# Gravitational wave triggered searches for high-energy neutrinos from binary
neutron star mergers: prospects for next generation detectors

Mainak Mukhopadhyay<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  Shigeo S. Kimura  Frontier Research Institute for
Interdisciplinary Sciences; Astronomical Institute, Graduate School of
Science, Tohoku University, Sendai 980-8578, Japan  Kohta Murase  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
University, Kyoto 606-8502, Japan  School of Natural Sciences, Institute for
Advanced Study, Princeton, NJ 08540, USA

###### Abstract

The next generation gravitational wave (GW) detectors - Einstein Telescope
(ET) and Cosmic Explorer (CE), will have distance horizons up to
$\mathcal{O}(10)$ Gpc for detecting binary neutron star (BNS) mergers. This
will make them ideal for triggering high-energy neutrino searches from BNS
mergers at the next generation neutrino detectors, such as IceCube-Gen2. We
calculate the distance limits as a function of the time window of neutrino
analysis, up to which meaningful triggers from the GW detectors can be used to
minimize backgrounds and collect a good sample of high-energy neutrino events
at the neutrino detectors, using the sky localization capabilities of the GW
detectors. We then discuss the prospects of the next generation detectors to
work in synergy to facilitate coincident neutrino detections or to constrain
the parameter space in the case of non-detection of neutrinos. We show that
good localization of GW events, which can be achieved by multiple third
generation GW detectors, is necessary to detect a GW-associated neutrino event
or put a meaningful constraint ($\sim 3\sigma$ confidence level) on neutrino
emission models. Such a model independent analysis can also help constrain
physical models and hence provide insights into neutrino production mechanisms
in binary neutron star mergers.

##  I Introduction

The era of multimessenger astronomy is rapidly progressing, with planned
upgrades to IceCube - IceCube-Gen2 [1], KM3NeT [2], the Pacific Ocean Neutrino
Experiment (P-ONE) [3], TRIDENT [4] and Baikal-GVD [5] bolstering the neutrino
sector and the next generation gravitational wave (GW) detectors like Einstein
Telescope (ET) [6] and Cosmic Explorer (CE) [7] complementing in the GW
sector. The former gives us an opportunity to look for high-energy neutrino
events, while the latter is expected to have a distance horizon of at least
$\sim$ a few $10$ Gpc for binary neutron star mergers. However, beyond the
individual capabilities of the detectors in regards to their respective
messengers, it is also important to consider their joint capabilities in
observing messengers from various astrophysical sources in order to maximize
our understanding through observational insights.

With the detection of the binary neutron star (BNS) merger event GW170817 by
the LIGO and Virgo collaborations [8] in the GW channel and subsequent
observations of a faint gamma-ray burst [9, 10] and other electromagnetic
signatures [11, 12, 13, 14, 15, 16], the power of multimessenger observations
using GW detectors and electromagnetic telescopes was well-established.
However no high-energy neutrinos (or high-energy gamma rays) were observed
from this event [17, 18, 19], even though BNS mergers are considered to be
sources of high-energy neutrinos [20, 21, 22, 23, 24, 25, 26, 27]. This
naturally leads to the question how will the landscape of joint GW and
neutrino observations look like in the next era with the upcoming powerful
detectors in both sectors. In particular, ideally even if a coincident
detection of high energy neutrino from a single nearby BNS merger event cannot
be made, the large distance horizons of future detectors might enable us to
stack signal events to eventually collect coincident neutrino events from
comparatively distant sources or in their absence help us constrain the
physical models.

This forms the basis of our present work. We discuss the possibilities of
synergic observations between the neutrino and GW detectors. Dedicated
searches for high-energy neutrinos coincident with BNS mergers have been
performed by the IceCube collaboration [28]. In [29] search techniques for
common sources of GW and high-energy neutrinos were discussed using a Bayesian
approach. An archival search for the $80$ confident events reported in
GWTC-2.1 and GWTC-3 catalogs was conducted by the IceCube collaboration using
an unbinned maximum likelihood analysis and a Bayesian analysis using
astrophysical priors, resulted in no significant neutrino associations [30]. A
dedicated triggered search pipeline called Low-Latency Algorithm for Multi-
messenger Astrophysics (LLAMA) [31] using LIGO/Virgo candidates was developed
to look for high-energy neutrinos from the candidates.

In principle, depending on the astrophysical system of interest, an analysis
can be performed where one of the neutrino or GW observations can be used as a
trigger for the other observation. For sources that are known to emit heavily
in neutrinos, e.g., core-collapse supernovae, a neutrino triggered search
might be more efficient [32, 33]. For the case of compact binary mergers, the
maximum energy is radiated in GWs, and the GW signals might contribute in
triggering neutrino searches. Binary neutron star mergers form an interesting
class of astrophysical sources that are known to emit heavily in GWs and may
also have significant high energy neutrino emission depending on the presence
of jet and ejecta (e.g., [23]). We particularly focus on BNS mergers for this
work.

The next generation GW detectors with large distance horizons would observe
several hundreds of BNS merger events per day, providing triggers for high-
energy neutrino searches in the next generation neutrino detectors. However,
with the increase in the number of triggers, the backgrounds for the neutrino
detectors can also be overwhelming. In this work, instead of using all the GW
events as triggers for neutrino searches, we calculate distance limits to
collect _meaningful_ triggers from, where meaningful in this context implies
triggers for which stacking the neutrino observations would lead to a
comparatively low background. We propose to do this by quantifying the
fraction of sky area covered using the sky localization capabilities of the GW
detectors. We present our results for the prospect of coincident detection of
high-energy neutrinos from BNS mergers along with the operation time of the
detectors. In case of a non-detection this would imply strong constraints on
the physical models associated with BNS mergers. We also give estimates of
backgrounds associated with such limiting distance triggered stacking
searches, which provides an optimal operation time and a threshold for the
fraction of sky area covered.

A related work was also performed for the low energy thermal neutrinos from
BNS mergers, where the potential of the next generation GW detectors was
examined in the context of searching for $\mathcal{O}(10)$ MeV neutrinos from
BNS merges at the next generation sub-megaton scale water Cherenkov detectors,
such as Hyper-Kamiokande [34, 35]. Besides being on the opposite end of the
energy spectrum, these have short emission timescales of $\sim 1-10$ s.
However, the timescales associated with high-energy neutrino emission from BNS
mergers can be diverse and as long as $\sim 10^{7}$ s, which would then
require a completely different analysis strategy to collect low background
coincident neutrino events from BNS mergers as described in this work. It is
also important to note that the current work is complimentary to the
triggered-stacking searches that the current neutrino detectors perform since
they use likelihood search techniques and do not include distance cuts as
proposed in this work.

The paper is organized as follows. In Sec. II, we discuss the main method and
the ingredients required to perform the analysis. Using the ingredients, the
main results including the detection prospects for coincident high energy
neutrino events using GW triggered searches and the effects of backgrounds are
discussed in Sec. III. We summarize in Sec. IV and discuss the implications of
our work in Sec. V.

##  II Strategy

Let us assume a given astrophysical source emits total energy
${\mathcal{E}}^{\rm tot}$. Consider the energy emitted in the GW sector is
${\mathcal{E}}_{\rm GW}$, in high energy neutrinos is $\mathcal{E}_{\nu}^{\rm
HE}$, and in all other channels $E^{\rm misc}$. Note that these other channels
($\mathcal{E}^{\rm misc}$) can also include electromagnetic and exotic
emissions. Thus, we have ${\mathcal{E}}^{\rm tot}={\mathcal{E}}_{\rm
GW}+\mathcal{E}_{\nu}^{\rm HE}+\mathcal{E}^{\rm misc}$. Note that independent
of the specific models, $\mathcal{E}^{\rm misc}>\mathcal{E}_{\nu}^{\rm HE}$
because the amount of generated gamma rays is comparable to that of high-
energy neutrinos whether one considers hadronuclear or photohadronic
interactions [36]. Since the focus of this work is to discuss the prospects of
GW triggered high energy neutrino observations, we consider that the source
emits strongly in gravitational waves and use that as the trigger. In
particular, BNS mergers, besides strongly emitting in GWs, might also have
considerable neutrino emissions [20, 21, 22, 23, 24, 25, 26, 27]. We focus on
BNS mergers and discuss the strategy in the context of such mergers.

For BNS mergers, typically a small fraction ($\alpha$) of the total energy is
emitted in GWs, that is, ${\mathcal{E}}_{\rm GW}\sim\alpha{\mathcal{E}}^{\rm
tot}$, where $\alpha\sim 1$% [37, 8, 38, 39, 40, 41, 42]. The total energy
$\mathcal{E}^{\rm tot}$ in this case refers to the orbital or the binding
energy of the BNS system111It is also important to note that the majority of
the energy budget does not go into radiation and is mostly utilized as the
spin energy of the compact object formed.. The energy in the GWs
(${\mathcal{E}}_{\rm GW}$) is the observed quantity and has all the
uncertainties based on the detector properties and the source. The quantity
${\mathcal{E}}_{\rm GW}$ will depend on the signal-to-noise ratio (SNR) of the
observation, which in turn depends on the noise curves or sensitivity of the
GW detector and the distance of the source.

The total energy emitted in high energy neutrinos can be assumed to be a
fraction of the gravitational-wave energy (${\mathcal{E}}_{\rm GW}$). Thus, we
can define

$\mathcal{E}_{\nu}^{\rm HE,true}=f_{\nu}{\mathcal{E}}_{\rm GW}\,,$ (1)

where $\mathcal{E}_{\nu}^{\rm HE,true}$ is the energy emitted in high-energy
neutrinos at the source and $f_{\nu}$ is an effective parameter and is defined
as the energy radiated in high energy neutrinos as a fraction of the GW energy
emitted by the same source. This will be an important parameter in our
analysis. A higher value of $f_{\nu}$ would imply a larger total energy
emitted in high-energy neutrinos leading to more optimistic prospects.

The high-energy neutrinos can be produced in the relativistic jets launched
after the merger, and we need to take into account the relativistic beaming
effect when discussing the neutrino emission from the jets. The isotropic-
equivalent energy emitted in high-energy neutrinos at the source is given by

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

Often $f_{\rm bm}\sim 1$% is used, which is roughly consistent with the
afterglow observations [43]. It is expected that $\mathcal{E}_{\nu}^{\rm
HE,true}$ is only a tiny fraction of ${\mathcal{E}}_{\rm GW}$. However the
neutrino emission and the gravitational wave emissions can have completely
different beaming factors. Finally, it is worth mentioning that the definition
of $f_{\nu}$ to be a ratio against the GW energy emitted in neutrinos is just
a definition. In general, the two emissions have different physical sources
and are not related, so $f_{\nu}>1$ is possible in principle but highly
unlikely for realistic astrophysical environment.

One can also choose $\mathcal{E}_{\nu}^{\rm HE,true}$ from physical models.
For example, according to neutrino emissions from BNS mergers considered in
[23, 25] one can find that $\mathcal{E}_{\nu}^{\rm HE,true}\sim 10^{48}$ erg.
Assuming ${\mathcal{E}}^{\rm tot}=5\times 10^{54}$ erg, then gives
$f_{\nu}\sim 10^{-4}$.

(a) (a)

(b) (b)

Figure 1:  (a) The right axis shows the the size of the localization area in
the sky for a single BNS merger event ($\Delta\Omega$) as a function of the
luminosity distance $d_{L}$ for CE (dashed dark blue line), ET (dashed pink
line), and ET+CE (solid orange line). The left axis shows the BNS merger rate
$R(z)$. The fiducial rate is shown as a solid teal line and the area between
the upper and the lower limits of $R(z=0)$ is shaded. (b) The fraction of sky
area covered $f_{\rm cov}$ as a function of $d_{L}$ for CE (dot-dashed purple
line), ET (dashed orange line), and ET+CE (solid dark blue line) (see Eq. 4
for details). The horizontal dotted lines show the threshold for the fraction
of sky area covered $f_{\rm th}$ equal to $10^{-2}$ (red), $10^{-3}$ (green),
and $10^{-4}$ (orange). The time duration assumed for the plot is $\delta
t=1000$ s.

Below we give brief details about the next generation GW and neutrino
detectors that we use for this work.

###  II.1 GW Detectors

The GW detectors we consider in this work are the upcoming third generation
detectors: Einstein Telescope (ET) and Cosmic Explorer (CE). Following Ref.
[44], we use the results of the geometrical configuration known as ET-D for
Einstein Telescope. This consists of 3 interferometers, each having an opening
angle of $60^{\circ}$ and are rotated by $120^{\circ}$ relative to the other,
resulting in a configuration of an equilateral triangle. The interferometer
arm length is $10$ km. Since the location of both ET and CE is undecided yet,
the results used assume that ET is located at (longitude,latitude)
$=(10.4^{\circ},43.7^{\circ})$ and CE at (longitude,latitude)
$=(-119.41^{\circ},46.45^{\circ})$. The corresponding antenna patterns assumed
for each detector is given in Ref. [45]. Both ET and CE are sensitive between
$10-100$ Hz and have an amplitude spectrum density of $\sim 10^{-25}\rm
Hz^{-1/2}$. The scenario of ET and CE working together (referred to as ET+CE
hereafter) is also considered to illustrate the prospects of triggered
searches.

Given the properties of the GW detector, the size of the localization area
$\Delta\Omega$ depends on the distance to the source [44]. In Fig. 1a on the
right axis, we show $\Delta\Omega$ as a function of the luminosity distance
$d_{L}$ for ET (dashed pink), CE (dotted dark-blue) and ET+CE (solid orange).
As the distance to the source increases, the localization area becomes larger
as expected. It is interesting to note that although CE has a longer arm and
hence is sensitive to higher distances for BNS mergers, it has a comparatively
poor sky localization capability and is limited to $<1$ Gpc for covering all
sky (sky localization area is $4\pi$) as compared to the ET which can use
triangulation techniques to have a much better $\sim\mathcal{O}(10^{3})\ \rm
deg^{2}$ at $1$ Gpc sky localization capability. The combination of ET+CE is
the most effective and has a localization capability $\sim\mathcal{O}(10^{3})\
\rm deg^{2}$ even at $d_{L}\sim 10$ Gpc, which is a couple of orders of
magnitude better than that of ET. Finally, the sky-localization areas can be
less optimistic [46] than what is considered in our current work.

###  II.2 Neutrino Detectors

For neutrino detectors, we consider the planned upgrade to IceCube - IceCube-
Gen2. The current IceCube detector has 86 strings deployed at a distance
spacing of 120 metres resulting in roughly a cubic-kilometre detector volume.
The planned IceCube-Gen2 will add $120$ strings to the existing configuration
at distances of $240$ metres resulting in a detector volume of $\sim 8\ \rm
km^{3}$. This combined with better PMTs will likely boost its sensitivity by a
factor of $10^{2/3}$ (roughly $5$). For this work, we use the publicly
available declination-dependent effective area used in the IceCube ten-year
point source (PS) search [47] and scale it by the factor ($10^{2/3}$) to
obtain our results for the future IceCube-Gen2.

###  II.3 Binary neutron star merger rates

In this section, we discuss the rate of BNS mergers. The redshift dependence
of the rate is adapted from that of short gamma-ray bursts (GRBs) [48]. This
is a reasonable assumption given that short GRBs might primarily originate
from NS-NS mergers222NS-BH (neutron star-black hole) mergers are also
suggested to produce short GRBs, but the NS-BH merger rate is lower than BNS
mergers based on GW observations [49].. The redshift dependent rate of BNS
mergers is given as [48]

$R(z)\equiv\mathcal{N}R(z=0)\begin{cases}{\rm
exp}\big{(}(z-0.9)/0.39\big{)}\,,z\leq 0.9\,,\\\ {\rm
exp}\big{(}-(z-0.9)/0.26\big{)}\,,z>0.9\,,\end{cases}$ (3)

where the normalization $\mathcal{N}$ is chosen such that the rate at $z=0$ is
given by the fiducial rate, $R(z=0)=300\ \rm Mpc^{-3}\rm yr^{-1}$. We choose a
conservative value for the fiducial case. However, this rate has associated
uncertainties and can be over a wide range between $10\ \rm Mpc^{-3}\rm
yr^{-1}$ (lower limit) and $1700\ \rm Mpc^{-3}\rm yr^{-1}$ (upper limit) [50].
In Fig. 1a, on the right axis, we show $R(z)$ as a function of luminosity
distance, $d_{L}$, where the fiducial rate ($R(z=0)=300$ Mpc-3 yr-1) is shown
as a thick solid line and the area between the upper and lower limits are
shaded with teal. We see the break at $z=0.9$ as the peak according to Eq.
(3). The cumulative rate is given by, $R_{\rm BNS}(z)=\int_{0}^{z}dz^{\prime}\
R(z^{\prime})$. Given that the rate decreases post $z=0.9$, the cumulative
rate is approximately constant for $z\gtrsim 0.9$.

###  II.4 Sky localization for GW detectors

With the next generation GW detectors reaching distance horizons of
$\mathcal{O}(10\ \rm Gpc)$, the number of events detected would be
$\mathcal{O}(100)$ a day [51]. Hence, the entire paradigm of performing
triggered neutrino searches based on GW detections depends on the localization
of the source based on the GW observations. We focus on quantifying the
possibilities and limitations based on the GW detectors and the sources in
this section.

The total fraction of the sky area covered by the error regions given by the
localization of BNS merger events by the GW detectors in a given time period
is

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

where $d_{\rm com}$ is the comoving distance, $f_{\rm cov}$ is the fraction of
the sky area covered333Note that $f_{\rm cov}=1$ would imply all sky ($4\pi$)
area being covered by the error regions as a result of the localization of BNS
merger events by the GW detectors in a given time period., $d_{\rm L}$ is the
luminosity distance which is a function of the redshift $z$ and comoving
distance $d_{\rm com}$, $d_{L}=(1+z)d_{\rm com}$, $\Delta\Omega$ is the size
of the localization area in the sky for a single BNS merger event for a given
GW detector, and $\delta t$ is the duration of neutrino emission in the
observer frame.

The integral is performed over $d_{\rm com}$ from $0$ to the given upper limit
$d_{\rm GW}^{\rm lim}$. For a chosen value of $f_{\rm cov}$, which we call the
threshold value $f_{\rm th}$, we evaluate the above equation to deduce the
upper limit, hence obtaining the limiting distance $d_{\rm lim}^{\rm GW}$.
Thus, $f_{\rm th}$ in this case is a parameter that then helps us in obtaining
distance limits such that a collection of signal events is not overwhelmed by
backgrounds. The identification of individual sources require nearby merger
events. However, for this work, we do not discuss identifying individual
sources. Instead, we focus on prospects for high-energy neutrino detection by
stacking BNS merger events using GW triggers.

The time duration of neutrino emission, $\delta t$, is an important quantity
and can have a wide range of variation. For example, high energy neutrino
emissions from a jet interacting with the ejecta post a merger was considered
in Ref. [23]. In this case, the neutrino emission peaks at around $\delta
t\sim 2$ s post the merger, and thus, this scenario would be very optimistic.
In contrast, a BNS merger may lead to the formation of a long-lived,
millisecond magnetar surrounded by a low-mass ejecta. A portion of the
magnetar’s rotational energy is deposited in the nebula behind the ejecta,
eventually leading to efficient pion production using the thermal and/or
nonthermal photons in the nebular region which can then produce high energy
neutrinos [25]. In this case the timescales associated with the peak in the
neutrino fluence is of the order $\delta t\sim 10^{6}-10^{7}$s, which would be
a conservative scenario.

As a fiducial value, we assume $\delta t=1000$ s and show the fraction of sky
area covered by CE (dot-dashed purple), ET (dashed red), and ET+CE (solid dark
blue) with the luminosity distance in Fig. 1b. The choice for the fiducial
value of $\delta t=1000$ s can be motivated from theoretical models [20, 52]
which suggest $\delta t\sim 100-1000$ s. Furthermore, current GW and neutrino
searches also use a typical time window of $\delta t=1000$ s for their
analysis [53, 30, 54]. Depending on the chosen value of $f_{\rm th}$, one can
estimate the distance limit for a given GW detector from which triggers can be
used to have reduced backgrounds. Similar to the localization capabilities, we
have the ET+CE combination to be the most effective followed by ET and CE. For
CE, the fraction of sky area covered needs to be relatively larger than ET and
ET+CE. For $f_{\rm th}\sim 1$% of the total sky area, CE is limited to $\sim
0.7$ Gpc. The case can be improved for ET where choosing a threshold of
$f_{\rm th}\sim 0.1$% of the total sky area puts the distance limit at $\sim
1.1$ Gpc and the combination of ET+CE can have a limit at $\sim 1.8$ Gpc where
the $f_{\rm th}$ has a very small value of $f_{\rm th}\sim 0.01$% of the total
sky area.

It is important to note here that the results shown are in the absence of
backgrounds. Including backgrounds and demanding a particular signal to
background ratio puts constraints on the chosen value of $f_{\rm th}$
corresponding to the operation timescales of the GW detectors. This will be
discussed in detail in Sec. III.3.

##  III Results

In this section, we outline the technique to highlight the prospects for the
next-generation GW detectors to act as triggers for the next-generation
neutrino detectors. Such observations also require some assumptions on the
neutrino emission from the source which is also discussed. We also present our
main results and give estimates for background events in the following
subsections.

###  III.1 Distance limit for triggers

Figure 2:  Density plot showing the distance upper limit for GW detectors
($d_{\rm GW}^{\rm UL}$) (see Eq. 5) for CE (_left_), ET (_middle_), and ET+CE
(_right_) on the $\delta t$ \- $f_{\rm th}$ plane.

From the previous section and Eqs. (3) and (4), it is evident that the choice
of the threshold for the fraction of sky area covered, $f_{\rm th}$, and the
duration of neutrino emission, $\delta t$, are two relevant parameters in
implementing the distance cuts for the triggers with the GW detectors. The
limiting distance $d^{\rm lim}_{\rm GW}$ is defined as the distance such that
from Eq. (4), $f_{\rm cov}(d^{\rm lim}_{\rm GW})=f_{\rm th}$, that is, it is
the upper limit of the integral in Eq. (4) that results in the desired value
of $f_{\rm th}$. There also exists a maximum distance horizon for the GW
detectors beyond which all triggers can be used. We define this as ($d_{\rm
GW}^{\rm hor}(z^{\rm hor}_{\rm GW})$) which is set to444This choice for the
horizon distance is motivated by the distance horizon of ET for BNS mergers
[51]. Although CE can be sensitive to BNS mergers from higher redshifts the
sky localization is poor. Furthermore, the rate of BNS mergers decreases post
$z\sim 0.9$ as can be seen from Fig. 1a, and thus, the choice of $z_{\rm
GW}^{\rm hor}$ has little influence on our conclusion. $z^{\rm hor}_{\rm
GW}\sim 3.5$.

Thus, the final limiting distance ($d_{\rm GW}^{\rm UL}$) is defined as the
minimum of the limiting distance ($d_{\rm GW}^{\rm lim}$) or the maximum
horizon distance considered for the GW detectors, that is,

$d_{\rm GW}^{\rm UL}={\rm min}\big{(}d_{\rm GW}^{\rm lim},d^{\rm hor}_{\rm
GW}\big{)}\,.$ (5)

To illustrate the dependence of the distance limits on $f_{\rm th}$ and
$\delta t$, we show a density plot for the final limiting distance $d^{\rm
UL}_{\rm GW}$ for CE (left), ET (middle) and the combination ET+CE (right) on
the $f_{\rm th}-\delta t$ plane in Fig. 2. The color map is shown as a rainbow
spectrum where the limiting distance increases as one goes upwards from violet
to red.

For CE even with the most optimistic choice of $f_{\rm th}$ and $\delta t$, we
are limited to $\sim$ Gpc for the limiting distance. For ET, considering an
optimistic scenario of $\delta t\sim 1$s, the distance limits can be a few Gpc
assuming $f_{\rm th}\sim 0.1$%. However for large values of $\delta t\sim
10^{6}$s, the limiting distance is reduced to $\sim$ a few $100$ Mpc for the
same value of $f_{\rm th}$. However, for the combination of ET+CE, $\delta
t\sim 1$s implies a limiting distance of $\sim 10$ Gpc for $f_{\rm th}\sim
0.01$%. In contrast to ET, even for the not so optimistic case of $\delta
t\sim 10^{6}$s, ET+CE can have limiting distances of up to a few Gpc. The area
colored in red indicates that the limiting distance obtained from Eq. (4) is
greater than the maximum distance horizon for the GW detectors, and hence, the
final limiting distance is fixed to be $d^{\rm hor}_{\rm GW}$.

###  III.2 Detection prospects

This section focuses on coincident detection of high-energy neutrinos from BNS
mergers at the next generation neutrino detectors based on triggers from the
next generation GW detectors, where the triggers are chosen based on the
distance cut calculated using the fraction of sky area covered, $f_{\rm th}$.

The probability to detect more than one neutrino associated with a GW signal
is given by [52]

$q\big{(}d^{\rm UL}_{\rm GW},T_{\rm op}\big{)}=1-{\rm exp}\bigg{(}-T_{\rm
op}I\big{(}d_{\rm GW}^{\rm UL}\big{)}\bigg{)}\,,$ (6)

where $T_{\rm op}$ is the operation time of the GW detector. The argument of
the exponential in the above expression $I$ is defined as

$I\big{(}d_{\rm GW}^{\rm UL}\big{)}=4\pi\int_{0}^{d_{\rm GW}^{\rm UL}}d(d_{\rm
com})\frac{T_{\rm op}}{\big{(}1+z\big{)}}R\big{(}z\big{)}d_{\rm
com}^{2}P_{n\geq 1}\big{(}d_{L}\big{)}$ (7)

The integral is performed over the comoving distance. The declination
integrated (or total) probability ($P_{n\geq 1}(d_{L})$) to detect at least
one neutrino as a function of distance is given by

$P_{n\geq 1}(d_{L})=\frac{1}{4\pi}\int d\Omega\ p_{n\geq 1}(\delta,d_{L})\,,$
(8)

where $p_{n\geq 1}(\delta,d_{L})$ is the declination ($\delta$) dependent
probabililty to detect at least one neutrino is given by the Poissonian
probability

$p_{n\geq
1}(\delta,d_{L})=1-\exp\bigg{(}-N_{\nu_{\mu}}(\delta,d_{L})\bigg{)}\,.$ (9)

In the above expression, $N_{\nu_{\mu}}(\delta,d_{L})$ is the number of
neutrino events from a source at a given luminosity distance $d_{L}$. The
expected number of neutrino events at a given declination is given as

$N_{\nu_{\mu}}(\delta,d_{L})=\int_{E_{\nu}^{\rm LL}}^{E_{\nu}^{\rm
UL}}dE_{\nu_{\mu}}\phi_{\nu_{\mu}}(E_{\nu_{\mu}},d_{L})\mathcal{A}_{\rm
eff}(E_{\nu_{\mu}},\delta)\,,$ (10)

where $E_{\nu_{\mu}}$ is the neutrino energy in the observer frame,
$E_{\nu}^{\rm LL}$ and $E_{\nu}^{\rm UL}$ gives the lower and the upper limits
of the integral and is decided based on the neutrino energy spectra from a
particular source given a production channel ($pp$ or $p\gamma$ emission
channels), $\phi_{\nu_{\mu}}$ gives the neutrino fluence from a given source
at a given luminosity distance $d_{L}$, and $\mathcal{A}_{\rm eff}$ is the
energy and declination dependent IceCube effective area.

In principle, one can consider either of the two channels for high energy
neutrino production - hadronuclear ($pp$) or photomesonic processes
($p\gamma$). In general in a BNS merger the nonthermal protons produced in the
collimation and internal shocks produce neutrinos through interactions with
the background photons and protons. Here, we choose the more optimistic
scenario where we focus on the $p\gamma$ interactions for neutrino production.
The dominant channel for such process is pion production. The charged pions
decay to muons and neutrinos whereas the neutral pions form gamma rays. The
muons further decay to form a corresponding charged lepton and neutrinos (and
antineutrinos). In this case, the upper and lower limits of the neutrino
energy spectra are given by Ref. [23], $\varepsilon_{\nu}^{\rm min}=10^{3}$
GeV and $\varepsilon_{\nu}^{\rm max}=10^{6}$ GeV, respectively. Note that
$\varepsilon_{\nu}^{\rm min}$ and $\varepsilon_{\nu}^{\rm max}$ are defined in
the source frame. The neutrino energy in the observer frame with the redshift
correction is then given by, $E_{\nu}=\varepsilon_{\nu}/(1+z)$. It is
important to note that this serves as a comparatively optimistic scenario.
Besides, the values chosen for $\varepsilon_{\nu}^{\rm min}$ and
$\varepsilon_{\nu}^{\rm max}$ can differ across various models. Considering
$pp$ interactions will make our results slightly more conservative. We assume
a power law neutrino spectrum with spectral index equal to 2, that is,
$E_{\nu}^{-2}$ spectrum for the neutrinos. Thus, the neutrino fluence is given
by

$\phi_{\nu}(\mathcal{E}_{\nu}^{\rm HE,iso},E_{\nu},d_{L})=\frac{(1+z)}{4\pi
d_{L}^{2}}\frac{\mathcal{E}_{\nu}^{\rm HE,iso}}{\rm
ln\big{(}\varepsilon_{\nu}^{\rm max}/\varepsilon_{\nu}^{\rm
min}\big{)}}E_{\nu}^{-2}\,.$ (11)

Recall from Sec. II that the total isotropic-equivalent energy emitted in
high-energy neutrinos is given by $\mathcal{E}_{\nu}^{\rm HE,iso}$. This
depends on the parameters $f_{\nu}$ and $f_{\rm bm}$. To illustrate our
results we choose $f_{\rm bm}=1$% and $f_{\nu}\sim\mathcal{O}(10^{-5})$. The
muon neutrino fluence is given by,
$\phi_{\nu_{\mu}}(E_{\nu},r)=(1/3)\phi_{\nu}(\mathcal{E}_{\nu}^{\rm iso-
HE},E_{\nu},r)$. Finally, for this work, we use the 10 years point source (PS)
effective area from IceCube [47], scaled with a factor of $10^{2/3}$ in
accordance with the estimates of IceCube-Gen2.

Figure 3:  Probability of neutrino detection ($q$) with the operation time
$T_{\rm op}$ for a range of $f_{\nu}$ (_top row_) and $\delta t$ (_bottom
row_) for the different GW detectors. The fiducial case for each panel is
shown as a solid line. The $2\sigma$ and $3\sigma$ C.L.s are also shown with
dashed lines. For each case, ${\mathcal{E}}_{\rm GW}=5\times 10^{52}$ erg and
$\lambda=10^{-5}$. See the text for details on parameters.

In summary, evaluating $q\big{(}d^{\rm lim}_{\rm GW},T_{\rm op}\big{)}$ from
Eq. (6) enables us to calculate the probability to detect high energy
neutrinos associated with GWs. However, it is important to understand the
time-scales over which the next generation GW detectors need to operate to
either enable coincident neutrino detections or constrain the physical
parameter space at a given confidence level in case of non-detections. We
address this question in Fig. 3, where we plot the probability of neutrino
detection at IceCube-Gen2 with $T_{\rm op}$ of the GW detectors, which shows
one of our main results. The two primary parameters are $f_{\nu}$ and $\delta
t$. While the former decides the total energy emitted in high energy
neutrinos, the latter allows us to put a distance limit to reduce backgrounds
to facilitate detections.

In the figure, we show the plots for $1-q\big{(}d^{\rm lim}_{\rm GW},T_{\rm
op}\big{)}$ with the operation time of the GW detectors in years. The fiducial
parameters for each case in the top panel is chosen as $\delta t=1000$ s and
for the bottom panel as $f_{\nu}=2.5\times 10^{-5}$. The choice of $f_{\nu}$
is motivated from physical models as discussed in Sec. II, where we
conservatively choose the fiducial case to be $20$% of that obtained from
physical models. The total energy emitted from the system of BNS merger is
fixed to be ${\mathcal{E}}^{\rm tot}=5\times 10^{54}$ erg, which is typical
for two $1.4M_{\odot}$ neutron star mergers [8]. The parameter $\alpha$ is
fixed to be $1$% implying $\mathcal{E}_{\rm GW}=5\times 10^{52}$ erg. The
fiducial cases are shown by the thick solid line in each panel. The dotted
horizontal lines show the confidence levels corresponding to $2\sigma$ (red)
and $3\sigma$ (green) respectively. For each of the GW detectors, we choose a
different value of $f_{\rm th}$. This is a reasonable choice since each
detector has a different localization capability as discussed in Sec. II.4 and
shown in Fig. 1b. For CE $f_{\rm th}=1$% is chosen due to its comparatively
poor localization capability, ET motivates $f_{\rm th}=0.1$% making the
situation better. Finally the combination ET+CE gives the best case of
choosing the fraction of sky localization to be, $f_{\rm th}=0.01$%. As
mentioned earlier, given a demand of signal to background ratio and an
operation time for the GW detectors these choices are not arbitrary as will be
discussed in Sec. III.3.

In the top row of Fig. 3, we vary $f_{\nu}$ between $10^{-5}$ and $5\times
10^{-5}$, where the lower limit is an order of magnitude less than the
predictions from physical models and the upper limit denotes a value that is
$50$% of the physical models to still remain in the conservative yet
optimistic regime. A lower value of $f_{\nu}$ implies less energy in emitted
in high-energy neutrinos, which implies a lower value of neutrino fluence.
Hence in all the top three panels, this gives the upper boundary of the shaded
region, that is not a so optimistic scenario, requiring longer timescales of
operation to reach the same confidence levels. A higher value of $f_{\nu}$
correspondingly gives the lower boundary of the shaded region. We see that for
the fiducial case CE requires an operation time of $\sim 20$ years to make a
constraint on the parameter space at the $3\sigma$ C.L. ET would take $\sim
14$ years to put constraints at the $3\sigma$ C.L., but can reach the
$2\sigma$ C.L. over a timescale of $\sim 17.5$ years even for the lowest value
of $f_{\nu}$ considered for this work. The combination ET+CE can reach the
$3\sigma$ level constraints in a timescale of $\lesssim 30$ years even for the
most conservative choice of $f_{\nu}$.

The bottom row of Fig. 3 shows the variation in $\delta t$ between $1$ s and
$10^{6}$ s. In this case, a longer $\delta t$ leads to less optimistic
results. This is because with larger values of $\delta t$ the number of
triggers within $\delta t$ increases, and hence, the distance limit for
reaching a fixed value of $f_{\rm th}$ is reduced (as seen from Eq. (4) and
Fig. 2). This results in a smaller value of the upper limit of the integral in
Eq. (7). Thus, a longer operation time is required to reach the same level of
sensitivity. For small values of $\delta t\sim 1$s, a $3\sigma$ constraint
based on non-detection is possible in $\lesssim 10$ years using any of the
detectors and/or the combination. However, for large values of $\delta t$, CE
may reach the $2\sigma$ level of constraints in a time scale $>20$ years.
While the combination of ET+CE can reach a $3\sigma$ level in around $20$
years, ET by itself would need $\sim 20$ years to reach the $2\sigma$ level of
constraints.

###  III.3 Backgrounds

Figure 4:  The fraction of sky area covered $f_{\rm th}$ with operation time
$T_{\rm op}$ for different values of triggered background events $N_{\rm
bkg}^{\rm trig}$ at IceCube-Gen2.

In all the discussions above we ignored the backgrounds in the context of the
neutrino detectors. Constraining the distance to collect triggers from using
sky-localization definitely helps in reducing the background, but it is also
important to roughly estimate the backgrounds associated with such triggered
searches including the distance limits. This is the focus of the current
section.

The relevant backgrounds for high energy neutrino detectors, such as IceCube
or KM3NeT, are the conventional and prompt atmospheric neutrinos and the
diffuse astrophysical neutrinos. In this work we provide a rough estimate of
resulting backgrounds from the search method presented here. For the
conventional atmospheric neutrino flux, we use the model in Ref. [55]. The
prompt atmospheric background neutrino fluence is taken from Ref. [56]. A more
advanced and realistic estimate for the atmospheric backgrounds was performed
in Refs. [57, 58], but we use the simple model to give a rough estimate of the
atmospheric backgrounds.

The diffuse astrophysical neutrino flux Ref. [59] is assumed to be of the form
$\Phi^{\rm bkg}_{\rm astro}=\Phi_{\rm astro}(E_{\nu}/100\rm TeV)^{-\gamma}$,
where the normalization $\Phi_{\rm astro}=1.44\times 10^{-18}\ \rm
GeV^{-1}cm^{-2}s^{-1}sr^{-1}$ and the power law index, $\gamma=2.37$. We
follow the same prescription outlined in Sec. III.2 to calculate the number of
triggered background events ($N_{\rm trig}^{\rm bkg}$) using the 10 years
point source effective area scaled by a factor of $10^{2/3}$ for IceCube-Gen2.

In Fig. 4 we show the contours for the number of triggered background events,
on the the operation time ($T_{\rm op}$) and the fraction of sky area covered
$f_{\rm th}$ plane. We assume that for large number of triggers the total time
in triggers for the neutrino detectors will be comparable to the operation
time of the GW detectors. This is a reasonable assumption owing to the next
generation of GW detectors having a large distance horizon. With the inclusion
of triggered background events, the figure presents an estimate for the choice
of fraction of sky area covered given an operation time and a desired level of
signal to background ratio. For example, fixing an operation of time of $10$
years and choosing $N_{\rm trig}^{\rm bkg}=1,0.1,0.01$ dictates that $f_{\rm
th}$ be chosen as $0.8$%, $0.08$%, $0.008$% respectively.

Let us now discuss the implications of Fig. 4 on the main results presented in
Fig. 3. In Fig. 3 we assumed $f_{\rm th}=1$%, $0.1$% and $0.01$% respectively
for CE, ET, and ET+CE. This translates to choosing $T_{\rm op}\sim 8$ years
corresponding to $N_{\rm trig}^{\rm bkg}=1,0.1,0.01$ for CE, ET and ET+CE
respectively. It is also important to highlight the fact that this implies
that the fraction of sky area covered $f_{\rm th}$ in the presence of
background is not an arbitrarily chosen value but is constrained by the
operating time of the GW detector and the choice of signal to background
ratio. For claiming a $3\sigma$ C.L. detection we should restrict ourselves to
$N_{\rm trig}^{\rm bkg}\lesssim 3\times 10^{-3}$. However, constraints on the
parameter space as a result of non-detection can be put using $N_{\rm
trig}^{\rm bkg}\sim 0.1-0.01$.

##  IV Summary

With the advent of the next generation of GW and neutrino detectors, it would
be crucial to lay down strategies to use their potential to perform
multimessenger studies of astrophysical sources. In this work, we examined the
prospects of performing triggered-stacking searches for coincident high-energy
neutrinos from BNS mergers with the next generation GW detectors like CE, ET,
and a combination of ET+CE in the context of the neutrino detector IceCube-
Gen2. However, the next generation GW detectors would be sensitive to large
distances, which makes it important to investigate which triggers would be
optimal to perform such searches for coincident high-energy neutrinos. This is
because sensitivities to large distances implies a large number of triggers in
turn leading to large backgrounds in the neutrino searches which is undesired.

In this work, we addressed this issue and showed that a limiting distance can
be used for the triggers given a chosen threshold of sky localization ($f_{\rm
th}$). We obtained the limiting distance based on $f_{\rm th}$ and the sky
localization capability for a given GW detector. In Fig. 1a, we show the size
of the localization area for a single BNS merger event with luminosity
distance for each of the detectors and infer that the combination ET+CE has
the best localization capability, followed by ET and CE. The redshift
dependent rate of BNS merger, shown in Fig. 1a, also helps in deciding the
number of triggers. We use a fiducial rate of $300\ \rm{Mpc^{-3}yr^{-1}}$ for
this work. The fraction of sky area covered obtained using Eq. (4) is shown in
Fig. 1b, where we see for CE choosing $f_{\rm th}\sim 1$% gives a limiting
distance of $\sim 0.72$ Gpc, for ET a choice of $f_{\rm th}\sim 0.1$% provides
a limiting distance of $\sim 1.12$ Gpc, and choosing $f_{\rm th}\sim 0.01$%
for ET+CE leads to a limiting distance of $\sim 1.85$ Gpc.

The limiting distance depends on the choice of $f_{\rm th}$ and also the time
interval between the merger and the neutrino emission peak $\delta t$. In Fig.
2, we show the limiting distance as a density plot on the $f_{\rm th}-\delta
t$ plane. We note that the optimistic scenario is given by small values of
$\delta t\sim 1-10$ s where a limiting distance of $\sim$ a few Gpc (a few ten
Gpc) is possible for ET (ET+CE), which is plausible for some physical models
for high-energy neutrino emission from BNS mergers. However, for large values
of $\delta t$, which can be realized in magnetar-powered scenarios, the
limiting distance reduces to $\sim$ a few hundred Mpc for ET and a few Gpc for
ET+CE.

Our main result for detection prospects is shown in Fig. 3. We considered a
variation in the total energy emitted in high-energy neutrinos by varying
$f_{\nu}$ between $10^{-5}$ and $5\times 10^{-5}$ and choose
$f_{\nu}=2.5\times 10^{-5}$ as the fiducial value. We also vary $\delta t$
between $1$ s and $10^{6}$ s while choosing $\delta t=1000$ s as the fiducial
case. For the fiducial parameters, we find that CE can make a $2\sigma$ level
constraint in case of non-detection in a time span of $\sim 11$ years. ET can
lead to $3\sigma$ level constraints in a time span of $14$ years, while the
combination of ET+CE does the same in an operation time scale of $\sim 10$
years. For the less optimistic cases, ET+CE, owing to its excellent sky
localization capabilities, can still lead to $3\sigma$ level of detection or
constraints in $\lesssim 20$ years. While ET would take $\sim 20$ years to
reach the $2\sigma$ C.L..

Even in the case of non-detection of coincident high-energy neutrino events,
our model independent analysis can constrain $f_{\nu}$, that is, the total
energy emitted in high-energy neutrinos. Such constraints can be very useful
for understanding the emission mechanisms associated with high-energy
neutrinos and gamma-rays. This provides further insights into understanding
neutrinos from choked jets, from the interaction of the jet with the ejecta
besides providing information about the neutrino emission sites and mechanisms
associated with BNS mergers. For example, in the optimistic scenario of Ref.
[23] where $f_{\nu}\sim 10^{-4}$ and $\delta t\sim 2$ s, a $3\sigma$ C.L.
constraint is possible using CE, ET, and ET+CE in a timescale of $\sim 7,4.5,$
and $2.5$ years respectively. However, the scenario considered in Ref. [23] is
optimistic and has been disfavored in subsequent works [60]. The method of
gravitational wave triggered searches for high energy neutrinos from BNS
mergers can thus provide observational evidence to confirm or constrain the
models. Another model for extended neutrino emission from short GRBs was
considered in Ref. [20]. In this case, $\delta t\sim 100$ s and $f_{\nu}\sim
10^{-5}$ for the optimistic scenario. This implies a $3\sigma$ level
constraint can be obtained in $\sim 4,2.5,$ and $1.5$ years, using CE, ET and
CE+ET respectively. Similarly, for the less optimistic case of Ref. [25] where
$f_{\nu}\sim 10^{-4}$ and $\delta\sim 10^{6}$ s, ET can give $2\sigma$ level
constraints on a timescale of $\sim 20$ years, while the combination of ET+CE
can lead to $3\sigma$ level constraints in an operating time of $\lesssim 30$
years.

We also provide an estimate of the backgrounds in IceCube-Gen2 associated with
the search method presented in this work. Considering backgrounds constrain
the choice of $f_{\rm th}$ given a desired value of signal to background ratio
and the desired operation time for the GW detectors. We show the contours of
the number of triggered background events in Fig. 4. From the figure it is
also evident that operation timescales of $>10$ years are not reasonable due
to the extremely small value of $f_{\rm th}$ required.

##  V Discussion and Implications

It is important to note that our current work is complimentary to the
likelihood searches performed by the high energy neutrino detectors which also
use triggered stacking searches to look for high energy neutrino events
associated with merger event signals from the GW detectors. This is because
our projections of a limiting distance for the next generation detectors may
allow for efficient likelihood searches to then be performed on the meaningful
triggers obtained from the distance cuts proposed in this work. Having an
appropriate distance cut will also help reduce the computational costs without
losing much of the signals and hence complement the likelihood analyses by
IceCube or KM3NeT.

The uncertainties in this work include the fiducial rate of BNS mergers since
it can vary over an order of magnitude compared to our fiducial rate.
Moreover, the redshift dependent rate for BNS mergers is assumed to be the
same as that of short GRBs. A higher fiducial rate would lead to even more
optimistic results, i.e., associated neutrino detection or constraint in
shorter operation timescales. Another important aspect that can further
improve our analysis is the quality of triggers obtained from the GW
detectors. For the present analysis, we treat every trigger to be of the same
quality which is far from reality. Since this search method need not be in
real time in principle detailed analysis of the signal events would be done by
CE and ET and one can then only select triggers above a threshold quality.
This would also lead to more improved results in shorter operation timescales.

We also ignore the downtime of the detectors for simplicity and assume the
duty cycle or detection efficiency to be unity. Realistically, the neutrino
detectors like IceCube would have a duty cycle of $100$% [61], whereas the
gravitational-wave detectors can generally have a duty cycle $\sim 70$% [62].
This would only affect our results in a minor way. Another important factor
that we neglect in this work is the beaming associated with the gravitational
wave emission itself. It is expected that a fraction of gravitational wave
events that are associated to on-axis jets can be much larger than that for
the isotropic case (see figure 4 in Ref. [63]). Such events will have a lower
associated error region. Since we do not consider this effect our estimates
are on the conservative side.

The potential of follow-up or simultaneous electromagnetic (EM) observation
can also help reduce the localization area for distant events, which can then
improve the results. A BNS merger can be followed by a short GRB, a kilonova,
afterglows, and magnetar wind nebulae [64, 42, 65]. A promising scenario would
be an optical detection of kilonova. Optical detectors like Rubin [66] have
very good angular resolution $\sim 1$ arcsecond, resulting in extremely good
sky localization. In case of sGRBs, the Gamma-ray Burst Monitor (GBM) onboard
Fermi satellite [67, 68] may detect the gamma-rays from the prompt phase. This
does not help with significant reduction in the sky localization area, because
the angular resolution can be comparable to that by GW detectors. Besides, the
fraction of BNS mergers estimated to have sGRB jets is low $\sim 1$%. A
possible next generation GRB detector on the satellite Galileo G2 [69] would
also significantly improve the localization prospects given its capability of
localization up to $\sim 1^{\circ}$ at $1\sigma$ C.L.. Another next generation
GRB detector, Space-based multi-band astronomical Variable Objects Monitor
(SVOM) [70] can help detect GRB’s upto very high redshifts of $z>5$
complementing the next-generation of gravitational wave detectors. Swift BAT
[71] and future wide-field soft x-ray detectors, such as Einstein Probe [72]
and HiZ-GUNDAM [73] will have minutes-scale angular resolution, which should
help reducing localization.

Electromagnetic triggered searches can also be used to look for associated
high-energy neutrino events from BNS mergers. The typical maximum redshift for
kilonova detection for the current optical, infrared telescopes like Roman
[74] is $z\sim 0.2$, the Vera C. Rubin Observatory’s Legacy Survey of Space
and Time (LSST) [75] is $z\sim 0.1$, the Zwicky Transient Facility (ZTF) [76]
is $z\sim 0.02$ (see [77] for details). In particular, Roman has a distance
horizon for kilonova detection comparable to LIGO A+ [78]. LSST’s kilonova
detection horizon is comparable to the planned upgrade of advanced LIGO
(aLIGO) [79]. The kilonova detection horizons using such wide-field
instruments vary significantly with its associated properties like ejecta
velocity and mass. However, these kilonova detection horizons from optical and
infrared telescopes, are less than the projected reach for the third
generation GW detectors: ET and CE.

Although we focus on the next generation GW detectors in the current work, the
upcoming improvements to the current generation of GW detectors might also
lead to some constraints for the optimistic scenarios. A detailed study
regarding the sky-localization capabilities of the next generation of ground
based detectors - LIGO, VIRGO, KAGRA, LIGO-India network was performed in
[80].

The second generation of the GW detectors would be able to detect mergers up
to $\sim 200-300$ Mpc, implying the detection horizon to be $\sim 3$ times
lower than what CE by itself is capable of. Furthermore, the current
generation of IceCube has an effective area that is $\sim 5$ times smaller
than the planned IceCube-Gen2. Plugging in the factors in Eqs. (6) and (7), we
can roughly estimate the required operation timescale for the second
generation of GW detectors along with current generation IceCube to be $\sim
15$ times longer than the timescales proposed using the next generation GW
detectors and IceCube-Gen2 used in this work. This explains the absence of
detection of high-energy neutrinos from BNS mergers using the current GW
detectors [17, 28, 54] and also illustrates the inability to place constraints
in reasonable timescales.

The distance sensitivities of the improved detectors are lesser than the
proposed next generation detectors like ET and CE, but they can still help
with the nearby detections reducing the sky-localization area and also
providing some level of constraints until ET and CE are operational. On the
other hand, for the neutrino detectors, a similar analysis can be performed
for other next generation of planned neutrino detectors - KM3NeT [2], Baikal
GVD [5], P-ONE [3], and TRIDENT [4]. Furthermore, various proposed radio
detectors like IceCube-Gen2 Radio [1], GRAND [81], RNO-G [82], and optical
air-shower ultra-high energy neutrino detectors like Trinity [83]. This can
also help with performing triggered stacking searches for high energy
neutrinos from BNS mergers.

We stress that since we did not focus on collecting neutrino events from
individual sources in this work, the sky localization area was a good quantity
to consider for deciding a distance threshold. In most cases, the BNS merger
can be identified as the high-energy neutrino source for nearby events,
because the probability of two error regions associated with the GW detection
have a low probability of overlapping. For the worst case, involving unlikely
but large overlaps, another strategy might have to be adapted to decide a
distance limit based on triggers from the GW detectors such that the total
number of sources at any given instant is less than $2$.

Ideally, a more complete strategy would be to consider any generic
astrophysical source and understand whether a gravitational or neutrino
trigger would be more useful in that particular context. Since the GW
detectors have a much larger horizon as compared to neutrino detectors even if
a source emits less energy in GWs, it might still be useful to use the GW
detectors as triggers. This would also help understand what sources are ideal
candidates for collecting a good sample of high energy neutrinos. Although a
universally approximate neutrino spectra can be constructed for such sources,
the GW templates differ between various sources depending on the physical
instabilities and existing numerical models. Thus, we leave such a generic and
evidently powerful analysis for future work.

With the next generation of planned detectors in neutrino, GW and
electromagnetic channels the multimessenger landscape will be filled with new
discovery potentials and detecting high-energy neutrino events coincident with
BNS mergers will definitely be a crucial goal. Our work provides a perspective
and lays down a strategy for optimum synergic operation of the GW detectors
along with the neutrino detectors.

###### Acknowledgements.

We thank Imre Bartos and Bangalore Sathyaprakash for useful comments and
discussion. M. M. wishes to thank the Astronomical Institute at Tohoku
University, the Yukawa Institute for Theoretical Physics (YITP), Kyoto
University for their hospitality where a major part of this work was
completed. 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. 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. The work of
K.M. is supported by the NSF Grant No. AST-2108467, and KAKENHI No. 20H01901
and No. 20H05852.

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# A Proof of Euler’s Theorem via Quandles

António Lages and Pedro Lopes
Center for Mathematical Analysis, Geometry, and Dynamical Systems,
Department of Mathematics,
Instituto Superior Técnico, Universidade de Lisboa
1049-001 Lisbon, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We prove Euler’s theorem of number theory developing an argument based on
quandles. A quandle is an algebraic structure whose axioms mimic the three
Reidemeister moves of knot theory.

Keywords: Euler’s theorem; quandles; permutations

Mathematics Subject Classification 2020: 11A99; 20N99

##  1 Introduction

In this article we present a different proof of Euler’s theorem whose
statement is as follows.

######  Theorem 1.1 (Euler).

Let $n$ and $m$ be coprime positive integers. Then

$m\quad\mid\quad\big{(}n^{\varphi(m)}-1\big{)}.$

In the statement of Theorem 1.1 above, $\varphi$ denotes Euler’s totient
function, i.e., the number of positive integers less than the argument that
are coprime to the argument. Our proof is based on quandles. A quandle ([5],
[11]) is an algebraic structure whose axioms mimic the three Reidemeister
moves in knot theory ([6]) and thus provides interesting invariants for knots
([1]). The three axioms of quandles (Section 2) are (i) idempotency, (ii)
right-invertibility, (iii) self-distributivity. Right-invertibility allows us
to conceive of each quandle as a sequence of permutations. Since permutations
can be broken down into disjoint cycles, we organize the study of quandles
around the lengths these cycles may have ([10], [7], [8]). In the course of
these considerations we arrive at a proof of Euler’s Theorem. Specifically, in
this article, we look at a particular class of quandles and for each one of
them we consider its permutations. Then we calculate the length of the
disjoint cycles of these permutations along with the number of elements that
make them up. Equating the length of these cycles with the number of elements
comprised by these cycles, we prove Euler’s Theorem for only one prime
(Theorem 2.1). Bringing together the contributions of the individual primes in
the general case, we prove the full Euler’s Theorem (Theorem 1.1).

The organization of the current article is as follows. The background material
is laid out in Section 2, as well as the proofs of the theorems. In Section 3
we collect information about the quandle dealt with in Section 2 to state some
facts about its structure. In Section 4 we compare our proof with other
algebraic proofs from the literature.

##  2 Quandles and proof of Euler’s Theorem

######  Definition 2.1.

A quandle, $(X,\ast)$, is a set, $X$, equipped with a binary operation,
$\ast$, such that, for any $a,b,c\in X$

  1. (i)

$a\ast a=a$ – idempotency;

  2. (ii)

there exists a unique $x\in X$ such that $x\ast b=a$ – right-invertibility;

  3. (iii)

$(a\ast b)\ast c=(a\ast c)\ast(b\ast c)$ – self-distributivity.

Here are some examples of quandles.

######  Example 2.1.

  1. 1.

Let $G$ be a group and consider the conjugation in the group, $a\ast
b:=bab^{-1}$. Then $(G,\ast)$ is a quandle.

  2. 2.

Let $n,m$ be positive integers with $m$ invertible modulo $n$. Consider $a\ast
b=ma+(1-m)b$ (mod $n$). Then $(\mathbf{Z}/n\mathbf{Z},\ast)$ is a quandle.
These quandles are called linear Alexander quandles. In the sequel, we call
$m$ the parameter of these quandles.

######  Definition 2.2.

For a quandle $(X,*)$ and for $y\in X$, we let $R_{y}:X\rightarrow X,x\mapsto
x*y$ be the right translation by $y$ in $(X,*)$. Axiom (ii) in Definition 2.1
guarantees that right translations are permutations of the elements of the
quandle under study.

We now prove a particular case of Euler’s Theorem in which there is only one
prime at stake.

######  Theorem 2.1.

Let $p$ be a prime. Let $n,k$ be positive integers with $n$ coprime with $p$.
Then,

$p^{k}\mid(n^{p^{k}-p^{k-1}}-1).$

###### Proof.

Since $n\cdot n^{p^{k}-1}\equiv n^{p^{k}}\equiv 1$ (mod $n^{p^{k}}-1$), it
follows that $n$ is invertible modulo $n^{p^{k}}-1$. Consider, then, the
linear Alexander quandle of order $n^{p^{k}}-1$ and parameter $n$ (cf. Example
2.1, $2.$). The underlying set is $\mathbf{Z}/(n^{p^{k}}-1)\mathbf{Z}$ and,
for any $a,b\in\mathbf{Z}$, $a\ast b=na+(1-n)b$ (mod $n^{p^{k}}-1$). We will
need the following lemma.

######  Lemma 2.1.

Let $a,b\in\mathbf{Z}$. Then, for any $l\in\mathbf{Z}^{+}$,

$R_{b}^{\,\,\,l}(a):=(\cdots((a\ast\underbrace{b)\ast b)\cdots)\ast
b}_{l}=n^{l}a-(n^{l}-1)b.$

###### Proof.

This is a straightforward exercise in induction that we leave to the reader. ∎

Resuming the proof of Theorem 2.1, setting $l=p^{k}$, we obtain

$R_{b}^{\,\,\,p^{k}}(a)=n^{p^{k}}a-(n^{p^{k}}-1)b\equiv a\qquad\text{ (mod
}n^{p^{k}}-1).$

We note that, modulo $n^{p^{k}}-1$, the sequence

$(R_{b}^{\,\,\,l}(a)\,:\,l\in\mathbf{Z}_{0}^{+})$

is a periodic sequence starting at $a$ and consisting of copies of the cycle
of $R_{b}$ that contains $a$. Then, $R_{b}^{\,\,\,p^{k}}(a)\equiv a$ implies
that the length of the disjoint cycle in $R_{b}$ containing $a$, divides
$p^{k}$. The possibilities for these lengths are then
$1,p,p^{2},\dots,p^{i},\dots,p^{k}$. We need to count the $a$’s that satisfy
$R_{b}^{\,\,\,p^{i}}(a)\equiv a$, for the different $i$’s. We call an $a$ that
satisfies $R_{b}^{\,\,\,p^{i}}(x)\equiv x$ but does not satisfy
$R_{b}^{\,\,\,p^{j}}(x)\equiv x$, for $j<i$, a proper solution of
$R_{b}^{\,\,\,p^{i}}(x)\equiv x$. We will need the following lemma.

######  Lemma 2.2.

The number of proper solutions, modulo $n^{p^{k}}-1$, of
$R_{b}^{\,\,\,p^{i}}(x)\equiv x$ is $n^{p^{i}}-n^{p^{i-1}}$, for $1\leq i\leq
k$, and $n-1$ for $i=0$.

###### Proof.

We use induction on $i$. For $i=0$, we are looking for the solutions of
$x\equiv R_{b}^{\,\,\,1}(x)=nx-(n-1)b$, i.e., $(n-1)(x-b)\equiv 0$ (mod
$n^{p^{k}}-1$). Now,

$n^{p^{k}}-1=(n-1)(n^{p^{k}-1}+\dots+n+1)$

so the proper solutions, modulo $n^{p^{k}}-1$, are:

$\displaystyle x_{0}\equiv b$ $\displaystyle x_{1}\equiv
b+(n^{p^{k}-1}+n^{p^{k}-2}+\cdots+n^{2}+n+1)$ $\displaystyle x_{2}\equiv
b+2(n^{p^{k}-1}+n^{p^{k}-2}+\cdots+n^{2}+n+1)$ $\displaystyle\vdots$
$\displaystyle x_{n-2}\equiv
b+(n-2)(n^{p^{k}-1}+n^{p^{k}-2}+\cdots+n^{2}+n+1).$

There are then $n-1$ proper solutions for $i=0$.

For $i=1$, we are looking for the solutions of $x\equiv
R_{b}^{\,\,\,p}(x)=n^{p}x-(n^{p}-1)b$, i.e., $(n^{p}-1)(x-b)\equiv 0$ (mod
$n^{p^{k}}-1$). Now,

$n^{p^{k}}-1=(n^{p}-1)((n^{p})^{p^{k-1}-1}+\dots+n^{p}+1)$

so the solutions are, modulo $n^{p^{k}}-1$:

$\displaystyle x_{0}\equiv b$ $\displaystyle x_{1}\equiv
b+((n^{p})^{p^{k-1}-1}+\dots+n^{p}+1)$ $\displaystyle x_{2}\equiv
b+2((n^{p})^{p^{k-1}-1}+\dots+n^{p}+1)$ $\displaystyle\vdots$ $\displaystyle
x_{n^{p}-2}\equiv b+(n^{p}-2)((n^{p})^{p^{k-1}-1}+\dots+n^{p}+1).$

There are then $n^{p}-1$ solutions for $i=1$. In order to obtain the number of
proper solutions of $x\equiv R_{b}^{\,\,\,p}(x)$ we subtract from $n^{p}-1$
the number of solutions of $x\equiv R_{b}^{\,\,\,1}(x)$. There are, then,
$n^{p}-1-(n-1)=n^{p}-n$ proper solutions of $x\equiv R_{b}^{\,\,\,p}(x)$.

We now assume that, modulo $n^{p^{k}}-1$, there is an integer $i\geq 1$ such
that the number of proper solutions of $R_{b}^{\,\,\,p^{j}}(x)\equiv x$, for
each $1\leq j\leq i-1$, is $n^{p^{j}}-n^{p^{j-1}}$, and $n-1$ for $j=0$.
Modulo $n^{p^{k}}-1$, the equation $R_{b}^{\,\,\,p^{i}}(x)\equiv x$ is
equivalent to $0\equiv(n^{p^{i}}-1)(x-b)$. Furthermore,

$n^{p^{k}}-1=\big{(}n^{p^{i}}\big{)}^{p^{k-i}}-1=\big{(}n^{p^{i}}-1\big{)}\big{(}\big{(}n^{p^{i}}\big{)}^{p^{k-i}-1}+\big{(}n^{p^{i}}\big{)}^{p^{k-i}-2}+\dots+n^{p^{i}}+1\big{)}.$

Then, the solutions, modulo $n^{p^{k}}-1$, are:

$x_{j}\equiv
b+j\big{(}\big{(}n^{p^{i}}\big{)}^{p^{k-i}-1}+\big{(}n^{p^{i}}\big{)}^{p^{k-i}-2}+\dots+n^{p^{i}}+1\big{)}\qquad\text{
for }\quad 0\leq j\leq n^{p^{i}}-2.$

There are, then, $n^{p^{i}}-1$ such solutions. In order to obtain the number
of proper solutions, we subtract from $n^{p^{i}}-1$, the number of proper
solutions of the equations of the sort $R_{b}^{\,\,\,p^{j}}(x)\equiv x$,
modulo $n^{p^{k}}-1$, for $0\leq j\leq i-1$:

$n^{p^{i}}-1-\sum_{j=1}^{i-1}\big{(}n^{p^{j}}-n^{p^{j-1}}\big{)}-(n-1)=n^{p^{i}}-n^{p^{i-1}}.$

This completes the proof of Lemma 2.2. ∎

Resuming the proof of Theorem 2.1, the number of proper solutions of
$R_{b}^{\,\,\,p^{k}}(x)\equiv x$, modulo $n^{p^{k}}-1$, is
$n^{p^{k}}-n^{p^{k-1}}$. But the set of these proper solutions coincides with
the union of all disjoint cycles of length $p^{k}$ of $R_{b}$. Therefore,

$p^{k}\mid\big{(}n^{p^{k}}-n^{p^{k-1}}\big{)}.$

Since $n^{p^{k}}-n^{p^{k-1}}=n^{p^{k-1}}\big{(}n^{p^{k}-p^{k-1}}-1\big{)}$ and
$n$ and $p$ are coprime, it follows that

$p^{k}\mid\big{(}n^{p^{k}-p^{k-1}}-1\big{)}.$

This completes the proof of Theorem 2.1. ∎

We are now ready to prove Euler’s Theorem. We rewrite the statement of Euler’s
Theorem here as Theorem 2.2 below. In the statement of Theorem 2.2 we
emphasize the role of the individual primes in order to bring out the
connection with Theorem 2.1.

######  Theorem 2.2 (Euler).

Let $p_{1},p_{2},\dots,p_{N}$ be a sequence of distinct primes,
$k_{1},k_{2},\dots,k_{N}$ a sequence of positive integers, and $n$ a positive
integer coprime with each of the primes. Then

$\prod_{j=1}^{N}p_{j}^{k_{j}}\quad\mid\quad\Bigg{(}n^{\,\,\displaystyle{\prod_{j=1}^{N}\big{(}p_{j}^{k_{j}}-p_{j}^{k_{j}-1}\big{)}}}-1\Bigg{)}.$

###### Proof.

Let $1\leq i\leq N$. By Theorem 2.1 we have

$p^{k_{i}}\,\mid\,\big{(}n^{p^{k_{i}}-p^{k_{i}-1}}-1\big{)}.$

Since

$m^{rs}-1=(m^{r})^{s}-1=(m^{r}-1)\big{(}(m^{r})^{s-1}+\cdots+m^{r}+1\big{)},$

it follows that

$(m^{r}-1)\,\mid\,(m^{rs}-1),$

and so

$p^{k_{i}}\,\mid\,\big{(}n^{p^{k_{i}}-p^{k_{i}-1}}-1\big{)}\,\mid\,\Bigg{(}n^{\,\,\displaystyle{\prod_{j=1}^{N}\big{(}p_{j}^{k_{j}}-p_{j}^{k_{j}-1}\big{)}}}-1\Bigg{)}.$

Moreover, since any two of the primes are distinct,

$\prod_{j=1}^{N}p_{j}^{k_{j}}\quad\Huge\mid\quad\Bigg{(}n^{\,\,\displaystyle{\prod_{j=1}^{N}\big{(}p_{j}^{k_{j}}-p_{j}^{k_{j}-1}\big{)}}}-1\Bigg{)}.$

This completes the proof of Euler’s Theorem. ∎

##  3 Extra information

In this section we gather some results we obtained so far (and come up with
another one) in order to prove two facts about the linear Alexander quandle we
have been dealing with. We need the following Definitions.

######  Definition 3.1.

The list of lengths of the disjoint cycles that make up a permutation is
called the pattern of the permutation. The list of patterns of the
permutations of a quandle is called the profile of the quandle. If the
profile, as a set, is a singleton, we identify the only pattern with the
profile.

######  Definition 3.2.

A quandle $(X,\ast)$ is connected if for any $i,j\in X$ there are finitely
many $k_{1},\dots,k_{l}\in X$ such that

$j=(\cdots((i\ast k_{1})\ast k_{2})\ast\cdots)\ast k_{l}.$

######  Theorem 3.1.

Let $p$ be a prime. Let $n,k$ be positive integers with $n$ coprime with $p$.

Consider the linear Alexander quandle of order $n^{p^{k}}-1$ and parameter
$n$. Then,

  1. (i)

the profile of this quandle is:

$\\{\underbrace{1,1,\dots,1}_{\displaystyle{n-1}},\underbrace{p,p,\dots,p}_{\displaystyle{\frac{n^{p}-n}{p}}},\dots,\underbrace{p^{i},p^{i},\dots,p^{i}}_{\displaystyle{\frac{n^{p^{i}}-n^{p^{i-1}}}{p^{i}}}},\dots,\underbrace{p^{k},p^{k},\dots,p^{k}}_{\displaystyle{\frac{n^{p^{k}}-n^{p^{k-1}}}{p^{k}}}}\\}$

  2. (ii)

this quandle is not connected.

###### Proof.

  1. (i)

In the course of the proof of Theorem 2.1 (between Lemmas 2.1 and 2.2) we
establish that the possible lengths of any disjoint cycle of any permutation
of the quandle at issue are

$1,p,p^{2},\dots,p^{i},\dots,p^{k}.$

In Lemma 2.2 we calculate the number of elements of the quandle that
constitute all cycles for each of these possibilities: cycles of length $1$:
$n-1$ elements; cycles of length $p$: $n^{p}-n$ elements; …cycles of length
$p^{i}$: $n^{p^{i}}-n^{p^{i-1}}$ elements; …cycles of length $p^{k}$:
$n^{p^{k}}-n^{p^{k-1}}$ elements. So there are $n-1$ cycles of length $1$;
$\frac{n^{p}-n}{p}$ cycles of length $p$;
…$\frac{n^{p^{i}}-n^{p^{i-1}}}{p^{i}}$ cycles of length $p^{i}$;
…$\frac{n^{p^{k}}-n^{p^{k-1}}}{p^{k}}$ cycles of length $p^{k}$.

  2. (ii)

We will show that successive quandle multiplication of $0$ by other elements
will only give rise to multiples of $n-1$.

######  Lemma 3.1.

Let $(b_{i})$ be a sequence of integers. Then, modulo $n^{p^{k}}-1$, we have

$(\cdots((0\ast b_{1})\ast b_{2})\ast\cdots)\ast
b_{l}=-(n-1)(n^{l-1}b_{1}+n^{l-2}b_{2}+\dots+nb_{l-1}+b_{l}).$

###### Proof.

This is a straightforward exercise in induction that we leave to the reader. ∎

According to Lemma 3.1, there are no $b_{1},\dots,b_{l}\in\mathbf{Z}$ such
that $1=(\cdots((0\ast b_{1})\ast b_{2}\dots)\ast b_{l}$ (mod $n^{p^{k}}-1$).
This completes the proof of Theorem 3.1.

∎

##  4 Final remarks

We would be remiss if we did not elaborate further about the totient function.
As is known, the value of the totient function at
$\prod_{j=1}^{N}p_{j}^{k_{j}}$ is
$\prod_{j=1}^{N}\big{(}p_{j}^{k_{j}}-p_{j}^{k_{j}-1}\big{)}$, its meaning
being the number of coprimes with $\prod_{j=1}^{N}p_{j}^{k_{j}}$, in the range
between $1$ and $\prod_{j=1}^{N}p_{j}^{k_{j}}-1$. But in the course of our
proofs, we do not need the interpretation of the totient function. This is why
the totient function has not been much mentioned this far.

To the best of our knowledge, the algebraic proofs of Euler’s Theorem found in
the literature seem to be the following two (we let $\varphi$ stand for the
totient function).

  1. 1.

([3], [4]) Given a positive integer $r$, $\mathbf{Z}/r\mathbf{Z}$ is
understood as a ring whose group of units has $\varphi(r)$ elements. Then
using a corollary of Lagrange’s theorem, any of these units (i.e.,
$[a]\in\mathbf{Z}/r\mathbf{Z}$ such that $a$ is coprime with $r$) satisfies
$[a]^{\varphi(r)}=[1]$, modulo $r$. Thus, $r\mid(a^{\varphi(r)}-1)$.

  2. 2.

([2], [9]) Given a positive integer $r$, $\mathbf{Z}/r\mathbf{Z}$ is
understood as a ring whose group of units has $\varphi(r)$ elements. These
$\varphi(r)$ elements have representatives in reduced form, as follows:

$1\leq r_{1}<r_{2}<\dots<r_{\varphi(r)}<r.$

Upon multiplication by any unit $[a]\in\mathbf{Z}/r\mathbf{Z}$ (i.e., such
that $a$ is coprime with $r$), we obtain:

$[ar_{1}],[ar_{2}],\dots,[ar_{\varphi(r)}].$

Now,

$[ar_{i}]=[ar_{j}]\Longleftrightarrow[a][r_{i}]=[a][r_{j}]\Longleftrightarrow[r_{i}]=[r_{j}]\Longleftrightarrow
r_{i}=r_{j}.$

Thus,

$\\{[ar_{1}],[ar_{2}],\dots,[ar_{\varphi(r)}]\\}=\\{[r_{1}],[r_{2}],\dots,[r_{\varphi(r)}]\\},$

so that

$\prod_{j=1}^{\varphi(r)}[r_{j}]=\prod_{j=1}^{\varphi(r)}[ar_{j}]=[a]^{\varphi(r)}\prod_{j=1}^{\varphi(r)}[r_{j}]\Longleftrightarrow[1]=[a]^{\varphi(r)},$

i.e.,

$r\mid(a^{\varphi(r)}-1).$

Both of these arguments rely on the existence of a function that counts the
units in $\mathbf{Z}/r\mathbf{Z}$ – the totient function. Also, they rely on
the structure of the group of units of the ring $\mathbf{Z}/r\mathbf{Z}$ and
the first one also relies on Lagrange’s Theorem.

On the other hand, our considerations set up the function

$\prod_{j=1}^{N}p_{j}^{k_{j}}\mapsto\prod_{j=1}^{N}\big{(}p_{j}^{k_{j}}-p_{j}^{k_{j}-1}\big{)},$

as the argument is developed. We do not associate any ulterior meaning with
this function. Furthermore, our proof has a simple interpretation related to
counting the elements in the cycles with the relevant length (for the prime
power case, Theorem 2.1).

Acknowledgements

Lages’s work is supported by FCT (Fundação para a Ciência e a Tecnologia),
Portugal, through LisMath fellowship PD/BD/150348/2019.

Lopes acknowledges support by FCT (Fundação para a Ciência e a Tecnologia),
Portugal, through project FCT PTDC/MAT-PUR/31089/2017, “Higher Structures and
Applications”

Both authors acknowledge support by FCT/Portugal through CAMGSD, IST-ID,
projects UIDB/04459/2020 and UIDP/04459/2020.

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