# Search for neutrino counterparts to the gravitational wave sources from
LIGO/Virgo O3 run with the ANTARES detector

A. Albert  S. Alves  M. André  M. Ardid  S. Ardid  J.-J. Aubert  J. Aublin  B.
Baret  S. Basa  Y. Becherini  B. Belhorma  M. Bendahman  F. Benfenati  V.
Bertin  S. Biagi  M. Bissinger  J. Boumaaza  M. Bouta  M.C. Bouwhuis  H.
Brânzaş  R. Bruijn  J. Brunner  J. Busto  B. Caiffi  D. Calvo  S. Campion  A.
Capone  L. Caramete  F. Carenini  J. Carr  V. Carretero  S. Celli  L. Cerisy
M. Chabab  T. N. Chau  R. Cherkaoui El Moursli  T. Chiarusi  M. Circella
J.A.B. Coelho  A. Coleiro  R. Coniglione  P. Coyle  A. Creusot  A.S.M. Cruz
A. F. Díaz  B. De Martino  C. Distefano  I. Di Palma  A. Domi  C. Donzaud  D.
Dornic  D. Drouhin  T. Eberl  T. van Eeden  D. van Eijk  S. El Hedri  N. El
Khayati  A. Enzenhöfer  P. Fermani  G. Ferrara  F. Filippini  L. Fusco  S.
Gagliardini  J. García  C. Gatius Oliver  P. Gay  N. Geißelbrecht  H. Glotin
R. Gozzini  R. Gracia Ruiz  K. Graf  C. Guidi  L. Haegel  S. Hallmann  H. van
Haren  A.J. Heijboer  Y. Hello  J.J. Hernández-Rey  J. Hößl  J. Hofestädt  F.
Huang  G. Illuminati  C. W. James  B. Jisse-Jung  M. de Jong  P. de Jong  M.
Kadler  O. Kalekin  U. Katz  A. Kouchner  I. Kreykenbohm  V. Kulikovskiy  R.
Lahmann  M. Lamoureux  A. Lazo  D. Lefèvre  E. Leonora  G. Levi  S. Le Stum
D. Lopez-Coto  S. Loucatos  L. Maderer  J. Manczak  M. Marcelin  A. Margiotta
A. Marinelli  J.A. Martínez-Mora  P. Migliozzi  A. Moussa  R. Muller  L. Nauta
S. Navas  E. Nezri  B. Ó Fearraigh  A. Păun  G.E. Păvălaş  M. Perrin-Terrin
V. Pestel  P. Piattelli  C. Poirè  V. Popa  T. Pradier  N. Randazzo  D. Real
S. Reck  G. Riccobene  A. Romanov  A. Sánchez-Losa  A. Saina  F. Salesa Greus
D. F. E. Samtleben  M. Sanguineti  P. Sapienza  J. Schnabel  J. Schumann  F.
Schüssler  J. Seneca  M. Spurio  Th. Stolarczyk  M. Taiuti  Y. Tayalati  S.J.
Tingay  B. Vallage  G. Vannoye  V. Van Elewyck  S. Viola  D. Vivolo  J. Wilms
S. Zavatarelli  A. Zegarelli  J.D. Zornoza  J. Zúñiga

###### Abstract

Since 2015 the LIGO and Virgo interferometers have detected gravitational
waves from almost one hundred coalescences of compact objects (black holes and
neutron stars). This article presents the results of a search performed with
data from the ANTARES telescope to identify neutrino counterparts to the
gravitational wave sources detected during the third LIGO/Virgo observing run
and reported in the catalogues GWTC-2, GWTC-2.1, and GWTC-3. This search is
sensitive to all-sky neutrinos of all flavours and of energies
$>$100\text{\,}\mathrm{GeV}$$, thanks to the inclusion of both track-like
events (mainly induced by $\nu_{\mu}$ charged-current interactions) and
shower-like events (induced by other interaction types). Neutrinos are
selected if they are detected within $\pm$500\text{\,}\mathrm{s}$$ from the GW
merger and with a reconstructed direction compatible with its sky
localisation. No significant excess is found for any of the 80 analysed GW
events, and upper limits on the neutrino emission are derived. Using the
information from the GW catalogues and assuming isotropic emission, upper
limits on the total energy $E_{\rm tot,\nu}$ emitted as neutrinos of all
flavours and on the ratio $f_{\nu}=E_{\rm tot,\nu}/E_{\rm GW}$ between
neutrino and GW emissions are also computed. Finally, a stacked analysis of
all the 72 binary black hole mergers (respectively the 7 neutron star - black
hole merger candidates) has been performed to constrain the typical neutrino
emission within this population, leading to the limits: $E_{\rm
tot,\nu}<$4.0\text{\times}{10}^{53}\text{\,}\mathrm{erg}$$ and $f_{\nu}<0.15$
(respectively, $E_{\rm
tot,\nu}<$3.2\text{\times}{10}^{53}\text{\,}\mathrm{erg}$$ and $f_{\nu}<0.88$)
for $E^{-2}$ spectrum and isotropic emission. Other assumptions including
softer spectra and non-isotropic scenarios have also been tested.

##  1 Introduction

Since the first detection of gravitational waves (GWs) from compact binary
mergers in 2015 [1], GW interferometers have opened a new window on the
Universe, complementary to the ones already being explored with other cosmic
messengers (cosmic rays, photons, neutrinos). These capabilities have already
allowed the association of the GW signal GW170817 emitted by the merger of a
binary neutron star system with the emission of a short gamma-ray burst (GRB),
GRB 170817A, detected by Fermi and INTEGRAL in gamma rays, and the related
afterglow across a wider range of the electromagnetic spectrum [2].

Neutrinos are also expected to be emitted from the relativistic outflows that
characterise such mergers: see e.g., [3] for binary neutron star (BNS)
mergers, [4] for neutron star-black hole (NSBH) mergers, and [5] for binary
black hole (BBH) mergers. Previous searches performed with ANTARES [6],
Baikal-GVD [7], IceCube [8], and Super-Kamiokande [9] have not been able to
identify an excess of neutrinos and upper limits have been reported.

This article presents an updated search using the latest GW catalogues
covering detections in 2019-2020 and the ANTARES data from the same period.
Besides the follow-up of individual events, performed in the same way as in
previous publications, first population studies are also presented by carrying
out a stacking analysis for binary mergers of the same nature and taking
advantage of the extensive catalogue reported by the GW community.

###  1.1 The GW catalogues

This paper focuses on GW sources detected during the third observing run (O3)
of the LIGO and Virgo detectors and reported in the following official
LIGO/Virgo catalogues:

  * •

GWTC-2 [10]: this catalogue reports detections made during the first half of
O3 (April - September 2019). It contains 39 candidates, including 1 BNS, 2
NSBH, and 36 BBH events.

  * •

GWTC-2.1 [11]: this is an update of GWTC-2, with 8 additional events not
reported in the previous catalogue and that have a high probability of
astrophysical origin. The catalogue also includes a larger selection of $\sim
1.2$ k events with a false alarm rate of less than $2$ per day but lower
astrophysical probability, which are not considered in this analysis.

  * •

GWTC-3 [12]: this catalogue covers the second half of O3 (November 2019 -
March 2020) and contains 35 objects, including 4 NSBH candidates. Seven
marginal candidates that do not fully satisfy the criteria for an
astrophysical origin are also reported, of which only GW200105_162426 is kept
as it has also been reported independently as a plausible NSBH candidate [13].

For each of these objects, the LIGO-Virgo collaboration provides a FITS file
[14] with the timing of the merger $t_{\rm GW}$ and the constraints on the
source direction $\Omega$ as a skymap $\mathcal{P}(\Omega)$, as well as
posterior samples containing all the correlations between source direction
$\Omega$, luminosity distance estimate $D_{L}$, and other source parameters
such as the masses of the two merging objects $m_{1,2}$ (with the convention
$m_{1}>m_{2}$), the energy radiated in gravitational waves $E_{\rm GW}$
(defined as the difference between the estimated mass of the final object and
the sum of the masses of the initial objects), and the inclination between the
total angular momentum and the line-of-sight $\theta_{\rm jn}$. The
classification among the different categories is made based on the mass
estimates: BNS if $m_{2}<m_{1}<$3\text{\,}\mathrm{M_{\odot}}$$, NSBH if
$m_{2}<$3\text{\,}\mathrm{M_{\odot}}$<m_{1}$, BBH otherwise. When considering
the different catalogues, 83 objects are selected, including 1 BNS and 7 NSBH
candidates.

###  1.2 The ANTARES telescope

The ANTARES neutrino telescope [15], located in the depths of the
Mediterranean Sea, offshore from Toulon (France), has been operating in its
final configuration between May 2008 and February 2022. It was composed of an
array of 885 photomultiplier tubes (PMTs) enclosed in pressure-proof glass
spheres, arranged in triplets over 12 vertical lines, spaced by
$\sim$70\text{\,}\mathrm{m}$$ and anchored at a depth of
$\sim$2475\text{\,}\mathrm{m}$$.

The PMTs detect the Cherenkov light induced by relativistic charged particles
originating in the interaction of a neutrino with matter surrounding the
detector; the space and time pattern of PMT signals (or hits) allows the
neutrino properties (direction and energy) to be inferred. Charged-current
interactions of muon (anti-)neutrinos are characterised by the presence of a
long muon track; the typical angular resolution for such events is
$<$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$ for
$E_{\nu}>$100\text{\,}\mathrm{GeV}$$ [16]. Other types of neutrino
interactions only produce hadronic (and, in the case of $\nu_{e}$ charged-
current interactions, electromagnetic) showers, with a more localised light
deposit and compact topology. Despite the shorter lever arm with respect to
the muon tracks, ANTARES still achieved a median angular resolution of a few
degrees for the neutrino direction [17]. In the following, the two event
categories described above are referred to as the track and shower samples
respectively.

ANTARES data were organised in consecutive runs of at most twelve hours. The
present analysis uses data from 2019 and 2020 to identify neutrino
counterparts to the GW emissions described in section 1.1. A larger dataset
including data from January 2018 to December 2020 is used for background
estimation.

Several searches for neutrino counterparts to GW events with ANTARES have been
carried out in the past. These studies were limited to event-by-event follow-
ups and reported null results. See [18] for GW170104, [19] for GW170817, and
[6] for other O2 events.

##  2 Analysis method

This search focuses on the selection of neutrino events in a time window of
$1000\text{\,}\mathrm{s}$ centred on the GW emission time $t_{\rm GW}$, as
motivated in [20], and in the region $\mathcal{R}_{90}$ containing 90% of the
source localisation probability, as built directly from the GW 2D skymaps
$\mathcal{P}(\Omega)$. As the reconstructed event direction does not match the
true neutrino direction perfectly due to the scattering angle and the finite
detector resolution, this region of interest (RoI) is extended by an angle
$\alpha$ to account for these effects, meaning that an event with direction
$\boldsymbol{x}$ would be selected if
$\min_{\boldsymbol{d}\in\mathcal{R}_{90}}(\arccos{\left(\boldsymbol{x}\cdot\boldsymbol{d}\right)})\leq\alpha$.
This extended angle $\alpha$ is a free parameter of the analysis, whose choice
is based on the optimisation method described below.

The ANTARES events are divided into four categories according to whether they
are classified as tracks or showers and whether their reconstructed direction
is upgoing or downgoing, each case based on a specific selection and
optimisation procedure, as described in sections 2.1 and 2.2.

The reconstructed data are largely dominated by atmospheric muons. A selection
is then applied to reduce it to an expected number of events
$B=$2.7\text{\times}{10}^{-3}$$, such that the detection of one event would
correspond to a $3\sigma$ excess ($3\sigma$ condition). This condition is
separately set for each category. For a given GW, the background expectation
is estimated using the dataset from 2018 to 2020. Only runs with similar data-
taking conditions as the ones during the ANTARES run $r_{\rm GW}$ overlapping
with the GW time, characterised by the mean burst fraction111This quantity is
defined by estimating how often, in a given run, the PMT counting rate is more
than 20% higher than the baseline value for this run. This quantity has been
found to be correlated with the detector noise level, whose evolution is
mainly driven by deep-sea bioluminescent emissions [21]., are selected. As
illustrated in the left panel of Figure 1, this procedure is found to allow
for a proper estimate of the background while ensuring a better
characterisation of the tails of the distribution as compared to a
statistically-limited estimation using only the data from the run $r_{\rm
GW}$.

A dedicated Monte Carlo (MC) simulation [22] has been produced to be used in
this optimisation process. Simulations are done on a run-by-run basis, where
each run has a specific simulation to reproduce the particular environmental
and detector conditions during this run. For GW events with precise sky
localisation ($\mathcal{R}_{90}$ region smaller than
$3000\text{\,}\mathrm{{}^{2}}\deg$), additional neutrinos generated solely
within this region have also been produced to accurately estimate the
corresponding detector acceptance for an $E^{-\gamma}$ spectrum ($=\int A_{\rm
eff}(E)E^{-\gamma}{\rm d}E$, where $A_{\rm eff}(E)$ is the effective area).
More details about the detector acceptance for a given neutrino spectrum are
in [23].

The final cuts are optimised to ensure that the expected number of selected
background events after all cuts is fulfilling the $3\sigma$ condition defined
above.

###  2.1 Track event selection

The track selection procedure is adapted from the one presented in [18]. The
upgoing and downgoing (respectively with reconstructed incoming direction
below or above the horizon) event selections are different, as the latter is
more likely to be contaminated by the atmospheric muon background and needs
extra care.

For upgoing tracks, a cut on the track reconstruction quality parameter,
$\Lambda$ [24], is applied. The values of $\alpha$ and of the cut on $\Lambda$
are optimised by ensuring the $3\sigma$ condition described above, as well as
maximising the signal acceptance in the hypothesis of an $E^{-2}$ spectrum.
For downgoing events, the procedure is similar except that an additional cut
on the number of hits employed for the reconstruction is also applied, as
illustrated in the right panel of Figure 1.

(a) Estimation of the background. (b) Optimisation of the cuts.

Figure 1: Illustration of the analysis procedure for the downgoing track
category for GW190521. The left panel shows the expected remaining background
after varying cuts on the track quality parameter $\Lambda$: the blue curve
shows the estimation based on the procedure described in the previous
paragraphs, the red histogram shows the background estimated using only the
ANTARES 12-hour run containing the signal time window with its statistical
uncertainty, and the dotted line is the required background to ensure the
$3\sigma$ condition. The right panel shows the average acceptance, assuming an
$E^{-2}$ neutrino spectrum, as a function of the cut on the number of hits and
the value of $\alpha$. For each bin, the cut on $\Lambda$ is fixed to the
value that satisfies exactly the $3\sigma$ condition. The optimal working
point, maximising the acceptance, is indicated by the red cross.

###  2.2 Shower event selection

The selection steps for neutrino interactions yielding showering events are
similar to the ones presented in [6]. Events must be contained within the
detector and must not be classified as a track by the selection described in
the previous section. The discrimination between neutrinos and atmospheric
muons is achieved thanks to an extended likelihood ratio $\mathcal{L}_{\mu}$
defined by comparing the neutrino and muon hypotheses for each hit associated
with the shower on the basis of its deposited charge, timing, and distance to
the reconstructed shower position [17]. Another parameter is used to further
reduce the background contamination: for upgoing events, this parameter is
defined from a Random Decision Forest (RDF) classifier [25] while the
downgoing selection exploits the number of hits used in the event fitting.

As for the track selection, the values of the cuts on these parameters and on
the extension of the RoI $\alpha$ are optimised to ensure the $3\sigma$
condition and to maximise the acceptance, as illustrated in Figure 2.

(a) Estimation of the background. (b) Optimisation of the cuts.

Figure 2: Illustration of the analysis procedure for the downgoing shower
category for GW190521. The left panel shows the expected remaining background
after varying cuts on $\mathcal{L}_{\mu}$ with the same colour code and line
style as in Figure 1. The right panel shows the average acceptance, assuming
an $E^{-2}$ neutrino spectrum, as a function of the cut on the RDF classifier
and the value of $\alpha$. For each bin, the cut on $\mathcal{L}_{\mu}$ is
fixed to the value that satisfies exactly the $3\sigma$ condition. The optimal
working point, maximising the acceptance, is indicated by the red cross.

###  2.3 Detector systematics

Several systematic effects may affect the detector performance, hence the
obtained constraints on the neutrino emission. Three sources of uncertainty,
found to be the dominant effects as already described e.g. in [6], are taken
into account and evaluated independently for the four event categories
(track/showers upgoing/downgoing):

  * •

The first one is related to the uncertainty on the PMT photon detection
efficiency and on the water absorption length; the related uncertainties have
been re-evaluated by varying these two quantities within a typical interval of
$\pm 10\%$ in dedicated MC simulations and estimating the overall impact on
the signal acceptance.

  * •

The second source is linked to the capability of the run-by-run MC simulations
to properly reproduce data conditions; the related error is estimated by
comparing the variability of event rates between data and simulations.

  * •

The last effect is the combined statistical and systematic uncertainty on the
background expectation. The related uncertainty is obtained by varying the
list of similar runs employed for its estimation.

The total uncertainties on the acceptance related to the first two sources are
18%, 14%, 21%, and 19% respectively for the upgoing tracks, downgoing tracks,
upgoing showers, and downgoing showers. The overall uncertainty on the
background is about 20% for all event categories.

##  3 Statistical analysis

For each GW event, the number of observed neutrino candidates in time and
spatial coincidence in each category can be converted into a significance of
the observation using Poisson statistics. In the absence of any excess of
neutrino events with respect to the background expectation, upper limits on
the neutrino emission are calculated. In the following section, several
assumptions are made:

  1. (i)

The source localisation is supposed to be within $\mathcal{R}_{90}$ (as this
is the region for which the selection has been optimised). Therefore, the GW
posterior samples are restricted to those with $\Omega\in\mathcal{R}_{90}$ and
the final constraints neglect the chances for the actual source to be
localised in the rest of the sky.

  2. (ii)

There is equipartition between the neutrino flavours at Earth due to the
averaging of oscillations over astronomical distances [26], starting from
$\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:2:0$ at production. This allows reporting
limits on the all-flavour neutrino emission.

  3. (iii)

The neutrino energy spectrum is described by a single power law ${\rm d}N/{\rm
d}E=\phi\cdot(E/$\mathrm{GeV}$)^{-\gamma}$ where $\phi$ is expressed in units
of ${\mathrm{GeV}}^{-1}\text{\,}{\mathrm{cm}}^{-2}$ and $\gamma$ is the
spectral index. The nominal case is $\gamma=2$ ($E^{-2}$ spectrum).

The ANTARES acceptance is estimated using the MC simulations described in
section 2. It depends on the shape of the assumed neutrino spectrum
(characterised by $\gamma)$, on the source direction $\Omega$, and on the
event category $c$. It is averaged over the neutrino flavours and can be
decomposed into a normalisation factor and a direction-dependent component:
$\mathcal{A}_{\gamma}^{(c)}(\Omega)=a_{\gamma}^{(c)}\cdot
f_{\gamma}^{(c)}(\Omega)$.

###  3.1 Constraints on the neutrino flux

This section presents the limits on the overall flux normalisation $\phi$
obtained for an all-flavour emission with $\gamma=2$. The cut-and-count
analysis described in the previous sections corresponds to a Poisson
likelihood

$\mathcal{L}\left(\\{N^{(c)}\\};\\{B^{(c)}\\},\\{a_{\gamma}^{(c)}\\},\Omega,\phi,\gamma\right)=\prod_{c\in\mathcal{C}}\textrm{Poisson}\left(N^{(c)};B^{(c)}+\phi\cdot
a_{\gamma}^{(c)}\cdot f_{\gamma}^{(c)}(\Omega)\right),$ (3.1)

where $N^{(c)}$ (resp. $B^{(c)}$) is the observed (resp. background-expected)
number of events in each category, and the product is performed over the set
of four event categories ($c\in\mathcal{C}$, $\mathcal{C}=$ {upgoing tracks,
downgoing tracks, upgoing showers, downgoing showers}). Given the selection
optimisation presented in section 2, the value of $B^{(c)}$ is fixed to
$2.7\text{\times}{10}^{-3}$, independently for all categories.

A Bayesian method is employed to obtain constraints on the neutrino flux
normalisation $\phi$ starting from this likelihood. A flat prior on $\phi$ is
employed, the systematic uncertainties described in section 2.3 are encoded in
Gaussian priors on $B^{(c)}$ and on $a_{\gamma}^{(c)}$ with the standard
deviations corresponding to the uncertainties reported there, and the GW
skymap $\mathcal{P}(\Omega)$ is used as a prior on $\Omega$.

The obtained posterior probability is then marginalised over all the nuisance
parameters (background, acceptance, direction) by using MC integration
techniques: toy samples (t.s.) are generated with values of $B^{(c)}$ and
$a_{\gamma}^{(c)}$ following the priors, and the posterior samples from GW
catalogues can be used directly for the sampling of $\Omega$. The marginalised
posterior probability distribution is computed as

$P(\phi)=C\sum_{s\in\rm
t.s.}\mathcal{L}\left(\\{N^{(c)}\\};\\{B^{(c)}\\}_{s},\\{a_{\gamma}^{(c)}\\}_{s},\Omega_{s},\phi,\gamma\right),$
(3.2)

where $C$ is a normalisation constant that can be determined numerically by
ensuring $\int_{0}^{\infty}P(\phi){\rm d}\phi=1$. The 90% upper limit
$\phi_{90}$ is finally obtained by solving $\int_{0}^{\phi_{90}}P(\phi){\rm
d}\phi=0.90$.

###  3.2 Constraints on the total energy

Similarly to the incoming neutrino flux on Earth, one may also constrain the
total energy $E_{\rm tot,\nu}$ emitted in neutrinos, correcting for the source
distance. This can be done under a specific assumption on the spatial
distribution of the neutrino emission around the source, e.g., either
isotropic or collimated into a jet. One may also consider the ratio between
the total energy emitted in neutrinos and the energy $E_{\rm GW}$ radiated in
GW: $f_{\nu}=E_{\rm tot,\nu}/E_{\rm GW}$.

#### Isotropic emission.

In the case of a source emitting isotropically, the total energy emitted in
neutrinos can be computed as

$E_{\rm tot,\nu}^{\rm iso}=4\pi
D_{L}^{2}\int_{E_{\min}}^{E_{\max}}E\times\dfrac{{\rm d}N}{{\rm d}E}\,{\rm
d}E=\phi\times 4\pi D_{L}^{2}\int_{E_{\min}}^{E_{\max}}E^{-\gamma+1}\,{\rm
d}E,$ (3.3)

where $D_{L}$ is the source luminosity distance, and $E_{\min}$, $E_{\max}$
are the integration bounds. For this analysis, these bounds are fixed to
$E_{\min}=$5\text{\,}\mathrm{GeV}$$ and
$E_{\max}=${10}^{8}\text{\,}\mathrm{GeV}$$, which is the typical range where
the emission is expected in most of the models (e.g., [27]).

Since the total energy $E^{\rm iso}_{\rm tot,\nu}$ is proportional to the flux
normalisation $\phi$, the likelihood from Equation (3.1) can be rewritten in
terms of $E^{\rm iso}_{\rm tot,\nu}$ instead of $\phi$. The marginalised
posterior probability and the upper limits are obtained similarly, where the
luminosity distance is also extracted from GW posterior samples to be used in
MC integration toy samples and a flat prior on $E^{\rm iso}_{\rm tot,\nu}$ is
assumed. A similar rewriting is possible to obtain the limits on $f^{\rm
iso}_{\nu}=E^{\rm iso}_{\rm tot,\nu}/E_{\rm GW}$. This is a relevant parameter
if it is assumed that the total energy emitted in neutrinos scales with the GW
emission.

#### Non-isotropic scenarios.

One may also consider the total energy $E_{\rm tot,\nu}$ for a given non-
isotropic model, and the corresponding likelihoods and posteriors can be
written using the relevant GW parameters for the marginalisation. For non-
isotropic emission, a simple von Mises [28] jet model is considered:

$p(\theta;\omega)=\dfrac{1}{4\pi\omega^{2}\sinh(1/\omega^{2})}\times\exp(\cos\theta/\omega^{2}),$
(3.4)

where $\omega$ characterises the jet opening and $\theta$ is the angle with
respect to the jet central direction. In the following, the jet is assumed to
be collinear with the total angular momentum of the merger, such that
$\theta=\theta_{\rm jn}$ from the GW data release. The total energy emitted in
neutrinos for a power-law spectrum is then

$E_{\rm tot,\nu}=\phi\times 2\pi
D_{L}^{2}\int_{E_{\min}}^{E_{\max}}E^{-\gamma+1}\,{\rm d}E\times\dfrac{\int
p(\theta;\omega)\sin\theta\,{\rm d}\theta}{p(\theta_{\rm jn};\omega)},$ (3.5)

where the last term is the correction corresponding to the jet visibility from
Earth (in the isotropic case, this term would be $2$, such that the Equation
(3.3) is retrieved). For a given jet opening $\omega$, the limits on $E_{\rm
tot,\nu}$ or the corresponding $f_{\nu}$ may be derived as described in the
isotropic case, including $\theta_{\rm jn}$ variable in MC integration toy
samples.

###  3.3 Stacking analysis

The GW catalogues published by LIGO/Virgo may contain populations of sources
with similar neutrino emissions, which could be more efficiently constrained
by performing stacking analyses. A first version of such an analysis is
presented here with two categories of GW events: the 72 BBH mergers on one
side and the 7 NSBH mergers on the other. While differing in its precise
implementation in this study, the method follows the ideas initially presented
in [29] and already applied in [9] with Super-Kamiokande data.

The stacking approach assumes that all objects in the selected population have
the same emission, either in terms of total energy $E_{\rm tot,\nu}$ or of
$f_{\nu}$. In the non-isotropic scenarios, GW sources in a given population
may have different jet inclinations but the shape of the jet (the parameter
$\omega$ for the von Mises model) is considered to be the same for all the
objects.

Assuming a flat prior on the signal parameters and considering all
observations as independent, the posterior distribution for a given population
$\mathcal{S}$ may be written as

$P_{\mathcal{S}}(X;Y)=C\prod_{i\in\mathcal{S}}P_{i}(X;Y,Z_{i}),$ (3.6)

where $C$ is a normalisation constant, $X$ is the signal parameter to be
constrained ($E_{\rm tot,\nu}$ or $f_{\nu}$), $Y$ may denote potential common
parameters (such as $\omega$), and $Z_{i}$ represents the parameters which may
differ from one GW event to another within $\mathcal{S}$. Upper limits on $X$
can then be obtained with the same method as before.

The stacking approach is particularly interesting to constrain the non-
isotropic models, as is detailed in the next section.

##  4 Results

A total of 80 GW events out of the 83 observed during O3, including all
candidates involving at least one neutron star, can be associated with
exploitable ANTARES data.

For each follow-up, the neutrino event selection is optimised according to the
procedure described in section 2 and the final number of selected events in
time and spatial coincidence with the GW event in each category is extracted.
No event has been selected for any of the follow-ups, which is fully
compatible with the expected background $\sum_{i\in{\rm
GWs}}\sum_{c\in\mathcal{C}}B_{i}^{(c)}\sim 0.82$. Therefore, only upper limits
on the neutrino flux and other related quantities are reported in the
following.

Table 1 displays the 90% upper limits on the incoming all-flavour neutrino
emission assuming an $E^{-2}$ spectrum, along with GW information, for the
events initially reported in the GWTC-2 catalogue. Similarly, the results for
the events in GWTC-2.1 and GWTC-3 are presented in Table 2 and Table 3,
respectively. The all-flavour flux limits assuming an $E^{-2}$ spectrum stands
mostly between $4$ and $60\text{\,}\mathrm{GeV}\text{\,}{\mathrm{cm}}^{-2}$.

Table 1: Summary of GWTC-2 follow-up results. The first four columns summarise the relevant GW information [10], including the most probable merger type, the estimated median luminosity distance $D_{L}$, and the size of the region $\mathcal{R}_{90}$ containing 90% of the localisation probability, while the three last ones show the 90% upper limits on the all-flavour neutrino emission assuming an $E^{-2}$ spectrum (this work), in terms of the flux normalisation $E^{2}{\rm d}N/{\rm d}E$, the total isotropic energy $E^{\rm iso}_{\rm tot,\nu}$, and the ratio $f_{\nu}^{\rm iso}$. GW name | Type | $D_{L}$ |  $\mathcal{R}_{90}$ area | Upper limits on neutrino emission
---|---|---|---|---
|  |  |  | $E^{2}{\rm d}N/{\rm d}E$ | $E^{\rm iso}_{\rm tot,\nu}$ | $f_{\nu}^{\rm iso}$
|  | $\mathrm{Mpc}$ | $\deg\mathrm{{}^{2}}$ | $\mathrm{GeV}\text{\,}{\mathrm{cm}}^{-2}$ |  $\mathrm{erg}$ |
GW190412 | BBH | $734$ | $24$ | $1.75\text{\times}{10}^{01}$ | $3.40\text{\times}{10}^{55}$ | $1.67\text{\times}{10}^{01}$
GW190413_052954 | BBH | $4190$ | $1383$ | $6.37\text{\times}{10}^{01}$ | $1.80\text{\times}{10}^{57}$ | $3.67\text{\times}{10}^{02}$
GW190413_134308 | BBH | $5182$ | $520$ | $5.35\text{\times}{10}^{00}$ | $5.90\text{\times}{10}^{56}$ | $9.06\text{\times}{10}^{01}$
GW190421_213856 | BBH | $3166$ | $1023$ | $1.04\text{\times}{10}^{01}$ | $3.35\text{\times}{10}^{56}$ | $6.12\text{\times}{10}^{01}$
GW190424_180648 | BBH | $2568$ | $25902$ | $4.62\text{\times}{10}^{01}$ | $1.23\text{\times}{10}^{57}$ | $2.02\text{\times}{10}^{02}$
GW190425 | BNS | $157$ | $9881$ | $2.13\text{\times}{10}^{01}$ | $2.31\text{\times}{10}^{54}$ | $8.56\text{\times}{10}^{00}$
GW190426_152155 | NSBH | $377$ | $1392$ | $2.61\text{\times}{10}^{01}$ | $2.44\text{\times}{10}^{55}$ | $8.69\text{\times}{10}^{01}$
GW190503_185404 | BBH | $1527$ | $97$ | $4.44\text{\times}{10}^{00}$ | $4.46\text{\times}{10}^{55}$ | $7.68\text{\times}{10}^{00}$
GW190512_180714 | BBH | $1462$ | $229$ | $4.62\text{\times}{10}^{00}$ | $3.99\text{\times}{10}^{55}$ | $1.63\text{\times}{10}^{01}$
GW190513_205428 | BBH | $2190$ | $494$ | $6.29\text{\times}{10}^{00}$ | $9.01\text{\times}{10}^{55}$ | $2.28\text{\times}{10}^{01}$
GW190514_065416 | BBH | $4988$ | $2402$ | $1.14\text{\times}{10}^{01}$ | $1.12\text{\times}{10}^{57}$ | $2.40\text{\times}{10}^{02}$
GW190517_055101 | BBH | $2270$ | $468$ | $1.75\text{\times}{10}^{01}$ | $6.73\text{\times}{10}^{56}$ | $9.47\text{\times}{10}^{01}$
GW190519_153544 | BBH | $3023$ | $770$ | $1.22\text{\times}{10}^{01}$ | $7.23\text{\times}{10}^{56}$ | $7.28\text{\times}{10}^{01}$
GW190521 | BBH | $4567$ | $937$ | $1.19\text{\times}{10}^{01}$ | $9.64\text{\times}{10}^{56}$ | $7.34\text{\times}{10}^{01}$
GW190521_074359 | BBH | $1244$ | $509$ | $1.75\text{\times}{10}^{01}$ | $9.87\text{\times}{10}^{55}$ | $1.65\text{\times}{10}^{01}$
GW190527_092055 | BBH | $3563$ | $3795$ | $4.11\text{\times}{10}^{01}$ | $4.56\text{\times}{10}^{57}$ | $6.38\text{\times}{10}^{02}$
GW190602_175927 | BBH | $3138$ | $721$ | $5.40\text{\times}{10}^{00}$ | $2.42\text{\times}{10}^{56}$ | $2.60\text{\times}{10}^{01}$
GW190620_030421 | BBH | $3211$ | $6674$ | $2.86\text{\times}{10}^{01}$ | $1.40\text{\times}{10}^{57}$ | $1.57\text{\times}{10}^{02}$
GW190630_185205 | BBH | $956$ | $1275$ | $2.65\text{\times}{10}^{01}$ | $9.92\text{\times}{10}^{55}$ | $2.23\text{\times}{10}^{01}$
GW190701_203306 | BBH | $2152$ | $47$ | $6.07\text{\times}{10}^{00}$ | $9.75\text{\times}{10}^{55}$ | $1.32\text{\times}{10}^{01}$
GW190706_222641 | BBH | $5184$ | $611$ | $9.72\text{\times}{10}^{00}$ | $1.25\text{\times}{10}^{57}$ | $1.26\text{\times}{10}^{02}$
GW190707_093326 | BBH | $791$ | $1343$ | $2.42\text{\times}{10}^{01}$ | $7.13\text{\times}{10}^{55}$ | $4.50\text{\times}{10}^{01}$
GW190708_232457 | BBH | $888$ | $13701$ | $2.60\text{\times}{10}^{01}$ | $7.59\text{\times}{10}^{55}$ | $3.17\text{\times}{10}^{01}$
GW190719_215514 | BBH | $4786$ | $2281$ | $5.19\text{\times}{10}^{03}$ | $2.14\text{\times}{10}^{59}$ | $3.01\text{\times}{10}^{04}$
GW190720_000836 | BBH | $906$ | $517$ | $2.52\text{\times}{10}^{01}$ | $5.98\text{\times}{10}^{55}$ | $3.62\text{\times}{10}^{01}$
GW190727_060333 | BBH | $3609$ | $861$ | $1.58\text{\times}{10}^{01}$ | $8.74\text{\times}{10}^{56}$ | $1.41\text{\times}{10}^{02}$
GW190731_140936 | BBH | $4034$ | $3042$ | $1.84\text{\times}{10}^{01}$ | $1.43\text{\times}{10}^{57}$ | $2.51\text{\times}{10}^{02}$
GW190803_022701 | BBH | $3750$ | $1538$ | $2.20\text{\times}{10}^{01}$ | $9.74\text{\times}{10}^{56}$ | $1.93\text{\times}{10}^{02}$
GW190814 | NSBH | $241$ | $20$ | $4.60\text{\times}{10}^{00}$ | $9.49\text{\times}{10}^{53}$ | $2.21\text{\times}{10}^{00}$
GW190828_063405 | BBH | $2160$ | $525$ | $8.73\text{\times}{10}^{00}$ | $1.35\text{\times}{10}^{56}$ | $2.61\text{\times}{10}^{01}$
GW190828_065509 | BBH | $1658$ | $637$ | $7.12\text{\times}{10}^{00}$ | $9.57\text{\times}{10}^{55}$ | $4.52\text{\times}{10}^{01}$
GW190909_114149 | BBH | $4924$ | $4071$ | $3.04\text{\times}{10}^{01}$ | $3.74\text{\times}{10}^{57}$ | $7.08\text{\times}{10}^{02}$
GW190910_112807 | BBH | $1670$ | $10014$ | $2.18\text{\times}{10}^{01}$ | $3.81\text{\times}{10}^{56}$ | $6.25\text{\times}{10}^{01}$
GW190915_235702 | BBH | $1715$ | $386$ | $5.37\text{\times}{10}^{00}$ | $7.21\text{\times}{10}^{55}$ | $1.54\text{\times}{10}^{01}$
GW190924_021846 | BBH | $572$ | $379$ | $8.89\text{\times}{10}^{00}$ | $1.05\text{\times}{10}^{55}$ | $1.13\text{\times}{10}^{01}$
GW190929_012149 | BBH | $3901$ | $1846$ | $1.44\text{\times}{10}^{01}$ | $1.44\text{\times}{10}^{57}$ | $2.25\text{\times}{10}^{02}$
GW190930_133541 | BBH | $786$ | $1829$ | $1.54\text{\times}{10}^{01}$ | $3.51\text{\times}{10}^{55}$ | $2.40\text{\times}{10}^{01}$
Table 2: Same as Table 1 for the additional events in the GWTC-2.1 catalogue [11]. GW name | Type | Distance |  $\mathcal{R}_{90}$ area | Upper limits on neutrino emission
---|---|---|---|---
|  |  |  | $E^{2}{\rm d}N/{\rm d}E$ | $E^{\rm iso}_{\rm tot,\nu}$ | $f_{\nu}^{\rm iso}$
|  | $\mathrm{Mpc}$ | $\deg\mathrm{{}^{2}}$ | $\mathrm{GeV}\text{\,}{\mathrm{cm}}^{-2}$ |  $\mathrm{erg}$ |
GW190403_051519 | BBH | $10853$ | $4250$ | $2.89\text{\times}{10}^{01}$ | $1.67\text{\times}{10}^{58}$ | $1.82\text{\times}{10}^{03}$
GW190426_190642 | BBH | $5995$ | $8030$ | $3.61\text{\times}{10}^{01}$ | $5.42\text{\times}{10}^{57}$ | $3.51\text{\times}{10}^{02}$
GW190725_174728 | BBH | $1128$ | $2435$ | $2.52\text{\times}{10}^{01}$ | $1.46\text{\times}{10}^{56}$ | $1.28\text{\times}{10}^{02}$
GW190805_211137 | BBH | $7235$ | $3089$ | $4.12\text{\times}{10}^{01}$ | $9.94\text{\times}{10}^{57}$ | $1.47\text{\times}{10}^{03}$
GW190916_200658 | BBH | $6180$ | $3573$ | $3.35\text{\times}{10}^{01}$ | $6.95\text{\times}{10}^{57}$ | $1.16\text{\times}{10}^{03}$
GW190917_114630 | NSBH | $741$ | $1801$ | $1.20\text{\times}{10}^{01}$ | $2.80\text{\times}{10}^{55}$ | $7.79\text{\times}{10}^{01}$
GW190925_232845 | BBH | $973$ | $955$ | $6.72\text{\times}{10}^{00}$ | $2.38\text{\times}{10}^{55}$ | $7.68\text{\times}{10}^{00}$
GW190926_050336 | BBH | $5073$ | $2212$ | $5.59\text{\times}{10}^{01}$ | $9.60\text{\times}{10}^{57}$ | $2.01\text{\times}{10}^{03}$
Table 3: Same as Table 1 for the events in the GWTC-3 catalogue [12]. GW name | Type | Distance |  $\mathcal{R}_{90}$ area | Upper limits on neutrino emission
---|---|---|---|---
|  |  |  | $E^{2}{\rm d}N/{\rm d}E$ | $E^{\rm iso}_{\rm tot,\nu}$ | $f_{\nu}^{\rm iso}$
|  | $\mathrm{Mpc}$ | $\deg\mathrm{{}^{2}}$ | $\mathrm{GeV}\text{\,}{\mathrm{cm}}^{-2}$ |  $\mathrm{erg}$ |
GW191103_012549 | BBH | $927$ | $2520$ | $4.34\text{\times}{10}^{01}$ | $1.61\text{\times}{10}^{56}$ | $9.95\text{\times}{10}^{01}$
GW191105_143521 | BBH | $1141$ | $730$ | $1.45\text{\times}{10}^{02}$ | $6.31\text{\times}{10}^{56}$ | $4.31\text{\times}{10}^{02}$
GW191109_010717 | BBH | $1386$ | $1784$ | $1.24\text{\times}{10}^{01}$ | $1.21\text{\times}{10}^{56}$ | $1.76\text{\times}{10}^{01}$
GW191113_071753 | BBH | $1469$ | $2993$ | $3.10\text{\times}{10}^{01}$ | $3.72\text{\times}{10}^{56}$ | $2.95\text{\times}{10}^{02}$
GW191126_115259 | BBH | $1618$ | $1514$ | $1.06\text{\times}{10}^{02}$ | $7.95\text{\times}{10}^{56}$ | $4.24\text{\times}{10}^{02}$
GW191127_050227 | BBH | $3614$ | $1499$ | $2.07\text{\times}{10}^{01}$ | $1.76\text{\times}{10}^{57}$ | $2.50\text{\times}{10}^{02}$
GW191129_134029 | BBH | $774$ | $851$ | $1.48\text{\times}{10}^{01}$ | $3.58\text{\times}{10}^{55}$ | $2.68\text{\times}{10}^{01}$
GW191204_110529 | BBH | $1975$ | $4747$ | $4.57\text{\times}{10}^{01}$ | $8.94\text{\times}{10}^{56}$ | $2.81\text{\times}{10}^{02}$
GW191204_171526 | BBH | $637$ | $365$ | $8.63\text{\times}{10}^{00}$ | $1.17\text{\times}{10}^{55}$ | $6.77\text{\times}{10}^{00}$
GW191215_223052 | BBH | $1937$ | $603$ | $1.30\text{\times}{10}^{01}$ | $2.50\text{\times}{10}^{56}$ | $7.75\text{\times}{10}^{01}$
GW191216_213338 | BBH | $339$ | $488$ | $1.13\text{\times}{10}^{01}$ | $5.34\text{\times}{10}^{54}$ | $3.51\text{\times}{10}^{00}$
GW191219_163120 | NSBH | $569$ | $2232$ | $1.74\text{\times}{10}^{01}$ | $2.78\text{\times}{10}^{55}$ | $1.90\text{\times}{10}^{02}$
GW191222_033537 | BBH | $2991$ | $2299$ | $5.62\text{\times}{10}^{01}$ | $2.69\text{\times}{10}^{57}$ | $4.60\text{\times}{10}^{02}$
GW191230_180458 | BBH | $4296$ | $1012$ | $1.15\text{\times}{10}^{01}$ | $9.18\text{\times}{10}^{56}$ | $1.40\text{\times}{10}^{02}$
GW200105_162426 | NSBH | $266$ | $7882$ | $1.75\text{\times}{10}^{01}$ | $6.11\text{\times}{10}^{54}$ | $1.68\text{\times}{10}^{01}$
GW200112_155838 | BBH | $1248$ | $4250$ | $1.81\text{\times}{10}^{01}$ | $1.36\text{\times}{10}^{56}$ | $2.57\text{\times}{10}^{01}$
GW200115_042309 | NSBH | $298$ | $519$ | $6.84\text{\times}{10}^{00}$ | $2.81\text{\times}{10}^{54}$ | $1.17\text{\times}{10}^{01}$
GW200128_022011 | BBH | $3405$ | $2677$ | $1.83\text{\times}{10}^{02}$ | $7.57\text{\times}{10}^{57}$ | $1.35\text{\times}{10}^{03}$
GW200129_065458 | BBH | $883$ | $87$ | $8.80\text{\times}{10}^{00}$ | $2.82\text{\times}{10}^{55}$ | $5.38\text{\times}{10}^{00}$
GW200202_154313 | BBH | $411$ | $160$ | $5.37\text{\times}{10}^{00}$ | $3.24\text{\times}{10}^{54}$ | $2.40\text{\times}{10}^{00}$
GW200208_130117 | BBH | $2258$ | $39$ | $7.22\text{\times}{10}^{00}$ | $1.51\text{\times}{10}^{56}$ | $2.99\text{\times}{10}^{01}$
GW200208_222617 | BBH | $4547$ | $1889$ | $1.57\text{\times}{10}^{01}$ | $2.52\text{\times}{10}^{57}$ | $4.90\text{\times}{10}^{02}$
GW200209_085452 | BBH | $3447$ | $925$ | $2.50\text{\times}{10}^{02}$ | $2.24\text{\times}{10}^{57}$ | $4.53\text{\times}{10}^{02}$
GW200210_092255 | NSBH | $959$ | $1830$ | $2.28\text{\times}{10}^{01}$ | $8.57\text{\times}{10}^{55}$ | $1.87\text{\times}{10}^{02}$
GW200216_220804 | BBH | $3997$ | $3010$ | $7.80\text{\times}{10}^{00}$ | $6.45\text{\times}{10}^{56}$ | $1.03\text{\times}{10}^{02}$
GW200219_094415 | BBH | $3447$ | $702$ | $6.41\text{\times}{10}^{00}$ | $3.74\text{\times}{10}^{56}$ | $8.05\text{\times}{10}^{01}$
GW200220_061928 | BBH | $6321$ | $3485$ | $1.87\text{\times}{10}^{01}$ | $5.89\text{\times}{10}^{57}$ | $6.59\text{\times}{10}^{02}$
GW200220_124850 | BBH | $4172$ | $3169$ | $4.53\text{\times}{10}^{01}$ | $4.26\text{\times}{10}^{57}$ | $9.34\text{\times}{10}^{02}$
GW200224_222234 | BBH | $1677$ | $51$ | $1.32\text{\times}{10}^{01}$ | $1.48\text{\times}{10}^{56}$ | $2.37\text{\times}{10}^{01}$
GW200225_060421 | BBH | $1144$ | $516$ | $2.15\text{\times}{10}^{01}$ | $1.05\text{\times}{10}^{56}$ | $4.26\text{\times}{10}^{01}$
GW200302_015811 | BBH | $1545$ | $7011$ | $2.77\text{\times}{10}^{01}$ | $3.06\text{\times}{10}^{56}$ | $7.44\text{\times}{10}^{01}$
GW200306_093714 | BBH | $2228$ | $4371$ | $1.08\text{\times}{10}^{01}$ | $2.66\text{\times}{10}^{56}$ | $7.37\text{\times}{10}^{01}$
GW200308_173609 | BBH | $8867$ | $18705$ | $3.50\text{\times}{10}^{01}$ | $3.27\text{\times}{10}^{58}$ | $1.75\text{\times}{10}^{04}$
GW200311_115853 | BBH | $1152$ | $37$ | $1.74\text{\times}{10}^{01}$ | $8.65\text{\times}{10}^{55}$ | $1.76\text{\times}{10}^{01}$
GW200322_091133 | BBH | $8302$ | $31571$ | $3.64\text{\times}{10}^{01}$ | $2.98\text{\times}{10}^{58}$ | $4.88\text{\times}{10}^{04}$

Figure 3 shows the limits on the total energy emitted in all-flavour neutrinos
and on $f^{\rm iso}_{\nu}=E^{\rm iso}_{\rm tot,\nu}/E_{\rm GW}$, assuming
isotropic emission, for events from the three catalogues. The limits range
from ${10}^{54}$ to ${10}^{59}\text{\,}\mathrm{erg}$ and are mostly following
the expected $D_{L}^{2}$ trend.

(a)  (b)

Figure 3: 90% upper limits on the total energy $E^{\rm iso}_{\rm tot,\nu}$
emitted in neutrinos of all flavours (left) and on $f^{\rm iso}_{\nu}=E^{\rm
iso}_{\rm tot,\nu}/E_{\rm GW}$ (right) as a function of the source luminosity
distance, assuming an $E^{-2}$ spectrum and isotropic emission. The horizontal
bars indicate the $5-95\%$ range of the luminosity distance estimate, and the
markers/colours correspond to the different source categories.

Population studies are performed by stacking (a) all 72 BBH events, and (b)
the 7 NSBH candidates. Figure 4 shows the stacked upper limits on the typical
total energy emitted in neutrinos (or the ratio $f_{\nu}$) within these two
categories, considering jetted emission with the von Mises model (as a
function of the jet opening $\omega$) as well as for isotropic emission, and
assuming $E^{-\gamma}$ spectra with $\gamma=2.0,\,2.5,\,3.0$.

For individual follow-ups, as $\theta_{\rm jn}$ is not strongly constrained by
GW detections, von Mises emission constraints are mainly dominated by
geometrical effects: when the jet opening angle is narrow, there is a very
small chance that the Earth is aligned with the jet, while a wide opening
leads to a larger spread in total energy, which is hence more difficult to
constrain. The limits on the total energy as a function of the jet opening are
thus expected to show a typical parabola shape corresponding to a trade-off
between these two effects, as illustrated by the envelope of individual limits
shown in Figure 4. While the individual results for jetted emission are of
little interest due to these limitations, the stacking analysis of this
scenario yields some benefit as, statistically, some of the sources are
expected to point toward Earth for large enough jet opening angles. This can
be seen on Figure 4, in particular for the BBH event sample, with the largest
statistics, where the neutrino emission can be constrained down to opening
angles as small as $\omega\sim 10-$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

For the BBH stacking, the stacked limits in the nominal scenario ($\gamma=2$
and isotropic emission) are $E^{\rm iso}_{\rm
tot,\nu}<$3.8\text{\times}{10}^{53}\text{\,}\mathrm{erg}$$ and $f_{\nu}<0.14$,
while the best limits from individual follow-ups (for GW200202_154313) are
respectively $3.2\text{\times}{10}^{54}\text{\,}\mathrm{erg}$ and $2.4$. For
the NSBH population, the corresponding limits are $E^{\rm iso}_{\rm
tot,\nu}<$3.2\text{\times}{10}^{53}\text{\,}\mathrm{erg}$$ and $f_{\nu}<0.88$,
with the best individual limits being (for GW190814)
$9.5\text{\times}{10}^{53}\text{\,}\mathrm{erg}$ and $2.2$, respectively.

Finally, it is worth noticing that the statistical power of the BBH stacking
already allows the exploration of reasonable values for $f_{\nu}$ (in the
context of the assumed simple power-law spectrum), while all individual limits
on this parameter are above 1, corresponding to a neutrino emission higher
than the GW one.

(a)  (b)

Figure 4: 90% upper limits on the per-source typical total energy emitted in
neutrinos (left) and on $f_{\nu}$ (right) as a function of the jet opening
angle (for von Mises model) and for the isotropic case (rightmost entry). The
black (orange) curves correspond to the BBH (NSBH) stacking of mergers
detected during O3. The different markers and line styles correspond to
different spectral indices $\gamma$. The filled areas with the corresponding
colours (merger category) and line styles (spectral index) indicate the
envelope of the best individual upper limits.

Supplemental material with all numerical results for individual follow-ups and
from the stacking analysis, for the various jet scenarios and assumed spectral
indices, can be provided under request.

##  5 Discussion and conclusions

The search for neutrino counterparts to GW signals detected during the O3 run
with the ANTARES detector yields no significant excess. This null result is
used to extract upper limits on the neutrino emission, both in terms of the
flux normalisation and of the total energy emitted in neutrinos, $E^{\rm
iso}_{\rm tot,\nu}$, in particular for the standard case with an $E^{-2}$
spectrum and isotropic emission.

The ANTARES analysis presented in this article benefits from the different
event topologies in the detector, leading to all-flavour limits for neutrino
energies above $100\text{\,}\mathrm{GeV}$. The constraints are also
interpreted in the context of population studies, for which limits can be put
on the typical neutrino emission of different categories of sources (here, BBH
and NSBH). As the most recent astrophysical models (e.g., [3, 27, 30]) seem to
be out of reach for current detectors, such stacking studies may be one of the
most promising approaches to identify joint emitters of neutrinos and GWs.
More refined stackings may be studied by considering e.g., sub-populations
among BBH mergers, such as the ones having similar source characteristics or
environments.

The Bayesian analysis performed here considers flat priors, such that the
stacking analysis is performed easily by multiplying the individual posterior
distributions. Different priors (e.g., non-informative Jeffreys prior [31])
introduce correlations between the GW follow-ups and would only be possible by
performing the posterior marginalisation over all GW events at once, therefore
requiring more sophisticated integration techniques than the one employed in
this paper.

First constraints on jetted emission for a von Mises model and for different
spectral indices are derived as well. In the absence of the determination of
any jet direction from the GW data or any electromagnetic observations, these
results are only indirectly deducted from the stacking analysis. In future
observation campaigns, any such measurement, even for one or a few sources,
would allow for deriving direct constraints, as well as considering more
complex jet models.

Though the ANTARES detector has been decommissioned in 2022, the KM3NeT
experiment [32] has taken over the observation of the neutrino sky from the
depths of the Mediterranean Sea. Data from the first few lines of KM3NeT/ORCA
has already been used to perform an initial search for neutrino counterparts
to the O3 events [33]. With more lines being deployed up to $\sim$ 2026,
KM3NeT/ARCA and ORCA are expected to provide complementary coverage of the
transient sky with respect to IceCube, increasing the chances to get
significant individual detections and creating new opportunities for joint
analyses.

The next GW observing period, O4, is expected to start in 2023, including the
KAGRA detector in addition to LIGO and Virgo [34]. With an expected number of
detections of the order of one hundred per year, population studies will
become even more relevant.

Lastly, combinations with ZTF [35] or Vera C. Rubin [36] surveys and Target of
Opportunity programs may help pinpoint candidate host galaxies for observed
binary mergers, and allow more detailed analyses using the electromagnetic
information and the environment characterisation as inputs for the prediction
and constraint of the neutrino emission. More generally, there are still only
a few models on the processes leading to neutrino production during binary
merger events in the literature [3, 4, 5, 27, 30], and further theoretical
developments in the coming years may bring new light to past, present, and
future observations.

## Acknowledgments

The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énergie
atomique et aux énergies alternatives (CEA), Commission Européenne (FEDER fund
and Marie Curie Program), LabEx UnivEarthS (ANR-10-LABX-0023 and
ANR-18-IDEX-0001), Région Alsace (contrat CPER), Région Provence-Alpes-Côte
d’Azur, Département du Var and Ville de La Seyne-sur-Mer, France;
Bundesministerium für Bildung und Forschung (BMBF), Germany; Istituto
Nazionale di Fisica Nucleare (INFN), the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 754496, Italy; Nederlandse organisatie voor Wetenschappelijk
Onderzoek (NWO), the Netherlands; Executive Unit for Financing Higher
Education, Research, Development and Innovation (UEFISCDI), Romania; Grants
PID2021-124591NB-C41, -C42, -C43 funded by MCIN/AEI/ 10.13039/501100011033
and, as appropriate, by “ERDF A way of making Europe”, by the “European Union”
or by the “European Union NextGenerationEU/PRTR”, Programa de Planes
Complementarios I+D+I (refs. ASFAE/2022/023, ASFAE/2022/014), Programa
Prometeo (PROMETEO/2020/019) and GenT (refs. CIDEGENT/2018/034, /2019/043,
/2020/049. /2021/23) of the Generalitat Valenciana, Junta de Andalucía (ref.
P18-FR-5057), EU: MSC program (ref. 101025085), Programa María Zambrano
(Spanish Ministry of Universities, funded by the European Union,
NextGenerationEU), Spain; Ministry of Higher Education, Scientific Research
and Innovation, Morocco, and the Arab Fund for Economic and Social
Development, Kuwait. We also acknowledge the technical support of Ifremer, AIM
and Foselev Marine for the sea operation and the CC-IN2P3 for the computing
facilities.

## References

  * [1] LIGO Scientific, Virgo collaboration, _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116 (2016) 061102 [1602.03837].
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