# Ultrahigh-energy neutrino searches using next-generation gravitational wave
detectors at radio neutrino detectors: GRAND, IceCube-Gen2 Radio, and RNO-G

Mainak Mukhopadhyay ID<EMAIL_ADDRESS>Department of Physics; Department of
Astronomy & Astrophysics; Center for Multimessenger Astrophysics, Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Kumiko Kotera ID Sorbonne Université, CNRS, UMR 7095,
Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
Astrophysical Institute, Vrije Universiteit Brussels, Pleinlaan 2, 1050
Brussels, Belgium  Department of Physics; Department of Astronomy &
Astrophysics; Center for Multimessenger Astrophysics, Institute for
Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Stephanie Wissel ID Department of Physics; Department of
Astronomy & Astrophysics; Center for Multimessenger Astrophysics, Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Kohta Murase ID Department of Physics; Department of
Astronomy & Astrophysics; Center for Multimessenger Astrophysics, Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto, Kyoto 606-8502 Japan  Shigeo
S. Kimura ID Frontier Research Institute for Interdisciplinary Sciences;
Astronomical Institute, Graduate School of Science, Tohoku University, Sendai
980-8578, Japan

###### Abstract

Binary neutron star (BNS) mergers can be sources of ultrahigh-energy (UHE)
cosmic rays and potential emitters of UHE neutrinos. The upcoming and current
radio neutrino detectors like the Giant Radio Array for Neutrino Detection
(GRAND), IceCube-Gen2 Radio, and the Radio Neutrino Observatory in Greenland
(RNO-G) are projected to reach the required sensitivities to search for these
neutrinos. In particular, in conjunction with the next-generation of
gravitational wave (GW) detectors like Cosmic Explorer (CE) and Einstein
Telescope (ET), GW-triggered stacking searches can be performed with the UHE
neutrino detectors. In this work, we explore the prospects of such searches by
implementing in our analysis an upper distance limit based on the sky-
localization capabilities of the GW detectors from which meaningful triggers
can be collected. We find that if each GW burst is associated with a total
isotropic-equivalent energy of $\sim 10^{50}-10^{51}$ erg emitted in UHE
neutrinos, along with a corresponding beaming fraction of $1$%, GRAND and
IceCube-Gen2 Radio have a large probability ($\sim 99$%) to detect a
coincident neutrino event using the joint combination of CE+ET in a timescale
of less than 15 years of operation for our fiducial choice of parameters. In
case of nondetections, the parameter spaces can be constrained at $3\sigma$
level in similar timescales of operation. We also highlight and discuss the
prospects of such joint radio neutrino detector network, their importance, and
their role in facilitating synergic GW and neutrino observations in the next
era of multimessenger astrophysics.

##  I Introduction

Cosmic accelerators provide us with a window to high-energy processes in the
Universe. Our understanding of these violent phenomena is based on
observations and evidences presented by the various messengers from these
sources, namely, neutrinos, gravitational waves (GWs), photons, and cosmic
rays (CRs). GW170817 detected in gravitational wave [1] and electromagnetic
channels [2, 3, 4, 5, 6, 7, 8, 9] serves as a prime example of the power of
multimessenger astronomy. With the advent of improved detectors across all the
messengers, the next era of multimessenger astronomy promises to be
revolutionary in our understanding of the Universe at the highest energy
scales.

In particular, neutrinos serve as one of the most prominent messengers in the
context of delivering intrinsic information from their sources. The
undeflected signatures provided by neutrinos play a key role in performing
precision time-domain astronomy. The first evidence of high-energy
astrophysical neutrinos from cosmic accelerators were presented by the the
IceCube Collaboration in 2013 [10, 11]. Since then, evidences for high-energy
neutrino emission have been found from multiple sources like NGC 1068 [12] and
the Galactic plane [13], which have also been observed in the electromagnetic
channel.

Binary neutron star (BNS) mergers are known to be rich GW sources (e.g., Refs.
[14, 1, 15, 16, 17, 18, 19]). BNS mergers may have relativistic jets leading
to short gamma-ray bursts (GRBs) and merger ejecta accompanied by shocks,
which can be sources of very and ultrahigh-energy cosmic rays (UHECRs) [20,
21, 22, 23, 24, 25]. High-energy neutrinos can be produced by jets and
winds[26, 27, 28, 29, 30]. Alternatively, analogous to pulsars from
collapsars, magnetars from low-mass BNS mergers could be UHECR and neutrino
emitters [31, 32, 33, 34, 35, 36, 37, 38]. A neutrino and GW channels have
jointly not yet yielded any significant results. However, even though no
neutrinos were observed from GW170817 [39, 40], BNS mergers may serve as an
ideal source class for future joint neutrino-GW observations. It presents us
with an opportunity to study and explore – what are the prospects of synergic
GW and neutrino observations in the upcoming future, given the next generation
of planned GW and neutrino detectors? How should we plan the joint searches
across multimessenger channels to study these sources?

In Ref. [41] we explored the above questions in the context of the planned
next-generation of IceCube – IceCube-Gen2 [42], to search for high-energy
neutrinos, using GW triggers from the next generation of GW detectors – Cosmic
Explorer (CE) [43] and Einstein Telescope (ET) [44]. We studied the prospects
by implementing _triggered-stacking searches_ for high-energy neutrinos. CE
and ET have large distance horizons for BNS mergers $\sim$ a few Gpc. The
large rate of BNS mergers at such distance scales and the uncertainty in the
neutrino emission timescales from such sources, present a challenge for
triggered-stacking searches, due to the unknown time window for the searches
combined with many triggers. This leads to the obvious question: how does one
collect _meaningful triggers_ in the era of the next generation of detectors?
We proposed to alleviate this problem using the sky localization capabilities
of the GW detectors, providing us with a distance limit for collecting
meaningful triggers. In the current work, we extend our analysis to search for
UHE neutrinos from BNS mergers at the next-generation of radio neutrino
detectors – the Giant Radio Detector for Neutrino Detection (GRAND) [45] and
IceCube-Gen2 Radio [42], and the currently operational Radio Neutrino
Observatory in Greenland (RNO-G) [46], using GW signals from BNS mergers in CE
and ET as triggers.

In the current era, a dedicated triggered search pipeline called LLAMA (Low-
Latency Algorithm for multimessenger Astrophysics) [47] was developed to
search for high-energy neutrinos from LIGO/Virgo candidates. This uses a fixed
time-window of $\pm 500$ s and the overlap between the GW and neutrino
localization areas to perform the analysis. Bayesian analysis based search
techniques for common sources of GW and high-energy neutrinos were discussed
in Ref. [48]. The IceCube collaboration also performed searches for high-
energy neutrinos from BNS mergers [49] including implementing unbinned maximum
likelihood and Bayesian analysis [50]. But no significant and strong evidence
has been found. GW triggered searches for low energy neutrinos in the context
of BNS mergers have also been explored in Ref. [51, 52]. GW triggered searches
have also been implemented to search for low-energy neutrinos from other
sources like core-collapse supernovae [53, 54]. Besides implementing the sky-
localization based distance cuts for triggers, the current work is also novel
in this way that, for the first time we propose synergic search strategies for
UHE neutrinos and investigate such prospects in the context of upcoming
detectors.

The paper is organized as follows. We discuss the radio neutrino detectors
relevant for UHE neutrino detection in Sec. II. In Sec. III we summarize the
formalism for performing GW triggered UHE neutrino searches. We present our
results and highlight the prospects for such searches in Sec. IV. We summarize
in Sec. V and discuss the implications of our work in Sec. VI.

##  II Ultrahigh-energy neutrino detectors

The low fluxes of UHE neutrinos require the development of cost effective
techniques to achieve their detection. Several next-generation experiments
have wagered on the observation of the radio signal emitted by cascades
generated, either in ice or in air, by the daughter particles of neutrinos,
once they have interacted in the ice or in the ground [55, 56]. Radio antennas
are indeed scalable, robust and usually cheaper than other more sophisticated
detectors, enabling their extensive deployment in large numbers, over large
areas.

Here we will focus on two outstanding instruments planning to reach a diffuse
source sensitivity of $\sim 2\times 10^{-10}\,{\rm GeV}\,{\rm cm}^{-2}\,{\rm
sr}^{-1}\,{\rm s}^{-1}$ over 10 years, in the long-term future: GRAND [45] and
IceCube-Gen2 Radio [42]. Their timeline is consistent with the construction of
next-generation GW instruments, opening the possibility of coincident
neutrino-GW detections.

In order to provide an insight of the up-coming possibilities at shorter
timescales, we also examine the case of RNO-G [46], which can be considered as
a precursor of IceCube-Gen2 Radio in terms of experimental concept. This
comparison will also highlight how the increase in sensitivity by more than an
order of magnitude between RNO-G and GRAND/IceCube-Gen2 Radio can make a
drastic difference on multimessenger detection prospects.

###  II.1 GRAND

GRAND is a proposal for a set of $\sim 20$ radio antenna sub-arrays of 10 000
km2 each deployed on several continents, totaling 200 000 antennas over 200
000 km2 by the 2030s [45, 57]. Its detection principle is as follows: Earth-
skimming tau neutrinos interact in the ground, producing a tau particle, which
in turn can emerge in the atmosphere and generate an extensive air-shower. The
electromagnetic signal emitted by this shower can be observed by radio antenna
arrays. By construction, the instantaneous field of view of each GRAND sub-
array is hence reduced to $\sim 6^{\circ}$ below the horizon, for a
$360^{\circ}$ azimuthal view, corresponding to $\sim 6\%$ of the sky per site.
The total instantaneous field of view hence scales roughly with the number of
sub-arrays. With sites located in both hemispheres, the day-averaged sky
coverage reaches 100% of the sky. In its ultimate configuration, the
instrument will reach $10^{-1}{\rm GeVcm}^{-2}$ instantaneous point source
sensitivity (assuming the zenith angle to be $90^{\circ}$) for neutrino
energies $\sim 10^{9}$ GeV. Thanks to the kilometer-size radio footprints on
the ground, an exquisite sub-degree angular resolution is expected to be
reached [58].

###  II.2 IceCube-Gen2 Radio and RNO-G

IceCube-Gen2 Radio and RNO-G are based on the radio-detection of the coherent
Askaryan emission initiated by any neutrino showers in ice. The in-ice
technique offers better performances at lower energies ($\sim 10^{17}\,$eV,
where the flux should be higher) than in-air techniques, which are strongly
limited in that range by the tau exit probability from the ground. This
enhancement comes however at the cost of a poorer angular resolution due to
radio propagation effects in polar ice. The in-ice technique is also sensitive
to all flavors of neutrinos, offering a complementary approach.

RNO-G is currently under construction. When completed, it will consist of 35
stations composed of 24 radio antennas, some installed at the surface of the
ice and some buried underneath at various depths. The peak effective area will
be the largest achieved to-date with a radio array, which is estimated to be
$\gtrsim 10^{8}{\rm cm}^{2}$ for neutrino energies $\gtrsim 10^{9}$ GeV. The
instantaneous field of view is mostly determined by the geometry of its
location in Greenland, corresponding to roughly $45^{\circ}\times
360^{\circ}$. Due to the high-latitude of the site, the day-average leads to a
coverage of half the Northern hemisphere sky. The preliminary estimates on the
angular reconstruction points towards a neutrino arrival direction resolution
at the few degree scale ($3^{\circ}\times 10^{\circ}$).

IceCube-Gen2 Radio plans to build upon the experience of RNO-G. Located in the
South Pole, it is planned to cover 500 km2 with 361 stations [59]. The peak
effective area in this case, is projected as $\gtrsim 10^{10}{\rm cm}^{2}$ for
neutrino energies $\gtrsim 10^{10}$ GeV. The instantaneous field of view is
expected to be of $55^{\circ}\times 360^{\circ}$, corresponding to 43% of the
sky. At the pole, the rotation of the Earth does not affect the daily field of
view. The angular resolution is expected to be similar to RNO-G, although
improvements due to better analysis and reconstruction of the electric field
polarization are expected.

##  III Formalism

In this section we will briefly discuss the formalism for performing GW
triggered searches for UHE neutrinos at the neutrino detectors using triggers
from CE, ET, and CE+ET. A more rigorous description of the formalism can be
found in Ref. [41]. The prospects of UHE neutrino detection will depend on the
properties of the source, the neutrino detector, and the relevant backgrounds.
Thus the key ingredients required for quantifying the detection prospects are
a typical profile for a flux of UHE neutrinos from BNS mergers, the redshift-
dependent rate of BNS mergers, the energy and declination-dependent effective
areas for the neutrino detectors, and background fluxes.

###  III.1 UHE neutrino flux from BNS mergers

BNS mergers, including short GRBs, can be accelerators of high-energy cosmic
rays. The cosmic rays can interact with the surrounding matter and photons
through hadronic interactions. In particular, the photomeson production
($p\gamma$) channel is relevant in jetted sources such as short GRBs. The
resultant charged pions decay to muons which further decay to neutrinos with a
typical energy of $1/20$ times the parent proton energy, although other mesons
such as kaons and charmed mesons may become dominant at the highest energies
[37].

In this work, for simplicity, we focus on the generic power-law assumption
without specifying neutrino production mechanisms. Then, the UHE neutrino
fluence from a typical BNS merger source can be written as

$\phi_{\nu}(\mathcal{E}_{\nu}^{\rm UHE,iso},E_{\nu},d_{L})=\frac{(1+z)}{4\pi
d_{L}^{2}}\frac{\mathcal{E}_{\nu}^{\rm UHE,iso}}{\rm
ln\big{(}\varepsilon_{\nu}^{\rm max}/\varepsilon_{\nu}^{\rm
min}\big{)}}E_{\nu}^{-2}\,,$ (1)

where we have assumed a power-law neutrino spectra with index $2$. The maximum
and minimum energy in the source frame are given as $\varepsilon_{\nu}^{\rm
min}=10^{3}$ GeV and $\varepsilon_{\nu}^{\rm max}=10^{8}$ GeV respectively.
The luminosity distance to the source is defined as $d_{L}$ and the neutrino
energy in observer frame is given by $E_{\nu}$ and gets a redshift correction
$E_{\nu}=\varepsilon_{\nu}/(1+z)$, where $z$ is the redshift. The total UHE
neutrino isotropic-equivalent energy is defined as $\mathcal{E}_{\nu}^{\rm
UHE,iso}$

$\mathcal{E}_{\nu}^{\rm UHE,iso}=\frac{\mathcal{E}_{\nu}^{\rm
UHE,true}}{f_{\rm bm}}\,,$ (2)

where $\mathcal{E}_{\nu}^{\rm UHE,true}$ is the true energy emitted in UHE
neutrinos at the source and $f_{\rm bm}$ is the beaming fraction. We assume
$f_{\rm bm}\sim 1$%, motivated from afterglow observations [60]. The quantity
$\mathcal{E}_{\nu}^{\rm UHE,true}$ is important and has a big influence on the
detection prospects. This is intuitive since the more energy emitted in UHE
neutrinos the better will be the detection prospects and lesser the detection
timescales. Thus, we will use $\mathcal{E}_{\nu}^{\rm UHE,iso}$ as a parameter
for this work. Neutrino searches will be sensitive to the isotropic-equivalent
energy rather than the true energy. So whether the model has a degeneracy with
the beaming factor is important and this can be solved with more detailed
observations in principle.

###  III.2 Triggered UHE neutrino searches

We outlined a typical UHE neutrino flux from a BNS merger in the previous
section, which now puts us in a position to consider the UHE neutrino
detection prospects and discuss a method to perform GW triggered neutrino
searches. The total number of neutrinos $N_{\nu}$ in a neutrino detector given
a flux $\phi_{\nu}(\mathcal{E}_{\nu}^{\rm UHE,iso},E_{\nu},d_{L})$ depends on
the neutrino energy and the instantaneous zenith (or declination)-dependent
effective area $\mathcal{A}_{\rm eff}(E_{\nu},\theta_{z})$

$N_{\nu}(\theta_{z},d_{L})=\int_{E_{\nu}^{\rm LL}}^{E_{\nu}^{\rm
UL}}dE_{\nu}\phi_{\nu}\big{(}\mathcal{E}_{\nu}^{\rm
UHE,iso},E_{\nu},d_{L}\big{)}\mathcal{A}_{\rm eff}(E_{\nu},\theta_{z})\,,$ (3)

where, the upper (lower) limits for neutrino energies are chosen to be $\
E_{\nu}^{\rm UL}=10^{11}$ GeV $\big{(}E_{\nu}^{\rm LL}=10^{7}$ GeV$\big{)}$.
There are a few important things to note here. Although we fix the limits,
GRAND becomes drastically less sensitive below energy $\lesssim 10^{8}$ GeV,
due to the combined effects of a drop in tau (daughter particle of the tau
neutrino) exit probability from the ground, and the radio signal becoming
dominated by the Galactic noise in this regime.

For this work, we use the zenith-dependent effective area for GRAND [45],
IceCube-Gen2 Radio [61, 59], and RNO-G [46]. In particular, for IceCube-Gen2
Radio the simulations for effective area were produced using NuRadioMC [62].
The effective areas are incorporated in calculation of the number of events
($N_{\nu}$) from the above equation which then gives the Poissonian
probability to detect at least one neutrino as, $p_{n\geq
1}(\theta_{z},d_{L})=1-{\rm exp}\big{(}-N_{\nu}(\theta_{z},d_{L})\big{)}$.
Integrating over the solid angle $\Omega$ leads to the total probability to
detect at least one neutrino as

$P_{n\geq 1}(d_{L})=\frac{1}{N_{\rm norm}}\int d\Omega\ p_{n\geq
1}\big{(}\theta_{z},d_{L}\big{)}\,,$ (4)

where $N_{\rm norm}$ is the solid angle over which the probability is
averaged. For all-sky, $N_{\rm norm}=4\pi$.

Until now we have not considered the information that could be derived from
the next-generation GW detectors. The BNS mergers detected by the GW detectors
can be used as triggers to search for UHE neutrinos in the neutrino detectors.
The probability to detect more than one neutrino associated with a GW signal
can be given as

$\displaystyle q\big{(}d^{\rm UL}_{\rm GW},T_{\rm op}\big{)}$
$\displaystyle=1-{\rm exp}\bigg{(}-T_{\rm op}I\big{(}d_{\rm GW}^{\rm
UL}\big{)}\bigg{)}\,,$ $\displaystyle I\big{(}d_{\rm GW}^{\rm
UL}\big{)}=4\pi\int_{0}^{d_{\rm GW}^{\rm UL}}d($ $\displaystyle d_{\rm
com})\frac{T_{\rm op}}{\big{(}1+z\big{)}}R_{\rm app}\big{(}z\big{)}d_{\rm
com}^{2}P_{n\geq 1}\big{(}d_{L}\big{)}\,,$ (5)

where $d^{\rm UL}_{\rm GW}$ is defined as the limiting distance for accepting
triggers from the GW detectors to perform the neutrino searches, $R_{\rm
app}(z)$ is the apparent redshift-dependent BNS merger rate, and $T_{\rm op}$
is the operation time of the GW detector. The integration is performed over
the comoving distance $d_{\rm com}$ which is related to $d_{L}$ and $z$,
$d_{L}=(1+z)d_{\rm com}$. For our analysis of triggered stacking searches,
given the large $d_{L}$ for sources, $N_{\nu}<<1$. This, combined with
accounting for the field of view (FOV) for the neutrino detectors the above
equation can be written as (see Appendix A for details and derivation)

$I\big{(}d_{\rm GW}^{\rm UL}\big{)}=4\pi\int_{0}^{d_{\rm GW}^{\rm UL}}d(d_{\rm
com})\frac{T_{\rm op}}{\big{(}1+z\big{)}}R_{\rm app}\big{(}z\big{)}d_{\rm
com}^{2}\left[\int_{E_{\nu}^{\rm LL}}^{E_{\nu}^{\rm
UL}}dE_{\nu}\phi_{\nu}\big{(}\mathcal{E}_{\nu}^{\rm
UHE,iso},E_{\nu},d_{L}\big{)}\frac{1}{2}\int d({\rm
cos}\theta_{z})\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})\right]\,.$ (6)

The redshift dependent rate of BNS mergers is derived from the rate of short
gamma-ray bursts (GRBs) [63]. This true rate involves two branches which split
at $z\approx 0.9$ and is proportional to the fiducial rate at $z=0,\ R_{\rm
true}(z=0)=R_{0}$. The fiducial rate is poorly constrained from the current
observations and can vary between $10\ {\rm Gpc}^{-3}{\rm yr}^{-1}-1700\ {\rm
Gpc}^{-3}{\rm yr}^{-1}$ [64, 65]. We mostly choose a conservative value of
$R_{0}=300\ {\rm Gpc}^{-3}{\rm yr}^{-1}$ unless otherwise stated. The GW
detectors provide the value of the true fiducial rate $R_{0}$. However the
neutrino detectors are sensitive to the apparent redshift-dependent rate
$R_{\rm app}(z)$ defined as

$R_{\rm app}(z)=f_{\rm bm}R_{\rm true}(z)=f_{\rm bm}R_{0}\mathcal{R}(z)=R_{\rm
app,0}\mathcal{R}(z)\,,$ (7)

where the fiducial apparent rate $R_{\rm app,0}=f_{\rm bm}R_{0}$ and
$\mathcal{R}(z)$ gives the redshift dependent rate [63, 41], such that $R_{\rm
true}(z)=R_{0}\mathcal{R}(z)$. We consistently take this into account in our
computations.

###  III.3 Distance limits from sky localization in GW detectors

In this section, we will focus on the role of the GW detectors in the
triggered search technique described above and how it quantifies $d^{\rm
UL}_{\rm GW}$ in Eqs. (III.2) or (6). The triggered search technique described
above would be reasonable if the time of neutrino emission peak post the BNS
merger was certain. However as discussed in Sec. III.1, this time window
$\delta t$ is highly uncertain. Moreover, the next generation GW detectors are
estimated to be sensitive to BNS mergers from very high redshifts. In
particular, CE and ET would be sensitive to BNS mergers from $z\sim 10$ and
$z\sim 3.5$ respectively [66]. Given the rate of the BNS mergers this implies
$\mathcal{O}(100)$ BNS mergers detected in a day [66]. Given the large number
of triggers, it then becomes important to have a selection criteria for
_meaningful triggers_ and quantify the same, such that, the triggered search
technique described above can still yield useful results.

There are multiple ways to address the above issue. For this work, we use the
sky localization capabilities of the next-generation GW detectors to achieve
the desired result. This in turn provides an upper limit on the distance from
which meaningful triggers can be collected - $d^{\rm UL}_{\rm GW}$. Let us now
briefly discuss the steps to evaluate $d^{\rm UL}_{\rm GW}$. Assume the GW
detector detects a BNS merger at some distance $d_{L}$ and provides a trigger
at some time $t_{0}$. The GW detector has a sky localization capability
quantified by the size of the error or uncertainty region
$\Delta\Omega(d_{L})$ given a distance $d_{L}$. This depends on the properties
of the detector. For example, ET has an equilateral configuration where the
signal events can be triangulated, leading smaller $\Delta\Omega$ as compared
to CE for the same distance $d_{L}$.

To search for a neutrino event corresponding to the trigger at $t_{0}$ one
needs a time-window $\delta t$. However within this time $\delta t$, given the
rate of BNS mergers $R_{\rm true}(z)$ there will be more signal detections by
the GW detector each with a corresponding distance and hence a corresponding
uncertainty region in the sky. Thus, at any point in time $\delta t$ post the
trigger at $t_{0}$ the total fraction of sky area covered ($f_{\rm cov}$) due
to the uncertainty regions of the subsequent detections by the GW detector,
within the time-window $\delta t$ can be given as

$\int_{0}^{d_{\rm GW}^{\rm lim}}d\big{(}d_{\rm
com}\big{)}\frac{\Delta\Omega\big{(}d_{L}\big{)}}{4\pi}R_{\rm
true}\big{(}z\big{)}4\pi d^{2}_{\rm com}\delta t=f_{\rm cov}(d_{\rm GW}^{\rm
lim})\,.$ (8)

The size of the uncertainty area $\Delta\Omega$ for CE, ET, and CE+ET is
implemented from Ref. [67]. Given the quality of data needed for the search
one can choose a threshold value for the fraction of sky-area covered, say
$f_{\rm th}$, that is, if at any given time post $t_{0}$, $f_{\rm cov}>f_{\rm
th}$ we stop accepting any more triggers. This fixes the limiting distance for
the GW detectors by demanding $d_{\rm GW}^{\rm lim}$ to be such that $f_{\rm
cov}(d_{\rm GW}^{\rm lim})=f_{\rm th}$. Thus the more signal events farther
away hence low in SNR and have larger uncertainty area, the quicker $f_{\rm
cov}$ reaches $f_{\rm th}$, the lesser the chances of such meaningless
triggers to be accepted. Smaller values of $f_{\rm th}$ would be suitable for
good quality data with lesser $d_{\rm GW}^{\rm lim}$, while larger values
would be suited for constraining various neutrino emission models. Thus for
this work, we will choose $f_{\rm th}$ depending on the properties of the
detector.

We also fix a maximum and minimum distance horizon for the GW detectors as
$d_{\rm GW}^{\rm hor,max}$ and $d_{\rm GW}^{\rm hor,min}$ respectively. The
maximum distance horizon is chosen such that beyond this distance all triggers
can be used. We fix this to the upper limit of the distance horizon for BNS
mergers for ET, $z_{\rm GW}^{\rm hor,max}\sim 3.5$. This choice has no effect
on our results since the BNS merger rate decreases for $z\gtrsim 0.9$. The
minimum distance horizon is chosen to $d_{\rm GW}^{\rm hor,min}\approx 500$
Mpc. This choice is motivated by the maximum horizon distance of LIGO A#
(A-sharp) [68]. This is reasonable because although we focus on the next-
generation of GW detectors in this work, the timescales for realizations of ET
and CE would be such that we will also have the improved next-generation of
LIGO detectors. Such detectors can help us improve the localization for nearby
sources. Besides, upcoming EM telescopes like the Vera C. Rubin Observatory’s
Legacy Survey of Space and Time (LSST) [69] or Roman [70] can detect kilonovae
as a result of BNS mergers from redshifts $z\sim 0.1$ and $z\sim 0.2$
respectively. This would greatly improve the sky localizations over just the
GW observations. This justifies our choice of the minimum distance horizon.
Finally, putting it altogether we have,

$d^{\rm UL}_{\rm GW}={\rm max}\left[d_{\rm GW}^{\rm hor,min},{\rm
min}\left(d_{\rm GW}^{\rm lim},d_{\rm GW}^{\rm hor,max}\right)\right]\,.$ (9)

##  IV Results

Figure 1:  Probability of coincident detection of GW-triggered UHE neutrinos
$q\big{(}d_{\rm GW}^{\rm UL},T_{\rm op}\big{)}$ (hence $1-q\big{(}d_{\rm
GW}^{\rm UL},T_{\rm op}\big{)}$ gives the probability to not see a coincident
neutrino event) with the operation timescale of GW detectors $T_{\rm op}$ for
GRAND-200k. The _left_ panels show the results when $\delta t=10^{2}$ s
(instantaneous) while the _right_ panels show the results when $\delta
t=10^{6}$ s. The _top_ panels show the results from using different GW
detectors whereas the _bottom_ panels highlight how the prospects depend on
$\mathcal{E}_{\nu}^{\rm UHE,iso}$. The fiducial case for each panel is shown
with a thick solid purple line, where we choose $\mathcal{E}_{\nu}^{\rm
UHE,iso}=2.5\times 10^{50}$ erg. In case of nondetection, the $2\sigma$ and
$3\sigma$ C.L.s with which the parameter space can be constrained are also
shown with dotted horizontal lines. We choose the true fiducial rate
$R_{0}=300\ {\rm Gpc}^{-3}{\rm yr}^{-1}$, however, the neutrino searches are
sensitive to $R_{\rm app}(z)$, see Eq. 7), for which we choose $f_{\rm bm}=1$%
and hence $R_{\rm app,0}=3\ {\rm Gpc}^{-3}{\rm yr}^{-1}$.

We discuss the main results of our work in this section. In particular, we
distinguish between the upcoming UHE neutrino detectors – GRAND and IceCube-
Gen2 Radio, and the currently operating detector RNO-G. The results are
presented with the aim to investigate the prospects in terms of timescale of
operation of the GW detectors for synergic GW triggered neutrino observations
or putting constraint at a given C.L. (confidence level) in case of
nondetections.

The results are shown as plots of ${\rm log}_{10}\big{(}1-q(d^{\rm UL}_{\rm
GW},T_{\rm op})\big{)}$ (see Eq. III.2) with the operation time of the GW
detectors in years. Recall that $q\big{(}d^{\rm UL}_{\rm GW},T_{\rm
op}\big{)}$ gives the probability to detect one or more _coincident_
neutrino(s) associated with a GW signal111Note that here we only consider the
probability of neutrino detections associated with the GW trigger. So it is
equivalent to saying that the neutrino detection significance is $\sim
5\sigma$ and we will consider this to be the case for the remaining text when
we describe _coincident neutrino detection_. However, the association of the
event with BNS mergers is non-trivial. This requires a separate analysis
pertaining to the probability of random associations between the neutrino
event and the BNS mergers. A $5\sigma$ C.L. coincident neutrino detection from
a BNS merger can only be claimed if _both_ the neutrino detection and
association significance are above $5\sigma$. For association significance,
the error region associated with the neutrino events will also play an
important role., thus when $q\big{(}d^{\rm UL}_{\rm GW},T_{\rm op}\big{)}$ is
0, we have ${\rm log}_{10}\big{(}1-q(d^{\rm UL}_{\rm GW},T_{\rm op})\big{)}$
to be 0 corresponding to 0 years of operation. As the operation time for the
GW detectors increase, $q\big{(}d^{\rm UL}_{\rm GW},T_{\rm op}\big{)}$
increases and ${\rm log}_{10}\big{(}1-q(d^{\rm UL}_{\rm GW},T_{\rm
op})\big{)}$ decreases. Thus lower values of ${\rm log}_{10}\big{(}1-q(d^{\rm
UL}_{\rm GW},T_{\rm op})\big{)}$ indicate higher probabilities of finding a
neutrino event corresponding to GW triggers. We also show the $95$%
($2\sigma$) and $99.7$% ($3\sigma$) C.L.s to constrain the associated
parameter space in case of a non-detection by dotted dark red and dark green
horizontal lines respectively.

We fix the true fiducial rate $R_{0}=300\ {\rm Gpc}^{-3}{\rm yr}^{-1}$ and
choose $f_{\rm bm}=1$%. This gives the fiducial apparent rate as $R_{\rm
app,0}=3\ {\rm Gpc}^{-3}{\rm yr}^{-1}$ (see Eq. 7). The next parameter that we
need to choose to perform our analysis is the threshold fraction of sky area
covered $f_{\rm th}$. This, in general, is motivated by the extent to which
the detectors can perform sky localization and is limited by the background in
the neutrino detector associated with triggered searches. At the neutrino
energies of interest for this work, the atmospheric background is minimal and
the astrophysical background is estimated to be low. Thus it is possible to
choose relatively large values of $f_{\rm th}$. In principle, we can choose
the most optimistic value of $f_{\rm th}\sim 10^{-2}$ for all the detectors.
However, to give conservative estimates we choose $f_{\rm
th}=10^{-2},10^{-2},$ and $10^{-3}$ for CE, ET, and the combination of CE+ET.
Finally, the two main parameters varied to compute the results are the
isotropic-equivalent energy emitted in UHE neutrinos ($\mathcal{E}_{\nu}^{\rm
UHE,iso}$) and the time duration post the merger ($\delta t$) when the
neutrino emission peak is reached, which is the same as the size of the search
window.

Figure 2:  Same as Fig. 1 but for IceCube-Gen2 Radio. For IceCube-Gen2 Radio
we choose the fiducial case as $\mathcal{E}_{\nu}^{\rm UHE,iso}=2.5\times
10^{51}$ erg.  Figure 3:  Contour plots highlighting the joint GW and neutrino
detector operation timescales ($T_{\rm op}$) in years for GW-triggered UHE
neutrino searches. The contours in the _upper_ panel denote $T_{\rm op}$ for
the GW and neutrino detectors required to have a $95$% probability to detect
coincident UHE neutrino events or constrain the parameter space at the
$2\sigma$ level in case of nondetections for GRAND (_left_ panels) and
IceCube-Gen2 Radio (_right_ panels). The _lower_ panels also show the same
contours for the same detectors, but for $99.7$% probability to detect
coincident UHE neutrino events or constrain the parameter space at the
$3\sigma$ level in case of nondetections. The search is performed assuming a
combination of CE+ET as the GW detectors with $f_{\rm th}=10^{-3}$ and the
true fiducial rate $R_{0}=300\ {\rm Gpc}^{-3}{\rm yr}^{-1}$ (this is the true
rate, however the neutrino searches are sensitive to $R_{\rm app}(z)$, see Eq.
7). We choose $f_{\rm bm}=1$%.

For the fiducial values of $\delta t$, we divide it into two classes –
instantaneous emission where $\delta t=10^{2}$ s and the case when the
neutrinos are emitted $\mathcal{O}(10)$ days post the merger, that is, $\delta
t=10^{6}$ s. The instantaneous case yields more optimistic results than the
case when $\delta t=10^{6}$ s. This is because as $\delta t$ increases, the
number of GW triggers within $\delta t$ also increases. This leads to a
decrease in the limiting distance $d_{\rm GW}^{\rm lim}$ for a fixed value of
$f_{\rm th}$. Thus the upper limit of in the integral in Eqs. (III.2) or (6)
goes down leading to the requirement of longer operation timescales. We give
the details of various parameters in Table 1.

$\delta t$ |  GW |  $f_{\rm th}$ |  $d_{\rm GW}^{\rm UL}[z]$ |  $N_{\rm trig}$
---|---|---|---|---
(s) |  Detectors |  |  (Gpc) |  (yr-1)
|  CE |  $10^{-2}$ |  $1.07\ [0.21]$ |  $9.6\times 10^{1}$
100 |  ET |  $10^{-2}$ |  $2.80\ [0.48]$ |  $1.6\times 10^{3}$
|  CE+ET |  $10^{-3}$ |  $4.72\ [0.74]$ |  $7.7\times 10^{3}$
|  CE+ET |  $10^{-2}$ |  $8.47\ [1.20]$ |  $3.3\times 10^{4}$
|  CE or ET |  - |  $0.50\ [0.10]$ |  $9.1\times 10^{0}$
$10^{6}$ |  CE+ET |  $10^{-3}$ |  $0.70\ [0.14]$ |  $2.6\times 10^{1}$
|  CE+ET |  $10^{-2}$ |  $1.14\ [0.22]$ |  $1.1\times 10^{2}$
Table 1:  Parameters relevant for computing $d_{\rm GW}^{\rm UL}$ (see Eq. 9)
for different GW detectors for the instantaneous $\delta t=100$ s and $\delta
t=10^{6}$ s scenarios. The true fiducial rate is chosen to be $R_{0}=300\ {\rm
Gpc}^{-3}{\rm yr}^{-1}$. The values of $d_{\rm GW}^{\rm UL}$ shown in the
table are used to compute the corresponding $q(d^{\rm UL}_{\rm GW},T_{\rm
op})$ (see Eq. III.2) shown in all the figures. The number of GW triggered
events per year ($N_{\rm trig}$) for a corresponding choice of detector and
$f_{\rm th}$ is also shown. A fraction $f_{\rm bm}=1$% of $N_{\rm trig}$ are
expected to contribute to on-axis sGRBs. Figure 4:  Same as Fig. 1 but for
RNO-G. For RNO-G we choose the fiducial case as $\mathcal{E}_{\nu}^{\rm
UHE,iso}=2.5\times 10^{52}$ erg. We only focus on the results of using
different GW detectors as triggers for the searches.  Figure 5:  Same as Fig.
3 but for RNO-G, where the operation timescales are shown in the
$\mathcal{E}_{\nu}^{\rm UHE,iso}-R_{\rm app,0}$ plane. For the apparent rate
we choose $f_{\rm bm}=1$% (see Eq. 7). The search is performed assuming a
combination of CE+ET as the GW detectors with $f_{\rm th}=10^{-2}$ and $\delta
t=1000$ s.

###  IV.1 Prospects for upcoming UHE neutrino detectors: GRAND and IceCube-
Gen2 Radio

We show the results for GRAND and IceCube-Gen2 Radio in Figs. 1 and 2
respectively. We choose the fiducial values of $\mathcal{E}_{\nu}^{\rm
UHE,iso}=2.5\times 10^{50}$ erg and $2.5\times 10^{51}$ erg for GRAND and
IceCube-Gen2 Radio respectively. This is within reasonable limits based on
predictions from various models of UHE neutrino emission from BNS mergers. The
lower values of instantaneous effective areas for IceCube-Gen2 Radio as
compared to GRAND requires us to choose $\mathcal{E}_{\nu}^{\rm UHE,iso}$ one
order of magnitude higher for the former (see Sec. III). The _top right_
panels of Figs. 1 and 2 show the results for the instantaneous fiducial case.
Both for GRAND and IceCube-Gen2 Radio, assuming the fiducial case, there is
$\sim 99$% probability to detect coincident neutrino event/events, using
triggers from CE+ET in a timescale of $\lesssim 15$ years. In case of a non-
detection $3\sigma$ level constraints can be put on the parameter space in
less than 15 years. Considering the most optimistic value of $f_{\rm
th}=10^{-2}$, increases the probability to $\sim 99$% for coincident
detections and decreases the time required to place $3\sigma$ level
constraints in case of non-detections to $\lesssim$ 10 years. Using ET and CE
individually, $3\sigma$ and $2\sigma$ level constraints can be put in
timescales $\lesssim$ 25 and 30 years respectively, in the case of non-
detection of UHE neutrinos. These results are optimistic and present
reasonable chances of coincident neutrino detection or the opportunity to
place constraints provided the neutrino emission is prompt post-merger.
However it is important to keep in mind that although the timescales required
for detection or placing constraints are similar for GRAND and IceCube-Gen2
Radio, our analysis based on the currently available effective areas, requires
larger values (by a factor of 10) of energy emitted in UHE neutrinos for
IceCube-Gen2 Radio to place the same constraints.

The _top left_ panels in Figs. 1 and 2 show the fiducial case but for $\delta
t=10^{6}$ s. As expected for this case the results are less optimistic. We see
that both for GRAND and IceCube-Gen2 Radio, the $90$% probability to detect
one or more coincident neutrino events requires a very long period of $\sim
20$ years of joint operation time even after assuming the most optimistic
threshold $f_{\rm th}=10^{-2}$ for CE+ET. Only $2\sigma$ level constraints in
case of nondetections are possible in a joint operation timescale of $\sim 25$
years using CE+ET. For the individual detectors we find $d_{\rm GW}^{\rm UL}$
to be fixed at $d_{\rm GW}^{\rm hor,min}=500$ Mpc (see Eq. 9). For this case
we can only have $\sim 80$% chance of coincident GW triggered UHE neutrino
detections.

The _bottom left_ and _bottom right_ panels of Fig. 1 (GRAND) and Fig. 2
(IceCube-Gen2 Radio) show the effects of varying the total energy emitted in
UHE neutrinos ($\mathcal{E}_{\nu}^{\rm UHE,true}$) for the instantaneous case
($\delta t=100$ s) and $\delta t=10^{6}$ s respectively. In these panels we
only show the results assuming CE+ET working together and choosing the
fiducial value of $f_{\rm th}=10^{-3}$. As a general trend we see that the
more the energy emitted in UHE neutrinos, the greater the number of events and
lesser the timescales of operation to have coincident neutrino detections or
stronger constraints in the case of nondetections. We vary
$\mathcal{E}_{\nu}^{\rm UHE,iso}$ between $10^{50}$ erg to $5\times 10^{50}$
erg $10^{51}$ erg to $5\times 10^{51}$ erg for GRAND and IceCube-Gen2 Radio
respectively.

For the upper limit of $\mathcal{E}_{\nu}^{\rm UHE,iso}$ we have the best case
scenario for the instantaneous case, where there is a $\sim 99$% probability
to detect a coincident neutrino event in $\sim 7$ years of joint operation of
the neutrino and GW detectors. The $3\sigma$ level constraints in case of non-
detection can be placed over similar timescales. Even for the least optimistic
scenario individually for GRAND and IceCube-Gen2 Radio, the probability of
coincident UHE neutrino detection is $\sim 99$% in $30$ years or non-detection
of the UHE neutrinos would lead to $3\sigma$ constraints on the parameter
space over similar timescales of operation. When $\delta t=10^{6}$ s, it would
take $\sim 20$ years of joint operation of the GW and the neutrino detectors
to have a $95$% probability to detect an event or place a $2\sigma$ level
constraint in the case of non-detection. If $\mathcal{E}_{\nu}^{\rm UHE,iso}$
is close to the lower limit we predict the instantaneous scenario might still
be able to have coincident neutrino observation in $\sim 15-20$ years with a
probability $\sim 95$%. However the prospects are not very promising if
$\delta t=10^{6}$ s, where the probability of such an observation is $<70$%.
This is to be expected, since this is a representation of a pessimistic
scenario where $\mathcal{E}_{\nu}^{\rm UHE,iso}$ is low and $\delta t$ is
large.

One of the main takeaways from this work is shown in Fig. 3 where we
investigate the prospects of coincident GW-triggered (using CE+ET with $f_{\rm
th}=10^{-3}$) UHE neutrino searches in the plane of $\mathcal{E}_{\nu}^{\rm
UHE,iso}$ and $\delta t$ – the two primary parameters in our analysis. The
contours in the _upper_ panels denote the joint operation timescale $T_{\rm
op}$ for the GW and neutrino detectors in years required to have a $95$%
probability to detect coincident UHE neutrino events or constrain the
parameter space at the $2\sigma$ level in case of nondetections for GRAND
(_left_ panel) and IceCube-Gen2 Radio (_right_ panel). The _lower_ panels also
show the same contours for the same detectors, but for $99.7$% probability to
detect coincident UHE neutrino events or constrain the parameter space at the
$3\sigma$ level in case of nondetections. We limit the timescale to a
reasonable value of $20$ years of operation, so any part of the parameter
space that would need more than $20$ years is shown in white.

The most optimistic parameters lie in the top left corner where
$\mathcal{E}_{\nu}^{\rm UHE,iso}\gtrsim 10^{51}$ erg and $\delta t\lesssim
100$ s. Note that for high-energy neutrinos this parameter space is ruled out
based on sGRB limits from IceCube [61]. The current limits from high energy
neutrinos are computed by considering $90$% C.L. limits, obtained from the
time-integrated flux at 1 TeV for all sGRBs assuming an injected spectrum of
$E_{\nu}^{-2.28}$. Considering an average sGRB redshift distance of $z_{\rm
GRB}^{\rm avg}=0.5$, one can convert the flux into $\mathcal{E}_{\nu}^{\rm
HE,iso}$. There are however some key differences in showing the same limits
for $\mathcal{E}_{\nu}^{\rm UHE,iso}$ based on this analysis. Firstly, our
study assumes a $E_{\nu}^{-2}$ spectrum which is harder than what was
considered in Ref. [61]. Second, the energy ranges for high-energy neutrinos
would be $10^{3}$ GeV to $10^{6}$ GeV which is different from the UHE case,
where the energy range is given by $10^{7}$ GeV to $10^{11}$ GeV.

Besides, the number of sGRBs obtained from implementing our current analysis
would be different from the one obtained in Ref. [61]. This is because we use
the GW triggers for calculating the number of sGRBs whereas, Ref. [61] used
the number of sGRBs observed in the electromagnetic band. The number of sGRBs
considered for the analysis in Ref. [61] was $317$ in 10 years, that is, $\sim
31$ per year. Let us look at some estimates related to the number of triggers
that we would obtain from our analysis. Assuming a distance horizon for BNS
mergers of $z\sim 3.2$ for the Einstein Telescope, gives $\sim 4.2\times
10^{4}$ triggers per year. Note that this is without implementing the distance
limits from considering the sky localization of the events. A fraction $f_{\rm
bm}=1$% of these triggers are expected to contribute to on-axis sGRBs, that
is, the number of such events are $\sim 4.2\times 10^{2}$ per year, which is
roughly an order of magnitude larger than what was used in Ref. [61]. On
implementing the distance cuts, the number of events based on this will be
even lower due to the reduced distance horizon. We show the number of GW-
triggered events per year ($N_{\rm trig}$) given a detector and a choice of
$f_{\rm th}$ in Table 1 to provide an estimate.

###  IV.2 Current prospects: RNO-G

The results for the currently operating RNO-G are shown in Fig. 4. We show the
prospects for the instantaneous case and $\delta t=10^{6}$ s in the _left_ and
the _right_ panels respectively. As discussed in Sec. II, RNO-G is aimed to be
a precursor to IceCube-Gen2 Radio and hence has comparatively smaller
effective areas. Due to this we choose the fiducial value of
$\mathcal{E}_{\nu}^{\rm UHE,iso}$ an order of magnitude higher than what we
chose for IceCube-Gen2 Radio, that is, $\mathcal{E}_{\nu}^{\rm
UHE,iso}=2.5\times 10^{52}$ erg. This of course, is a very optimistic choice
and is unlikely. However, the goal of the results in this section is just to
provide an estimate of the performance of RNO-G for the purpose of GW-
triggered UHE neutrino searches.

For the most optimistic choice of $f_{\rm th}=10^{-2}$ using CE+ET, RNO-G has
a $99.7$% probability to detect a coincident UHE neutrino event or constrain
the parameter space at $3\sigma$ in the case of non-detection in a timescale
of $\sim 10$ years for the instantaneous case when $\delta t=100$ s. If
$\delta t\sim 10^{6}$ s, the joint operation timescales required to have a
$95$% probability to detect a coincident UHE neutrino event is $\gtrsim 30$
years, which is similar to the timescale required to place constraints at the
$2\sigma$ level in case of non-detections.

It is evident that given the fiducial estimates, RNO-G due its small size
requires impractical timescales of joint operation to detect coincident
neutrino events or constrain the parameter spaces. However recall that the
fiducial true rate of BNS mergers have a huge uncertainty window ranging from
$10\ {\rm GeV}^{-3}{\rm yr}^{-1}$ to $1700\ {\rm GeV}^{-3}{\rm yr}^{-1}$. Thus
one can speculate assuming the energy emitted in UHE neutrinos is indeed
optimistic, what sort constraints can RNO-G place in the plane of isotropic-
equivalent energy emitted in UHE neutrinos ($\mathcal{E}_{\nu}^{\rm UHE,iso}$)
and the apparent fiducial rate ($R_{\rm app,0}$)? This is what we show in Fig.
5. We show the results in the apparent rate and the isotropic-equivalent
energy emitted in UHE neutrinos plane, assuming the beaming factor to be
$f_{\rm bm}=1$%222 Note that the beaming factor important for neutrino
production is not well-constrained and thus the relations between the
isotropic-equivalent and the true energy and the true and the apparent rate
could be non-trivial.. We also assume that the neutrinos are emitted at a
typical value of $\delta t\lesssim 1000$ s post-merger hence fixing the size
of the time-window for the triggered search. The results manifest that given
the assumptions, RNO-G might be able to put constraints on the apparent rates
based on non-detections at the $2\sigma$ level in a joint operation timescale
of $\gtrsim 8$ years. In the meantime, if more precise measurements of the
true rate if BNS mergers are reported by the GW detectors,
$\mathcal{E}_{\nu}^{\rm UHE,iso}$ can be constrained.

##  V Summary

(a) (a) (b) (b)

Figure 6: The instantaneous field of view (FOV) for GRAND, IceCube-Gen2 Radio, and RNO-G for an arbitrarily chosen date of December 15, 2032 and time of the day 12:00:00 (_left_ panel) and 23:00:00 (_right_ panel) (this choice is arbitrary but reflects the complementary nature of the radio neutrino detectors). For IceCube-Gen2 Radio we only show the FOV which has the highest effective areas for neutrino energies $E_{\nu}\gtrsim 10^{7}$ GeV. The locations used for summit stations of the detectors are provided in Table 2.  Neutrino Detectors |  Latitude |  Longitude |  Elevation (m) |  Zenith Angle
---|---|---|---|---
GRAND |  $43^{\circ}00^{\prime}00.00^{\prime\prime}$ N |  $80^{\circ}00^{\prime}00.00^{\prime\prime}$ E |  1300 |  $85^{\circ}-95^{\circ}$
GRAND (Auger site) |  $37^{\circ}12^{\prime}48.62^{\prime\prime}$ S |  $55^{\circ}06^{\prime}35.10^{\prime\prime}$ W |  1000 |  $85^{\circ}-95^{\circ}$
IceCube-Gen2 Radio |  $90^{\circ}00^{\prime}00.00^{\prime\prime}$ S |  $45^{\circ}00^{\prime}00.00^{\prime\prime}$ E |  2835 |  $50^{\circ}-90^{\circ}$
RNO-G |  $72^{\circ}34^{\prime}27.89^{\prime\prime}$ N |  $38^{\circ}27^{\prime}19.83^{\prime\prime}$ W |  3216 |  $46^{\circ}-83^{\circ}$
Table 2:  Location of the various radio neutrino detectors along with their
zenith angle covergae, used to show the instantaneous FOV in Fig. 6.

The search for ultrahigh-energy neutrinos from BNS mergers is timely because
of the upcoming radio neutrino detectors. To maximize our chances of detecting
UHE neutrinos from BNS mergers the GW-triggers associated with the latter can
be utilized. In this work, we explore the prospects of GW-triggered stacking
searches of UHE neutrinos at the radio neutrino detectors using GW triggers
from the next-generation GW detectors like Cosmic Explorer (CE), Einstein
Telescope (ET) and the combination of the two (CE+ET).

The probability of detect one or more coincident UHE neutrino event is given
by Eq. (6). This depends on the joint operation time of the GW and neutrino
detectors, the redshift dependent rate of BNS mergers, the flux of UHE
neutrinos, the zenith- and energy-dependent effective areas of the neutrino
detectors, and their respective FOVs. For our work, we use the consistent
zenith- and energy-dependent effective areas along with FOVs to compute our
results. Furthermore, for the next generation of GW detectors, given their
large distance horizon for BNS mergers, it is also important to collect
_meaningful_ triggers to perform the triggered-stacking searches we present in
this work. To achieve this, we use the sky-localization capabilities of the GW
detectors and impose a maximum threshold of sky area covered ($f_{\rm th}$) to
get an upper limit on the maximum distance ($d_{\rm GW}^{\rm UL}$) from which
meaningful triggers can be collected (see Eqs. III.2 or 6).

We present our results in the form of probability to detect coincident UHE
neutrinos ($q(d_{\rm GW}^{\rm UL},T_{\rm op})$) with the timescale of joint
operation for the GW and neutrino detectors ($T_{\rm op}$) in years. We also
show in case of non-detections the time it would take to put $2\sigma$ and
$3\sigma$ level constraints in the parameter space. We analyze two different
scenarios of UHE neutrino emission timescales motivated by different physical
models - an instantaneous emission with $\delta t=100$ s and when the emission
happens $\delta t=10^{6}$ s post-merger. The former presents us with the most
optimistic scenarios while the latter leads to far less optimistic results.

In Fig. 1 we show the results for GRAND. For the fiducial case, is UHE
neutrino emission is instantaneous, assuming a very reasonable energy emitted
in UHE neutrinos $\mathcal{E}_{\nu}^{\rm UHE,iso}=2.5\times 10^{50}$ erg, we
find that the probability to see coincident UHE neutrino events is $99.7$% in
less than 15 years of joint operation with CE+ET detectors. Constraints at the
$3\sigma$ level can be imposed in the case of non-detections in similar
timescales using CE+ET. For $\delta t=10^{6}$ s however, to have a $95$%
probability for a coincident detection or to place constraints at the
$2\sigma$ level in the case of non-detection, roughly $25$ years oj joint
operation is required, assuming CE+ET working with the most optimistic $f_{\rm
th}=10^{-2}$. For IceCube-Gen2 Radio, we show our results in Fig. 2. The exact
same conclusions can be arrived at by choosing the fiducial energy emitted in
UHE neutrinos to be one order of magnitude higher, that is,
$\mathcal{E}_{\nu}^{\rm UHE,iso}=2.5\times 10^{51}$ erg. This is a result of
the instantaneous effective areas of GRAND being higher than that of IceCube-
Gen2 Radio. We also show how the conclusions change if $\mathcal{E}_{\nu}^{\rm
UHE,iso}$ is varied across a range.

Probably our most important takeaway from this work is shown in Fig. 3. We
present the prospects of joint GW-triggered UHE neutrino searches using CE+ET
and choosing $f_{\rm th}=10^{-3}$. We plot the contours for the joint
operation timescales ($T_{\rm op}$) required to have a $95$% and $99$%
probabilities to detect coincident UHE neutrino events or constrain the
parameter space at the $2\sigma$ and $3\sigma$ levels in case of nondetections
for GRAND and IceCube-Gen2 Radio. We note that GRAND, for
$\mathcal{E}_{\nu}^{\rm UHE,iso}\gtrsim 2\times 10^{51}$ erg, can probe the
parameter space across all values of $\delta t$ for the desired probabilities
and level of constraints in an operation timescale of $\sim 20$ years.

We also investigate the prospects for the current generation detector RNO-G.
Since RNO-G is operating as a precursor to IceCube-Gen2 Radio, it has
comparatively smaller effective areas than the latter. This mandates the
fiducial choice of $\mathcal{E}_{\nu}^{\rm UHE,iso}=2.5\times 10^{52}$ erg, an
order of magnitude higher than that chosen for IceCube-Gen2 Radio and
extremely optimistic and unlikely. Our results are shown in Fig. 4. Using
RNO-G and CE+ET with the most optimistic $f_{\rm th}=10^{-2}$, the probability
for a coincident UHE neutrino detection is $\sim 10$ years of joint operation,
assuming the instantaneous UHE neutrino emission scenario. Constraints at the
$2\sigma$ level can be placed in slightly more than $7$ years in the same
scenario. If $\delta t=10^{6}$ s, 30 years of joint operation time is required
to place $2\sigma$ level constraints or to have a $95$% chance of coincident
detection probability. We also show the prospects of RNO-G to constrain the
apparent rates assuming optimistic values of energy emitted in UHE neutrinos
in Fig. 5.

##  VI Discussion and outlook

In this section we discuss the shortcomings of our current work and the
implications. We aim to partially answer the questions we posed in Sec. I,
that is, what could the joint GW and UHE neutrino detectors add to the
multimessenger paradigm in the context of searching for UHE neutrinos from BNS
mergers.

Let us first discuss the possible improvements and caveats of our current
work. Perhaps, the most important shortcoming of this work is the fact that in
our analysis, we ignore the trial factors that would be associated with
performing a triggered search with an unknown time-window. For each time
window we choose to perform our analysis, a weighting needs to be done for the
associated trial factor. This is beyond the scope of our current work. The
assumption here is that the BNS mergers uniformly emit UHE neutrinos with the
same $\delta t$, which is crude. However, this is still useful, since this can
enable us to test for and constrain physical models that predict a particular
order of magnitude for $\delta t$. Accounting for the trial factors can
possibly increase the required timescales for achieving the desired
probability of detection or placing constraints in the case of nondetection.

There are other factors that contribute to uncertainties as well. The true
fiducial rate has a very large range and can vary between $10\ {\rm
Gpc}^{-3}{\rm yr}^{-1}$ to $1700\ {\rm Gpc}^{-3}{\rm yr}^{-1}$. The lower
limit would increase the timescale by roughly an order magnitude whereas the
upper limits can reduce the same by an order of magnitude. Next, we ignore the
duty cycles of the radio neutrino and the GW detectors and assume it to be
unity. In reality, the GW detectors have a duty cycle of $\sim 70$% [71]. The
radio neutrino detectors can generally have a duty cycle of $\sim 100$% [45,
59] but the duty-cycle for RNO-G is $\sim 70$% [46]. This can have minor
effects in our results. Furthermore, GW events from on-axis jets can have a
larger strain and hence a lower associated error region. Since we do not
assume this, our results are conservative in this respect. Finally, it is also
important to note that we choose $f_{\rm th}=10^{-3}$ for the combination of
CE+ET to make our predictions conservative. A more optimistic prediction can
be made by choosing $f_{\rm th}=10^{-2}$ for CE+ET. Moreover, the current
configurations for both the Einstein Telescope and Cosmic Explorer are
uncertain. Depending on how the detectors are oriented their respective sky
localization capabilities are subject to change.

The main theme of this work was to highlight the possibilities that the next-
generation GW and the radio neutrino detectors can provide us with, in
particular to look for UHE neutrinos from BNS mergers at the radio neutrino
detectors. The complementarity of the current and the upcoming radio neutrino
detectors is quite powerful in this regard. In Fig. 6 we show the
instantaneous field of view (FOV) for GRAND, IceCube-Gen2 Radio, and RNO-G for
an arbitrarily chosen date of December 15, 2032 and two different instances
during the day corresponding to $12:00:00$ and $23:00:00$. The information on
the location and the zenith angle coverage for these detectors is provided in
Table 2. In particular, we assume that GRAND will be composed of at least two
sub-arrays, for which we show the instantaneous FOV: one in the Northern
hemisphere located near the current 300-antenna prototype site, in Xiao
Dushan, Gansu, China [57], and a second in the Southern hemisphere. For the
latter, we have chosen as an example, the location of the GRAND@Auger
prototype [57] at the Pierre Auger Observatory, in Argentina, but assuming
that the array will be located at an elevation of 1000 m333The motivation for
this site comes in part from the Beamforming Elevated Array for COsmic
Neutrinos (BEACON) experiment [72, 73].. We see that the instantaneous FOVs
help cover large portions in the sky both in the northern ($\delta>0^{\circ}$)
and the southern ($\delta<0^{\circ}$) hemispheres. This is truly remarkable
since the instantaneous FOV is important for short timescale transients in
general, and in particular for this work when $\delta t=100$ s. This can thus
ensure that the radio neutrino detectors can collectively have chance to
detect UHE neutrinos from mostly all BNS mergers, that is, the mergers are in
the combined FOV.

There are a few subtle things associated with the effective areas that are
important to note. Firstly, we use the GRAND-200k effective areas. However, if
GRAND is built in two different sites, each individual site will have a
reduced effective area (by roughly a factor of $1/2$, keeping a constant
overall effective area for the full detector) and the FOV of the combined
locations would go up by a factor of $2$. Thus one cannot directly map our
results assuming GRAND is built in two different sites. Using this reasoning,
one can make a general remark that for triggered-stacking searches as proposed
in this work, the $A_{\rm eff}$ times the FOV is a conserved quantity for
various cluster configurations. A narrower FOV may mean that certain events
will perhaps be missed. However the deeper effective areas resulting from
beamed experiments like GRAND or BEACON, lead to higher rates of neutrino
signals. On the other hand, experiments with a broader FOV like IceCube-Gen2
Radio or RNO-G, imply that there is a greater chance that a specific
multimessenger event will be seen in the neutrino telescope, although this is
not the focus of the current work. Secondly, the apparent difference between
the sensitivities of IceCube-Gen2 Radio and GRAND for this analysis is a
result of the relevant quantity here being the direction-averaged effective
area times the FOV. We stress that for a nearby source this is not true and
hence GRAND and IceCube-Gen2 will have different sensitivities and imply
different results than what we find from this work. This does not affect our
study, since here we focus on far away sources to implement triggered-stacking
searches.

Although we only considered joint GW and neutrino observations, EM
observations can significantly help improve our results. BNS mergers can be
followed by a short GRB or kilonova afterglows. Detection of short GRBs by the
satellite Galileo G2 [74] or the Space- based multi-band astronomical Variable
Objects Monitor (SVOM) [75] and x-ray detectors like Swift BAT [76], Einstein
Probe [77], and HiZ-GUNDAM [78] can help with improving the sky-localization
and hence yield higher values of $d_{\rm GW}^{\rm UL}$ which can then improve
our results. The kilonova detection horizons for the EM detectors is even more
promising. For example, in the optical and infrared bands, the Vera C. Rubin
Observatory’s Legacy Survey of Space and Time (LSST) [79, 69], Roman [70] are
$z\sim 0.1$ and $0.2$ respectively and thus can significantly reduce the sky
localization areas and lead to more optimistic results than what we present
here. In fact we do take this into account in some cases where $d_{\rm
GW}^{\rm UL}<0.5$ Gpc. This is also motivated by the improved second
generation of GW detectors like LIGO A+ [80] and LIGO A# (A-sharp) [68].

The addition of radio neutrino detectors to the existing GW, neutrino, and EM
detectors will bolster the search strategies for UHE neutrino events from BNS
mergers. The implications of a coincident UHE neutrino event from a BNS merger
or constraining the parameter space at $2\sigma$ or $3\sigma$ C.L.s in the
case of nondetections, in reasonably short timescales of joint operation are
major. Our current work presents an effective way and elaborates on a search
technique that is timely and useful for the upcoming era of multimessenger
astrophysics, particularly in the context of BNS mergers and stacking searches
from transient sources.

###### Acknowledgements.

We thank Rafael Alves Batista, Mauricio Bustamante, Cosmin Deaconu, Christian
Glaser, Kaeli Hughes, Marco Stein Muzio, and Foteini Oikonomou for comments
and useful discussions. We are grateful to Christian Glaser for providing us
with the effective volumes for IceCube-Gen2 Radio, Foteini Oikonomou and the
GRAND Collaboration for providing the effective areas for GRAND. M. M. and K.
M. are supported by NSF Grant No. AST-2108466. M. M. also acknowledges support
from the Institute for Gravitation and the Cosmos (IGC) Postdoctoral
Fellowship. K. K. acknowledges support from the Fulbright-France program, the
CNRS Programme Blanc MITI (”GRAND” 2023.1 268448; France), and the CNRS
Programme AMORCE (”GRAND” 258540; France). S.W. acknowledges support from NSF
Grant Nos. 2033500, 2111232, and 2310122. The work of K.M. is supported by the
NSF Grant Nos. AST-2108467, and AST-2308021, and KAKENHI Nos. 20H01901 and
20H05852. S.S.K. acknowledges the support by KAKENHI No. 22K14028, No.
21H04487, No. 23H04899, and the Tohoku Initiative for Fostering Global
Researchers for Interdisciplinary Sciences (TI-FRIS) of MEXT’s Strategic
Professional Development Program for Young Researchers.

##  Appendix A Effective volumes and effective areas

Unlike optical neutrino telescopes like IceCube or KM3NeT, radio neutrino
detectors are sensitive to cascades $\gtrsim 10^{16}$ eV. The prospects for
detection of a cascade thus depends on the orientation of the Cherenkov cone.
Besides, the sparse configuration of the detectors necessitates the use of
effective volumes instead effective areas. To achieve this the effective
volumes $\mathcal{V}_{\rm eff}(E_{\nu},\theta_{z})$ are computed by simulating
neutrino interactions for a given neutrino energy $E_{\nu}$ and zenith angle
$\theta_{z}$ in a volume $V$. Assuming a thin-target this effective volume is
converted to effective area $\mathcal{A}_{\rm eff}(E_{\nu},\theta_{z})$

$\mathcal{A}_{\rm eff}(E_{\nu},\theta_{z})=\frac{\mathcal{V}_{\rm
eff}(E_{\nu},\theta_{z})}{\mathcal{L}_{\rm int}(E_{\nu})}\,,$ (10)

where the interaction length of a neutrino in the Earth is defined as
$\mathcal{L}_{\rm int}(E_{\nu})$, which depends on cross-sections which in
turn depends on the neutrino energies.

There are different ways to estimate an effective area for a given neutrino
detector

  * •

_instantaneous effective area_ \- which is relevant for very short periods of
time $\sim 1-1000$ s,

  * •

_day-averaged effective area_ \- which is relevant searching for neutrinos
from long transients $\sim 10^{5}-10^{7}$ s,

  * •

_direction-dependent effective area_ \- which can depend on the right
ascension (RA) and/or the declination (dec),

  * •

_direction-averaged effective area_ \- which is computed by integrating over
the complete solid angle.

IceCube-Gen2 Radio’s location at the south pole makes it such that the
instantaneous and the day-averaged effective area is the same. However, the
same is not true for GRAND and RNO-G. In this work, we use the zenith (or
declination)-dependent effective area for each of the neutrino detectors.

Assume a radio neutrino detector has an effective area which is sensitive
between zenith angles $\theta_{z}=[\theta_{z}^{\rm min},\theta_{z}^{\rm max}]$
at any instant. The direction-averaged effective area can be defined as,

$\mathcal{A}_{\rm eff}^{\rm avg.}=\frac{1}{N_{\rm norm}}\int
d\Omega\mathcal{A}_{\rm eff}(\phi,\theta_{z},E_{\nu})\,,$ (11)

where $N_{\rm norm}$ is the normalization, $\mathcal{A}_{\rm
eff}(\phi,\theta_{z},E_{\nu})$ is the instantaneous azimuth, zenith444The
zenith angle $\theta_{z}$ can be converted to altitude angle $\alpha$ by
$\alpha=90^{\circ}-\theta_{z}$. However conversion to declination angle
$\delta$ is non-trivial and depends on the location (latitude and longitude)
of the detector and the time of the day., and neutrino energy dependent
effective area, $\phi$ is the azimuthal angle and $\phi=[0,2\pi]$ and
$d\Omega$ is the integration over solid angle given as $d\Omega=d\phi\ d({\rm
cos}\theta_{z})$. Since GRAND, IceCube-Gen2 Radio, and RNO-G have
sensitivities to the whole azimuth at any instant and
$\mathcal{A}(\phi,\theta_{z},E_{\nu})=\mathcal{A}(\theta_{z},E_{\nu})$. Thus
the integration over $d\phi$ is trivial and gives a factor of $2\pi$. The
choice of $N_{\rm norm}$ will depend on whether we want to average over all
sky, that is, $4\pi$ or over the field of view of the detector, in which case,
$N_{\rm norm}=2\pi\left|\big{(}{\rm cos}\ \theta_{z}^{\rm min}-{\rm cos}\
\theta_{z}^{\rm min}\big{)}\right|$. Note that when the full zenith is
covered, as is the case for IceCube-Gen2 $N_{\rm norm}=4\pi$, which gives the
all-sky average. In other words, Eq. (11) provides us with the direction-
averaged neutrino energy dependent effective area in a given zenith band.

For the purposes of triggered searches from distances $\gtrsim 500$ Mpc, the
number of neutrinos in the detector $N_{\nu}<<1$ (see Eq. 3). In this scenario
Eq. (4) can be re-written as

$P_{n\geq 1}(d_{L})=\frac{1}{N_{\rm norm}}\int d\Omega\
N_{\nu}(\theta_{z},d_{L})\,.$ (12)

The field of view (FOV) for the detector can be defined as the fraction of
total sky area covered by the detector and can be given as

${\rm FOV}=\frac{2\pi\left|\big{(}{\rm cos}\ \theta_{z}^{\rm min}-{\rm cos}\
\theta_{z}^{\rm min}\big{)}\right|}{4\pi}\,.$ (13)

The radio neutrino detector will only detect a neutrino if it is in its FOV.
Thus the number of sources in the FOV is given by

$N_{\rm src}={\rm FOV}\int_{0}^{d^{\prime}_{\rm com}}d(d_{\rm com})\big{(}4\pi
d_{\rm com}^{2}R_{\rm app}(z)\big{)}\,.$ (14)

Using Eqns. (11) and (11), Eq. III.2 can be re-written as

$\displaystyle q\big{(}d^{\rm UL}_{\rm GW},T_{\rm op}\big{)}$
$\displaystyle=1-{\rm exp}\bigg{(}-T_{\rm op}I\big{(}d_{\rm GW}^{\rm
UL}\big{)}\bigg{)}\,,$ $\displaystyle I\big{(}d_{\rm GW}^{\rm
UL}\big{)}=4\pi\int_{0}^{d_{\rm GW}^{\rm UL}}d($ $\displaystyle d_{\rm
com})\frac{T_{\rm op}}{\big{(}1+z\big{)}}\left[\frac{2\pi\big{(}{\rm cos}\
\theta_{z}^{\rm min}-{\rm cos}\ \theta_{z}^{\rm
min}\big{)}}{4\pi}\right]R_{\rm app}\big{(}z\big{)}d_{\rm com}^{2}$
$\displaystyle\left[\frac{2\pi\int_{E_{\nu}^{\rm LL}}^{E_{\nu}^{\rm
UL}}dE_{\nu}\phi_{\nu}\big{(}\mathcal{E}_{\nu}^{\rm
UHE,iso},E_{\nu},d_{L}\big{)}\int d({\rm cos}\theta_{z})\mathcal{A}_{\rm
eff}(\theta_{z},E_{\nu})}{2\pi\big{(}{\rm cos}\ \theta_{z}^{\rm min}-{\rm
cos}\ \theta_{z}^{\rm min}\big{)}}\right]\,,$ (15) $\displaystyle
I\big{(}d_{\rm GW}^{\rm UL}\big{)}=4\pi\int_{0}^{d_{\rm GW}^{\rm UL}}d($
$\displaystyle d_{\rm com})\frac{T_{\rm op}}{\big{(}1+z\big{)}}R_{\rm
app}\big{(}z\big{)}d_{\rm com}^{2}\left[\int_{E_{\nu}^{\rm LL}}^{E_{\nu}^{\rm
UL}}dE_{\nu}\phi_{\nu}\big{(}\mathcal{E}_{\nu}^{\rm
UHE,iso},E_{\nu},d_{L}\big{)}\frac{1}{2}\int d({\rm
cos}\theta_{z})\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})\right]\,.$

The above equation is universally true for triggered-searches when the sources
are very far away, that is, $N_{\nu}<<1$, which is what we consider in this
paper. Recall that $\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})$ is the
instantaneous zenith-dependent effective area. As evident from above Eq. (A),
the results depend _only_ on the _instantaneous zenith-dependent effective
area_ $\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})$, no matter the size of the
triggered-search time window ($\delta t$). This is because in the above
equations the product of $\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})\times{\rm
FOV}$ is a constant and is equal to $(1/2)\int d({\rm
cos}\theta_{z})\mathcal{A}_{\rm eff}(\theta_{z},E_{\nu})$. This highlights the
fact that larger instantaneous effective areas will lead to better results
irrespective of the FOV for this analysis and hence explains why GRAND yields
better results than IceCube-Gen2 Radio.

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