# Tests of General Relativity with GW230529:
a neutron star merging with a lower mass-gap compact object

Elise M. Sänger<EMAIL_ADDRESS>Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam,
14476, Germany  Soumen Roy<EMAIL_ADDRESS>Nikhef – National Institute
for Subatomic Physics, Science Park 105, 1098 XG Amsterdam, The Netherlands
Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University,
Princetonplein 1, 3584 CC Utrecht, The Netherlands  Michalis Agathos  Queen
Mary University of London, Mile End Road, London, E1 4NS, UK  Department of
Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road CB3 0WA Cambridge, United Kingdom  Ofek Birnholtz  Bar-Ilan
University, Ramat Gan, 5290002, Israel  Alessandra Buonanno  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg
1, Potsdam, 14476, Germany  Department of Physics, University of Maryland,
College Park, MD 20742, USA  Tim Dietrich  Institut für Physik und Astronomie,
Universität Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam,
Germany  Max Planck Institute for Gravitational Physics (Albert Einstein
Institute), Am Mühlenberg 1, Potsdam, 14476, Germany  Maria Haney  Nikhef –
National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam,
The Netherlands  Félix-Louis Julié  Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam, 14476, Germany
Geraint Pratten  School of Physics and Astronomy and Institute for
Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham,
B15 2TT, United Kingdom  Jan Steinhoff  Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam, 14476, Germany
Chris Van Den Broeck  Nikhef – National Institute for Subatomic Physics,
Science Park 105, 1098 XG Amsterdam, The Netherlands  Institute for
Gravitational and Subatomic Physics (GRASP), Utrecht University,
Princetonplein 1, 3584 CC Utrecht, The Netherlands  Sylvia Biscoveanu  Center
for Interdisciplinary Exploration and Research in Astrophysics (CIERA),
Northwestern University, Evanston, IL 60201, USA  Prasanta Char  Université de
Liège, B-4000 Liège, Belgium  Anna Heffernan  IAC3–IEEC, Universitat de les
Illes Balears, E-07122 Palma, Spain  Prathamesh Joshi  The Pennsylvania State
University, University Park, PA 16802, USA  Atul Kedia  Center for
Computational Relativity and Gravitation, Rochester Institute of Technology,
Rochester, NY 14623, USA  R. M. S. Schofield  University of Oregon, Eugene, OR
97403, USA  M. Trevor  Department of Physics, University of Maryland, College
Park, MD 20742, USA  Michael Zevin  Adler Planetarium, 1300 South DuSable Lake
Shore Drive, Chicago, IL, 60605, USA

###### Abstract

On 29 May 2023, the LIGO Livingston observatory detected the gravitational-
wave signal GW230529_181500 from the merger of a neutron star with a lower
mass-gap compact object. Its long inspiral signal provides a unique
opportunity to test General Relativity (GR) in a parameter space previously
unexplored by strong-field tests. In this work, we performed parameterized
inspiral tests of GR with GW230529_181500. Specifically, we search for
deviations in the frequency-domain GW phase by allowing for agnostic
corrections to the post-Newtonian coefficients. We performed tests with the
Flexible Theory Independent (FTI) and Test Infrastructure for General
Relativity (TIGER) frameworks using several quasi-circular waveform models
that capture different physical effects (higher modes, spins, tides). We find
that the signal is consistent with GR for all deviation parameters. Assuming
the primary object is a black hole, we obtain particularly tight constraints
on the dipole radiation at $-1$PN order of $|\delta\hat{\varphi}_{-2}|\lesssim
8\times 10^{-5}$, which is a factor $\sim 17$ times more stringent than
previous bounds from the neutron star–black hole merger GW200115_042309, as
well as on the 0.5PN and 1PN deviation parameters. We discuss some challenges
that arise when analyzing this signal, namely biases due to correlations with
tidal effects and the degeneracy between the 0PN deviation parameter and the
chirp mass. To illustrate the importance of GW230529_181500 for tests of GR,
we mapped the agnostic $-1$PN results to a class of Einstein-scalar-Gauss-
Bonnet (ESGB) theories of gravity. We also conducted an analysis probing the
specific phase deviation expected in ESGB theory and obtain an upper bound on
the Gauss-Bonnet coupling of $\ell_{\rm GB}\lesssim 0.51\leavevmode\nobreak\
\rm{M}_{\odot}$ ($\sqrt{\alpha_{\rm GB}}\lesssim 0.28$ km), which is better
than any previously reported constraint.

##  I Introduction

On 29 May 2023, at 18:15:00 UTC, a gravitational wave (GW) signal emitted from
the merger of two compact objects, most likely a black hole (BH) and a neutron
star (NS) Abac  _et al._ (2024), was observed by the Advanced LIGO detector
Abbott  _et al._ (2018); Aasi  _et al._ (2015). This event was named
GW230529_181500 and will be referred to as GW230529 for brevity. It was
observed only by the LIGO Livingston observatory, with a signal-to-noise ratio
(SNR) of 11.6 and a false-alarm rate of less than 1 in 1000 years. Assuming
that General Relativity (GR) is the correct theory to describe the GW230529
signal, the follow-up analyses estimated the source component masses to be
$3.6_{-1.1}^{+0.8}\leavevmode\nobreak\ \rm{M}_{\odot}$ and
$1.4_{-0.2}^{+0.6}\leavevmode\nobreak\ \rm{M}_{\odot}$ with a 90% credible
interval. This puts the primary squarely in the lower mass gap of $\sim
2\textup{ -- }5\leavevmode\nobreak\ \rm{M}_{\odot}$, where few compact objects
were observed Fishbach  _et al._ (2020); Farah  _et al._ (2022); Abbott  _et
al._ (2020a, 2023a, 2023b). This event, with the compact object within the
hypothesized mass gap and with a long inspiral signal, provides an opportunity
to test GR in a region of parameter space previously unexplored by strong-
field tests of GR.

GW230529 is not the first observation to find support for compact objects in
the lower mass gap. A number of recent studies that have found evidence for
compact objects in the mass gap include observations of noninteracting binary
systems Thompson  _et al._ (2018); Jayasinghe  _et al._ (2021), radio pulsar
surveys Barr  _et al._ (2024), and GWs from compact binary coalescences Abbott
_et al._ (2017, 2020b, 2020a, 2023b). When a binary is observed with component
masses measured in the mass-gap range of $\sim 2\textup{ --
}5\leavevmode\nobreak\ \rm{M}_{\odot}$, the immediate challenge is to
determine whether the component in the gap is a NS or a black hole (BH). Such
an identification would be remarkable because it would either increase the
known maximum mass for a NS Kalogera and Baym (1996); Rezzolla  _et al._
(2018) or decrease the known minimum mass for a BH from low-mass X-ray binary
observations Bailyn  _et al._ (1998); Farr  _et al._ (2011); Ozel  _et al._
(2010). However, it has not been conclusively proven whether these components
are BHs, NSs, or something else.

Recent theoretical research has suggested several potential exotic compact
objects, such as gravastars Mazur and Mottola (2004), boson stars Liebling and
Palenzuela (2023), or Planck-scale modifications of BH horizons Lunin and
Mathur (2002); Cardoso  _et al._ (2016), which could also fall into this gap.
Primordial BHs formed from overdense regions in the early Universe could also
fill the lower mass gap Carr  _et al._ (2016). Besides strophysical and
primordial BHs in GR and exotic compact objects, NSs in alternative GR
theories could also inhabit this gap. For example, axionic scalar-tensor
theory with viable phenomenological equations-of-state (EOS) can produce NSs
with maximum masses larger than $2.5\leavevmode\nobreak\ \rm{M}_{\odot}$, but
below the $3\leavevmode\nobreak\ \rm{M}_{\odot}$ threshold Oikonomou (2024).
Therefore, the GW230529 signal provides an opportunity to perform tests of GR
to uncover new physics, as well as pointing to the existence of potential
exotic compact objects.

General Relativity is the simplest and most successful theory of gravity to
date, and it has been rigorously validated by various experimental tests in
our Solar System Will (2014), observations of binary pulsars Wex (2014); Shao
_et al._ (2017), cosmological data Ferreira (2019), GWs Abbott  _et al._
(2016a, b, 2019a, 2019b, 2021a, 2021b), and massive BHs Genzel  _et al._
(2024). Recently, the evidence of a stochastic GW background with pulsar
timing arrays Bernardo and Ng (2023); Cannizzaro  _et al._ (2023) has also led
to some new gravitational tests. Despite its overall success, we do not know
how to reconcile GR with quantum mechanics and how (or if) to employ GR to
explain certain cosmological phenomena, such as dark matter and dark energy.
These limitations have motivated the development of alternative theories of
gravity, such as the Brans-Dicke Brans and Dicke (1961), Horndeski scalar-
tensor Horndeski (1974), Aether Jacobson and Mattingly (2004), Einstein–Gauss-
Bonnet Nojiri  _et al._ (2005); De Felice and Tsujikawa (2010), Chern-Simons
Jackiw and Pi (2003); Alexander and Yunes (2009) theories, and the effective-
field-theory extension of GR Endlich  _et al._ (2017), all of which have
survived a wide range of experimental tests Will (2014); Yunes  _et al._
(2016); Lyu  _et al._ (2022); Takeda  _et al._ (2023); Silva  _et al._ (2023).

The observation of GW signals from coalescences of compact binary systems has
opened the avenue for studying the two-body dynamics in strong gravitational
fields with high velocities approaching the speed of light. In contrast, other
observations typically probe states of either weak gravitational fields —
where spacetime curvature is negligible — or where velocities are well below
the speed of light. However, in binary pulsars, the strongly gravitating
bodies allow for strong-field tests of gravity, albeit for low velocities.
Developing an accurate GW waveform model that captures the entire evolution of
compact binaries is essential for determining source characteristics,
validating the predictions of GR, and exploring potential alternative theories
of gravity. However, the development of waveforms within specific alternative
theories has not yet reached a level of maturity sufficient to perform model
comparisons with GR. While numerical-relativity simulations in beyond-GR
theories are becoming more prevalent, either by solving the full equations of
motions East and Ripley (2021); Bezares  _et al._ (2022); Figueras and França
(2022); Aresté Saló  _et al._ (2022, 2023a, 2023b); Corman  _et al._ (2023);
Corman and East (2024) or through approximate treatments Okounkova  _et al._
(2019); Okounkova (2020); Cayuso  _et al._ (2023), there are not yet
sufficient simulations in one specific theory to calibrate a semi-analytic
inspiral-merger-ringdown model in that theory. Therefore, semi-analytical
inspiral-merger-ringdown models in alternative theories of gravity that are
compared to GW data either leave the merger like GR Carson and Yagi (2020a, b)
or they introduce agnostic parameters in the merger that are then marginalized
over during parameter estimation Julié  _et al._ (2024). Another possibility
is to directly compare numerical-relativity simulations in beyond-GR theories
with observational data Okounkova (2020), but those tests are limited by the
lengths of the numerical waveforms and the region of parameter space where
simulations are available. Nevertheless, there may be a more accurate
alternative theory that we are not aware of.

The alternative models often propose variances in the post-Newtonian (PN)
phase coefficients of the GW signal, which are derived solving perturbatively
the two-body dynamics and gravitational radiation by a series expansion in the
orbital velocity, intertwined with the strength of the gravitational field.
This has led to the development of theory-agnostic approaches to parameterized
tests of GR Blanchet and Sathyaprakash (1995); Arun  _et al._ (2006a, b);
Yunes and Pretorius (2009); Mishra  _et al._ (2010). The same framework can be
used with search pipelines to detect non-GR signals Narola  _et al._ (2023);
Sharma  _et al._ (2023). In these parameterized tests, an additional parameter
is introduced into the PN coefficients to represent generic deviations from GR
and a series of Bayesian parameter estimation analyses is performed for each
of these parameters. The resulting posterior distribution of a deviation
parameter indicates whether generic features in the GW signal suggest that GR
is not the most probable theory of gravity. Importantly, claiming a GR-
violation would require performing extensive tests that prove the GR deviation
is not due to systematics due to waveforms’ inaccuracy, lack of physical
effects, environmental astrophysical effects, and gravitational lensing Gupta
_et al._ (2024). The parameterized test in the Bayesian framework has been
extensively developed for the LIGO-Virgo-KAGRA (LVK) collaboration to test GR
as part of the GW transient catalogs (GWTC) Abbott  _et al._ (2019b, 2021a,
2021b). It has been performed using two implementations: the Flexible Theory-
Independent (FTI) approach Abbott  _et al._ (2019a); Mehta  _et al._ (2023),
which can be applied to any aligned-spin frequency-domain waveform model, and
the Test Infrastructure for General Relativity (TIGER) approach Li  _et al._
(2012); Agathos  _et al._ (2014); Meidam  _et al._ (2018); Roy  _et al._
(2024), which is based on a (frequency-domain) phenomenological waveform
family. In this study, we perform analyses of GW230529 using both approaches
with the SEOBNRv4 Bohé  _et al._ (2017); Cotesta  _et al._ (2018, 2020);
Dietrich  _et al._ (2019); Matas  _et al._ (2020) and IMRPhenomX Pratten  _et
al._ (2021); García-Quirós  _et al._ (2020); Colleoni  _et al._ (2023)
families, respectively.

Since alternative theories often predict the functional form of the beyond-GR
correction as a parametric variation of the PN terms, the measurements of
phenomenological deviation parameters can be mapped onto specific alternative
theories of gravity Yunes  _et al._ (2016); Berti  _et al._ (2018); Tahura and
Yagi (2018). By using the leading order correction of an alternative gravity
theory, we typically provide a theory-agnostic bound on the beyond-GR
parameters predicted by that theory. In this work, we use this method to
provide bounds on the Gauss-Bonnet coupling in the class of Einstein-scalar-
Gauss-Bonnet (ESGB) theories of gravity with $f^{\prime}(0)\neq 0$. We also
perform a theory-specific test for ESGB by implementing all known corrections
to the PN terms Yagi  _et al._ (2012); Sennett  _et al._ (2016); Julié and
Berti (2019); Shiralilou  _et al._ (2021, 2022); Bernard  _et al._ (2022);
Julié  _et al._ (2022, 2023). This includes newly computed corrections at
1.5PN order, as given in Appendix C. The corrections to ESGB are larger for
smaller BH masses, so GW230529 provides an excellent opportunity to probe
relatively small values of the coupling. Previous tests for ESGB have placed
constraints on the coupling of $\sqrt{\alpha_{\mathrm{GB}}}\lesssim 1.18$ km
by combining bounds from multiple events Lyu  _et al._ (2022).

The rest of this paper is organized as follows. Section II explains the
parameterized inspiral tests of GR used. Section III describes the setup of
the analyses performed. The results and constraints on deviations from GR
obtained for GW230529 are presented in Sec. IV. Finally, in Sec. V we map the
agnostic constraints to a specific modified gravity theory, namely ESGB
gravity. In Appendix A, we list the priors used in the analyses. Appendix B
discusses possible false violations of GR when using too wide priors on the
deviation parameters, which can cause wrap-around in the waveform. Lastly, we
list in Appendix C the ESGB corrections up to 1.5PN order in the GW phase.
Throughout this paper we follow the convention of $G=c=1$.

##  II Parameterized inspiral tests of GR

In GR, the GW signal from the early inspiral of a quasi-circular compact
binary can be approximated using the PN formalism, which expands the waveform
in powers of the velocity $v$, where $\mathcal{O}(v^{2n})$ relative to the
leading order term is called the $n$PN order Blanchet (2014); Buonanno  _et
al._ (2009). The frequency domain phase of a GW signal can then be obtained
using the stationary phase approximation Sathyaprakash and Dhurandhar (1991).
In GR, it is given by

$\begin{split}\Psi_{\ell m}^{\text{GR}}(f)=&2\pi
ft_{c}-\phi_{c}-\frac{\pi}{4}\\\ &+\frac{3}{128\eta
v^{5}}\frac{m}{2}\sum_{n=0}^{7}\left(\psi_{n}^{\text{GR}}+\psi_{n(l)}^{\text{GR}}\log
v\right)v^{n}\end{split}$ (1)

where $t_{c},\phi_{c}$ are the time and phase at coalescence, $v=(2\pi
fM/m)^{1/3}$ with $f$ the GW frequency, $M=m_{1}+m_{2}$ is the total mass, and
$\eta=m_{1}m_{2}/M^{2}$ is the symmetric mass ratio. Here,
$\psi_{n}^{\text{GR}}$ and $\psi_{n(l)}^{\text{GR}}$ represent the $(n/2)$PN
coefficients in GR, and they depend only on the intrinsic parameters of the
binary. The subscript $(l)$ indicates the logarithmic terms that enter at
2.5PN and 3PN orders. It is also important to note that the 0.5PN term is
absent in GR. The subscript $\ell m$ denotes the $(\ell,m)$-mode in the mode
decomposition of the GW signal into spherical harmonics (not to be confused
with the subscript $(l)$ denoting the coefficients of the $\log$ terms). The
dominant mode is the $(2,2)$-mode, and other modes are referred to as higher
modes.

In beyond-GR theories, these PN coefficients can be different from the ones in
GR Yunes  _et al._ (2016); Berti  _et al._ (2018); Tahura and Yagi (2018). To
test for deviations from GR during the inspiral, we therefore add a correction
to the frequency domain phase of the form

$\delta\Psi_{\ell m}(f)=\frac{3}{128\eta
v^{5}}\frac{m}{2}\left(\sum_{n=-2}^{7}\delta\psi_{n}v^{n}+\sum_{n=5}^{6}\delta\psi_{n(l)}v^{n}\log
v\right),$ (2)

where $\delta\psi_{n}$ and $\delta\psi_{n(l)}$ are the deviations in the
$(n/2)$PN coefficients.

For parameterized inspiral tests, we introduce deviation parameters
$\delta\hat{\varphi}_{n}$ and $\delta\hat{\varphi}_{n(l)}$ that are the
fractional deviations of the corresponding PN coefficients in GR. We thus have
that

$\displaystyle\delta\psi_{n}$
$\displaystyle=\delta\hat{\varphi}_{n}\psi_{n}^{\mathrm{GR}},$ (3)
$\displaystyle\delta\psi_{n(l)}$
$\displaystyle=\delta\hat{\varphi}_{n(l)}\psi_{n(l)}^{\mathrm{GR}}.$ (4)

When the $(n/2)$PN coefficient vanishes in GR (i.e., for $n=-2,1$), we instead
let $\delta\psi_{n}=\delta\hat{\varphi}_{n}$ so that it is an absolute
deviation normalized to the Newtonian coefficient. We do not test for
$\delta\hat{\varphi}_{5}$ since this would give a constant phase shift and
therefore be degenerate with the phase at coalescence. We also do not perform
tests at $-0.5$PN. In this parameterization, GR is recovered in the limit
$\delta\hat{\varphi}_{n},\delta\hat{\varphi}_{n(l)}\rightarrow 0$.

We employ two different frameworks that can perform this type of test of GR:
FTI Mehta  _et al._ (2023) and TIGER Agathos  _et al._ (2014); Meidam  _et
al._ (2018). These two frameworks differ in the GR waveform models used and
the exact implementation of the inspiral test. FTI uses the SEOBNRv4 waveform
family Bohé  _et al._ (2017); Cotesta  _et al._ (2018, 2020); Dietrich  _et
al._ (2019); Matas  _et al._ (2020) and is only available for aligned spins,
while TIGER uses the IMRPhenomX waveform family Pratten  _et al._ (2021);
García-Quirós  _et al._ (2020); Colleoni  _et al._ (2023) and can use
precessing spin waveforms. The main difference in the implementation occurs at
the transition from a non-GR inspiral to the GR merger ringdown. In FTI, the
testing GR corrections are added to the GW phase of an inspiral-merger-
ringdown frequency-domain waveform from GR. The corrections to the frequency-
domain phase are computed using Eq. (2), and then smoothly tapered off to zero
using a windowing function so that the merger-ringdown signal coincides with
the GR one (for details on how this is done, see Ref. Mehta  _et al._ (2023)).
This is done for every mode separately and the corrections are then added
together. In the TIGER framework, the parameterized deviations are
incorporated into the GR phase coefficients of the IMRPhenomXAS model. This
model represents the primary spherical harmonic radiation mode for the
coalescence of nonprecessing binary black hole (BBH) systems. The inspiral
phase of higher-order modes in the time domain is modeled as a scaling of the
dominant quadrupole mode, allowing any deviations in the inspiral phase to
propagate to the higher-order modes. In the phenomenological model, the
separate inspiral and merger-ringdown segments are smoothly connected by
ensuring $\mathcal{C}^{1}$ continuous condition in both phase and amplitude.
When we introduce the variations in the phase coefficients, we simultaneously
adjust the phase derivative to preserve the continuity. Deviations in the
inspiral phase coefficients do not affect the post-inspiral portion of the
waveform. Although TIGER also has the option to introduce deviations in the
post-inspiral part of the waveform, we do not use that in this work.

##  III Analysis setup

As baseline GR models for the FTI framework, we use the waveform models based
on SEOBNRv4HM_ROM Cotesta  _et al._ (2020), which is the reduced order,
frequency-domain version of the time-domain model SEOBNRv4HM Bohé  _et al._
(2017); Cotesta  _et al._ (2018). It is an effective-one-body (EOB) waveform
model for BBHs, that assumes quasi-circular orbits and aligned spins, and
includes the modes: (2,2), (2,1), (3,3), (4,4), and (5,5)111We do not include
the (5,5)-mode in our FTI analyses, but do use all other modes available.. Its
neutron star–black hole (NSBH) version SEOBNRv4_ROM_NRTidalv2_NSBH Dietrich
_et al._ (2019); Matas  _et al._ (2020) contains only the dominant
$(2,2)$-mode, and allows for tidal deformability of the secondary object and
tidal disruption. We also use its binary neutron star (BNS) version
SEOBNRv4_ROM_NRTidalv2, which has tidal effects on both components and
contains the $(2,2)$-mode only. With the TIGER framework, we use the models
based on IMRPhenomXPHM Pratten  _et al._ (2021); García-Quirós  _et al._
(2020), which is a phenomenological waveform model that allows for precessing
spins and includes the higher modes: (2,2), (2,1), (3,3), (3,2), and (4,4).
Its BNS version IMRPhenomXP_NRTidalv2 Dietrich  _et al._ (2019); Colleoni  _et
al._ (2023) allows for tides on both component objects and is $(2,2)$-mode
only. To investigate the effect of waveform systematics due to the presence of
precession, higher harmonics, or difference in waveform treatment, we also
perform the parametrized test using SEOBNRv4_ROM, IMRPhenomXP, and
IMRPhenomXHM models. An overview of the models used and the physics they
include is given in Table 1. The waveforms are generated using Bilby TGR
Ashton  _et al._ (2024) and LALSimulation LIGO Scientific Collaboration  _et
al._ (2018); Wette (2020).

Waveform model | Short name | Color | Higher modes | Spin precession | Tides
---|---|---|---|---|---
SEOBNRv4HM_ROM Bohé  _et al._ (2017); Cotesta  _et al._ (2018, 2020) | SEOBHM | $\bullet$ | ✓ | - | -
SEOBNRv4_ROM Bohé  _et al._ (2017) | SEOB |  | - | - | -
SEOBNRv4_ROM_NRTidalv2_NSBH Dietrich  _et al._ (2019); Matas  _et al._ (2020) | SEOBNSBH | $\bullet$ | - | - | ✓(NSBH)
SEOBNRv4_ROM_NRTidalv2 Dietrich  _et al._ (2019); Cotesta  _et al._ (2020) | SEOBT | $\bullet$ | - | - | ✓(BNS)
IMRPhenomXPHM Pratten  _et al._ (2021) | PhenomXPHM | $\bullet$ | ✓ | ✓ | -
IMRPhenomXHM García-Quirós  _et al._ (2020) | PhenomXHM |  | ✓ | - | -
IMRPhenomXP Pratten  _et al._ (2021) | PhenomXP |  | - | ✓ | -
IMRPhenomXP_NRTidalv2 Dietrich  _et al._ (2019); Colleoni  _et al._ (2023) | PhenomXPT | $\bullet$ | - | ✓ | ✓(BNS)
Table 1: An overview of the different waveform models that we use in this work
and the physics they include (higher modes or (2,2)-mode only, spin precession
or aligned spins only, allowing for tidal effects). We will refer to them
throughout this paper using the shortened name and use the color indicated in
the plots.

For parameter estimation, we employ Bayesian inference using Bilby Ashton  _et
al._ (2019); Romero-Shaw  _et al._ (2020); Smith  _et al._ (2020). We use the
standard likelihood function assuming additive noise that is stationary and
Gaussian. The one dimensional posteriors for the deviation parameters are
obtained by marginalizing over all other parameters. The sampling algorithm
used is nested sampling with the Dynesty sampler Speagle (2020). We only allow
one deviation parameter to vary at a time and repeat the analysis for each
deviation parameter. We use uniform priors for the deviation parameters. The
priors used are listed in Appendix A.

We analyze 128 seconds of data from the LIGO Livingston Observatory with a
sampling frequency of 4096 Hz. The analysis is done over the frequency range
20 Hz to 1792 Hz when calculating the likelihood. The power spectral density
used to describe the noise was produced with BayesWave Cornish and Littenberg
(2015); Littenberg and Cornish (2015).

##  IV Results

Figure 1: The posterior distributions for the different deviation parameters $\delta\hat{\varphi}_{n},\delta\hat{\varphi}_{n(l)}$ for GW230529. The blue histograms are obtained with TIGER using the IMRPhenomX waveform family. The orange posteriors are results from FTI using the SEOBNRv4 waveform family. The filled violins are for BBH waveforms models, while the dashed lines are BNS models which include tidal effects on both components, and the red solid line is an NSBH model with tides only on the secondary. GR is recovered at $\delta\hat{\varphi}_{i}=0$, which is indicated by the horizontal gray line. All results are consistent with GR except for 0PN, which will be discussed in more detail in Sec. IV.3. Parameter | SEOBHM | SEOBNSBH | PhenomXPHM | PhenomXPT
---|---|---|---|---
$\delta\hat{\varphi}_{i}$ | $Q_{\mathrm{GR}}$ | $\delta\hat{\varphi}_{n}$ | $Q_{\mathrm{GR}}$ | $\delta\hat{\varphi}_{i}$ | $Q_{\mathrm{GR}}$ | $\delta\hat{\varphi}_{i}$ | $Q_{\mathrm{GR}}$
$\delta\hat{\varphi}_{-2}$ | $2.5_{-6.1}^{+7.2}\times 10^{-5}$ | 23% | $4.6_{-6.8}^{+22}\times 10^{-5}$ | 14% | $0.1_{-6.8}^{+8.7}\times 10^{-5}$ | 49% | $14_{-12}^{+27}\times 10^{-5}$ | 3%
$\delta\hat{\varphi}_{0}$ | $4.2_{-1.2}^{+0.6}$ | 0% | $1.6_{-0.8}^{+1.7}$ | 0% | $2.0_{-0.4}^{+1.2}$ | 0% | $3.4_{-1.0}^{+1.1}$ | 0%
$\delta\hat{\varphi}_{1}$ | $-0.04_{-0.10}^{+0.11}$ | 75% | $-0.12_{-0.21}^{+0.15}$ | 90% | $-0.04_{-0.15}^{+0.15}$ | 67% | $-0.25_{-0.22}^{+0.17}$ | 99%
$\delta\hat{\varphi}_{2}$ | $-0.06_{-0.11}^{+0.11}$ | 82% | $-0.14_{-0.15}^{+0.16}$ | 93% | $-0.04_{-0.14}^{+0.15}$ | 69% | $-0.23_{-0.16}^{+0.13}$ | 99%
$\delta\hat{\varphi}_{3}$ | $0.11_{-0.16}^{+0.15}$ | 13% | $0.17_{-0.21}^{+0.18}$ | 9% | $0.07_{-0.23}^{+0.21}$ | 30% | $0.23_{-0.17}^{+0.15}$ | 2%
$\delta\hat{\varphi}_{4}$ | $-0.4_{-2.3}^{+2.7}$ | 61% | $0.5_{-2.6}^{+3.7}$ | 40% | $-0.1_{-2.9}^{+2.8}$ | 51% | $1.0_{-3.0}^{+3.0}$ | 32%
$\delta\hat{\varphi}_{5l}$ | $0.01_{-0.60}^{+0.79}$ | 49% | $-0.27_{-0.68}^{+0.74}$ | 73% | $-0.03_{-0.68}^{+0.81}$ | 54% | $-1.1_{-1.7}^{+1.2}$ | 94%
$\delta\hat{\varphi}_{6}$ | $0.0_{-1.3}^{+1.6}$ | 49% | $0.7_{-1.4}^{+1.8}$ | 21% | $0.1_{-1.2}^{+1.4}$ | 46% | $2.4_{-2.1}^{+2.9}$ | 3%
$\delta\hat{\varphi}_{6l}$ | $-0.9_{-6.1}^{+8.1}$ | 59% | $-3.0_{-6.7}^{+7.3}$ | 75% | $-0.2_{-4.0}^{+4.5}$ | 54% | $-7.0_{-9.9}^{+7.2}$ | 95%
$\delta\hat{\varphi}_{7}$ | $0.6_{-4.3}^{+3.3}$ | 40% | $-1.5_{-4.7}^{+3.4}$ | 77% | $-0.1_{-4.7}^{+3.7}$ | 52% | $-8.9_{-10}^{+7.7}$ | 97%
Table 2: The values for the different deviation parameters
$\delta\hat{\varphi}_{n},\delta\hat{\varphi}_{n(l)}$ obtained with GW230529.
Displayed are the median values and 90% confidence intervals inferred. We also
display the quantile $Q_{\mathrm{GR}}$ corresponding to the GR value
$\delta\hat{\varphi}_{i}=0$.

The posteriors for the different deviation parameters are shown in Fig. 1. The
blue violins on the left are obtained with TIGER, while the orange violins on
the right are results from FTI. The filled violins do not have tidal effects,
while the unfilled violins with dash and continuous lines are for waveform
models including tidal effects. Table 2 shows the values for the deviation
parameters found for GW230529 and the quantile corresponding to GR. We see
that all results obtained are consistent with GR, except for the 0PN results.
We will first discuss in Sec. IV.1 in more detail the results obtained with
BBH waveforms. Then, in Sec. IV.2, we explain the effects that tides in the
NSBH and BNS models have on the results. Finally, we will discuss the 0PN
results in more detail in Sec. IV.3.

###  IV.1 Binary black hole waveforms

The results for the aligned-spin PhenomXHM, spin-precessing (2,2)-mode
PhenomXP, and aligned-spin (2,2)-mode SEOB are not shown in the violin plot,
but they are similar to the spin-precessing PhenomXPHM and aligned-spin SEOBHM
results, respectively. It is not surprising that the results for spin-
precessing PhenomXPHM and aligned-spin PhenomXHM agree, since for GW230529
there was no evidence found for spin-precession (the posteriors on the
effective precessing spin were uninformative Abac  _et al._ (2024)). It was
also expected that the results for PhenomXPHM and PhenomXP agree well, because
for the SNR and source parameters of GW230529, the higher modes are not
expected to strongly contribute. The posteriors for SEOB are slightly wider
than for SEOBHM, but otherwise agree well. The increase in width is most
likely due to a slight loss in SNR when not including the higher-order modes.

Let us compare the FTI and TIGER results for BBH waveform models by comparing
PhenomXPHM (blue filled) and SEOBHM (orange filled) results with each other.
Differences between these results can come from waveform systematics, for
example the details of how the waveforms are constructed and the
approximations made therein, or from differences in the implementation of the
inspiral test of GR in FTI and TIGER. This is a good consistency check to see
that both pipelines work as intended. Looking at the filled violins in Fig. 1,
we see that the results are consistent with each other (except for
$\delta\hat{\varphi}_{0}$). There are sometimes some minor differences between
the SEOBHM and PhenomXHM, e.g. one posterior is slightly wider than the other,
but there are no major differences.

###  IV.2 Waveforms with tidal effects

When looking at the waveforms with tidal effects in Fig. 1, we see that there
are no major differences between PhenomXPT (blue dashed line) and SEOBT
(orange dashed line). However, we notice that there are significant
differences compared to the waveforms without tidal effects (filled violins).
The posteriors for PhenomXPT and SEOBT, which have tides on both compact
objects, are shifted away from GR and are wider compared to the results
without tidal effects. This also happens to a lesser amount for SEOBNSBH (red
solid line), which only has tides on the secondary object.

The NRTidalv2 models Dietrich  _et al._ (2019) modify the BBH waveforms to
account for differences in the waveform coming from tides and other NS matter
effects. Both the phase and amplitude are modified due to tidal effects, which
are modeled using the tidal deformabilities $\Lambda_{i}$ of the two objects.
For BHs, the tidal deformability is zero, so the NSBH model SEOBNSBH enforces
$\Lambda_{1}=0$. We do not expect changes in the amplitude to significantly
influence our results because, in the parameterized inspiral tests, we only
modify the frequency domain phase. We therefore only expect the change in
phase evolution to influence the posteriors on the deviation parameters for
the BNS and NSBH models.

The merger-ringdown is modeled differently for BNS and NSBH models. The merger
frequency of the PhenomXPT and SEOBT models is calibrated to NR. Above this
frequency, the amplitude is smoothly tapered off to zero in an agnostic way
Dietrich  _et al._ (2019). The reason for this is that the merger-ringdown of
BNSs is not modeled well enough, and, more importantly, the merger frequency
is typically high enough that the post-merger signal is outside the sensitive
frequency band of current detectors, hence, no bias is expected during
parameter estimation Dudi  _et al._ (2018). When using these models for
GW230529, the merger frequency is lower and inside the frequency band
analyzed. To check that the amplitude tapering is not influencing our results,
we also did an analysis with a maximum frequency of 400 Hz, which is below the
merger frequency. No differences were found between the posteriors, so the
amplitude tapering is not causing the shift away form GR that we observe when
using PhenomXPT and SEOBT.

For SEOBNSBH, the merger-ringdown is modeled differently. For NSBHs, tidal
disruption of the NS can happen when the tidal disruption frequency is lower
than the ringdown frequency Matas  _et al._ (2020). If this happens, then the
amplitude is smoothly tapered off above the tidal disruption frequency.
Otherwise, the merger-ringdown is modeled similarly to the merger-ringdown of
BBHs, with a slightly suppressed amplitude. For GW230529, the NS most likely
gets tidally disrupted Abac  _et al._ (2024); Chandra  _et al._ (2024).
However, the tidal disruption frequency is above the maximum frequency used in
our analysis Matas  _et al._ (2020); Abac  _et al._ (2024). We therefore do
not expect tidal disruption to influence our results.

During inspiral, both the phase and amplitude are modified due to tidal
effects. Because the tidal deformability is always positive (at least for
NSs), the change in phase is always in the same direction. The deviation
parameters try to cancel this effect by changing the phase in the opposite
way. This means that a nonzero $\Lambda$ will always lead to a specific
$\delta\hat{\varphi}_{n}$ to shift towards either positive or negative values
(depending on the sign of the PN coefficient in GR), but not both. This
explains why the deviation parameters all shift away from zero for the BNS and
NSBH models compared to the GR models. The shift is less strong for the NSBH
model since there is only one tidal parameter that is nonzero so the tides
have less of an effect. The extra tidal parameters also lead to broadening of
the posteriors, especially for low SNR signals where their values are not well
measured. Of course, this explanation is oversimplified due to correlations
between the tides and other GR parameters as well as the deviation parameters,
but nonetheless it explains why there is a shift away from GR and not just a
broadening of the posteriors.

To check that this reasoning is correct, we look at the $\Lambda_{i}$ –
$\delta\hat{\varphi}_{n}$ posteriors in Fig. 2. The top plot shows the
posteriors from an NSBH run for the 0.5PN deviation parameter (red). We see in
the 2D posterior that $\delta\hat{\varphi}_{1}$ shifts away from 0 for larger
values of $\Lambda_{2}$. This can be seen in the histogram where it has more
support for negative values of $\delta\hat{\varphi}_{1}$ than is the case for
the corresponding BBH run (orange). If we restrict the tidal deformability to
$\Lambda_{2}<1000$ (green shaded area), which is a more realistic range based
on the mass of the secondary and current constraints on the NS EOS Huth  _et
al._ (2022) and it is consistent with what was found using an astrophysically-
motivated prior in Ref. Abac  _et al._ (2024), then the shift away from GR
goes away and the posterior (green histogram) becomes similar to the BBH one.

In the middle and bottom plots of Fig. 2 we look at BNS runs (dark blue) for
the $-1$PN deviation parameter (middle) and 3.5PN deviation parameter
(bottom). Here we see the same effect where $\delta\hat{\varphi}_{-2}$ and
$\delta\hat{\varphi}_{7}$ are correlated with $\Lambda_{1}$ and they shift
away from GR for large values of $\Lambda_{1}$. If we now restrict the tides
to small values of $\Lambda_{1}<300$222More realistically, using the EOS set
of Huth  _et al._ (2022), the tidal deformability for a NS with masses above
$2.5\rm{M}_{\odot}$ (if such masses are supported) would be $\Lambda\lesssim
5$. Restricting $\Lambda_{1}$ to such low values would however leave us with
too few samples to say anything sensible. We therefore went with a larger cut
off value that is a balance between the number of samples left and keeping the
tides realistic. and the more realistic values of $\Lambda_{2}<1000$ (green
shaded area) Huth  _et al._ (2022); Abac  _et al._ (2024), we once again see
that this leads to the posterior (green histogram) shifting to the BBH one
(light blue). This behavior is also seen for the other deviation parameters.

Figure 2: The 2D posteriors between the tidal deformability $\Lambda_{i}$ and
deviation parameters $\delta\hat{\varphi}_{n}$ (red and dark blue). The green
shaded regions are more realistic ranges for the tides, which, when
restricting the posterior to those regions, give the green histograms. The
orange and light blue histograms are the posterior obtained with the
comparable BBH models. The top plot is for an NSBH waveform, while the other
two are with a BNS waveform. We see that there is a correlation between the
tides and the deviation parameters, which leads to a shift away from GR. When
restricting the tides to more reasonable values, this shift disappears.

We conclude that the shift away from GR of the posteriors on the deviation
parameters for the SEOBNSBH and especially the PhenomXPT and SEOBT waveform
models is most likely due to correlations between the deviation parameters and
the tidal deformability combined with the tidal deformability not being well
constrained. Constraining the tidal deformability of the two objects to more
realistic values based on their masses removes this shift away from GR. Since
the NSBH and BNS results with the tides constrained to realistic values are
similar to the BBH results, we will use the BBH results from now on (unless
otherwise specified), even though GW230529 is most likely a NSBH Abac  _et
al._ (2024).

###  IV.3 Degeneracy between 0PN-deviation parameter and chirp mass

At low frequencies (i.e., during early inspiral), the phase of a GR waveform
is dominated by the 0PN term. In our testing GR waveform model, the phase
contribution from the 0PN term is given by

$\Psi_{\ell m}^{0\mathrm{PN}}(f)=\frac{3}{128\eta
v^{5}}\frac{m}{2}\left(1+\delta\hat{\varphi}_{0}\right)\psi_{0}^{\mathrm{GR}}=C_{0}\frac{1+\delta\hat{\varphi}_{0}}{\mathcal{M}_{c}^{5/3}}f^{-5/3},$
(5)

where $C_{0}$ is a constant that does not depend on the intrinsic parameters
of the binary and $\mathcal{M}_{c}=\eta^{3/5}M$ is the chirp mass. This means
that in the frequency regime where the 0PN term is dominant, the deviation
parameter $\delta\hat{\varphi}_{0}$ is degenerate with the chirp mass of the
binary, so the two are strongly correlated (see also Sec. IVC in Ref. Mehta
_et al._ (2023)). Assuming that the binary is consistent with GR and that the
true chirp mass is equal to $\mathcal{M}_{c}^{\mathrm{GR}}$, this correlation
is given by

$\delta\hat{\varphi}_{0}=\left(\frac{\mathcal{M}_{c}}{\mathcal{M}_{c}^{\mathrm{GR}}}\right)^{5/3}-1.$
(6)

For low chirp-mass systems, the late-inspiral and merger-ringdown are outside
the frequency band observed by the GW detectors. For such systems, this
degeneracy at 0PN means that, when doing inspiral tests of GR at 0PN, the
chirp mass can take any value and $\delta\hat{\varphi}_{0}$ can be used to
compensate without significantly altering the waveform. For high enough chirp
mass, the late inspiral and merger-ringdown move into the frequency band used
for analysis. This eventually breaks the degeneracy, and such high values of
the chirp mass would then be disfavored. This means that the $\mathcal{M}_{c}$
– $\delta\hat{\varphi}_{0}$ posterior is more or less free to move along the
line of the degeneracy between the two, until it reaches too high values of
the chirp mass.

Figure 3 shows the 2D posterior of the chirp mass $\mathcal{M}_{c}$ and 0PN
deviation parameter $\delta\hat{\varphi}_{0}$ for GW230529 obtained with
SEOBHM (orange). The maximum likelihood sample from a GR run with SEOBHM is
highlighted in green. This is also the value used in Eq. (6) to compute the
correlation (dark green). We see that the posterior closely follows the
correlation that is expected from the 0PN degeneracy.

Figure 3: The 2D posterior between the chirp mass $\mathcal{M}_{c}$ and the
0PN deviation parameter $\delta\hat{\varphi}_{0}$ using the SEOBHM waveform
model (orange). The green dot indicates the maximum likelihood values obtained
with a GR run. The dark green line is the expected degeneracy between
$\mathcal{M}_{c}$ and $\delta\hat{\varphi}_{0}$ based on the GR maximum
likelihood chirp mass $\mathcal{M}_{c}^{\mathrm{GR}}$ as per Eq. (6). We see
that the posterior follows the expected correlation.

To check that this degeneracy should be present in GW230529-like signals, we
compute the mismatch between the maximum likelihood GR waveform and non-GR
waveforms using PhenomXPHM. We vary the chirp mass and 0PN deviation
coefficient, but keep all other parameters the same, and the mismatch is
minimized over time and phase. Figure 4 gives a map of these mismatches. We
see that there is a line along which the mismatch is small. This line
corresponds to the correlation in Eq. (6). We notice that the low mismatch
region becomes wider for larger chirp mass (see also the insets).

Figure 4: The mismatch between the maximum likelihood waveform from a GR run
using PhenomXPHM and non-GR waveforms as a function of chirp mass
$\mathcal{M}_{c}$ and the 0PN deviation parameter $\delta\hat{\varphi}_{0}$.
We clearly see the presence of the degeneracy from Eq. (6). The insets show a
zoom-in on two regions around the degeneracy line. We notice that the low
mismatch region becomes wider for larger chirp mass.

As an extra check that this strong correlation should indeed be present in a
GR signal with similar source parameters as GW230925, we inject the maximum
likelihood GR waveform using SEOBHM into a zero-noise realization of the LIGO
Livingston detector with the same power spectral density as at the time of
GW230529. We then evaluate the likelihood for random values of the chirp mass
and 0PN deviation parameter, while keeping the other parameters the same as
for the injected waveform. This gives the likelihood map shown in Fig. 5. The
red star indicates the injected values, and the red line shows the expected
degeneracy in the waveform according to Eq. (6). We see that the maximum
likelihood region (yellow colors) indeed follows this line. There is no
visible gradient along the line, which confirms that the waveform does not
significantly change along the line (at least not within this chirp mass
range).

Figure 5: The likelihood for non-GR waveforms as a function of chirp mass
$\mathcal{M}_{c}$ and the 0PN deviation parameter $\delta\hat{\varphi}_{0}$
when injecting the maximum likelihood waveform from a GR run using SEOBHM. The
injected values are indicated by the red star, and the red line is the
expected degeneracy according to Eq. (6). We see that the high likelihood
region is sharply peaked and follows the expected correlation.

When looking at Fig. 3, we also notice that the posterior is away from the GR
value of $\delta\hat{\varphi}_{0}=0$. There could be multiple causes for this
shift away from GR, which we will explain below. Most likely, a combination of
them are playing a role for GW230529.

Because the likelihood barely changes along the line of the correlation, small
changes in the likelihood due to, for example, noise could push the posterior
to certain values. Normally when having multiple detectors observing a signal,
the expectation is that such fluctuations due to noise cancel each other out
between the detectors since the noise is mostly uncorrelated. For single-
detector events, the correlation between detectors cannot be used to prevent
extra testing GR parameters being fitted to noise features. Because this event
is single detector, the noise might indeed be playing a part in the shift of
the posterior towards higher chirp mass and $\delta\hat{\varphi}_{0}$.

It is known that noise effects such as non-stationarity, non-gaussianity, and
glitches can impact tests of GR Kwok  _et al._ (2022); Abbott  _et al._
(2021b); Gupta  _et al._ (2024); Maggio  _et al._ (2023) and even parameter
estimation in GR Chatziioannou  _et al._ (2019); Cornish  _et al._ (2021); Edy
_et al._ (2021); Macas  _et al._ (2022). More studies would be needed to see
if biases like the shift away from GR at 0PN can be produced by noise and to
see if biases in tests of GR due to noise effects are indeed enhanced for
single-detector events. There was no transient noise found around GW230529
Abac  _et al._ (2024), so we do not expect the noise to be majorly impacting
our results.

The choice of prior could also significantly influence the posterior when
there is such a degeneracy. For testing GR parameters that are strongly
correlated with astrophysical parameters, this is a known issue, and it is
therefore important to choose the correct priors Payne  _et al._ (2023). We
used a prior for the chirp mass that is uniform in the component masses, which
translates to a preference for higher chirp-mass values. Since the waveform
(and thus the likelihood) does not change much along this degeneracy, the
posterior is likely shifted towards higher chirp mass and
$\delta\hat{\varphi}_{0}$ because of this choice of prior.

Payne  _et al._ (2023) show that an astrophysically informed prior can help
mitigate biases in tests of GR. To check the effect the choice of prior has on
the 0PN results for GW230529, we also performed an analysis with a prior that
resembles their astrophysical prior for the primary mass and mass ratio (see
Eqs. (5) and (6) in Ref. Payne  _et al._ (2023)). Instead of inferring the
powers $\alpha$ and $\beta$, we fix their values to $\alpha=3$ and $\beta=3$,
which is consistent with their results for the 0PN test. Note that we extend
their prior to lower masses, where the power law is no longer representative
of the astrophysical population, but does exemplify a simple analytic prior
that prefers lower masses and more equal masses. We also performed an analysis
where we use a prior that is uniform in the chirp mass and mass ratio. The
results are shown in Fig. 6. We see that the low-mass-inclined power law prior
(blue) gives results that are consistent with GR. The results also show more
equal masses, which is expected considering the prior prefers this, and a
negative effective spin. This shift towards more equal masses and negative
effective spin also happens for GR when using certain population-informed
priors Abac  _et al._ (2024). The results with the prior uniform in chirp mass
and mass ratio (red) agree well with the results using our default prior that
is uniform in the component masses (orange). This confirms that the apparent
deviation from GR for $\delta\hat{\varphi}_{0}$ is prior dependent and most
likely not a true deviation.

Figure 6: The results for the 0PN analysis using different priors for the
masses obtained with SEOBHM. We see that the low-mass-inclined power law prior
(blue) gives results that are consistent with GR as opposed to the default
uniform in component masses prior (orange) and the uniform in chirp mass and
mass ratio prior (red). We also show the posteriors for a GR run using the
default prior and SEOBHM.

Another effect that could be causing the shift of the posterior towards higher
chirp mass is some sampling issue. As can be seen in Fig. 5, the likelihood is
sharply peaked around the correlation line (i.e., it is extremely narrow and
elongated). It is often difficult for the sampler to find such narrow features
in the likelihood. We see in Fig. 4 that the low mismatch region becomes wider
for high chirp mass, which means that the high-likelihood region also becomes
wider for high chirp mass. This makes it easier for the sampler to find that
part of the high-likelihood region. This could be part of the cause for the
shift of the posterior towards high chirp mass and $\delta\hat{\varphi}_{0}$.
Changing the parameterization and priors used could make it easier for the
sampler to find the high likelihood region in the lower chirp mass regime.

Lastly, GW230529 being detected by only the LIGO Livingston detector might
also lead to biases. For single-detector events, the extrinsic parameters are
not well measured. Correlations between parameters could then mean that
intrinsic parameters are also less well measured than would be the case when
an event is detected by multiple detectors. The chirp mass, luminosity
distance, and inclination are correlated through the amplitude of the signal.
In GR, the chirp mass is being well measured in the phase and the amplitude is
not needed to constrain it. Since the chirp mass is free to change together
with $\delta\hat{\varphi}_{0}$ in the phasing for low-mass systems, the
amplitude would help constrain the chirp mass. But for single-detector events,
the inclination and distance are not well measured (multiple detectors are
needed for that). Indeed, we also see that the luminosity distance is wider
and shifted towards larger values compared to the GR run. This increase in
correlations for single-detector events could be contributing to the bias in
chirp mass and $\delta\hat{\varphi}_{0}$. This could also explain why this
shift away from GR is not observed as strongly for the BNS GW170817 Abbott
_et al._ (2019a) and the NSBH GW200115_042309 Abbott  _et al._ (2021b), which
were both observed in three detectors.

As a check to see that this shift away from GR is not a true deviation from
GR, we performed parameter estimation on zero-noise injections of the maximum
likelihood waveform from the corresponding GR run using SEOBHM and PhenomXPHM.
Figure 7 shows that the posteriors obtained for the chirp mass and
$\delta\hat{\varphi}_{0}$ (green) are again shifted towards higher values away
from GR and the true chirp mass (green dashed lines), but they are different
from the ones obtained for GW230529 using the same waveform models (light
orange and light blue). The differences are of the same order as the
differences we get between different waveform models. From this we can
conclude that the noise is most likely not the dominant factor in the shift
away from GR, but we cannot fully exclude that it might be playing a part.

Figure 7: The posteriors for the chirp mass $\mathcal{M}_{c}$ and 0PN
deviation parameter $\delta\hat{\varphi}_{0}$ for zero-noise injections of the
maximum likelihood waveform from the corresponding GR run (green), compared to
the posteriors obtained for GW230529 with the different waveform models. Left
shows the posteriors obtained with FTI and right shows the results from TIGER.
We see that the results from the GR injections are biased towards higher
$\mathcal{M}_{c}$ and positive $\delta\hat{\varphi}_{0}$ and are similar to
the GW230529 results.

We conclude that for GW230529 the results show a strong correlation between
the 0PN deviation parameter and the chirp mass, and that this correlation
matches the expected degeneracy that would be there for a GR signal with such
low masses. Even though the posterior excludes the GR values, we do not
consider the 0PN results for GW230529 as evidence against consistency with GR,
because the shift away from zero is along the line of this degeneracy. We also
observe a similar shift away from GR for a zero-noise injection with the
maximum likelihood GR waveform, which further confirms that the 0PN results
for GW230529 are most likely not a true deviation from GR.

###  IV.4 Comparison with previous events

Figure 8: The bounds for the different deviation parameters
$\delta\hat{\varphi}_{n},\delta\hat{\varphi}_{n(l)}$ obtained with GW230529
(green stars) compared to previously obtained bounds. We show the results
obtained with the BBH waveforms as a proxy for an NSBH with realistic tides,
as discussed in Sec. IV.2. The orange triangles and red diamonds are bounds
from GWTC-2 and GWTC-3 respectively, obtained by combining the posteriors for
different events together (using SEOBNRv4_ROM and IMRPhenomPv2). The blue
circles are the bounds obtained with binary NS GW170817 (where they used
SEOBNRv4_ROM_NRTidalv2 and IMRPhenomPv2_NRTidalv2). The grey stripes are
bounds from individual events in GWTC-2 and GWTC-3. We see that GW230529 gives
exceptionally tight bounds for the $-1$PN, 0.5PN, and 1PN deviation
parameters. (Adapted from Ref. Abbott  _et al._ (2021b).)

Figure 8 shows the 90% upper bounds on the deviation coefficients found for
GW230529 (green stars), as well as the combined bounds from testing GR with
GWTC-2 Abbott  _et al._ (2021a) (orange triangles) and GWTC-3 Abbott  _et al._
(2021b) (red diamonds). The plot also shows for comparison the bounds from
GW170817 Abbott  _et al._ (2019a) (blue circles) and other bounds from
individual events (grey stripes). Most noteworthy is the bound on the $-1$PN
deviation of $|\delta\hat{\varphi}_{-2}|\lesssim 8\times 10^{-5}$. This is a
factor $\sim 17$ better than previous bounds from the NSBH GW200115_042309. It
is also tighter than the combined constraint from GWTC-3 of
$|\delta\hat{\varphi}_{-2}|\lesssim 8\times 10^{-4}$, which combined the
results from all NSBHs and BBHs in GWTC-3 for which the LVK performed tests of
GR. The only previous $-1$PN bound that is tighter than our constraints from
GW230529, is the results for BNS GW170817, from which a bound of
$|\delta\hat{\varphi}_{-2}|\lesssim 2\times 10^{-5}$ was obtained. The bounds
found at 0.5PN and 1PN are also relatively tight, with
$|\delta\hat{\varphi}_{1}|\lesssim 0.2$ even being similar to the previous
combined bounds.

We would expect the degeneracy at 0PN to also be there for other low-mass
events. This is indeed the case for the BNS GW170817 Abbott  _et al._ (2019a)
and the NSBH GW200115_042309 Abbott  _et al._ (2021b). Those two events also
show the shift away from GR towards positive $\delta\hat{\varphi}_{0}$,
however, not as strong as for GW230529 and they still have support for GR.
This is most likely due to those events being observed in three detectors, so
there are less degeneracies with extrinsic parameters.

##  V Tests of ESGB gravity

To illustrate the importance of the $-1$PN bounds from GW230529 for
constraints on specific alternative theories of gravity, we have here a look
at ESGB Nojiri  _et al._ (2005); De Felice and Tsujikawa (2010). ESGB is a
modified gravity theory where a scalar field is coupled to the Gauss-Bonnet
density. It is described by the action

$S=\frac{1}{16\pi}\int\mathrm{d}x^{4}\sqrt{-g}\left(R-2(\partial\phi)^{2}+\ell_{\mathrm{GB}}^{2}f(\phi)\mathcal{G}\right).$
(7)

Here, $g$ is the metric determinant, $R$ is the Ricci scalar, $\phi$ is a
scalar field with kinetic term
$(\partial\phi)^{2}=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, and
$\mathcal{G}$ is the Gauss-Bonnet invariant

$\mathcal{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^{2},$
(8)

where $R^{\mu\nu\rho\sigma}$ is the Riemann tensor and $R^{\mu\nu}$ the Ricci
tensor. The integral over a four-dimensional spacetime $\int
d^{4}x\sqrt{-g}\,\mathcal{G}$ is a boundary term Myers (1987). The function
$f(\phi)$, thus defined modulo a constant, specifies the theory. As for
$\ell_{\mathrm{GB}}$, it is the Gauss-Bonnet coupling strength with dimension
of length. We assume that matter fields are minimally coupled to $g_{\mu\nu}$.

In this paper, we consider the class of theories such that $f^{\prime}(0)\neq
0$. Without loss of generality, the function $f(\phi)$ can be expanded as
follows:

$f(\phi)=2\phi+\mathcal{O}(\phi^{2}).$ (9)

At leading order in $\phi$, the action (7) reduces to that of shift-symmetric
ESGB theories Julié and Berti (2019), which is invariant under the shift
$\phi\rightarrow\phi+\Delta\phi$, where $\Delta\phi$ is a constant. This is
equivalent to the small-$\phi$ approximation of Einstein-dilaton-Gauss-Bonnet
for which $f(\phi)=e^{\gamma\phi}$ with a constant $\gamma$ (the value of
$\gamma$ can be absorbed into the coupling constant). Here, we use the
convention of Refs. Julié and Berti (2019); Julié  _et al._ (2022, 2023).
Another convention denotes the Gauss-Bonnet coupling by $\alpha_{\mathrm{GB}}$
Lyu  _et al._ (2022), which is related to our $\ell_{\mathrm{GB}}$ as
$\alpha^{2}_{\mathrm{GB}}=\ell_{\mathrm{GB}}^{4}/(16\pi)$.

It is possible to find the ESGB corrections to the GW phase during inspiral
using the PN formalism. The leading-order correction to the frequency domain
phase in ESGB appears at $-1$PN due to scalar dipole radiation. In the class
of ESGB theories (9) and at $\mathcal{O}\left(\ell_{\mathrm{GB}}^{4}\right)$,
it reads Lyu  _et al._ (2022)

$\delta\hat{\varphi}_{-2}=-5\ell_{\mathrm{GB}}^{4}\frac{\left(m_{1}^{2}s_{2}-m_{2}^{2}s_{1}\right)^{2}}{168m_{1}^{4}m_{2}^{4}},$
(10)

where $m_{1,2}$ are the masses of the compact objects, while $s_{1,2}$ were
introduced in Refs. Yunes  _et al._ (2016); Lyu  _et al._ (2022). Here, $s$ is
defined as the scalar monopole of a body at $\mathcal{O}(\ell_{\rm GB}^{2})$,
divided by that of a nonspinning BHs with the same mass Julié and Berti
(2019). The higher-order corrections have been fully computed up to 1PN (2PN
relative order) for nonspinning binaries in the past Yagi  _et al._ (2012);
Sennett  _et al._ (2016); Julié and Berti (2019); Shiralilou  _et al._ (2021,
2022); Julié  _et al._ (2022, 2023). These higher-order corrections are given
in Appendix C. We also provide the full 1.5PN correction in Eq. (14e), which
is new to this paper.

In the class of ESGB theories (9), NSs cannot carry scalar monopoles at
$\mathcal{O}(\ell_{\rm GB}^{2})$ so $s=0$ Yagi  _et al._ (2012, 2016); Lyu
_et al._ (2022), and for nonspinning BHs, $s=1$ Julié and Berti (2019). So for
a binary consisting of a NS and a nonspinning BH, Eq. (10) can be simplified
to

$\delta\hat{\varphi}_{-2}=\frac{-5\ell_{\mathrm{GB}}^{4}}{168m_{1}^{4}},$ (11)

where $m_{1}$ is the mass of the BH. When computing the ESGB corrections for
GW230529, we assume the secondary to be a NS independently of the waveform
model used (remember that we use the BBH results as a proxy for the NSBH
results with realistic tides).

###  V.1 Constraints on ESGB by mapping agnostic results

It is possible to get approximate constraints on the ESGB coupling from
theory-agnostic tests. This is done by mapping the $-1$PN agnostic constraints
to the ESGB coupling using the leading order correction. Inverting Eq. (11)
gives

$\ell_{\mathrm{GB}}=\left(\frac{-168m_{1}^{4}\delta\hat{\varphi}_{-2}}{5}\right)^{1/4}$
(12)

Note that we need to restrict ourselves to negative values of
$\delta\hat{\varphi}_{-2}$. When mapping posteriors, we also need to reweight
the samples to account for differing priors. To map from a prior uniform in
$\delta\hat{\varphi}_{-2}$ to a prior uniform in $\ell_{\mathrm{GB}}$, the new
sample weights $w$ become

$w=\frac{\partial\ell_{\mathrm{GB}}}{\partial(\delta\hat{\varphi}_{-2})}=\frac{\ell_{\mathrm{GB}}}{4\delta\hat{\varphi}_{-2}}.$
(13)

The posteriors obtained by mapping the agnostic posteriors for
$\delta\hat{\varphi}_{-2}$ to $\ell_{\mathrm{GB}}$ are shown in the top panel
in Fig. 9. We see that the posteriors for results obtained with SEOBHM
(orange), SEOBNSBH (red), and PhenomXPHM (blue) do not significantly differ.
The gap between $\ell_{\mathrm{GB}}\sim 0.1$ and 0 is not due to lack of
support for GR. It is rather an effect of the sampling and reweighting due to
the $\ell_{\mathrm{GB}}^{4}$ dependence of $\delta\hat{\varphi}_{-2}$. Values
of $\ell_{\mathrm{GB}}\lesssim 0.1$ in this case require
$\delta\varphi_{-2}\lesssim 10^{-8}$, which is four orders of magnitude
smaller than the bound found. The limited number of samples is therefore
causing a gap for small values. This is also the reason why the lower end of
the posterior looks so ragged; there is only a few samples that have a
relatively large weight. We would need many more samples to close this gap and
make the posterior smooth for small $\ell_{\mathrm{GB}}$.

The 90% upper bound on the ESGB coupling (indicated with dashed lines in Fig.
9) is $\ell_{\mathrm{GB}}\lesssim 0.67\leavevmode\nobreak\ \rm{M}_{\odot}$ or
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.37$ km (for SEOBHM and SEOBNSBH, for
PhenomXPHM the bound is $\ell_{\mathrm{GB}}\lesssim 0.72\leavevmode\nobreak\
\rm{M}_{\odot}$ or $\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.40$ km). This bound
can be improved by including higher PN order effects in ESGB. In the next
section, we include these effects in a theory-specific inspiral test for ESGB.

Figure 9:  Top: Posteriors for the ESGB coupling $\ell_{\mathrm{GB}}$ obtained
by reweighting the $\delta\hat{\varphi}_{-2}$ posterior. Bottom: Posteriors
for the Einstine-scalar-Gauss Bonnet coupling $\ell_{\mathrm{GB}}$ from the
theory-specific test. The top axes show a slightly different definition of the
ESGB coupling
$\sqrt{\alpha_{\mathrm{GB}}}={\ell_{\mathrm{GB}}}/{(2\pi^{1/4})}$ that is also
commonly used.

###  V.2 Theory-specific test for ESGB

Due to the flexibility of the FTI framework, we can not only use it to do
theory-agnostic inspiral test, but also for theory-specific inspiral tests
Mehta  _et al._ (2023). This is done by explicitly using the corrections to
the frequency-domain phase from a specific theory. This means that we can not
only use the leading-order corrections, but also the higher-order corrections.
For ESGB, we can thus use all corrections up to 1.5PN, as given in Appendix C.
We only add corrections to the dominant $(2,2)$-mode. We can then do Bayesian
inference sampling over the ESGB coupling $\ell_{\mathrm{GB}}$, where we use a
prior uniform in $\ell_{\mathrm{GB}}$. A posteriori, we check that the
requirement $\left(\ell_{\mathrm{GB}}/m_{1}\right)^{4}\ll 1$ holds, which
comes from the small coupling approximation that is made in the derivation of
the coefficients.

The results from the theory-specific ESGB test are shown in the bottom panel
of Fig. 9. Note that for this case the posterior looks much smoother for small
$\ell_{\mathrm{GB}}$ than was the case for the mapping and there is no gap
close to zero. That is because we are now sampling directly on
$\ell_{\mathrm{GB}}$ so there is no problems with under-sampling low values of
the coupling.

The constraint on the ESGB coupling found using SEOBHM (orange) is
$\ell_{\mathrm{GB}}\lesssim 0.51\leavevmode\nobreak\ \rm{M}_{\odot}$ or
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.28$ km, and using SEOBNSBH gives
slightly worse constraints of $\ell_{\mathrm{GB}}\lesssim
0.62\leavevmode\nobreak\ \rm{M}_{\odot}$ or
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.35$ km (red). We notice that the bounds
obtained with the theory-specific test are better than the bounds found from
mapping the agnostic $-1$PN results. We note that in this work, the higher-
order corrections to the phase for testing ESGB theory is incorporated only in
the FTI framework and not in the waveforms for TIGER. This is because the
implementation with FTI is more flexible, and we do not expect the results to
change, as the $-1$PN mapping-based results are consistent.

These constraints on the ESGB coupling are better than any previously obtained
constraints using GWs. From NSBHs, the best constraint so far was
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 1.33$ km for GW200115_042309 Lyu  _et
al._ (2022). The best overall constraint was obtained with GW190814 and was
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.37$ km (assuming it is a BBH; its
secondary lies in the lower mass gap) Lyu  _et al._ (2022). These bounds are
also better than constraints obtained using low-mass x-ray binaries of
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 1.9$ km Yagi (2012) and NS EOS of
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 1.29$ km Saffer and Yagi (2021).
Laboratory tests for ESGB in the weak field limit provide bounds that are at
least 12 orders of magnitude weaker than strong-field tests Elder and Sakstein
(2023).

##  VI Conclusion

In this work we have performed inspiral tests of GR with GW230529, the merger
of two compact objects with masses $3.6_{-1.1}^{+0.8}M_{\odot}$ and
$1.4_{-0.2}^{+0.6}M_{\odot}$. The tests used are parameterized inspiral tests
that modify the phase in the frequency domain by allowing generic
modifications of the PN coefficients.

We used two different versions of the parameterized inspiral test that employ
different waveform approximants: FTI Abbott  _et al._ (2019a); Mehta  _et al._
(2023) using the SEOBNRv4 waveform family Bohé  _et al._ (2017); Cotesta  _et
al._ (2018, 2020); Dietrich  _et al._ (2019); Matas  _et al._ (2020) and TIGER
Agathos  _et al._ (2014); Meidam  _et al._ (2018) using the IMRPhenomX
waveform family Pratten  _et al._ (2021); García-Quirós  _et al._ (2020);
Colleoni  _et al._ (2023). We compared the results of both tests. The various
BBH waveform approximants gave consistent results, all of which are also
consistent with GR, except for the 0PN deviation coefficient. The results are
shown in Fig. 1.

At 0PN, there is a degeneracy between the chirp mass and the deviation
parameter. This means that the chirp mass and the 0PN deviation parameter are
free to move along the correlation given by Eq. (6) without significantly
changing the waveform. For higher mass systems, this degeneracy is broken by
higher PN contributions in the late inspiral and by the merger-ringdown. For
GW230529, however, the merger-ringdown is outside the analyzed frequency band,
so there is a strong correlation between $\mathcal{M}_{c}$ and
$\delta\hat{\varphi}_{0}$ in the results, as can be seen in Fig. 3. We notice
that the posterior is shifted away from GR and towards higher chirp mass.
Since this is also observed in a zero-noise GR injection, it is very likely a
false deviation from GR. The cause of this shift is probably a combination of
the choice of priors, sampling issues due to a sharply peaked likelihood, the
event being single detector, and noise features.

Since the source of GW230529 is most likely an NSBH, we also performed
analyses with waveforms that include tidal effects. The deviation-parameter
posteriors obtained with these waveforms are broader and shifted away from GR
compared to their BBH counterparts. This is due to correlations between the
tidal deformability and the deviation parameters. For GW230529, the tidal
deformabilities are not well constrained and even allow unrealistically high
values Abac  _et al._ (2024), leading to the broad and shifted posteriors of
the $\delta\hat{\varphi}_{i}$. By constraining the tides to more realistic
values, the shift and broadening (mostly) disappear (see Fig. 2). We therefore
use the results from the BBH waveform models when quoting bounds below.

The bounds obtained on the low-PN deviation parameters (except for 0PN) are
some of the best to date. The bound obtained on dipole radiation ($-1$PN
deviation) for GW230529 of $|\delta\hat{\varphi}_{-2}|\lesssim 8\times
10^{-5}$ is an order of magnitude tighter than the combined bounds from GWTC-2
Abbott  _et al._ (2021a) and GWTC-3 Abbott  _et al._ (2021b). The only tighter
bound obtained with GWs is for the binary NS GW170817 Abbott  _et al._
(2019a). The bounds obtained at 0.5PN and 1PN are also relatively tight
compared to previous constraints. Figure 8 compares the bounds for GW230529
with previous constraints from GWs.

The bounds on $-1$PN deviations allow us to constrain scalar-tensor theories
that have their leading-order modification at $-1$PN. In particular, we have
mapped the posteriors obtained for $\delta\hat{\varphi}_{-2}$ to the ESGB
coupling $\ell_{\mathrm{GB}}$ using Eq. (10). This gives us an upper bound on
the ESGB coupling of $\ell_{\mathrm{GB}}\lesssim 0.67\leavevmode\nobreak\
\rm{M}_{\odot}$. We have also performed a theory-specific test for ESGB
gravity by implementing all corrections up to 1.5PN in the FTI framework. This
allows us to sample directly over the ESGB coupling $\ell_{\mathrm{GB}}$ and
obtain a slightly better bound of $\ell_{\mathrm{GB}}\lesssim
0.51\leavevmode\nobreak\ \rm{M}_{\odot}$. Figure 9 shows the posteriors
obtained from both the mapping and the theory-specific test. These constraints
on the ESGB coupling are the best obtained from GW signals to date.

In this work, we found no evidence for deviations from GR for GW230529. We
obtained particularly tight constraints on deviations from GR at low-PN orders
(with the exception of 0PN), demonstrating the importance of signals similar
to GW230529 for tests of GR. We also discussed some challenges that arise when
analyzing signals from low-mass systems such as GW230529, namely tidal effects
and the degeneracy between the chirp mass and the 0PN deviation parameter.
Future, more detailed studies of these effects will hopefully lead to ways to
better account for, and perhaps even mitigate, their systematic biases.

When completing this paper, we became aware of an independent study Gao  _et
al._ (2024) which obtained constraints on Einstein-dilaton-Gauss-Bonnet
gravity using GW230529. They obtain a bound of
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.260$ km when including higher-order
corrections and using IMRPhenomXPHM, which is similar to our bound of
$\sqrt{\alpha_{\mathrm{GB}}}\lesssim 0.28$ km using the theory-specific test
for ESGB with SEOBNRv4HM_ROM.

## Acknowledgments

The authors would like to thank N.V. Krishnendu for reviewing the manuscript
and providing useful feedback. We also thank Aditya Vijaykumar for his work on
developing and maintaining the Bilby TGR package. S.R. was supported by the
research program of the Netherlands Organization for Scientific Research
(NWO). The authors are grateful for computational resources provided by the
LIGO Laboratory and supported by National Science Foundation Grants
PHY-0757058 and PHY-0823459. The authors are also grateful for the
computational resources provided by Cardiff University supported by STFC grant
ST/I006285/1, as well as computational resources from the Max Planck Institute
for Gravitational Physics in Potsdam. This research has made use of data or
software obtained from the Gravitational Wave Open Science Center
(https://www.gwosc.org), a service of the LIGO Scientific Collaboration, the
Virgo Collaboration, and KAGRA. This material is based upon work supported by
NSF’s LIGO Laboratory which is a major facility fully funded by the National
Science Foundation, as well as the Science and Technology Facilities Council
(STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO and
construction and operation of the GEO600 detector. Additional support for
Advanced LIGO was provided by the Australian Research Council. Virgo is
funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education,
Culture, Sports, Science and Technology (MEXT), Japan Society for the
Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and
Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National
Science and Technology Council (NSTC) in Taiwan.

##  Appendix A Priors used

The priors used in the analyses of GW230529 are given in the tables below. The
names of the priors refer to the names in the Bilby package Ashton  _et al._
(2019). The numbers indicate the range used. The priors are chosen to be
agnostic, which means most of them are uniform in the parameter sampled or
uniform in volume (e.g. for angles). The priors used for the deviation
parameters and chirp masses are listed in Table 3. The increase in width of
the chirp-mass priors for low-PN deviations is because the chirp-mass
posterior becomes broader due to correlations with the deviation parameters at
low-PN order. The priors used for the spins are listed in Table 4 and differ
between different waveform approximants. This is due to differing spin
descriptions (aligned spins versus precessing spins) and due to limitations in
the validity of the models. Lastly, all other priors are listed in Table 5.

Deviation parameter | Prior | Chirp mass prior
---|---|---
$\varphi_{-2}$ | $\mathtt{Uniform}(-0.1,0.1)$ | $\mathtt{UniformInComponentsChirpMass}(1.9,2.2)$
$\varphi_{0}$ |  $\mathtt{Uniform}(-5,5)$333The prior was limited to $\mathtt{Uniform}(-0.5,5)$ for PhenomXPT due to problems with waveform wrap-around like described in Appendix B. | $\mathtt{UniformInComponentsChirpMass}(1.8,6.0)$
$\varphi_{1}$ | $\mathtt{Uniform}(-2,2)$ | $\mathtt{UniformInComponentsChirpMass}(1.9,2.2)$
$\varphi_{2}$ | $\mathtt{Uniform}(-2,2)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{3}$ | $\mathtt{Uniform}(-2,2)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{4}$ | $\mathtt{Uniform}(-20,20)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{5l}$ | $\mathtt{Uniform}(-20,20)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{6}$ | $\mathtt{Uniform}(-20,20)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{6l}$ | $\mathtt{Uniform}(-30,30)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
$\varphi_{7}$ | $\mathtt{Uniform}(-30,30)$ | $\mathtt{UniformInComponentsChirpMass}(2.00,2.05)$
Table 3: The priors used for the deviation parameter and the chirp mass for the different runs. The priors on the deviation parameter are uniform between the indicated range. The priors on the chirp mass are uniform in the component masses and between the indicated range. Waveform approximant | Parameter | Prior
---|---|---
SEOBHM, SEOB, PhenomXHM | $\chi_{1}$ | $\mathtt{AlignedSpin}(0,0.99)$
$\chi_{2}$ | $\mathtt{AlignedSpin}(0,0.99)$
SEOBNSBH | $\chi_{1}$ | $\mathtt{AlignedSpin}(0,0.9)$
$\chi_{2}$ | $\mathtt{AlignedSpin}(0,0.05)$
PhenomXPHM | $a_{1}$ | $\mathtt{Uniform}(0,0.99)$
$a_{2}$ | $\mathtt{Uniform}(0,0.99)$
$\theta_{1}$ | $\mathtt{Sine}(0,\pi)$
$\theta_{2}$ | $\mathtt{Sine}(0,\pi)$
$\phi_{12}$ | $\mathtt{Uniform}(0,2\pi)$
$\phi_{jl}$ | $\mathtt{Uniform}(0,2\pi)$
PhenomXP, PhenomXPT | $a_{1}$ | $\mathtt{Uniform}(0,0.99)$
$a_{2}$ | $\mathtt{Uniform}(0,0.05)$
$\theta_{1}$ | $\mathtt{Sine}(0,\pi)$
$\theta_{2}$ | $\mathtt{Sine}(0,\pi)$
$\phi_{12}$ | $\mathtt{Uniform}(0,2\pi)$
$\phi_{jl}$ | $\mathtt{Uniform}(0,2\pi)$
Table 4: The priors on the spin parameters used for the different waveform approximants. The AlignedSpin prior gives the prior distribution of the aligned spin component based on the generic spin priors. Parameter | Prior
---|---
$q$ | $\mathtt{UniformInComponentsMassRatio}(0.125,1)$
$d_{L}$ | $\mathtt{UniformSourceFrame}(1,1000)$
$\delta$ | $\mathtt{Cosine}(-\pi/2,\pi/2)$
$\alpha$ | $\mathtt{Uniform}(0,2\pi)$
$\theta_{jn}$ | $\mathtt{Sine}(0,\pi)$
$\psi$ | $\mathtt{Uniform}(0,\pi)$
$\phi_{\mathrm{ref}}$ | $\mathtt{Uniform}(0,2\pi)$
$t_{c}$ | $\mathtt{Uniform}(1369419318.6460938,1369419318.8460937)$
$\Lambda_{1,2}$ | $\mathtt{Uniform}(0,5000)$
Table 5: The priors used for the other GR parameters for all runs and
waveforms. The prior on the mass ratio is such that it is uniform in the
component masses. The prior on the geocentric time is a window of 0.2 seconds
around the merger time. The tidal deformabilities are set to zero for BBH
waveforms, and $\Lambda_{1}=0$ for SEOBNSBH.

##  Appendix B False violations of GR when using too wide priors

Large values of the deviation parameters can lead to significant changes in
the length of the waveform. For more extreme cases, this can lead to the
waveform wrapping around due to the choice of segment length of the data
analyzed. In the case of GW230529, this problem can arise for, e.g., the 0.5PN
deviation coefficient, where, for extreme negative values of
$\delta\hat{\varphi}_{1}$, the post-trigger duration becomes more than the 2
seconds used in the analysis. When using wider priors on the deviation
parameters, this can lead to false deviations of GR. For example, the
posteriors for the 0.5PN deviation parameter would be peaking around
$\delta\hat{\varphi}_{1}\sim-9$. Normally, these unphysical waveforms that
lead to wraparound would be penalized enough by having a low likelihood so as
not to get any support in the posterior. Because GW230529 is a single-detector
event, we suspect it instead picked up on some noise feature or artifact at
the edge of the data segment analyzed, which lead to a high likelihood. When
increasing the post-trigger duration to include a longer segment of data after
the signal ends, this false violation of GR disappears. For the results
presented in this work, we instead restricted the priors on the deviation
parameters to avoid the unphysical waveforms that lead to wraparound.

##  Appendix C Higher-order ESGB corrections in the GW phase

The corrections to the frequency-domain phase in ESGB can be computed using
the PN formalism. The leading order corrections appear at $-1$PN (compared to
GR), and the phase corrections have been fully computed up to 1PN (2PN
relative order) for nonspinning binaries Sennett  _et al._ (2016); Julié and
Berti (2019); Shiralilou  _et al._ (2021, 2022); Lyu  _et al._ (2022); Julié
_et al._ (2022). We calculated as well the complete 1.5PN corrections (14e),
which are new to this paper, using the recent fluxes at relative 2.5PN order
of Ref. Bernard  _et al._ (2022). In the formalism described in Sec. II, the
corrections are given by

$\displaystyle\delta\hat{\varphi}_{-2}$
$\displaystyle=-5\ell_{\mathrm{GB}}^{4}\frac{\left(m_{1}^{2}s_{2}-m_{2}^{2}s_{1}\right)^{2}}{168m_{1}^{4}m_{2}^{4}},$
(14a) $\displaystyle\delta\hat{\varphi}_{0}$
$\displaystyle=-5\ell_{\mathrm{GB}}^{4}\frac{659m_{1}^{4}s_{2}^{2}+1370m_{1}^{2}m_{2}^{2}s_{1}s_{2}+659m_{2}^{4}s_{1}^{2}+728\eta\left(m_{1}^{2}s_{2}-m_{2}^{2}s_{1}\right)^{2}}{16128m_{1}^{4}m_{2}^{4}},$
(14b) $\displaystyle\delta\hat{\varphi}_{1}$
$\displaystyle=25\pi\ell_{\mathrm{GB}}^{4}\frac{\left(m_{1}^{2}s_{2}-m_{2}^{2}s_{1}\right)^{2}}{56m_{1}^{4}m_{2}^{4}},$
(14c)
$\displaystyle\begin{split}\delta\hat{\varphi}_{2}&=\ell_{\mathrm{GB}}^{4}\left\\{\frac{5m_{1}^{4}s_{2}^{2}\left[-13792267+5588352\delta-17640\eta(743+594\eta)\right]}{290304m_{1}^{4}m_{2}^{4}(743+924\eta)}-\frac{5m_{2}^{4}s_{1}^{2}\left[13792267+5588352\delta+17640\eta(743+594\eta)\right]}{290304m_{1}^{4}m_{2}^{4}(743+924\eta)}\right.\\\
&+\left.\frac{2m_{1}^{2}m_{2}^{2}s_{1}s_{2}\left[56018615+3528\eta(12239+14850\eta)\right]}{290304m_{1}^{4}m_{2}^{4}(743+924\eta)}\right\\},\end{split}$
(14d) $\displaystyle\delta\hat{\varphi}_{3}$
$\displaystyle=\ell_{\mathrm{GB}}^{4}\frac{m_{1}^{4}s_{2}^{2}(-14363+1792\delta-4564\eta)+2m_{1}^{2}m_{2}^{2}s_{1}s_{2}(-3557+4564\eta)-m_{2}^{4}s_{1}^{2}(14363+1792\delta+4564\eta)}{43008m_{1}^{4}m_{2}^{4}}.$
(14e)

Here, $\ell_{\mathrm{GB}}$ is the ESGB coupling, $m_{1},m_{2}$ are the masses
of the two objects which also appear in the combinations
$\eta=m_{1}m_{2}/(m_{1}+m_{2})^{2}$ and $\delta=(m_{1}-m_{2})/(m_{1}+m_{2})$.
For NSs, $s=0$ Lyu  _et al._ (2022) and for nonspinning BHs, $s=1$ Julié and
Berti (2019); Julié  _et al._ (2022).

The ESGB corrections to the phase above are complete up to 1.5PN for
nonspinning binaries. By contrast, Ref. Gao  _et al._ (2024) uses the
corrections presented in Ref. Lyu  _et al._ (2022), which are complete up to
1PN only. Indeed, at 1.5PN, the corrections of Ref. Lyu  _et al._ (2022)
depend on the free parameter $f^{\rm GB}_{3}$ to represent the contributions
from the fluxes at relative 2.5PN order, which were unknown at the time.
Similarly, the 2PN corrections of Ref. Lyu  _et al._ (2022) depend on $f^{\rm
GB}_{4}$ to represent contributions from the binding energy at 3PN, and from
the (still unknown) fluxes at relative 3PN order. Note that in practice, both
parameters are set to zero in Ref. Lyu  _et al._ (2022).

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