PASA 2024

# Multimessenger astronomy with a kHz-band gravitational-wave observatory

Nikhil Sarin1,2,3,4 and Paul D. Lasky1,2
nsarin.astro<EMAIL_ADDRESS>1School of Physics and Astronomy,
Monash University, Clayton VIC 3800, Australia  2OzGrav: The ARC Centre of
Excellence for Gravitational Wave Discovery, Clayton VIC 3800, Australia
3Nordita, KTH Royal Institute of Technology and Stockholm University
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden  4The Oskar Klein Centre,
Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm,
Sweden

###### Abstract

Proposed next-generation networks of gravitational-wave observatories include
dedicated kilohertz instruments that target neutron star science, such as the
proposed Neutron Star Extreme Matter Observatory, NEMO. The original proposal
for NEMO highlighted the need for it to exist in a network of gravitational-
wave observatories to ensure detection confidence and sky localisation of
sources. We show that NEMO-like observatories have significant utility on
their own as coincident electromagnetic observations can provide the detection
significance and sky localisation. We show that, with a single NEMO-like
detector and expected electromagnetic observatories in the late 2020s and
early 2030s such as the Vera C. Rubin observatory and SVOM, approximately
$40\%$ of all binary neutron star mergers detected with gravitational waves
could be confidently identified as coincident multimessenger detections. We
show that we expect $2^{+10}_{-1}\,\mathrm{yr^{-1}}$ coincident observations
of gravitational-wave mergers with gamma-ray burst prompt emission,
$13^{+23}_{-10}\,\mathrm{yr^{-1}}$ detections with kilonova observations, and
$4^{+18}_{-3}\,\mathrm{yr^{-1}}$ with broadband afterglows and kilonovae,
where the uncertainties are $90\%$ confidence intervals arising from
uncertainty in current merger-rate estimates. Combined, this implies a
coincident detection rate of $14^{+25}_{-11}\,\mathrm{yr^{-1}}$ out to
$300\,\mathrm{Mpc}$. These numbers indicate significant science potential for
a single kilohertz gravitational-wave detector operating without a global
network of other gravitational-wave observatories.

###### doi:

10.1017/pas.2024.xxx

##  1 Introduction

Multimessenger gravitational-wave astronomy is a new field that already boasts
a wealth of new scientific achievements, despite only having one detected
event to date. The future of this field relies on continued improvement in
both gravitational-wave and traditional astronomical instruments. Third-
generation gravitational-wave observatories such as the Einstein Telescope
(Punturo et al., 2010) and Cosmic Explorer (Abbott et al., 2017a) are billion-
dollar scale instruments with approximately a factor ten improvement in
sensitivity over current, second-generation observatories like Advanced LIGO
(Aasi et al., 2015), Advanced Virgo (Acernese et al., 2015), and KAGRA (Akutsu
et al., 2019). Slated for the mid-2030s, these third-generation observatories
still require significant technology development to see them realised. To this
end, so-called 2.5-generation observatories have been proposed that serve as
both dedicated astronomy observatories as well as technology drivers and
facilitators for third-generation observatories. These 2.5-generation
observatories include broadband interferometers such as A+ (Miller et al.,
2015; Abbott et al., 2018) and Voyager (Adhikari et al., 2020), as well as
dedicated kilohertz observatories (Martynov et al., 2019) such as the newly-
proposed Neutron Star Extreme Matter Observatory, NEMO (Ackley et al., 2020).

The NEMO design proposal held neutron star physics from binary neutron star
mergers as the primary science driver. It proposed comparable high-frequency
($\gtrsim 1\,\mathrm{kHz}$) sensitivity to Einstein Telescope and Cosmic
Explorer, sacrificing low-frequency sensitivity ($\lesssim 500\,\mathrm{Hz}$)
to significantly reduce costs and the requirement for new technologies. The
original proposal argued that NEMO must co-exist with 2.5-generation
instruments to maximise scientific impact—those instruments would provide
source localisation with their superior low-frequency sensitivity. However,
during the proposed period of operation (late 2020s/early 2030s), there may be
significant periods of time when NEMO must operate without a global network of
gravitational-wave observatories. In this work, we show that NEMO-like
observatories can operate in isolation without a heterogeneous global array
because of the presence of electromagnetic telescopes with survey and/or all-
sky capabilities that will make routine detections of the electromagnetic
counterparts of binary neutron star mergers out to relevant distances.

The Vera C. Rubin observatory (Ivezić et al., 2019) is an 8.4-m optical/UV
telescope capable of detecting kilonovae associated with binary neutron star
mergers. Providing they are in the correct hemisphere, Vera Rubin will detect
100% of kilonovae using only $30\,\mathrm{s}$ exposures out to $\sim
450\,\mathrm{Mpc}$ (Chen et al., 2021), approximately equivalent to the
maximum horizon distance of NEMO (Ackley et al., 2020). In optical/UV, Vera
Rubin is the tip of the iceberg; other likely facilities with even better
sensitivity include the James Webb Space Telescope (Gardner et al., 2006) and
Keck Wide-Field imager (Gillingham et al., 2020) as well as the thirty-metre
class of ground-based facilities such as the Giant Magellan Telescope (GMT;
Johns et al., 2012), and the Thirty-Metre Telescope (TMT; Skidmore et al.,
2015). These will be complemented with all-sky survey facilities including
Zwicky Transient Facility (ZTF; Bellm et al., 2019) and Pan-STARRS (Chambers
et al., 2016), capable of providing nightly all-sky surveys to observe
transient kilonovae and gamma-ray burst afterglows. Observatories like
Evryscope may also provide continuous all-sky coverage (Law et al., 2015).

By the late 2020s there will likely be multiple successor x-ray and gamma-ray
telescopes to the Neil Gehrel’s Swift Observatory (Gehrels et al., 2004) and
Chandra (Weisskopf et al., 2000), including proposed instruments such as
ATHENA (Barcons et al., 2017) and Theseus (Amati et al., 2018), and currently
operational or upcoming facilities like SVOM (Wei et al., 2016) and eROSITA
(Merloni et al., 2012). Current instruments have the capability to detect on-
axis bursts well beyond the horizon of NEMO (e.g., Howell et al., 2019), and
proposed missions will be more sensitive with equivalent or better sky
coverage, and faster slew times. These instruments even offer the possibility
of detecting binary neutron star pre-cursor emission (e.g., Most & Philippov,
2020; Sridhar et al., 2021; Ascenzi et al., 2021), as well as potential
emission from the central engine following the merger (e.g., Sarin & Lasky,
2021). Afterglows in radio will be detectable by instruments like the Square
Kilometre Array beyond the horizon distance of NEMO (Dobie et al., 2021),
which will facilitate very-long baseline interferometry follow-up to determine
jet structure and orientation of some short gamma-ray bursts from binary
neutron star mergers (Duque et al., 2019). Observatories like the Cherenkov
Telescope Array will also provide all-sky high energy gamma-ray survey
capabilities (Cherenkov Telescope Array Consortium et al., 2019).

With all of these detections across multiple wavebands and with high-frequency
gravitational waves, the problem becomes: how confidently can we associate any
two measurements as coming from the same source? Without this association, we
cannot fulfil the rich promise that multimessenger gravitational-wave science
brings. In this Paper, we calculate predictions for the rate of binary neutron
star mergers NEMO alone can detect and that can be confidently associated with
their various electromagnetic counterparts. We show the gamma-ray prompt
emission will be confidently associated with the gravitational-wave emission
at an overall rate of $2^{+10}_{-1}\,\mathrm{yr^{-1}}$, where the
uncertainties are $90\%$ confidence intervals arising from uncertainty in
current merger-rate estimates. Gravitational waves and kilonovae will be
associated at $13^{+23}_{-10}\,\mathrm{yr^{-1}}$ with broadband afterglows at
$4^{+18}_{-3}\,\mathrm{yr^{-1}}$. Combined, this implies a coincident
detection rate of $14^{+25}_{-11}\,\mathrm{yr^{-1}}$ out to
$300\,\mathrm{Mpc}$.

In this work, we discuss only neutron star binary mergers. The aLIGO/aVirgo
network have already detected $2^{+2}_{-2}$ (100% confidence interval) neutron
star-black hole binaries (Abbott et al., 2020, 2021b, 2021a, 2021d), however
we note that each of these purported mergers have black hole masses and spins
such that the neutron stars are not expected to be tidally disrupted, and
therefore no electromagnetic counterparts are expected (e.g., Foucart, 2020).
The rates of neutron star-black hole binary mergers with tidal disruption are
therefore unknown (e.g., Sarin et al., 2022), and we do not speculate further
on the expected science outcomes of those systems in this work. We further
note that NEMO-like observatories are potentially good for detecting
gravitational-wave signals from supernovae (e.g., Powell & Müller, 2020;
Mezzacappa et al., 2020; Pan et al., 2021), however the rates of
gravitational-wave only, and coincident gravitational-wave and electromagnetic
detections, are highly uncertain. Finally, NEMO-like observatories may also be
able to detect nearly-monochromatic signals from millisecond pulsars, however
there the gravitational-wave amplitude is highly uncertain (e.g., see Lasky,
2015; Haskell & Schwenzer, 2021, and references therein; although see Woan et
al. (2018) for evidence of a minimum ellipticity).

This paper is set out as follows. In Sec. 2 we detail the salient aspects of
future gravitational-wave and electromagnetic facilities for this study and
describe the modelling of the gravitational-wave and electromagnetic signal.
In Sec. 3 we provide the main results of the paper, using a Bayesian method
for determining coincidences between multiple different observations which is
presented in Appendix A. Section 3 focuses first on a hypothetical ‘jackpot’
event (Sec. 3.1) that is both on axis and nearby at 40 Mpc—the distance of the
first gravitational-wave multimessenger event GW170817. In Sec. 3.2 and 3.3 we
provide more realistic scenarios by looking at populations of events with
varying distances and inclination angles to the observer’s line of sight,
respectively. In Sec. 4 we conclude by discussing the promising multimessenger
science that can be learned from these future coincident detections.

##  2 Observations in the 2030s

###  2.1 Gravitational-wave observations

In this work, we assume only a single ground-based gravitational-wave
interferometer is operating around the globe with NEMO-like sensitivity
(Ackley et al., 2020). While we hope this situation never eventuates, we are
mindful that there may be a data gap between the end-of-life stage of
2.5-generation interferometers such as A+ and Virgo+ (Abbott et al., 2018),
and the beginning of third-generation observatories such as Einstein Telescope
and Cosmic Explorer (Punturo et al., 2010; Abbott et al., 2017a).

We simulate a realistic population of binary neutron star gravitational-wave
signals at fixed distances but with the inclination angle $\iota$ drawn from a
uniform in $\cos\iota$ distribution. We inject all signals with masses drawn
from a Gaussian mixture model motivated by analysis of the mass distribution
of all neutron stars in our Galaxy (Alsing et al., 2018). This mass
distribution mixture model is

$\displaystyle
p(M)=\left(1-\epsilon\right)\mathcal{N}\left(\mu_{1},\,\sigma_{1}\right)+\epsilon\mathcal{N}\left(\mu_{2},\,\sigma_{2}\right),$
(1)

where $\mathcal{N}\mu,\,\sigma)$ denotes a normal distribution of mean $\mu$
and standard deviation $\sigma$, $\mu_{1}=1.32M_{\odot}$ and
$\sigma_{1}=0.11$, $\mu_{2}=1.80M_{\odot}$, $\sigma_{2}=0.21M_{\odot}$, and
mixing fraction $\epsilon=0.35$. We inject these signals isotropically over
the sky.

We inject these signals into a NEMO detector located in Gingin, near Perth,
Australia, to determine the fraction of events detectable at a given distance.
We define a gravitational-wave detection as having an optimal matched-filter
signal-to-noise ratio greater than eight. This threshold is somewhat
arbitrarily defined—it is approximately the single-detector threshold
currently used for the LIGO/Virgo Collaborations—however a more nuanced
signal-to-noise threshold would require a detailed understanding of the
instrument’s noise properties. Regardless, the precise numerical values for
the fraction of detectable events is not important, but the overall
approximate numbers are useful to understand for the context of this paper. We
note that the number of events is $\propto 1/\textrm{SNR}^{3}$.

In the top right panel of Fig. 1, we plot the the total number of detectable
events in a single NEMO as a function of distance as the solid coloured band.
We use the current best estimate for the binary neutron star merger event rate
of $320^{+490}_{-240}\,\mathrm{Gpc^{-3}yr^{-1}}$ (Abbott et al., 2021c), where
the uncertainties are the 90% confidence intervals, which are shown as the
dashed curves in each panel of Fig. 1. This figure indicates that NEMO will
detect 95%, 65% and 5% of binary neutron star mergers at $40\,\mathrm{Mpc}$,
$100\,\mathrm{Mpc}$, and $300\,\mathrm{Mpc}$, respectively.

###  2.2 Electromagnetic observations

The 2030s will see the full promise of several current and proposed
electromagnetic observatories with all-sky or survey capabilities. These
observatories will enable routine and confident detections of kilonovae, short
gamma-ray bursts, and their afterglows beyond the horizon distance of NEMO
(e.g., Amati et al., 2018; Ivezić et al., 2019). These observatories may also
enable detection of precursor electromagnetic emission from binary neutron
star mergers as well as their post-merger central engines out to distances
relevant to NEMO (see Ascenzi et al., 2021; Sarin & Lasky, 2021, and
references therein).

In this work, we focus on whether the electromagnetic counterpart of binary
neutron star mergers are detectable, and how confidently such signals can be
associated with their gravitational-wave counterpart. We consider an
electromagnetic signal to be detectable if it is above a pre-defined threshold
for each instrument (described below). For gamma-ray burst prompt emission,
their afterglows, and kilonovae, these thresholds are relatively well
estimated for various electromagnetic telescopes (e.g., Coward et al., 2011;
Kanner et al., 2012; Siellez et al., 2014). However, the thresholds, rates,
and rate of false positives for precursor emission and post-merger emission
from the central engine are unknown (e.g., Ascenzi et al., 2021; Sridhar et
al., 2021). We therefore only focus on the relatively well observed prompt
gamma-ray emission, broadband afterglow, and kilonova.

The prompt flash of gamma rays in short gamma-ray bursts is typically followed
by a long-lasting, broadband afterglow. While the mechanism responsible for
producing the prompt gamma-ray emission is ill understood (e.g., Kumar &
Zhang, 2015), the broadband afterglow is relatively simple to understand. Most
afterglows are believed to be from the interaction of a relativistic jet with
the surrounding interstellar medium (Sari et al., 1998). There are additional
types of afterglows that we do not consider in detail here, such as those
suggesting an additional contribution from a neutron star (e.g., Zhang &
Mészáros, 2001; Rowlinson et al., 2013; Lü et al., 2015; Sarin et al., 2019).

The multi-wavelength observations of GRB170817A confirmed the long-held
suspicion that gamma-ray burst jets are structured, i.e., the energy and
Lorentz factor have a non-trivial angular dependence (Troja et al., 2017;
Alexander et al., 2018; Mooley et al., 2018a, b). While the true jet structure
is unknown, several phenomenological models have been developed and fit to
gamma-ray burst observations. One such commonly used model is the Gaussian
structured jet (e.g., Lamb et al., 2018; Ryan et al., 2019)

$E(\theta)=E_{0}e^{-\frac{\theta^{2}}{2\theta_{c}^{2}}}.$ (2)

Here, $E_{0}$ is the isotropic equivalent kinetic energy in the afterglow,
$\theta$ is the angle from the jet axis, and $\theta_{c}$ is the half-opening
angle of the ultra-relativistic core or, equivalently, the opening angle of
the jet. None of these parameters are well constrained empirically, but multi-
wavelength observations of GRB170817A and other gamma-ray bursts offer some
clues.

The ultra-relativistic jet emanating from a neutron star merger interacts with
the surrounding interstellar medium accelerating a fraction of electrons,
$\xi_{n}$, with some fraction of the total energy of the jet, $\epsilon_{e}$,
and some fraction of the energy in the magnetic field, $\epsilon_{b}$. The
radiation produced by these electrons is responsible for the observed
broadband afterglow. This interaction is relatively easy to model, but
computationally expensive. Fortunately, semi-analytic approximations make
modeling these broadband afterglows tractable. We use `afterglowpy` (Ryan et
al., 2019) with the above jet structure profile to model our afterglow
lightcurves.

The prompt emission is even less understood than the broadband afterglow.
There is no robust generative model that allows predictions or modeling of the
prompt emission energetics. Observations of the off-axis prompt emission from
GRB170817A (e.g., Matsumoto et al., 2019; Ioka & Nakamura, 2019) further
highlighted our lack of understanding of the angular structure (e.g., Howell
et al., 2019; Ioka & Nakamura, 2019). By the late 2020s/early 2030s, we will
undoubtedly have more insight into prompt energetics and angular structure. We
therefore plough forward in our endeavour, acknowledging these known unknowns.

In light of GRB170817A, one common approach is to compute the peak flux at a
given observer viewing angle using a similar structured jet profile used to
describe the afterglow (e.g., Howell et al., 2019; Salafia et al., 2019; Amati
et al., 2018). Such analyses makes two assumptions: 1) the jet structure in
the afterglow and prompt emission phase are the same, and 2) the jet is
transparent to the gamma rays at all viewing angles. Both these assumptions
could be incorrect. Alternatively, one may use a range of estimated gamma-ray
efficiencies for prompt emission inside the opening angle of the jet (e.g.,
Fong et al., 2015), and model the emission outside the jet as a cocoon-shock
breakout (e.g., Gottlieb et al., 2018). This gives a qualitatively similar
answer for prompt-emission energetics outside the jet opening angle as the
approach of Howell et al. (2019); Salafia et al. (2019). We therefore adopt
the latter approach. By the 2030s, we may better understand the energetics of
off-axis emission if we observe more multimessenger events like GW170817
(Biscoveanu et al., 2020).

All of the above ignores complications surrounding jet-launching itself.
Numerical simulations suggest that no neutron star remnant can launch a jet
that can produce a short gamma-ray burst (Murguia-Berthier et al., 2017).
However, this is in tension with x-ray afterglow observations of many short
gamma-ray bursts (e.g., Rowlinson et al., 2013). In this work, we stick with
numerical results, placing a constraint that a gamma-ray burst and
subsequently the afterglow emission is only possible when the remnant mass is
above the threshold to form a black hole i.e., $M_{\rm{rem}}\gtrsim
1.2M_{\rm{TOV}}$, where $M_{\rm{TOV}}$ is the maximum non-rotating neutron
star mass; a property of the nuclear equation of state. For each merger, we
estimate the remnant mass $M_{\rm{rem}}$ using the relationship described in
Gao et al. (2016). We then calculate whether that merger would produce a
gamma-ray burst and afterglow marginalising over piecewise polytropic
equations of state with $M_{\rm{TOV}}$ between $2.2-2.4M_{\odot}$ motivated by
current estimates (e.g., Ai et al., 2020).

Beyond the prompt gamma-ray emission and broadband afterglow, r-process
nucleosynthesis from the neutron-rich ejecta of a binary neutron star merger
is expected to power a week-long thermal transient, i.e., a kilonova. This was
likely first seen in GRB130603B (Tanvir et al., 2013; Berger et al., 2013),
but has since been studied in considerably more detail through multi-
wavelength observations of AT2017gfo, the kilonova counterpart to GW170817
(e.g., Evans et al., 2017; Smartt et al., 2017; Villar et al., 2017). There
are more kilonova models in the literature than there are observations. Many
models do a reasonable job of explaining the observations of AT2017gfo (see
Breschi et al., 2021, for a quantitative model comparison). Some models are
predominantly phenomenological while others attempt to connect the
progenitor’s parameters to the resulting lightcurve. We use the latter class
of models. In particular, we use the kilonova model from Metzger (2017)
relating it to the intrinsic binary parameters through fits of the ejecta
properties derived from numerical-relativity simulations (Dietrich & Ujevic,
2017) with a piecewise polytropic equation of state (using the software
presented in Hernandez Vivanco et al., 2020) and a baryonic mass relation
(e.g., Coughlin et al., 2017; Gao et al., 2020).

The afterglow emission is broadband, and will be detectable with various
electromagnetic telescopes and surveys at different wavelengths. However, for
the purposes of detectability here, we focus specifically on the optical
afterglow with the Vera Rubin Observatory. The $5\sigma$ magnitude threshold
for single images in Vera Rubin in $r$-band is $24.5$ with an equivalent flux
density of $4.79\times 10^{-4}\,\mathrm{mJy}$, which requires a
$30\,\mathrm{s}$ integration time (Ivezić et al., 2019). Therefore, we
consider any optical afterglow signal with an $r$-band peak flux greater than
the single-image threshold to be detectable. We use the same threshold for a
kilonova signal. For gamma-ray prompt emission, we consider a gamma-ray burst
to be detectable if the peak gamma-ray flux in the $30-150\,\mathrm{keV}$
regime is above $3\times 10^{-8}\,\mathrm{erg~{}cm^{-2}~{}s^{-1}}$. This is
comparable to the expected sensitivity of Theseus (Amati et al., 2018). We
could also use Theseus to study the detectability of the x-ray afterglow,
however the x-ray instrument proposed for Theseus does not have the wide-field
survey capability of Vera Rubin and will not serendipitously detect as many
afterglows.

We simulate the expected electromagnetic emission using the models described
above using the population of binary neutron star mergers described in Sec.
2.1. The prior ranges of the various parameters in the relevant models are
motivated by analyses of GRB170817A (Ryan et al., 2019) and AT2017gfo
(Coughlin et al., 2017) and summarised in Table 2 in Appendix B. We calculate
the number of these events that produce a detectable electromagnetic
counterpart at a given distance, which is shown in the top left panel of Fig.
1 for kilonovae, bottom left panel for prompt emission, and bottom right panel
for the broadband afterglows. The bands correspond to the uncertainties in
detectable fraction due to the uncertain model parameters, nuclear equation of
state, and binary neutron star merger rate. The absolute merger rate is shown
in each panel as the dashed curves.

Figure 1: The rate of detectable events as a function of distance with
different messengers. The dashed curves indicate the $90\%$ credible interval
of the binary neutron star merger rate (Abbott et al., 2021c), while the
coloured band represents the $90\%$ credible interval from marginalising over
the model and systematic uncertainties for each messenger.

##  3 Associating gravitational-wave and electromagnetic signals

In the following we calculate coincident detection rates under different
scenarios. We generalise the Bayesian calculation of coincidences between any
two data sets first presented in Ashton et al. (2018) to multiple data sets.
We provide details of the formalism in Appendix A.

###  3.1 A GW170817-like event

We first walk through a fiducial example of a coincident multimessenger event
akin to GW170817. We include a single NEMO-like gravitational-wave detector,
with electromagnetic observations that provide prompt gamma-ray burst
identification, broadband afterglow and kilonovae observations, and host-
galaxy identification. This is truly a jackpot event; in subsequent sections
we consider scenarios with only a subset of these observations.

We simulate an event coined GW301116, which is a face-on (inclination angle
$\iota=0$) binary neutron star merger at luminosity distance
$D_{L}=40\,\mathrm{Mpc}$ with component masses $m_{1}=1.5\,\mathrm{M_{\odot}}$
and $m_{2}=1.4\,\mathrm{M_{\odot}}$, dimensionless tidal deformabilities
$\Lambda_{1}=400$ and $\Lambda_{2}=450$, dimensionless in-plane spins
$\chi_{1,2}=0.02$, at ${\rm RA}=1.375$ and ${\rm Dec}=0.55$. We inject this
signal into a single NEMO detector using the `IMRPhenomPv2_NRTidal` (Dietrich
et al., 2019) waveform, and recover the system parameters using the same
waveform. We utilise `Parallel Bilby` (Ashton et al., 2019; Romero-Shaw et
al., 2020; Smith et al., 2020) with the `dynesty` nested sampler (Speagle,
2020). The sky map showing the $50\%$ and $90\%$ posterior credible intervals
is shown in Fig. 2, and the one-dimensional marginalised posteriors for the
time of coalescence and luminosity distance are shown in green in Figs. 3 and
4, respectively.

Figure 2: Sky localisation posterior distribution of GW301116, a GW170817-like
on-axis binary neutron star merger at 40 Mpc. The main plot shows the sky
localisation posterior from a single gravitational-wave detector, NEMO. The
inset shows the sky localisation posterior for the same event but from the
detection of the prompt gamma-ray emission form a Theseus-like observatory.

The event GW301116 being on axis and at $40\,\mathrm{Mpc}$ implies the prompt
gamma-ray signal would be observed by any telescope not occulted by the Earth
(e.g., Howell et al., 2019). We consider capabilities of future gamma-ray
telescopes SVOM (Wei et al., 2016), and Theseus (Amati et al., 2018), which
are all capable of localising the gamma-ray burst to $\lesssim 15$ arcmin
precision (Amati et al., 2018). We thus model the sky-localisation posterior
as a two-dimensional distribution mimicking current gamma-ray detectors such
as Swift using the `GWCelery` package (Singer et al., 2020). This sky map is
shown as the inset in Fig. 2.

A prompt gamma-ray detection also provides a strong constraint on the merger
time, even accounting for the uncertainty in the delay between the merger and
prompt emission signal. By the late 2020s/early 2030s, we estimate that the
uncertainty on the merger time from the identification of a prompt gamma-ray
signal will be $\mathcal{O}(2\,\mathrm{s})$. We therefore draw samples for the
time of coalescence from a uniform distribution spanning 2 s to derive a
posterior distribution for $t_{c}$, which is shown as the purple curve in Fig.
3.

Figure 3: Posterior distribution for the merger time, $t_{c}$ for GW301116
from NEMO (green) and the prompt emission (purple).

An on-axis binary neutron star merger at $40\,\mathrm{Mpc}$ will also be
detectable as a kilonova with observatories such as Vera Rubin (Ivezić et al.,
2019) and the Zwicky Transient Facility (Bellm et al., 2019), and as an
afterglow with various broadband electromagnetic telescopes. These
observations will enable the identification of the host galaxy and redshift,
as well as an indirect measurement of the merger time. The precision for sky
localisation of such an event will therefore be sub-arcsecond (Wei et al.,
2016; Ivezić et al., 2019; Amati et al., 2018). This is such a small
localisation area by comparison to gravitational waves that it is not visible
in Fig. 2. We estimate that by the late 2020s/early 2030s, the error on the
redshift will be $\Delta z\sim\mathcal{O}({10^{-4}})$. We note that the error
on the redshift for the host galaxy of GW170817, NGC4993, is
$\mathcal{O}({10^{-3}})$ (Hjorth et al., 2017; Abbott et al., 2017b), and
improvements in electromagnetic surveys and better understanding of systematic
uncertainties like peculiar velocities will improve this limit (Howlett &
Davis, 2020). We further note that for on-axis events it may be difficult to
disentangle the optical afterglow from the kilonova but we ignore this
complication. We assume a Gaussian posterior of width $\sigma_{z}=10^{-4}$ for
the redshift, which we convert to luminosity distance assuming Planck 2018
cosmology (Aghanim et al., 2020). The luminosity distance posterior is shown
in Fig. 4 as the red curve.

Figure 4: Posterior distribution for the luminosity distance $D_{L}$ for
GW301116 from NEMO (green) and a host galaxy localisation (red) with a
redshift uncertainty of $10^{-4}$.

Using these various posterior distributions, we calculate the odds using Eqs.
3 and 8 as derived in App. A. For a jackpot event like GW301116, the Bayes
factor is $4.3\times 10^{11}$. To estimate the prior odds, we consider how
many events with a short gamma-ray burst, afterglow, and kilonova we expect in
the four-dimensional combined uncertainty region (i.e., $t_{c}$, $D_{L}$, RA,
and Dec.) given the binary neutron star merger rate. For a merger rate of
$320^{+490}_{-240}\,\mathrm{Gpc^{-3}yr^{-1}}$, and uncertainties on the
coalescence time, luminosity distance, and sky position, we expect one merger
every $\approx 12$ years in the four-dimensional uncertainty region, implying
a prior odds of $9.8\times 10^{11}$. This implicitly assumes that every binary
neutron star merger at $D_{L}=40\,\mathrm{Mpc}$ will produce a detectable
prompt gamma-ray emission, afterglow or kilonova, which is true for an on-axis
merger such as GW301116, but may not be for other mergers. We explore the
effect of the inclination in Sec. 3.3. For prior odds of $9.8\times 10^{11}$,
the odds is therefore $4.2\times 10^{23}$, indicating it is $4.2\times
10^{23}$ times more likely that the observations come from the same
astrophysical event than they do not. We note that in more realistic
scenarios, the prior odds will be dictated by the rate of false positives for
a given observatory, the true astrophysical rate, and the potential for event
misidentification. We elaborate on these more realistic prior odds
calculations in the next section. We note that the above scenario of observing
an on-axis event at $D_{L}=40\,\mathrm{Mpc}$ is going to be exceedingly rare.
In the next sections we consider more common types of events.

###  3.2 Coincident detections as a function of distance

The expected detection rate of GW301116-like events that include coincident
gravitational-wave, prompt emission, afterglow, and kilonovae observations as
well as host-galaxy identification is far less than one per year. More
realistically, we expect most multimessenger gravitational-wave events to be
observed only with one of the aforementioned emissions and at varying
distances. To wit, we repeat the calculation of Sec. 3.1 but with a number of
events at different distances from $D_{L}=40\,\mathrm{Mpc}$ to
$400\,\mathrm{Mpc}$. All other parameters and the calculation remain the same
as detailed in the previous section.

The afterglow and kilonova provide only an indirect measure on the time of
coalescence $t_{c}$. The peak timescales for an on-axis merger are dictated by
the deceleration of the relativistic jet for the afterglow, and the timescale
for the expansion of the ejecta to balance the diffusion timescale for the
kilonova. These peak timescales are $\mathcal{O}(\mathrm{1-2}\,\mathrm{s})$
and $\mathcal{O}(1\,\mathrm{d})$, respectively. By the late 2020s/early 2030s,
we predict that advancements in cadence of optical telescopes and theoretical
modelling of these phenomena will enable identification of afterglows and
kilonovae within approximately $3\,\mathrm{hr}$ and $6\,\mathrm{hr}$ from the
merger time, respectively. We emphasize that this is speculation, and
realistic values will depend on detector design and survey choices too
difficult to predict. Depending on the scenario used, we take this time
uncertainty as the width of the posterior on the time of coalescence $t_{c}$.

The prior odds depend on the scenario being considered. For a jackpot event
like GW170817, i.e., an event with an observation of a short gamma-ray burst,
multi-wavelength afterglow and kilonova, the prior odds is the inverse of the
number of events expected in the given four-dimensional combined uncertainty
region given the binary neutron star merger rate itself. Given the low merger
rate, this implies a priori that if we see a confident short gamma-ray burst,
kilonova, gravitational-wave signal, and afterglow in the same four-
dimensional combined uncertainty region, it is overwhelmingly likely they are
coincident (see previous section for the calculation). The same is not true
for other scenarios. For example, the prior odds on a gamma-ray burst in
coincidence with a gravitational-wave signal is dictated by the rate of
triggers (astrophysical or otherwise) from gamma-ray telescopes. For Theseus,
this is predicted to be $1-8\,\mathrm{week^{-1}}$ (Amati et al., 2018), which
dwarfs the astrophysical rate of detectable short gamma-ray bursts themselves.

Optical telescopes will find many potential candidates for both kilonovae and
afterglows. Even if these candidates are astrophysical, they could easily be
mistaken as supernovae or afterglows of long gamma-ray bursts for example. A
realistic rate of triggers in an observatory such as Vera Rubin is not known
and will depend on the survey strategy (Setzer et al., 2019; Cowperthwaite et
al., 2019). We therefore take estimates from current wide-field optical
telescopes, using particularly the Gravitational-Wave Optical Transient
Observer (GOTO; Dyer et al., 2020), which has recently announced results from
a survey that searched for the optical afterglows of gamma-ray bursts, and had
a trigger rate of $\approx 1-20\,\mathrm{week^{-1}}$ (Mong et al., 2021). This
is significantly larger than the astrophysical rate of kilonova-like or
afterglow-like optical transients, and can be used to determine a conservative
estimate of the prior odds for scenarios involving kilonovae and afterglows.
We use these trigger estimates, alongside the rate of binary neutron star
mergers and potential contaminants such as long gamma-ray bursts and their
afterglows (Ghirlanda et al., 2021) and supernovae and fast-blue optical
transients (Ho et al., 2020), to calculate the prior odds.

We note that alongside binary neutron star mergers, some fraction of neutron
star-black hole mergers are expected to produce a kilonova, short gamma-ray
burst and afterglow. In fact, there may already be evidence of this
contamination in the observed sample of short gamma-ray bursts (Siellez et
al., 2016; Gompertz et al., 2020; Hamburg et al., 2020). However, given the
astrophysical rate of neutron star-black hole mergers is small (Abbott et al.,
2021e), ignoring contamination from neutron star-black hole mergers does not
affect our result.

The green “GW170817-like” curve in Fig. 5 shows the Bayes factor (top panel)
and odds (bottom panel) as a function of distance for events where we detect
the gravitational-wave signal as well as the prompt emission, broadband
afterglow and kilonovae, as well as identify the host-galaxy. At all
distances, the odds is overwhelmingly in favour of coincidence, going from
$8.3\times 10^{27}$ at $D_{L}=40\,\mathrm{Mpc}$, to $1.7\times 10^{24}$ at
$D_{L}=400\,\mathrm{Mpc}$, largely due to the strong a priori odds for a
coincident event in that scenario. The purple “GRB + Afterglow” curve shows
the same, however assuming that the kilonova was not observed. The red
“Afterglow + kilonova” curve assumes only the prompt emission is missed, while
the turquoise “Kilonova only” curve assumes only the gravitational-wave and
kilonova signal are detected (i.e., the prompt and afterglow are missed).

Figure 5: Bayes factor (top panel) and odds (bottom panel) as a function of
distance for coincident detections of gravitational waves and various
electromagnetic messengers. All events are on axis, with $\iota=0$. For
example, the turquoise curves show the Bayes factor and odds for coincident
detections of gravitational waves with NEMO and a kilonova signal. The blue
curve shows the coincident detection for a GW170817-like event; that is, the
coincident detection of the prompt emission, broadband afterglow, and kilonova
signal, as well as the gravitational-wave signal with NEMO. The width of the
band represents the $90\%$ credible interval from marginalising over the prior
odds.

The horizontal grey band in the bottom panel shows an odds of
$\log_{10}\mathcal{O}_{C/R}=6$, corresponding approximately to a five-sigma
threshold for the association of the gravitational-wave signal with the
electromagnetic counterpart. Using this relatively arbitrary threshold, we see
that all face on binary neutron star mergers will have confident coincidences
between the gravitational-wave and electromagnetic counterpart, with the
exception of signals that are only detected through their kilonova at distance
$D_{L}\gtrsim 250\,\mathrm{Mpc}$. We note that this threshold is somewhat
arbitrary, and deriving a realistic threshold requires understanding the
distribution of $\log_{10}\mathcal{O}_{C/R}$.

###  3.3 The effect of inclination angle

The previous two sections assume the gravitational-wave and electromagnetic
signal(s) are observed, and quantifies the confidence we could get that those
signals originate from the same source. Not all electromagnetic signals will
be detectable, in part due to the inclination angle of the source implying,
for example, the gamma-ray burst jet is pointed away from Earth. To understand
this, we look at the number of detectable gamma-ray burst, afterglow,
kilonovae, and gravitational-wave sources as a function of both distance and
inclination angle.

At fixed values of distance, we create a population of neutron star mergers
that are isotropically distributed in inclination angle (i.e., uniform in
$\cos\iota$) with the same intrinsic parameters as described in Sec. 2. We
assume the same jet structure and kilonova models as described in the previous
sections. For each event, we calculate whether the signal is detectable with
each messenger and, if detectable, whether any two or more messengers can be
confidently associated. We marginalise over systematic uncertainties due to
the unknown equation of state, jet structure and prior odds discussed in
Sec.2.2 and 3, which directly affect the expected kilonova, gamma-ray burst
signature and odds calculation, respectively.

In Fig. 6, we show the total coincident detection rate per year as a function
of luminosity distance. As an example, the top left panel shows in the
turquoise-coloured band the total detection rate for which the gravitational-
wave signal and kilonova can be confidently identified (above a $5-\sigma$
threshold) as coming from the same source, marginalising over all previously-
mentioned uncertainties, as well as the uncertainty in the overall merger
rate. This coloured band can be contrasted with the total binary neutron star
merger rate, where the 90% confidence intervals are shown as the dashed curves
in each panel.

Figure 6: The number of coincident events as a function of distance with
$\log_{10}\mathcal{O_{C/R}}\gtrsim 6$ corresponding to five sigma confidence.
For example, the top left panel shows the total rate of confident coincident
detections that can be made between gravitational-wave and kilonova signals.
The dashed curves indicate the $90\%$ credible interval of the binary neutron
star merger rate, and the coloured band represents the $90\%$ credible
interval from marginalising over the model uncertainty and prior odds.

One can inspect the curves in Fig. 6 to derive an overall merger fraction
$\mathcal{F}$ of mergers that will be detected with gravitational waves and
various electromagnetic messengers, as well as the overall rate $\mathcal{R}$
of detections. These numbers are provided in Tab. 1. These numbers show
$14^{+25}_{-11}\,\mathrm{yr^{-1}}$ (or $\approx 40\%$) of binary neutron star
mergers per year will have confident associations between their independently-
identified electromagnetic signatures and gravitational-wave signal with only
a NEMO detector.

We note that the above numbers, in particular the fraction of binary neutron
star mergers with any confident association, should be taken with some
caution. These numbers are sensitive to three assumptions: 1) The true
distribution of $\log_{10}\mathcal{O}_{C/R}$ is distributed in a way such that
our threshold for a confident association is too weak. 2) The modelling
assumptions and priors for the electromagnetic models are too optimistic. 3)
The fraction $\mathcal{F}$ represents the fraction of mergers detectable with
both an electromagnetic telescope and with NEMO that can also be confidently
associated with one another. However, whether a given electromagnetic
telescope can actually detect a counterpart (even if it is above the threshold
for detection) will depend on the survey strategy and specifics of the
detector. These strategies and detector specifics are too difficult to predict
for observatories that do not yet exist, and a calculation of the true
distribution of $\log_{10}\mathcal{O}_{C/R}$ is not computationally feasible
as it requires simulating the complete population multiple times to create a
distribution of $\log_{10}\mathcal{O}_{C/R}$ for each binary neutron star
merger. However, both of these factors will likely be better understood in the
near future, accompanied by a better understanding of electromagnetic models.

| prompt | prompt |  |
---|---|---|---|---
| afterglow | afterglow | afterglow |
| kilonova |  | kilonova | kilonova
$\mathcal{F}$ ($<300\,\mathrm{Mpc}$) | $0.096^{+0.072}_{-0.068}$ | $0.096^{+0.072}_{-0.068}$ | $0.13^{+0.13}_{-0.09}$ | $0.38^{+0.3}_{-0.01}$
$\mathcal{R}$ (${\rm yr}^{-1}$) | $3^{+11}_{-2}\,\mathrm{yr^{-1}}$ | $3^{+11}_{-2}\,\mathrm{yr^{-1}}$ | $4^{+18}_{-3}\,\mathrm{yr^{-1}}$ | $13^{+23}_{-10}\,\mathrm{yr^{-1}}$
Table 1: Expected fraction $\mathcal{F}$ and rate $\mathcal{R}$ of coincident
gravitational-wave and electromagnetic observations using a single NEMO-like
detector. The fraction measures the fraction of binary neutron star mergers
occurring at a luminosity distance of less than $300\,\mathrm{Mpc}$ that have
an identified electromagnetic counterpart. For example, the first column shows
the fraction (first row) and number (second row) of expected mergers for which
the gravitational-wave detection can be confidently associated with
observations of both prompt emission, afterglow, and kilonova. We note that
all systems detected through their prompt emission will also be detected as
afterglows. For comparison, a single NEMO-like detector will detect the
gravitational-wave signal from $\mathcal{F}=0.45$ mergers out to
$300\,\mathrm{Mpc}$, corresponding to an event rate of
$16^{+26}_{-12}\,\mathrm{yr^{-1}}$.

##  4 Conclusions

The future of gravitational-wave astronomy relies on sequential technological
and infrastructure upgrades, as well as the development of objectively
expensive new facilities (albeit far less than some space-based infrared
telescopes). One plausible scenario sees a period of time for which a
designated kilohertz gravitational-wave detector like the proposed NEMO
instrument (Ackley et al., 2020) is the only operational ground-based
observatory. In this work, we show that such an instrument has significant
scientific utility on its own due to the multimessenger capabilities expected
in the same era. For example, in operation with the Vera Rubin observatory
(Ivezić et al., 2019) and Theseus (Amati et al., 2018), we show that
approximately $40\%$ of binary neutron star mergers out to $300\,\mathrm{Mpc}$
can be confidently identified as multimessenger coincident observations. Given
current estimates for the local merger rate of neutron stars (Abbott et al.,
2021c), this implies a total number of $14^{+25}_{-11}\,\mathrm{yr^{-1}}$
multimessenger detections per year.

Coincident multimessenger observations of binary neutron star mergers will
enable several precise tests into the nature of nuclear matter. The $\gtrsim
500\,\mathrm{Hz}$ gravitational-wave sensitivity enables simultaneous
measurements of the neutron star’s masses and tidal deformability, which can
be further constrained when coupled with kilonovae observations that inform
the ejecta mass, and hence the progenitor’s mass ratio and complementary
information about the equation of state (e.g., Metzger, 2017). For nearby
mergers (see Ackley et al., 2020) this will be combined with measurements of
gravitational waves from the hot post-merger remnant, which can further inform
the equation of state (e.g., Bauswein et al., 2019), including the potential
to discern temperature-dependent phase transitions (Bauswein et al., 2016).
Combining gravitational-wave information from the inspiral and post-merger
remnant with constraints on the jet-launching timescale from timing of the
prompt emission (Ren et al., 2020; Beniamini et al., 2020b) can constrain the
maximum mass of hot and cold neutron stars (e.g., Chatziioannou, 2020; Sarin &
Lasky, 2021); also a key indicator of the nuclear equation of state. This
information can also be determined from complementary observations of, for
example, the colour and energetics of the kilonova (Margalit & Metzger, 2019)
and the duration of x-ray plateaus from surviving supramassive neutron stars
(Rowlinson et al., 2013; Ravi & Lasky, 2014; Sarin et al., 2020).

Associating gamma-ray bursts and kilonovae with gravitational-wave signals
will also enrich our understanding of the physics driving these enigmatic
transients. In particular, constraints on the delay time between merger and
prompt emission will deepen our understanding of ultra-relativistic jet
launching and propagation (Ren et al., 2020). This will also shed insight into
the prompt emission mechanism. Detection of post-merger gravitational waves
combined with prompt emission observations will provide a smoking-gun
observation into the contentious issue of whether a black-hole central engine
is required to launch an ultra-relativistic jet capable of generating short
gamma-ray burst emission. This will also provide an opportunity to understand
the impact of the remnant on r-process enrichment (e.g., Perego et al., 2014;
Metzger, 2017; Bernuzzi, 2020). Coincident multimessenger observations will
also improve our understanding of the jet structure and beaming of gamma-ray
burst jets (e.g., Beniamini et al., 2020a; Biscoveanu et al., 2020).

The capabilities of survey telescopes in the 2030s will enable host-galaxy
identification of neutron star mergers, and hence precise redshift
measurements (e.g., Howlett et al., 2017). Required peculiar-velocity
measurements will be enabled by next-generation spectroscopic surveys (e.g.,
Palmese et al., 2019; Howlett & Davis, 2020), and possibly percent-level
measurements of the Hubble constant with single events (Coughlin et al., 2020;
Calderón Bustillo et al., 2021).

Dedicated kilohertz gravitational-wave observatories have the power to produce
valuable science without the requirement of living in a global, heterogeneous
network of other gravitational-wave observatories, provided the capabilities
of electromagnetic telescopes do not get worse. In this paper, we show this is
true only considering the guaranteed science case of binary neutron star
mergers. Neutron star-black hole binaries, millisecond pulsars, and supernovae
are all targets for NEMO-like observatories that would provide rich
multimessenger information about the physics and astrophysics of these extreme
objects. However, the unknown neutron star-black hole tidal-disruption rate,
the unknown ellipticity of rapidly-rotating isolated neutron stars, and the
unknown amplitude of gravitational waves from supernovae prohibits us from
making concrete statements on the likelihood of coincident detections.
Regardless, we know that the science outcomes from such events would likely be
bountiful.

###### Acknowledgements.

We are grateful to David Ottaway and Eric Howell for their thoughtful comments
on the manuscript. This work was supported through Australian Research Council
(ARC) Centre of Excellence CE170100004, ARC Future Fellowship FT160100112, and
ARC Discovery Project DP180103155. This work was performed on the OzSTAR
national facility at Swinburne University of Technology. The OzSTAR program
receives funding in part from the Astronomy National Collaborative Research
Infrastructure Strategy (NCRIS) allocation provided by the Australian
Government.

##  Appendix A Nitty-gritty formalism

We wish to determine the probability that observed electromagnetic and
gravitational-wave signals are coming from the same source. This involves two
questions: 1) are the two events coincident in space and time, and 2) is it
physically possible/probable that the signals were generated in the same
event. In a Bayesian sense, the former question is quantified by the Bayes
factor, while the latter is quantified by the prior odds. Together, they
enable us to calculate the odds that any two or more pieces of data were
generated by the same source.

To determine whether two or more independent observations are coincident
(i.e., come from the same physical source), we generalise the Bayesian
formalism of Ashton et al. (2018) from two to $N$ coincident data sets. We
calculate the odds $\mathcal{O}_{C/R}$ (Ashton et al., 2018; Ashton et al.,
2020), where $C$ refers to the hypothesis that the two independent detections
share a common origin and $R$ refers to the hypothesis that they are random.
This is given by

$\mathcal{O}_{C/R}\equiv\mathcal{B}_{C/R}\pi_{C/R},$ (3)

where

$\mathcal{B}_{C/R}\equiv\frac{P\left(C\mid
d_{1},d_{2},\ldots,d_{N},I\right)}{P\left(R\mid
d_{1},d_{2},\ldots,d_{N},I\right)},$ (4)

is the Bayes factor, ${P\left(C\mid d_{1},d_{2},\ldots,d_{N},I\right)}$ is the
probability of a common hypothesis conditional on data sets
$d_{1},d_{2},\ldots,d_{N}$, while the denominator refers to the random
hypothesis conditioned on the same data. The second term in Eq. 3, $\pi_{C/R}$
is the a priori probability of a coincident event, which conservatively can be
estimated as one over the number of expected events in the given overlapping
region.

For a given gravitational-wave and electromagnetic transient to share a common
origin, they must have occurred at the same location in the sky, at the same
distance, and within some coincident time-span as specified by theoretical
models. This implies they must have the same sky-location parameters $\Omega$,
same luminosity distance $D_{L}$, and same inferred time of coalescence
$t_{c}$. In the case of observations with a common source, some other
parameters will also be the same (e.g., source inclination). However, given
theoretical limitations in current electromagnetic models, particularly in
connecting to the progenitor parameters measured by the gravitational-wave
signal, we ignore these common parameters. The confidence gained by including
these additional parameters is proportional to the information gained on these
parameters relative to the prior.

Following the framework from Ashton et al. (2018) the numerator of the Bayes
factor is,

$\begin{split}P\Big{(}C\mid&d_{1},d_{2},\ldots,d_{N},I\Big{)}\\\ &=\int
P\left(C,\theta\mid d_{1},d_{2},\ldots,d_{N},I\right)\pi(\theta\mid
C)~{}d\theta.\end{split}$ (5)

Here, $\theta$ is the set of parameters common to all data sets,
$\pi(\theta\mid C)$ is the common-hypothesis prior on $\theta$, and

$P\left(C,\theta\mid
d_{1},d_{2},\ldots,d_{N},I\right)=\prod\limits_{i=1}^{N}\frac{P\left(d_{i}\mid
C\right)P\left(\theta\mid d_{i},C\right)}{P\left(\theta\mid C\right)}.$ (6)

Substituting Eq. 6 into Eq. 5 gives,

$\mathcal{O}_{C/R}=\pi_{C/R}\mathcal{I}_{\theta},$ (7)

where $\mathcal{I}_{\theta}$ is the posterior overlap integral for the
parameter $\theta$ generalised for $N$ different data sets

$\mathcal{I}_{\theta}=\int\prod_{i=1}^{N}p\left(\theta\mid
d_{i},C\right)\pi(\theta\mid C)^{1-N}d\theta.$ (8)

We now move from the generalised abstract formalism described above to our
specific study. For example, we consider the scenario of three different data
sets: a gravitational-wave signal, kilonova, and afterglow. The overlap
integral can be written as

$\mathcal{I}_{\theta}=\int\frac{p\left(\theta\mid
d_{\mathrm{gw}},C\right)p\left(\theta\mid
d_{\mathrm{kn}},C\right)p\left(\theta\mid
d_{\mathrm{ag}},C\right)}{\pi(\theta\mid C)^{2}}d\theta,$ (9)

where $d_{\mathrm{gw}}$, $d_{\mathrm{kn}}$, and $d_{\mathrm{ag}}$ are the
gravitational-wave, kilonova and afterglow data, respectively. The odds given
our common parameters $\theta=\\{D_{L},\,\Omega,\,t_{c}\\}$ is

$\mathcal{O}_{C/R}\approx\pi_{C/R}\mathcal{I}_{D_{L}}\mathcal{I}_{\Omega}\mathcal{I}_{t_{c}},$
(10)

In main body of the Paper we discuss what constraints a detection of
kilonovae, prompt gamma-ray and afterglow emission and a gravitational-wave
signal detected by a kHz-band gravitational-wave observatory can provide on
these joint parameters. We use the framework described above to determine how
confidently we can associate a given electromagnetic counterpart to the
gravitational-wave signal. We note that since these transients are
independently identified, we do not rely on low-latency estimates to calculate
the odds and associate events.

##  Appendix B Priors

Here we list the priors used to model the prompt emission, afterglow, and
kilonova signal from a binary neutron star merger. These priors are motivated
by analysis of the afterglow of GRB170817A (Ryan et al., 2019) and the
kilonova AT2017gfo (Coughlin et al., 2017). The priors are displayed in Tab.
2.

Table 2: Parameters used in our modelling of the electromagnetic counterparts to binary neutron star mergers along with a brief description and the prior. Parameter [unit] | Description | Prior
---|---|---
$\Gamma_{0}$ | initial Lorentz factor | $\textrm{Uniform}[100,1000]$
$\log_{10}(E_{\textrm{iso}}/\rm{erg})$ | isotropic-equivalent energy | $\textrm{Uniform}[52,54]$
$\theta_{\textrm{core}}$ [rad] | half-width of jet core | $\textrm{Uniform}[0.1,0.2]$
$\log_{10}(n_{\textrm{ism}}/\mathrm{cm^{-3}})$ | number density of ISM | $\textrm{Uniform}[-4,1]$
$p$ | electron distribution power-law index | $\textrm{Uniform}[2,3]$
$\log_{10}\epsilon_{e}$ | thermal energy fraction in electrons | $\textrm{Uniform}[-3,0]$
$\log_{10}\epsilon_{b}$ | thermal energy fraction in magnetic field | $\textrm{Uniform}[-4,0]$
$\xi_{N}$ | fraction of accelerated electrons | $\textrm{Uniform}[0,1]$
$\log_{10}p_{c}$ | neutron star central pressure | $\textrm{Uniform}[32.6,33.6]$
$\Gamma_{1,2,3}$ | polytropic indices | $\textrm{Uniform}[1.1,4.5]$
$\beta$ | ejecta velocity distribution power-law slope | $\textrm{Uniform}[3,5]$
$\kappa$ | kilonova ejecta opacity | $\textrm{Uniform}[1,10]$
$M_{\rm{TOV}}$ [$M_{\odot}$] | Maximum non-rotating neutron star mass | $\textrm{Uniform}[2.2,2.4]$

## References

  * Aasi et al. (2015) Aasi J., Abbott B. P., Abbott R., Abbott T., Abernathy M. R., et al., 2015, Classical and Quantum Gravity, 32, 074001
  * Abbott et al. (2017a) Abbott B. P., Abbott R., Abbott T. D., Abernathy M. R., Ackley K., et al., 2017a, Classical and Quantum Gravity, 34, 044001
  * Abbott et al. (2017b) Abbott B. P., Abbott R., Abbott T. D., Acernese F., Ackley K., et al., 2017b, Nature, 551, 85
  * Abbott et al. (2018) Abbott B. P., Abbott R., Abbott T. D., Abernathy M. R., et al. 2018, Living Reviews in Relativity, 21, 3
  * Abbott et al. (2020) Abbott R., Abbott T. D., Abraham S., Acernese F., Ackley K., Adams C., Adhikari R. X., et al., 2020, ApJ, 896, L44
  * Abbott et al. (2021a) Abbott T. D., Acernese F., Ackley K., Adams C., Adhikari N., Adhikari R. X., Adya V. B., et al., 2021a, arXiv e-prints, p. arXiv:2108.01045
  * Abbott et al. (2021b) Abbott R., et al., 2021b, Physical Review X, 11, 021053
  * Abbott et al. (2021c) Abbott R., Abbott T. D., Abraham S., Acernese F., et al. 2021c, Physical Review X, 11, 021053
  * Abbott et al. (2021d) Abbott R., et al., 2021d, ApJ, 915, L5
  * Abbott et al. (2021e) Abbott R., et al., 2021e, Astrophys. J. Lett., 915, L5
  * Acernese et al. (2015) Acernese F., Agathos M., Agatsuma K., Aisa D., Allemandou N., et al., 2015, Classical and Quantum Gravity, 32, 024001
  * Ackley et al. (2020) Ackley K., et al., 2020, PASA, 37, e047
  * Adhikari et al. (2020) Adhikari R. X., et al., 2020, Classical and Quantum Gravity, 37, 165003
  * Aghanim et al. (2020) Aghanim N., Akrami Y., Ashdown M., Aumont J., Baccigalupi C., Ballardini M., Banday A. J., et al., 2020, A&A, 641, A6
  * Ai et al. (2020) Ai S., Gao H., Zhang B., 2020, ApJ, 893, 146
  * Akutsu et al. (2019) Akutsu T., Ando M., Arai K., Arai Y., Araki S., Araya A. o., 2019, Nature Astronomy, 3, 35
  * Alexander et al. (2018) Alexander K. D., Margutti R., Blanchard P. K., Fong W., Berger E., others. 2018, ApJ, 863, L18
  * Alsing et al. (2018) Alsing J., Silva H. O., Berti E., 2018, MNRAS, 478, 1377
  * Amati et al. (2018) Amati L., et al., 2018, Advances in Space Research, 62, 191
  * Ascenzi et al. (2021) Ascenzi S., Oganesyan G., Branchesi M., Ciolfi R., 2021, Journal of Plasma Physics, 87, 845870102
  * Ashton et al. (2018) Ashton G., et al., 2018, ApJ, 860, 6
  * Ashton et al. (2019) Ashton G., et al., 2019, ApJS, 241, 27
  * Ashton et al. (2020) Ashton G., Ackley K., Magaña Hernandez I., Piotrzkowski B., 2020, arXiv e-prints, p. arXiv:2009.12346
  * Barcons et al. (2017) Barcons X., Barret D., Decourchelle A., den Herder J. W., et al. 2017, Astronomische Nachrichten, 338, 153
  * Bauswein et al. (2016) Bauswein A., Stergioulas N., Janka H.-T., 2016, European Physical Journal A, 52, 56
  * Bauswein et al. (2019) Bauswein A., Bastian N.-U. F., Blaschke D. B., Chatziioannou K., Clark J. A., et al., 2019, Phys. Rev. Lett., 122, 061102
  * Bellm et al. (2019) Bellm E. C., Kulkarni S. R., Graham M. J., Dekany R., et al. 2019, PASP, 131, 018002
  * Beniamini et al. (2020a) Beniamini P., Duque R., Daigne F., Mochkovitch R., 2020a, MNRAS, 492, 2847
  * Beniamini et al. (2020b) Beniamini P., Duran R. B., Petropoulou M., Giannios D., 2020b, ApJ, 895, L33
  * Berger et al. (2013) Berger E., Fong W., Chornock R., 2013, ApJ, 774, L23
  * Bernuzzi (2020) Bernuzzi S., 2020, arXiv e-prints, p. arXiv:2004.06419
  * Biscoveanu et al. (2020) Biscoveanu S., Thrane E., Vitale S., 2020, ApJ, 893, 38
  * Breschi et al. (2021) Breschi M., Perego A., Bernuzzi S., Del Pozzo W., Nedora V., Radice D., Vescovi D., 2021, MNRAS, 505, 1661
  * Calderón Bustillo et al. (2021) Calderón Bustillo J., Leong S. H. W., Dietrich T., Lasky P. D., 2021, ApJ, 912, L10
  * Chambers et al. (2016) Chambers K. C., Magnier E. A., Metcalfe N., Flewelling H. A., et al. 2016, arXiv e-prints, p. arXiv:1612.05560
  * Chatziioannou (2020) Chatziioannou K., 2020, General Relativity and Gravitation, 52, 109
  * Chen et al. (2021) Chen H.-Y., Cowperthwaite P. S., Metzger B. D., Berger E., 2021, ApJ, 908, L4
  * Cherenkov Telescope Array Consortium et al. (2019) Cherenkov Telescope Array Consortium Acharya B. S., Agudo I., Al Samarai I., et al. 2019, Science with the Cherenkov Telescope Array, doi:10.1142/10986.
  * Coughlin et al. (2017) Coughlin M., Dietrich T., Kawaguchi K., Smartt S., Stubbs C., Ujevic M., 2017, ApJ, 849, 12
  * Coughlin et al. (2020) Coughlin M. W., et al., 2020, Nature Communications, 11, 4129
  * Coward et al. (2011) Coward D. M., et al., 2011, MNRAS, 415, L26
  * Cowperthwaite et al. (2019) Cowperthwaite P. S., Villar V. A., Scolnic D. M., Berger E., 2019, ApJ, 874, 88
  * Dietrich & Ujevic (2017) Dietrich T., Ujevic M., 2017, Classical and Quantum Gravity, 34, 105014
  * Dietrich et al. (2019) Dietrich T., Khan S., Dudi R., Kapadia S. J., et al. 2019, Phys. Rev. D, 99, 024029
  * Dobie et al. (2021) Dobie D., Murphy T., Kaplan D. L., Hotokezaka K., Bonilla Ataides J. P., Mahony E. K., Sadler E. M., 2021, MNRAS, 505, 2647
  * Duque et al. (2019) Duque R., Daigne F., Mochkovitch R., 2019, A&A, 631, A39
  * Dyer et al. (2020) Dyer M. J., et al., 2020, Ground-based and Airborne Telescopes VIII
  * Evans et al. (2017) Evans P. A., et al., 2017, Science, 358, 1565
  * Fong et al. (2015) Fong W., Berger E., Margutti R., Zauderer B. A., 2015, ApJ, 815, 102
  * Foucart (2020) Foucart F., 2020, Frontiers in Astronomy and Space Sciences, 7, 46
  * Gao et al. (2016) Gao H., Zhang B., Lü H.-J., 2016, Phys. Rev. D, 93, 044065
  * Gao et al. (2020) Gao H., Ai S.-K., Cao Z.-J., Zhang B., Zhu Z.-Y., Li A., Zhang N.-B., Bauswein A., 2020, Frontiers of Physics, 15, 24603
  * Gardner et al. (2006) Gardner J. P., Mather J. C., Clampin M., Doyon R., et al. 2006, Space Sci. Rev., 123, 485
  * Gehrels et al. (2004) Gehrels N., Chincarini G., Giommi P., Mason K. O., et al. 2004, ApJ, 611, 1005
  * Ghirlanda et al. (2021) Ghirlanda G., Salvaterra R., Toffano M., Ronchini S., et al. 2021, Experimental Astronomy,
  * Gillingham et al. (2020) Gillingham P., Cooke J., Glazebrook K., Mould J., Smith R., Steidel C., 2020, in Advances in Optical Astronomical Instrumentation 2019\. p. 112030F, doi:10.1117/12.2540717
  * Gompertz et al. (2020) Gompertz B. P., Levan A. J., Tanvir N. R., 2020, ApJ, 895, 58
  * Gottlieb et al. (2018) Gottlieb O., Nakar E., Piran T., Hotokezaka K., 2018, MNRAS, 479, 588
  * Hamburg et al. (2020) Hamburg R., Fletcher C., Burns E., Goldstein A., et al. 2020, ApJ, 893, 100
  * Haskell & Schwenzer (2021) Haskell B., Schwenzer K., 2021, arXiv e-prints, p. arXiv:2104.03137
  * Hernandez Vivanco et al. (2020) Hernandez Vivanco F., Smith R., Thrane E., Lasky P. D., 2020, MNRAS, 499, 5972
  * Hjorth et al. (2017) Hjorth J., et al., 2017, ApJ, 848, L31
  * Ho et al. (2020) Ho A. Y. Q., Perley D. A., Kulkarni S. R., Dong D. Z. J., et al. 2020, ApJ, 895, 49
  * Howell et al. (2019) Howell E. J., Ackley K., Rowlinson A., Coward D., 2019, MNRAS, 485, 1435
  * Howlett & Davis (2020) Howlett C., Davis T. M., 2020, MNRAS, 492, 3803
  * Howlett et al. (2017) Howlett C., Staveley-Smith L., Blake C., 2017, MNRAS, 464, 2517
  * Ioka & Nakamura (2019) Ioka K., Nakamura T., 2019, MNRAS, 487, 4884
  * Ivezić et al. (2019) Ivezić Ž., et al., 2019, ApJ, 873, 111
  * Johns et al. (2012) Johns M., et al., 2012, in Stepp L. M., Gilmozzi R., Hall H. J., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 8444, Ground-based and Airborne Telescopes IV. p. 84441H, doi:10.1117/12.926716
  * Kanner et al. (2012) Kanner J., Camp J., Racusin J., Gehrels N., White D., 2012, ApJ, 759, 22
  * Kumar & Zhang (2015) Kumar P., Zhang B., 2015, Phys. Rep., 561, 1
  * Lamb et al. (2018) Lamb G. P., Mandel I., Resmi L., 2018, MNRAS, 481, 2581
  * Lasky (2015) Lasky P. D., 2015, PASA, 32, e034
  * Law et al. (2015) Law N. M., Fors O., Ratzloff J., Wulfken P., et al. 2015, PASP, 127, 234
  * Lü et al. (2015) Lü H.-J., Zhang B., Lei W.-H., Li Y., Lasky P. D., 2015, ApJ, 805, 89
  * Margalit & Metzger (2019) Margalit B., Metzger B. D., 2019, ApJ, 880, L15
  * Martynov et al. (2019) Martynov D., Miao H., Yang H., Vivanco F. H., Thrane E., et al., 2019, arXiv e-prints,
  * Matsumoto et al. (2019) Matsumoto T., Nakar E., Piran T., 2019, MNRAS, 483, 1247
  * Merloni et al. (2012) Merloni A., et al., 2012, arXiv e-prints, p. arXiv:1209.3114
  * Metzger (2017) Metzger B. D., 2017, Living Reviews in Relativity, 20, 3
  * Mezzacappa et al. (2020) Mezzacappa A., Marronetti P., Landfield R. E., Lentz E. J., et al. 2020, Phys. Rev. D, 102, 023027
  * Miller et al. (2015) Miller J., Barsotti L., Vitale S., Fritschel P., Evans M., Sigg D., 2015, Phys. Rev. D, 91, 062005
  * Mong et al. (2021) Mong Y. L., Ackley K., Galloway D. K., Dyer M., et al. 2021, arXiv e-prints, p. arXiv:2108.11802
  * Mooley et al. (2018a) Mooley K. P., Nakar E., Hotokezaka K., Hallinan G., Corsi A., et al., 2018a, Nature, 554, 207
  * Mooley et al. (2018b) Mooley K. P., Deller A. T., Gottlieb O., Nakar E., Hallinan G., et al., 2018b, Nature, 561, 355
  * Most & Philippov (2020) Most E. R., Philippov A. A., 2020, ApJ, 893, L6
  * Murguia-Berthier et al. (2017) Murguia-Berthier A., et al., 2017, ApJ, 835, L34
  * Palmese et al. (2019) Palmese A., et al., 2019, BAAS, 51, 310
  * Pan et al. (2021) Pan K.-C., Liebendörfer M., Couch S. M., Thielemann F.-K., 2021, ApJ, 914, 140
  * Perego et al. (2014) Perego A., Rosswog S., Cabezón R. M., Korobkin O., Käppeli R., Arcones A., Liebendörfer M., 2014, MNRAS, 443, 3134
  * Powell & Müller (2020) Powell J., Müller B., 2020, MNRAS, 494, 4665
  * Punturo et al. (2010) Punturo M., Abernathy M., Acernese F., Allen B., Andersson N., Arun K., et al., 2010, Classical and Quantum Gravity, 27, 194002
  * Ravi & Lasky (2014) Ravi V., Lasky P. D., 2014, MNRAS, 441, 2433
  * Ren et al. (2020) Ren J., Lin D.-B., Zhang L.-L., Wang K., Li X.-Y., Wang X.-G., Liang E.-W., 2020, ApJ, 901, L26
  * Romero-Shaw et al. (2020) Romero-Shaw I. M., Talbot C., Biscoveanu S., D’Emilio V., et al. 2020, MNRAS, 499, 3295
  * Rowlinson et al. (2013) Rowlinson A., O’Brien P. T., Metzger B. D., Tanvir N. R., Levan A. J., 2013, MNRAS, 430, 1061
  * Ryan et al. (2019) Ryan G., van Eerten H., Piro L., Troja E., 2019, arXiv e-prints, p. arXiv:1909.11691
  * Salafia et al. (2019) Salafia O. S., Ghirlanda G., Ascenzi S., Ghisellini G., 2019, A&A, 628, A18
  * Sari et al. (1998) Sari R., Piran T., Narayan R., 1998, ApJ, 497, L17
  * Sarin & Lasky (2021) Sarin N., Lasky P. D., 2021, General Relativity and Gravitation, 53, 59
  * Sarin et al. (2019) Sarin N., Lasky P. D., Ashton G., 2019, ApJ, 872, 114
  * Sarin et al. (2020) Sarin N., Lasky P. D., Ashton G., 2020, Phys. Rev. D, 101, 063021
  * Sarin et al. (2022) Sarin N., Lasky P. D., Vivanco F. H., Stevenson S. P., Chattopadhyay D., Smith R., Thrane E., 2022, arXiv e-prints, p. arXiv:2201.08491
  * Setzer et al. (2019) Setzer C. N., Biswas R., Peiris H. V., Rosswog S., Korobkin O., Wollaeger R. T., LSST Dark Energy Science Collaboration 2019, MNRAS, 485, 4260
  * Siellez et al. (2014) Siellez K., Boër M., Gendre B., 2014, MNRAS, 437, 649
  * Siellez et al. (2016) Siellez K., Boer M., Gendre B., Regimbau T., 2016, arXiv e-prints, p. arXiv:1606.03043
  * Singer et al. (2020) Singer L. P., et al., 2020, GWCelery, https://git.ligo.org/emfollow/gwcelery
  * Skidmore et al. (2015) Skidmore W., TMT International Science Development Teams Science Advisory Committee T., 2015, Research in Astronomy and Astrophysics, 15, 1945
  * Smartt et al. (2017) Smartt S. J., Chen T. W., Jerkstrand A., Coughlin M., Kankare E., et al., 2017, Nature, 551, 75
  * Smith et al. (2020) Smith R. J. E., Ashton G., Vajpeyi A., Talbot C., 2020, MNRAS, 498, 4492
  * Speagle (2020) Speagle J. S., 2020, MNRAS, 493, 3132
  * Sridhar et al. (2021) Sridhar N., Zrake J., Metzger B. D., Sironi L., Giannios D., 2021, MNRAS, 501, 3184
  * Tanvir et al. (2013) Tanvir N. R., Levan A. J., Fruchter A. S., Hjorth J., Hounsell R. A., Wiersema K., Tunnicliffe R. L., 2013, Nature, 500, 547
  * Troja et al. (2017) Troja E., Piro L., van Eerten H., Wollaeger R. T., Im M., et al., 2017, Nature, 551, 71
  * Villar et al. (2017) Villar V. A., Guillochon J., Berger E., Metzger B. D., Cowperthwaite P. S., et al., 2017, ApJ, 851, L21
  * Wei et al. (2016) Wei J., Cordier B., Antier S., Antilogus P., et al. 2016, arXiv e-prints, p. arXiv:1610.06892
  * Weisskopf et al. (2000) Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O’Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2–16 (arXiv:astro-ph/0004127), doi:10.1117/12.391545
  * Woan et al. (2018) Woan G., Pitkin M. D., Haskell B., Jones D. I., Lasky P. D., 2018, ApJ, 863, L40
  * Zhang & Mészáros (2001) Zhang B., Mészáros P., 2001, ApJ, 552, L35