# Radio Afterglows from Compact Binary Coalescences: Prospects for Next-
Generation Telescopes

Dougal Dobie,1,2,3,4 Tara Murphy1,3 David L. Kaplan,5 Kenta Hotokezaka,6,7
Juan Pablo Bonilla Ataides,1 Elizabeth K. Mahony,2 Elaine M. Sadler1,2
1Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW
2006, Australia
2ATNF, CSIRO Astronomy and Space Science, PO Box 76, Epping, NSW 1710,
Australia
3ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav), Hawthorn,
Victoria, Australia
4Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria, Australia
5Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI
53201, USA
6Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane,
Princeton, NJ 08544, USA
7Research Center for the Early Universe, Graduate School of Science,
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
E-mail<EMAIL_ADDRESS>(DD)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The detection of gravitational waves from a neutron star merger, GW170817,
marked the dawn of a new era in time-domain astronomy. Monitoring of the radio
emission produced by the merger, including high-resolution radio imaging,
enabled measurements of merger properties including the energetics and
inclination angle. In this work we compare the capabilities of current and
future gravitational wave facilities to the sensitivity of radio facilities to
quantify the prospects for detecting the radio afterglows of gravitational
wave events. We consider three observing strategies to identify future mergers
– widefield follow-up, targeting galaxies within the merger localisation and
deep monitoring of known counterparts. We find that while planned radio
facilities like the Square Kilometre Array will be capable of detecting
mergers at gigaparsec distances, no facilities are sufficiently sensitive to
detect mergers at the range of proposed third-generation gravitational wave
detectors that would operate starting in the 2030s.

###### keywords:

gravitational waves – radio continuum: transients – stars: neutron –
telescopes

††pubyear: 2021††pagerange: Radio Afterglows from Compact Binary Coalescences:
Prospects for Next-Generation Telescopes–References

##  1 Introduction

The 2015 discovery of gravitational waves produced by a binary black hole
merger by the Laser Interferometer Gravitational Wave Observatory (LIGO)
kickstarted a new era of astronomy (Abbott et al., 2016a). Despite
comprehensive follow-up observations carried out using telescopes across the
electromagnetic spectrum, no electromagnetic counterpart was detected (Abbott
et al., 2016c, d), nor were any coincident neutrinos (Adrián-Martínez et al.,
2016). Almost two years later LIGO/Virgo detected gravitational waves from a
neutron star merger (Abbott et al., 2017b), which was accompanied by the
contemporaneous detection of a short gamma-ray burst (Abbott et al., 2017c;
Goldstein et al., 2017). Approximately 10 hours later a candidate optical
counterpart was discovered (Coulter et al., 2017; Soares-Santos et al., 2017;
Valenti et al., 2017; Arcavi et al., 2017a; Tanvir et al., 2017; Lipunov et
al., 2017), while X-ray and radio emission was detected 9 and 16 days post-
merger respectively (Troja et al., 2017; Hallinan et al., 2017). The delay in
detecting the X-ray and radio emission stems from different physical origins –
the optical kilonova is produced by the decay of elements produced via
r-process nucleosynthesis in the merger, while the X-ray and radio emission
originates from the relativistic outflow launched by the merger interacting
with the surrounding medium (Metzger & Berger, 2012). Although none have been
detected to-date, the slow moving material in the kilonova may eventually give
rise to late-time radio emission (e.g. Hotokezaka et al., 2018; Kathirgamaraju
et al., 2019), while the synchrotron emission from the relativistic outflow
was later detected at optical wavelengths (Lyman et al., 2018; Fong et al.,
2019; Lamb et al., 2019a).

Continued radio monitoring of GW170817 revealed a lightcurve that gradually
rose over the following months (Alexander et al., 2017b; Mooley et al., 2018a;
Margutti et al., 2018), before peaking approximately 150 days post-merger
(Dobie et al., 2018) and rapidly fading (Alexander et al., 2018; Troja et al.,
2018; Mooley et al., 2018d). This comprehensive monitoring campaign with
radio, optical and X-ray telescopes has allowed tight constraints to be placed
on the spectral and temporal evolution of the non-thermal afterglow (Hajela et
al., 2019; Makhathini et al., 2020; Troja et al., 2020; Hajela et al., 2020).
In turn, these constraints have enabled physical properties of the merger,
including the energetics, circum-merger density and inclination angle, to be
inferred (e.g. Nakar et al., 2018). Further constraints on the outflow
geometry were obtained through Very Long Baseline Interferometry (VLBI)
observations (Mooley et al., 2018b; Ghirlanda et al., 2019), while the non-
detection of linearly polarised radio emission constrains properties of the
magnetic field of the outflow (Corsi et al., 2018).

There has been extensive research into the detectability of the radio
afterglow of compact object mergers (e.g. Feng et al., 2014; Hotokezaka et
al., 2016; Lazzati et al., 2017; Duque et al., 2019; Gottlieb et al., 2019;
Lin & Totani, 2020). The joint detection of GW170817 and GRB170817A confirmed
the relationship between neutron star mergers and short GRBs (Beniamini et
al., 2019; Wu & MacFadyen, 2019) that has been predicted for decades (Eichler
et al., 1989; Narayan et al., 1992), and therefore studies into the
detectability of short GRB radio afterglows are also of relevance (e.g.
Ghirlanda et al., 2013, 2014; Metzger et al., 2015; Zhang et al., 2015). That
being said, the intricacies of this relationship are not yet clear, as the GRB
accompanying GW170817 was an outlier compared to the standard population and
the gamma-ray emission likely did not originate from the core of the jet
(Matsumoto et al., 2019b, a; Salafia et al., 2019).

In this work we build upon previous studies by quantifying prospects for
detecting afterglows with all major current and future GHz-frequency radio
facilities using observing strategies tailored to the specifications of each
facility. We have expanded our study beyond current gravitational wave
detectors, and compare the maximum detectable distance to the range of
detectors that will be built over the coming decades. In Section 2 we describe
the afterglow model we use in this work, and summarise the other forms of
radio emission that may be produced by mergers. In Section 3 we summarise
potential observing strategies for radio follow-up and the ability of existing
and planned radio telescopes to carry out those observations. In Section 4 we
outline the detection and localisation capabilities of existing and planned
gravitational wave detectors. Section 5 includes a discussion of the benefits
and limitations of radio follow-up, and quantifies prospects for the detection
of radio counterparts using the previously described afterglow model and
observing strategies.

##  2 Afterglow Physics

###  2.1 Emission from a power-law jet

To assess the detectability of radio afterglows we use the same power-law jet
model as Dobie et al. (2020). The kinetic energy is distributed according to

$E(\theta)=\frac{E_{\rm iso}}{1+(\theta/\theta_{j,c})^{3.5}},$ (1)

where $\theta$ is the polar angle from the jet axis, $\theta_{j,c}$ and
$E_{\rm iso}$ are the half opening angle and isotropic-equivalent energy of
the core of the jet respectively. Here we assume $\theta_{j,c}=0.05\,{\rm
rad}$, $E_{\rm iso}=10^{52}\,{\rm erg}$ based on the afterglow of GW170817
(Mooley et al., 2018b; Ghirlanda et al., 2019; Makhathini et al., 2020; Troja
et al., 2020). The power law profile of $E\propto\theta^{-3.5}$ is motivated
by the result of a hydrodynamic simulation by Gottlieb et al. (2021). We use
the standard synchrotron afterglow model of Sari et al. (1998) to calculate
the radio flux produced by the radial expansion of a jet into a uniform
medium. The initial Lorentz factor of the jet is given by

$\Gamma(\theta)=1+\frac{\Gamma_{c}}{1+(\theta/\theta_{j,c})^{5}},$ (2)

where $\Gamma_{c}$ is the Lorentz factor of the jet’s core. Based on the
population of short GRB afterglows (Fong et al., 2015) and the afterglow of
GW170817 (e.g. Troja et al., 2020; Hajela et al., 2019; Lamb et al., 2020) we
assume that the electron energy distribution follows a power law with index
$p=2.16$. We also assume the fraction of shock energy distributed to the
electrons and magnetic field of the jet to be $\epsilon_{e}=0.1$ and
$\epsilon_{B}=0.01$ respectively, based on the canonical GRB microphysics
parameters (e.g. Berger, 2014, and references therein), which we note are
higher than those typically found for GW170817 (see Table 1 of Dobie et al.,
2020). Our results rely on these parameter assumptions, which are biased
towards GW170817-like events and therefore may not represent the general
population of compact binary coalescences.

We initially prepared a set of afterglow lightcurves $S_{\nu,0}$ as a function
of observer time $t_{0}$ for observing angles from 10–90 $\deg$ for fixed
$E_{\rm iso}$, $n_{0}$, $\theta_{j,c}$ and microphysics parameters. We then
calculate $S_{\nu}(t)$ for different values of $n$ by using the scaling
relations

$S_{\nu}(t)=\left(\frac{n}{n_{0}}\right)^{(p+1)/4}S_{\nu,0},~{}~{}~{}~{}~{}~{}~{}~{}t=\left(\frac{n}{n_{0}}\right)^{-1/3}t_{0}$
(3)

(see Granot & Sari, 2002; van Eerten et al., 2012). Here we have assumed that
the emission frequency lies below the synchrotron cooling frequency and above
the characteristic frequency of the lowest energy electrons and self-
absorption frequency (i.e. $\nu_{m},\nu_{s}\ll\nu\ll\nu_{c}$). Population
modelling (e.g. Duque et al., 2019) suggests that this is a valid assumption
for the GHz-frequency range, which is the focus of this work.

While we only explicitly consider fixed $E_{\rm iso}$ and microphysics
parameters in this work, for completeness we note that the lightcurves obey

$S_{\nu}(t)\propto E_{\rm
iso}(\epsilon_{B}n)^{(p+1)/4}\epsilon_{e}^{p-1},~{}~{}~{}~{}~{}t=\left(\frac{n_{0}E_{\rm
iso}}{nE_{{\rm iso},0}}\right)^{1/3}t_{0},$ (4)

which can be used to scale the results of this work to generalised energetics
and microphysics parameters. We note that this model does not consider
synchrotron self-absorption, and as such, the predicted flux density is
overestimated for $n>1$ cm-3 and $\theta_{\rm obs}<10\,\deg$. We therefore do
not consider parameters outside of these limits in this work.

To scale the flux density from the nominal emission frequency of our model,
$\nu_{0}$, to a general observing frequency, $\nu$, we assume the emission
obeys a simple power law with spectral index $\alpha=(1-p)/2$ (Sari et al.,
1998). Since the model flux density, $S_{0}$, is calculated in the rest frame
of the merger we also apply the standard K-correction (see e.g. Novak et al.,
2017), resulting in a frequency-corrected flux density of

$S=\frac{S_{0}}{(1+z)^{\alpha}}\left(\frac{\nu}{\nu_{0}}\right)^{\alpha}$ (5)

where we convert between redshift, $z$, and luminosity distance, $D_{L}$,
using the cosmological parameters in Hinshaw et al. (2013) implemented in
astropy.cosmology.WMAP9. We emphasise that this model is by no means
comprehensive – it is a single, generalised, model for one of many components
of the radio emission that may be produced by compact binary mergers (e.g.
Hotokezaka & Piran, 2015).

###  2.2 Other forms of emission

At early times ($t\lesssim 10$ days post-merger) the dominant source of radio
emission may be the ‘reverse shock’ (Sari & Piran, 1995; Resmi & Zhang, 2016;
Lamb & Kobayashi, 2019), which propagates from the outflow towards the site of
the merger. This emission is dependent on similar parameters to the forward
shock, including the inclination angle. Reverse-shock emission has been
detected in long GRBs (e.g. Kulkarni et al., 1999; Kobayashi, 2000; Kobayashi
& Sari, 2000; Laskar et al., 2013; Perley et al., 2014; Alexander et al.,
2017a; Laskar et al., 2019; Rhodes et al., 2020) but may be difficult to
detect in short GRBs due to the required follow-up latency (Lloyd-Ronning,
2018). However, observations of GRB 160821B suggest that a reverse shock
component is responsible for part of the early time radio emission (Troja et
al., 2019; Lamb et al., 2019b). We stress that even if reverse shock emission
may be difficult to detect as a distinct component, its contribution to the
overall emission may be significant enough that forward-shock jet models
(including the one we use in this paper) alone may not accurately describe the
early-time lightcurve evolution.

While most mergers likely launch a jet, the fate of the jet is dependent on
factors including the energetics of the merger and the density of the
surrounding environment. As the jet propagates into the surrounding ejecta, it
may form a cocoon, which will produce a distinct signal as it expands and
breaks out of the ejecta (Nagakura et al., 2014; Murguia-Berthier et al.,
2014; Lazzati et al., 2017; Nakar & Piran, 2017; Gottlieb et al., 2018).
Beyond the initial rise and decline, this geometry may also produce a double-
peak in the lightcurve (Barkov et al., 2018).

Radio emission may also be detectable from kilonovae, which produced the
early-time optical and infrared emission associated with GW170817 (Pian et
al., 2017; Kasen et al., 2017; Smartt et al., 2017). This emission is produced
by the ejecta associated with the kilonova which expands at sub-relativistic
velocity and is therefore expected to be fainter, and peak at later times,
than the non-thermal synchrotron emission. The timescale of this peak ranges
from months to decades and the peak luminosity is similarly uncertain (e.g.,
Nakar & Piran, 2011; Piran et al., 2013; Hotokezaka et al., 2018;
Kathirgamaraju et al., 2019; Margalit & Piran, 2020). The kilonova ejecta are
expected to expand quasi-isotropically, and therefore this form of radio
emission may be detectable from future mergers even when the emission
associated with the relativistic ejecta is not detected due to beaming.

Table 1: Capabilities of existing and planned radio facilities including
observing frequency ($\nu$), bandwidth ($\Delta\nu$), field of view
($\Omega$), angular resolution ($\theta$). See Section 3 for details.

Facility | Band | $\nu$ | $\Delta\nu$ | $\Omega$ | $\theta$ | Dec. limit
---|---|---|---|---|---|---
|  | (GHz) | (GHz) | (deg2) | (arcsec) | (deg)
ATCAa | C/X | $8.0$ | $8.0$ | $0.01$ | $2$ | $<+30$
GMRT | B3 | $0.4$ | $0.2$ | $1.4$ | $8$ | $>-50$
| B4 | $0.7$ | $0.3$ | $0.4$ | $4$ | $>-50$
VLAb | L | $1.5$ | $1.0$ | $0.12$ | $6$ | $>-30$
| S | $3.0$ | $1.5$ | $0.06$ | $2.7$ | $>-30$
| C | $6.0$ | $4.0$ | $0.01$ | $1.3$ | $>-30$
Apertif | L | $1.4$ | $0.3$ | $6$ | $15$ | $>-20$
ASKAP | Band 1 | $0.9$ | $0.3$ | $30$ | $15$ | $<+30$
MeerKAT | L | $1.4$ | $0.7$ | $0.8$ | $7$ | $<+30$
DSA-2000 | – | $1.35$ | $1.3$ | $10.6$ | $3.5$ | $>-30$
SKA-1 | Band 2 | $1.43$ | $0.4$ | $0.8$ | $0.6$ | $<+30$
ngVLA | Band 1 | $2.4$ | $2.3$ | $0.13$ | $0.002$ | $>-30$
SKA-2 | Band 2 | $1.43$ | $0.4$ | $0.8$ | $0.6$ | $<+30$

  * a

Assumes a 6 km array configuration

  * b

Assumes B configuration

##  3 Radio Telescope Capabilities

In this section we outline the follow-up capabilities of radio
interferometers. We exclude low-frequency ($\leq 300$ MHz) telescopes from
this discussion as our model does not account for synchrotron self-absorption
and is therefore not accurate at these frequencies. Table 1 lists the
specifications of each telescope in our analysis.

Figure 1: Comparison of lightcurve properties for a range of inclination
angle, $\theta$, and circum-merger density, $n$ (in units of cm-3), assuming a
merger distance of 40 Mpc and an observing frequency of 1.4 GHz for the merger
parameters outlined in Section 2. Top: Lightcurves for a range of inclination
angles, with $n=10^{-2}\,$cm-3, corresponding to typical short GRB circum-
burst density. Bottom: Lightcurves for a range of circum-merger densities,
with $\theta=30\,\deg$.

###  3.1 Follow-up Strategies

The localisation areas of previously detected gravitational wave events span
tens to thousands of square degrees (Abbott et al., 2019a, 2020a) and while
the addition of more detectors to the network will lead to smaller
localisation areas for events at comparable distances, improvements to
detector sensitivity will also lead to the detection of more distant events
with comparable localisation areas (Abbott et al., 2020b; Hall & Evans, 2019).
Therefore the largest impediment to detecting an electromagnetic counterpart
is the ability to cover sufficient area to a sufficient depth. Below we
outline possible observing strategies for detecting a long-lasting (months-
years), sub-mJy radio afterglow as predicted by theoretical models (e.g.
Hotokezaka et al., 2016) and observed from GW170817 (e.g. Hajela et al., 2019;
Makhathini et al., 2020; Troja et al., 2020). The search for other forms of
emission (e.g. an optical counterpart or a coherent radio burst) may require
different strategies.

####  3.1.1 Targeting known galaxies

One survey strategy adopted in the follow-up of GW170817 was to target known
galaxies in the localisation volume (e.g., Andreoni et al., 2017; Arcavi et
al., 2017a; Coulter et al., 2017; Díaz et al., 2017; Evans et al., 2017;
Valenti et al., 2017). This was also the predominant strategy used in radio
follow-up, as at the time no facilities had sufficient survey speed to perform
an unbiased search for a radio counterpart.

Currently the best catalogues for this purpose include the Galaxy List for the
Advanced Detector Era (GLADE; Dálya et al., 2018), Census of the Local
Universe (CLU; Cook et al., 2019), Photometric Redshifts for the Legacy
Surveys (PRLS; Zhou et al., 2021), the WISE $\times$ SuperCOSMOS Photometric
Redshift Catalog (Bilicki et al., 2016), and the 2MASS photometric redshift
galaxy catalogue (2MPZ; Bilicki et al., 2014). GLADE and CLU are compilations
of existing surveys (and encompass observations from 2MPZ), while PRLS uses
the Dark Energy Camera Legacy Survey (Dey et al., 2019) and data from the
Wide-field Infrared Survey Explorer (WISE; Wright et al., 2010). Additionally,
the Mass AssociatioN for GRavitational waves ObserVations Efficiency
(MANGROVE; Ducoin et al., 2020) combines GLADE with estimates of stellar mass
from WISE, and Artale et al. (2020b) compute host probabilities from estimates
of stellar mass and star formation rate.

Numerous works explore the prospects for this search method – for example,
Gehrels et al. (2016) demonstrates that simple galaxy targeting can reduce the
required number of pointings by a factor of 10–100 compared to an unbiased
search, while Coughlin et al. (2018) find that search efficiency can further
be improved by a factor of 2. Some strategies expand on basic galaxy targeting
by preferring to target galaxies that are closer (Arcavi et al., 2017b), or by
using assumptions on afterglow emission (Coughlin & Stubbs, 2016; Salafia et
al., 2017). Telescopes with wider fields of view may be able to observe
multiple galaxies in a single pointing, and this follow-up may be optimised by
convolving the gravitational wave skymap with a galaxy catalogue (Evans et
al., 2016).

Once we have obtained a larger sample of localised mergers we will be able to
use the population of known host galaxies to weight galaxy catalogues by more
robust metrics than simple mass-analogues. While observational constraints are
currently limited by a lack of events, detailed studies of the host galaxy of
GW170817 were carried out (Blanchard et al., 2017; Im et al., 2017; Levan et
al., 2017; Pan et al., 2017; Ebrová et al., 2020; Contini, 2018) and
simulations are already revealing potential relationships between merger and
host properties (e.g. Toffano et al., 2019; Artale et al., 2019; Artale et
al., 2020a). Radio observations will have a role to play in these efforts, as
they are useful in determining the presence of AGN and thereby determine
whether stellar mass or star formation is the driving factor of merger
formation (Zheng & Ramirez-Ruiz, 2007; Corsi et al., 2017). They can also
measure star formation rate in galaxies with optical obscuration (Perley &
Perley, 2013).

Ultimately, the limiting factor of this technique is the completeness of the
survey catalogue, which for current surveys (e.g. Dálya et al., 2018; Cook et
al., 2019) is lacking at distances comparable to the planned LIGO detector
range of $330\,$Mpc (Abbott et al., 2020b). While PRLS is deeper than the
other wide surveys (with a limiting magnitude of $\sim 20.4$ subject to other
selection criteria), it only covers $\sim 16\%$ of the sky and therefore
cannot be utilised for all events. We discuss the issue of galaxy catalogue
completeness and its impact on follow-up in 5.1.4. For more distant events it
is therefore natural to consider searches that only use the localisation
obtained from the gravitational wave signal.

####  3.1.2 Unbiased searches

Telescopes with larger survey speeds are capable of performing unbiased
searches for radio counterparts by observing entire localisation regions. Here
we consider unbiased searches to be feasible for telescopes with a survey
speed larger than $10\,\deg^{2}\,$hr-1 for a detection threshold of 0.5 mJy.
This choice reflects the requirement to cover a substantial fraction of a
localisation area (typically tens–hundreds of square degrees) to a
constraining sensitivity (given that afterglows are expected to peak at
tens–hundreds of $\mu$Jy) in a reasonable amount of time. This technique has
the obvious advantage of not being limited by the completeness of existing
galaxy catalogues, while on the other hand, can result in coverage of a large
amount of extraneous area and an increase in false-positives (which we discuss
in Section 5.1.2). However, this can be partly mitigated by restricting
candidates to those associated with known galaxies within the localisation
volume.

There are multiple ways to optimise an unbiased search (e.g. Ghosh et al.,
2016, 2017; Dobie et al., 2019a; Gupte & Bartos, 2020) but for the purposes of
this work we simply assume that a telescope can observe a given sky area
without any consideration of exact observing strategies.

####  3.1.3 Monitoring known electromagnetic counterparts

The final type of follow-up involves searching for radio emission from known
counterparts detected at other wavelengths, which allows for maximised
sensitivity as all available telescope time can be allocated to a single
pointing.

The detection of a known counterpart can also help inform the required cadence
and sensitivity of radio observations. Localising the counterpart to a galaxy
provides accurate distance measurements (Hjorth et al., 2017) which can also
be used to infer tighter constraints on the merger inclination angle
independent of any afterglow modelling (Mandel, 2018). The detection of a
short GRB can loosely constrain the merger energetics (Abbott et al., 2017c;
Goldstein et al., 2017), and a combination of multi-wavelength detections can
be used to broadly infer the overall geometry of the merger (Kasliwal et al.,
2017).

Even if a radio counterpart is not detected, radio observations are still
useful in placing constraints on the spectral properties of the synchrotron
afterglow (in combination with optical and X-ray follow-up) and inferring
properties of the host galaxy. Single-dish spectral line observations can also
be used to constrain the average density of the host galaxy and infer the
density of the environment surrounding the merger (Hallinan et al., 2017).
X-ray observations can also be used as independent confirmation of these
estimates (e.g. Hajela et al., 2019).

###  3.2 Existing Facilities

####  3.2.1 Australia Telescope Compact Array (ATCA)

The Australia Telescope Compact Array is an East-West array consisting of six
22-m dishes with a maximum baseline of 6 km (depending on the array
configuration). Each dish is equipped with a set of receivers that can sample
frequencies from 1.1 to 105 GHz, with the System Equivalent Flux Density
(SEFD) increasing towards higher frequencies (Wilson et al., 2011). In this
paper we consider observations in the C/X band (4–12 GHz) which is generally
the most sensitive due to the high Radio Frequency Interference (RFI)
occupancy at lower
frequencies111https://www.narrabri.atnf.csiro.au/observing/rfi/monitor/rfi_monitor.html##atca.

Currently correlation is performed using the Compact Array Broadband Backend
(Wilson et al., 2011), which allows observations in $2\times 2048$ MHz
windows. However, the Broadband Integrated-GPU Correlator for the Australia
Telescope (BIGCAT) upgrade, which will be completed by December 2021, will
double the available bandwidth to 8 GHz. As this upgrade will occur on a
similar timeline to the start of the next LIGO/Virgo observing run, in this
paper we assume that the upgrade is successfully commissioned and therefore
the sensitivity will improve by a factor of $\sqrt{2}$ compared to the current
level.

It is not feasible to carry out unbiased follow-up with the ATCA due to its
low survey speed, even using mosaicing. However, it is useful for targeted
follow-up, either of galaxies within the localisation volume of the
gravitational wave event, or of counterparts detected at other wavelengths.
Taking into account observing overheads, it’s possible to observe $\sim 50$
galaxies to a detection threshold of $<70\,\mu$Jy in a typical 12-hour
observation222The ATCA is an East-West array and requires a full 12 hour track
for complete uv coverage. (12 minutes per source), while a detection threshold
of $\sim 15\,\mu$Jy can be achieved for follow-up of a single source. We note
that in many cases this can be improved by optimising pointings such that
multiple galaxies are observed within the same field of
view333https://github.com/ddobie/atca-ligo, but to simplify the estimates in
this paper we assume that we can simply observe 50 galaxies with 1 per
pointing.

####  3.2.2 Karl. G. Jansky Very Large Array (VLA)

The Karl. G. Jansky Very Large Array (VLA) consists of twenty seven 25-m
antennas with maximum baselines ranging from 36.4 km (A configuration) to 1.03
km (D configuration). In this paper we consider the B configuration which has
baselines spanning 0.21–11.1 km, although the A and C (which have maximum
baselines of 36.4 and 3.4 km respectively) configurations would also be
suitable . The telescope can observe at frequencies from 58 MHz to 50 GHz, but
in this case we consider observations at S-band (3 GHz; offering the best
sensitivity for the expected negative spectral index) and L-band (1.5 GHz;
offering lower sensitivity but a twice-larger field of view).

Rana & Mooley (2019) propose an optimised galaxy targeting strategy for the
VLA, improving the probability of detecting a radio counterpart by a factor of
two compared to a simple approach. This strategy will enable observers to
cover approximately 200 galaxies to a sensitivity of $15\,\mu$Jy. The VLA can
also carry out unbiased searches using on-the-fly mosaicing, where the
antennas are driven at a constant rate along a strip of sky, eliminating slew
overheads which are dominated by the settling time of $\sim 7$ seconds per
pointing. This technique has already been applied to follow-up of two
gravitational wave events, GW151226 (Abbott et al., 2016b) and GW190814
(Abbott et al., 2020c) covering $100\,\deg^{2}$/10% (Mooley et al., 2018c) and
$5\,\deg^{2}$/50% (Mooley et al., 2019) of the localisation regions
respectively.

####  3.2.3 Giant Metrewave Radio Telescope (GMRT)

The Giant Metrewave Radio Telescope (GMRT) consists of thirty 45-m antennas
with a maximum baseline of 25 km. Recent upgrades to the receivers give the
GMRT unparalleled sensitivity at sub-GHz frequencies, and here we consider
observations in bands 3 and 4, centered on 400 and 700 MHz respectively.
Observations in these lower bands are vital in understanding the spectral
evolution of the afterglow, in particular, placing constraints on the
evolution of the synchrotron self-absorption frequency.

Here we consider the GMRT as a dedicated follow-up instrument for localised
mergers, as its combination of sensitivity and field of view is not conducive
to unbiased searches. We consider 3 hour observations in both Band 3 and 4,
corresponding to an image sensitivity of 15 and 20$\,\mu$Jy respectively. This
choice of exposure time is to ensure both bands can be observed in a typical
GMRT session of $\sim 6$ hours.

####  3.2.4 Arcminute Microkelvin Imager Large Array (AMI-LA)

The Arcminute Microkelvin Imager Large Array (AMI-LA) consists of eight 12.8m
antennas with a maximum baseline of 110 m operating at an observing frequency
of 15 GHz. The AMI-LA Rapid Response Mode (ALARRM; Staley et al., 2013)
enables the telescope to respond to GRB alerts within 2 minutes of the burst,
allowing for tight constraints to be placed on early-time emission (e.g. Lamb
& Kobayashi, 2019). However, the AMI-LA field of view limits the practicality
of early-time follow-up of neutron star mergers to those with a simultaneous
detection of a GRB by the Neil Gehrels Swift Observatory. While AMI-LA can
also be used to monitor known counterparts, providing additional spectral
coverage and achieving a sensitivity of $25\,\mu$Jy in a 3 hour observation,
we do not consider it in our detectability analysis in Section 5.2, and omit
it from Table 1, due to the superior sensitivity of the VLA at these
frequencies.

####  3.2.5 Australian Square Kilometre Array Pathfinder (ASKAP)

The Australian Square Kilometre Array Pathfinder (ASKAP; Johnston et al.,
2008; Hotan et al., 2021) is an array of thirty-six 12-m dishes with baselines
ranging from 37 m to 6 km. ASKAP is designed for all-sky surveys between
700-1800 MHz (in this case we consider observations at 900 MHz to maximise the
sensitivity) and uses MkII phased-array feeds (Hampson et al., 2012)
consisting of 36 beams resulting in a $30\,\deg^{2}$ field of view. While the
SEFD is higher than other comparable telescopes, the large field of view
results in a high survey speed, making it possible to search large areas of
sky for a gravitational wave counterpart. However, this does mean that ASKAP
is not useful for monitoring events where an electromagnetic counterpart has
already been discovered. We therefore only consider the utility of ASKAP for
localising events, and not monitoring them.

Similarly, ASKAP is not particularly useful for a galaxy-targeted search
strategy since tens–hundreds of candidate host galaxies can be covered with a
single pointing. Some widefield telescopes select their pointing strategy by
convolving the localisation skymap with galaxy catalogues (e.g., Evans et al.,
2016) but in Dobie et al. (2019a) we demonstrated that this strategy does not
produce any appreciable benefits for ASKAP follow-up. In this work we consider
two strategies, a single deep $\sim 10$ hour pointing (achieving a noise of
$35\,\mu$Jy) or a widefield strategy consisting of four $\sim 3$ hour
pointings (achieving a noise of $110\,\mu$Jy) for events localised to
$<100\,\deg^{2}$ (corresponding to $\sim 60\%$ of neutron star mergers during
O4, and the majority of mergers detected with 3G detectors Abbott et al.,
2020b; Hall & Evans, 2019). So far ASKAP has performed follow-up of S190510g
and GW190814 (Dobie et al., 2019b, Stewart et al. in prep.) and both times the
latter pointing strategy was used. We note that due to the comparably low
angular resolution of ASKAP, some afterglows may be contaminated by emission
from the host galaxy. We discuss this problem in Section 5.1.3.

####  3.2.6 Apertif

The Westerbork Synthesis Radio Telescope (WSRT) is an East-West array
consisting of fourteen 25-m antennas with a maximum baseline of 2.7 km. Twelve
of the antennas have been fitted with L-band (operating from 1–1.75 GHz) PAFs
as part of the APERture Tile In Focus (Apertif) project, improving the
telescopes field of view to $9.5\,\deg^{2}$ (Oosterloo et al., 2010; Adams &
van Leeuwen, 2019). Similar to ASKAP, Apertif’s wide field of view means it is
more suited to widefield searches rather than targeted follow-up, while the
observing frequency and maximum baseline mean it is also subject to the same
host galaxy contamination issues. The sensitivity of Apertif is comparable to
ASKAP, and we consider follow-up with one deep 12 hour pointing achieving a
sensitivity of 25 $\mu$Jy or four pointings achieving a sensitivity of 50
$\mu$Jy.

####  3.2.7 MeerKAT

MeerKAT (Jonas & MeerKAT Team, 2016) consists of sixty four 13.5-m diameter
antennas with a maximum baseline of 8 km fitted with 1.4 GHz receivers and has
a $\sim 1\,\deg^{2}$ field of view. The highly sensitive receivers have a
system equivalent flux density of $\sim 430\,$Jy (Mauch et al., 2020), making
MeerKAT a suitable facility for searching for emission from known
counterparts. Here we consider two follow-up strategies – a single deep 10
hour pointing achieving a sensitivity of $\sim 2\,\mu$Jy and ten 1 hour
pointings covering $10\,\deg^{2}$ to a sensitivity of $\sim 7\,\mu$Jy. The
former strategy applies to very well localised events (comparable to
expectations for third generation detectors, see 4.2) and monitoring of known
counterparts, while the latter strategy applies to unbiased follow-up of
events localised to $\lesssim 10\,\deg^{2}$ or targeted follow-up of less-
localised events using a galaxy catalogue convolution strategy (Evans et al.,
2016).

###  3.3 Future Facilities

####  3.3.1 Square Kilometre Array (SKA)

The Square Kilometre Array (SKA) will be the worlds largest radio telescope,
with a mid-frequency array in South Africa and a low-frequency array in
Western Australia. In this paper we focus on the mid-frequency array, which
will be split into two stages.

SKA-1 mid will consist of the existing MeerKAT array and an additional one
hundred and thirty three 15-m dishes, with the array expected to come online
in the mid-2020s. Braun et al. (2019) outlines the anticipated array
performance, and for the purposes of this paper we consider observations in
Band 2, centered on 1.43 GHz as a compromise between maximising sensitivity
and the expected negative spectral index of gravitational wave afterglows. We
assume the same observing strategy as MeerKAT, with a 1 hour continuum
sensitivity of $2\,\mu$Jy.

We also consider observations with the SKA-2. While the design specifications
for the SKA-2 are still uncertain, here we assume an order of magnitude
sensitivity improvement over SKA-1 for the same observing strategies.

####  3.3.2 Next Generation Very Large Array (ngVLA)

The Next Generation Very Large Array (ngVLA; Murphy et al., 2018) is a planned
replacement for the VLA operating between 1.2 and 116 GHz. The main array will
consist of two hundred and fourteen 18-m dishes on baselines of up to 1000 km.
We consider observations in the lowest frequency band, centered on 2.4 GHz due
to the larger field of view and higher relative sensitivity (Selina et al.,
2018). It is currently anticipated that Early Science observations will begin
in 2028 with full operation from 2034
onwards444https://ngvla.nrao.edu/page/faq#faq_16_content.

The ngVLA will be capable of performing unbiased follow-up of well-localised
events, observing $10\,\deg^{2}$ to a sensitivity of $1\,\mu$Jy in 10 hours
(Corsi et al., 2019). The ngVLA has comparable instantaneous sensitivity to
the SKA, making it the premier northern hemisphere facility for targeted
follow-up of known counterparts. The ngVLA is also capable of observing at
frequencies up to 93 GHz, compared to the SKA which will only observe up to
12.5 GHz (Braun et al., 2019). This will allow a better characterisation of
the spectral properties of afterglows in conjunction with optical and X-ray
telescopes, as was performed for GW170817 (Makhathini et al., 2020; Troja et
al., 2020).

####  3.3.3 Deep Synoptic Array-2000 (DSA-2000)

The Deep Synoptic Array (DSA-2000) is a proposed telescope that will consist
of two thousand 5-m dishes capable of simultaneously observing between 0.7-2
GHz and is scheduled to be fully operational by 2026 (Hallinan et al., 2019).
The telescope is optimised for survey speed, with a 10.6 deg2 instantaneous
field of view and 2.5 Jy SEFD. Approximately 1 hour per day will be allocated
to follow-up of gravitational wave events, with the focus being on well-
localised events that can be covered with a single pointing. In these cases,
the entire localisation region can be covered to a $5\sigma$ detection
threshold of $\sim 5\,\mu$Jy. Similar to ASKAP, we also consider a widefield
strategy of twelve $\sim 5$ minute pointings for follow-up of events localised
to $\sim 100\,\deg^{2}$.

While the DSA-2000 can cover a $10\,\deg^{2}$ localisation ten times faster
than the SKA or ngVLA, it is important to note that both telescopes can cover
that area more efficiently. Gravitational wave localisations are generally
irregular shapes (although this will be less true as we move to arrays with >3
detectors) and often multi-modal, while telescope fields of view are either
circular or rectangular. Ghosh et al. (2016) find that it is significantly
more efficient to cover localisation regions with a distributed group of
multiple small-FoV telescopes than a single widefield telescope due to the
lower extraneous coverage. Similarly, the SKA/ngVLA will use many small-FoV
pointings and will therefore achieve more efficient coverage than the DSA-2000
strategy of fewer large-FoV pointings.

###  3.4 Serendipitous Observations

Advances in radio telescope technology will allow for numerous widefield
surveys of the radio sky to be undertaken in the coming decades. Deep all-sky
surveys will provide sensitive reference images for transient follow-up, and
widefield transient searches will likely provide serendipitous coverage of
gravitational wave events.

####  3.4.1 ASKAP

The Evolutionary Map of the Universe (EMU; Norris et al., 2011) will cover the
sky South of $+30\,\deg$ declination to a design sensitivity ($1\sigma$ noise)
of $\sim 10\,\mu$Jy at 1.3 GHz. This will provide the most sensitive map of
the Southern radio sky to-date and be useful as a reference image for
transient searches. The Rapid ASKAP Continuum Survey (RACS; McConnell et al.,
2020) achieves a typical sensitivity of $250\,\mu$Jy at $900\,$MHz and has
already been used as a reference image in follow-up of GW190814 (Dobie et al.,
2019b).

The proposed Variables And Slow Transients (VAST; Murphy et al., 2013) survey
is split into three main components, Wide, Deep and Galactic, and will span at
least 5 years. VAST-Wide will observe an area of $10000\,\deg^{2}$ to a
detection threshold of $2.5\,$mJy on a daily cadence. VAST-Deep will achieve a
detection threshold of $250\,\mu$Jy and will observe $10000\,\deg^{2}$ 7
times, and a single $30\,\deg^{2}$ field daily. VAST-Galactic will observe
$750\,\deg^{2}$ of the galactic plane 64 times to a detection threshold of
$500\,\mu$Jy, which will be useful for events where optical follow-up is
hindered by extinction and a high rate of unrelated transients. Here we
consider the VAST-Wide and low cadence VAST-Deep surveys, as the wide areal
coverage makes them more conducive to this kind of search. We also note that
the ASKAP Survey Science observing strategy has not yet been finalised

####  3.4.2 MeerKAT

While MeerKAT has a higher instantaneous sensitivity than ASKAP, there are
currently no plans to use it for a dedicated widefield untargeted transients
survey (Fender et al., 2017). Instead, transient searches will be conducted
using commensal data from other surveys, which cover relatively small areas of
sky (generally less than 30$\deg^{2}$, see Holwerda et al., 2012; Serra et
al., 2016; de Blok et al., 2016; Bailes et al., 2018). Therefore the proposed
transients search with MeerKAT will search a smaller area of sky to a greater
sensitivity compared to ASKAP. As this is less conducive to serendipitous
coverage of multi-messenger events, we instead consider an idealised
untargeted survey for radio transients covering 5000 $\deg^{2}$ observed to a
sensitivity of $20\,\mu$Jy with 9 observations separated by 4 months. This
corresponds to a total observing time of 3750 hours.

####  3.4.3 Deep Synoptic Array

The Cadenced All-Sky Survey (Hallinan et al., 2019) will observe 16 epochs of
the sky North of $-30\,\deg$ on a 4-month cadence to a detection threshold of
$10\,\mu$Jy, providing coverage of the majority of gravitational wave events.
It will also ultimately provide a reference image with an rms noise of 500
nJy.

Table 2: Capabilities of gravitational wave detector networks made of the
Hanford (H), Livingston (L), Virgo (V), Kagra (K), LIGO-India (I) detectors.
Detectors improved by the A+ upgrade are denoted by a subscript $+$ while
LIGO-Voyager detectors are denoted by a subscript $V$.

Epoch | Facilities | Timeline | Rangea | Localisationb | Ratec
---|---|---|---|---|---
|  |  | (Mpc) | ($\deg^{2}$) | (yr-1)
O4 | HLVK | 2022–23 | 190d | 35 | 10
O5 | H+L+V+K | 2025–26 | 330d | 35 | 50
2G | H+L+V+KI+ | 2026 | 330 | 35 | 50
Voy. | HVLVVV | 2030 | 1100 | 70 | 1800
3G | ET, CE, Voy | 2040 | $5\times 10^{4}$ | 10 | $10^{8}$
| ET, 2CE |  | $5\times 10^{4}$ | 1 | $10^{8}$

  * a

Maximum range of any detector in the network

  * b

Order of magnitude estimate for typical localisation

  * c

Number of detections per year assuming a merger rate of $320\,$Gpc-3yr-1 (The
LIGO Scientific Collaboration & the Virgo Collaboration, 2020)

  * d

We assume a total network range of 160 Mpc and 300 Mpc for O4 and O5
respectively

##  4 Gravitational Wave Detectors

###  4.1 Second Generation Detectors

####  4.1.1 Fourth Observing Run (O4; 2022–2023)

The Fourth Observing Run (O4) will run for one year with both LIGO detectors
close to the design sensitivity of a sky and inclination angle averaged
detection range of 190 Mpc (Abbott et al., 2020b). Advanced Virgo will have a
binary neutron star range of 90–120 Mpc (comparable to the LIGO detector
ranges during O3), while the sensitivity of KAGRA (KAGRA Collaboration et al.,
2019, 2020) has a large uncertainty and estimates for its binary neutron star
range is 25–130 Mpc. Assuming a KAGRA range of 80 Mpc the estimated number of
detections is $10^{+52}_{-10}$, with a median 90% localisation of
$33\,\deg^{2}$. We adopt a detection range of 160 Mpc based on the minimum
specifications for the LIGO detectors and reflecting the lower range of the
other two detectors.

####  4.1.2 Fifth Observing Run (O5; 2024–2025)

The fifth observing run will begin after the A+ upgrade, which will increase
the LIGO detector range to 330 Mpc Abbott et al. (2020b). The Virgo detector
will also undergo significant upgrades, and will operate with a binary neutron
star range of 150–260 Mpc and KAGRA will operate with a range of at least 130
Mpc and possibly as high as 155 Mpc. For the purposes of this paper we assume
a sky and inclination angle averaged detection range of 300 Mpc.

####  4.1.3 A Five Detector Network (2025+)

LIGO-India (Iyer et al., 2011) is expected to join operations in 2025 and will
eventually reach design specifications with a range of 330 Mpc. The
geographical location of LIGO-India improves the localisation capabilities of
the global network by a factor of 5–10, and in some cases may result in
mergers being localised to areas as small as $1\,\deg^{2}$.

###  4.2 Third Generation Detectors

LIGO Voyager (LIGO Scientific Collaboration, 2019) is a planned upgrade to the
three existing LIGO facilities that will increase their range to 1.1 Gpc, and
also improve the localisation of closer events. This upgrade is expected to
occur by the end of the decade, and we assume an operational start date of
2030. Voyager is an intermediate step between the Advanced LIGO detectors and
third generation detectors, however due to the significant improvements in the
nominal detector range and the uncertainty in the design and timeline of the
detector, we consider it a third generation detector for the purposes of this
work.

The proposed Neutron Star Extreme Matter Observatory (NEMO; Ackley et al.,
2020a) will also bridge the gap between detectors like A+ and true third-
generation detectors. While the addition of this detector to the network will
not significantly improve detector horizons or merger localisations, it will
make the detection of post-merger gravitational waves feasible by extending
the observable frequency range. This will provide insight into the nature of
merger remnants (Abbott et al., 2019b), and the properties of the jet produced
by the merger (e.g., Beniamini et al., 2020). Constraints on both of these
will inform radio follow-up efforts.

True third generation gravitational wave detectors will be an order of
magnitude more sensitive than current detectors due to reduced quantum shot
noise, improved mirror coatings and the placement of the detectors deep
underground to reduce Newtonian noise. Both proposed detectors will be triple
Michelson interferometers (Freise et al., 2009), as opposed to the L-shaped
interferometers used in second generation detectors, which enables the
measurement of the gravitational wave polarisation.

The Einstein Telescope (ET; Punturo et al., 2010) will have 10 km arms, while
Cosmic Explorer (CE) will have 40 km arms resulting in a higher sensitivity
(Abbott et al., 2017a). While a single third generation detector will be
capable of detecting compact binary coalescences, precise localisation
requires multiple detectors. Hall & Evans (2019) outline various observing
scenarios based on combinations of Voyager, ET and CE detectors. For the
purposes of this work we consider three simplified scenarios - the proposed
network of three Voyager detectors, one Voyager detector with two 3G detectors
(likely both ET and CE) and three 3G detectors (likely ET and two CEs). A
network of one ET detector and two CE detectors will localise $>10\%$ of
neutron star coalescences at redshift $z=0.3$ (corresponding to $\gtrsim 50$
events per year) to $\sim 0.1\,\deg^{2}$ with a median localisation of
$1\,\deg^{2}$ (Hall & Evans, 2019).

##  5 Searching for Radio Afterglows

###  5.1 Benefits and Limitations

####  5.1.1 Advantages over other wavelengths

The ultraviolet/optical/infrared luminosity of kilonovae is dependent on
parameters including the mass and velocity of the ejecta and the fraction of
lanthanides produced (e.g. see Metzger, 2019, and references therein).
Additionally, in higher mass mergers (including neutron star-black hole
mergers), the blue component of the kilonova that was vital in localising
GW170817 may not be produced and even the red component may be suppressed
(Hotokezaka et al., 2013; Kasen et al., 2017). The lack of optical
counterparts detected during O3 suggests that not all events will produce
kilonovae comparable to GW170817 (Kasliwal et al., 2020). Radio afterglows are
mostly independent of these parameters, and therefore probe an independent
part of the merger parameter space to optical observations.

While the flux density of a radio afterglow is dependent on viewing angle,
even significantly off-axis mergers will produce detectable emission (see
Figure 1). In comparison, short gamma-ray bursts are highly anisotropic and
therefore most mergers will not produce a detectable GRB counterpart. In fact,
the true rate of GRBs may be as much as $10^{4}$ times higher than what we
detect (Soderberg et al., 2006), and radio observations present a promising
method of detecting afterglows from off-axis events (Levinson et al., 2002).
However, the inclination angle dependence of GW detector sensitivity means
that the fraction of GW events with a GRB counterpart is higher than the
fraction of detectable GRBs from the general population, and the rate of joint
GW-GRB detections will be a few per year for O4 and tens per year for designed
LIGO specifications (Howell et al., 2019; Saleem, 2020).

Extrinsic factors such as dust extinction and solar angle may also limit
follow-up at other wavelengths. For example, comprehensive optical follow-up
of GW170817 would not have been possible had the merger occured a month later,
and even X-ray monitoring was hindered by $\sim 90$ days of sun avoidance.
Radio telescopes are not limited by either of these factors in general,
although some may have worse performance pointing near the Sun.

####  5.1.2 False-positive rate

Radio searches are limited by the discovery of false positives in the form of
variable sources manifesting as transients, as well as unrelated transients.
Previous untargeted searches for radio transients over long ($>1\,$d)
timescales (Mooley et al., 2013; Mooley et al., 2016, O’Brien et al. in prep.,
Stewart et al. in prep.) have been dominated by variable Active Galactic
Nuclei (AGN). Some variable AGN will be straightforward to immediately
classify, but there exist many that cannot be classified without comprehensive
broadband observations (e.g. Sadler et al., 2006; Lovell et al., 2008; Nyland
et al., 2021). Searches will also be hindered by sources exhibiting extrinsic
variability caused by interstellar scintillation, which can cause variabiliy
of tens of percent at GHz frequencies (Cordes & Lazio, 2002), resulting in the
same compact source being undetected in one epoch and detected in the next.
This can be mitigated by comparing candidate counterparts to galaxy catalogues
and ruling out any that are spatially consistent with the nucleus – neutron
star mergers are likely to occur away from the galaxy nucleus. However, some
telescopes may not provide sufficient astrometric accuracy to do this,
necessitating follow-up observations with other facilities.

Searches consisting of multiple short-integration pointings may also discover
short-duration radio transients such as Fast Radio Bursts, or flare stars
(Helfand et al., 1999; Villadsen & Hallinan, 2019; Pritchard et al., 2021),
although these will be easily ruled out as unrelated by comparison to archival
data along with subsequent follow-up observations.

As radio follow-up observations become more sensitive, we will also discover
afterglows from other transients including tidal disruption events and a
variety of GRBs. At a detection threshold of $10\,\mu$Jy the expected areal
density of radio transients is a few per $\deg^{2}$ at GHz frequencies
(Metzger et al., 2015). Since the emission from most radio transients
originates from a synchrotron blast wave, even broadband radio observations
may not be sufficient to classify transient types, and instead long-term
monitoring to determine the temporal evolution of the source, along with
multi-wavelength follow-up, will be required.

While widefield optical searches discover thousands of false positives, ruling
them out is made easier by having a large sample of known optical transients
with more distinct spectra and underlying physics, enabling the use of machine
learning techniques for immediate and automatic classification (e.g. Mahabal
et al., 2008; Bloom et al., 2012; Goldstein et al., 2015; Mahabal et al.,
2019; Stachie et al., 2020). Comparably few true radio transients have been
detected in untargeted searches to-date (Stewart et al., 2016; Law et al.,
2018) and current radio transient surveys rely on manual inspection of
candidates and classification using follow-up observations and archival data.
Planned widefield transients searches will allow the better characterisation
of transient properties, which will ultimately enable the use of automated
classification algorithms.

It is naturally preferable to confirm the association between any detected
radio transient as soon as possible. It is possible (albeit, unlikely) that
the radio afterglow may be detected early enough that optical emission from
the kilonova is still detectable, observations of which are vital in
constraining merger properties like the ejecta mass. More importantly,
comprehensive broadband monitoring of the non-thermal afterglow that traces
the rise, peak and decline of the lightcurve, is vital in constraining merger
properties.

Overall, we strongly emphasise the importance of designing follow-up
strategies that are not only sufficient to detect afterglows, but also use a
cadence that enables false-positives to be ruled out in a timely manner.

####  5.1.3 Mergers with radio-loud hosts

Resolving the afterglow from any host galaxy emission may be a decisive factor
in whether the afterglow is detectable, particularly in a widefield unbiased
search. GW170817 occurred in a radio-loud host galaxy and was offset by
$10.31\,$arcsec (Blanchard et al., 2017; Levan et al., 2017) and therefore
non-standard sourcefinding techniques may have been required to find it in an
unbiased search depending on the angular resolution of the data. Host galaxy
offsets for short GRBs range from 0.5-75 kpc with a median of 5 kpc (Fong et
al., 2010; Fong & Berger, 2013; Berger, 2014). At 200(500) Mpc this
corresponds to an angular offset of 10(4) arcsec, comparable to the angular
resolution of at least some current radio telescopes. For events occuring at
cosmological distances, detectable with third generation detectors (see 4.2),
typical offsets will be comparable to the angular resolution of the SKA.
Therefore the presence of nuclear emission from the host may complicate
searches, but not make them impossible. However, radio emission from star
formation regions can span a much larger volume (Linden et al., 2020) and may
pose more of a problem.

By comparing the expected afterglow luminosity to the typical luminosity of
both AGN and star forming galaxies, Hotokezaka et al. (2016) found that most
mergers will not occur in galaxies that are sufficiently radio-bright to
hinder radio follow-up efforts. This is true even for telescopes with angular
resolution $\gtrsim 10\,$arcsec. The detectability metrics we use in this work
assume that the detectability of an event is not limited by host emission.

####  5.1.4 Galaxy catalogue completeness

Most current radio telescopes lack the field of view and survey speed to carry
out unbiased searches and are therefore restricted to the galaxy-targeted
approach outlined in Section 3.1.1. The effectiveness of this strategy relies
upon having a complete catalogue of galaxies within the localisation volume of
each event.

There is currently no all-sky galaxy catalogue that approaches completeness at
the current LIGO horizon. GLADE (Dálya et al., 2018) is complete to a distance
of $\sim 40\,$Mpc, $\sim 50$% complete at the nominal O4 range of $170\,$Mpc
and $<40$% complete at the design range of 330 Mpc all based on cumulative
blue luminosity (which is an approximate analog for star formation, and
therefore, merger rate). Additionally the galaxies in GLADE are not
isotropically distributed – the median line of sight density is $\sim
10\,\deg^{-2}$, compared to $<1\,\deg^{-2}$ in the Galactic plane and
$>10^{3}\,\deg^{-2}$ in fields covered by the HyperLEDA survey (Makarov et
al., 2014).

Other existing surveys provide more complete samples along particular lines of
sight. These smaller catalogues may still be useful for follow-up of specific
events – e.g. the galaxy-targeted follow-up of GW190814 carried out by Gomez
et al. (2019) and Ackley et al. (2020b) could have used the PRLS survey rather
than GLADE. However, they do not present a general solution for follow-up of
all events.

Planned all-sky surveys (e.g. LSST Science Collaboration et al., 2009; Doré et
al., 2016, among others) will drastically improve upon existing catalogues and
make it feasible to target galaxies for events at Gpc distances. However,
these surveys will likely not be finished within the 5 years, by which time
facilities like ASKAP and MeerKAT will be fully operational and the DSA-2000
and SKA may be coming online. These telescopes are not suited to galaxy
targeting due to their wide fields of view (compared to most existing
facilities), and therefore it’s unlikely that future galaxy catalogues will
have a tangible impact on the pointing strategies used in radio follow-up.
However, they will remain extremely useful for ruling out false-positive radio
transient candidates and in determining whether discovered transients are
associated with host galaxies within the localisation volume.

###  5.2 Follow-up of GW-identified mergers

To quantify prospects for detecting radio afterglows we use a simple detection
metric – the flux density at the observing frequency must exceed five times
the expected thermal noise, $\sigma$. This estimate does not include the noise
due to source confusion, which will not be the limiting factor for the
observations discussed here.

We also compare our results to the range of gravitational wave detectors which
is dependent on inclination angle, scaling as

$\mathcal{R}(\theta_{\rm obs})\approx
0.589\overline{\mathcal{R}}\sqrt{1+6\cos^{2}\theta_{\rm
obs}+\cos^{4}\theta_{\rm obs}}$ (6)

where $\mathcal{R}(\theta_{\rm obs})$ is the inclination angle dependent range
and $\overline{\mathcal{R}}$ is the gravitational wave detector range found in
Table 2. The variable terms in this equation are obtained by averaging eq.
3.31 of Finn & Chernoff (1993) across the antenna pattern terms555We note that
this is a crude approximation to the true survey volume, which can be more
accurately calculated via a numerical Monte Carlo integral (Mandel, private
communication). We have compared both methods and find the above equation
underestimates the range by $\sim$10%. However, we have chosen to use this
method as it is sufficiently accurate for our purposes and is far easier to
reproduce., while the normalisation constant is calculated by requiring
$\langle\mathcal{R}(\theta_{\rm obs})\rangle=\overline{\mathcal{R}}$. Duque et
al. (2019) obtain the same result by instead computing the gravitational wave
detector horizon. While Finn & Chernoff (1993) only calculate antenna patterns
for L-shaped interferometers, the above equation is a sufficient approximation
to the 3G detector configurations for our purposes as any discrepancy is
negligible compared to the current uncertainty in detector specifications and
sensitivity.

Figure 2: Maximum distance at which gravitational wave afterglows can be
detected as a function of inclination angle for a range of circum-merger
densities. Solid lines denote observations of 200 galaxies to a detection
threshold of 75 $\mu$Jy at 3 GHz with the VLA, while dashed lines denote
observations with the ATCA targeting 50 galaxies to a detection threshold of
70 $\mu$Jy at 8 GHz. The LIGO range for O4 and design specifications is shaded
in dark and light grey respectively. The 50% completeness of the GLADE
catalogue is also labelled.

####  5.2.1 Galaxy Targeting

Figure 2 shows the maximum distance at which an afterglow can be detected for
a range of circum-merger densities spanning $10^{-4}$–$1\,$ cm-3, using a
galaxy-targeting approach with the ATCA and the VLA as outlined in Section
3.2.

We note that while this strategy will allow most events to be detected at
distances comparable to the LIGO range, the incompleteness of existing galaxy
catalogues at these distances makes this strategy only feasible for the
closest mergers ($D_{L}\ll 100\,$Mpc). We do not consider applying this
approach to any next generation facilities as their fields of view are large
enough that they contain multiple candidate hosts per pointing, and their
survey speeds are large enough that an unbiased search is generally feasible.

Table 3: Capabilities of unbiased searches for radio afterglows for a range of telescopes and observing strategies, including observing frequency ($\nu$), bandwidth ($\Delta\nu$), total areal coverage ($\Omega_{\rm total}$) and required observing time ($T_{\rm total}$) Telescope | $\nu$ | Strategy | $\Omega_{\rm total}$ | $S_{\rm detect}$ | $T_{\rm total}$
---|---|---|---|---|---
|  | (GHz) | (deg2) | ($\mu$Jy) | (hr)
Apertif | 1.4 | deep | $10$ | $125$ | $12$
|  | wide | $40$ | $250$ | $12$
ASKAP | 0.9 | deep | $30$ | $175$ | $10$
|  | wide | $300$ | $550$ | $10$
DSA | 1.35 | deep | $10$ | $5$ | $1$
|  | wide | $100$ | $5$ | $2.5$
MeerKAT | 1.4 | wide | $10$ | $35$ | $12$
ngVLA | 2.4 | wide | $10$ | $5$ | $10$
|  | ultra-wide | $100$ | $25$ | $10$
SKA-1 | 1.43 | wide | $10$ | $10$ | $10$
|  | ultra-wide | $100$ | $40$ | $10$
SKA-2 | 1.43 | wide | $10$ | $1$ | $10$
|  | ultra-wide | $100$ | $4$ | $10$
VLA | 1.5 | wide | $5$ | $75$ | $12$

Figure 3: Similar to Figure 2. Left: unbiased observations of events localised
to $\leq 10\,\deg^{2}$ with MeerKAT, Apertif and the VLA. Right: unbiased
observations with Apertif (covering $40\,\deg^{2}$) and ASKAP (deep covering
$30\,\deg^{2}$ and wide covering $300\,\deg^{2}$). The gravitational wave
detector range for O4, A+ and Voyager are shown in increasingly light tones of
grey.

Figure 4: Similar to Figure 2. Unbiased observations of events localised to
$\leq 10\,\deg^{2}$ (left) and $\leq 100\,\deg^{2}$ (right) with the SKA-1,
DSA-2000 and ngVLA. The range of a nominal SKA-2 design is $\sim 3$ times
further than the SKA-1. The detector range of Voyager, a preliminary 3G
detector with a 5 Gpc range, and 3G detectors are shown in increasingly light
tones of grey.

####  5.2.2 Unbiased Searches

We therefore turn our focus to the unbiased searches described in Table 3,
which we split into four broad categories.

Figure 3 shows the detectability of events in unbiased searches with current
facilities. Most on-axis mergers, as well as most off-axis mergers occuring in
dense environments, localised to $\leq 10\,\deg^{2}$ detected with current
gravitational wave facilities and the A+ upgrade will be detectable with
MeerKAT and some will be detectable with the VLA and Apertif. However, we note
that only a small fraction of events will be localised this well with these
detectors. LIGO Voyager will have better localisation capabilities, and some
events will produce afterglows that are detectable out to the detector
horizon.

For events that are localised to tens of square degrees, comparable to the
median localisation for 2G detectors (Abbott et al., 2020b), we consider
follow-up with ASKAP and Apertif. Figure 3 also shows the maximum distance at
which afterglows will be detected, and we find that it is feasible to detect
the afterglow produced by most on-axis mergers with current gravitational wave
detectors.

Figure 4 shows the same metrics applied to the DSA-2000, ngVLA and SKA-1
compared to the range of Voyager and 3G detectors for events localised to
$\leq 10\,\deg^{2}$ and $\leq 100\,\deg^{2}$. We find that while the majority
of events detected with Voyager will be accompanied by detectable afterglows,
it will not be possible to detect afterglows in widefield follow-up of the
most distant events discovered by a complete 3G network. However, we note that
the median localisation achievable with a complete 3G network is $\sim
1\,\deg^{2}$. Therefore widefield searches will not be necessary for most
events, and the targeted single-pointing strategy outlined in Section 5.2.3
may be a more useful metric.

Figure 5: Similar to Figure 2 showing targeted single-pointing observations
with existing radio telescopes. The range of gravitational wave detectors for
O4, A+ and Voyager are shown in increasingly light tones of grey. Figure 6:
Similar to Figure 2 showing targeted single-pointing observations with planned
radio telescopes. The range of the DSA-2000 is comparable to the range of the
SKA-1. The range of Voyager, a preliminary 3G detector with a 5 Gpc range, and
3G detectors are shown in increasingly light tones of grey.

####  5.2.3 Monitoring electromagnetic counterparts

Figure 5 shows prospects for detecting radio emission with current radio
facilities from events that have been localised by the detection of an
electromagnetic counterpart. Assuming that neutron star mergers occur in
comparably dense environments to short GRBs ($n\sim 10^{-2}\,{\rm cm}^{-3}$
Fong et al., 2015) we find that most neutron star mergers detected during O4,
and a large fraction with the A+ configuration, should produce radio emission
that is detectable with deep single pointing observations. However current
facilities will not be sufficient for a comprehensive census of radio
afterglows as we move towards the 3G era – only on-axis mergers occuring in
dense environments ($n\gtrsim 10^{-1}\,{\rm cm}^{-3}$) will be detectable at
the Voyager horizon.

The sensitivity of future radio telescopes will partially address this
problem. Figure 6 shows the detectability horizon for the ngVLA and both
phases of the SKA (with the DSA-2000 horizon comparable to that of the SKA-1).
While most events detected with Voyager will be within range of future radio
telescopes, the most distant events detected with 3G detectors will be well
beyond the range of even the SKA-2. However, these detectors will detect
thousands of events per year and therefore the limiting factor in obtaining a
census of radio afterglows will be the amount of telescope time available
rather than the current scenario which is limited by a lack of events.

Figure 7: Detectability of radio afterglows for planned (left) and potential future (right) untargeted transients surveys. The detector range for O4, A+ and Voyager are shown in increasingly light shades of grey. Figure 8: Detectability of a complete sample of radio afterglows (i.e. where the signal remains detectable for longer than the survey cadence) in planned and potential future untargered transients surveys. The detector range for O4, A+ and Voyager are shown in increasingly light shades of grey. As the inclination angle of the merger decreases the detectability becomes limited by the survey cadence rather than sensitivity. This effect is more pronounced for mergers occurring in denser environments. Table 4: Ongoing, upcoming and idealised transients surveys. The fraction of the total sky covered by the survey is in column 3, while $S_{\rm det}$ corresponds to a $5\sigma$ detection threshold based on expected image noise. The MeerKAT survey is a theoretical idealised survey and there are no current plans to undertake it. Survey | $\nu$ | Sky coverage | Cadence | $S_{\rm det}$
---|---|---|---|---
| (GHz) |  | (months) | ($\mu$Jy)
VLASS | 3 | 0.82 | 32 | 600
VAST-Wide | 0.9 | 0.23 | daily | 2500
VAST-Deep | 0.9 | 0.23 | 8 | 250
MeerKAT | 1.4 | 0.12 | 4 | 20
DSA (CASS) | 1.35 | 0.75 | 4 | 10

###  5.3 Serendipitous detections and orphan afterglows

We also consider the detection of afterglows from known gravitational wave
events and orphan afterglows (i.e. events with no previous EM/GW detection),
in the transients surveys outlined in 3.4. Table 4 shows the properties of the
surveys, and Figure 7 shows the application of the same detectability metrics
as above.

To determine the capability of surveys to obtain a complete sample of mergers
occuring within their footprint, we also apply an additional constraint of the
afterglow remaining detectable for a time corresponding to the survey cadence.
This ensures that the afterglow will be detected in at least one epoch of the
survey, although we note that multiple detections and multi-wavelength follow-
up will be required to confirm the detection and classify it. The results of
this change are shown in Figure 8, where we have excluded VLASS (as its slow
cadence results in effectively zero range) and VAST-Wide (as its daily cadence
results in no significant changes to the result above). Sensitivity remains
the dominant limiting factor for off-axis events, while on-axis events
occurring in denser environments are limited by survey cadence as their
emission peaks at earlier times with a shorter turnover period. We note that
these results cannot be scaled using equation 4, as the rise and decay of the
lightcurve are also dependent on our assumptions of $E_{\rm iso}$ and
$\Gamma(\theta)$.

While planned transients searches will not provide complete samples of
afterglows due to their limited sky coverage, these results demonstrate that
it is worthwhile carrying out targeted searches for afterglows within those
datasets. These surveys may also make it feasible to search for counterparts
to poorly localised events that do not individually warrant follow-up
observations. We also note that once Voyager begins operations very few orphan
afterglows will be detected, as only on-axis events occuring in the most dense
environments will be detectable at distances beyond the Voyager horizon,
although this does not consider the duty cycle of the detector network.

##  6 Conclusions

In this work we have summarised the synergies between radio telescopes and
gravitational wave detectors, encompassing both existing facilities and
planned or proposed facilities spanning the coming decades. We demonstrate
that while targeting potential host galaxies proved useful in the follow-up of
GW170817, this method will be less feasible in future follow-up due to limited
catalogue completeness at distances comparable to the range of gravitational
wave detectors. Additionally, the larger field of view of future telescopes is
more conducive to unbiased widefield searches that target the localisation
region of the merger. We find that these searches with current facilities will
be capable of detecting mergers at hundreds of Mpc, while future facilities
will be able to detect mergers at Gpc distances. Widefield transients surveys
will provide serendipitous coverage of events and may also detect afterglows
of events beyond the detector horizon, and those that occur during detector
downtime.

Radio observations can be used to place constraints on properties of the
merger outflow and the circum-merger environment, and for events with
counterparts detected at other wavelengths, we find that current radio
facilities are capable of detecting some afterglows at the Voyager horizon,
while future facilities will detect afterglows at distances up to tens of Gpc
for the most on-axis events. However, lightcurve monitoring alone is
insufficient to completely constrain the geometry of the merger, and we also
discuss possible ways of breaking model degeneracies.

## Acknowledgements

We thank Rob Fender, Samaya Nissanke, Andrew Zic, and Ilya Mandel for useful
discussions. DD was supported by an Australian Government Research Training
Program Scholarship. TM acknowledges the support of the Australian Research
Council through grant DP190100561. DLK was supported by NSF grant AST-1816492.
Parts of this research were conducted by the Australian Research Council
Centre of Excellence for Gravitational Wave Discovery (OzGrav), project number
CE170100004. This research has made use of NASA’s Astrophysics Data System
Bibliographic Services and the FRB Theory wiki (Platts et al., 2019).

Software: Astropy (Astropy Collaboration et al., 2018), Jupyter (Kluyver et
al., 2016), Matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), SciPy
(Virtanen et al., 2020)

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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