$\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{2.5}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{2.5}{GW230529ay_{i}mrphen_{n}sbh}{2.8}{GW230529ay_{i}mrseobv4_{n}sbh}{2.5}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{2.3}}$–$\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{4.5}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{4.3}{GW230529ay_{i}mrphen_{n}sbh}{4.8}{GW230529ay_{i}mrseobv4_{n}sbh}{4.8}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{4.3}}~{}M_{\odot}$
compact object and a compact object consistent with neutron star masses. The
more massive component in the merger provides evidence that compact objects in
the hypothesized lower mass gap exist in merging binaries. Based on mass
estimates of the two components in the merger and current constraints on the
supranuclear EoS, we find the most probable interpretation of the GW230529
source to be a NSBH coalescence. In this scenario, the source binary of
GW230529 is the most symmetric-mass NSBH merger yet observed, and the primary
component of the merger is the lowest-mass primary black hole (BH) observed in
GWs to date. Because NSBHs with more symmetric masses are more susceptible to
tidal disruption, the observation of GW230529 implies that more NSBHs than
previously inferred may produce EM counterparts. However, we cannot rule out
the contrasting scenario that the source of GW230529 consisted of two neutron
stars rather than a neutron star and a black hole. In this case, the source of
GW230529 would be the only BNS coalescence observed to have strong support for
non-zero and anti-aligned spins, as well as the highest-mass BNSs system
observed to date. Regardless of the true nature of the GW230529 source, it is
a novel addition to the growing population of CBCs observed via their GW
emission and highlights the importance of continued exploration of the CBC
parameter space in O4 and beyond.

Strain data containing the signal from the LIGO Livingston observatory is
available from the Gravitational Wave Open Science
Center.111https://doi.org/10.7935/6k89-7q62 Specifically, we release the
L1:GDS-CALIB_STRAIN_CLEAN_AR channel, where the textttAR designation means the
strain data is analysis ready; this strain channel was also released at the
end of O3. Samples from the posterior distributions of the source parameters,
hyperposterior distributions from population analyses, and notebooks for
reproducing all results and figures in this paper are available on Zenodo
(Abbott et al., 2024a). The software packages used in our analyses are open-
source.

This material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. The authors
also gratefully acknowledge the support of 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 GEO 600 detector.
Additional support for Advanced LIGO was provided by the Australian Research
Council. The authors gratefully acknowledge the Italian Istituto Nazionale di
Fisica Nucleare (INFN), the French Centre National de la Recherche
Scientifique (CNRS) and the Netherlands Organization for Scientific Research
(NWO), for the construction and operation of the Virgo detector and the
creation and support of the EGO consortium. The authors also gratefully
acknowledge research support from these agencies as well as by the Council of
Scientific and Industrial Research of India, the Department of Science and
Technology, India, the Science & Engineering Research Board (SERB), India, the
Ministry of Human Resource Development, India, the Spanish Agencia Estatal de
Investigación (AEI), the Spanish Ministerio de Ciencia, Innovación y
Universidades, the European Union NextGenerationEU/PRTR (PRTR-C17.I1), the
Comunitat Autonòma de les Illes Balears through the Direcció General de
Recerca, Innovació i Transformació Digital with funds from the Tourist Stay
Tax Law ITS 2017-006, the Conselleria d’Economia, Hisenda i Innovació, the
FEDER Operational Program 2021-2027 of the Balearic Islands, the Conselleria
d’Innovació, Universitats, Ciència i Societat Digital de la Generalitat
Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the
National Science Centre of Poland and the European Union – European Regional
Development Fund; Foundation for Polish Science (FNP), the Polish Ministry of
Science and Higher Education, the Swiss National Science Foundation (SNSF),
the Russian Science Foundation, the European Commission, the European Social
Funds (ESF), the European Regional Development Funds (ERDF), the Royal
Society, the Scottish Funding Council, the Scottish Universities Physics
Alliance, the Hungarian Scientific Research Fund (OTKA), the French Lyon
Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique
(FRS-FNRS), Actions de Recherche Concertées (ARC) and Fonds Wetenschappelijk
Onderzoek – Vlaanderen (FWO), Belgium, the Paris Île-de-France Region, the
National Research, Development and Innovation Office Hungary (NKFIH), the
National Research Foundation of Korea, the Natural Science and Engineering
Research Council Canada, Canadian Foundation for Innovation (CFI), the
Brazilian Ministry of Science, Technology, and Innovations, the International
Center for Theoretical Physics South American Institute for Fundamental
Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National
Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research
Corporation, the National Science and Technology Council (NSTC), Taiwan, the
United States Department of Energy, and the Kavli Foundation. The authors
gratefully acknowledge the support of the NSF, STFC, INFN and CNRS for
provision of computational resources. This work was supported by MEXT, JSPS
Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially
Promoted Research 26000005, JSPS Grant-in-Aid for Scientific Research on
Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Core-to-
Core Program A. Advanced Research Networks, JSPS Grant-in-Aid for Scientific
Research (S) 17H06133 and 20H05639 , JSPS Grant-in-Aid for Transformative
Research Areas (A) 20A203: JP20H05854, the joint research program of the
Institute for Cosmic Ray Research, University of Tokyo, National Research
Foundation (NRF), Computing Infrastructure Project of Global Science
experimental Data hub Center (GSDC) at KISTI, Korea Astronomy and Space
Science Institute (KASI), and Ministry of Science and ICT (MSIT) in Korea,
Academia Sinica (AS), AS Grid Center (ASGC) and the National Science and
Technology Council (NSTC) in Taiwan under grants including the Rising Star
Program and Science Vanguard Research Program, Advanced Technology Center
(ATC) of NAOJ, and Mechanical Engineering Center of KEK.

##  Appendix A Upgrades to the Detector Network for O4

The Advanced LIGO, Advanced Virgo, and KAGRA detectors have all undergone a
series of upgrades to improve the network sensitivity in preparation for O4.
In this section, we focus on upgrades made to the LIGO Livingston observatory
as it was the only detector to observe GW230529; similar upgrades were made at
LIGO Hanford. Some of these improvements are a part of the Advanced LIGO Plus
(A+) detector upgrades (Abbott et al., 2020b).

The principal commissioning work done in preparation for O4 was to enhance the
optics systems in the detector. The pre-stabilized laser was redesigned for O4
with amplifiers allowing input power up to 125 W (Bode et al., 2020; Cahillane
& Mansell, 2022). LIGO Livingston operated at 63 W in O4a. Both end test
masses at LIGO Livingston were replaced because their mirror coatings
contained small, point-like absorbers that limited detector sensitivity by
degrading the power-recycling gain (Brooks et al., 2021). For O4, frequency-
dependent squeezing was implemented with the addition of a 300 m filter cavity
to rotate the squeezed vacuum quadrature across the bandwidth of the detector
(Dwyer et al., 2022; McCuller et al., 2020). With frequency-dependent
squeezing, uncertainty is reduced in both the amplitude and phase quadratures,
allowing for a broadband reduction in quantum noise (Ganapathy et al., 2023).
Squeezing was implemented independently of frequency in O3, which only reduced
the quantum noise at high frequencies (Tse et al., 2019).

Other commissioning work was done to improve data quality and reduce noise at
low frequencies. For O4, scattered light noise was mitigated by removing some
problematic scattering surfaces, and improving the resonant damping of other
scatterers (Soni et al., 2020; Davis et al., 2021). Low-frequency technical
noise (i.e., noise that is not intrinsic to the detector design, but can limit
the detector sensitivity and performance) was reduced with the commissioning
of feedback control loops, noise subtraction, and better electronics. Overall,
the upgrades made during the commissioning period for O4 led to a broadband
reduction in noise and improvement in detector sensitivity and performance.

##  Appendix B Detection Timeline and Circulars

The LVK issues low-latency alerts to facilitate prompt follow-up of GW
candidates (Chaudhary et al., 2023). GW230529 was initially identified in a
low-latency search using data from the LIGO Livingston observatory. The
candidate was given the name S230529ay in the Gravitational-Wave Candidate
Event Database (GraceDB).222https://gracedb.ligo.org/

After the detection of GW230529, an initial notice was sent out to astronomers
through NASA’s General Coordinates Network (GCN) (LIGO Scientific, Virgo, and
KAGRA Collaboration, 2023a). The sky localization computed in low latency by
BAYESTAR (Singer & Price, 2016; Singer et al., 2016) had a $90\%$ credible
area of $\approx 24200~{}{\rm{deg}^{2}}$; the large credible area was due to
GW230529 only being observed by a single detector. The initial alert also
included low-latency estimates of the mass-based source classification (BNS,
BBH, and NSBH) produced by the PyCBC pipeline (Villa-Ortega et al., 2022).
Additional estimates that the source binary of GW230529 contained at least one
neutron star, contained at least one compact object in the lower mass gap
$3\text{--}5~{}M_{\odot}$, and had neutron-star matter ejected outside the
final compact object were produced by a machine learning method trained to
infer source properties from the search pipeline results (Chatterjee et al.,
2020). The estimated probability that the source binary of GW230529 included
at least one neutron star was $>$ 99.9$\%$.

Low-latency parameter estimation was performed with Bilby (Ashton et al.,
2019; Romero-Shaw et al., 2020) and used to produce an updated sky
localization and estimates of source properties sent out in another alert
(LIGO Scientific, Virgo, and KAGRA Collaboration, 2023b). The updated sky
localization had a $90\%$ credible area of $\approx 25600~{}{\rm{deg}^{2}}$.
The updated estimate of source properties led to a slight decrease in the
probability that the source binary included at least one neutron star, which
was updated to 98.5$\%$.

##  Appendix C Quantifying Significance for a Single-Detector Candidate Event

Each search pipeline has its own method for calculating the FAR of a single-
detector GW candidate. In this section we will summarize these methods.

###  C.1 GstLAL

Figure 9: The noise background for the LIGO Livingston observatory collected
by the GstLAL pipeline in offline mode, with the S/N–$\xi^{2}$ values of
GW230529 overlaid. The test statistic $\xi^{2}$ measures signal consistency.
The background is collected from templates that are part of the same bin as
the template that recovered GW230529, and is consequently the background used
to rank GW230529 and calculate its significance. The background distribution
has effectively no support at the position of GW230529. The colorbar
represents the probability of producing a certain S/N and $\xi^{2}$ from noise
triggers.

The GstLAL pipeline assigns a likelihood ratio to all of its candidate signals
as the ranking statistic to estimate their significance. The likelihood ratio
is the ratio of the probability of obtaining the candidate under the signal
hypothesis to the probability of obtaining the candidate under the noise
hypothesis. The latter probability is largely calculated by collecting S/N and
$\xi^{2}$ statistics of noise triggers during the analysis, where $\xi^{2}$ is
a signal consistency test statistic (Messick et al., 2017). Similar templates
in the template bank are grouped together on the basis of the post-Newtonian
(PN) expansion of their waveforms (Morisaki & Raymond, 2020; Sakon et al.,
2024). Each template bin collects its own S/N and $\xi^{2}$ statistics, which
get used to rank candidates recovered by templates in that bin. Further
details about the calculation of the probability of obtaining the candidate
under the noise hypothesis, as well as the GstLAL likelihood ratio in general
can be found in Tsukada et al. (2023). The S/N–$\xi^{2}$ statistics collected
from noise triggers from the same template bin as GW230529 with GW230529
superimposed are shown in Figure 9. GW230529 clearly stands out from the
background, and there are no noise triggers at its position in S/N–$\xi^{2}$
space.

Since non-stationary noise transients known as glitches (Nuttall, 2018) are
not expected to be correlated across detectors, single-detector GW candidates,
whether during times when a single or multiple detectors are observing, are
more likely to be glitches as compared to coincident GW candidates. To account
for this, the GstLAL pipeline downweights the logarithm of the likelihood
ratio of single-detector candidates by subtracting an empirically-tuned factor
of 13, which is optimized to maximize the recovery of candidates with an
astrophysical origin while minimizing the recovery of glitches (Abbott et al.,
2020a). The FAR of the GstLAL pipeline for GW230529 quoted in Table 1 is
calculated after the application of this penalty.

Finally, the FAR for a candidate in the search is computed by comparing its
likelihood ratio to the likelihood ratios of noise triggers not found in
coincidence, after accounting for the live-time of the analysis. These noise
triggers not found in coincidence allow the pipeline to extrapolate the
background distribution of likelihood ratios to large values, enabling the
inverse FAR of a candidate to be greater than the duration of the analysis.
While in the low-latency mode, the GstLAL FAR calculation uses the background
from the start of the analysis period until the time of the candidate. In
contrast, for its offline analysis, the GstLAL pipeline uses background
collected from the full analysis period including the entirety of O4a to rank
candidates. This differs from the other search pipelines presented in this
work which only use background from the first two weeks of O4a for their
offline search results presented here. The GstLAL offline analysis was
performed using the same template bank as the online analysis (Sakon et al.,
2024; Ewing et al., 2023). Details are liable to change for future offline
GstLAL analyses in O4a. However, for such a significant candidate as GW230529,
we do not expect these changes to impact its interpretation as highly likely
being of astrophysical origin.

###  C.2 MBTA

Figure 10: Ranking statistic distribution of MBTA single-detector triggers in
the LIGO Livingston observatory during the first two weeks of O4, excluding
significant public alerts aside from GW230529.

The MBTA single-detector candidate search for O4a focuses on a subset of the
full MBTA parameter space. The goal is to focus on BNS and NSBH signals, which
are most interesting because of their rarity and possible EM counterparts.
Their long signal duration also makes them easier to identify and allows for a
better sensitivity to be reached when compared to a single-detector candidate
search using the whole MBTA parameter space. The parameter space for the O4a
online single-detector search was defined based on the detector-frame
(redshifted) primary mass and chirp mass of the templates and motivated by the
computation of the probability for whether a non-zero amount of neutron star
material remained outside the final remnant compact object (Chatterjee et al.,
2020) introduced in Appendix B. The considered space is
$(1+z)\,m_{1}<50~{}M_{\odot}$ and $(1+z)\,\mathcal{M}<5~{}M_{\odot}$ (Juste,
2023). For the offline analysis, this space was adapted and we consider only
$(1+z)\,\mathcal{M}<7~{}M_{\odot}$. MBTA online also applied selection
criteria based on data-quality tests to its single-detector candidates (Juste,
2023). This was changed to a reweighting of the ranking statistic for the
offline analysis.

The ranking statistic for MBTA single-detector triggers is a reweighted S/N.
The reweighting is based on the computation of a quantity called
$\texttt{auto}~{}\chi^{2}$ (Aubin et al., 2021), which tests the consistency
of the time evolution of the S/N time series. The additional reweighting used
in the offline analysis relies on the computation of a quantity that
identifies an excess of S/N for single-detector triggers. The excess of S/N is
larger for triggers that have a noise origin compared to astrophysical signals
or injections, and therefore allows for discrimination between astrophysical
signals and noise. The ranking statistic distribution of the MBTA offline
analysis for single-detector triggers during the first two weeks of O4a is
shown in Figure 10. Other triggers which produced significant public alerts
have been removed from the plotted distribution. GW230529 stands out from the
background with a high ranking statistic that reflects the
$\texttt{auto}~{}\chi^{2}$ value being completely consistent with a signal
origin.

The FAR in MBTA is a function of $p_{\mathrm{astro}}$ (Andres et al., 2022),
the probability of astrophysical origin of the candidate. It is derived from
the combined parametrizations of the FAR as a function of the ranking
statistic and of $p_{\mathrm{astro}}$ as a function of the ranking statistic.
Inverting the latter gives the ranking statistic as a function of
$p_{\mathrm{astro}}$ and eventually the FAR as a function of
$p_{\mathrm{astro}}$. The background estimation for MBTA single-detector
triggers was computed differently during the online and offline analyses. The
online analysis relies on the computation of simulated single-detector
triggers obtained through random combinations of single frequency-band data
(Juste, 2023). O4a online analysis and the method of randomly combining
single-frequency band data showed that the background for MBTA single-detector
triggers follows an exponential distribution and is stable in time beyond
statistical fluctuations. This prompted us to update the model we use for the
offline (and future) analyses to reach greater sensitivity. This new model
involves extrapolating the observed distribution of single-detector triggers
and removing the significant triggers. A safety margin is used in the
extrapolation such that the FAR is over-estimated relative to the best-fit
extrapolated distribution. This change in methods, in addition to the
difference in handling of single-detector triggers online and offline,
explains the difference in inverse FAR for the online ($1.1~{}\text{yr}$) and
offline ($>$1000 yr) analyses.

###  C.3 PyCBC

Figure 11: Ranking statistic distribution of PyCBC offline single-detector
triggers in the LIGO Livingston observatory during the first two weeks of O4.
The plotted distribution may include triggers from other less-significant
signals arriving during the period analyzed.

In PyCBC, each GW candidate is assigned a ranking statistic, and then a FAR is
calculated by comparing the ranking statistic of the candidate to the
background distribution. The details of this procedure differ between the
online and offline versions of the PyCBC search.

In the online pipeline, we consider single-detector candidates only from
templates with duration greater than $7~{}\mathrm{s}$ above a starting
frequency of $17~{}\mathrm{Hz}$, corresponding to a range of masses and spins
for which EM emission due to neutron star ejecta might be expected. Low-
latency data quality timeseries produced by iDQ, a machine learning framework
for autonomous detection of noise artifacts using only auxiliary data channels
insensitive to GWs (Essick et al., 2020a), are used to veto candidates at
times when a glitch is likely to be present in the data. The remaining single-
detector candidates are ranked by the reweighted S/N described in Nitz (2018).
Only candidates with a $\chi^{2}$ statistic (Allen, 2005) $<2.0$ and
reweighted S/N $>6.75$ are kept. The rate density of single-detector triggers
above the reweighted S/N threshold is fit with a decreasing exponential.

In offline PyCBC, we only consider single-detector candidates from templates
with duration greater than $0.3~{}\mathrm{s}$ above a starting frequency of
$15~{}\mathrm{Hz}$. We also require that candidates have a $\chi^{2}$
statistic $<10.0$, a reweighted S/N $>5.5$, and a PSD variation statistic
(Mozzon et al., 2020) $<10.0$, in order to exclude high-amplitude noise
transients. Each candidate is assigned a ranking statistic equal to the
logarithm of the ratio of astrophysical signal likelihood to detector noise
likelihood. The PyCBC O4 offline search introduced two changes to the ranking
statistic relative to the O3 calculation. First, an explicit model of the
signal distribution covering the space of binary masses and spins is included
(Kumar & Dent, 2024). The model is designed to maximize the number of detected
signals from the known compact binary distribution, while also maintaining
sensitivity to signals in previously unpopulated regions. Second, the model of
the rate density of events caused by detector noise now includes a term
describing the variation in rate during times of heightened detector noise
(Davis et al., 2022). The rate density of single-detector candidates above a
ranking statistic threshold of $0$ is fit with a decreasing exponential.

In both the online and offline versions of PyCBC, the exponential fit of
trigger rate density above the relevant ranking statistic threshold is used to
extrapolate the FAR of single-detector candidates beyond the observing time of
the search (Cabourn Davies & Harry, 2022). The FAR calculations for GW230529
used triggers from the first two weeks of O4 at the LIGO Livingston
observatory. The distribution of ranking statistics for offline single-
detector candidates at the LIGO Livingston observatory is shown in Figure 11.
Similar to the other searches, GW230529 clearly stands out from the background
distribution.

##  Appendix D Priors, Waveform Systematics, and Bayes Factors

Table 4: Summary of parameter-estimation analysis choices for GW230529, and
the physical content of each waveform model used. Here, tides refers to
modeling of neutron star tidal deformability and disruption when tidal forces
overcome the self-gravity of the neutron star. The spin prior denotes any
restrictions on the spin magnitude of the binary components.

Waveform Model | Precession | Higher Multipoles | Tides | Disruption | Spin Prior
---|---|---|---|---|---
IMRPhenomNSBH | $-$ | $-$ | ✓ | ✓ | $\chi_{1}<0.50,\chi_{2}<0.05$
IMRPhenomPv2_NRTidalv2 | $\checkmark$ | $-$ | ✓ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.05$
IMRPhenomXPHM | $\checkmark$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.99$
SEOBNRv5PHM | $\checkmark$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.99$
SEOBNRv4_ROM_NRTidalv2_NSBH | $-$ | $-$ | ✓ | ✓ | $\chi_{1}<0.90,\chi_{2}<0.05$
IMRPhenomXPHM | $\checkmark$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.05$
IMRPhenomXP | $\checkmark$ | $-$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.99$
IMRPhenomXHM | $-$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.99$
IMRPhenomXAS | $-$ | $-$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.99$
IMRPhenomXAS | $-$ | $-$ | $-$ | $-$ | $\chi_{1}<0.50,\chi_{2}<0.05$
IMRPhenomPv2_NRTidalv2 | $\checkmark$ | $-$ | $\checkmark$ | $-$ | $\chi_{1}<0.05,\chi_{2}<0.05$
IMRPhenomXPHM | $\checkmark$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.05,\chi_{2}<0.05$
SEOBNRv5PHM | $\checkmark$ | $\checkmark$ | $-$ | $-$ | $\chi_{1}<0.99,\chi_{2}<0.05$

Given the uncertain nature of the compact objects, we analyze GW230529 with a
suite of models that incorporate a number of key physical effects. The choices
of data duration ($128~{}\mathrm{s}$) and frequency bandwidth
($20\text{--}1792~{}\mathrm{Hz}$) analyzed were informed by comparing
waveforms spanning the mass range recovered in preliminary analyses to the
detector noise PSD at the time of the event. The high-frequency cutoff is
chosen to avoid loss of power at high frequencies due to low-pass filtering of
the data. For a subset of the signal models below, we employ a range of
techniques to speed up the evaluation of the likelihood including heterodyning
(also known as relative binning; Cornish 2010; Zackay et al. 2018; Cornish
2021; Krishna et al. 2023), multibanding (García-Quirós et al., 2021;
Morisaki, 2021), and reduced order quadratures (Canizares et al., 2015; Smith
et al., 2016; Morisaki et al., 2023).

The physical effects included and the spin prior ranges for each waveform
model considered in this work are shown in Table 4. All analyses use mass
priors that are flat in the redshifted component masses with chirp masses,
$\mathcal{M}=(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}\in[2.0214,2.0331]~{}M_{\odot}$
and mass ratios $q\in[0.125,1]$. The luminosity distance prior is uniform in
comoving volume and source-frame time on the range
$D_{\mathrm{L}}\in[1,500]~{}\mathrm{Mpc}$. The priors on the tidal
deformability parameters are chosen to be uniform in the component tidal
deformabilities over $\Lambda\in[0,5000]$. Standard priors (e.g., Romero-Shaw
et al., 2020; Abbott et al., 2024b) are used for all the other extrinsic
binary parameters. Below we provide further motivation for considering this
suite of waveform models.

The primary analysis in this work uses the IMRPhenomXPHM and SEOBNRv5PHM BBH
waveform models (corresponding to the third and fourth rows in Table 4), which
provide an accurate description of BBHs but do not incorporate tidal effects
or the potential tidal disruption of a companion neutron star. We did not
consider an extension of IMRPhenomXPHM that incorporates additional physics
beyond the two precessing higher-order multipole moment BBH models used here
as these effects only enter at high frequencies and are not relevant for this
event (Thompson et al., 2024). In order to quantify the impact of neglecting
tidal physics, we analyze GW230529 with NSBH and BNS waveform models that
incorporate different tidal information. The NSBH models only incorporate
tidal information from the less massive compact object, are restricted to
spins aligned with the orbital angular momentum, and only model the dominant
$\ell=m=2$ multipole moment. However, they do model the possible tidal
disruption of the neutron star, which occurs when tidal forces of the black
hole dominate over the self-gravity of the neutron star.

We use two NSBH models, a frequency-domain phenomenological model
IMRPhenomNSBH (Thompson et al., 2020a), and a frequency-domain effective-one-
body surrogate model SEOBNRv4_ROM_NRTidalv2_NSBH (Matas et al., 2020).
IMRPhenomNSBH uses the BBH models IMRPhenomC (Santamaria et al., 2010) and
IMRPhenomD (Husa et al., 2016; Khan et al., 2016) as baselines for the
amplitude and phase, respectively, incorporating corrections to the phase due
to tidal effects following the NRTidal model (Dietrich et al., 2017, 2019b,
2019a) and to the amplitude following Pannarale et al. (2013, 2015).
SEOBNRv4_ROM_NRTidalv2_NSBH uses the SEOBNRv4 BBH model as a baseline (Bohé et
al., 2017) and applies the same corrections to account for tidal deformability
and disruption as IMRPhenomNSBH but is additionally calibrated against
numerical relativity (NR) simulations of NSBH mergers (Foucart et al., 2013,
2014, 2019; Kyutoku et al., 2010, 2011). As the NSBH waveform models do not
capture spin-induced orbital precession, we analyze GW230529 with the aligned-
spin BBH waveform models IMRPhenomXAS (Pratten et al., 2020), which only
contains the dominant harmonic, and IMRPhenomXHM (García-Quirós et al., 2020),
which includes higher-order multipole moments. Models for NSBH binaries that
contain both higher-order multipole moments and precession are under active
development (Gonzalez et al., 2023) and could potentially allow for a more
consistent model selection between the different source categories.

Given that the mass of the primary overlaps with current estimates of the
maximum known neutron star mass (Landry & Read, 2021; Romani et al., 2022), we
also analyze GW230529 with a BNS waveform model that allows for tidal
interactions on both the primary and the secondary components of the binary.
We use IMRPhenomPv2_NRTidalv2 (Dietrich et al., 2019a), which adds a model for
the tidal phase that is calibrated against both effective-one-body (EOB) and
NR simulations to the underlying precessing-spin point-particle waveform model
IMRPhenomPv2 (Hannam et al., 2014). A limitation is that
IMRPhenomPv2_NRTidalv2 is calibrated against a suite of equal-mass, non-
spinning BNS NR simulations, where the maximum neutron star mass is only
$1.372~{}M_{\odot}$. Nevertheless, the model has been validated against a
catalog of EOB–NR hybrid waveforms that includes asymmetric configurations and
heavier neutron star masses (Dietrich et al., 2019a; Abac et al., 2024).
Because of large uncertainties in BNS postmerger waveform modeling,
IMRPhenomPv2_NRTidalv2 includes an amplitude taper from $1\text{--}1.2$ times
the estimated merger frequency (Dietrich et al., 2019b). The resulting
suppression of S/N at these frequencies may be responsible for the posterior
differences between IMRPhenomPv2_NRTidalv2 and the BBH and NSBH waveform
models we consider (see Figure 12).

From the mass estimates alone, the secondary object is consistent with being a
neutron star. As such, we follow earlier analyses and analyze GW230529 with
two spin priors (Abbott et al., 2021a): an agnostic high-spin prior and an
astrophysically motivated low-spin prior. The high-spin prior assumes that the
spins on both objects are isotropically oriented and have dimensionless spin
magnitudes that are uniformly distributed up to $\chi_{1},\chi_{2}\leq 0.99$.
The low-spin prior, on the other hand, is inspired by the extrapolated maximum
spin observed in Galactic BNSs that will merge within a Hubble time (Burgay et
al., 2003; Stovall et al., 2018), and restricts the spin magnitude of the
secondary to be $\chi_{2}\leq 0.05$ while keeping $\chi_{1}\leq 0.99$. As the
nature of the primary as a black hole or neutron star is uncertain, we also
use a third choice of prior where both spins are restricted to
$\chi_{1},\chi_{2}\leq 0.05$ for the IMRPhenomPv2_NRTidalv2 and IMRPhenomXPHM
waveform models (rows 11 and 12 of Table 4, respectively). The purpose of
using multiple priors is to help gauge whether astrophysical assumptions on
the neutron star spin impact any of the statements on the probability that the
primary object lies in the lower mass gap. In Figure 12, we show the impact of
waveform systematics and prior assumptions on the strongly correlated mass
ratio–effective inspiral spin distribution.

The degree to which a given waveform model under certain assumptions matches
the data can be gauged using the Bayes factor. However, Bayes factors penalize
extra degrees of freedom in models that do not improve the fit when those
extra degrees of freedom are constrained by the data (Mackay, 2003). We find
no significant preference ($\log_{10}{\mathcal{B}}\lesssim 0.5$) between the
high-spin and low-spin prior on the secondary; we similarly do not find a
statistical preference for or against the effects of precession, higher-order
multipole moments, or tidal deformation or disruption of the secondary object.

Figure 12:  The two-dimensional $q$–$\chi_{\rm eff}$ posterior probability
distributions for GW230529 using various BBH, NSBH, and BNS signal models. The
vertical dotted lines indicate primary masses that have been mapped to the
mass ratio given the median chirp mass estimated from the IMRPhenomXPHM high-
spin analysis.

##  Appendix E Additional source properties

###  E.1 Component Spins and Precession

Assuming a high-spin prior, the posteriors for the primary spin magnitude are
only weakly informative, disfavoring zero and extremal spins with
$\chi_{1}=\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.07}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.10}{GW230529ay_{i}mrphen_{n}sbh}{0.00}{GW230529ay_{i}mrseobv4_{n}sbh}{0.01}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.09}}\text{--}\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.85}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.84}{GW230529ay_{i}mrphen_{n}sbh}{0.32}{GW230529ay_{i}mrseobv4_{n}sbh}{0.41}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.86}}$.
For the secondary, the posteriors are even less informative. Under the low-
spin prior, the primary spin magnitude posterior is peaked at slightly higher
values with zero and extremal spins being disfavored to a larger degree,
$\chi_{1}=\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.07}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.10}{GW230529ay_{i}mrphen_{n}sbh}{0.00}{GW230529ay_{i}mrseobv4_{n}sbh}{0.01}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.09}}\text{--}\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.85}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.84}{GW230529ay_{i}mrphen_{n}sbh}{0.32}{GW230529ay_{i}mrseobv4_{n}sbh}{0.41}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.86}}$.
The secondary spin is completely uninformative over the restricted range of
spin magnitudes. The joint two-dimensional posterior probability distribution
of the dimensionless spin magnitude and the spin tilt are shown in Figure 13.
Regions of high (low) probability are denoted by a darker (lighter) shade.

When the spins are misaligned with the orbital angular momentum, relativistic
spin-orbit and spin-spin couplings drive the evolution of the orbital plane
and the spins themselves (Apostolatos et al., 1994; Kidder, 1995). The
leading-order effect can be captured by an effective precession spin
parameter, $0\leq\chi_{\rm p}\leq 1$, which approximately measures the degree
of in-plane spin and can be used to parameterize the rate of precession of the
orbital plane (Schmidt et al., 2015)

$\displaystyle\chi_{p}$
$\displaystyle=\max\left[\chi_{1}\,\sin\theta_{1},\left(\frac{3+4q}{4+3q}\right)\,q\,\chi_{2}\,\sin\theta_{2}\right].$
(E1)

We find the constraints on $\chi_{\mathrm{p}}$ to be uninformative, and we are
not able to make any significant statements on precession. The uninformative
nature of these results is corroborated by the Bayes factor between the
precessing and non-precessing phenomenological waveform models,
$\log_{10}\mathcal{B}^{\mathrm{XP}}_{\mathrm{XAS}}=0.22$.

Figure 13:  Two-dimensional posterior probability distributions for the spin
magnitude $\chi_{i}$ and the spin-tilt angle $\theta_{i}$ for the primary
(left) and secondary (right) compact objects at a reference frequency of
$20~{}\mathrm{Hz}$. A spin-tilt angle of $0^{\circ}$ ($180^{\circ}$) indicates
a spin that is perfectly aligned (anti-aligned) with the orbital angular
momentum $\hat{L}$. The pixels have equal prior probability, and shading
denotes the posterior probability of each pixel of the high-spin prior
analysis, after marginalizing over azimuthal angles. The solid (dashed)
contours denote the $90\%$ credible region for the high-spin (low-spin) prior
analyses respectively. The probability distributions are marginalized over the
azimuthal spin angles.

###  E.2 Source Location and Distance

As GW230529 was a single-detector observation, the sky localization is poor
and covers a sky area of $\approx 24100~{}{\rm{deg}^{2}}$ at the $90\%$
credible level. The luminosity distance is inferred to be
$D_{\mathrm{L}}=\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{201}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{197}{GW230529ay_{i}mrphen_{n}sbh}{202}{GW230529ay_{i}mrseobv4_{n}sbh}{194}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{196}}^{+\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{102}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{108}{GW230529ay_{i}mrphen_{n}sbh}{110}{GW230529ay_{i}mrseobv4_{n}sbh}{109}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{111}}}_{-\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{96}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{96}{GW230529ay_{i}mrphen_{n}sbh}{101}{GW230529ay_{i}mrseobv4_{n}sbh}{101}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{98}}}~{}\mathrm{Mpc}$,
corresponding to a redshift of
$z=\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.04}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.04}{GW230529ay_{i}mrphen_{n}sbh}{0.04}{GW230529ay_{i}mrseobv4_{n}sbh}{0.04}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.04}}^{+\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.02}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.02}{GW230529ay_{i}mrphen_{n}sbh}{0.02}{GW230529ay_{i}mrseobv4_{n}sbh}{0.02}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.02}}}_{-\IfEqCase{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{{GW230529ay_{c}ombined_{i}mrphm_{h}igh_{s}pin}{0.02}{GW230529ay_{c}ombined_{i}mrphm_{l}ow_{s}pin}{0.02}{GW230529ay_{i}mrphen_{n}sbh}{0.02}{GW230529ay_{i}mrseobv4_{n}sbh}{0.02}{GW230529ay_{i}mrpv2_{l}ow_{s}pin}{0.02}}}$
computed using the Planck 2015 cosmological parameters (Ade et al., 2016). The
luminosity distance has a degeneracy with the inclination angle of the
binary’s total angular momentum with respect to the line of sight,
$\theta_{\rm JN}$ (e.g., Cutler & Flanagan, 1994; Nissanke et al., 2010; Aasi
et al., 2013; Vitale & Chen, 2018). The posterior distribution on
$\theta_{\mathrm{JN}}$ is broadly unconstrained, showing no preference for a
total angular momentum vector that is pointed towards or away from the line of
sight.

###  E.3 Tidal Deformability

The tidal deformability of a neutron star is defined by

$\displaystyle\lambda$ $\displaystyle=\frac{2}{3}k_{2}R^{5},$ (E2)

where $k_{2}$ is the gravitational Love number of the object and $R$ is its
radius (Damour et al., 1992; Mora & Will, 2004; Flanagan & Hinderer, 2008;
Damour & Nagar, 2009). However, it is often convenient to introduce a
dimensionless tidal deformability

$\displaystyle\Lambda$
$\displaystyle=\frac{\lambda}{m^{5}}=\frac{2}{3}k_{2}C^{-5},$ (E3)

where $C=m/R$ is the compactness of the neutron star in geometrized units.

Using the IMRPhenomPv2_NRTidalv2 model, which allows for tidal deformability
on both objects but does not model tidal disruption, we infer $\Lambda_{1}\leq
1462$ at 90% credibility with the $\chi_{1}<0.99,\chi_{2}<0.05$ spin prior;
the posterior peaks at $\Lambda_{1}=0$. We do not constrain $\Lambda_{2}$
relative to the prior. Constraints on the tidal deformability $\Lambda$ of
both components of GW230529 are shown in Figure 14, along with the tidal
deformabilities predicted for the secondary object under the BSK24 neutron
star EoS (Goriely et al., 2013; Pearson et al., 2018; Perot et al., 2019) and
using the Gaussian process (GP)-EoS constraints introduced in Section 6. BSK24
is chosen as an illustrative EoS that is thermodynamically consistent and
falls within the range of support of constraints from nuclear physics and
astrophysics, including GW170817. The uninformative $\Lambda_{2}$ posterior is
consistent with both of these predictions. In addition to the tidal
deformability posteriors, we use two distinct likelihood interpolation schemes
(Ray et al., 2023a; Landry & Essick, 2019) to directly constrain the neutron
star EoS using both spectral (Lindblom, 2010; Lindblom & Indik, 2012, 2014)
and GP (Landry & Essick, 2019; Essick et al., 2020b) representations. Both
methods incorporate consistency with the observed heavy pulsars PSR J0740+6620
(Fonseca et al., 2021) and PSR J0348+0432 (Antoniadis et al., 2013) as priors
on the EoS and enforce thermodynamic stability and causality. Unlike other
analyses where the GP representations are conditioned on both GW and pulsar
data, here we only include the pulsar constraints in the prior, in order to
distinguish the EoS information provided by GW230529 alone. These direct EoS
inferences are also uninformative, returning their respective priors.

Figure 14: Constraints on dimensionless tidal deformability $\Lambda$ for the
primary (green dashed) and secondary (green solid) components of GW230529. We
also show tidal deformabilities predicted for the secondary object under the
BSK24 neutron star EoS (navy; Goriely et al. 2013; Pearson et al. 2018; Perot
et al. 2019) and constraints obtained from BNS and pulsar observations using
the GP model (GP-EoS; red).

##  Appendix F Testing General Relativity

To verify whether GW230529 is consistent with general relativity, we perform
parameterized tests searching for deviations in the PN coefficients that
determine the phase evolution of the GW signal (Blanchet & Sathyaprakash,
1994, 1995; Arun et al., 2006a, b; Yunes & Pretorius, 2009; Mishra et al.,
2010; Li et al., 2012a, b). Specifically, we search for parametric deviations
to the GW inspiral phasing applied to the frequency-domain waveform models
IMRPhenomXP_NRTidalv2 (Colleoni et al., 2023) and IMRPhenomXPHM (Pratten et
al., 2020; García-Quirós et al., 2020; Pratten et al., 2021) using the TIGER
framework (Agathos et al., 2014; Meidam et al., 2018), and
SEOBNRv4_ROM_NRTidalv2_NSBH (Matas et al., 2020) and SEOBNRv4HM_ROM (Bohé et
al., 2017; Cotesta et al., 2018, 2020) using the FTI framework (Mehta et al.,
2023a). These waveform models each capture different physical effects
(precession, higher-order multipole moments, and neutron star tidal
deformability) to determine whether their absence in the model leads to
inferred inconsistencies with general relativity.

For all waveform models and PN orders whose analyses have been completed, we
find that GW230529 is consistent with general relativity within the inferred
uncertainties on the deviation parameters. We do not yet include results about
0PN deviations, which are being examined separately due to technical issues
related to the degeneracy between the 0PN deviation parameter and the chirp
mass (Payne et al., 2023). The constraints obtained at $-1$PN are an order of
magnitude tighter than previously reported bounds for NSBH and BBH (Abbott et
al., 2021b). The previously reported bounds using GW170817 remain the tightest
constraints on $-1$PN deviations obtained with GWs (Abbott et al., 2019c).
Further details about the tests of general relativity performed will be
presented in a follow-up paper.

##  Appendix G Compact Binary Merger Rates Methods

A key component of the event-based rate estimation described in Section 5 is
the sensitive time–volume of our GW searches. The sensitivities to GW200105-,
GW200115-, and GW230529-like events are computed from O1 through the first two
weeks of O4a according to the corresponding mass and spin posteriors from the
IMRPhenomXPHM high-spin analyses of these signals. The posterior samples
chosen for this analysis are consistent with previous event-based rate
estimates presented for NSBH mergers (Abbott et al., 2021a). Simulated
signals, whose binary parameters are drawn from the mass and spin posteriors
for each signal, are distributed uniformly in comoving volume and following
the other extrinsic parameter distributions given in Appendix D, and are added
to simulated detector noise characterized by representative PSDs for each
observing run. The detectability of these injections is then calculated semi-
analytically to determine the sensitive time–volume by calculating network
responses to the simulated signals and applying a threshold on the network
optimal $\mathrm{S}/\mathrm{N}$ of $>10$. This choice of threshold was
previously tuned to the results of matched filter searches (Abbott et al.,
2021c), and is comparable to threshold statistics calculated for semi-analytic
sensitivity estimates (Essick, 2023).

For the population-based approach, we estimate the merger rate of three
astrophysical populations (BBH, BNS, and NSBH) by aggregating triggers found
by GstLAL from O1 through the first two weeks of O4, while accounting for the
possibility that some of these triggers are of terrestrial origin. Our
population analyses do not, however, include data from the engineering run
preceding the start of O4, during which another NSBH candidate was identified
(LIGO Scientific, Virgo, and KAGRA Collaboration, 2023c). As outlined and
implemented in previous LVK results (Abbott et al., 2021a, 2023b), we
construct the joint likelihood on the Poisson parameters of the astrophysical
populations
($\Lambda_{\mathrm{BBH}},\Lambda_{\mathrm{BNS}},\Lambda_{\mathrm{NSBH}}$) and
of the terrestrial triggers ($\Lambda_{\mathrm{background}}$; Farr et al.,
2015; Kapadia et al., 2020). We then extrapolate the sensitive time–volume to
each astrophysical population ($\langle VT\rangle_{\mathrm{BBH}},\langle
VT\rangle_{\mathrm{BNS}},\langle VT\rangle_{\mathrm{NSBH}}$) obtained at the
end of O3 (Abbott et al., 2023b) to account for the additional two weeks of
observation time during O4 during which GW230529 was detected. Using a uniform
prior on the merger rates $\mathcal{R}_{\alpha}=\Lambda_{\alpha}/\langle
VT\rangle_{\alpha}$, we infer their posterior distributions. The three
astrophysical populations are defined by dividing up the space of compact
binary component masses and spins into disjoint regions. We consider all
components with masses between 1 and 3 $M_{\odot}$ to be neutron stars, and
assume a maximum dimensionless spin of 0.05 for such components; all
components that are more massive are assumed to be black holes with a maximum
dimensionless spin of 0.99. The distribution of component masses within each
region is assumed to be a power law in primary mass with index $-2.35$, and
uniform in secondary mass.

Even though we infer the posterior distributions of the merger rates of all
three populations, we only present $\mathcal{R}_{\mathrm{NSBH}}$ given that
GW230529 contributes negligibly to $\mathcal{R}_{\mathrm{BBH}}$ and
$\mathcal{R}_{\mathrm{BNS}}$.

##  Appendix H Additional Details of Population Analyses

We use three different models to analyze the population of compact-object
binaries with and without GW230529.

###  H.1 Binned Gaussian Process

Table 5: Summary of Binned Gaussian Process model parameters (Ray et al.,
2023b; Mandel et al., 2017; Mohite, 2022). Arguments in the priors specify the
mean and standard deviation in a normal (N) or half-normal (HN) distribution.

Parameter | Description | Prior
---|---|---
$\mu$ | Mean $\log\left(\displaystyle\mathrm{Rate}\over\displaystyle\text{Gpc}^{-3}\,\text{yr}^{-1}\,M_{\odot}^{-2}\right)$ in each bin | $\mathrm{N}(0,10)$
$\sigma$ | Amplitude of the covariance kernel | $\mathrm{HN}(0,10)$
$\mathrm{log}(l)$ |  $\log\left(\displaystyle\mathrm{Length\ scale}\over\displaystyle\log(M/M_{\odot})\right)$ of the covariance kernel | $\mathrm{N}(-0.085,0.93)$

The Binned Gaussian Process method models the merger rate density per log
component masses as a piecewise constant function. By inferring the merger
rate density in each bin, we reconstruct the shape of the CBC mass spectrum up
to the resolution limit imposed by our choice of binning. A GP prior with an
exponential quadratic kernel is imposed on the logarithmic rate densities to
smooth out the inferred shapes over sparse regions of the parameter space. To
assess the impact of GW230529 on the shape of the CBC mass spectrum, we use
the same bin locations and priors on the GP hyper-paramters as the GWTC-3
analysis (Abbott et al., 2023b). The means ($\mu$), correlation length ($l$),
and covariance amplitude ($\sigma$) of the GP are drawn from normal, log-
normal, and half-normal priors, respectively. Further details of these priors
are summarized in Table 5.

Unlike more recent implementations of the Binned Gaussian Process model which
also fit for the redshift distribution (Ray et al., 2023b), we assume a
redshift distribution such that the overall merger rate of compact binaries is
uniform in comoving volume and source-frame time, so as to facilitate a direct
comparison with the findings of the GWTC-3 analysis (Abbott et al., 2023b) and
to avoid any systematic biases in Binned Gaussian Process-based redshift
distribution models originating from the inclusion of low-mass events, which
remains an active area of study. As in previous analyses (Abbott et al.,
2023b; Ray et al., 2023b), we fix the spin distributions for each component to
be isotropic in direction and uniform in spin magnitude.

###  H.2 Power law + Dip + Break

The Power law + Dip + Break model is designed to search for a separation in
masses between neutron stars and black holes by employing a broken power law
with a dip at the location of the power-law break. The dip is modeled by a
notch filter with depth $A$, which is fit along with other model parameters in
order to determine the existence and depth of a potential mass gap (Farah et
al., 2022). A value $A=0$ corresponds to no gap, whereas $A=1$ corresponds to
zero merger rate over the interval of the gap, i.e., a maximally deep gap.
Power law + Dip + Break also employs a low-pass filter at high black hole
masses to allow for a tapering of the mass spectrum, which has the effect of
adding a smooth second break to the power law.

The component mass distributions in this model are both fit by the same broken
power law with exponents $\alpha_{1}$ between $m_{\rm min}$ and
$M^{\mathrm{gap}}_{\rm low}$ and $\alpha_{2}$ between $M^{\mathrm{gap}}_{\rm
low}$ and $m_{\rm max}$. The model additionally includes a power-law pairing
function in mass ratio (Fishbach & Holz, 2020), assumed to be the same for all
component masses. The parameters for this mass model are summarized in Table
6. Like the Binned Gaussian Process model, the Power law + Dip + Break model
additionally assumes that CBCs are uniformly distributed in comoving volume
and source-frame time and that component spins are isotropically oriented and
uniformly distributed in magnitude. These assumptions are made for simplicity
and consistency with GWTC-3 analyses (Abbott et al., 2023b). Components with
$m<2.5~{}M_{\odot}$ are limited to spin magnitudes $<0.4$ while components
with $m>2.5~{}M_{\odot}$ can have spin magnitudes up to 0.99.

Table 6: Summary of Power law + Dip + Break model parameters (Farah et al.,
2022). Arguments in the priors column specify the lower and upper bounds of a
uniform (U) distribution. The first several entries describe the mass-
distribution parameters, and the last two entries describe the spin-
distribution parameters.

Parameter | Description | Prior
---|---|---
$\alpha_{1}$ | Spectral index for the power law of the mass distribution below $M^{\mathrm{gap}}_{\rm low}$ | U($-8$, $2$)
$\alpha_{2}$ | Spectral index for the power law of the mass distribution above $M^{\mathrm{gap}}_{\rm low}$ | U($-3$, $2$)
$A$ | Lower mass gap depth | U($0,1$)
$M^{\mathrm{gap}}_{\rm low}$ [$M_{\odot}$] | Location of the lower end of the mass gap | U($1.4$, $3$)
$M^{\mathrm{gap}}_{\rm high}$ [$M_{\odot}$] | Location of the upper end of the mass gap | U($3.4$, $9$)
$\eta_{\rm low}$ | Parameter controlling how the rate tapers at the low end of the mass gap | 50
$\eta_{\rm high}$ | Parameter controlling how the rate tapers at the high end of the mass gap | 50
$\eta$ | Parameter controlling tapering the power law at high black hole mass | U($-4$, $12$)
$\beta$ | Spectral index for the power law-in-mass-ratio pairing function | U($-2$, $7$)
$m_{\mathrm{min}}$ [$M_{\odot}$] | Minimum mass of the mass distribution | U($1\,$, $1.4$)
$m_{\mathrm{max}}$ [$M_{\odot}$] | Maximum mass of the mass distribution | U($35$, $100$)
$\chi_{\mathrm{max,NS}}$ | Maximum allowed component spin for objects with mass $<2.5M_{\odot}$ | $0.4$
$\chi_{\mathrm{max,BH}}$ | Maximum allowed component spin for objects with mass $\geq 2.5M_{\odot}$ | $0.99$

###  H.3 NSBH-pop

The NSBH-pop model introduced in Biscoveanu et al. (2022) is designed to
capture the mass and spin distributions of the NSBH population. The black hole
mass distribution is fit by a truncated power law, and the conditional mass
ratio distribution, $p(q|m_{1})$, is fit by a truncated Gaussian between
$q_{\min}=1~{}M_{\odot}/m_{1}$ and
$q_{\max}=\min(m_{\mathrm{NS},\max}/m_{\mathrm{1}},1)$. The black hole spin
magnitude distribution is modeled as a Beta distribution (including singular
distributions that peak at the edges of the $\chi_{\mathrm{1}}\in[0,0.99]$
space), while the neutron star spin magnitude distribution is assumed to be
uniform over $\chi_{\mathrm{2}}\in[0,0.7]$ to encapsulate the effect of the
neutron star breakup spin at the mass-shedding limit (Shao et al., 2020; Most
et al., 2020a). We assume the redshift distribution is uniform in comoving
volume and source-frame time and that spins are isotropically oriented. The
parameters for this model are summarized in Table 7.

###  H.4 Population reweighting

We use the population-level mass and spin distributions inferred using all
three population models to reweight (e.g., Payne et al. 2019) the high-spin
combined parameter estimation samples from the default priors described in
Appendix D to three different astrophysically-motivated priors given by the
one-left-out posterior predictive distributions (Galaudage et al., 2020;
Callister, 2021; Essick & Fishbach, 2021) inferred for each model. A
comparison of the posteriors under the default and astrophysically-motivated
priors is shown in Figure 7.

Because of the $q-\chi_{\mathrm{eff}}$ degeneracy, the spin distribution
assumptions also impact the inferred component masses, and hence
classification, for this source. However, the fact that the Binned Gaussian
Process and NSBH-pop model priors recover similar mass posteriors indicates
that the different spin distribution assumptions have a subdominant effect
compared to the different mass distribution assumptions.

###  H.5 Selection effects

For our population analyses, we account for search selection bias to measure
the underlying astrophysical mass and spin distributions of the NSBH and CBC
populations (Loredo, 2004; Mandel et al., 2019; Farr, 2019; Vitale et al.,
2020). To estimate the sensitivity of the searches over the data recorded by
the detector network across the binary parameter space, we use a large suite
of injections and impose a significance threshold on the FAR. In contrast to
the two methods for calculating merger rates described in Appendix G, we use
the same set of injections used for GWTC-3 population studies recovered using
matched-filter searches (Abbott et al., 2023b). By using the combined
injection sets from the first three observing runs (Abbott et al., 2021c), we
are effectively assuming that GW230529 occurred at the end of O3. Without
accounting for the extra time–volume provided by the first two weeks of O4,
this introduces a bias in our inferred estimates of the merger rate in the
mass gap (Figure 5), rate of gamma-ray bursts with NSBH progenitors, and total
contribution of NSBH mergers to the production of heavy elements (Section 7).
However this effect is negligible given that GW230529 occurred in the first
two weeks of O4a and the detector sensitivity during this period was not
drastically different from O3 (see Section 2), meaning that the additional
time–volume unaccounted for in the injection sets is small.

To generate samples from the hierarchical likelihood for our population
analyses we use the Dynesty nested sampler (Speagle, 2020) as implemented in
GWPopulation (Talbot et al., 2019) for the Power law + Dip + Break and NSBH-
pop analyses and the Hamiltonian Monte Carlo sampler implemented in the PyMC
package (Salvatier et al., 2016) for the Binned Gaussian Process analysis.

Table 7: Summary of NSBH-pop model parameters (Biscoveanu et al., 2022).
Arguments in the priors column specify the lower and upper bounds of a uniform
(U) distribution.

Parameter | Description | Prior
---|---|---
$\alpha$ | Black hole mass power-law index | U$(-4,12)$
$m_{\mathrm{BH},\min}$ [$M_{\odot}$] | Minimum black hole mass | U$(2,10)$
$m_{\mathrm{BH},\max}$ [$M_{\odot}$] | Maximum black hole mass | U$(8,20)$
$m_{\mathrm{NS},\max}$ [$M_{\odot}$] | Maximum neutron star mass | U$(1.97,2.7)$
$\mu$ | Mass ratio mean | U$(0.1,0.6)$
$\sigma$ | Mass ratio standard deviation | U$(0.1,1)$
$\alpha_{\chi}$ | Beta distribution shape parameter ($\alpha$) for black hole spin distribution | U$(0.1,10)$
$\beta_{\chi}$ | Beta distribution shape parameter ($\beta$) for black hole spin distribution | U$(0.1,10)$

##  Appendix I Equation of State

To assess the effect of the EoS constraint used for source classification
(Section 6) and calculation of $f_{\mathrm{EM\text{-}bright}}$ (Section 7), we
repeat these analyses using GP-EoS constraints additionally conditioned on
NICER observations (Legred et al., 2021, 2022) of J0030+0451 (Miller et al.,
2019; Riley et al., 2019; Vinciguerra et al., 2024) and J0740+6620 (Miller et
al., 2021; Riley et al., 2021; Salmi et al., 2022, 2023).

We find that including the NICER observations only changes the source
classification results by at most a few percent. This is because the
constraint on the maximum neutron star mass used in the source classification
analysis is primarily driven by the observation of massive pulsars. Changing
the information included within the EoS constraint leads to a smaller effect
than the choice of astrophysical mass and spin distribution, as discussed in
Section 6.

The inference on the fraction of NSBH systems that may have EM counterparts
changes more significantly when NICER observations are included in the EoS
constraint. This is because the GW and pulsar-only results provide much more
support for compact neutron stars, which inhibits the probability that the
neutron star is disrupted and suppresses the probability of an EM counterpart.
The constraints including the NICER observations favor stiffer EoSs, enhancing
the inferred EM-bright fraction,
$f_{\mathrm{EM\text{-}bright}}=0.13^{+0.19}_{-0.11}$, and pulling the
posterior peak away from zero.

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  *[O4]: fourth observing run
  *[GW]: gravitational-wave
  *[CBC]: compact binary coalescence
  *[LVK]: LIGO Scientific, Virgo, and KAGRA Collaboration
  *[O3]: third observing run
  *[EM]: electromagnetic
  *[CBCs]: compact binary coalescence
  *[GWs]: gravitational-wave
  *[NSBHs]: neutron star–black hole
  *[NSBH]: neutron star–black hole
  *[BNS]: binary neutron star
  *[S/N]: signal-to-noise ratio
  *[FAR]: false alarm rate
  *[FARs]: false alarm rate
  *[O4a]: first half of the fourth observing run
  *[PSD]: power spectral density
  *[BBH]: binary black hole
  *[BNSs]: binary neutron star
  *[EoS]: equation of state
  *[O1]: first observing run
  *[BBHs]: binary black hole
  *[GP]: Gaussian process
  *[EoSs]: equation of state
  *[BH]: black hole
  *[GCN]: General Coordinates Network
  *[PN]: post-Newtonian
  *[NR]: numerical relativity
  *[EOB]: effective-one-body
  *[PSDs]: power spectral density