11institutetext: Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy
22institutetext: INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi,
Italy 33institutetext: INAF—Osservatorio Astronomico di Brera, Merate, Italy
44institutetext: Università degli Studi di Milano-Bicocca, Dip. di Fisica “G.
Occhialini”, Milano, Italy 55institutetext: Dipartimento di Fisica e
Astronomia ’G. Galilei’, Università degli studi di Padova, vicolo
dell’Osservatorio 3, I-35122, Padova, Italia 66institutetext: INFN, Sezione di
Padova, via Marzolo 8, I-35131, Padova, Italia

# Perspectives for multi-messenger astronomy
with the next generation of gravitational-wave detectors and high-energy
satellites

S. Ronchini 1122 M. Branchesi 1122 G. Oganesyan 1122 B. Banerjee 1122 U.
Dupletsa 1122 G. Ghirlanda 3344 J. Harms 1122 M. Mapelli 5566 F. Santoliquido
5566

The Einstein Telescope (ET) is going to bring a revolution for the future of
multi-messenger astrophysics. In order to detect the counterparts of binary
neutron star (BNS) mergers at high redshift, the high-energy observations will
play a crucial role. Here, we explore the perspectives of ET, as single
observatory and in a network of gravitational-wave (GW) detectors, operating
in synergy with future $\gamma$-ray and X-ray satellites. We predict the high-
energy emission of BNS mergers and its detectability in a theoretical
framework which is able to reproduce the properties of the current sample of
observed short GRBs (SGRB). We estimate the joint GW and high-energy detection
rate for both the prompt and afterglow emissions, testing several combinations
of instruments and observational strategies. We find that the vast majority of
SGRBs detected in $\gamma$-rays will have a detectable GW counterpart; the
joint detection efficiency approaches $100\%$ considering a network of third
generation GW observatories. The probability of identifying the
electromagnetic counterpart of BNS mergers is significantly enhanced if the
sky localization provided by GW instruments is observed by wide field X-ray
monitors. We emphasize that the role of the future X-ray observatories will be
very crucial for the detection of the fainter emission outside the jet core,
which will allow us to probe the yet unexplored population of low-luminosity
SGRBs in the nearby Universe, as well as to unveil the nature of the jet
structure and the connections with the progenitor properties.

###### Key Words.:

(stars:) gamma-ray bursts - general – gravitational waves – astroparticle
physics

##  1 Introduction

The first detection of gravitational-waves from the merger of two neutron
stars, GW 170817, by the Advanced LIGO (LIGO Scientific Collaboration & Aasi
2015) and Virgo (Virgo Collaboration & Acernese 2015) interferometers and the
observed electromagnetic (EM) counterparts represented a milestone for multi-
messenger astronomy, in particular for probing the nature of EM transients
originated from BNS mergers (Abbott et al. 2017b, c, d). The extensive study
of the EM emission allowed us to obtain an exhaustive picture of the outflow
properties of both the relativistic structured jet (Troja et al. 2017,
Hallinan et al. 2017, Mooley et al. 2018, Ghirlanda et al. 2019a) and the
isotropic sub-relativistic kilonova outflow (Pian et al. 2017, Smartt et al.
2017, Kasen et al. 2017). Considering the distance range expected to be
reached by LIGO, Virgo and KAGRA in the next runs (Abbott et al. 2020) for the
detection of BNS mergers and the astrophysical rate of this type of system
$R_{BNS}=[10-1700]$ Gpc-3 yr-1 (The LIGO Scientific Collaboration et al.
2021a) derived in the first three runs of observations by the Advanced GW
detectors, the detection rate results to be from a few tens to a few hundreds
per year in the next observational runs.Therefore, taking into account that
only a fraction of these produces a jet and only a fraction of them points
along the line of sight, the probability of joint GW and SGRB detection with
current instruments is quite limited (Metzger & Berger 2012, Chen & Holz 2013,
Clark et al. 2015, Patricelli et al. 2016, Ghirlanda et al. 2016, Beniamini et
al. 2019a, Saleem et al. 2018a, Saleem et al. 2018b, Saleem 2020, Perna et al.
2022,Patricelli et al. 2022,Colombo et al. 2022). The advent of third
generation GW observatories, such as the Einstein Telescope (ET) (Punturo et
al. 2010) and Cosmic Explorer (CE) (Abbott et al. 2017a), will be a
substantial leap forward compared to the current instruments for the
cosmological census of GW sources. ET is an underground observatory, designed
as three low and three high frequency interferometers nested in a triangular
shape forming three 10 km arms. CE, instead, is an L-shape surface
interferometer of 40 km length. ET and CE are expected to detect $\sim 10^{5}$
BNS per year, reaching redshifts well above the star formation peak (Maggiore
et al. 2020, Evans et al. 2021a); for optimally located and oriented systems,
ET will be able to detect BNS up to $z\sim 5$, while CE up to $z\sim 10$. For
comparison, the current GW detectors are expected to reach $z\sim 0.2$, even
considering new upgrades planned for the fifth run of observations in 2025
(Abbott et al. 2020). Furthermore, accessing low frequencies makes ET capable
to detect BNSs well before the merger and follow their inspiral not only for
minutes but also for several hours in the case of closeby events (Chan et al.
2018, Grimm & Harms 2020, Nitz & Dal Canton 2021, Li et al. 2021). This
enables us to use the Earth rotation to determine the sky localization even
when ET is operating as a single detector (Chan et al. 2018; Grimm & Harms
2020; Nitz & Dal Canton 2021; Li et al. 2021). These authors found that a
fraction of a few percents of BNS detected by ET will have a sky localization
uncertainity $\Delta\Omega<100$ deg2. This fraction increases to a few tens of
percents when ET is operating in network with CE. Considering a network of
three third generation detectors located in Europe, USA, and Australia a few
tens of percents of detected BNS will be localized within 10 deg2 and almost
the totality within 100 deg2 (Li et al. 2021,Borhanian & Sathyaprakash 2022,
Mills et al. 2018). A detailed study of performance metrics of several
configurations of network including CE, ET, and the second generation
detectors and their upgrade, are given in Borhanian & Sathyaprakash 2022.
Another proposed GW observatory to be mentioned is the Neutron star Extreme
Matter Observatory (NEMO, Ackley et al. 2020), a 4-km detector whose
sensitivity at high frequencies will be relevant for the investigation of
neutron star physics and for multi-messenger purposes (Sarin & Lasky 2022).
Wide Field Of View (FOV) $\gamma$-ray and X-ray observatories will become more
and more crucial to work in synergy with gravitational-wave detectors which
will make it possible to reach even larger distances. They represent a unique
way to detect high-z counterparts. Being intrinsically fainter than the high
energy emission, the kilonova emission will be hardly detected at distances
larger than a redshift of 0.2-0.3 even with next generation optical wide FOV
observatories, such as the Vera Rubin Observatory (see e.g. Maggiore et al.
2020). Larger distances can be reached with extremely large telescopes, such
as ELT, but only arcmin-arcsec localized GW sources can be effectively pointed
by these telescopes. The next generation of GW detectors observing with
sensitive wide FOV high-energy satellites will enable i) to build large
samples of joint detections, ii) to have joint detections at high redshift,
and iii) to observe, in the local Universe, counterparts from larger viewing
angles with respect to the jet axis which will enable precise parameter
estimation of the source progenitor properties. This will allow us to deeply
investigate the connection between BNS mergers and short GRBs, to unveil the
jet structure (Hayes et al. 2020; Beniamini et al. 2019a; Biscoveanu et al.
2020), to estimate how many BNS mergers produce jets (Salafia et al. 2022),
how the production of the jet is connected to the progenitor properties, and
what is the relation between merger remnants and the properties of the high-
energy emission. Accessing a large sample of joint detections at high-redshift
will enable us to evaluate cosmological parameters, including the Hubble
constant and the dark energy equation of state (Zhao & Wen 2018; Mendonça
Soares de Souza et al. 2021; Yu et al. 2021; Chen et al. 2021), and test
modified gravity at cosmological distances by comparing the GW luminosity
distance and the one derived from the electromagnetic side (Belgacem et al.
2019; Mancarella et al. 2021). Identifying the BNS host galaxies and having a
redshift will be of primary importance for these science cases. $\gamma$-ray
and X-ray telescopes can potentially give precise (arcmin-arcsec) sky
localization to drive the ground-based follow-up.

To maximize the scientific return of the next generation GW detectors in the
multi-messenger context it is of paramount importance to evaluate the expected
joint detections, to identify instrument scientific requirements and optimal
observation strategy. In this work, we focus on the synergy of third
generation (3G) GW detectors with high-energy ($\gamma$-ray and X-ray)
telescopes. The novelty of our study consists in the development of a
theoretical framework that starting from an astrophysically-motivated
population of BNSs and the modelling of the associated high-energy EM signal
is able to reproduce the statistical properties of current SGRB catalogs. This
approach, based on calibrating the BNS population and the jet properties using
observed SGRBs, allows us to be less dependent on the uncertainties related to
the BNS population synthesis models about the local merger rate and its
redshift distribution. On the GW side, we evaluate the detection efficiency of
3G GW detectors, as well as their capabilities in localizing the source. On
the EM side, we provide a comprehensive overview of the detectability of
prompt and afterglow emission associated with the SGRB.

The population synthesis model for the cosmic BNS merger rate density, the GW
detector configurations and the modelling of the GW signal, the modelling and
calibration of the prompt and afterglow emission are described in Sec. 2. In
Sec. 3 we show the predictions for joint detection of GW along with prompt and
afterglow emission exploring several observational strategies. We consider ET
as single observatory and in different network of 3G GW detectors observing
together with future high-energy facilities, which can operate in survey and
pointing modes. We conclude highlighting the perspectives for joint GW and
high-energy detections, discussing the optimal instruments and observational
strategies to completely exploit the synergy between 3G GW detectors and
$\gamma-$ and X-ray satellites, and the impact on the multi-messenger science
case.

##  2 Modelling and data analysis

###  2.1 The BNS population model

The population of BNS mergers is produced as detailed in Santoliquido et al.
(2021). The catalog of isolated compact binaries is generated with MOBSE
(Mapelli et al. 2017; Giacobbo et al. 2018; Giacobbo & Mapelli 2018), which is
a population synthesis-code taking into account the metallicity dependence of
mass-loss rate of massive stars, the uncertainty related to the common
envelope evolution, natal kicks and mass transfer efficiency. The BNS merger
rate density as a function of redshift is obtained with the
cosmo$\mathcal{R}$ate code (Santoliquido et al. 2020). Here,
cosmo$\mathcal{R}$ate adopts the star formation rate density and average
metallicity evolution of the Universe from Madau & Fragos (2017). We assume a
metallicity spread $\sigma_{\rm Z}=0.3$. Redshift evolution and overall
normalisation of BNS merger density depend on several assumptions regarding
the common envelope prescription (parametrised by the common envelope ejection
efficiency $\alpha_{\rm CE}$), the natal kick model, the supernova explosion,
the mass transfer via accretion, and the shape of the initial mass function.
Here, we describe electron-capture supernovae as in Giacobbo & Mapelli (2019)
and assume the delayed supernova model (Fryer et al. 2012) to decide whether a
core-collapse supernova produces a black hole or a neutron star. When a
neutron star forms from either a core-collapse or an electron-capture
supernova, we randomly draw its mass according to an uniform distribution
between 1 and 2.5 M⊙. The assumption of a uniform and broad mass distribution
is consistent with the last results from GW observations (The LIGO Scientific
Collaboration et al. 2021c). In the Appendix we discuss how using a mass
distribution sharply peaked around 1.33 M⊙ as observed in Galactic double
neutron-star systems (Özel & Freire 2016) can impact our results. We generate
the natal kicks according to $v_{\rm kick}\propto{}m_{\rm ej}/m_{\rm
rem}\,{}v_{\rm H05}$, where $m_{\rm ej}$ is the mass of the ejecta, $m_{\rm
rem}$ is the mass of the compact remnant (neutron star or black hole) and
$v_{\rm H05}$ is a random number extracted from a Maxwellian distribution with
one-dimensional root-mean square $\sigma=265$ km s-1 (Hobbs et al. 2005). This
formalism ensures low kicks for most electron-capture and ultra-stripped
supernovae (Giacobbo & Mapelli 2020). We assume a value of $\alpha_{\rm CE}=3$
for common envelope, we calculate the concentration parameter $\lambda$ as in
Claeys et al. (2014) and the mass transfer formalism from Hurley et al.
(2002). We draw the mass of the progenitor primary star from a Kroupa mass
function (Kroupa 2001). For the initial mass ratios, orbital periods and
eccentricities we use the distributions inferred by Sana et al. (2012). With
such combination of assumptions the entire population consists of $\sim
9\times 10^{5}$ BNS mergers per year (observed at Earth by an ideal
instrument), and the local BNS merger rate ($R_{BNS}=365\,\rm
Gpc^{-3}yr^{-1}$) is consistent with the most updated constraints provided by
GW observations during the first, second and third run of observations by the
Advanced LIGO and Virgo detectors ($R_{BNS}\in[10-1700]\rm Gpc^{-3}yr^{-1}$,
The LIGO Scientific Collaboration et al. 2021b, c; Abbott et al. 2021a, b).
Moreover, the local rate assumed here is perfectly consistent with the
collection of local BNS merger rates given by Mandel & Broekgaarden (2021) and
including both observed (from GW, kilonovae, SGRBs, and Galactic pulsar
binaries’ observations) and theoretically predicted rates. The BNS systems are
isotropically distributed in the sky and randomly oriented.

###  2.2 GW detector configuration, GW signal modeling and parameter
estimation

In this work, we consider three GW detector configurations: 1) ET as a single
detector located in Sardinia (Italy), 2) ET in a network with CE (40 km arms)
located in the LIGO-Livingston site, 3) ET in a network with 2CE of 40 km
arms, with the second CE located in Australia. We assume the full sensitivity
configuration for ET (referred as ET-D configuration, Hild et al. 2011). For
CE we use the sensitivity of the 40 km arms detector given by Evans et al.
(2021b). For CE as well as for each of the three combinations of high and low
frequency interferometers of ET we assume a duty cycle of 0.85. Starting from
the BNS population described in Section 2.1, we inject GW signals constructed
using a post-Newtonian formalism, in particular assuming a TaylorF2 waveform
(Buonanno et al. 2009). Since the spin is expected to be small in binaries of
neutron stars that merge within a Hubble time (Burgay et al. 2003), the
effects due to spins are neglected. The signal to noise ratio (SNR) of each
inspiral is then computed through a matched-filter. A network SNR threshold
equal to 8 is used to select GW detections. The parameter estimation is
obtained using the GWFish code (Harms et al. 2022) and computing the elements
of the inverse of the Fisher matrix (Grimm & Harms 2020). The code is publicly
available at this repository111https://github.com/janosch314/GWFish. In GWFish
the waveforms, modeled in frequency domain, are implemented from scratch and
the code includes an interface with
LALSimulation222https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/.
Detector networks consider Earth’s rotation, which has an impact on the
localization capabilities. The most important computational aspects (waveform
derivatives and Fisher matrix inversion) are carried out using an hybrid
approach for derivatives (analytic + numeric) and a combination of matrix
normalization and singular value decomposition for matrix inversion (which
takes care of the quasi singularity of the Fisher matrices). Such an approach
approximates the true likelihood with a Gaussian profile, which is valid in
the limit of a high information content in the data. In this regime the
likelihood is highly peaked and the role of priors is negligible. Hereafter,
the uncertainty on the sky localization $\Delta\Omega$ is given as 90$\%$
credible region, while the uncertainty on other GW parameters, such as
$\theta_{v}$ and luminosity distance, is given at $1\sigma$ confidence level.

###  2.3 Prompt emission modeling

For the estimation of the high-energy EM emission from the BNS merger we
assume that only a fraction $f_{j}$ of the mergers produces a jet which is
able to penetrate the post-merger ejecta and to successfully break-out. In
principle, $f_{j}$ could depend on the properties of the binary system, but
here we do not assume any dependency. The EM counterpart is evaluated in the
assumption of a universal jet structure (Lipunov et al. 2001, Salafia et al.
2015, Salafia et al. 2019, Salafia et al. 2020), which is the same of GRB
170817A. In particular, we adopt:

$\epsilon(\theta)=\frac{dE}{d\Omega}\propto\frac{1}{1+(\theta/\theta_{c})^{s_{\epsilon}}}$
(1)

and

$\Gamma(\theta)=1+\frac{\Gamma_{0}-1}{1+(\theta/\theta_{c})^{s_{\Gamma}}}$ (2)

where $\epsilon(\theta)$ is the angular structure of the local emissivity and
$\Gamma(\theta)$ the bulk Lorentz factor profile ($\theta$ is the polar angle
measured from the jet symmetry axis). The choice of $\theta_{c}$ and
$\Gamma_{0}$ influences the number of prompt emission detections, while the
off-core specific shape of the jet has more impact on the brightness of the
afterglow component and the prompt tail. From the modeling of radio afterglow
of GRB 170817A (Ghirlanda et al. 2019b), the best fit parameters of the jet
structure are:

$s_{\epsilon}=5.5^{+1.3}_{-1.4},\,s_{\Gamma}=3.5^{+2.1}_{-1.7},\,\theta_{c}=(3.4\pm
1.0)\text{ deg}$ (3)

For our modeling, we fix for simplicity $s_{\epsilon}=s_{\Gamma}=s$ and we
adopt:

$s=4,\,\theta_{c}=3.4\text{ deg}$ (4)

Moreover, we fix $\Gamma_{0}=500$, which is in agreement with usual values
inferred for GRBs (e.g. Ghirlanda et al. 2018). These parameters are the same
for the modeling of both prompt and afterglow emission. A broader structure
with $s=2$ and same $\theta_{c}$ is also tested (Rossi et al. 2002). Hereafter
we refer to:

  1. 1.

_Stru1_ for the case $s=4$ and $\theta_{c}=3.4$ deg

  2. 2.

_Stru2_ for the case $s=2$ and $\theta_{c}=3.4$ deg

The dependency of our results on the choice of the structure parameters is
discussed later in the paper. The aperture angle assumed here is consistent
with the values found in SGRBs with well constrained jet break (typical values
of $\theta_{c}$ are in the interval $\sim 1^{\circ}-8^{\circ}$, see Fong et
al. 2014, Jin et al. 2018 and references within). Similar values are found in
hydrodynamical simulations of jet propagation in NS mergers, e.g. Lazzati et
al. 2018 predicts a jet aperture angle of $\sim 5^{\circ}$ (see also Geng et
al. 2019, Wu & MacFadyen 2018 ). Also Lamb et al. (2022) find a rather small
jet core ($\theta_{c}\sim 2.2^{\circ}$) and a jet profile compatible with eq.
1 and 2, with best fit values $s_{\epsilon}\sim 3.1$ and $s_{\Gamma}\sim 1.8$.

We define $E_{t}$ the beaming corrected energy budget of the jet, which is
connected to the isotropic energy measured by an observer aligned to the jet
(on-axis) in the following way:

$E_{\rm iso}=\frac{4\pi}{\Delta\omega}E_{t}$ (5)

where $\Delta\omega\sim 4\pi(1-\cos(\theta_{c}))$. $E_{t}$ is related to
$\epsilon(\theta)$ through the following relation:

$E_{t}=\int_{\ \Omega}\epsilon(\theta)d\Omega.$

$E_{t}$ does not represent the energy content of the jet, but rather the
energy which is converted into radiation through dissipation mechanisms (such
as internal shocks or magnetic reconnection). We extract the value of $E_{t}$
from the following probability distribution:

$P\left(E_{t}/E^{*}\right)\propto\left(E_{t}/E^{*}\right)^{-\lambda_{E}}e^{-E^{*}/E_{t}}$
(6)

which is a power law with a low-energy exponential cut-off. The prompt
emission spectrum is assumed to be angle-independent in the jet comoving frame
and follows a smoothly broken power law, with low- and high- energy photon
indexes of $2/3$ and 2.5, respectively. These are typical values measured for
SGRBs (Nava et al. 2011). In the frame of an on-axis observer, we assume that
the spectral peak energy has a log-normal distribution, i.e.:

$P(\log(E_{p}))\propto\exp\left(-\frac{(\log(E_{p})-\log(\mu_{E}))^{2}}{\sigma_{E}^{2}}\right)$
(7)

where $\mu_{E}$ and $\sigma_{E}$ are the mean value and the standard deviation
of the distribution, respectively. The assumed distribution function
corresponds to that adopted by Ghirlanda et al. (2016). Such distribution in
the observer frame is equivalent to assume a monochromatic distribution of
$E_{p}^{\prime}$ in the jet comoving frame and a log-normal distribution of
$\Gamma_{0}$ (indeed $E_{p}=\Gamma_{0}E_{p}^{\prime}$). For the computation of
the burst duration, we compute the time $t_{90}$ such that the energy emitted
from 0 to $t_{90}$ is equal to the 90$\%$ of the total energy released by the
GRB. The light curve is approximated with a linear rising phase from $t=0$ to
the peak time $t=t_{p}$, followed by a decreasing phase which evolves
according to high latitude emission (HLE). Therefore we assume that a single
pulse dominates the SGRB emission. For $t_{p}$ we consider a log-normal
distribution as well, as measured in the rest frame of the burst:

$P(\log(t_{p}))\propto\exp\left(-\frac{(\log(t_{p})-\log(\mu_{\tau}))^{2}}{\sigma_{\tau}^{2}}\right)$
(8)

The angle-dependent prompt emission light curve is computed adopting the
approach of Ascenzi et al. (2020). Specifically, we work in the approximation
of an infinitesimal duration emission at a fixed radius $R_{0}$, namely the
burst duration in the rest frame is negligible compared to the time needed by
the jet to propagate up to $R_{0}$. The jet moves with a bulk Lorentz factor
$\Gamma_{0}$. As shown by Oganesyan et al. (2020), the presence of a jet
structure like the one adopted here produces an X-ray light curve with a steep
decay followed by a shallower phase, known as plateau. The timescale
$\tau_{p}=R_{0}/2c\Gamma_{0}^{2}$, which corresponds to the typical prompt
emission timescale, determines the duration of the plateau and larger values
of $\tau_{p}$ produce a longer plateau. In order to find a fiducial
combination of $R_{0}$ and $\Gamma_{0}$, we considered all the SGRBs from the
_Swift_ -XRT catalog with a plateau phase. We verified that the end time of
the plateau typically spans in the range $10^{2}-10^{3}$ s. Having fixed
$\Gamma_{0}=500$, we therefore adopted $R_{0}=5\times 10^{14}$ cm in order to
obtain a typical plateau duration around few hundred seconds. With this
approach we are able to predict the prompt emission output, as well as the
contribution at later time given by the jet wings due to HLE effects, which
can be relevant for the detectability of the SGRB afterglow, especially in the
X-rays and for off-axis observers.

###  2.4 Calibration of the prompt emission model

As illustrated in the previous section, our prompt emission model depends on
seven free parameters which determine the observational properties of our SGRB
sample. Here, we show how our prompt emission model is calibrated such that
the BNS population is able to reproduce the number of SGRBs detected by
_Fermi_ -GBM and their statistical properties reported in the _Fermi_ -GBM
catalog333https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html. In
particular, the choice of our fiducial parameters is done comparing our
simulated observables with the distributions of observed peak energy,
duration, peak flux and fluence reported by Ghirlanda et al. (2016). This
calibration sample is obtained by selecting SGRBs in the _Fermi_ catalog with
peak flux $F_{p}>5\text{ ph }\text{cm}^{-2}\text{ s}^{-1}$ to ensure the
completeness of the sample. For the parameter estimation we use Goodman $\&$
Weare’s Affine Invariant Markov chain Monte Carlo (MCMC) Ensemble sampler
444We adopted 16 walkers and 30000 steps. The likelihood function is
$\log{\mathcal{L}}=\log(P_{C}(N_{\rm pre}|N_{\rm
det}))+\sum_{i=1}^{4}\log(P_{\rm KS,i})$, where $P_{C}(N_{\rm pre}|N_{\rm
det})$ is the Poissonian probability that _Fermi_ -GBM detects $N_{pre}$ SGRBs
per year given that the average observed number is $N_{det}$, while $P_{KS,i}$
is, for each observable $i$, the Kolmogorov-Smirnov probability that the
predicted and observed distributions come from the same population. The
adopted likelihood is valid in the approximation that each observable is
independent from the others.

The detection rate of SGRBs $R_{\rm SGRB,det}$ can be related to the BNS rate
$R_{\rm BNS}$ as:

$R_{\rm SGRB,det}=\int[\frac{dN_{\rm SGRB}}{dzdt}|_{\rm det}]dz$ (9)

where

$\frac{dN_{\rm SGRB}}{dzdt}|_{\rm det}=\expectationvalue{\Omega_{\rm
obs}}\times\phi_{\rm det}(z)\times f_{j}\frac{dN_{\rm BNS}}{dzdt}.$ (10)

$\expectationvalue{\Omega_{\rm obs}}$ is the time-averaged sky coverage of the
instrument and $f_{j}$ is the fraction of BNS able to form a jet. The quantity
$\frac{dN_{\rm BNS}}{dzdt}$ is the number of BNS mergers observed at Earth per
unit time and redshift and it is provided by the population synthesis model.
$\phi_{\rm det}(z)$ is the variable that combines the detection efficiency of
the instrument (which takes into account the k-correction) and the beaming of
the SGRB emission. Namely, $\phi_{\rm det}(z)$ represents the probability that
a SGRB at redshift $z$ is detectable by a given instrument at Earth, i.e.:

$\phi_{\rm det}(z)=\frac{N_{\rm SGRB}(F>F_{\rm lim},z)}{N_{SGRB}(z)}$

where $N_{\rm SGRB}(z)=\dfrac{dN_{\rm SGRB}}{dzdt}$, $N_{\rm SGRB}(F>F_{\rm
lim},z)=\dfrac{dN_{\rm SGRB}}{dzdt}(F>F_{\rm lim})$ and $F_{\rm lim}$ is the
limiting flux in the band of the instrument. In turn, $N_{\rm SGRB}(F>F_{\rm
lim},z)$ depends on the luminosity function of SGRBs, their average spectral
properties, as well as on the jet structure.

The inclusion of $f_{j}$ as free parameter of our model makes our estimates of
absolute numbers of detections independent on the overall normalization of the
BNS population model which is still subject to large uncertainties
(Santoliquido et al. 2021). Indeed the likelihood function is written in such
a way that, whatever the BNS merger rate normalisation is, the value of
$f_{j}$ is optimized to reproduce the current average rate of _Fermi_ -GBM
detections. Namely, in eq. 10 the quantity $\frac{dN_{BNS}}{dzdt}$ is uniquely
determined by the BNS population model, $\phi_{det}(z)$ depends on the prompt
emission model and therefore the MCMC converges to that value of $f_{j}$ that
reproduces $R_{\rm SGRB,det}$, which is known.

The MCMC sampling has been performed for two different jet structures _Stru1_
and _Stru2_. For both structures, we obtain the posterior distribution of the
model parameters. The confidence intervals of the parameters are reported in
Tab. 1, while the corner plots of the posterior distributions are reported in
Figs. 14 and 15 for _Stru1_ and _Stru2_ , respectively. The modification of
the off-core slope of the jet structure does not influence the best fit
parameters, except for $f_{j}$, which tends to be smaller for $s=2$. Such
result is justified by the fact that a broader structure implies a larger
detectability of the prompt emission at larger viewing angles, which
corresponds to a larger number of off-axis detections. Therefore, for a given
BNS population and an average SGRB detection rate, a broader structure
requires a smaller fraction of BNS to be able to produce SGRB. The confidence
interval we obtain for $f_{j}$ is not particularly tight and it is compatible
with other works which combine EM observations of SGRBs and BNS rates from GW
observations (Ghirlanda et al. 2019b; Salafia et al. 2022; Sarin et al. 2022;
Beniamini et al. 2019b; Lamb & Kobayashi 2017).

parameter | Stru 1 | Stru 2
---|---|---
$\lambda_{E}$ | $6.7^{+3.6}_{-3.3}$ | $6.4^{+3.7}_{-3.5}$
$\log_{10}{(E_{t}/10^{49}\text{erg/s})}$ | $-0.30^{+0.40}_{-0.34}$ | $-0.23^{+0.48}_{-0.69}$
$\log_{10}{(\mu_{E}/\text{keV})}$ | $3.2^{+0.1}_{-0.1}$ | $3.2^{+0.1}_{-0.1}$
$\sigma_{E}$ | $0.36^{+0.10}_{-0.09}$ | $0.37^{+0.10}_{-0.10}$
$\log_{10}{(\mu_{\tau}/s)}$ | $0.05^{+0.24}_{-0.31}$ | $0.02^{+0.09}_{-0.10}$
$\sigma_{\tau}$ | $0.60^{+0.10}_{-0.11}$ | $0.59^{+0.09}_{-0.10}$
$\log_{10}{f_{j}}$ | $-0.59^{+0.37}_{-0.32}$ | $-0.95^{+0.52}_{-0.41}$
Table 1: Prompt emission model parameters from the MCMC sampling for the two
jet structures analysed in the present work. The interval of confidence from
the posterior distribution of the prompt emission parameters is given as
1$\sigma$.

###  2.5 Forward shock modeling

Figure 1: X-ray light curves of SGRBs with viewing angle
$0^{\circ}<\theta_{v}<15^{\circ}$ predicted by our model. The lines are
produced with a random extraction from the posterior distribution of prompt
parameters. In the computation we assume _Stru1_ and we include both HLE and
FS (configuration _FS-SGRB_). The color of each line indicates the viewing
angle. The X-ray luminosity is computed at 5 keV.

The forward shock (FS) light curves are produced using the Python package
_afterglowpy_ (Ryan et al. 2020). For the jet kinetic structure we assume the
same profile adopted for the structure of the comoving frame emissivity
reported in the prompt emission modeling (see eq. 1). This implies that we are
assuming that the conversion efficiency from kinetic energy to radiated energy
has no angular dependence. The detectability of X-ray emission is evaluated
for both _Stru1_ and _Stru2_. _Afterglowpy_ includes the parameter
$\theta_{w}$ which is the angular extension of the jet wings. For _Stru1_ we
assume $\theta_{w}=15$ deg, while for _Stru2_ we assume $\theta_{w}=30$ deg.
The impact of the choice of $\theta_{w}$ on the detectability of X-ray
emission is discussed later in the text. The model depends also on the micro-
physical parameters $n_{0},p,\epsilon_{e}$ and $\epsilon_{B}$ which are the
ISM density, the slope of the electron energy distribution, the fraction of
energy carried by electrons and the fraction of energy carried by the magnetic
field, respectively. In the following, we perform our simulation using two
setups of parameters:

  1. 1.

$n_{0}=2.5\times 10^{-4}\text{ cm}^{-3}$, $p=2.2$, $\epsilon_{e}=0.1$ and
$\epsilon_{B}=1.3\times 10^{-4}$, which are the best fit values obtained from
the multi-wavelength (X-ray, optical, radio) modeling of the X-ray afterglow
of GRB 170817A (Ghirlanda et al. 2019b). These values are also consistent with
the confidence intervals reported by Wu & MacFadyen (2019). We call this
configuration _FS-GW17_.

  2. 2.

For the ISM density we take the fiducial interval derived by Fong et al.
(2015) $n_{0}\in(3-15)\times 10^{-3}\text{ cm}^{-3}$. The other parameters are
fixed to $p=2.2$, $\epsilon_{e}=0.1$ and $\epsilon_{B}\in[0.01-0.1]$. $n_{0}$
and $\epsilon_{B}$ are uniformly extracted from the confidence intervals
reported above. This configuration is more representative for SGRB population
with respect to _FS-GW17_ , and we call it _FS-SGRB_.

If we call $\eta$ the fraction of kinetic energy which is transformed into
radiation, then $E_{\rm rad}/E_{\rm kin}=(1-\eta)/\eta=\tilde{\eta}$ and we
assume $\tilde{\eta}$ randomly distributed in the interval $[0.01-0.1]$. Fig.
1 shows a collection of X-ray light curves produced with our model at
different viewing angles, including both HLE and FS (configuration _FS-SGRB_).
The curves are obtained with a random extraction from the probability
distributions of the prompt parameters described in sec. 2.3 and the parameter
of each distribution is extracted from the Monte Carlo posterior described in
sec. 2.4.

##  3 Results

###  3.1 Joint detection of GWs and the prompt emission

Figure 2: Histogram of the joint ET+_Fermi_ -GBM detections (red), together with the distribution of injected BNS (blue), the mergers detected by ET (orange), the mergers detected by _Fermi_ -GBM (green). The histogram is normalised to $N_{inj}=10^{5}$, which is the number of injected BNS mergers in the angle range $0^{\circ}<\theta_{v}<15^{\circ}$ and in the redshift range $0<z<4$. For visualization purposes, the fraction of BNS producing a jet, $f_{j}$, has been assumed to be one. Figure 3: Same as Fig. 2, but considering _Fermi_ -GBM+(ET+CE). INSTRUMENT | band | $F_{\rm lim}$ | FOV/$4\pi$ | loc. acc. | Joint ET | $N_{JD}/N_{\gamma}$ | Joint (ET+CE) | $N_{JD}/N_{\gamma}$
---|---|---|---|---|---|---|---|---
MeV | erg cm-2 s-1 | +$\gamma$-ray | +$\gamma$-ray
_Fermi_ -GBM | 0.01 - 25 | 0.5$(^{*})$ | 0.75 | 5 deg $(^{a})$ | $33^{+14}_{-11}$ | $68_{-18}^{+13}\%$ | $47_{-14}^{+14}$ | $95_{-7}^{+5}\%$
_Swift_ -BAT | 0.015 - 0.15 | $2\times 10^{-8}$ | 0.11 | 1-3 arcmin | $10^{+3}_{-3}$ | $62_{-14}^{+11}\%$ | $13_{-4}^{+5}$ | $94_{-7}^{+6}\%$
GECAM | 0.006 - 5 | $2\times 10^{-8}$ | 1.0 | 1 deg | $121_{-48}^{+84}$ | $57_{-10}^{+8}\%$ | $205_{-72}^{+145}$ | $92_{-5}^{+4}\%$
SVOM-ECLAIRs | 0.004 - 0.250 | 1.792(*) | 0.16 |  $<10$ arcmin | $3_{-1}^{+1}$ | $69_{-9}^{+10}\%$ | $4_{-1}^{+1}$ | $95_{-4}^{+5}\%$
SVOM-GRM | 0.03 - 5 | 0.23(*) | 0.16 |  $\sim 5$ deg | $9_{-3}^{+4}$ | $59_{-6}^{+6}\%$ | $14_{-4}^{+6}$ | $92_{-3}^{+3}\%$
THESEUS-XGIS | 0.002 - 10 | $3\times 10^{-8}$ | 0.16 |  $<15$ arcmin | $10_{-4}^{+5}$ | $63_{-13}^{+13}\%$ | $15_{-4}^{+6}$ | $94_{-7}^{+6}\%$
HERMES | 0.05 - 0.3 | 0.2$(^{*})$ | 1.0 | 1 deg | $84_{-30}^{+42}$ | $61_{-11}^{+10}\%$ | $139_{-36}^{+54}$ | $94_{-6}^{+6}\%$
TAP-GTM | 0.01 - 1 | 1$(^{*})$ | 1.0 | 20 deg | $60_{-24}^{+24}$ | $67_{-14}^{+13}\%$ | $84_{-24}^{+30}$ | $95_{-6}^{+5}\%$
INSTRUMENT | band | $F_{\rm lim}$ | FOV/$4\pi$ | loc. acc. | Joint ET | $N_{JD}/N_{\gamma}$ | Joint (ET+CE) | $N_{JD}/N_{\gamma}$
---|---|---|---|---|---|---|---|---
MeV | erg cm-2 s-1 | +$\gamma$-ray | +$\gamma$-ray
_Fermi_ -GBM | 0.01 - 25 | 0.5$(^{*})$ | 0.75 | 5 deg $(^{a})$ | $36^{+14}_{-14}$ | $70_{-14}^{+16}\%$ | $47_{-14}^{+18}$ | $95_{-7}^{+5}\%$
_Swift_ -BAT | 0.015 - 0.15 | $2\times 10^{-8}$ | 0.11 | 1-3 arcmin | $9^{+6}_{-3}$ | $64_{-13}^{+11}\%$ | $14_{-4}^{+7}$ | $94_{-7}^{+6}\%$
GECAM | 0.006 - 5 | $2\times 10^{-8}$ | 1.0 | 1 deg | $163_{-60}^{+175}$ | $60_{-8}^{+7}\%$ | $253_{-96}^{+271}$ | $92_{-4}^{+3}\%$
SVOM-ECLAIRs | 0.004 - 0.250 | 1.792(*) | 0.16 |  $<10$ arcmin | $3_{-1}^{+1}$ | $71_{-10}^{+15}\%$ | $4_{-1}^{+1}$ | $95_{-4}^{+5}\%$
SVOM-GRM | 0.03 - 5 | 0.23(*) | 0.16 |  $\sim 5$ deg | $11_{-4}^{+6}$ | $60_{-6}^{+13}\%$ | $15_{-5}^{+10}$ | $92_{-3}^{+3}\%$
THESEUS-XGIS | 0.002 - 10 | $3\times 10^{-8}$ | 0.16 |  $<15$ arcmin | $11_{-4}^{+7}$ | $66_{-16}^{+12}\%$ | $16_{-5}^{+7}$ | $94_{-7}^{+6}\%$
HERMES | 0.05 - 0.3 | 0.2$(^{*})$ | 1.0 | 1 deg | $96_{-31}^{+60}$ | $64_{-12}^{+12}\%$ | $151_{-48}^{+66}$ | $94_{-6}^{+4}\%$
TAP-GTM | 0.01 - 1 | 1$(^{*})$ | 1.0 | 20 deg | $66_{-24}^{+24}$ | $69_{-14}^{+15}\%$ | $90_{-24}^{+30}$ | $95_{-7}^{+5}\%$
Table 3: As in Tab. LABEL:tab_joint, assuming the jet structure _Stru2_.
Figure 4: Angle dependency of the joint GW+$\gamma$-ray detections for three
redshift bins, considering ET+_Fermi_ -GBM. The bands are obtained with random
realizations extracted from the MCMC posterior samples. For each redshift bin,
the y-axis gives the number of joint detections normalised to $5\times 10^{4}$
BNS injections with viewing angle $0^{\circ}<\theta_{v}<15^{\circ}$. We assume
that all BNS produce a jet ($f_{j}=1$).

For the detection of the prompt emission from our SGRBs population, we
consider the following $\gamma$-ray instruments: the Gamma-ray Burst Monitor
on board of _Fermi_ (_Fermi_ -GBM, Meegan et al. 2009), the _Swift_ -Burst
Alert Telescope (_Swift_ -BAT, Barthelmy et al. 2005), the Gravitational wave
high-energy Electromagnetic Counterpart All-sky Monitor (GECAM, Li et al.
2020), the ECLAIRs telescope and the Gamma-Ray burst Monitor (GRM) on board of
the Space-based multi-band astronomical Variable Objects Monitor (SVOM,
Cordier et al. 2015; Götz & SVOM Collaboration 2012), the X/Gamma-ray Imaging
Spectrometer (XGIS) on board of the Transient High-Energy Sky and Early
Universe Surveyor (THESEUS, Amati et al. 2018, 2021; Ciolfi et al. 2021;
Rosati et al. 2021), the Gamma-ray Transient Monitor (GTM) on board of the
Transient Astrophysical Probe (TAP, Camp et al. 2019), and the High Energy
Rapid Modular Ensemble of Satellites (HERMES, Fiore et al. 2020). The first
three are currently operating missions; _Swift_ and _Fermi_ have been
observing for more than 10 years, and GECAM was launched in 2020. SVOM is
expected to be launched in 2023. Nominally they are not expected to be
operative in mid ’30s and ’40s when ET and CE will start observations, but we
take them as reference instruments. THESEUS, TAP, and HERMES are mission
concepts. The HERMES-Pathfinder project consisting in the deployment of six
cube-satellites is expected to be operative in the next few years (Fiore et
al. 2021). In this work we consider the final HERMES as a full constellation
of cube-satellites. In order to establish whether the prompt emission is
detected, we compute the peak flux during 1 second exposure and we compare
this value with the detection threshold of the instrument. The viewing angle
of each injection is assumed to be exactly the same used for the evaluation of
the GW signal, and thus extracted from the randomly oriented BNS population
(see Sect. 2.1).
In order to estimate the number of joint detections, we simulate the high-
energy emission (keV-MeV) according to the model described in Sect. 2.3. The
results relative to one year of joint GW+$\gamma$-ray detections are reported
in Tabs. LABEL:tab_joint and 3 for _Stru1_ and _Stru2_ , respectively, where
we considered ET and ET+CE as GW networks. The uncertainty intervals are
computed simulating 1000 years of observations and for each year the prompt
emission parameters are randomly extracted from the Monte Carlo posterior
distribution. In the computation of the absolute numbers, we took into account
the FOV of each instrument, assuming that the probability of detection is
FOV$/4\pi$, which implicitly assumes an optimistic duty cycle of 100$\%$. In
the case of _Fermi_ -GBM, _Swift_ -BAT and SVOM, we adopt a more realistic
value of sky-averaged FOV, which takes into account Sun and Moon occultations
(Burns et al. 2016). In the case of THESEUS-XGIS the FOV we assumed ($\sim 2$
sr) is relative to the energy band 2-150 keV. Since the FOV in the energy band
above 150 keV is 2$\pi$ sr, the reported number of joint detections are a
conservative estimate. Our results show that, already with ET alone, $\sim
60-70\%$ of all the SGRBs will have a detectable GW counterpart, and this
fraction approaches 100$\%$ if we consider ET operating with CE. The less
steep structure profile of _Stru2_ increases the number of joint detections
for the majority of the satellites. Depending on the properties of each
satellite, Tabs. LABEL:tab_joint and 3 show instruments giving a few tens of
detections per year and other hundreds of detections per year. While
instruments like GECAM are optimal for statistical studies by giving a large
number of detections, instruments with a smaller number of detections but able
to localize the source, such as THESEUS-XGIS, are crucial for science cases
requiring the identification of the host galaxy, the knowledge of the source
redshift, and the complete multi-wavelength characterization of the source
emission. Furthermore, the sensitivity of the instruments considered for the
detection of the prompt emission maximizes in different energy bands. While
instruments like GECAM are appropriate for detecting GRBs with harder
spectrum, GRBs less energetic (intrinsically or because viewed off-axis) will
peak at lower energies (soft/hard X-rays) and for them instruments such as
XGIS are more suitable. The presence of multiple instruments will be extremely
valuable to cover the entire energy range typical of the prompt emission of
short GRBs.

Figs. 2 and 3 show the distribution in redshift of the joint detections, in
the specific case of _Fermi_ -GBM in synergy with ET and ET+CE, respectively.
These figures are produced injecting $10^{5}$ BNS mergers (extracted from the
BNS population, and thus following the astrophysical merger rate evolution
with z), with the viewing angle uniformly distributed in the range
$0^{\circ}-15^{\circ}$. Even if we distribute the injections over a wider
range of angles, the number of joint detections (given as fraction of
$\gamma$-ray detections having an associated GW counterpart) does not change,
since, while there are GW sources detectable at $\theta_{v}>15^{\circ}$, there
are no $\gamma$-ray detections for $\theta_{v}>15^{\circ}$ (see Fig 4 and Fig
11). The $\gamma$-ray detections are computed considering the best fit
configuration of parameters derived from MCMC posterior distribution, except
for $f_{j}$, which has been assumed to be one, for better visibility. The
value of $f_{j}$ shifts the green and red histograms vertically but does not
change their relative ratio. In the case of ET alone, the GW detector is so
sensitive that up to $z\simeq 0.8-1.0$ the probability that a SGRB has a
detectable GW counterpart is close to 100$\%$. Adding CE to the network, the
GW detection efficiency remains close to 100$\%$ for redshifts above the BNS
merger peak. In this section we do not include the results for ET+2CE, since
the addition of another CE would not further increase the joint detection
efficiency in the redshift range accessible to $\gamma$-ray instruments for
SGRBs. The $\gamma$-ray missions that are more suited to maximize the joint
detection rates are those with large FOV and best sensitivity around MeV
energies. However, another parameter to take into account is the localization
accuracy by the $\gamma$-ray detectors (given in Tables LABEL:tab_joint and
3), which defines what are the instruments that are able to drive the follow-
up observations by ground-based telescopes, crucial for obtaining the source
redshift and to completely characterize the multi-wavelength emission of the
source.

Fig. 4 shows the distribution of the ET+$\gamma$-ray joint detections as a
function of the viewing angle, assuming _Stru1_ , for three redshift bins:
$z<0.5,0.5<z<1.5,z>1.5$. Fig. 11 in Appendix shows the same, but assuming
_Stru2_. Again, as example, we consider _Fermi_ -GBM as $\gamma$-ray detector,
but we obtain consistent results for the other $\gamma$-ray satellites
considered in the present paper. On the y axis we report the ratio $N_{\rm
JD}/N_{\rm inj}$ between the joint detections over the total injected BNS
(assuming $f_{j}=1$) per redshift bin and viewing angle bin. For the viewing
angle we consider a linearly spaced grid, where the width of the single bin is
$\theta_{c}/3$. The uncertainty bands are obtained with 15 random extractions
of the prompt parameters and each realization considers $5\times 10^{4}$ BNS
injections with viewing angle $0^{\circ}<\theta_{v}<15^{\circ}$. As before, a
different assumed value of $f_{j}$ just shifts the bands vertically. The plot
shows a mild decrease of the ratio $N_{\rm JD}/N_{\rm inj}$ for very small
viewing angles ($\theta_{v}\lesssim\theta_{c}/3$). Such effect is due to the
fact that SGRB viewed on-axis have very high values of peak energy, meaning
that the bulk of the flux is above the _Fermi_ -GBM band. Therefore,
considering two identical SGRBs at the same redshift, the one viewed at
$\theta_{v}\sim\theta_{c}$ appears slightly brighter than the one viewed at
$\theta_{v}\sim 0$. Moreover, the number of sources per unit angle scales as
$\sin(\theta_{v})$, therefore the small decrease of $N_{\rm JD}/N_{\rm inj}$
around $\theta_{v}\sim 0$ is related to the paucity of sources contained in
the first angle bin.

####  3.1.1 Joint detection of GWs and $\gamma$-rays from cocoon shock break-
out

instrument |  $D_{\rm L,max}$ (Mpc) | $N_{\rm det}$
---|---|---
_Fermi_ -GBM | 76.4 | $<1$
_Swift_ -BAT | 123 | 3
GECAM | 210 | 12
THESEUS-XGIS | 114 | 2
HERMES | 121 | 3
TAP-GTM | 80 | 1
Table 4: Maximum luminosity distance $D_{\rm L,max}$ and number of joint
GW+$\gamma$-ray detections of a SBO with the same properties of the
$\gamma$-ray flash observed in coincidence with GW 170817.

In the derivation of the joint GW+$\gamma$-ray detections, we have assumed
that all the SGRBs share a common jet structure and that the $\gamma$-ray
emission is given by the dissipation of the internal energy of the jet.
However, for GRBs observed at large viewing angles, i.e.
$\theta_{v}\gg\theta_{c}$, the $\gamma$-ray emission from the shock break-out
(SBO) of the jet should be taken into consideration. Indeed, even if the
formation and successful break-out of a relativistic jet through the post-
merger ejecta is not guaranteed, the formation of a cocoon at very wide angles
is commonly expected in the BNS mergers (Ramirez-Ruiz et al. 2002; Nakar &
Piran 2017; Gottlieb et al. 2018a). The shocks driven by the mildly
relativistic expansion of the cocoon can produce $\gamma$-ray emission which
is potentially detectable at least in the local Universe. Moreover, several
studies found a remarkable agreement among the properties of the gamma-ray
emission of GW 170817 and the shock-breakout model (Kasliwal et al. 2017;
Gottlieb et al. 2018b; Nakar et al. 2018; Bromberg et al. 2018; Pozanenko et
al. 2018). The $\gamma$-ray emission detected is not emitted by the wings of a
structured jet, but rather by the shock breakout of the cocoon produced by the
interaction of the jet with the NS merger ejecta (Gottlieb et al. 2018b;
Bromberg et al. 2018). In order to investigate this scenario, as a reference,
we take the spectrum (cut-off power law with photon index $\alpha=0.62\pm
0.40$ and cut-off energy $E_{c}=185\pm 62$ keV) and the _Fermi_ -GBM peak flux
($(3.1\pm 0.7)\times 10^{-7}$ erg cm-2 s-1) of the $\gamma$-ray flash
associated to GW 170817 (Goldstein et al. 2017) and we compute the maximum
distance $D_{\rm L,max}$ at which this emission can be detected by the
$\gamma$-ray instruments considered in the present work. In a first order
approximation, the SBO emission can be considered isotropic, therefore we do
not include any angle dependency in our treatment. In order to evaluate the
number of $\gamma$-ray detections associated to SBO, we compute the number of
BNS at a luminosity distance $D_{L}<D_{\rm L,max}$. In our derivation, we
assume that each BNS produces a SBO. Since the detection efficiency of any 3G
GW detector is $\sim 100\%$ for distances $D_{L}<D_{\rm L,max}$, the number of
$\gamma$-ray detections per year that we report corresponds also to the number
of joint GW+$\gamma$-ray detections associated to SBO from BNS mergers. The
results are reported in Tab. 4.

(a) ET, all angles

(b) ET, $\theta_{v}<15^{\circ}$

(c) ET+CE, all angles

(d) ET+CE, $\theta_{v}<15^{\circ}$

(e) ET+2CE, all angles

(f) ET+2CE, $\theta_{v}<15^{\circ}$

Figure 5: Redshift distribution of the sky-localization uncertainty (given as 90$\%$ credible region) for three detector configurations: ET, ET+CE and ET+2CE. The absolute numbers are relative to one year of observation and assuming a duty cycle of 0.85 as described in the text. Plots on the right show BNS systems with a viewing angle $0<\theta_{v}<15^{\circ}$. | ET | ET+CE | ET+2CE
---|---|---|---
$N_{\rm det}$ | 12970 | 23600 | 25668
$N_{\text{det}}(\Delta\Omega<1\text{ deg}^{2})$ | 0 | 20 | 636
$N_{\text{det}}(\Delta\Omega<10\text{ deg}^{2})$ | 2 | 845 | 13673
$N_{\text{det}}(\Delta\Omega<100\text{ deg}^{2})$ | 69 | 17049 | 23935
$N_{\text{det}}(\Delta\Omega<1000\text{ deg}^{2})$ | 526 | 21564 | 25367
Table 5: Number of GW detections per year with sky localization better than 1, 10, 100 and 1000 $\text{deg}^{2}$. The reported numbers are relative to BNS mergers with $\theta_{v}<15^{\circ}$. The numbers of GW detections per year are obtained assuming a duty cycle of 0.85 as described in the text. | ET | ET+CE | ET+2CE
---|---|---|---
$N_{\rm det}$ | 143970 | 458801 | 592565
$N_{\text{det}}(\Delta\Omega<1\text{ deg}^{2})$ | 2 | 184 | 5009
$N_{\text{det}}(\Delta\Omega<10\text{ deg}^{2})$ | 10 | 6797 | 154167
$N_{\text{det}}(\Delta\Omega<100\text{ deg}^{2})$ | 370 | 192468 | 493819
$N_{\text{det}}(\Delta\Omega<1000\text{ deg}^{2})$ | 2791 | 428484 | 585317
Table 6: Same as Tab. 5, but without any selection on $\theta_{v}$

###  3.2 Joint detection of GWs and X-ray emission

In this section, we evaluate the detectability of the afterglow emission with
future X-ray telescopes. Specifically, we show the expected rate of BNS
mergers which will have both a detectable GW signal and an X-ray emission
associated to the afterglow phase of the SGRB. For the X-ray emission, we
consider two components: 1) the standard forward shock (FS) emission, 2) the
HLE associated to the last photons emitted during the prompt phase. Under the
assumption of a structured jet, the HLE produces an X-ray flux which can be
comparable or even dominant with respect to the FS emission (Oganesyan et al.
2020; Panaitescu 2020). We evaluate the joint X-ray and GW detections
considering satellites in survey mode as well as satellites pointing to the GW
source. For the pointing strategy, we use the localization capabilities of ET
alone and in the network of 3G observatories. Since the number of sources
suitable for pointing observations result to be high (especially for network
of 3G detectors), we also evaluate other source parameters estimated from the
GW signals, such as viewing angle and distance, which can be used to down-
select sources to be observed.

####  3.2.1 GW sky localization

Fig. 5 shows the sky localization of ET, ET+CE and ET+2CE. The figures show
the error on sky position, expressed in deg2, as a function of redshift for
one year of BNS injections. The injections are extracted from the BNS
population described in Sect. 2.1, thus following the same redshift evolution
of the merger rate. We show the sky localization both for sources with
$0<\theta_{v}<15^{\circ}$ and for sources with no selection on $\theta_{v}$.
We chose $\theta_{v}$ maximum equal to $15^{\circ}$ as it is consistent with
the largest viewing angle up to which the X-ray emission from a _Stru1_ jet is
observable by the analysed WFX-ray satellites (see Fig. 8). For the
combination ET+CE (ET+2CE), a considerable fraction of sources detected at
$z<1$ ($z<3$) has a sky localization smaller than $10$ deg2, which is small
enough to enable prompt and efficient multi-wavelength search for EM
counterparts.

In Tab. 5 and 6 we summarize the sky localization capabilities of ET, ET+CE
and ET+2CE for systems observed with $\theta_{v}<15^{\circ}$ and for systems
observed at all angles, respectively. The tables list the number of detections
per year with sky localization uncertainty $\Delta\Omega<1,10,100,1000$ deg2.
Fig. 10 in Appendix shows the redshift distribution of $\Delta\Omega$ as
scatter plot with a color code indicating the SNR for ET, ET+CE and ET+2CE,
considering cases with $\theta_{v}<15^{\circ}$ and cases with no selection on
$\theta_{v}$.

Fig. 12 in Appendix shows the cumulative distribution of the SNR for BNS
detected with ET, ET+CE and ET+2CE with a sky uncertainty $\Delta\Omega<100$
deg2, for $\theta_{v}<15^{\circ}$ (panel a) and without selection on
$\theta_{v}$ (panel b). The SNR distribution shows that, even adopting a
larger SNR threshold, e.g. SNR$>12$, the number of sources with $8<\rm SNR<12$
is negligible and therefore not relevant for the evaluation of the joint
GW+X-ray detection rate at least for well localized sources.

####  3.2.2 Viewing angle and luminosity distance from GWs

(a)

(b)

(c)

Figure 6: Distribution of $\Delta\theta_{v}$ vs $\theta_{v}$ relative to one
year of observation, in the case of ET (a), ET+CE (b) and ET+2CE (c). The
color bar indicates the redshift. We report only BNS mergers detected at $z<1$
and with a $\Delta\theta_{v}<40^{\circ}$. A duty cycle of 0.85 has been
assumed for the GW detectors as described in the text.

(a)

(b)

(c)

Figure 7: Distribution of the relative error on the luminosity distance
$D_{L}$ as a function of redshift relative to one year of observation, in the
case of ET (a), ET+CE (b) and ET+2CE (c). The color bar indicates the viewing
angle. Only cases with $\Delta D_{L}/D_{L}<1$ and at $z<3$ are shown. A duty
cycle of 0.85 has been assumed for the GW detectors as described in the text.

Here, we explore how the GW observations can give constraints about the
viewing angle with respect to the axis perpendicular to the BNS orbital plane
(assumed to be parallel to the jet axis) and the luminosity distance of the
system. Fig. 6 shows how the detected sources are distributed in the plane
$\Delta\theta_{v}-\theta_{v}$, considering ET, ET+CE and ET+2CE.
$\Delta\theta_{v}$ is the 1$\sigma$ error on the viewing angle. Only sources
at $z<1$ and with $\Delta\theta_{v}<40$ deg are shown. For all the networks, a
considerable fraction of sources have $\Delta\theta_{v}/\theta_{v}<0.1$.
Moreover, since the gradient of the GW amplitude is maximum at $\theta_{v}=90$
deg, the $\theta_{v}$ parameter is better constrained when the source is seen
edge-on. This result is particularly important to prioritize the BNS events to
be followed-up. Indeed, in the era of 3G GW detectors we expect hundreds of
BNS detections per day. Due to the limited observational time at each
observatory, with a pointing strategy only a limited fraction of GW-detected
signals can be followed up. For example, aiming at detecting the high-energy
emission, one can exclude all the sources edge-on for which the viewing angle
estimate is more precise from the GW parameter estimation.

Fig. 7 shows the distribution in redshift of $\Delta D_{L}/D_{L}$, where
$\Delta D_{L}$ is 1$\sigma$ error on the luminosity distance $D_{L}$. We
consider ET, ET+CE and ET+2CE. Only sources with $\Delta D_{L}/D_{L}<1$ are
shown. For ET operating alone the relative error can be below $10\%$ only for
sources with $z\lesssim 0.2$, while for ET+CE and ET+2CE the same is true for
$z\lesssim 1.5$ and $z\lesssim 3$, respectively. The plots also show that the
constraint on distance is tighter for sources seen edge on. This is due to the
strong degeneracy between luminosity distance and viewing angle and therefore
the error on the first is significantly reduced when the second is well
constrained.

####  3.2.3 Wide-FOV X-ray telescopes: survey and pointing observations

For the detection of the afterglow emission from our SGRBs population, we
consider the following X-ray instruments: the Einstein Probe (EP, Yuan et al.
2018) which is scheduled for launch by the end of 2022, the ECLAIRs instrument
on board of SVOM (Cordier et al. 2015) which is scheduled for launch in 2023,
and three mission concepts, THESEUS (Amati et al. 2018, 2021), the Wide Field
Imager on board of TAP (Camp et al. 2019), and the Gamow Explorer (White et
al. 2021). The launch date of this last is expected to be in 2028-2032.
If the instrument operates in survey mode, the probability of detecting the
source (if the flux is above the sensitivity limit) is given by
$\sim$FOV/$4\pi$. Instead, if the instrument operates only in pointing mode,
the light curve is monitored only after a time $t_{\rm resp}$ from the trigger
time, which is the sum of the time to respond to a trigger from ground and the
time to re-point the instrument. In order to evaluate how each instrument will
sample the afterglow temporal evolution, we use the sensitivity curve (The
sensitivity curve of EP is taken
here555https://sci.esa.int/documents/34375/36249/1567258027270-ESA-CAS-
workshop1_20140225_7__Einstein_Probe-exploring_the_dynamic_X-
ray_Universe__Yuan.pdf, while for all the other X-ray missions, we use the
sensitivity curves reported in the references cited at the beginning of this
section), which relates the minimum detectable flux as a function of the
exposure time. The temporal sampling is computed using an iterative approach.
Starting from the beginning of the light curve, we determine the length
$\Delta T$ of each bin at time $t_{i}$ such that

$\frac{1}{\Delta T}\int_{t_{i}}^{t_{i}+\Delta T}F(t)dt>F_{\rm lim}(\Delta T)$
(11)

where $F_{\rm lim}(\Delta T)$ is derived from the sensitivity curve. The
afterglow light curve is classified as undetectable if, even increasing
$\Delta T$, the eq. 11 is never satisfied. For _Gamow_ , THESEUS and EP we
consider sensitivity curves including a median Galactic absorption
$N_{H}=5\times 10^{20}\rm cm^{-2}$. For TAP the published sensitivity curve
does not include this information.

For all the results reported in this section, we simulate 25 realizations of
one year of BNS mergers, each with parameters extracted randomly from the MCMC
posterior distributions. For each realization we consider a number of
injections $N_{\rm inj}=\min(N_{\rm req},10^{4})$, where $N_{\rm req}$ is the
number of BNS in one year that satisfy the selection criteria (e.g. about SNR,
$\theta_{v}$ and $\Delta\Omega$). The choice $N_{\rm inj}=\min(N_{\rm
req},10^{4})$ is made in order to have enough statistics and such that the
full parameter space is correctly covered. Then the number of detections
correspondent to one year is obtained re-scaling by the factor $N_{\rm
req}/N_{\rm inj}$. The parameters of the prompt model influence the afterglow
light curve because both the FS and HLE brightness depend on the $E_{\rm iso}$
which is predicted by our model. First we evaluate the expected rate of joint
GW+X-ray detections, considering the telescopes operating in survey mode. The
results are reported in Tab. 8. Due to the good sensitivity down to 2 keV and
the large FOV, THESEUS-XGIS can contribute substantially to the total amount
of X-ray detections. Thanks to the larger FOV (1.1 sr) and the better
sensitivity, EP gives a larger number of detections compared to THESEUS-SXI
and TAP.

The probability of detection can be increased considering the possibility to
point GW sources localized with enough precision through the GW signal. In
such case, wide field X-ray (WFX-ray) telescopes can point to the sky error
box and start the search of the X-ray counterpart. We explore this scenario
simulating one year of BNS mergers and selecting only those that are detected
with an error on sky position better than 100 deg2. The typical FOV of WFX-ray
telescopes is larger than 1 sr, so an error region of 100 deg2 can be well
contained inside the FOV of the instrument. A sky region of 100 deg2 is around
1/10 of a typical FOV of a WFX-ray telescope. If, instead, we select sources
localized better than 1000 $\text{deg}^{2}$ and considering that the bulk of
sources detectable in the X-rays are at $z\lesssim 1$, we expect, with respect
to the cases with $\Delta\Omega<100$ deg${}^{2}:$

  1. 1.

a factor $\sim$ 10 more joint GW+X-ray detections with ET

  2. 2.

no substantial variation for ET+CE, since sources with $100\text{
deg}^{2}<\Delta\Omega<1000\text{ deg}^{2}$ are located at $z\gtrsim 1$

  3. 3.

no substantial variation for ET+2CE, since $N(\Delta\Omega<100$
deg${}^{2})\sim N(\Delta\Omega<1000$ deg2) for $z\lesssim 1$.

For the selected sub-sample of well localized sources, we predict the X-ray
emission and we check if it is detectable with the telescopes listed before.
The numbers of THESEUS-SXI and Gamow are identical since we consider for them
the same sensitivity, and the slightly different FOV does not influence
pointing observations. In Tab. 9 we show the number of expected joint GW+X-ray
detections for one year of observation with ET, ET+CE and ET+2CE. We consider
the configuration _FS-SGRB_ for the FS parameters and we assume that the X-ray
telescopes are able to point towards the sky position provided by GW detectors
within 100 s from the BNS merger. A response time of 100 s is a short amount
of time to communicate the trigger from the ground to the satellites, but the
possibility of accessing lower frequencies by the next generation GW detectors
will make it possible pre-merger alerts. We evaluated that, in the case of ET,
a fraction of $\sim$30% and $\sim$60% of sources localized better than 100
deg2 at the merger are above the detection threshold (SNR=8) and with a sky-
localization better than 1000 deg2, which is well within the FOV of the
satellites analyzed here, 20 and 10 minutes before the merger, respectively
(see tab. 14). For ET+CE (ET+2CE) this fraction is $\sim$5% (6%) 10 minutes
before the merger, leaving the absolute number of triggers to be followed up
of order of several hundreds. The typical exposure time for detection in
X-rays is $t_{\rm exp}=66_{-42}^{+249}$ s for SXI and _Gamow_ , $t_{\rm
exp}=96_{-29}^{+219}$ s for TAP, $t_{\rm exp}=239_{-173}^{+171}$ s for EP,
where the uncertainties are reported at $1\sigma$ level of confidence and they
are computed considering a random sampling of the MCMC posterior distribution.

| FOV (sr) | loc. accuracy (arcmin)
---|---|---
Einstein Probe | 1.1 | 5
Gamow | 0.4 | 1-2
THESEUS-SXI | 0.5 | 1-2
TAP-WFI | 0.4 | 1
Table 7: Field of view and localization accuracy of the WFX-ray telescopes considered in this work. | ET | ET+2CE
---|---|---
SVOM-ECLAIRs | $4\pm 2$ | $5\pm 2$
Einstein Probe | $50^{+15}_{-16}$ | $64^{+12}_{-20}$
Gamow | $9^{+2}_{-2}$ | $10^{+3}_{-3}$
THESEUS-SXI | $11^{+3}_{-3}$ | $13^{+4}_{-3}$
THESEUS-(SXI+XGIS) | $23^{+6}_{-5}$ | $27^{+7}_{-5}$
TAP-WFI | $16^{+3}_{-4}$ | $17^{+6}_{-3}$
Table 8: Expected number of joint GW+X-ray detections in one year of observations, for different X-ray instruments operating in survey mode. The assumed structure is _Stru1_ and the FS configuration is _FS-SGRB_. | ET | ET+CE | ET+2CE
---|---|---|---
Einstein Probe | $9^{+5}_{-3}$ | $294^{+80}_{-59}$ | $359^{+168}_{-110}$
THESEUS-SXI/ | $7^{+5}_{-3}$ | $95^{+43}_{-14}$ | $122^{+41}_{-23}$
Gamow
TAP-WFI | $8^{+5}_{-3}$ | $182^{+43}_{-31}$ | $225^{+76}_{-72}$
Table 9: Expected number of joint GW+X-ray detections in one year of observations, for different X-ray instruments operating in pointing mode. The reported numbers correspond to BNS mergers with sky-localization uncertainty $\Delta\Omega<100$ deg2 and with a detectable X-ray emission, assuming a $t_{\rm resp}=100$ s. The assumed structure is _Stru1_ and the FS configuration is _FS-SGRB_. | 100 s | 1 hr | 4 hr
---|---|---|---
Einstein Probe | $359^{+168}_{-110}$ | $48^{+24}_{-15}$ | $17_{-10}^{+15}$
THESEUS-SXI/ | $122^{+41}_{-23}$ | $12\pm 7$ | $<9$
Gamow
TAP-WFI | $225^{+76}_{-72}$ | $50^{+20}_{-10}$ | $17_{-5}^{+10}$
Table 10: As in Tab. 9, but considering different delay times between the merger and the beginning of X-ray observation. In this table, we consider ET+2CE as GW network. | HLE | FS-SGRB | HLE
---|---|---|---
| +FS-SGRB |  | +FS-GW17
Einstein Probe | $359^{+168}_{-110}$ | $344^{+106}_{-95}$ | $67^{+14}_{-17}$
THESEUS-SXI/ | $122^{+41}_{-23}$ | $98^{+34}_{-31}$ | $22^{+7}_{-6}$
Gamow
TAP-WFI | $225^{+76}_{-72}$ | $174^{+94}_{-27}$ | $33^{+12}_{-8}$
Table 11: As in Tab. 9, but considering different assumptions for the computation of the X-ray light curve. In this table we consider ET+2CE as GW network and $t_{\rm resp}=100$ s. | _Stru1_ | _Stru2_
---|---|---
Einstein Probe | $359^{+168}_{-110}$ | $383^{+131}_{-112}$
THESEUS-SXI/ | $122^{+41}_{-23}$ | $128^{+73}_{-61}$
Gamow
TAP-WFI | $225^{+76}_{-72}$ | $219^{+128}_{-73}$
Table 12: Comparison between _Stru1_ and _Stru2_ for the prediction of joint
GW+X-ray detections in pointing mode, considering ET+2CE. Both HLE and FS are
included. The FS configuration is _FS-SGRB_ and we assume $t_{\rm resp}=100$
s.

(a)

(b)

Figure 8: Panel (a): Distribution of the viewing angle as a function of
redshift for a sample of 1000 joint GW+X-ray detections, considering ET+2CE
and SXI and under the assumption _Stru1_. Only BNS detections with GW sky
localization $\Delta\Omega<100$ deg2 are selected. The assumed jet core angle
is $\theta_{c}=3.4$ deg. The color bar indicates the GW sky localization
uncertainty of each detection. Panel (b): distribution in redshift of the same
sample, where we distinguish between detections with $\theta_{v}<5^{\circ}$
and $\theta_{v}>5^{\circ}$. The histogram is normalized to the number of total
joint detections with $\Delta\Omega<100$ deg2.

(a)

(b)

Figure 9: Same as in Fig. 8, but for _Stru2_.

Considering ET alone and FS configuration (_FS-SGRB_), Tab. 8 and 9 show that
the survey mode is more promising than the pointing mode in terms of number of
GW+X-ray detections per year. For ET+CE and ET+2CE, due to the larger number
of well localized sources, the number of joint detections is considerably
larger than the survey mode case. Hence, the exploitation of sky localization
provided by GW detectors can enhance substantially the probability of
identifying the EM counterpart of BNS mergers. We point out that the reported
numbers for pointing mode do not take into account the duty factor of the
instruments, which is assumed to be 100$\%$. Moreover, each X-ray mission,
depending on the scientific objectives, can dedicate only a limited amount of
time for the follow-up of GW triggers. Therefore, if we call $\epsilon_{\rm
GW}$ the fraction of observational time dedicated to GW follow-up and
$\epsilon_{\rm DC}$ the duty cycle, a more realistic estimate of the number of
joint detections would be $N_{\rm JD,real}=\epsilon_{\rm GW}\epsilon_{\rm
DC}N_{\rm JD}$, where $N_{\rm JD}$ are the numbers reported in this section
about the pointing mode. Regarding the survey mode, instead we just have
$N_{\rm JD,real}=\epsilon_{\rm DC}N_{\rm JD}$. Another pivotal point of the
future pointing strategy, which can strongly influence the real joint
detection efficiency, is the trigger selection. As shown in Table 5, the
number of sources localized better than 100 deg2 are of order of 100 for ET
and of 105 for ET+CE and ET+2CE. In order to not lose observational time, the
triggers to be observed will require to be prioritized. A strategy based on
the knowledge of the satellite detection efficiency, the distance and viewing
angle from the GW signals (see Sect. 3.2.2) will be mandatory to select the
triggers with a higher chance of joint detection. Therefore the reported
numbers for pointing mode should be considered as potential detections, which
can be achieved only provided that an efficient selection of the GW triggers
is performed to restrict the follow-up to cases with highest probability of
detection.

In Tab. 10 we consider longer response time (relaxing the assumption of
$t_{\rm resp}=100$ s) and we show how the detection rate significantly
decreases if the telescope needs more time to respond to the trigger and point
the GW localization. The significant decrease of the chance to detect the
afterglow emission pointing with one hour delay is due to a combination of the
rapid decline of the GRB X-ray flux and the sensitivity of wide field X-ray
instruments. Among the population of detected BNS, nearby events similar to
GW170817 are a tiny fraction, while the large majority of joint detections
comes from on-axis (or slightly off-axis) short GRBs distributed at
cosmological distances, with an afterglow light curve which peaks at early
times and followed by rapid drop of the flux. For these cases a rapid response
is fundamental to catch the first decaying part of the X-ray emission. In Tab.
11, we show how the prediction of joint GW+X-ray detections depends on the
assumption of FS parameters and on the inclusion of HLE. In this case, we
assume ET+2CE as GW network and $t_{\rm resp}=100$ s. In the specific, we test
three possible setups, where we compute the X-ray light curve:

  1. 1.

including HLE + FS emission, with _FS-SGRB_ configuration

  2. 2.

including HLE + FS emission, with _FS-GW17_ configuration

  3. 3.

including only FS emission, with _FS-SGRB_ configuration.

Our results show that the assumption of FS parameters like GW 170817 leads to
less bright FS emission, mainly due to lower value of $\epsilon_{B}$.
Moreover, the tables show that the inclusion of HLE is non negligible in the
estimation of detectability of the X-ray afterglow. The reasons are mainly
twofold: 1) the HLE produces an initial steep decay phase which is well
visible in X-rays and exceeds the FS contribution at least in the first tens
of seconds, 2) the HLE associated to a structured jet typically produces a
plateau phase for sources observed at $\theta_{v}\lesssim\theta_{c}$ and this
contribution can be as relevant as the FS emission itself.

Finally, we test how the expected joint detection rate of GW+X-rays from BNS
mergers depends on the assumption of the jet structure. The specific structure
_Stru1_ assumed so far, namely the one derived from GW 170817, has a very
steep off-core profile, meaning that the probability of detecting prompt
and/or afterglow emission from SGBRs viewed at $\theta_{v}>\theta_{c}$ is
lower than the one of detecting a shallower off-core jet profile, such as
_Stru2_. Tab. 12 shows the difference in joint GW+X-ray detections in pointing
mode assuming _Stru1_ and _Stru2_. The overall rate of joint GW+X-ray
detections for _Stru2_ is, within the uncertainties, comparable with the one
of _Stru1_. This is due to the fact that, though the X-ray emission is
detectable at larger viewing angles for _Stru2_ , the best fit value for the
fraction of BNS producing a jet, $f_{j}$, derived from the Monte Carlo is
smaller for _Stru2_ than the one of _Stru1_. Fig. 8 and 9 show the
distribution of viewing angle of GW+X-rays joint detections as a function of
redshift, adopting the jet structure _Stru1_ and _Stru2_ , respectively. For
each detection, we use a color code for the sky-localization uncertainty. The
plot is realized simulating 1000 GW+X-rays detections (ET+2CE + SXI) which
satisfy the requirement $\Delta\Omega<100$ deg2, assuming $t_{\rm resp}=100$ s
and _FS-SGRB_ configuration. Below $z\sim 0.5-1$, for both the structures, the
X-ray emission is detectable up to viewing angles much larger than
$\theta_{c}=3.4$ deg. In the bottom panel of Fig. 8, we show how the GW+X-rays
joint detections are distributed in redshift and we distinguish between cases
detected at $\theta_{v}<5^{\circ}$ and $\theta_{v}>5^{\circ}$. We adopt
$\theta_{v}=5^{\circ}$ as a threshold because, according to Fig. 4, this is
the maximum angle up to which prompt emission is detectable by _Fermi_ -GBM
(similarly for XGIS onboard of THESEUS), considering an average on redshift.
This implies that all the sources detected in X-rays at $\theta_{v}>5^{\circ}$
have a non-detectable prompt emission in $\gamma$-ray. In the case of _Stru1_
, sources with $\theta_{v}>5^{\circ}$ represent $\sim 40\%$ of all the
GW+X-rays joint detections. Similarly, for _Stru2_ the threshold angle for
detectability of prompt emission is $\theta\sim 10^{\circ}$ and $\sim 50\%$ of
all the GW+X-rays joint detections is above this limit. Independently on the
assumption on the jet structure, this result means that there is a significant
number of off-axis events concentrated below $z\sim 0.5-1$ which are only
detectable with WFX-ray instruments. In terms of absolute numbers of off-axis
GW+X-rays joint detections, a combination of ET+2CE + SXI would observe few a
tens of such events per year. For TAP and Einstein Probe a similar fraction
($\sim 40-50\%$) of events are expected to be observed off-axis. This result
demonstrates the fundamental role of WFX-ray telescopes in synergy with GW
detectors for the detection of off-axis emission from BNS mergers that would
be otherwise undetactable for $\gamma$-ray instruments. We finally evaluated
how the choice of a larger value of the angular extension of the jet wings,
$\theta_{w}$ (used in _Afterglowpy_), could increase the number of detections.
Increasing $\theta_{w}$ also the maximum viewing angle at which the source is
detectable increases. We estimated that increasing $\theta_{w}$ from 15 deg
(30 deg) to 45 deg, the number of joint detections increases of a factor
$<5\%$ ($\sim 10\%$) for _Stru1_ (_Stru2_).

####  3.2.4 Medium and small-FOV sensitive X-ray observatories

When the GW sources are localized at the level of arcminutes the X-ray
afterglow can be directly detected by pointing small-FOV instruments. However,
there are no BNSs localized at arcminute precision through the GW signal when
ET is operating as a single detector, and the number is also negligible when
ET is in a network of 3G detectors. When the GW localization is of order of
1-10 deg2, medium-FOV X-ray instruments can scan the entire localization
region by performing mosaic observations. Considering the GW sources localized
better than 10 deg2 from Table 6, the sources that can be followed-up by
medium-FOV instruments are around 10 when ET is operating as a single
detector, but several thousands (hundred of thousands) when the ET+CE (ET+2CE)
network is observing. However, in the case of ET+CE, the distances of these
sources are limited to $z<1.2$. For larger $z$ the way to detect the X-ray
counterpart remains through the use of wide-FOV $\gamma$-ray and/or X-ray
satellites. In this case, the more sensitive small-FOV satellites can operate
at a later time by pointing the precise localization provided by the wide-FOV
satellites, following-up and characterizing the detected counterparts.

During the activity period of 3G GW detectors, the mission Athena is expected
to be operative (Nandra et al. 2013). The exceptional sensitivity of Athena
X-ray telescope and its focal plane instruments will allow us to detect the
faint afterglow emission from SGRBs and GW counterparts hours to years (for
closer events) after the merger and to make high resolution spectra (Piro et
al. 2021). Athena will host the Wide Field Imager (WFI) with a FOV of 0.4 deg2
and the X-IFU high resolution spectrometer with a FOV of 5 arcmin equivalent
diameter. The WFI is able to directly point GW signals with sky localization
uncertainties smaller than its FOV (a few tens per year for ET+CE and
thousands for ET+2CE considering all the orientations of the binary systems),
and in addition is capable to carry out a mosaic of sky regions extended up to
10 deg2. On the other hand, X-IFU requires a precise localization ($\simeq$2
arcmin) to directly point the source. Even considering a GW network with
ET+2CE, there are no sources with such a small uncertainty on the GW sky
location. Therefore, instead of being directly triggered by the GW detection,
X-IFU can be used either for following up precise localizations delivered by
WFX-ray telescopes or by the Athena WFI itself. Taking into account that
Athena will likely have a Time of Opportunity (TOO) response time of 4 hours,
we find that the totality of the sources jointly detected by ET+2CE and the
WFX-ray telescopes can be detected and followed-up by Athena X-IFU.

In order to evaluate the Athena-WFI capabilities to detect GW counterparts, we
consider the sources localized by the GW detectors within a region of 10 deg2.
We assume an average number of exposures to fully cover the GW sky region
equal to $N_{\rm exp}\sim\Delta\Omega_{\rm GW}$/FOVWFI, where FOVWFI = 0.4
deg2, and an exposure time $T_{\rm exp}=10^{4}$ s for each mosaic observation.
For simplicity, we work in the approximation that the source is randomly
located within the 90$\%$ GW error region and that each position is equi-
probable. This implies that, if the GW region is tiled in $N_{\rm exp}$ sub-
regions, the probability that the source is located inside the $k^{th}$ sub-
region is $P(k)=1/N_{\rm exp}$ and is the same for all the sub-regions. We
consider a TOO response time $T_{\rm TOO}$ of 4 hr and we include a dead time
interval $T_{d}$ of 30 minutes corresponding to the time necessary to slew
from one sub-region of the mosaic to the adjacent
one666https://www.cosmos.esa.int/documents/400752/400864/Athena+Mission+Proposal/18b4a058-5d43-4065-b135-7fe651307c46.
Hence, the time necessary to point and detect the source is $T(k)=T_{\rm
TOO}+k\times(T_{\rm exp}+T_{d})$. Operatively, for each source we extract a
random number $k$ in the range $[0,N_{\rm exp}]$ and we check if at time
$T(k)$ after the GW trigger the X-ray afterglow emission is above the
sensitivity of Athena-WFI corresponding to $10^{4}$ s of exposure.

Given the several science goals of Athena, only a fraction of its
observational time can be realistically dedicated to the follow-up of GWs;
specifically we assume one month over one year of observations. While for ET
alone the number of sources with $\Delta\Omega<10$ deg2 is around 10, and all
of them can in principle be followed up by Athena in this total amount of
time, the number of the sources detected by ET+CE with $\Delta\Omega<10$ deg2
is of several thousands. In order to maximize the effectiveness of the search
and not lose time on sources without any chance of detection, a further
selection of the sources to be followed-up results to be necessary. We select
the sources by excluding all those with $\theta_{v}>50^{\circ}$ (for which the
error on $\theta_{v}$ is relatively small, see Fig. 6, and thus can be
reliably considered off-axis) and $z>0.5$. We select randomly among these
sources ($\sim 130$) up to reach a total amount of time of one month of
observation, and find that Athena-WFI is able to detect $N_{\rm
GW+Athena}=\epsilon_{\rm FOR}\times(6_{-3}^{+2})$ events per year, where
$\epsilon_{\rm FOR}$ is a factor that takes into account the TOO efficiency
and it corresponds to a field of regard of $\sim 50\%$ (i.e. only a fraction
$\epsilon_{\rm FOR}$ of the events can be successfully followed-up by Athena).
Including another CE in the GW network does not increase significantly the
number of Athena detections ($N_{\rm GW+Athena}=\epsilon_{\rm FOR}\times
8_{-2}^{+3}$ detections per year), because even increasing the number of
sources well localized, the maximum number of Athena detections is anyway
limited by the time dedicated to TOO. If we exclude more sources by lowering
the threshold on the viewing angle to $\theta_{v}<30^{\circ}$ instead of
$\theta_{v}<50^{\circ}$, the number of joint detection increases to
$\epsilon_{\rm FOR}\times 14^{+4}_{-3}$/yr ($\epsilon_{\rm FOR}\times
27^{+3}_{-3}$/yr) for ET+CE (ET+2CE). The knowledge about the luminosity
distance and the viewing angle coming from GW analysis can considerably help
for the selection of a golden sample of events that have a larger probability
to be detected, giving the possibility to maximize the GW+X-ray joint
detection efficiency.

##  4 Discussion and conclusions

Our study evaluates the perspectives for multi-messenger astronomy in the ET
era focusing on the search of high-energy signals, which is a unique way to
detect EM counterparts of GW signals at high redshift. We estimate the
expected rate of joint detections of GWs and $\gamma$-ray and/or X-ray signals
from BNS mergers considering prompt and afterglow emissions, different GW
detector configurations (ET, ET+CE, ET+2CE), and different observational
strategies (survey and pointing mode) for several satellites. Our theoretical
framework starts from an astrophysically-motivated population of BNS mergers
and predicts the high-energy signals associated with these mergers making the
BNS population able to reproduce all the statistical properties of currently
observed SGRBs. This approach makes it possibile to reproduce not only the
observational features of SGRBs, such as luminosity, duration and spectral
properties, but also the average detection rate with current $\gamma$-ray
instruments, such as _Fermi_ -GBM. This enables us to normalize the number of
joint detections, evaluate the fraction of BNS producing a jet, and be less
dependent of the BNS merger absolute number, which is still largely uncertain.

Our main results are summarized as follows:

  * •

Regarding the joint detection of GWs and $\gamma$-ray emission, we find that
already considering ET alone, more than $60\%$ of all the SGRBs having a
detectable prompt emission will also have a detectable GW counterpart. This
percentage approaches $100\%$ if we include CE as an additional interferometer
in the network. From a few tens to hundred joint detections are expected per
year, mainly depending on the FOV of the high energy-satellites. However,
although increasing the number of joint detections is important for
statistical studies, characterizing the sources and identifying the host
galaxies is of primary importance. As shown by _Swift_ in many years of
observations, this is possible and effective when the source is localized at
arcmin level enabling to drive the follow-up of ground-based telescopes.
Mission concepts like THESEUS-XGIS, will be essential to detect and localize
high-energy counterpart of BNS mergers with enough precision (arcmin
uncertainty). This is particularly important when the sources are located at
large redshift, namely $z>1$ where also in the optimistic scenario of a
network of 3G GW detectors, the GW sky localization will be larger than 1 deg2
from the GW signals. Instruments such as HERMES (full constellation of cube-
satellites) can also be a good compromise of number of detections and sky
localization uncertainties which are however order of a few deg2.

  * •

Considering the possible shock breakout of the cocoon produced by the
interaction of the jet with the NS merger ejecta, we show that sensitive (more
than the current ones) and wide FOV $\gamma$-ray are required.

  * •

Regarding the joint detection of GWs and X-ray emission, we demonstrate the
important role of WFX-ray missions, such as SVOM, Einstein Probe, Gamow,
THESEUS, and TAP. We predict tens of detections per year when these
instruments operates in survey mode. The joint detection rate in survey mode
is limited by the chance of having the source inside the FOV at the moment of
the merger. We propose an additional observational strategy, the pointing
strategy, which could in principle enhance the probability of joint detection
exploiting the information about sky localization provided by GW instruments
(see below).

  * •

ET operating as a single observatory can detect several hundreds of BNS merger
per year with $\Delta\Omega<100$ deg2, among them about $\sim 70$ BNS/yr have
a viewing angle $\theta_{v}<15^{\circ}$, i.e. sources with potentially
detectable high-energy emission. The inclusion of CE in the network
significantly improves the localization capabilities by reaching several
thousands (hundreds with $\theta_{v}<15^{\circ}$) of BNS detections per year
with $\Delta\Omega<10$ deg2. The network of CE+ET is able to localize a few
hundreds (a few tens with $\theta_{v}<15^{\circ}$) with $\Delta\Omega<1$ deg2
up to redshift 0.3. This number increases to thousands (hundreds on-axis) per
year up to redshift 1 with the network of ET+2CE.

  * •

Considering WFX-ray telescopes slewing to the sky position provided by GW
detectors for sources with a good sky localization (better than 100 deg2), we
evaluate that the X-ray afterglow emission from ten to hundreds SGRBs could be
detected during one year of observation assuming a response time $t_{\rm
resp}=$100 s.

  * •

A rapid communication of the GW sky location and a rapid repointing of the
high-energy satellites is crucial to maximize the efficiency of the poiting
strategy. Indeed, due to the typical rapid decline of the X-ray flux of SGRBs,
we showed that a longer delay time between the merger and the beginning of
X-ray monitoring would decrease the probability of detection by a factor
$\sim$ 10 or more going from $t_{\rm resp}=$100 s to $t_{\rm resp}=$ 1 hr. The
low-frequency sensitivity of ET will make it possible to provide a good sky
localization even several minutes before the merger (see Chan et al. 2018; Li
et al. 2021; Nitz & Dal Canton 2021, Banarjee et al. in prep.). This will
allow us to send information about the sky location of the source well before
the merger and X-ray telescope can realistically start earlier the slewing
procedure. We evaluate that a not negligible fraction of GW events, localized
within 100 deg2 at merger, are detectable tens of minutes before the BNS
merger with a sky-localization well within the FOV of the WFX-ray satellites.

  * •

While the number of detections for WFX-ray satellites in survey mode is
reliable, the number of detections in pointing is optimistic and relies on a
perfect prioritization of the trigger to be followed (see below).

  * •

We show that using WFX-ray instruments makes it possible to detect a
significant fraction of BNS merger which are too off-axis to have a detectable
$\gamma$-ray emission and that would be otherwise missed in absence of these
instruments. The presence of WFX-ray monitors is also fundamental to provide
the spectral coverage at smaller energies than $\gamma$-rays and to precisely
localize the source. The WFX-ray telescopes can provide sky location accuracy
of $\sim$ arcmin, which enables to trigger follow-up observations by ground-
based optical telescopes, radio arrays and exceptional sensitive X-ray
instruments, such as the X-IFU instrument onboard of Athena. We evaluate that
the totality of the sources jointly detected by the GW detectors and the WFX-
ray telescopes can be detected and followed-up by the Athena X-IFU.

  * •

The network of ET+CE and ET+2CE is expected to localize a large number of BNS
signals with $\Delta\Omega<10$ deg2. An instrument such as the WFI (FOV of
$0.4$ deg2) on-board of Athena is able to carry out a mosaic of these sky
regions by detecting from a few to a few dozen signals by using 1 month of
Athena observations. The joint detection number comes from an observational
strategy which maximizes the chance of detections by removing the BNS signals
with smaller probability to have observable jet. In particular, this is done
by removing BNS with larger viewing angles and distances.

  * •

Due to the high number of BNS events expected to be released by the next
generation GW detectors, for the pointing observational strategy, it will be
crucial to select and prioritize the ones to be followed-up. The selection
based on sky-localization will not be enough to avoid to lose observational
time of the high-energy satellites when the network ET+CE and ET+2CE will
operate. On the basis of the different scientific science goals, specific
selection based on the GW source parameters such as distance and viewing
angles, but also pre-merger sky-localization, will be mandatory to maximize
the joint detection efficiency. This shows the importance in the ET era to
send in low-latency (updating) information about GW parameter estimation.

Our work shows that to maximize the science return of 3G GW detectors in the
multi-messenger context, it is necessary to develop sensitive wide-FOV high-
energy instruments able to detect but also localize a large number of short
GRBs. Instruments, such as THESEUS, combining $\gamma$-ray and WFX-ray
telescopes, are of primary importance to guarantee the observation of
$\gamma$-ray and X-ray counterparts, by detecting $\sim 10-100$ well localized
SGRBs up to high redshifts, for which also the host galaxy can be identified.
This is of primary importance for evaluating the cosmological parameters and
testing general relativity at cosmological scales. WFX-ray telescopes are
required to detect GRB off-axis and make it possible to explore the jet
structure, its interconnection with the ejecta, and the nature of low-
luminosity SGRBs with a unique level of detail. Medium-size X-ray instruments,
such-as Athena WFI, will become effective (operating mosaic of the GW sky-
localization) for the X-ray counterpart search when a network of GW detector
will observe. Mosaic observations with deep exposure by Athena WFI can enhance
the possibility to detect off-axis X-ray emission. Sensitive instruments such
as Athena X-IFU will be crucial to follow-up and characterize the X-ray
emission at later times with respect to the merger.

##  5 Acknowledgements

We thank Yann Bouffanais, Boris Goncharov, Andrea Maselli, Om Sharan Salafia,
Giulia Stratta for valuable discussions. We thank Michele Maggiore, Francesco
Iacovelli, Michele Mancarella, and Stefano Foffa for checking the gravitional-
wave results and for very helpful discussions. We acknowledge Stefano
Bagnasco, Federica Legger, Sara Vallero, and the INFN Computing Center of
Turin for providing support and computational resources. We thank Ssohrab
Borhanian, Bangalore Sathyaprakash, Man Leong Chan, Yufeng Li, Ik Siong Heng
for useful information and cross-check on the gravitational-wave data
analysis. We thank Lorenzo Amati, Fabrizio Fiore, Paul O’Brien and Nicholas
White for useful information about some of the X-ray instruments. We thank
Maria Grazia Bernardini for the information about SVOM. MB and SR acknowledge
financial support from MIUR (PRIN 2020 grant 2020KB33TP$\\_$001). MB and GO
acknowledge financial support from the AHEAD2020 project (grant agreement n.
871158). BB, GG and MB acknowledge financial support from MIUR (PRIN 2017
grant 20179ZF5KS). GG acknowledges ASI-INAF agreement n. 2018-29-HH. MM and FS
acknowledge financial support from the European Research Council for the ERC
Consolidator grant DEMOBLACK, under contract no. 770017. The research leading
to these results has been conceived and developed within the Einstein
Telescope Observational Science Board.

## Code availability

The latest public version of mobse can be downloaded from this repository.
GWFish is publicly available at this
repository777https://github.com/janosch314/GWFish

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##  Appendix A Computation of the observables in the structured jet scenario

(a)

(b)

(c)

Figure 10: Sky localization (90$\%$ credible region in deg2) as a function of
the redshift for one year of BNS injections with viewing angle in the range
$0^{\circ}<\theta_{v}<15^{\circ}$ and considering ET (a), ET+CE (b) and ET+2CE
(c). The color bar indicates the SNR of each detection. A duty cycle of 0.85
has been assumed for the GW detectors as described in the text. Figure 11:
Same as Fig. 4, but for _Stru2_.

(a)

(b)

Figure 12: Cumulative distribution of the SNR for BNS mergers localized better than $100$ deg2 for the three GW networks considered in this work. Panel (a) shows cases with $\theta_{v}<15^{\circ}$, while in panel (b) all angles are considered. Figure 13: The plot shows the detection efficiency of ET as a function of redshift assuming a Gaussian distribution ($\expectationvalue{M}=1.33M_{\odot}$, $\sigma_{M}=0.1M_{\odot}$) and a uniform distribution ($M\in[1.0-2.5]M_{\odot}$). Only cases with $\theta_{v}<15^{\circ}$ are considered. A duty cycle of 0.85 has been assumed for the GW detectors as described in the text. The ratio between the orange bars and the blue bars corresponds to the function $\phi(z)$ defined in the text. parameter | prior interval
---|---
$\lambda_{E}$ | [1,12]
$\log_{10}{(E_{t}/10^{49}\text{erg/s})}$ | [-2,$\log_{10}(50)$]
$\log_{10}{(\mu_{E}/\text{keV})}$ | $[\log_{10}(700),\log_{10}(3000)]$
$\sigma_{E}$ | [0.05,1]
$\log_{10}{(\mu_{\tau}/s)}$ | $[-2,\log_{10}(4)]$
$\sigma_{\tau}$ | [0.05,1]
$\log_{10}{f_{j}}$ | [-3,0]
Table 13: Prior intervals for the prompt emission model. For $\lambda_{E}$, $\sigma_{E}$ and $\sigma_{\tau}$ we consider a flat prior distribution, while for the others the distribution is flat in logarithm. | $N_{\Delta\Omega<1000\rm\,deg^{2},t_{\rm pre}}/N_{\Delta\Omega<100\rm\,deg^{2},t_{\rm merger}}$
---|---
| SNR$>8$ | SNR$>7$
$t_{\rm pre}$ | 10 min | 20 min | 10 min | 20 min
ET | $63\%$ | $29\%$ | $68\%$ | $37\%$
ET+CE | $5\%$ | $0.5\%$ | $5\%$ | $0.7\%$
ET+2CE | $6\%$ | $0.5\%$ | $6\%$ | $0.6\%$
Table 14: We show the fraction $N_{\Delta\Omega<1000\rm\,deg^{2},t_{\rm
pre}}/N_{\Delta\Omega<100\rm\,deg^{2},t_{\rm merger}}$ of BNS localized better
than 100 deg2 at the merger that are localized better than 1000 deg2 at a pre-
merger time $t_{\rm pre}$, for ET, ET+CE and ET+2CE. We show both cases that
are detected with SNR$>8$ and SNR$>7$ at the pre-merger time. We consider only
BNS observed at a viewing angle $\theta_{v}<15^{\circ}$.

In this section we show how the peak flux, the fluence and the isotropic
energy are derived in the structured jet framework. The derivation is done
assuming a unique jet structure, with fixed values of $R_{0}$ and $\Gamma$,
defined in the main text. For a given viewing angle $\theta$, we write the
rest frame photon spectrum as

$N(E,\theta,t)=N_{0}(\theta,t)f(E/E_{0}(\theta,t))$ (12)

where we included the time dependency only in the normalisation, assuming that
there is not a strong spectral evolution during the GRB pulse.
$f(E/E_{0}(\theta))$ is the spectral shape and it is assumed to be a smoothly
broken power law, whose spectral slopes are assumed to be angle independent.
$f$ is normalised in such a way that $f(1)=1$. $E_{0}(\theta,t)$ is the peak
energy and we express it as

$E_{0}(\theta,t)=E_{0}(\theta,t=t_{p})P_{E}(t/t_{p},\theta)$ (13)

where $t_{p}$ is the peak time of the pulse and $P(t/t_{p}\theta)$ describes
the temporal profile, in principle angle-dependent. Analogously, we express
the time behaviour of the photon spectrum normalisation as:

$N_{0}(\theta,t)=N_{0}^{p}(\theta)P_{N}(t/t_{p},\theta)$ (14)

In both cases, we approximate the temporal profiles of peak energy and
normalisation as

$P_{X}(t)=\begin{cases}t/t_{p}&0<t<t_{p}\\\
(t/t_{p})^{-\alpha_{X}(\theta)}&t>t_{p}\end{cases}\quad X=E,N$ (15)

At the peak time we write the photon spectrum as

$N^{p}(E,\theta)=N_{0}^{p}(\theta)f(E/E_{0}(\theta))$ (16)

We define also

$R_{E}(\theta)=E_{0}(\theta)/E_{0}(\theta=0)=E_{0}(\theta)/\hat{E}_{0}$ (17)

and

$R_{F}(\theta)=\frac{N^{p}(\theta)|_{E=E_{0}(\theta)}}{\hat{N}^{p}|_{E=\hat{E}_{0}}}$
(18)

The functions $\alpha_{E}(\theta)$, $\alpha_{N}(\theta)$, $R_{E}(\theta)$ and
$R_{F}(\theta)$ are computed numerically using the structured jet model.
Hereafter the notation $\hat{x}$ indicates that the quantity $x$ is evaluated
at viewing angle $\theta=0$. In the observer frame, the peak photon spectrum
is:

$N^{p}(E,\theta)=(1+z)^{2}N_{0}^{p}(\theta)f\left(\frac{1}{R_{E}}\frac{(1+z)E}{\hat{E}_{0}}\right)$
(19)

and using eq. 18

$N^{p}(E,\theta)=(1+z)^{2}\hat{N}_{0}^{p}R_{F}f\left(\frac{1}{R_{E}}\frac{(1+z)E}{\hat{E}_{0}}\right)$
(20)

The value of $\hat{N}_{0}^{p}$ is derived from the rest frame on-axis
isotropic energy $\hat{E}_{\rm iso}=E_{t}/(1-\cos{\theta_{c}})$, where $E_{t}$
is extracted from the assumed probability distribution defined in the main
text. In the specific,

$\hat{E}_{\rm iso}=4\pi D_{L}^{2}\int dt\int_{1\text{ keV}}^{10^{4}\text{
keV}}F_{\nu}(t)dE$ (21)

where $D_{L}$ is the luminosity distance and $F_{\nu}$ is the flux density.
Using eq. 16:

$\hat{E}_{\rm iso}=4\pi D_{L}^{2}\int
dt\hat{N}_{0}(t)\hat{E}_{0}^{2}(t)\int_{1\text{
keV}/\hat{E}_{0}(t)}^{10^{4}\text{ keV}/\hat{E}_{0}(t)}xf(x)dx$ (22)

which can be written as

$\hat{E}_{\rm iso}=4\pi
D_{L}^{2}\hat{N}_{0}^{p}\int\hat{P}_{N}(t/t_{p})\hat{E}_{0}^{2}(t)I_{E}(t)dt$
(23)

where $I_{E}(t)=\int_{1\text{ keV}/\hat{E}_{0}(t)}^{10^{4}\text{
keV}/\hat{E}_{0}(t)}xf(x)dx$

This procedure allows us to write $\hat{N}_{0}^{p}$ in terms of known
quantities, namely

$\hat{N}_{0}^{p}=\frac{\hat{E}_{\rm iso}}{4\pi
D_{L}^{2}\int\hat{P}_{N}(t/t_{p})\hat{E}_{0}^{2}(t)I_{E}(t)dt}$ (24)

Once $\hat{N}_{0}^{p}$ is determined, the peak photon flux integrated in the
energy band of a given instrument is

$\displaystyle F_{p}(\theta)=\int_{(1+z)E_{1}}^{(1+z)E_{2}}N^{p}(E,\theta)dE$
(25)
$\displaystyle=(1+z)\hat{N}_{0}^{p}R_{F}(R_{E}\hat{E}_{0})\int_{(1+z)E_{1}/R_{E}\hat{E}_{0}}^{(1+z)E_{2}/R_{E}\hat{E}_{0}}f(x)dx$
(26)

The duration of the burst $t_{90}$ is computed imposing that the 10-1000 keV
fluence accumulated up to $t_{90}$ is 90$\%$ of the total fluence, namely

$\displaystyle\int_{0}^{t_{90}}dt\int_{(1+z)\text{ keV}}^{(1+z)10^{4}\text{
keV}}EN(E,\theta,t)dE$ (27)
$\displaystyle=90\%\int_{0}^{\infty}dt\int_{(1+z)\text{
keV}}^{(1+z)10^{4}\text{ keV}/}EN(E,\theta,t)dE$ (28)

For computational reasons, the last integral in time is performed up to a time
$t_{max}$, which is the maximum time for which the $\gamma$-ray emission is
still detectable by the instrument. Finally the isotropic luminosity is
defined as $E_{\rm iso}/t_{90}$.

##  Appendix B Detection efficiency for different neutron-star mass
distribution

In Fig. 13 we show the detection efficiency (detected signals over injected
ones) of ET as a function of redshift for a Gaussian mass distribution as
observed in Galactic double neutron-star systems in radio (Özel & Freire 2016)
and a uniform distribution of NS mass consistent with gravitational-waves
observations (The LIGO Scientific Collaboration et al. 2021c). For the
Gaussian distribution we fix the mean value to
$\expectationvalue{M}=1.33M_{\odot}$ and the standard deviation to
$\sigma_{m}=0.1M_{\odot}$. The detection efficiency obtained with the two
distributions is consistent up to a redshift equal to 0.5, and then the
Gaussian distribution leads to a lower number of GW detections accountable by
the fact that the GW signal amplitude scales with the mass.

In order to evaluate the impact on joint detections, we estimate, as example,
the loss of joint GW+$\gamma$-rays detections with ET+_Fermi-GBM_. We assume
the probability of having a joint detection as

$P_{JD}(z)=P_{EM}(z)P_{GW}(z)$

where $P_{EM}(z)$ is the probability of detecting the EM counterpart, and
$P_{GW}(z)$ the probability of detecting the associated GW signal. Each
probability corresponds to the ratio between the number of detected and
injected sources.

$P_{i}(z)=\frac{N_{det,i}(z)}{N_{inj}(z)},\,i=GW,EM$

The assumption of NS mass distribution only affects $P_{GW}(z)$, since we
specified in the main text that we neglect any dependence on the NS mass for
the formation of a jet. We call $P_{GW,flat}(z)$ and $P_{GW,Gauss}(z)$ the GW
detection efficiency for flat and Gaussian distributions, respectively. The
variation on the rate of joint detections can be computed as

$\frac{dN_{JD,Gauss}}{dz}(z)=\frac{dN_{inj}(z)}{dz}P_{JD,Gauss}(z)=$
$=\phi(z)\frac{dN_{inj}(z)}{dz}P_{EM}(z)P_{GW,flat}(z)=\phi(z)\frac{N_{JD,flat}}{dz}$

where we wrote $P_{GW,Gauss}(z)=\phi(z)P_{GW,flat}(z)$. The total variation,
integrated in redshift, is

$N_{JD,Gauss}=\int\phi(z)\frac{N_{JD,flat}}{dz}dz$

In Fig.13 the function $\phi(z)$ corresponds to the ratio between the two
histograms. Finally, knowing the function $\phi(z)$ and the redshift
distribution of joint detections, we derive the variation in joint detection
rate for a Gaussian mass distribution. We obtain that the joint
GW+$\gamma$-rays detections with ET+_Fermi-GBM_ decreases of a factor $\sim
27\%$ (from 33/yr to 24/yr), assuming a Gaussian mass distribution.

Figure 14: Corner plot of the posterior distribution of the parameters adopted
for the prompt emission model. The assumed structure is _Stru1_. The number of
MCMC steps is chosen such that the auto-correlation time reaches a plateau, as
described in Foreman-Mackey et al. (2013). Figure 15: Same as Fig. 14,
adopting _Stru2_