, (ii)
$\theta_{1}=90^{\circ},\theta_{2}\approx 115^{\circ}$, (iii)
$\theta_{1}=180^{\circ},\theta_{2}=150^{\circ}$. We fix $f_{0}=20\,$Hz and
vary the total mass from $10M_{\odot}$ to $150M_{\odot}$ in steps of
$10M_{\odot}$. We further set $\vartheta_{\text{LN},0}=0$ and
$d_{\mathrm{L}}=500\,\mathrm{Mpc}$. We run each case 1000 times per model and
quote the median values of the results. For the time-domain models, we use a
sample rate of $16384\,$Hz and for the frequency domain XPHM we use
$df=1/32\,$Hz. All runs were performed on an AMD EPYC 7453 server, running
under virtualisation with `OMP_NUM_THREADS` and `NUMBA_NUM_THREADS` both set
to one. We present the timing results in Fig. 28.

Figure 28: Timing benchmarks for SEOB, TEOB, TPHM and XPHM. Each panel gives
results from runs with key parameters displayed above with
$Q=2,\vartheta_{\text{LN},0}=0,f_{0}=20\,\text{Hz},d_{\text{L}}=500\,$Mpc.
Each individual case was run 1000 times per model in a virtual environment on
an AMD EPYC 7453 server.

## References

  * Abbott  _et al._ (2016a) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. Lett. 116, 241102 (2016a), arXiv:1602.03840 [gr-qc] .
  * Abbott  _et al._ (2016b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. D 93, 122003 (2016b), arXiv:1602.03839 [gr-qc] .
  * Abbott  _et al._ (2016c) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. D 93, 122004 (2016c), [Addendum: Phys.Rev.D 94, 069903 (2016)], arXiv:1602.03843 [gr-qc] .
  * Abbott  _et al._ (2016d) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. X6, 041015 (2016d), arXiv:1606.04856 [gr-qc] .
  * Abbott  _et al._ (2016e) B. P. Abbott _et al._ (LIGO Scientific, Virgo, ASKAP, BOOTES, DES, Fermi-GBM, Fermi-LAT, GRAWITA, INTEGRAL, iPTF, InterPlanetary Network, J-GEM, La Silla-QUEST Survey, Liverpool Telescope, LOFAR, MASTER, MAXI, MWA, Pan-STARRS, PESSTO, Pi of the Sky, SkyMapper, Swift, TAROT, Zadko, Algerian National Observatory, C2PU, TOROS, VISTA), Astrophys. J. Suppl. 225, 8 (2016e), arXiv:1604.07864 [astro-ph.HE] .
  * Abbott  _et al._ (2016f) T. D. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. X6, 041014 (2016f), arXiv:1606.01210 [gr-qc] .
  * Abbott  _et al._ (2016g) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. D94, 064035 (2016g), arXiv:1606.01262 [gr-qc] .
  * Abbott  _et al._ (2016h) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. Lett. 116, 241103 (2016h), arXiv:1606.04855 [gr-qc] .
  * Abbott  _et al._ (2017a) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Class. Quant. Grav. 34, 104002 (2017a), arXiv:1611.07531 [gr-qc] .
  * Abbott  _et al._ (2017b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Annalen Phys. 529, 1600209 (2017b), arXiv:1608.01940 [gr-qc] .
  * Abbott  _et al._ (2017c) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Astrophys. J. 851, L35 (2017c), arXiv:1711.05578 [astro-ph.HE] .
  * Abbott  _et al._ (2017d) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. Lett. 119, 141101 (2017d), arXiv:1709.09660 [gr-qc] .
  * Abbott  _et al._ (2017e) B. P. Abbott _et al._ (VIRGO, LIGO Scientific), Phys. Rev. Lett. 118, 221101 (2017e), arXiv:1706.01812 [gr-qc] .
  * Abbott  _et al._ (2019) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE] .
  * Venumadhav  _et al._ (2019) T. Venumadhav, B. Zackay, J. Roulet, L. Dai, and M. Zaldarriaga, Phys. Rev. D 100, 023011 (2019), arXiv:1902.10341 [astro-ph.IM] .
  * 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] .
  * Nitz  _et al._ (2019a) A. H. Nitz, C. Capano, A. B. Nielsen, S. Reyes, R. White, D. A. Brown, and B. Krishnan, Astrophys. J. 872, 195 (2019a), arXiv:1811.01921 [gr-qc] .
  * 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, Astrophys. J. 891, 123 (2020), arXiv:1910.05331 [astro-ph.HE] .
  * Abbott  _et al._ (2020a) R. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 896, L44 (2020a), arXiv:2006.12611 [astro-ph.HE] .
  * Abbott  _et al._ (2020b) R. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. D 102, 043015 (2020b), arXiv:2004.08342 [astro-ph.HE] .
  * Abbott  _et al._ (2020c) R. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 125, 101102 (2020c), arXiv:2009.01075 [gr-qc] .
  * Abbott  _et al._ (2021a) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), (2021a), arXiv:2111.03606 [gr-qc] .
  * Nitz  _et al._ (2021) A. H. Nitz, S. Kumar, Y.-F. Wang, S. Kastha, S. Wu, M. Schäfer, R. Dhurkunde, and C. D. Capano, (2021), arXiv:2112.06878 [astro-ph.HE] .
  * Abbott  _et al._ (2017f) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Phys. Rev. Lett. 119, 161101 (2017f), arXiv:1710.05832 [gr-qc] .
  * Abbott  _et al._ (2020d) B. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 892, L3 (2020d), arXiv:2001.01761 [astro-ph.HE] .
  * Abbott  _et al._ (2020e) R. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 900, L13 (2020e), arXiv:2009.01190 [astro-ph.HE] .
  * Abbott  _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, KAGRA, VIRGO), Astrophys. J. Lett. 915, L5 (2021b), arXiv:2106.15163 [astro-ph.HE] .
  * Olsen  _et al._ (2022) S. Olsen, T. Venumadhav, J. Mushkin, J. Roulet, B. Zackay, and M. Zaldarriaga, Phys. Rev. D 106, 043009 (2022), arXiv:2201.02252 [astro-ph.HE] .
  * Williams (2024) D. Williams, (2024), arXiv:2401.08709 [astro-ph.HE] .
  * Abbott  _et al._ (2021c) R. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X 11, 021053 (2021c), arXiv:2010.14527 [gr-qc] .
  * Mandel and Farmer (2022) I. Mandel and A. Farmer, Phys. Rept. 955, 1 (2022), arXiv:1806.05820 [astro-ph.HE] .
  * Mapelli (2021) M. Mapelli, “Formation Channels of Single and Binary Stellar-Mass Black Holes,” (2021) arXiv:2106.00699 [astro-ph.HE] .
  * Gerosa  _et al._ (2013) D. Gerosa, M. Kesden, E. Berti, R. O’Shaughnessy, and U. Sperhake, Phys. Rev. D 87, 104028 (2013), arXiv:1302.4442 [gr-qc] .
  * Vitale  _et al._ (2017) S. Vitale, R. Lynch, R. Sturani, and P. Graff, Class. Quant. Grav. 34, 03LT01 (2017), arXiv:1503.04307 [gr-qc] .
  * Rodriguez  _et al._ (2016) C. L. Rodriguez, M. Zevin, C. Pankow, V. Kalogera, and F. A. Rasio, Astrophys. J. Lett. 832, L2 (2016), arXiv:1609.05916 [astro-ph.HE] .
  * Stevenson  _et al._ (2017) S. Stevenson, C. P. L. Berry, and I. Mandel, Mon. Not. Roy. Astron. Soc. 471, 2801 (2017), arXiv:1703.06873 [astro-ph.HE] .
  * Apostolatos  _et al._ (1994) T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Phys. Rev. D49, 6274 (1994).
  * Kidder (1995) L. E. Kidder, Phys.Rev. D52, 821 (1995), arXiv:gr-qc/9506022 [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. D52, 848 (1995), arXiv:gr-qc/9502040 [gr-qc] .
  * Abbott  _et al._ (2016i) B. P. Abbott _et al._ (Virgo, LIGO Scientific), Astrophys. J. 818, L22 (2016i), arXiv:1602.03846 [astro-ph.HE] .
  * Abbott  _et al._ (2016j) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 241103 (2016j), arXiv:1606.04855 [gr-qc] .
  * Abbott  _et al._ (2017g) B. . P. . Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 851, L35 (2017g), arXiv:1711.05578 [astro-ph.HE] .
  * Abbott  _et al._ (2021d) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), (2021d), arXiv:2111.03634 [astro-ph.HE] .
  * 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] .
  * Damour (2001) T. Damour, Phys. Rev. D64, 124013 (2001), arXiv:gr-qc/0103018 .
  * Racine (2008) E. Racine, Phys. Rev. D78, 044021 (2008), arXiv:0803.1820 [gr-qc] .
  * Ajith (2011) P. Ajith, Phys.Rev. D84, 084037 (2011), arXiv:1107.1267 [gr-qc] .
  * Schmidt  _et al._ (2011) P. Schmidt, M. Hannam, S. Husa, and P. Ajith, Phys. Rev. D84, 024046 (2011), arXiv:1012.2879 [gr-qc] .
  * Pürrer  _et al._ (2013) M. Pürrer, M. Hannam, P. Ajith, and S. Husa, Phys. Rev. D 88, 064007 (2013), arXiv:1306.2320 [gr-qc] .
  * Baird  _et al._ (2013) E. Baird, S. Fairhurst, M. Hannam, and P. Murphy, Phys. Rev. D 87, 024035 (2013), arXiv:1211.0546 [gr-qc] .
  * Hannam  _et al._ (2013) M. Hannam, D. A. Brown, S. Fairhurst, C. L. Fryer, and I. W. Harry, Astrophys.J. 766, L14 (2013), arXiv:1301.5616 [gr-qc] .
  * Galaudage  _et al._ (2021) S. Galaudage _et al._ , Astrophys. J. Lett. 921, L15 (2021), [Erratum: Astrophys.J.Lett. 936, L18 (2022), Erratum: Astrophys.J. 936, L18 (2022)], arXiv:2109.02424 [gr-qc] .
  * Golomb and Talbot (2022) J. Golomb and C. Talbot, (2022), arXiv:2210.12287 [astro-ph.HE] .
  * Adamcewicz and Thrane (2022) C. Adamcewicz and E. Thrane,  (2022), 10.1093/mnras/stac2961, arXiv:2208.03405 [astro-ph.HE] .
  * Tong  _et al._ (2022) H. Tong, S. Galaudage, and E. Thrane, Phys. Rev. D 106, 103019 (2022), arXiv:2209.02206 [astro-ph.HE] .
  * Biscoveanu  _et al._ (2022) S. Biscoveanu, T. A. Callister, C.-J. Haster, K. K. Y. Ng, S. Vitale, and W. M. Farr, Astrophys. J. Lett. 932, L19 (2022), arXiv:2204.01578 [astro-ph.HE] .
  * Callister  _et al._ (2022) T. A. Callister, S. J. Miller, K. Chatziioannou, and W. M. Farr, Astrophys. J. Lett. 937, L13 (2022), arXiv:2205.08574 [astro-ph.HE] .
  * Baibhav  _et al._ (2022) V. Baibhav, Z. Doctor, and V. Kalogera, (2022), arXiv:2212.12113 [astro-ph.HE] .
  * Schmidt  _et al._ (2015) P. Schmidt, F. Ohme, and M. Hannam, Phys. Rev. D91, 024043 (2015), arXiv:1408.1810 [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] .
  * Schmidt  _et al._ (2012) P. Schmidt, M. Hannam, and S. Husa, Phys. Rev. D86, 104063 (2012), arXiv:1207.3088 [gr-qc] .
  * Miller  _et al._ (2023) S. J. Miller, M. Isi, K. Chatziioannou, V. Varma, and I. Mandel, (2023), arXiv:2310.01544 [astro-ph.HE] .
  * Hannam  _et al._ (2022) M. Hannam, C. Hoy, J. E. Thompson, S. Fairhurst, V. Raymond, _et al._ , Nature (2022), 10.1038/s41586-022-05212-z, arXiv:2112.11300 [gr-qc] .
  * Payne  _et al._ (2022) E. Payne, S. Hourihane, J. Golomb, R. Udall, R. Udall, D. Davis, and K. Chatziioannou, Phys. Rev. D 106, 104017 (2022), arXiv:2206.11932 [gr-qc] .
  * Macas  _et al._ (2023) R. Macas, A. Lundgren, and G. Ashton, (2023), arXiv:2311.09921 [gr-qc] .
  * Cotesta  _et al._ (2018) R. Cotesta, A. Buonanno, A. Bohé, A. Taracchini, I. Hinder, and S. Ossokine, Phys. Rev. D98, 084028 (2018), arXiv:1803.10701 [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. D99, 064045 (2019a), arXiv:1812.07865 [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, Phys. Rev. D 105, 084039 (2022a), arXiv:2012.11923 [gr-qc] .
  * Pratten  _et al._ (2020) G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. Ramos-Buades, H. Estelles, and R. Jaume, Phys. Rev. D 102, 064001 (2020), 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, Phys. Rev. D 102, 064002 (2020), arXiv:2001.10914 [gr-qc] .
  * Riemenschneider  _et al._ (2021) G. Riemenschneider, P. Rettegno, M. Breschi, A. Albertini, R. Gamba, S. Bernuzzi, and A. Nagar, Phys. Rev. D 104, 104045 (2021), arXiv:2104.07533 [gr-qc] .
  * Pompili  _et al._ (2023) L. Pompili _et al._ , (2023), arXiv:2303.18039 [gr-qc] .
  * Nagar  _et al._ (2023) A. Nagar, P. Rettegno, R. Gamba, A. Albertini, and S. Bernuzzi, (2023), arXiv:2304.09662 [gr-qc] .
  * Flanagan and Hughes (1998a) E. E. Flanagan and S. A. Hughes, Phys. Rev. D 57, 4566 (1998a), arXiv:gr-qc/9710129 .
  * Pürrer and Haster (2020) M. Pürrer and C.-J. Haster, Phys. Rev. Res. 2, 023151 (2020), arXiv:1912.10055 [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] .
  * Blackman  _et al._ (2014) J. Blackman, B. Szilagyi, C. R. Galley, and M. Tiglio, Phys. Rev. Lett. 113, 021101 (2014), arXiv:1401.7038 [gr-qc] .
  * Blackman  _et al._ (2017) J. Blackman, S. E. Field, M. A. Scheel, C. R. Galley, D. A. Hemberger, P. Schmidt, and R. Smith, Phys. Rev. D95, 104023 (2017), arXiv:1701.00550 [gr-qc] .
  * Williams  _et al._ (2019) D. Williams, I. S. Heng, J. Gair, J. A. Clark, and B. Khamesra, (2019), arXiv:1903.09204 [gr-qc] .
  * Walker  _et al._ (2022) M. Walker, V. Varma, G. Lovelace, and M. A. Scheel, (2022), arXiv:2208.02927 [gr-qc] .
  * Yoo  _et al._ (2023) J. Yoo _et al._ , (2023), arXiv:2306.03148 [gr-qc] .
  * Islam  _et al._ (2023) T. Islam, A. Vajpeyi, F. H. Shaik, C.-J. Haster, V. Varma, S. E. Field, J. Lange, R. O’Shaughnessy, and R. Smith, (2023), arXiv:2309.14473 [gr-qc] .
  * Buonanno and Damour (1999) A. Buonanno and T. Damour, Phys. Rev. D59, 084006 (1999), arXiv:gr-qc/9811091 .
  * Buonanno and Damour (2000) A. Buonanno and T. Damour, Phys. Rev. D62, 064015 (2000), arXiv:gr-qc/0001013 .
  * Damour  _et al._ (2000) T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D62, 084011 (2000), arXiv:gr-qc/0005034 [gr-qc] .
  * Buonanno  _et al._ (2006) A. Buonanno, Y. Chen, and T. Damour, Phys. Rev. D74, 104005 (2006), arXiv:gr-qc/0508067 .
  * Damour  _et al._ (2015) T. Damour, P. Jaranowski, and G. Schäfer, Phys. Rev. D91, 084024 (2015), arXiv:1502.07245 [gr-qc] .
  * Bohé  _et al._ (2017) A. Bohé _et al._ , Phys. Rev. D95, 044028 (2017), arXiv:1611.03703 [gr-qc] .
  * Babak  _et al._ (2017) S. Babak, A. Taracchini, and A. Buonanno, Phys. Rev. D95, 024010 (2017), arXiv:1607.05661 [gr-qc] .
  * Hinderer  _et al._ (2016) T. Hinderer _et al._ , Phys. Rev. Lett. 116, 181101 (2016), arXiv:1602.00599 [gr-qc] .
  * Steinhoff  _et al._ (2016) J. Steinhoff, T. Hinderer, A. Buonanno, and A. Taracchini, Phys. Rev. D94, 104028 (2016), arXiv:1608.01907 [gr-qc] .
  * Lackey  _et al._ (2019) B. D. Lackey, M. Pürrer, A. Taracchini, and S. Marsat, Phys. Rev. D 100, 024002 (2019), arXiv:1812.08643 [gr-qc] .
  * Matas  _et al._ (2020) A. Matas _et al._ , Phys. Rev. D 102, 043023 (2020), arXiv:2004.10001 [gr-qc] .
  * Ossokine  _et al._ (2020) S. Ossokine _et al._ , Phys. Rev. D 102, 044055 (2020), arXiv:2004.09442 [gr-qc] .
  * Gadre  _et al._ (2022) B. Gadre, M. Pürrer, S. E. Field, S. Ossokine, and V. Varma, (2022), arXiv:2203.00381 [gr-qc] .
  * Thomas  _et al._ (2022) L. M. Thomas, G. Pratten, and P. Schmidt, Phys. Rev. D 106, 104029 (2022), arXiv:2205.14066 [gr-qc] .
  * Ramos-Buades  _et al._ (2023) A. Ramos-Buades, A. Buonanno, H. Estellés, M. Khalil, D. P. Mihaylov, S. Ossokine, L. Pompili, and M. Shiferaw, (2023), arXiv:2303.18046 [gr-qc] .
  * van de Meent  _et al._ (2023) M. van de Meent, A. Buonanno, D. P. Mihaylov, S. Ossokine, L. Pompili, N. Warburton, A. Pound, B. Wardell, L. Durkan, and J. Miller, (2023), arXiv:2303.18026 [gr-qc] .
  * Khalil  _et al._ (2023) M. Khalil, A. Buonanno, H. Estelles, D. P. Mihaylov, S. Ossokine, L. Pompili, and A. Ramos-Buades, (2023), arXiv:2303.18143 [gr-qc] .
  * Mihaylov  _et al._ (2023) D. P. Mihaylov, S. Ossokine, A. Buonanno, H. Estelles, L. Pompili, M. Pürrer, and A. Ramos-Buades, (2023), arXiv:2303.18203 [gr-qc] .
  * Damour and Nagar (2014) T. Damour and A. Nagar, Phys.Rev. D90, 024054 (2014), arXiv:1406.0401 [gr-qc] .
  * Bernuzzi  _et al._ (2015) S. Bernuzzi, A. Nagar, T. Dietrich, and T. Damour, Phys.Rev.Lett. 114, 161103 (2015), arXiv:1412.4553 [gr-qc] .
  * Nagar  _et al._ (2017) A. Nagar, G. Riemenschneider, and G. Pratten, Phys. Rev. D96, 084045 (2017), arXiv:1703.06814 [gr-qc] .
  * Nagar  _et al._ (2018) A. Nagar _et al._ , Phys. Rev. D98, 104052 (2018), arXiv:1806.01772 [gr-qc] .
  * Akcay  _et al._ (2019) S. Akcay, S. Bernuzzi, F. Messina, A. Nagar, N. Ortiz, and P. Rettegno, Phys. Rev. D99, 044051 (2019), arXiv:1812.02744 [gr-qc] .
  * Akcay  _et al._ (2021) S. Akcay, R. Gamba, and S. Bernuzzi, Phys. Rev. D 103, 024014 (2021), arXiv:2005.05338 [gr-qc] .
  * Nagar  _et al._ (2020) A. Nagar, G. Riemenschneider, G. Pratten, P. Rettegno, and F. Messina, Phys. Rev. D 102, 024077 (2020), arXiv:2001.09082 [gr-qc] .
  * Chiaramello and Nagar (2020) D. Chiaramello and A. Nagar, Phys. Rev. D 101, 101501 (2020), arXiv:2001.11736 [gr-qc] .
  * Gamba  _et al._ (2022) R. Gamba, S. Akçay, S. Bernuzzi, and J. Williams, Phys. Rev. D 106, 024020 (2022), arXiv:2111.03675 [gr-qc] .
  * Nagar and Rettegno (2021) A. Nagar and P. Rettegno, (2021), arXiv:2108.02043 [gr-qc] .
  * Albertini  _et al._ (2021) A. Albertini, A. Nagar, P. Rettegno, S. Albanesi, and R. Gamba, (2021), arXiv:2111.14149 [gr-qc] .
  * Nagar and Albanesi (2022) A. Nagar and S. Albanesi, Phys. Rev. D 106, 064049 (2022), arXiv:2207.14002 [gr-qc] .
  * Albertini  _et al._ (2022) A. Albertini, A. Nagar, A. Pound, N. Warburton, B. Wardell, L. Durkan, and J. Miller, Phys. Rev. D 106, 084061 (2022), arXiv:2208.01049 [gr-qc] .
  * Gonzalez  _et al._ (2022) A. Gonzalez, R. Gamba, M. Breschi, F. Zappa, G. Carullo, S. Bernuzzi, and A. Nagar, (2022), arXiv:2212.03909 [gr-qc] .
  * Andrade  _et al._ (2023) T. Andrade _et al._ , (2023), arXiv:2307.08697 [gr-qc] .
  * Ajith  _et al._ (2007) P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, _et al._ , Class.Quant.Grav. 24, S689 (2007), arXiv:0704.3764 [gr-qc] .
  * Ajith  _et al._ (2008) P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, _et al._ , Phys.Rev. D77, 104017 (2008), arXiv:0710.2335 [gr-qc] .
  * Ajith  _et al._ (2011) P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Brügmann, _et al._ , Phys.Rev.Lett. 106, 241101 (2011), arXiv:0909.2867 [gr-qc] .
  * Santamaria  _et al._ (2010) L. Santamaria, F. Ohme, P. Ajith, B. Brügmann, N. Dorband, _et al._ , Phys.Rev. D82, 064016 (2010), arXiv:1005.3306 [gr-qc] .
  * Husa  _et al._ (2016) S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. Jiménez Forteza, and A. Bohé, Phys. Rev. D93, 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. D93, 044007 (2016), arXiv:1508.07253 [gr-qc] .
  * Dietrich  _et al._ (2017) T. Dietrich, S. Bernuzzi, and W. Tichy, Phys. Rev. D96, 121501 (2017), arXiv:1706.02969 [gr-qc] .
  * London  _et al._ (2018) L. London, S. Khan, E. Fauchon-Jones, X. J. Forteza, M. Hannam, S. Husa, C. Kalaghatgi, F. Ohme, and F. Pannarale, Phys. Rev. Lett. 120, 161102 (2018), arXiv:1708.00404 [gr-qc] .
  * Khan  _et al._ (2019a) S. Khan, K. Chatziioannou, M. Hannam, and F. Ohme, Phys. Rev. D100, 024059 (2019a), arXiv:1809.10113 [gr-qc] .
  * Dietrich  _et al._ (2019) T. Dietrich, A. Samajdar, S. Khan, N. K. Johnson-McDaniel, R. Dudi, and W. Tichy, Phys. Rev. D100, 044003 (2019), arXiv:1905.06011 [gr-qc] .
  * Khan  _et al._ (2019b) S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam, (2019b), arXiv:1911.06050 [gr-qc] .
  * Pratten  _et al._ (2021) G. Pratten _et al._ , Phys. Rev. D 103, 104056 (2021), 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._ (2021) H. Estellés, A. Ramos-Buades, S. Husa, C. García-Quirós, M. Colleoni, L. Haegel, and R. Jaume, Phys. Rev. D 103, 124060 (2021), arXiv:2004.08302 [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, 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, (2021), arXiv:2107.08876 [gr-qc] .
  * Yu  _et al._ (2023) H. Yu, J. Roulet, T. Venumadhav, B. Zackay, and M. Zaldarriaga, (2023), arXiv:2306.08774 [gr-qc] .
  * (134) “SEOBNRv5 code repository,” https://git.ligo.org/waveforms/software/pyseobnr.
  * Thompson  _et al._ (2023) J. E. Thompson, E. Hamilton, L. London, S. Ghosh, P. Kolitsidou, C. Hoy, and M. Hannam, (2023), arXiv:2312.10025 [gr-qc] .
  * Colleoni  _et al._ (2024) M. Colleoni _et al._ , “in preparation,” (2024).
  * Hamilton  _et al._ (2023) E. Hamilton _et al._ , (2023), arXiv:2303.05419 [gr-qc] .
  * Husa  _et al._ (2008) S. Husa, J. A. González, M. Hannam, B. Brügmann, and U. Sperhake, Class. Quant. Grav. 25, 105006 (2008), arXiv:0706.0740 [gr-qc] .
  * Brügmann  _et al._ (2008) B. Brügmann, J. A. Gonzalez, M. Hannam, S. Husa, U. Sperhake, _et al._ , Phys.Rev. D77, 024027 (2008), arXiv:gr-qc/0610128 [gr-qc] .
  * Puecher  _et al._ (2023) A. Puecher, A. Samajdar, G. Ashton, C. Van Den Broeck, and T. Dietrich, (2023), arXiv:2310.03555 [gr-qc] .
  * Boyle  _et al._ (2019) M. Boyle _et al._ , Class. Quant. Grav. 36, 195006 (2019), arXiv:1904.04831 [gr-qc] .
  * Hoy (2022) C. Hoy, Phys. Rev. D 106, 083003 (2022), arXiv:2208.00106 [gr-qc] .
  * Peters and Mathews (1963) P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963).
  * Romero-Shaw  _et al._ (2019) I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Mon. Not. Roy. Astron. Soc. 490, 5210 (2019), arXiv:1909.05466 [astro-ph.HE] .
  * Nitz  _et al._ (2019b) A. H. Nitz, A. Lenon, and D. A. Brown, Astrophys. J. 890, 1 (2019b), arXiv:1912.05464 [astro-ph.HE] .
  * Yun  _et al._ (2020) Q.-Y. Yun, W.-B. Han, G. Wang, and S.-C. Yang, (2020), arXiv:2002.08682 [gr-qc] .
  * Ramos-Buades  _et al._ (2020a) A. Ramos-Buades, S. Tiwari, M. Haney, and S. Husa, Phys. Rev. D 102, 043005 (2020a), arXiv:2005.14016 [gr-qc] .
  * Abbott  _et al._ (2021e) R. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 913, L7 (2021e), arXiv:2010.14533 [astro-ph.HE] .
  * Lenon  _et al._ (2020) A. K. Lenon, A. H. Nitz, and D. A. Brown, Mon. Not. Roy. Astron. Soc. 497, 1966 (2020), arXiv:2005.14146 [astro-ph.HE] .
  * Wu  _et al._ (2020) S. Wu, Z. Cao, and Z.-H. Zhu, Mon. Not. Roy. Astron. Soc. 495, 466 (2020), arXiv:2002.05528 [astro-ph.IM] .
  * Iglesias  _et al._ (2022) H. L. Iglesias _et al._ , (2022), arXiv:2208.01766 [gr-qc] .
  * Romero-Shaw  _et al._ (2020a) I. M. Romero-Shaw, P. D. Lasky, E. Thrane, and J. C. Bustillo, Astrophys. J. Lett. 903, L5 (2020a), arXiv:2009.04771 [astro-ph.HE] .
  * Romero-Shaw  _et al._ (2022) I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Astrophys. J. 940, 171 (2022), arXiv:2206.14695 [astro-ph.HE] .
  * Breivik  _et al._ (2016) K. Breivik, C. L. Rodriguez, S. L. Larson, V. Kalogera, and F. A. Rasio, Astrophys. J. Lett. 830, L18 (2016), arXiv:1606.09558 [astro-ph.GA] .
  * Samsing and Ramirez-Ruiz (2017) J. Samsing and E. Ramirez-Ruiz, Astrophys. J. Lett. 840, L14 (2017), arXiv:1703.09703 [astro-ph.HE] .
  * Samsing  _et al._ (2018) J. Samsing, A. Askar, and M. Giersz, Astrophys. J. 855, 124 (2018), arXiv:1712.06186 [astro-ph.HE] .
  * Rodriguez  _et al._ (2018) C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F. A. Rasio, Phys. Rev. Lett. 120, 151101 (2018), arXiv:1712.04937 [astro-ph.HE] .
  * Gondán and Kocsis (2019) L. Gondán and B. Kocsis, Astrophys. J. 871, 178 (2019), arXiv:1809.00672 [astro-ph.HE] .
  * Banerjee (2018) S. Banerjee, Mon. Not. Roy. Astron. Soc. 481, 5123 (2018), arXiv:1805.06466 [astro-ph.HE] .
  * Liu  _et al._ (2019) B. Liu, D. Lai, and Y.-H. Wang,  (2019), 10.3847/1538-4357/ab2dfb, arXiv:1905.00427 [astro-ph.HE] .
  * Zevin  _et al._ (2021) M. Zevin, S. S. Bavera, C. P. L. Berry, V. Kalogera, T. Fragos, P. Marchant, C. L. Rodriguez, F. Antonini, D. E. Holz, and C. Pankow, Astrophys. J. 910, 152 (2021), arXiv:2011.10057 [astro-ph.HE] .
  * Michaely and Perets (2020) E. Michaely and H. G. Perets, Mon. Not. Roy. Astron. Soc. 498, 4924 (2020), arXiv:2008.01094 [astro-ph.HE] .
  * Hamers  _et al._ (2021) A. S. Hamers, G. Fragione, P. Neunteufel, and B. Kocsis, Mon. Not. Roy. Astron. Soc. 506, 5345 (2021), arXiv:2103.03782 [astro-ph.HE] .
  * Spera  _et al._ (2022) M. Spera, A. A. Trani, and M. Mencagli, Galaxies 10, 76 (2022), arXiv:2206.15392 [astro-ph.HE] .
  * Samsing  _et al._ (2022) J. Samsing, I. Bartos, D. J. D’Orazio, Z. Haiman, B. Kocsis, N. W. C. Leigh, B. Liu, M. E. Pessah, and H. Tagawa, Nature 603, 237 (2022), arXiv:2010.09765 [astro-ph.HE] .
  * Hellström  _et al._ (2022) L. Hellström, A. Askar, A. A. Trani, M. Giersz, R. P. Church, and J. Samsing, Mon. Not. Roy. Astron. Soc. 517, 1695 (2022), arXiv:2203.00034 [astro-ph.SR] .
  * Dall’Amico  _et al._ (2023) M. Dall’Amico, M. Mapelli, S. Torniamenti, and M. A. Sedda, (2023), arXiv:2303.07421 [astro-ph.HE] .
  * Schmidt  _et al._ (2017) P. Schmidt, I. W. Harry, and H. P. Pfeiffer, (2017), arXiv:1703.01076 [gr-qc] .
  * Maggiore (2007) M. Maggiore,  _Gravitational Waves. Vol. 1: Theory and Experiments_, Oxford Master Series in Physics (Oxford University Press, 2007).
  * Boyle  _et al._ (2011) M. Boyle, R. Owen, and H. P. Pfeiffer, Phys. Rev. D84, 124011 (2011), arXiv:1110.2965 [gr-qc] .
  * Sakurai (1994) J. J. Sakurai,  _Modern quantum mechanics; rev. ed._ (Addison-Wesley, Reading, MA, 1994).
  * Brown  _et al._ (2007) D. Brown, S. Fairhurst, B. Krishnan, R. A. Mercer, R. K. Kopparapu, L. Santamaria, and J. T. Whelan, (2007), arXiv:0709.0093 [gr-qc] .
  * Gerosa and Kesden (2016) D. Gerosa and M. Kesden, Phys. Rev. D 93, 124066 (2016), arXiv:1605.01067 [astro-ph.HE] .
  * Gerosa  _et al._ (2023) D. Gerosa, G. Fumagalli, M. Mould, G. Cavallotto, D. P. Monroy, D. Gangardt, and V. De Renzis, (2023), arXiv:2304.04801 [gr-qc] .
  * 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] .
  * Gerosa  _et al._ (2015) D. Gerosa, M. Kesden, U. Sperhake, E. Berti, and R. O’Shaughnessy, Phys. Rev. D92, 064016 (2015), arXiv:1506.03492 [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] .
  * Buonanno  _et al._ (2003) A. Buonanno, Y.-b. Chen, and M. Vallisneri, Phys. Rev. D67, 104025 (2003), [Erratum: Phys. Rev.D74,029904(2006)], arXiv:gr-qc/0211087 [gr-qc] .
  * Buonanno  _et al._ (2009) A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. Sathyaprakash, Phys.Rev. D80, 084043 (2009), arXiv:0907.0700 [gr-qc] .
  * Chatziioannou  _et al._ (2013) K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish, Phys. Rev. D88, 063011 (2013), arXiv:1307.4418 [gr-qc] .
  * Ramos-Buades  _et al._ (2020b) A. Ramos-Buades, P. Schmidt, G. Pratten, and S. Husa, Phys. Rev. D 101, 103014 (2020b), arXiv:2001.10936 [gr-qc] .
  * Keppel  _et al._ (2009) D. Keppel, D. A. Nichols, Y. Chen, and K. S. Thorne, Phys. Rev. D 80, 124015 (2009), arXiv:0902.4077 [gr-qc] .
  * Bruegmann  _et al._ (2008) B. Bruegmann, J. A. Gonzalez, M. Hannam, S. Husa, and U. Sperhake, Phys. Rev. D 77, 124047 (2008), arXiv:0707.0135 [gr-qc] .
  * Kalaghatgi and Hannam (2021) C. Kalaghatgi and M. Hannam, Phys. Rev. D 103, 024024 (2021), arXiv:2008.09957 [gr-qc] .
  * Kolitsidou  _et al._ (2024) P. Kolitsidou, J. E. Thompson, and M. Hannam, (2024), arXiv:2402.00813 [gr-qc] .
  * Ghosh  _et al._ (2023) S. Ghosh, P. Kolitsidou, and M. Hannam, (2023), arXiv:2310.16980 [gr-qc] .
  * Gerosa  _et al._ (2019) D. Gerosa, A. Lima, E. Berti, U. Sperhake, M. Kesden, and R. O’Shaughnessy, Class. Quant. Grav. 36, 105003 (2019), arXiv:1811.05979 [gr-qc] .
  * Gangardt  _et al._ (2022) D. Gangardt, D. Gerosa, M. Kesden, V. De Renzis, and N. Steinle, Phys. Rev. D 106, 024019 (2022), arXiv:2204.00026 [gr-qc] .
  * Gerosa  _et al._ (2021) D. Gerosa, M. Mould, D. Gangardt, P. Schmidt, G. Pratten, and L. M. Thomas, Phys. Rev. D 103, 064067 (2021), arXiv:2011.11948 [gr-qc] .
  * De Renzis  _et al._ (2022) V. De Renzis, D. Gerosa, G. Pratten, P. Schmidt, and M. Mould, Phys. Rev. D 106, 084040 (2022), arXiv:2207.00030 [gr-qc] .
  * Thomas  _et al._ (2021) L. M. Thomas, P. Schmidt, and G. Pratten, Phys. Rev. D 103, 083022 (2021), arXiv:2012.02209 [gr-qc] .
  * Johnson-McDaniel  _et al._ (2022) N. K. Johnson-McDaniel, S. Kulkarni, and A. Gupta, Phys. Rev. D 106, 023001 (2022), arXiv:2107.11902 [astro-ph.HE] .
  * Mould and Gerosa (2022) M. Mould and D. Gerosa, Phys. Rev. D 105, 024076 (2022), arXiv:2110.05507 [astro-ph.HE] .
  * Aasi  _et al._ (2015) J. Aasi _et al._ (LIGO Scientific), Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc] .
  * aLI (a) “Updated Advanced LIGO sensitivity design curve,” https://dcc.ligo.org/LIGO-T1800044/public (a).
  * aLI (b) “L1 Calibrated Sensitivity Spectra Jan 04 2020,” https://dcc.ligo.org/LIGO-G2100675 (b).
  * Harry  _et al._ (2016) I. Harry, S. Privitera, A. Bohé, and A. Buonanno, Phys. Rev. D94, 024012 (2016), arXiv:1603.02444 [gr-qc] .
  * Owen (1996) B. J. Owen, Phys. Rev. D53, 6749 (1996), arXiv:gr-qc/9511032 [gr-qc] .
  * Flanagan and Hughes (1998b) E. E. Flanagan and S. A. Hughes, Phys. Rev. D57, 4535 (1998b), arXiv:gr-qc/9701039 [gr-qc] .
  * Waskom (2021) M. L. Waskom, Journal of Open Source Software 6, 3021 (2021).
  * (201) “SXS collaboration catalog tools,” https://github.com/sxs-collaboration/catalog_tools.
  * LIGO Scientific Collaboration  _et al._ (2018) LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “LVK Algorithm Library - LALSuite,” Free software (GPL) (2018).
  * Boyle and Scheel (2023) M. Boyle and M. Scheel, “The sxs package,”  (2023).
  * Boyle (2013) M. Boyle, Phys. Rev. D87, 104006 (2013), arXiv:1302.2919 [gr-qc] .
  * Boyle  _et al._ (2014) M. Boyle, L. E. Kidder, S. Ossokine, and H. P. Pfeiffer, (2014), arXiv:1409.4431 [gr-qc] .
  * Boyle (2016) M. Boyle, Phys. Rev. D 93, 084031 (2016), arXiv:1509.00862 [gr-qc] .
  * Boyle  _et al._ (2020) M. Boyle, D. Iozzo, and L. C. Stein, “moble/scri: v1.2,”  (2020).
  * Breschi  _et al._ (2021) M. Breschi, R. Gamba, and S. Bernuzzi, Phys. Rev. D 104, 042001 (2021), arXiv:2102.00017 [gr-qc] .
  * aLI (2022) “Noise curves used for Simulations in the update of the Observing Scenarios Paper,” https://dcc.ligo.org/LIGO-T2000012/public (2022).
  * Fairhurst  _et al._ (2019) S. Fairhurst, R. Green, C. Hoy, M. Hannam, and A. Muir, (2019), arXiv:1908.05707 [gr-qc] .
  * Ashton  _et al._ (2019) G. Ashton _et al._ , Astrophys. J. Suppl. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM] .
  * Romero-Shaw  _et al._ (2020b) I. M. Romero-Shaw _et al._ , Mon. Not. Roy. Astron. Soc. 499, 3295 (2020b), arXiv:2006.00714 [astro-ph.IM] .
  * bil (2020) “bilby_pipe,” https://lscsoft.docs.ligo.org/bilby_pipe/master/index.html (2020).
  * Skilling (2004) J. Skilling, in  _Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering_, American Institute of Physics Conference Series, Vol. 735, edited by R. Fischer, R. Preuss, and U. V. Toussaint (2004) pp. 395–405.
  * Skilling (2006) J. Skilling, Bayesian Analysis 1, 833 (2006).
  * Speagle (2020) J. S. Speagle, Mon. Not. Roy. Astron. Soc. 493, 3132 (2020), arXiv:1904.02180 [astro-ph.IM] .
  * Koposov  _et al._ (2023) S. Koposov, J. Speagle, K. Barbary, G. Ashton, E. Bennett, J. Buchner, C. Scheffler, B. Cook, C. Talbot, J. Guillochon, P. Cubillos, A. A. Ramos, B. Johnson, D. Lang, Ilya, M. Dartiailh, A. Nitz, A. McCluskey, and A. Archibald, “joshspeagle/dynesty: v2.1.3,” (2023).
  * Ade  _et al._ (2016) P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 [astro-ph.CO] .
  * Ng  _et al._ (2018) K. K. Y. Ng, S. Vitale, A. Zimmerman, K. Chatziioannou, D. Gerosa, and C.-J. Haster, Phys. Rev. D 98, 083007 (2018), arXiv:1805.03046 [gr-qc] .
  * Lin (1991) J. Lin, IEEE Transactions on Information Theory 37, 145 (1991).
  * Ochsner and O’Shaughnessy (2012) E. Ochsner and R. O’Shaughnessy, Phys. Rev. D 86, 104037 (2012), arXiv:1205.2287 [gr-qc] .
  * Hamilton and Hannam (2018) E. Hamilton and M. Hannam, Phys. Rev. D 98, 084018 (2018), arXiv:1807.06331 [gr-qc] .
  * Wikipedia contributors (2023) Wikipedia contributors, “List of gravitational wave observations — Wikipedia, the free encyclopedia,” https://en.wikipedia.org/w/index.php?title=List_of_gravitational_wave_observations&oldid=1188064092 (2023), [Online; accessed 5-December-2023].
  * (224) “Black Hole Perturbation Toolkit,” (bhptoolkit.org).
  * Hoy and Raymond (2021) C. Hoy and V. Raymond, SoftwareX 15, 100765 (2021), arXiv:2006.06639 [astro-ph.IM] .
  * Abbott  _et al._ (2021f) R. Abbott _et al._ (LIGO Scientific, Virgo), SoftwareX 13, 100658 (2021f), arXiv:1912.11716 [gr-qc] .
  * Harris  _et al._ (2020) C. R. Harris _et al._ , Nature 585, 357 (2020), arXiv:2006.10256 [cs.MS] .
  * Hunter (2007) J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007).
  * The pandas development team (2020) The pandas development team, “pandas-dev/pandas: Pandas,”  (2020).
  * Wes McKinney (2010) Wes McKinney, in  _Proceedings of the 9th Python in Science Conference_, edited by Stéfan van der Walt and Jarrod Millman (2010) pp. 56 – 61\.
  * Veitch  _et al._ (2015) J. Veitch _et al._ , Phys. Rev. D91, 042003 (2015), arXiv:1409.7215 [gr-qc] .
  * Biwer  _et al._ (2019) C. Biwer, C. D. Capano, S. De, M. Cabero, D. A. Brown, A. H. Nitz, and V. Raymond, Publ. Astron. Soc. Pac. 131, 024503 (2019), arXiv:1807.10312 [astro-ph.IM] .