# Testing beyond-Kerr spacetimes with GWTC-3

Rafael M. Santos<EMAIL_ADDRESS>Divisão de Astrofísica, Instituto
Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos
Campos, 12227-010, SP, Brazil  Rafael C. Nunes<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970
Porto Alegre RS, Brazil  Divisão de Astrofísica, Instituto Nacional de
Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos Campos,
12227-010, SP, Brazil  Jose C. N. de Araujo<EMAIL_ADDRESS>Divisão
de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos
Astronautas 1758, São José dos Campos, 12227-010, SP, Brazil

###### Abstract

The Kerr spacetime is a fundamental solution of general relativity (GR),
describing the gravitational field around a rotating, uncharged black hole
(BH). Kerr spacetime has been crucial in modern astrophysics and it serves as
a foundation for the study of gravitational waves (GWs). Possible deviations
in Kerr geometry may indicate deviations from GR predictions. In this work, we
consider the Johannsen-Psaltis metric, which is a beyond-Kerr metric
characterized by a single free parameter, and then we probe this theory
framework using several GWs observations from the third Gravitational-wave
Transient Catalog (GWTC-3). We find that, for most of the events analyzed,
there are no significant deviations from the null hypothesis, i.e. the Kerr
metric. Our main findings demonstrate alignment and certain enhancements when
compared to previous estimates documented in the literature.

##  I Introduction

Since the first direct detection of gravitational waves (GWs) from the
incredible observation of GW150914 event Abbott  _et al._ (2016), there have
been several other new detections from binary mergers, providing ever-
increasing data and information on these extreme gravity phenomena. At the
current date, almost one hundred coalescing compact binary events have been
detected, being of this total, 35 events identified in the third
Gravitational-wave Transient Catalog (GWTC-3) Collaboration  _et al._ (2021a).
These GW detections provide us the possibility to perform ever more robust and
novel tests of the general relativity (GR) framework in strong field regimes
(see et al (2015) for a review).

It is a common understanding that GR is the standard theory of gravity. Not
only did it withstand a variety of experimental tests in different settings
Will (2014); Turyshev (2008), but one may argue that one of its greatest
achievements was the prediction of GWs that are being detected today.
Nevertheless, it is also a general agreement that GR has its drawbacks and is
not a complete theory of gravity, giving rise to the need to take into
consideration alternative theories that extend GR beyond its limitations.

There are theoretical and observational reasons to believe that GR should be
modified when gravitational fields are strong and/or on the universe at large
scales. From an observational point of view, the physical mechanism
responsible for accelerating the Universe at late times is still an open
question, and new degrees of freedom of the gravitational origin are
alternatives to explain such an accelerated stage (see Ishak (2019);
Heisenberg (2019); Saridakis  _et al._ (2021) for review). Theories beyond GR
can serve as alternatives to explain the current tension in the Hubble
constant that persists in the framework of the $\Lambda$CDM model Di Valentino
_et al._ (2021). Also, modified gravity models are motivated to drive the
accelerating expansion of the Universe at early times (inflation era). See et
al (2015) and references therein for motivation of modified gravity (MG)
scenarios under the regime of strong gravitational field.

However, concerning tests of GR, the choice of an MG theory against which GR
should be tested is another problem entirely. In strong field regimes, this
endeavor can be facilitated with the possibility of performing theory-agnostic
tests, putting entire sets of alternative metric theories of gravity under
scrutiny, through the parameterized post-Einsteinian framework (ppE) Yunes and
Pretorius (2009). Extra parameters introduced by some MG theories can be
mapped to generalized ppE parameters, such that parameter estimation using the
ppE framework can establish constraints to specific MG parameters Yunes  _et
al._ (2016). The ppE framework has been widely and efficiently used for test
of GR Chatziioannou  _et al._ (2012); Barausse  _et al._ (2016); Yunes and
Pretorius (2009); Yunes  _et al._ (2016); Carson and Yagi (2020a). Until the
present date, regarding GW detection data, no significant evidence for
deviations from the GR was found under generic modifications Collaboration
_et al._ (2021b); et al (2021).

To test the very physical nature of space-time metrics is to test GR itself.
Efforts in this direction have recently been carried out regarding GR’s
description of an astrophysical black hole — the Kerr solution Kerr (1963). In
Gair and Yunes (2011); Xin  _et al._ (2019); Carson and Yagi (2020b) beyond-
Kerr solutions span from the construction of waveforms to be further tested
against some forecast bounds in LISA sensitivity up to derivation of
corrections to inspiral and ringdown waveforms. Regarding the latter, an
approach is outlined in Hussain and Zimmerman (2022) for the computation of
deviations to the quasinormal modes of the ringdown phase for a broad class of
beyond-GR theories, including beyond-Kerr. Furthermore, a Bayesian analysis
was carried in Dey  _et al._ (2022) with the event signal GW150914 to estimate
posterior constraints to beyond-Kerr parameters in the ringdown regime.
Deviations from the Kerr geometry have also been tested on electromagnetic
signals Kong  _et al._ (2014); Tripathi  _et al._ (2022); Glampedakis and
Pappas (2021); Liu  _et al._ (2019); Wang  _et al._ (2020); Krawczynski
(2018); Cardoso and Gualtieri (2016); Bambi  _et al._ (2017); Cárdenas-
Avendaño  _et al._ (2016); Psaltis  _et al._ (2021); Völkel and Barausse
(2020). Notably, it has been suggested that GW data can provide more robust
constraints on deformation parameters, surpassing the limitations of X-ray
instruments, in tests evaluating deviations from the Schwarzschild metric
Cardenas-Avendano  _et al._ (2020) and challenging the Kerr hypothesis
Shashank and Bambi (2022).

In this work, our main aim is to search and probe beyond Kerr spacetimes with
real data from binary black holes (BBH) coalescence systems from GWTC-3. For
this purpose, we will consider the Johannsen-Psaltis (JP) metric Johannsen and
Psaltis (2011), which is a beyond-Kerr metric, with one single deviation
parameter, $\epsilon_{3}$, that is regular and free of unphysical features,
such as closed timelike curves outside of the event horizon or naked
singularities. Although it is a simple beyond-GR model, taking into account at
most 2 PN corrections, the JP metric also preserves core features of the Kerr
solution — being also axisymmetric, stationary, and asymptotically flat. In
addition, the fact that there is only one extra free parameter analyzes
several GW events more viable statistically.

Through Bayesian parameter estimation using data from the GWTC-3 catalog, we
find constraints on $\epsilon_{3}$ to be of $\mathcal{O}(1)$, which is in
agreement with Collaboration  _et al._ (2021b). Furthermore, our constraints
present an improvement over the ones found through x-ray observations of black
hole accretion disks, which impose $\epsilon_{3}<5$ Kong  _et al._ (2014);
Bambi  _et al._ (2016). For most of the events analyzed, we did not find
significant deviations of Kerr spacetimes. For certain specific events, we
observe a slight statistical trend where $\epsilon_{3}>0$. We explore
potential factors contributing to this trend.

This paper is structured as follows: In Section II we present the theoretical
model that will be used in our analysis, whose methodology is described in
Section III; in Section IV we present our results and, in Section V, we
outline our final considerations and perspectives.

##  II Beyond-Kerr spacetimes and modified gravitational wave

In this section, we begin our discussion on the ppE framework introduced in
Yunes and Pretorius (2009), and how it is used for this paper. The inspiral
phase of the waveform, based on the ppE phenomenological approach, in the
frequency domain, is given by

$h(f)=A_{\rm GR}(f)(1+\alpha_{\rm ppE}u^{a_{\rm ppE}})\exp(i[\Phi_{\rm
GR}(f)+\beta_{\rm ppE}u^{b_{\rm ppE}}]),$ (1)

where $A_{\rm GR}$ and $\Phi_{\rm GR}$ are the GR predictions for the
amplitude and phase of the waveform, $u=(\pi\mathcal{M}f)^{1/3}$,
$\mathcal{M}=M_{t}\eta^{3/5}$ is the chirp mass, $M_{t}=m_{1}+m_{2}$ is the
total mass, $m_{1}$ and $m_{2}$ are the masses of the two black holes (BHs),
and $\eta=m_{1}m_{2}/M_{t}^{2}$ is the symmetric mass ratio. The various
parameters with the “ppE” subscripts describe potential deviations from the GR
predictions.

Following Carson and Yagi (2020b), we now briefly review how the ppE
parameters can be computed for beyond-GR metrics, whose dominant corrections
to the inspiral phase can be traced to its $(t,t)$ component. Starting from
the Ansatz

$g_{tt}=-1+\frac{2M}{r}\left(1+A\frac{M^{p}}{r^{p}}\right)+\mathcal{O}\left(\frac{M^{2}}{r^{2}}\right),$
(2)

where the parameters $A$ and $p$ characterize the aforementioned dominant
corrections and $M$ is the mass of a BH, one can compute corrections to the
gravitational binding energy,

$E_{b}=-\frac{1}{2}\mathcal{M}u^{2}\left[1-\frac{2(2p-1)}{3}Av^{2p}\right],$
(3)

and to the GW luminosity,

$\mathcal{L}_{\rm
GW}=\frac{32}{5}\eta^{2}v^{10}\left[1+\frac{4(p+1)}{3}Av^{2p}\right],$ (4)

where $v$ is the relative velocity.

Eqs. (3) and (4) lead to expressions relating the ppE parameters with the
general deviation parameters $A$ and $p$ introduced in eq. (2) (see Tahura and
Yagi (2018) for details). For the phase ppE parameter, we have

$\beta_{\rm ppE}=-\frac{15}{16(2p-8)(2p-5)}\gamma_{\dot{f}},$ (5)

where

$\gamma_{\dot{f}}=\frac{2}{3}\frac{(p+1)(2p+1)}{\eta^{2p/5}}A.$ (6)

The $b_{ppE}$ term is given by

$b_{\rm ppE}=2p-5$ (7)

Additionally, the amplitude correction is given by

$\alpha_{\rm ppE}=-\frac{1}{3}(p+1)(2p-1)\frac{A}{\eta^{2p/5}},$ (8)

while the $a_{\rm ppE}$ term becomes

$a_{\rm ppE}=2p.$ (9)

Eqs. (5)-(9) are fixed once a particular metric is chosen — in this case, the
JP metric, which parameterizes deviations from the Kerr metric. In Boyer-
Lindquist coordinates, the latter’s metric components are given by

$\displaystyle g^{\rm K}_{tt}$
$\displaystyle=-\left(1-\frac{2Mr}{\Sigma}\right),$ $\displaystyle g^{\rm
K}_{rr}$ $\displaystyle=\frac{\Sigma}{\Delta},$ $\displaystyle g^{\rm
K}_{\theta\theta}$ $\displaystyle=\Sigma,$ $\displaystyle g^{\rm
K}_{\phi\phi}$
$\displaystyle=\left(r^{2}+a^{2}+\frac{2Ma^{2}r\sin^{2}\theta}{\Sigma}\right),$
$\displaystyle g^{\rm K}_{t\phi}$
$\displaystyle=-\frac{2Mar\sin^{2}\theta}{\Sigma},$

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$ and $\Delta=r^{2}-2Mr+a^{2}$. The
deviation from the Kerr solution proposed by the JP metric can be summarized
by the deviation function $h(r,\theta)$ Johannsen and Psaltis (2011), which is
given by

$h(r,\theta)=\sum_{k=0}^{\infty}\left(\epsilon_{2k}+\epsilon_{2k+1}\frac{Mr}{\Sigma}\right)\left(\frac{M^{2}}{\Sigma}\right)^{k},$
(10)

where $\epsilon_{k}$ represents the set of deviation parameters. These are
constrained by the requirement that the metric be asymptotically flat, which
implies $\epsilon_{0}=\epsilon_{1}=0$, and by observational constraints
obtained through weak-field tests of GR in the parameterized post-Newtonian
(PPN) framework Will (2014), which implies $|\epsilon_{2}|\lesssim 10^{-4}$.
Constraints for the next leading order parameter, $\epsilon_{3}$, can be found
to be $\epsilon_{3}<4$ through the analysis of the thermal spectrum of
geometrically thin BH accretion disks Kong  _et al._ (2014). Stronger bounds
such as $\epsilon_{3}\lesssim 10^{-3}$ can also be found through IMR
consistency tests simulating data of extreme events expected to be detected by
LISA Carson and Yagi (2020b). These constraints share the common bound
$\epsilon_{3}>0$, which is found in Dey  _et al._ (2022) through Bayesian
analysis with current and future GW data to test deviations from the multipole
structure of the Kerr solution.

Figure 1: Left panel: GW polarisation signal GW150914-like BH binaries in the
detector frame from the GR and ppE model under consideration in this work
assuming some deviations from the GR prediction. Right panel: Same as in the
left panel, but for the waveform amplitudes in the frequency domain.

Nevertheless, the case considered here is the JP metric such that, except
$\epsilon_{3}$, all other deviation parameters are null. Hence, the deviation
function $h(r,\theta)$ given by Eq. (10) becomes

$h(r,\theta)=\epsilon_{3}\frac{M^{3}r}{\Sigma^{2}}.$ (11)

Considering this correction, the JP metric is given by

$\displaystyle g^{\rm JP}_{tt}$
$\displaystyle=-\left(1-\frac{2Mr}{\Sigma}\right)-\epsilon_{3}\frac{M^{3}(r-2M)}{r^{4}},$
$\displaystyle g^{\rm JP}_{rr}$
$\displaystyle=\frac{\Sigma}{\Delta}+\epsilon_{3}\frac{M^{3}(r-2M)}{\Delta^{2}},$
$\displaystyle g^{\rm JP}_{\theta\theta}$ $\displaystyle=\Sigma,$
$\displaystyle g^{\rm JP}_{\phi\phi}$
$\displaystyle=\left(r^{2}+a^{2}+\frac{2Ma^{2}r\sin^{2}\theta}{\Sigma}\right)+\epsilon_{3}\frac{a^{2}M^{3}(r+2M)}{r^{4}},$
$\displaystyle g^{\rm JP}_{t\phi}$
$\displaystyle=-\frac{2Mar\sin^{2}\theta}{\Sigma}-\epsilon_{3}\frac{2aM^{4}}{r^{4}}.$

Hence, the general deviation parameters can be identified as $p=2$ and
$A=-\frac{\epsilon_{3}}{2}$, yielding the following values for the ppE
parameters:

$\beta_{\rm ppE}=\frac{75\epsilon_{3}}{64\eta^{4/5}},\,\,\,\,\,b_{\rm
ppE}=-1,$ (12) $\alpha_{\rm
ppE}=\frac{3\epsilon_{3}}{2\eta^{4/5}},\,\,\,\,\,a_{\rm ppE}=4;$ (13)

the corrections were entered in 2PN order.

Figure 1 shows the waveform as predicted by GR together with waveforms from
the ppE framework under the beyond-Kerr model. To draw the figure we use the
pyCBC Biwer  _et al._ (2019) package with waveform IMRPhenomXPHM. For a
quantitative test, we set $\epsilon_{3}=1,10^{-1}$, and the other 15 baseline
parameters to the GW150914 event. We observe a relative phase difference at
the order of $\mathcal{O}(10^{-3})$ when comparing the predicted waveforms
between the GR and beyond-Kerr frameworks. Additionally, we identify peaks in
the amplitude difference occurring at around 250 Hz.

In what follows, we introduce our methodology to be used to probe the full
parametric space of the model presented here.

##  III Methodology and GWTC-3

We employ a standard Gaussian noise likelihood $\mathcal{L}$ for the strain
data $d$ given the source parameters $\theta$ Ashton  _et al._ (2019)

$\ln\mathcal{L}(d|\theta)=-\frac{1}{2}\sum_{k}\Big{\\{}\frac{[d_{k}-h_{k}(\theta)]^{2}}{\sigma^{2}_{k}}+\ln(2\pi\sigma^{2}_{k})\Big{\\}},$
(14)

where $k$ is the frequency bin index, $\sigma$ is the noise amplitude spectral
density, and $h(\theta)$ is the waveform.

The GW signal emitted from a BBH coalescence depends on intrinsic parameters
that directly characterize the binary’s dynamics and emitted waveform, and
extrinsic parameters that encode the relation of the source to the detector
network. An isolated BH is uniquely described by its mass, spin, and electric
charge. Here let us assume that the electric charge is negligible. A BBH
undergoing quasi-circular inspiral can be described by eight intrinsic
parameters: the primary mass $m_{1}$, the secondary mass $m_{2}$ and the
three-dimensional spin vectors of the primary spin vector $\vec{S}_{1}$, and
the secondary spin vector $\vec{S}_{2}$, defined at a reference frequency.
Also, we have seven additional extrinsic parameters that are needed to
describe a BH binary: the sky location (right ascension $\alpha$ and
declination $\delta$); the luminosity distance $d_{L}$; the orbital
inclination $\iota$; the polarization angle $\psi$; the time $t_{c}$ and phase
$\phi_{c}$ at coalescence. For precessing binaries, the orbital angular
momentum is not a stable direction, and it is preferable to describe the
source inclination by the angle $\theta_{JN}$ between the total angular
momentum and the line-of-sight instead of the orbital inclination angle
$\iota$.

On the other hand, as reviewed in the previous section II, the beyond-Kerr
spacetimes to be analyzed in this work are fully characterized by the new and
free parameter $\epsilon_{3}$, which quantifies possible deviations from GR.
Therefore, our full parameter space (free parameter baseline) can be defined
as

$\displaystyle\theta\equiv\\!\Bigl{\\{}m_{1},m_{2},a_{1},a_{2},{\rm
tilt}_{1},{\rm tilt}_{2},\phi_{12},$
$\displaystyle\phi_{JL},\alpha,\delta,\theta_{JN},\psi,\phi_{c},\epsilon_{3}\Bigr{\\}},$

for all BBH mergers. In the definition above, $a_{i}$ is the dimensionless
spin magnitude of the ith object and ${\rm tilt}_{i}$ is the zenith angle
between the spin and orbital angular momenta for the ith object.

We analyze binary black hole (BBH) coalescence systems from the Gravitational-
Wave Transient Catalog 3 (GWTC-3). To conduct this analysis, we utilize the
PyCBC Bayesian inference subpackage for gravitational-wave astronomy Biwer
_et al._ (2019). We have customized this code to integrate our theoretical
model as described previously. Additionally, we cross-validate our results for
consistency using the bilby code Ashton  _et al._ (2019).

To test our main motivation, our event sample consists mainly in: i) We do not
consider events where the secondary component, $m_{2}$, is a neutron star.
Thus, we only take into account events where both compact objects are BH. ii)
Because we are carrying out a GR test with additional free parameters that can
considerably change the predictions on the signal, we only consider events
with (signal-to-noise ratio) SNR $>$ 10\. This point can be interpreted that
we only consider events with very significant signals in their detection.

Table 1: Constraints on the $\epsilon_{3}$ parameters at 95% CL obtained from various BBH coalescence events. Event | $\epsilon_{3}$
---|---
GW150914 | $0.13^{+0.45}_{-0.27}$
GW191129 | $-0.17^{+0.23}_{-0.23}$
GW191215 | $0.46^{+1.01}_{-0.61}$
GW191222 | $0.42^{+0.48}_{-0.23}$
GW191230 | $-0.19^{+0.28}_{-0.28}$
GW200112 | $0.13^{+0.61}_{-0.17}$
GW200128 | $0.12^{+0.14}_{-0.39}$
GW200129 | $-0.01^{+0.32}_{-0.29}$
GW200202 | $-0.1^{+0.24}_{-0.13}$
GW200208 | $0.22^{+0.75}_{-0.73}$
GW200219 | $0.3^{+0.6}_{-0.6}$
GW200224 | $0.14^{+0.47}_{-0.47}$
GW200302 | $0.13^{+0.56}_{-0.45}$
GW200311 | $0.06^{+0.61}_{-0.55}$
Figure 2: Posterior probability densities for $\epsilon_{3}$ as recovered for
all the events analyzed.The GR value, $\epsilon_{3}=0$, is represented by the
vertical red line. Figure 3: Relative difference at 95% CL for the chirp mass,
$\Delta_{\mathcal{M}}=\mathcal{M}_{\rm GR}-\mathcal{M}_{\rm ppE}$, between GR
and the model under consideration in this work. Figure 4: Relative difference
at 95% CL for the luminosity distance $\Delta_{d_{L}}=d_{L_{\rm GR}}-d_{L_{\rm
ppE}}$, between GR and the model under consideration in this work.

##  IV Main Results

We analyze 15 events of GWTC-3, using the data available on GWOSC
111https://gwosc.org/. For our analysis runs, the waveform templates are
generated using the IMRPhenomXPHM model, starting at $f_{\rm low}$ = 20 Hz.
All free parameters were evaluated in the detector frame.

In all analyses carried out with phase and amplitude correction, we choose a
flat prior $\epsilon_{3}\in[-10,10]$. For all other additional intrinsic and
extrinsic parameters of the binary system, we follow the prior range by
default corresponding to each event. Before analyzing the GWTC-3 events, we
first check the model on the GW150914 event. We find
$\epsilon_{3}=0.13^{+0.45}_{-0.27}$ at 95% confidence level (CL). The
constraint on $\epsilon_{3}$ is fully compatible with the null hypothesis,
i.e., $\epsilon_{3}=0$, for greater statistical significance.

Table 1 summarizes our key results, where we show the constraints at 95% CL on
the parameter $\epsilon_{3}$ for all events analyzed in this work. We note
that the analyzed signals impose constraints of approximately $\mathcal{O}(1)$
magnitude on the beyond-Kerr distortion parameter, considering the expected
error bar in interpretation. These results are consistent with the anticipated
behavior of the $\phi_{4}$ parameter, corresponding to corrections at the 2PN
order, as presented in Collaboration  _et al._ (2021b). It is noteworthy that
these observational constraints align with the findings presented in Carson
and Yagi (2020b), which adopts a different observational approach (focusing on
events within the LISA sensitivity range). Additionally, in Dey  _et al._
(2022) (see also Ahmed  _et al._ (2024)), using solely the ringdown signal of
GW150914, the parameter $\epsilon_{3}$ is constrained at $\mathcal{O}(10)$,
demonstrating compatibility with GR and exhibiting a statistical tendency
towards positive values rather than negative ranges.

The parameter $\epsilon_{3}$, constrained by x-ray observations of black hole
accretion disks, is bounded by $\epsilon_{3}<5$ Kong  _et al._ (2014); Bambi
_et al._ (2016). Consequently, our results indicate a substantial refinement
of these constraints when examining GWCT-3 events.

Figure 2 shows the posterior probability densities for $\epsilon_{3}$ as
recovered for all the events analyzed. As mentioned, most events are
compatible with $\epsilon_{3}=0$. We emphasize that for most of the events
analyzed, the parameter $\epsilon_{3}$ tends to exhibit positive values, yet
remains statistically compatible with GR at a high level of significance. This
tendency is also noted in Dey  _et al._ (2022) in the context of ringdown
measurements, where it is argued that the region of parameter space
$\epsilon_{3}>0$ leads to frequencies and damping times that satisfy their
respective bounds for a wide range of remnant masses. We verify that
$\epsilon_{3}$ exhibits no significant correlation with the masses or chirp
masses of the events. Consequently, it becomes plausible that during parameter
estimation, waveforms with amplitudes exceeding those predicted by GR may
receive marginal preference, a scenario also consistent with higher values of
$\epsilon_{3}$, but yet ultimately remains compatible with GR at a high
significance level exceeding 2$\sigma$.

To assess changes in key baseline parameters such as masses and distance, we
examine the model’s ability to approximate GR. In Figures 3 and 4, we present
the disparities in chirp mass and luminosity distance between the GR
predictions obtained from the GWOSC database and our analyses based on the ppE
framework. Notably, in these figures, subscripts GR and ppE denote values
derived from the GWOSC database and our ppE framework analyses, respectively.
Consistent with expectations and findings in previous studies, e.g.,
Collaboration  _et al._ (2021b), we observe no significant deviations for the
most of events analyzed. In particular, we highlight the event GW200202,
characterized by high-mass sources, which potentially renders it relevant in
the search for intermediate-mass black hole binaries. This event occurred at a
luminosity distance of $d_{L}=410^{+150}_{-160}$ Mpc, as predicted by GR. We
observe a minor deviation towards $\Delta d_{L}<0$ when compared with the
predictions of the model considered in this work. This deviation may suggest
the presence of some extra cosmological effects on GW200202 if $\Delta
d_{L}<0$ is not merely a systematical effect/artifact. It is well established
that alterations in the gravitational framework can influence predictions of
luminosity distance (see Nishizawa (2018); D’Agostino and Nunes (2019); Finke
_et al._ (2021) for a few examples). However, in our main baseline parameters,
we do not consider this possibility. Thus, we interpret that the
$\epsilon_{3}$ correction on the amplitude of the waveform may systematically
mimic some cosmological effect, despite it not being considered. The GW200202
event has also recently been investigated as a dark siren event for measuring
the Hubble constant Alfradique  _et al._ (2023). This intriguing case warrants
further investigation, which we plan to address in future communications.

This conclusion extends to all other physical parameters of the binary
systems. While additional parameters beyond GR may modulate signals
differently, there are no statistically significant deviations observed
between the intrinsic and extrinsic parameters of binary systems under both GR
and the ppE framework beyond the Kerr model.

##  V Final remarks

The detection of GWs not only provided direct evidence for the existence of
these waves but also validated various aspects of GR. The observed waveforms
have consistently matched the predictions of GR’s equations, affirming its
accuracy in describing gravitational behavior. Subsequently, observatories
like LIGO, Virgo, and KAGRA have continuously detected GWs from diverse
sources, ushering in a new era of astronomy and opportunities for additional
tests of GR. In this paper, we investigate the JP metric in light of several
GW events from the GWTC-3. Our analysis reveals that, for the majority of the
scrutinized signals, there is no discernible evidence supporting deviations
from a Kerr spacetime. It is important to emphasize that the adjustments made
in this study align with the findings outlined in Collaboration  _et al._
(2021b). However, it is statistically unrealistic to expect a seamless
integration of the parameter $\epsilon_{3}$ into the extensive phase-only
correction analysis described in Collaboration  _et al._ (2021b), due to the
complexity of the statistical parametric space and the unique functional form
of the modified waveform. Nevertheless, we anticipate that the additional free
parameters will be of comparable magnitude, although exact replication of
statistical outcomes may not be essential. Therefore, within the framework
presented here, it is imperative to employ more comprehensive methodologies to
thoroughly investigate these signals, offering opportunities for future
research.

On the other hand, considerable efforts have been dedicated to probing the
nature of spacetime in the presence of strong gravitational fields. Future
investigations should extend beyond the specific beyond-Kerr spacetimes
considered in this study. Notably, exploring scenarios that incorporate
higher-order post-Newtonian (PN) corrections would be valuable. This becomes
particularly relevant for forecast analyses targeting upcoming detectors such
as the Einstein Telescope and Cosmic Explorer. Additionally, as real data from
the fourth Gravitational-Wave Transient Catalog (GWTC-4) becomes available
shortly, incorporating these higher-order corrections would provide a more
comprehensive understanding.

In forthcoming works, it is essential to develop strategies that involve joint
signal analysis in both the electromagnetic and gravitational bands. This
approach is crucial for achieving tests with greater consistency in the
context of deviations from General Relativity’s predictions. The potential
integration of data from both domains could offer more robust insights into
the behavior of gravitational waves. These avenues for future research
represent exciting prospects and will undoubtedly contribute to advancing our
understanding of gravitational physics.

###### Acknowledgements.

The authors express their gratitude to the referee for providing valuable
comments and suggestions aimed at enhancing the overall quality of the work.
R.M.S. thanks CNPq (141351/2023-3) for partial financial support. R.C.N thanks
the financial support from the Conselho Nacional de Desenvolvimento Científico
e Tecnologico (CNPq, National Council for Scientific and Technological
Development) under the project No. 304306/2022-3, and the Fundação de Amparo à
pesquisa do Estado do RS (FAPERGS, Research Support Foundation of the State of
RS) for partial financial support under the project No. 23/2551-0000848-3.
J.C.N.A. thanks FAPESP (2013/26258-4) and CNPq (308367/2019-7) for partial
financial support.

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