# Limits on Fast Radio Burst-like Counterparts to Gamma-ray Bursts using
CHIME/FRB

Alice P. Curtin Department of Physics, McGill University, 3600 rue University,
Montréal, QC H3A 2T8, Canada  Trottier Space Institute, McGill University,
3550 rue University, Montréal, QC H3A 2A7, Canada  Shriharsh P. Tendulkar
Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, Mumbai, 400005, India  National Centre for Radio Astrophysics, Post
Bag 3, Ganeshkhind, Pune, 411007, India  Alexander Josephy Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
Trottier Space Institute, McGill University, 3550 rue University, Montréal, QC
H3A 2A7, Canada  Pragya Chawla Anton Pannekoek Institute for Astronomy,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Bridget Andersen Department of Physics, McGill University, 3600 rue
University, Montréal, QC H3A 2T8, Canada  Trottier Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Victoria M.
Kaspi Department of Physics, McGill University, 3600 rue University, Montréal,
QC H3A 2T8, Canada  Trottier Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  Mohit Bhardwaj Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
Trottier Space Institute, McGill University, 3550 rue University, Montréal, QC
H3A 2A7, Canada  Tomas Cassanelli Department of Electrical Engineering,
Universidad de Chile, Av. Tupper 2007, Santiago 8370451, Chile  Amanda Cook
Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON M5S 3H4, Canada  David A. Dunlap Department of
Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  Fengqiu Adam Dong Department of Physics and
Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,
BC V6T 1Z1 Canada  Emmanuel Fonseca Department of Physics and Astronomy, West
Virginia University, PO Box 6315, Morgantown, WV 26506, USA  Center for
Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge
Research Building, Morgantown, WV 26505, USA  B. M. Gaensler Dunlap Institute
for Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  David A. Dunlap Department of Astronomy &
Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S
3H4, Canada  Jane F. Kaczmarek Dominion Radio Astrophysical Observatory,
Herzberg Research Centre for Astronomy and Astrophysics, National Research
Council Canada, PO Box 248, Penticton, BC V2A 6J9, Canada  Adam E. Lanmnan
Department of Physics, McGill University, 3600 rue University, Montréal, QC
H3A 2T8, Canada  Trottier Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  Calvin Leung MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139, USA  Department of Physics,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA  Aaron B. Pearlman Department of Physics, McGill University, 3600
rue University, Montréal, QC H3A 2T8, Canada  Trottier Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Emily Petroff
Department of Physics, McGill University, 3600 rue University, Montréal, QC
H3A 2T8, Canada  Trottier Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  Anton Pannekoek Institute for
Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands  Ziggy Pleunis Dunlap Institute for Astronomy & Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
Masoud Rafiei-Ravandi Department of Physics, McGill University, 3600 rue
University, Montréal, QC H3A 2T8, Canada  Trottier Space Institute, McGill
University, 3550 rue University, Montréal, QC H3A 2A7, Canada  Scott M. Ransom
National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA
22903, USA  Kaitlyn Shin MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139, USA  Department of Physics, Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA  Paul Scholz Dunlap
Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON M5S 3H4, Canada  Kendrick Smith Perimeter Institute for
Theoretical Physics, 31 Caroline Street N, Waterloo, ON N25 2YL, Canada
Ingrid Stairs Department of Physics and Astronomy, University of British
Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1 Canada Alice P. Curtin
<EMAIL_ADDRESS>
###### Abstract

Fast Radio Bursts (FRBs) are a class of highly energetic, mostly extragalactic
radio transients lasting for $\sim$milliseconds. While over 600 FRBs have been
published so far, their origins are presently unclear, with some theories for
extragalactic FRBs predicting accompanying high-energy emission. In this work,
we use the Canadian Hydrogen Intensity Mapping Experiment (CHIME) Fast Radio
Burst (CHIME/FRB) Project to explore whether any FRB-like radio emission
coincides in space and time with 81 gamma-ray bursts (GRBs) detected between
2018 July 17 and 2019 July 8 by Swift/BAT and Fermi/GBM. We do not find any
statistically significant coincident pairs within 3$\sigma$ of each other’s
spatial localization regions and within a time difference of up to one week.
In addition to searching for spatial matches between known FRBs and known
GRBs, we use CHIME/FRB to constrain FRB-like ($\sim$1-10 ms) radio emission
before, at the time of, or after the reported high-energy emission at the
position of 39 GRBs. For short gamma-ray bursts (SGRBs), we constrain the
radio flux in the 400- to 800-MHz band to be under a few kJy for $\sim$5.5 to
12.5 hr post-high-energy burst. We use these limits to constrain models that
predict FRB-like prompt radio emission after SGRBs. For long gamma-ray bursts
(LGRBs), we constrain the radio flux to be under a few kJy from $\sim$6 hours
pre-high-energy burst to $\sim$12 hours post-high-energy burst.

Fast Radio Bursts, Gamma-ray Bursts, Radio transient sources

ApJ

##  1 Introduction

###  1.1 FRBs

Fast radio bursts (FRBs), which were first reported in Lorimer et al. (2007),
are a class of highly energetic, transient events lasting for a few
milliseconds and originating from extragalactic distances. While the original
discovery consisted of a “one-off” burst, some FRBs have been shown to repeat
(Spitler et al., 2016; CHIME/FRB Collaboration et al., 2019a, b; Fonseca et
al., 2020), with two showing periodic burst activity (CHIME/FRB Collaboration
et al., 2020a; Rajwade et al., 2020; Cruces et al., 2021).

FRBs are energetic phenomena, with spectral luminosities $\sim 10^{32}$ erg
s-1 Hz-1 over bandwidths of $\sim 0.1$ to 1 GHz. While many models for FRB
progenitors involve neutron stars (NS), these NS-based models either require
substantial scaling up of previous models for NS emission mechanisms, e.g.,
curvature radiation in Kumar et al. (2017) or synchrotron radiation in Sironi
& Spitkovsky (2011), or require new astrophysical mechanisms to provide the
necessary energy. Proposed new mechanisms include interactions of magnetized
relativistic shockwaves (Metzger et al., 2019), a stream of particles combing
through the magnetosphere of the NS (Zhang, 2017), an asteroid belt
surrounding a pulsar (Dai et al., 2016), or an expanding fireball (Ioka,
2020). In addition to radio emission, these mechanisms might also produce
accompanying, detectable high-energy emission (e.g., Zhang, 2017; Metzger et
al., 2019) as the radio emission is expected to be a small fraction of the
total energy output.

###  1.2 High-energy Emission from FRBs

The earliest observational evidence for accompanying high-energy emission for
an FRB came when Bannister et al. (2012) detected single radio pulses after
two gamma-ray bursts (GRBs) using the 12-m dish at the Parkes Observatory.
However, they were unable to rule out radio frequency interference (RFI) as
the cause of these pulses, and their simulations/modeling predicted that there
was only a 2$\%$ chance that the pulses were actually associated with the
GRBs. Additionally, DeLaunay et al. (2016) claimed an association between FRB
20131104A and an untriggered GRB found in archival Swift/Burst Alert Telescope
(BAT) data. However, a more recent analysis done by Sakamoto et al. (2021) of
the same Swift/BAT dataset found no significant gamma-ray emission at the time
of FRB 20131104A.

More conclusive observational evidence for the association of FRBs with high-
energy emission came in April 2020 when the Canadian Hydrogen Intensity
Mapping Experiment Fast Radio Burst (CHIME/FRB) Project detected an ultra-
bright FRB-like pulse from the Galactic magnetar SGR 1935+2154 (CHIME/FRB
Collaboration et al., 2020b). The burst was detected at radio frequencies by
CHIME/FRB and STARE2 (Bochenek et al., 2020), and in the hard X-ray/soft
gamma-ray regimes by the Hard X-ray Modulation Telescope (Insight-HXMT)
Satellite (Li et al., 2021), the Konus-Wind Experiment (Ridnaia et al., 2020),
and the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL, Mereghetti
et al., 2020). The burst detected at CHIME/FRB had a radio-to-gamma-ray (20 to
200 keV) fluence ratio of $9^{+9}_{-5}\times 10^{11}$ Jy ms erg-1 cm2
($4^{+4}_{-1.8}\times 10^{-6}$ in dimensionless units assuming a 400-MHz
bandwidth), with the dispersion-corrected arrival times of the subcomponents
of the burst at CHIME/FRB leading the high-energy emission at Insight-HXMT by
$\sim$3 ms (Mereghetti et al., 2020).

Numerous simultaneous X-ray and high-energy observations, as well as follow-up
observations and searches through archival data, have been carried out to find
detected high-energy associations with an extragalactic FRB (e.g., Scholz et
al., 2017; Cunningham et al., 2019; Martone et al., 2019; Cunningham et al.,
2019; Scholz et al., 2020; Guidorzi et al., 2020; Tohuvavohu et al., 2020;
Casentini et al., 2020; Anumarlapudi et al., 2020; Verrecchia et al., 2021;
Principe et al., 2021; Mereghetti et al., 2021). Despite these many efforts,
no other unambiguous detections have been made.

Possibilities for the aforementioned lack of detections include limited
sensitivity of high-energy instruments, observations which do not overlap with
FRB bursts (e.g., Mereghetti et al., 2021; Guidorzi et al., 2020) and
different beaming angles for the radio and high-energy emission (e.g., Sridhar
et al., 2021). One way of overcoming some of these challenges is to search for
radio bursts associated with high-energy emission, rather than searching for
high-energy bursts associated with radio emission. For the remainder of this
paper, we specifically focus on searching for FRB-like radio emission from
GRBs.

###  1.3 GRBs

GRBs are extremely energetic, extragalactic sources of gamma-ray emission with
luminosities $>10^{53}$ erg s-1 (Klebesadel et al., 1973; Gehrels et al.,
2009). Since their initial discovery in 1967, over 7000 GRBs have been
detected, with GRBs now classified as either short GRBs (SGRBs), whose 90$\%$
burst duration (T90) is less than 2 s, or long GRBs (LGRBs), whose T90 is
longer than 2 s. However, making an unambiguous distinction between the two
classes requires more than just duration as SGRBs typically also have a harder
spectrum than LGRBs (Kouveliotou et al., 1993; Ghirlanda et al., 2009;
Jespersen et al., 2020).

Most SGRBs are likely produced by compact object mergers such as black hole
(BH)-NS mergers or NS-NS mergers (Berger, 2014). Some nearby (e.g., less than
a few tens of Mpc) SGRBs can be produced by magnetar giant flares though
(Burns et al., 2021; Fermi-LAT Collaboration et al., 2021; Svinkin et al.,
2021). However, since SGRBs produced by magnetar giant flares are a very small
fraction of the observable SGRB population due to the limited distance range
to which we can detect them, we do not focus on them in detail when discussing
models for FRB-like emission from GRBs, and instead only focus on SGRBs
produced by NS-NS or NS-BH mergers.

LGRBs are traditionally thought to be produced during the collapse of a
rapidly rotating massive star (Gal-Yam, 2019). However, a LGRB was recently
associated with a kilonova, suggesting that some LGRBs can similarly be
produced by compact object mergers (Rastinejad et al., 2022).

###  1.4 Theoretical Predictions for Radio Emission from GRBs

Assuming that SGRBs originate from the merger of compact objects, then an FRB-
like burst might be produced prior to the SGRB from precursor winds (Lyutikov,
2013; Sridhar et al., 2021), interactions between a highly magnetized NS and a
less magnetized NS under a unipolar inductor model (Piro, 2012; Wang et al.,
2016), synchronization of two NS’s magnetic fields to the binary orbital
period (Totani, 2013), magnetic flares produced by the twisting of
magnetospheres (Most & Philippov, 2020), an induced electric field due to the
motion of a magnetized NS (Wada et al., 2020), or abrupt magnetic
reconnections due to interactions between two NS’s magnetospheres (Zhang,
2020). Pre-merger emission is also possible for BH-NS binaries, e.g., from a
BH-NS battery (Mingarelli et al., 2015), from an induced electric field due to
the motion of a magnetized NS around a BH (Wada et al., 2020), or from
extraction of orbital kinetic energy due to the NS orbiting the BH (Carrasco
et al., 2021). Emission at the approximate time of the SGRB is also possible
from interactions between a highly magnetized wind produced by the merger and
the surrounding medium (Usov & Katz, 2000). However, although this emission
would be produced approximately simultaneously, its detection would be
significantly delayed from the time of the high-energy emission due to the
intergalactic dispersion of radio waves. This intergalactic dispersion is
quantified using the dispersion measure (DM) delay of the radio emission, and
allows rapid radio follow-up of GRBs to possibly detect radio emission that
was emitted simultaneously with the high-energy emission.

FRB-like emission may also be possible after a NS-NS or NS-BH merger. If a NS-
NS merger results in another NS, then given certain NS parameters (e.g., high
magnetic field strength, fast spin period) pulsar-like emission from the NS
could be on the same energy scale as that of FRBs (Pshirkov & Postnov, 2010;
Totani, 2013). If the NS is stable (dependent on its mass, spin, spin-down,
and the NS equation of state), then this emission could theoretically last
indefinitely. However, if the NS is unstable and collapses to form a BH, then
an additional FRB-like burst might be produced from the transfer of the NS’s
magnetic field to the BH (Falcke & Rezzolla, 2014; Zhang, 2014). In a NS-BH
merger, an FRB-like burst might be produced in a similar manner when the NS’s
magnetic field is transferred to the BH (Mingarelli et al., 2015).

There are fewer theoretical predictions for prompt FRB-like radio emission
from LGRBs. The host environment of the first detected repeating FRB, FRB
20121102A, is very similar to that of LGRBs as it is a star-forming dwarf
galaxy with low metallicity (Tendulkar et al., 2017). This provides some
evidence for an association between FRBs and LGRBs (Metzger et al., 2017).
However, FRBs have since been found in other host galaxy environments (e.g.,
Niu et al., 2021; Bhardwaj et al., 2021; Kirsten et al., 2021; Bhandari et
al., 2022), and no definitive association has been made between an LGRB site
and an FRB site. Additionally, while the gamma-ray emission after a supernova
could escape, the surrounding region would likely be opaque to radio emission
on the timescale of approximately years to decades, and thus not
simultaneously detectable. It is possible that if an LGRB were produced in a
unconventional mechanism, e.g., a merger-driven explosion induced by a compact
object as suggested by Dong et al. (2021), then there might be the possibility
of pre-LGRB radio emission, but this has yet to be theoretically explored.
Additionally, a recent detection of an LGRB associated with a kilonova
suggests that LGRBs can be produced by compact object mergers, and hence some
of the above mentioned mechanisms for producing radio emission at the time of
SGRBs could be applied to some LGRBs (Rastinejad et al., 2022; Yang et al.,
2022).

###  1.5 Observational Efforts to Search for Radio Emission from GRBs

There have already been a number of searches for FRB-like radio emission
associated with SGRBs and LGRBs, but none has been successful111During the
preparation of this manuscript, a tentative association was claimed between
FRB 20190425A and the binary neutron star merger GW 20190425 (Moroianu et al.,
2022). There is one candidate SGRB that may be related to this event, but its
significance is low and its uncertainty region high (Pozanenko et al., 2020).
No definitive association has been made between the FRB and this candidate
SGRB. . Bannister et al. (2012), Staley et al. (2013), Palaniswamy et al.
(2014), Kaplan et al. (2015), Anderson et al. (2021), Rowlinson et al. (2021)
all performed rapid radio follow-up of GRB locations, while Madison et al.
(2019), Men et al. (2019), Hilmarsson et al. (2020), and Bruni et al. (2021)
performed late-time follow-up of GRB sites. Some of the GRBs followed-up by
Madison et al. (2019) and Men et al. (2019) were notable for possibly being
associated with NS remnants due to anomalous X-ray activity and their X-ray
light curves (Fong et al., 2014; Metzger & Piro, 2014; Lü et al., 2017; Troja
et al., 2019; Lamb et al., 2019). However, no FRB-like radio emission was
detected in any of the above searches. In addition to the above follow-up,
Obenberger et al. (2014) performed radio observations that were simultaneous
with high-energy detections, yet no radio emission was seen. Tendulkar et al.
(2016) also performed radio observations of the SGR 1806–20 magnetar giant
flare (relevant as some magnetar giant flares produce SGRBs), but again no
FRB-like radio burst was detected.

It is possible that the sources followed up in the above studies either do not
produce FRBs or produced FRBs at a time different than that of the follow-up
observations. To further explore whether FRBs and GRBs might be related, we
take advantage of the unique wide field of view and large exposure of
CHIME/FRB and search for radio emission before, at the time of, and after 81
GRBs. Below, in Section 2, we provide a basic overview of the CHIME telescope
and the CHIME/FRB backend. In Section 3, we describe our GRB and FRB samples.
Then, in Section 4, we outline our search for temporally and spatially
coincident GRBs and FRBs and present our results. In Section 5, we outline our
pipeline for calculating radio upper limits at times of non-detections and
discuss the results from this pipeline. In Section 6, we discuss the
implications of our results for different SGRB and LGRB models. We end in
Section 7 by summarizing our work.

##  2 Overview of CHIME/FRB

CHIME consists of four, $100$-m $\times$ 20-m, cylindrical, parabolic
reflectors with the cylinder axes oriented in the north-south (N-S) direction.
It has a N-S field of view (FOV) of $\sim 120\degree$ and an east-west (E-W)
FOV of $\sim 1.3$ to 2.5$\degree$ (dependent on frequency, with 1.3$\degree$
corresponding to 800-MHz) for a total FOV of $\sim 250\degree$ (Ng et al.,
2017). Each of the four reflectors has 256 dual-polarization feeds hanging
along its axis for a total of 2048 antenna signals operating between 400- and
800-MHz.

The 2048 signals are fed into a hybrid FX correlator where the F-engine
digitizes and transforms the antenna signals and the X-engine performs spatial
correlation. More specifically, the X-engine forms 256 N-S beams through a
spatial fast Fourier transform (FFT) of the antenna signals, and then forms
three more sets of 256 N-S beams using exact phasing for a total of 1024
stationary formed beams. For more details on the formed beams, see Ng et al.
(2017) and Masui et al. (2019).

After the X-engine, the data are sent to the CHIME/FRB pipeline, which is
split into four processing levels: L1, L2, L3, and L4. First, L1 performs RFI
mitigation (Rafiei-Ravandi & Smith, 2022), performs a fast dedispersion
transform, and identifies the candidates with FRB-like signals. Then, L2/L3
determine whether a candidate is RFI, a known source, or an unknown source,
and decide which set of actions to perform for the candidate based on that
classification. L4 then implements the actions that are decided in L2/L3. For
more details on CHIME/FRB and its pipelines, see CHIME/FRB Collaboration et
al. (2018). More details on CHIME and its cosmology experiment can be found in
The CHIME Collaboration et al. (2022).

##  3 Sources

###  3.1 FRBs

The FRB sample used in this analysis consists of 536 FRBs detected between
2018 July 25 and 2019 July 1 from the first CHIME/FRB catalog. This includes
62 bursts from 18 previously reported repeating FRBs, and 474 thus far one-off
FRBs. A detailed analysis and description of these FRBs can be found in
CHIME/FRB Collaboration et al. (2021).

###  3.2 GRBs

The GRB sample used in this analysis consists of 81 GRBs detected between 2018
July 17 and 2019 July 8. The time-frame for these GRBs is chosen to be the
same as that of the first CHIME/FRB Catalog. However, we also include GRBs
from 7 days before and 7 days after the time of the catalog, with the
reasoning for this discussed below in Section 4. The GRBs are chosen using the
Gamma-ray Coordination Network (GCN)222www.gcn.gsfc.nasa.gov circulars from
four different high-energy instruments: the Fermi Gamma-ray Burst Monitor,
(Fermi/GBM; Meegan et al., 2009), the Neil Gehrels Swift Observatory Burst
Alert Telescope (Swift/BAT, Gehrels et al., 2004; Barthelmy et al., 2005),
INTEGRAL (Winkler et al., 2003), and Konus-Wind (Aptekar et al., 1995). We
also restrict our analysis to GRBs having 1$\sigma$ localization errors in RA
and DEC less than $0.5\degree$ as it is not possible to make conclusive
statements on spatial coincidences or calculate radio flux limits for GRBs
with either unknown or large uncertainty regions.

Many of the GCN circulars contain false positives, and we manually remove all
of these from our sample along with all bursts for which either the T90 or
flux/fluence could not be determined. The high-energy fluxes and fluences,
along with the T90s, were not readily available in the GCNs, and were instead
determined using the respective online GRB catalogs (Lien et al., 2016; Gruber
et al., 2014; von Kienlin et al., 2014; Narayana Bhat et al., 2016; von
Kienlin et al., 2020). To best differentiate SGRBs from LGRBs, we would
ideally have both the T90 and the spectral hardness of each burst. However, in
this work, we limit ourselves to classifications based solely on the T90s of
the bursts, with GRBs whose T90 is less than 2 s classified as SGRBs, and
those whose T90 is greater than 2 s classified as LGRBs. We do manually
compare our classifications with those found online though. The majority of
the GRBs we classify as LGRBs do not have available online classifications,
but their durations are $\gg 2$ s. Of the SGRBs in our sample, two SGRBs, GRB
190326A and GRB 190627A, are unique in that while they are short, they are
also spectrally soft, and SGRBs are expected to be spectrally hard (Markwardt
et al., 2019; Barthelmy et al., 2019). Thus, while we still classify them as
SGRBs, we note that they may be exceptions. Additionally, while GRB 190331A’s
T90 is 4.03 s, it is noted to likely be a SGRB in a GCN
Notice.333https://www.mpe.mpg.de/ jcg/grb190331A.html Thus, we include GRB
190331A in our SGRB sample.

##  4 Searching for Temporally and Spatially Coincident FRBs and GRBs

###  4.1 Search

Our primary interest is searching for GRBs and FRBs that are spatially and
temporally coincident. To search for spatial coincidence between a GRB and
FRB, we look for FRB and GRB pairs where the GRB’s 3$\sigma$ localization
overlaps with part of the 3$\sigma$ localization region of the FRB. Note that
the average 1$\sigma$ localization radius of our sample of GRBs is
0.04$\degree$ and the average 1$\sigma$ localization radius of our sample of
FRBs is 0.27$\degree$. The localization regions used in this analysis for our
FRB sample are determined using CHIME/FRB Metadata headers (for more details
on CHIME/FRB Metadata headers, see CHIME/FRB Collaboration et al., 2021). For
the 3$\sigma$ localization regions for both the GRBs and FRBs, we assume
$3\times 1\sigma$ localization. For our FRBs, we refer to this localization
radius as the 3$\sigma$ CHIME/FRB localization uncertainty. However, this
assumes that the localization regions are Gaussian, which is not true for
CHIME/FRB FRBs (see CHIME/FRB Collaboration et al. (2021) for more details on
this). The true localization regions are, however, highly complex multi-region
geometries (e.g., see Figure 6 in CHIME/FRB Collaboration et al., 2021) and
non-trivial to search over. Thus, rather than searching over the true
localization region for each FRB, we instead perform an additional, more
conservative search in which we re-do our joint temporal and spatial
coincidence search using a radius of 1$\degree$ for our localization regions
for all our FRBs (and hence a 3$\sigma$ radius of 3$\degree$). Then, for any
coincidences when using this method, we manually check the coincidences using
the true CHIME/FRB localization regions presented in CHIME/FRB Collaboration
et al. (2021). As a reference, the FWHM of the CHIME/FRB primary beam used in
this work is $\sim$1.6$\degree$ at 400-MHz, and the FWTM is $\sim$3$\degree$
at 400-MHz. Thus, searching within a 3$\degree$ radius is equivalent to
searching within a radius equal to the FWTM of the primary beam.

Temporally, we search for FRBs and GRBs that are coincident within periods up
to seven days. While the upper limit of seven days is chosen slightly
arbitrarily, we are primarily interested in coincident high-energy and radio
emission and hence a small time window around the high-energy emission.
Additionally, as shown in Fig. 1, the probability of having at least one
chance coincidence association for periods greater than 7 days is $\gtrapprox
5\%$ (the method for calculating this is discussed in detail below in Section
4.2) and thus it would be difficult to conclude that the association is
significant. The search of $\pm$ 7 days is the reason for the GRB sample
window extending from 2018 July 17 to 2019 July 8 rather than from 2018 July
25 to 2019 July 1.

Figure 1: Probability of having at least one temporal and spatial chance
coincidence (Pcc) as a function of the time difference in days between a given
GRB and FRB. Simulation is performed using our true sample of GRBs and a
simulated sample of FRBs. We assume a 3$\sigma$ localization radius of
0.81$\degree$ for our simulated FRBs.

When applying the joint temporal (up to seven days) and spatial criteria
(overlapping 3$\sigma$ CHIME/FRB localization radius), there are no GRBs
within our sample that are coincident with any of the FRBs from the first
CHIME/FRB catalog. The same result holds when using a 3$\sigma$ localization
radius of 3$\degree$ for the FRBs. However, when considering solely spatial
coincidence, and using the 3$\sigma$ CHIME/FRB localization uncertainties for
our FRBs, there are two GRBs in our sample that are consistent with two FRBs
from the first CHIME/FRB Catalog. These GRBs are GRB 180914B and GRB 181231A
with corresponding FRBs FRB 20190614B and FRB 20190109B, respectively. The
time differences, $\textrm{t}_{\textrm{FRB}}-\textrm{t}_{\textrm{GRB}}$, for
the events are 273 days and 10 days, respectively, and the spatial differences
are $0.45\degree$ and $1.29\degree$ between the most likely positions. The
total quadrature summed 1$\sigma$ uncertainties on the spatial positions for
the two pairs are $0.33\degree$ and $0.46\degree$, respectively.

###  4.2 Calculating Chance Probability of FRB-GRB Associations

To determine whether or not these two spatial coincidences are astrophysically
significant, we perform two different tests of the chance probability of two
coincident FRB-GRB pairs within our sample. First, we simulate a set of 81
GRBs uniformly distributed on the sphere. We also randomly distribute these
simulated GRBs over the time interval covered by the first CHIME/FRB catalog.
For the spatial error on this set of GRBs, we use the average 3$\sigma$
spatial error from our true sample of 81 GRBs, 0.12°. We then cross-check the
spatial positions of this sample of simulated GRBs with our real sample of 536
FRBs. For our real sample of FRBs, we use the 3$\sigma$ CHIME/FRB localization
uncertainties. Performing 1000 different simulations, we find there is a 41%
chance of detecting two or more pairs of spatially coincident GRBs and FRBs
within our given samples.

For our second test, instead of simulating our GRBs, we simulate a set of 536
FRBs. Using the first CHIME/FRB catalog, we generate a 2D probability
distribution in declination and time. We independently sample from the
declination and time axis in order to produce a set of declinations and
timestamps, and use our timestamps to determine a distribution of simulated
RAs within $\sim$1.6$\degree$ of the meridian as this is the FWHM of the
primary beam at 400-MHz. We combine the sampled declinations with these RAs to
produce a simulated set of FRBs. For the spatial error on each FRB, we use a
3$\sigma$ localization radius of 0.81$\degree$. We then cross-check the
spatial positions of this sample of 536 simulated FRBs with our true sample of
81 GRBs. Performing 1000 simulations, we find there is a 65$\%$ chance of
detecting two or more spatially coincident FRBs and GRBs within our given
samples. In Figure 2, we show the chance probability of having at least two
spatial FRB-GRB pairs as a function of the 1$\sigma$ FRB uncertainty. Our
1$\sigma$ localization uncertainties would need to be $\ll 1\degree$ in order
for an association to be significant.

This method of simulating a set of FRBs and comparing it with the known sample
of GRBs is also used to determine the chance probability of having both a
temporal and spatial association as a function of the time differences (e.g.,
$|\textrm{t}_{\textrm{FRB}}-\textrm{t}_{\textrm{GRB}}|$) as discussed earlier
in Section 4.1 and shown in Figure 1.

Figure 2: Probability of having at least two spatial chance coincidences (Pcc)
as a function of the 1$\sigma$ uncertainty on the FRB’s position. Simulation
is performed using our true sample of GRBs for which the average 3$\sigma$
uncertainty is $0.12\degree$, and a simulated sample of FRBs. The leftmost red
line indicates the average positional uncertainty of the future
CHIME/Outriggers Experiment ($\sim$ 50 mas; Cassanelli et al., 2022), the
second leftmost red line indicates the average positional uncertainty of the
CHIME/FRB baseband system ($<1$ arcmin; Michilli et al., 2021), and the
rightmost red line indicates the average positional uncertainty of the
CHIME/FRB header localizations ($\sim 0.27\degree$).

Both of these methods argue that the two spatially-coincident pairs found are
not astrophysically significant. We therefore conclude that there are no
significant detections of coincident FRBs and GRBs within our sample.

##  5 Radio Upper Limits

###  5.1 Determining Radio Upper Limits

In addition to searching for coincident FRB and GRB pairs, we also place
limits on the possible 400- to 800-MHz radio flux for GRBs which are within
the FOV of CHIME/FRB at a point either before, at the time of, or after their
high-energy emission. There are seven GRBs that are within the FOV of
CHIME/FRB at the time of their high-energy emission, and another 32 that are
within the FOV either six hours before or 12 hours after the high-energy
emission (these time windows are discussed in detail in Section 5.2). These 39
GRBs are also detected on nominal operational days, and have radio flux limits
(discussed in detail below) $<100$ kJy. Here, we limit the FOV to within
$\sim$17$\degree$ of CHIME’s meridian, as the CHIME beam is well-modeled
within this range, and it is outside the scope of this work to model the beam
at locations much further from meridian (CHIME/FRB Collaboration et al., 2021;
CHIME Collaboration et al., 2022; The CHIME Collaboration et al., 2022). We
calculate an upper limit on the FRB-like radio emission by combining our beam
sensitivity at the location of the GRB with the flux to signal-to-noise ratio
(S/N) of a spatially nearby calibrator FRB.

More specifically, to calculate an upper limit on radio emission at a specific
time and sky position, we would ideally have a near-simultaneous and near-on-
the-sky source for our flux calibration. As described in CHIME/FRB
Collaboration et al. (2018), CHIME/FRB performs flux calibration using steady
sources. These sources are used to determine a conversion between the
beamformed intensity units (e.g., the output units after the FFT beamforming)
and the conventional flux units of Jy. However, this process is not dependent
on, nor does it account for, the CHIME/FRB detection S/N. As our goal is to
convert a given minimum detection S/N to a flux or fluence, a new calibration
process is needed for this work.

Instead of using steady-sources, we use previously detected and calibrated
FRBs as our GRB-flux calibrators. Ideally, these FRBs would be near-
simultaneous and near-on-the-sky to the GRB. However, as CHIME/FRB only
detects $\sim 2$ to 3 FRBs per day, it is not possible to satisfy both
criteria. Instead, we choose an FRB nearby in declination (e.g., $<1\degree$
away444This angular separation is equivalent to the width of $\sim$2-3 N-S
CHIME/FRB beams.) to a given GRB and use models of our beam shape to account
for slight differences in position between the two sources (Ng et al., 2017;
Masui et al., 2019; CHIME/FRB Collaboration et al., 2021; The CHIME
Collaboration et al., 2022).

We do not restrict the timestamp of the calibrator FRB and instead use system
sensitivity metrics to account for temporal differences by scaling our radio
limits by the ratio of the system sensitivity at the time of the FRB with that
at the time of the GRB. These system sensitivity metrics are determined using
the daily root mean square (RMS) noise of pulsar detections with CHIME/FRB,
with a detailed description of these metrics provided by Josephy et al. (2019)
and Fonseca et al. (2020). We do not account for fluctuations in the system
sensitivity on a timescale smaller than one day as these metrics were not
consistently available. However, as discussed by Josephy et al. (2021),
bandwidth effects due to RFI and correlator cluster health typically only
result in average sensitivity losses of $\sim$10$\%$. While CHIME/FRB has
developed a parallel injections system (Merryfield et al., 2022) to quantify
further aspects of the system’s sensitivity to given parameters, this has not
been applied in this work. However, as discussed by CHIME/FRB Collaboration et
al. (2021), the only known major bias to-date of the CHIME/FRB pipeline is
against highly scattered bursts. This does not present a large bias against
redshift though, as the two main contributors to the observed scattering
timescales of FRBs are the circumburst environment and intervening
galaxies/galaxy halos (Chawla et al., 2021), and scattering in both locations
is inversely proportional to redshift (Cordes et al., 2022).

There is an additional, unaccounted DM delay between any detection of high-
energy emission and the time at which a simultaneous radio burst would be
detected by CHIME/FRB. Such a DM delay can, in principle, be estimated if the
distance or redshift of the GRB is known. Unfortunately, only eight of our 81
GRBs have readily available redshifts. For the GRBs without redshifts, we try
three different redshift/DM estimates. First, we use the median redshift of
the respective SGRB or LGRB redshift distribution as an approximate redshift
for each GRB ($z=0.82$ for SGRBs and $z=1.71$ for LGRBs). The distributions
are created using the online GRB catalog GRBWeb (Coppin, 2022). Second, we use
the minimum redshift from the SGRB and LGRB redshift distributions (0.038 for
SGRBs and and 0.0085 for LGRBs). Lastly, instead of using a redshift, we use
1.2$\times$ the maximum DM detected in the first CHIME/FRB catalog (3645 pc
cm-3). The extra 20% was chosen arbitrarily to allow for an event detected
further out into the tail of the DM distribution than CHIME/FRB has sampled
thus far.

For cases in which we have a redshift rather than a DM, e.g., either the true
redshift or an approximate redshift, we convert the redshift to a DM by
assuming:

$\textrm{DM}_{\textrm{tot}}=\textrm{DM}_{\textrm{MWdisk}}+\textrm{DM}_{\textrm{MWhalo}}+\textrm{DM}_{\textrm{IGM}}+\textrm{DM}_{\textrm{Host}}$
(1)

where $\textrm{DM}_{\textrm{MWdisk}}$ is the Milky Way disk (MW) contribution,
$\textrm{DM}_{\textrm{MWhalo}}$ is the MW halo contribution,
$\textrm{DM}_{\textrm{IGM}}$ is the intergalactic medium contribution (IGM),
and $\textrm{DM}_{\textrm{Host}}$ is the contribution from the host galaxy. We
calculate $\textrm{DM}_{\textrm{MW}}$ for each individual GRB by using the
average of the NE2001 (Cordes & Lazio, 2002) and the YMW16 (Yao et al., 2017)
electron density models along the respective LOS. We assume a
$\textrm{MW}_{\textrm{Halo}}$ DM of 50 pc cm-3 (Prochaska & Zheng, 2019),
estimate $\textrm{DM}_{\textrm{IGM}}$ using the Macquart relation (Macquart et
al., 2020), and assume a host galaxy DM of 175 pc cm-3/(1+z) as suggested by
Chawla et al. (2021). By using the minimum of the redshift distribution and
1.2$\times$ the maximum DM from the first CHIME/FRB catalog in addition to the
median redshift, our DM ranges should cover uncertainties in the host galaxy
DM and MW halo DM.

While $\textrm{DM}_{\textrm{tot}}$ depends on the Galactic longitude, Galactic
latitude, and redshift of the GRB, the average $\textrm{DM}_{\textrm{tot}}$
when using the median redshift of the respective distribution is $\sim$1142 pc
cm-3 for SGRBs and $\sim$1845 pc cm-3 for LGRBs corresponding to delays of
$\sim$29 s and $\sim$47 s at 400-MHz, respectively. The average
$\textrm{DM}_{\textrm{tot}}$ when using the minimum redshifts are $\sim$300 pc
cm-3 for SGRBs and $\sim$400 pc cm-3 for LGRBs. The upper limits we present
are the most conservative limits associated with the timestamps from the three
possible DM delays. Note that in the vast majority of cases ($>$95$\%$), the
upper limits calculated using either the minimum redshift or maximum DM are
still within the 3$\sigma$ uncertainty of the upper limit calculated using the
median redshift.

Due to the differences in position between our calibrator FRB and the GRB of
interest, we must scale the flux of the FRB by differences in CHIME/FRB’s beam
sensitivity at the two positions. CHIME/FRB’s spatial sensitivity can be
described by a combination of two different beam models: that of the
individual antennas which we will refer to as the primary beam model
(discussed in detail by CHIME/FRB Collaboration et al., 2021; The CHIME
Collaboration et al., 2022), and that of the interferometric formed beams
(discussed in Ng et al., 2017). The beam response of CHIME/FRB in the far
sidelobes is still being characterized, and thus an FRB detected in the
sidelobes of CHIME/FRB could both have an inaccurately determined position and
a significantly underestimated flux. In order to avoid using FRBs detected in
the sidelobes as our calibrator FRBs, we impose the condition that the
bandwidth of our calibrator FRBs be sufficiently large (e.g., larger than
$3/4$ of our 400-MHz bandwidth), as the bandwidth of an FRB detected in the
sidelobe would be significantly modulated (Hsiu-Hsien et al. in prep.).

We use our calibrator FRB to scale the FRB’s measured flux/fluence (see
CHIME/FRB Collaboration et al., 2021, for details on the flux/fluence
measurements from the first CHIME/FRB catalog) to detection S/N. However, as
described in detail by CHIME/FRB Collaboration et al. (2021), the flux and
fluence for each FRB is calculated assuming the FRB is located on the
meridian. Thus, the flux/fluence is the estimated flux/fluence for a given S/N
burst on the meridian. Rather than correcting the flux/fluence to S/N for the
true position of the FRB, and then scaling it to the location of the GRB, we
instead directly scale the flux/fluence to S/N of the calibrator FRB by the
ratio of the primary beam sensitivity at meridian with the primary beam
sensitivity at the location of the GRB. Additionally, we must scale this ratio
by the interferometric formed beam sensitivity at the location of the GRB. The
published fluxes and fluences for FRBs also assume the FRBs are at the centers
of the formed beams, so we take the formed beam sensitivity for the FRB to be
unity.

As both the primary and formed beams are frequency dependent, we must make
assumptions on the spectral shape of the emission for the calibrator FRB and
the GRB. As in the first CHIME/FRB catalog (CHIME/FRB Collaboration et al.,
2021), we assume a flat spectral index for all of our calibrator FRBs. We
similarly assume a flat spectrum for all of our GRBs, as there is not yet any
observational evidence to guide this. We note that the spectral indices used
in the emission models presented by Rowlinson & Anderson (2019) all assume a
negative spectral index, although the magnitude of the spectral index is not
well constrained in any model. Additionally, while our FRB sample has measured
bandwidths555We do not correct these bandwidths for instrumental effects in
this work., we assume a bandwidth of 400-MHz for the GRBs. This assumption
affects our estimates of our beam sensitivity, as these sensitivities are
highly frequency dependent.

Assuming conservatively that CHIME/FRB is sensitive to bursts with a S/N
threshold of 10, we scale the flux/fluence to detection S/N of our calibrator
FRB to a S/N of 10. Combining the calibrator FRB’s flux to S/N ratio, the
system sensitivities, and the beam sensitivities, the upper limit on the radio
flux in the 400- to 800-MHz CHIME/FRB band is given by:

$\textrm{Flux}_{\textrm{GRB}}=10\times\frac{\textrm{Flux}_{\textrm{FRB}}}{\textrm{S/N}_{\textrm{FRB}}}\times\frac{\textrm{B}_{\textrm{M}}}{\textrm{B}_{\textrm{GRB}}}\times\frac{1}{\textrm{F}_{\textrm{GRB}}}\times\frac{\Delta\textrm{S}_{\textrm{Sys,GRB}}}{\Delta\textrm{S}_{\textrm{Sys,FRB}}}$
(2)

where B is the primary beam response, F is the formed beam response,
$\Delta\textrm{S}_{\textrm{Sys}}$ is the system sensitivity666The system
sensitivity is determined by RMS noise, so a higher value for the system
sensitivity means we were less sensitive on that day, thus the ratio is
flipped for this one quantity., and M stands for meridian. A similar equation
can be applied for an upper limit on the fluence of a radio burst from a GRB.
While not shown in Eq. (2), we scale our limits to an assumed radio burst
width of 10-ms. Additionally, for all GRBs (or FRB calibrators) within
10$\degree$ of the Galactic plane, we scale Eq. (2) by the ratio of the sky
temperature at the two locations using the 2014 Haslam all-sky continuum map
at 408 MHz (Remazeilles et al., 2015). We assume an observing frequency of
600-MHz for CHIME/FRB and a brightness temperature spectral index of $-2.5$
for the Galactic synchrotron emission when calculating the sky temperatures
(Bennett et al., 2003).

The uncertainties on the upper limits are dominated by the uncertainties of
the fluxes of our FRB calibrators, but are also determined using the
uncertainties on our primary beam model, our system sensitivity, and the GRB
localization. To quantify the effect of the GRB localization uncertainties on
our final flux and fluence upper limits, we run a Monte Carlo (MC) in which
for every timestamp for a given GRB, we re-calculate the formed beam and
primary beam sensitivities at 1000 different locations within the 3$\sigma$
uncertainty region of the GRB. We quantify the effect that this MC has on our
final quoted upper limits by calculating the difference in the GRB’s flux
density limit uncertainty before and after this MC step as compared with the
GRB’s flux density limit e.g.,
$|\sigma_{\textrm{initial}}-\sigma_{\textrm{final}}|/\textrm{flux}_{\textrm{GRB}}$.
In all but two cases (discussed below) this difference is negligible e.g.,
$|\sigma_{\textrm{initial}}-\sigma_{\textrm{final}}|/\textrm{flux}_{\textrm{GRB}}<0.05$.
This is expected since the average 3$\sigma$ localization uncertainty of our
GRB sample is small e.g., 0.12$\degree$. The two cases for which the above
does not hold are GRB 181120A which has the largest 1$\sigma$ positional error
of our sample of 0.5$\degree$ and GRB 180914B which has the second largest
1$\sigma$ positional error of 0.16$\degree$. Thus, for these two GRBs, we
include an additional MC step when determining the flux and fluence upper
limit uncertainties.

###  5.2 Results

While we do not find any GRBs that are spatially and temporally coincident
with our FRBs, ten GRBs are within the FOV of CHIME at the time of their high-
energy detection (after accounting for the additional dispersion delay).
However, two of these high-energy bursts occurred during periods of non-
nominal telescope operations (GRB 181002A and GRB 180930A), and are removed
from this analysis. A third burst is located at the edge of CHIME’s FOV (GRB
190326A), and is removed due to an estimated flux limit $>$100 kJy due to the
lack of constraining power from such a high flux density limit. In addition to
the remaining seven bursts, there are 32 bursts which are within the FOV
(assumed to be within $\sim$17$\degree$ of CHIME’s meridian as discussed above
in Section 5.1) at points either before or after the time of the high-energy
emission, occur during nominal telescope operations, and for which we can
calculate radio flux limits $<100$ kJy.

We restrict our radio limits to six hours (21.6 ks) before the high-energy
emission from each GRB and 12 hours (43.2 ks) after. We choose this time frame
based firstly on the models for FRB-like bursts, which predict most (but not
all) pre-GRB radio emission in the seconds before merger and post-GRB emission
that could last indefinitely, and secondly on the times at which GRBs are
within the FOV of CHIME/FRB. We do not have any SGRBs that are within the FOV
of CHIME/FRB in the period of $\sim 4$ hours prior to their high-energy
emission. However, given that pre-SGRB radio emission could commence
decades/centuries prior to the merger (Zhang, 2020), we arbitrarily extend our
range to 6 hours pre-burst. We do not extend our search to the time frame of
weeks or years as this is outside the scope of this work.

We list the seven GRBs for which we can constrain the radio emission at the
time of the high-energy burst in Table 1. Note that while these GRBs are
within the FOV of CHIME/FRB, they are not necessarily close to the meridian at
the time of their high-energy emission, and thus the limits can be $>$1 kJy.
In Table LABEL:table:_individual_GRBs, we list our most constraining limits
for these seven GRBs at times not necessarily equal to that of the high-energy
emission, along with the the most constraining limits for the 32 other GRBs.

Table 1: CHIME/FRB Upper Limits on Coincident Radio Emission

Name | Type aaSignifies whether the GRB is a SGRB (S) or LGRB (L). | Flux bb End time for this specific time segment for which we can constrain the flux to be less than that listed in the first column of the table e.g., 10 Jy, 100 Jy, or 1 kJy. A positive time indicates a time post-high-energy emission, while a negative time indicates a time pre-high-energy emission. | Fluence Ratio ccfootnotemark:  |  $\eta$ ddConstraints on $\epsilon_{r}$ in the model by Totani (2013) assuming $z<1$ for the SGRBs used to determine the limits. For this model, we also assume that the GRB would be have the following properties: $B_{15}=1$, $R_{6}=1$, and $P_{-3}=1$.
---|---|---|---|---
|  | (Jy) | ($10^{8}$ Jy ms | ($10^{-11}$)
|  |  | erg-1 cm2) |
GRB 180721A | L |  $<$120 |  $<$20 |  $<$700
GRB 181213A | L |  $<$1200 |  $<$80 |  $<$3000
GRB 190219A | L |  $<$3000 |  $<$40 |  $<$2000
GRB 190613A | L |  $<$800 |  $<$30 |  $<$1000
GRB 190613B | L |  $<$600 |  $<$20 |  $<$700
GRB 180914A | L |  $<$6000 |  $<$5 |  $<$200
GRB 181120A | L |  $<$2000 |  $<$50 |  $<$2000

bb End time for this specific time segment for which we can constrain the flux
to be less than that listed in the first column of the table e.g., 10 Jy, 100
Jy, or 1 kJy. A positive time indicates a time post-high-energy emission,
while a negative time indicates a time pre-high-energy emission.

Upper limit on the 400- to 800-MHz radio flux at the 99$\%$ confidence level
for a 10-ms radio burst at the time of the high-energy emission (including the
dispersion delay).

ccfootnotemark:

Upper limit on the 400- to 800-MHz radio-to-high-energy fluence ratio at the
99$\%$ confidence level for a 10-ms radio burst at the time of the high-energy
emission (including the dispersion delay).

ddConstraints on $\epsilon_{r}$ in the model by Totani (2013) assuming $z<1$
for the SGRBs used to determine the limits. For this model, we also assume
that the GRB would be have the following properties: $B_{15}=1$, $R_{6}=1$,
and $P_{-3}=1$.

Upper limit on $\eta$ (unitless radio-to-high-energy fluence ratio assuming a
400-MHz radio bandwidth) at the 99$\%$ confidence level for a 10-ms radio
burst at the time of the high-energy emission (including the dispersion
delay). The energy range for GRBs detected by Swift/BAT is 15 to 150 keV, and
that for those detected by Fermi/GBM is 10 to 1000 keV.

bb End time for this specific time segment for which we can constrain the flux
to be less than that listed in the first column of the table e.g., 10 Jy, 100
Jy, or 1 kJy. A positive time indicates a time post-high-energy emission,
while a negative time indicates a time pre-high-energy emission.

Time (in ks) before (negative time) or after the detected high-energy emission
for which the radio flux/fluence limits apply. These times include an
estimated dispersion delay.

ccfootnotemark:

Upper limit on the 400- to 800-MHz radio flux at the 99$\%$ confidence level
for a 10-ms radio burst.

ddConstraints on $\epsilon_{r}$ in the model by Totani (2013) assuming $z<1$
for the SGRBs used to determine the limits. For this model, we also assume
that the GRB would be have the following properties: $B_{15}=1$, $R_{6}=1$,
and $P_{-3}=1$.

Upper limit on the 400- to 800-MHz radio-to-high-energy fluence ratio at the
99$\%$ confidence level for a 10-ms radio burst.

eeSame as in c except for $z<0.1$.

Upper limit on $\eta$ (unitless radio-to-high-energy fluence ratio assuming a
400-MHz radio bandwidth) at the 99$\%$ confidence level for a 10-ms radio
burst at the time of the high-energy emission. The energy range for GRBs
detected by Swift/BAT is 15 to 150 keV, and that for those detected by
Fermi/GBM is 10 to 1000 keV.

Table 2: Most Constraining CHIME/FRB Upper Limits on Possible Radio Emission from 39 SGRBs/LGRBs Name | TypeaaSignifies whether the GRB is a SGRB (S) or LGRB (L). | Timebb End time for this specific time segment for which we can constrain the flux to be less than that listed in the first column of the table e.g., 10 Jy, 100 Jy, or 1 kJy. A positive time indicates a time post-high-energy emission, while a negative time indicates a time pre-high-energy emission. | Fluxccfootnotemark:  | Fluence RatioddConstraints on $\epsilon_{r}$ in the model by Totani (2013) assuming $z<1$ for the SGRBs used to determine the limits. For this model, we also assume that the GRB would be have the following properties: $B_{15}=1$, $R_{6}=1$, and $P_{-3}=1$. |  $\eta$eeSame as in c except for $z<0.1$.
---|---|---|---|---|---
|  | (ks) | (Jy) | ($10^{8}$ Jy ms | ($10^{-11}$)
|  |  |  | erg-1 cm2) |
GRB 181123B | S | $38.016$ |  $<$2 |  $<$1 |  $<$50
GRB 190326A | S | $39.168$ |  $<$2 |  $<$4 |  $<$140
GRB 190427A | S | $27.774$ |  $<$1.6 |  $<$0.9 |  $<$40
GRB 190610A | S | $22.806$ |  $<$20 |  $<$3 |  $<$110
GRB 190627A | S | $-19.236$ |  $<$20 |  $<$13 |  $<$500
GRB 180720B | L | $-7.992$ |  $<$30 |  $<$0.02 |  $<$0.6
GRB 180721A | L | $0.396$ |  $<$10 |  $<$1 |  $<$60
GRB 180805B | L | $-0.972$ |  $<$80000 |  $<$5000 |  $<$200000
GRB 180812A | L | $-19.47$ |  $<$2 |  $<$0.11 |  $<$4
GRB 180818B | L | $16.692$ |  $<$2 |  $<$0.05 |  $<$2
GRB 180823A | L | $17.232$ |  $<$1.5 |  $<$0.015 |  $<$0.6
GRB 180904A | L | $20.994$ |  $<$2 |  $<$0.8 |  $<$30
GRB 180905A | L | $3.924$ |  $<$20 |  $<$0.3 |  $<$11
GRB 181013A | L | $-21.6$ |  $<$400 |  $<$130 |  $<$5000
GRB 181023A | L | $-13.11$ |  $<$4 |  $<$0.6 |  $<$20
GRB 181027A | L | $-18.54$ |  $<$3 |  $<$0.2 |  $<$7
GRB 181125A | L | $42.726$ |  $<$6000 |  $<$900 |  $<$40000
GRB 181213A | L | $21.708$ |  $<$2 |  $<$0.13 |  $<$5
GRB 181224A | L | $-12.318$ |  $<$6 |  $<$0.8 |  $<$30
GRB 181228A | L | $7.032$ |  $<$5 |  $<$0.04 |  $<$1
GRB 190103B | L | $-20.892$ |  $<$3 |  $<$0.02 |  $<$0.7
GRB 190109A | L | $-8.028$ |  $<$4 |  $<$0.06 |  $<$3
GRB 190109B | L | $-21.078$ |  $<$1200 |  $<$200 |  $<$9000
GRB 190114A | L | $5.766$ |  $<$8 |  $<$0.6 |  $<$20
GRB 190202A | L | $30.99$ |  $<$3 |  $<$0.03 |  $<$1
GRB 190203A | L | $31.764$ |  $<$3 |  $<$0.014 |  $<$0.6
GRB 190204A | L | $-21.084$ |  $<$1100 |  $<$7 |  $<$300
GRB 190211A | L | $-18.828$ |  $<$2 |  $<$0.3 |  $<$10
GRB 190219A | L | $10.83$ |  $<$11 |  $<$0.1 |  $<$6
GRB 190331A | L | $-17.568$ |  $<$5 |  $<$1 |  $<$40
GRB 190424A | L | $39.678$ |  $<$7 |  $<$0.4 |  $<$14
GRB 190515B | L | $25.002$ |  $<$3 |  $<$0.2 |  $<$7
GRB 190613A | L | $-4.998$ |  $<$3 |  $<$0.11 |  $<$5
GRB 190613B | L | $0.492$ |  $<$30 |  $<$0.8 |  $<$30
GRB 190630B | L | $-6.006$ |  $<$2 |  $<$0.8 |  $<$30
GRB 190701A | L | $19.698$ |  $<$4 |  $<$0.3 |  $<$11
GRB 180914A | L | $-2.13$ |  $<$20 |  $<$0.013 |  $<$0.5
GRB 181120A | L | $-3.534$ |  $<$10 |  $<$0.15 |  $<$6
GRB 190530A | L | $43.188$ |  $<$500 |  $<$0.12 |  $<$5
GRB 180914B | L | $43.194$ |  $<$100 |  $<$0.006 |  $<$0.2
|  |  |  |  |

In Figure 3, we show an example of the 99$\%$ confidence radio flux upper
limits for one of our GRBs, GRB 190613B. The limit at the time of the high-
energy emission (dispersion delay accounted for) is shown as a red star (and
is given in Table 1), while limits before and after the GRB (again, dispersion
delay accounted for) are depicted using a solid black line. The shape of the
limits time series in Figure 3 is solely due to the transit of the GRB
overhead CHIME and our beam shape, with the central minimum value of Figure 3
given in Table LABEL:table:_individual_GRBs.

The evolution of the limits for other GRBs is highly dependent on both the
time of the high-energy emission relative to when the source transited
directly above CHIME, and the declination of the GRB. Sources closer to the
North Celestial Pole are above the horizon at CHIME/FRB for longer than
sources with lower declinations e.g., a source at 10$\degree$ in declination
is within the primary beam for five minutes while a source at 80$\degree$ is
within the primary beam for an hour777This source would be visible for 30
minutes during an upper transit and 30 minutes during a lower transit for a
total of one hour.. Thus, CHIME/FRB has more exposure to, and thus can
calculate limits for longer periods of time, for GRBs at higher rather than
lower declinations.

Figure 3: Upper limits on the 400- to 800-MHz radio flux for GRB 190613B
assuming a 10-ms radio burst. Radio flux limits (99$\%$ confidence) before and
after the high-energy emission (dispersion delay accounted for) are calculated
every six seconds. Due to the frequency and continuity of these limits, we
show a black solid line rather than showing each individual upper limit. The
radio flux limit at the time of the high-energy emission (dispersion delay
accounted for) is shown as a red star. All flux density limits are for 2019
June 13, with the hour, minute, and second for which the limit applies shown
on the x-axis. The high-energy burst was detected at 10:47:02.05 UT on 2019
June 13 (no dispersion correction) by the Swift/BAT telescope.

Of the 39 GRBs for which we can constrain the radio emission, five are SGRBs
detected by Swift/BAT, 30 are LGRBs detected by Swift/BAT, and four are LGRBs
detected by Fermi/GBM. Note these classifications are made solely based on the
T90s of the bursts. For Swift/BAT, we separate the SGRBs from the LGRBs, and
show our 3$\sigma$ flux limits (calculated every six seconds) as a function of
time relative to the GRB arrival time in Figures 4 and 5. Rather than showing
the flux limit for all GRBs at each relative timestamp, we only show our most
constraining FRB-like radio limit at each six second timestamp. Thus, each
plot only contains one limit per timestamp with multiple GRBs used to
determine the full set of limits shown. Note that a given GRB is often used to
determine a handful of radio flux limits for adjacent timestamps, e.g., GRB
181123B is used to place the most constraining SGRB Swift/BAT flux limits for
$-21.6$ ks to $-17.4$ ks while GRB 190326A is used to place the most
constraining flux limits from 20.6 ks to 22.2 ks. Which GRB’s limits are the
most constraining at a given timestamp, along with how many timestamps the GRB
is used for, is dependent on the declination of the GRB and the location of
the GRB relative to the meridian at the time of the high-energy emission.

For Figure 4 (and Figure 7; discussed below), all five SGRBs in our sample
detected by Swift/BAT are used to calculate the most constraining flux (and
fluence ratio; discussed below) limits for at least one timestamp. However,
while all LGRBs in our sample detected by Swift/BAT are considered for Figure
5 (and Figure 8; discussed below), only a subset of the 30 LGRBs contribute to
the most conservative flux (and fluence ratio) limits. These LGRBs are: GRB
190219A, GRB 190701A, GRB 190613B, GRB 181213A, GRB 180818B, GRB 180904A, GRB
180721A, GRB 190211A, GRB 190109A, GRB 190203A, GRB 190613A, GRB 190114A, GRB
190515B, GRB 181228A, GRB 181027A, GRB 181224A, GRB 190331A, GRB 190424A, GRB
190630B, GRB 180823A, GRB 180812A, GRB 190103B, GRB 190202A, GRB 180905A, and
GRB 181023A.

Additionally, rather than listing all the limits shown in Figures 4 and 5 in
Tables LABEL:table:_SGRBs_Swift and LABEL:table:_swift_long (presented in
Appendix A), we give the range of flux limits for a given 1 ks time bin e.g.,
the minima and maxima of the limits within that time bin. These are the minima
and maxima for a given time bin considering all GRBs of that given category
(e.g., Swift/BAT SGRBs).

We show our 3$\sigma$ flux limits for LGRBs detected by Fermi/GBM888There are
no SGRBs in our sample detected by Fermi/GBM. in Figure 6 and list the limits
in Table 6 in Appendix A. For each timestamp relative to the GRB, we again
only show our most constraining limit from the respective sample of GRBs
within a 1 ks time period. Similar to Figure 4, all LGRBs detected by
Fermi/GBM in our sample are used to calculate the flux (and fluence ratio)
limits for at least one timestamp in Figure 6 (and Figure 9; discussed below).
Note that the large gaps in our upper limits in Figures 4, 6, 7, and 9 are
solely due to a lack of GRBs within the CHIME/FRB FOV at certain timestamps.

In addition to flux limits, we also calculate radio-to-high-energy fluence
ratios (both in units of Jy ms erg-1 cm2 and as a unitless quantity $\eta$
assuming a 400-MHz bandwidth) using the available high-energy fluences from
the online Swift/BAT999https://swift.gsfc.nasa.gov/archive/grb_table/ and
Fermi/GBM101010https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html
catalogs. Similar to our flux limits, we show these ratios for Swift/BAT
SGRBs, Swift/BAT LGRBs, and Fermi/GBM LGRBs in Figures 7, 8 and 9, and present
them in Tables LABEL:table:_SGRBs_Swift, LABEL:table:_swift_long, and 6,
respectively. Here, which GRB’s limits are the most constraining at a given
timestamp, along with how many timestamps the GRB is used for, is also
dependent on the high-energy fluence of the GRB.

Figure 4: Radio flux limits at the 99$\%$ confidence level for SGRBs within
our sample detected by Swift/BAT. Upper limits are calculated every six
seconds starting six hours (21.6 ks) prior to the GRB until 12 hours (43.2 ks)
after the GRB, with the time shifted to account for the necessary DM delay.
For each timestamp, we show our most constraining limit from all SGRBs
detected by Swift/BAT within the FOV of CHIME/FRB at that relative time.
Figure 5: Same as in Figure 4, except for LGRBs within our sample that are
detected by Swift/BAT.  Figure 6: Same as in Figure 4, except for LGRBs within
our sample that are detected by Fermi/GBM. Figure 7: Same as in Figure 4,
except the radio-to-high-energy fluence ratio limits for SGRBs within our
sample that are detected by Swift/BAT. We show the fluence ratios in units of
Jy ms erg-1 cm2 and as unitless quantities ($\eta$) assuming a 400-MHz
bandwidth.  Figure 8: Same as in Figure 4, except the radio-to-high-energy
fluence ratio limits for LGRBs within our sample that are detected by
Swift/BAT. Figure 9: Same as in Figure 4, except the radio-to-high-energy
fluence ratio limits for LGRBs within our sample that are detected by
Fermi/GBM. We show the fluence ratios in units of Jy ms erg-1 cm2 and as
unitless quantities ($\eta$) assuming a 400-MHz bandwidth.

##  6 Discussion

###  6.1 SGRBs

There are a number of models predicting FRB-like radio emission prior to a
SGRB, but these models predict radio emission in the $\sim$ms before the NS-NS
or NS-BH merger111111We can not place any radio limits at these times as no
SGRBs were close enough to the meridian. (e.g., Mingarelli et al., 2015;
Yamasaki et al., 2018; Wada et al., 2020; Sridhar et al., 2021; Carrasco et
al., 2021), predict FRB-like radio emission that would be misaligned with the
GRB jet (e.g., Piro, 2012; Wang et al., 2016; Sridhar et al., 2021), or
predict radio emission that would be too weak to be visible by CHIME/FRB
(e.g., Piro, 2012; Wang et al., 2016). The model that predicts FRB-like radio
emission that might be detectable by CHIME/FRB is that by Zhang (2020) in
which FRB-like radio emission is generated prior to the merger of two NS due
to interactions between their magnetospheres. This emission would start
decades to centuries prior to the merger, with the bursting activity
increasing as the two NSs inspiral. While there is not any quantitative
prediction for the radio flux of this model, it is possible that our limits
could constrain it, with our best pre-SGRB limit of $<20$ Jy at $\sim$5 hours
(18.6 ks) prior to the high-energy emission (see Table
LABEL:table:_SGRBs_Swift).

After the SGRB, if the merger results in a NS, then both Pshirkov & Postnov
(2010) and Totani (2013) predict that FRB-like emission is possible from
magnetic braking in a manner similar to that of pulsar emission. As derived by
Totani (2013) and then expanded on by Rowlinson & Anderson (2019), the
predicted flux density from this pulsar-like emission is:

$F_{\nu}=8\times
10^{7}\nu_{\textrm{obs}}^{-1}\epsilon_{r}D^{-2}B_{15}^{2}R_{6}^{6}P_{-3}^{-4}\textrm{Jy}$
(3)

where $\nu_{\textrm{obs}}$ is the observing frequency in MHz, $\epsilon_{r}$
the efficiency of converting energy into radio emission, $D$ the distance in
Gpc, $B_{15}$ the magnetic field strength in units of $10^{15}$ G, $R_{6}$ the
radius of the NS in units of $10^{6}$ cm, and $P_{-3}$ the NS spin period in
units of $10^{-3}$ s. While all of these factors except for observing
frequency are unknown for our sample of GRBs121212One of the SGRBs in our
sample, GRB 190627A, has a redshift of 1.94. However, this SGRB was only
within the FOV of CHIME/FRB prior to the high-energy emission, and thus cannot
be used to constrain post-SGRB radio emission models., we assume a millisecond
magnetar and hence set $B_{15}=1$, $R_{6}=1$, and $P_{-3}=1$. We show the
predicted flux density for a range of redshifts and $\epsilon_{r}$ in Figure
10. While our chosen parameters describe a millisecond magnetar, a slower
magnetar could generate the same level of radio emission if the magnetic field
strength were appropriately scaled up. Note that our limits assume a burst
width of 10-ms, which is unrealistically long for a millisecond magnetar.
However, as the flux scales by the square root of the width, this will only
result in a factor of $\sim$3 difference in our flux/fluence limits.

Assuming the surrounding environment is transparent to MHz radio emission at
$\sim$8 to 12 hours (28.8 to 43.2 ks) post-SGRB, we set limits on the radio
efficiency, $\epsilon_{r}$, of Eq. (3) using some of our most constraining
post-SGRB limits. We list the time periods for which we can constrain the
radio emission to $<$ 10 Jy, $<$ 100 Jy, and $<$ 1 kJy in Table 3.
Additionally, using our above assumptions, we calculate constraints on
$\epsilon_{r}$ for this model assuming two different redshifts for our SGRBs,
$z<1$ and $z<0.1$. Note that of the 74 SGRBs in the online GRB catalog GRBWeb
(Coppin, 2022), 46 have redshifts $<1$ and six have redshifts $<0.1$. While we
are able to constrain $\epsilon_{r}$ over $\sim$hour-long timescales, it is
possible that a radio burst from a GRB could either be a) fainter than our
limits or b) occur outside the time frame of our limits. Thus $\epsilon_{r}$
is only constrained by our work for very specific timestamps.

Table 3: Duration of SGRB Flux Limits and Model Constraints

Flux Limit | Start timeaaStart time for a specific time segment for which we can constrain the flux to be less than that listed in the first column of the table e.g., 10 Jy, 100 Jy, or 1 kJy. A positive time indicates a time post-high-energy emission, while a negative time indicates a time pre-high-energy emission. | End timebb End time for this specific time segment for which we can constrain the flux to be less than that listed in the first column of the table e.g., 10 Jy, 100 Jy, or 1 kJy. A positive time indicates a time post-high-energy emission, while a negative time indicates a time pre-high-energy emission. | Total time ccfootnotemark:  |  $\epsilon_{r}$ ($z<1$) ddConstraints on $\epsilon_{r}$ in the model by Totani (2013) assuming $z<1$ for the SGRBs used to determine the limits. For this model, we also assume that the GRB would be have the following properties: $B_{15}=1$, $R_{6}=1$, and $P_{-3}=1$. |  $\epsilon_{r}$ ($z<0.1$) eeSame as in c except for $z<0.1$. |  $\epsilon_{r}$ ($z<1$) ffConstraints on $\epsilon_{r}$ in the model by Zhang (2014) assuming the redshifts of the SGRBs used to determine the limits are $z<1$. We list the $\alpha$ used for the calculation in the final column. For this model, we also assume the following properties: $E_{B}=1.7\times 10^{47}$ erg, $\tau=1$ ms, and $\nu_{p}=1$ kHz, and $\nu_{\textrm{obs}}=600$ MHz for the other model parameters. |  $\epsilon_{r}$ ($z<0.1$) ggSame as in e except for $z<0.1$. |  $\alpha$hhThe value for $\alpha$ in the model by Zhang (2014) which is used to calculate the $\epsilon_{r}$ values in the previous two columns.
---|---|---|---|---|---|---|---|---
(Jy) | (ks) | (ks) | (min.) | (Totani model) | (Totani model) | (Zhang model) | (Zhang model) |
$<10$ |  | 27.498
---
37.746
38.976
| 28.200
---
38.208
39.558
29.2 | $<0.0013$ | $<1.3\times 10^{-5}$ |  | $<9.5\times 10^{-10}$
---
$<2.8\times 10^{-4}$
$<110$
| $<5.2\times 10^{-12}$
---
$<1.6\times 10^{-6}$
$<0.6$
| $-2$
---
$-3$
$-4$
$<100$ |  | $-19.446$
---
22.662
27.096
37.386
| $-19.002$
---
23.094
28.590
39.936
82.1 | $<0.013$ | $<1.6\times 10^{-4}$ |  | $<9.5\times 10^{-9}$
---
$<2.8\times 10^{-3}$
$<1100$
| $<5.2\times 10^{-11}$
---
$<1.6\times 10^{-5}$
$<6$
| $-2$
---
$-3$
$-4$
$<1000$ |  | $-19.770$
---
22.350
26.544
29.628
34.446
41.916
42.672
| $-18.660$
---
26.154
29.148
32.224
41.622
42.384
43.002
294.9 | $<0.13$ | $<1.6\times 10^{-3}$ |  | $<9.5\times 10^{-8}$
---
$<2.8\times 10^{-2}$
$<11000$
| $<5.2\times 10^{-10}$
---
$<1.6\times 10^{-4}$
$<60$
| $-2$
---
$-3$
$-4$

ccfootnotemark:

Total time over the entire time period of six hours prior and 12 hours after
the high-energy emission for which we can constrain the flux to be less than
the limit given in the first column.

It is also possible that at our times of interest the surrounding environment
was not yet transparent to radio emission. This has been suggested by Yamasaki
et al. (2018), who find that the ejecta surrounding a remnant magnetar from a
NS-NS merger would not be transparent to GHz radio emission for $\sim$years
post-merger (although there is significant uncertainty on this estimation).
Additionally, while it is suggested that the FRB and GRB emission would be
emitted along the same axis in this model (Rowlinson & Anderson, 2019), it is
possible that the two are emitted along different axes, and hence the radio
and high-energy emission would not both be observable from the same source.
Lastly, it is possible that the duration of the radio burst could be longer
than the CHIME/FRB parameter search space, and hence would not be detectable
by CHIME/FRB. While CHIME/FRB searches for bursts out to $\sim$100 ms (e.g.,
see CHIME/FRB Collaboration et al., 2018), the sensitivity is greatly reduced
for bursts of this duration (see Figure 7 of Merryfield et al., 2022), and
hence it is more reasonable to assume CHIME/FRB is senstive to bursts up to
$\sim$10-30 ms.

Figure 10: Radio flux densities predicted by Totani (2013) for FRB-like
emission after a NS-NS merger. Flux densities at 600-MHz are shown as a
function of redshift, with different diagonal lines corresponding to different
radio efficiencies in Eq. (3). The three red, dashed, horizontal lines show
flux limits for an FRB-like radio burst of 10 Jy, 100 Jy, and 1 kJy. The blue
line is placed at the median SGRB redshift, $\textrm{z}=0.82$, from the online
GRB catalog GRBWeb (Coppin, 2022).

In addition to pulsar-like radio emission from a NS-NS merger, radio emission
might also be produced if the resulting NS is unstable and collapses to form a
BH (Falcke & Rezzolla, 2014; Zhang, 2014). This might occur hundreds to
thousands of seconds post merger, which is within the time frame of our SGRB
limits.

The flux density of the radio emission predicted by Zhang (2014) and further
expanded on by Rowlinson & Anderson (2019) can be described by:

$F_{\nu}=\frac{-10^{-23}\epsilon_{r}E_{B}}{4\pi
D^{2}\tau}(\alpha+1)\nu_{p}^{-(\alpha+1)}\frac{\nu_{\textrm{obs}}^{\alpha}}{(1+z)}\textrm{Jy}$
(4)

where $\epsilon_{r}$ is the efficiency of converting the magnetic energy into
radio emission, $E_{B}$ is the predicted magnetic energy released during the
magnetic reconnection event responsible for the FRB, $D$ is the distance in
Gpc, $\tau$ is the time over which the magnetic energy is released, $\nu_{p}$
is the plasma frequency in MHz, $\alpha$ is the spectral index of the
emission, $\nu_{\textrm{obs}}$ is the observing frequency in MHz, and $z$ is
the redshift of the GRB. Note that for the flux to be positive, $\alpha<-1$.

Again, almost all of these parameters are unknown for our sample of GRBs.
Thus, we assume the parameters presented by Zhang (2014) and Rowlinson &
Anderson (2019) of $E_{B}=1.7\times 10^{47}$ erg, $\tau=1$ ms, and $\nu_{p}=1$
kHz. We assume $\nu_{\textrm{obs}}=600$ MHz for CHIME/FRB. The three remaining
unknowns are $\alpha$, $z$, and $\epsilon_{r}$.

We list in Table 3 the limits on $\epsilon_{r}$ for our 10 Jy, 100 Jy, and 1
kJy limits. We originally calculate all the constraints using $\alpha=-3$ as
used by Rowlinson & Anderson (2019), and show the predicted flux as a function
of redshift for $\alpha=-3$ as the solid lines in Figure 11. However, as there
is no strong evidence for $\alpha=-3$, we also calculate the limits for
$\alpha=-2$ and $\alpha=-4$. Assuming $\alpha=-2$, our limits become even more
constraining (e.g., see the dotted line in Figure 11). Conversely, if
$\alpha=-4$, our limits are significantly less constraining (e.g., see the
dashed dotted line in Figure 11).

Figure 11: Radio flux densities predicted by Zhang (2014) for FRB-like
emission during the collapse of a NS to a BH. Flux densities are shown as a
function of redshift, with different diagonal lines corresponding to different
radio efficiencies in Eq. (4). The solid black lines show the flux densities
for $\alpha=-3$, while the dotted line is for $\alpha=-2$, and the dashed
dotted line is for $\alpha=-4$. The three red dashed, horizontal lines show
flux limits for an FRB-like radio burst of 10 Jy, 100 Jy, and 1 kJy. The blue
line is placed at the median SGRB redshift, $\textrm{z}=0.82$ from the online
GRB catalog GRBWeb (Coppin, 2022).

If the SGRB is produced by a BH-NS merger rather than a NS-NS merger, then
Mingarelli et al. (2015) predict emission similar to that of Zhang (2014) from
the transfer of the NS’s magnetic field to the BH during the merger. While
this emission might occur as soon as $\sim$milliseconds after the merger, it
is possible that the BH might be able to maintain the magnetic fields of the
NS for a significant period of time, and thus the FRB-like burst might be
produced a short while after the merger131313The maximum length of time for
which the BH could maintain the magnetic fields of the NS has not yet been
quantified, but it is at least 0.5 ms (Mingarelli et al., 2015).. However,
with a predicted luminosity of $10^{41}$ erg s-1, the radio flux densities
from this model would be significantly below the limits for our current sample
unless the SGRB redshifts were $<0.08$. Of the 74 SGRBs with redshifts in the
online GRB catalog GRBWeb (Coppin, 2022), only three have redshifts $<0.08$,
and thus this is a very small fraction of the SGRB population.

###  6.2 LGRBs

There are fewer theoretical predictions for radio emission coincident with
LGRBs. The recent detection of an LGRB associated with a kilonova (and hence a
compact object merger) by Rastinejad et al. (2022) and Yang et al. (2022)
suggests that some of the emission mechanisms mentioned in Section 6.1 might
be relevant for LGRBs. However, we leave this to future work, as hopefully
more LGRBs associated with compact object mergers will be detected and a
distinction can be made between the LGRBs that are produced in this manner
versus in the more classical manner of a core-collapse supernova.

For LGRBs not associated with compact object mergers, it is likely that the
surrounding ejecta of the young, magnetized NS produced by the supernova would
not be transparent to radio emission for $\sim$ years to decades post-LGRB,
and thus the radio emission would not have been detectable during our
observing periods (Metzger et al., 2017).

However, given our sample of LGRBs for which we can place radio emission
limits is significantly larger than our sample of SGRBs141414There are 34
LGRBs for which we can constrain the radio flux and only 5 SGRBs for which we
can do this., we still calculate radio flux limits for our set of LGRBs. These
limits span the full period of 6 hours (21.6 ks) prior to and 12 hours (43.2
ks) after the time of the GRB, with limits as small as 2 Jy in the $\sim$5 ks
before the high-energy emission (see Table LABEL:table:_swift_long and 6).

##  7 Summary and Future Work

In this work, we have searched for temporally and spatially coincident FRBs
and GRBs using the first CHIME/FRB catalog along with all well-localized GRBs
detected between 2018 July 17 and 2019 July 8. We do not find any temporal and
spatial coincidences within the combined 3$\sigma$ uncertainty regions of the
FRBs and GRBs, and a time range of one week. There are two GRBs that are
solely spatially coincident with two FRBs in our sample within a time frame
longer than one week, but the chance of such a coincidence is high, and thus
we do not find these coincidences to be significant.

In addition to checking for coincident GRBs and FRBs, we also determine upper
limits on possible radio emission for 39 GRBs either before, at the time of,
or after the time of high-energy emission. We can constrain the radio emission
for SGRBs to be $<10$ Jy for a total of 29.2 minutes, $<100$ Jy for a total of
82.1 minutes, and $<1$ kJy for a total of 294.9 minutes.

Unfortunately, there are no SGRBs within the FOV of CHIME/FRB at the time of
the high-energy emission or within a few ks of it. Assuming our SGRB sample is
associated with NS-NS mergers, our 10 Jy limits can constrain the radio
efficiency of the emission produced by a magnetar in the model of Totani
(2013) to be $<0.0013$ if our GRB is at $z<1$ and $<1.3\times 10^{-5}$ if the
GRB is at $z<0.1$. Additionally, if the remnant magnetar is unstable and
collapses to form a BH in a manner similar to that outlined by Zhang (2014),
then using our 10 Jy limits we can additionally constrain the radio efficiency
of the emission to be $<2.8\times 10^{-4}$ for $\alpha=-3$ if the GRB is at
$z<1$, or $<1.6\times 10^{-6}$ for $\alpha=-3$ if the GRB is at $z<0.1$. Our
larger radio limits are less constraining, but still interesting as they are
relevant over longer periods of time. However, we note that it is still
possible for radio bursts to occur just outside our time windows with
unconstrained $\epsilon_{r}$.

For LGRBs, we can constrain FRB-like radio emission to $<10$ Jy for a total of
91.3 minutes pre-LGRB and a total of 98.7 minutes post-LGRB, but there are no
strong theoretical predictions for coincident FRB-like radio emission for
LGRBs. Within 0.1 ks of the high-energy emission, our best constraint is $<$25
Jy for a 10-ms radio burst (with this limit valid for $\sim$ 4.4 minutes).
However, given the limited number of theories predicting radio emission at the
time of an LGRB, we do not use our LGRB limits to constrain any radio-emission
mechanisms. Future work which could explore using LGRB limits to possibly
constrain radio emission from compact object mergers.

Given the large number of FRBs that have been and will be detected by the
CHIME/FRB collaboration, future work will continue to search for coincident
FRBs and GRBs using the most-recent GRB catalogs along with all the verified
FRBs detected by CHIME/FRB. Additionally, it is possible that more recent
SGRBS (e.g., outside the time frame considered here) have been within the main
lobe of CHIME/FRB at the time of their high-energy emission. Whether or not
future radio emission will be detected from SGRBs by CHIME/FRB depends both on
the chance of an SGRB being within the mainlobe of CHIME/FRB at the time of
the high-energy detection (which did not occur for the sample considered here)
and the SGRB emitting radio emission that is temporally coincident and beamed
in the same direction as the high-energy emission. It is possible that a
detection of radio emission at a time different than that of the high-energy
emission is also possible, but it will be harder to associate with a given GRB
due to high chance coincidences. Nonetheless, it remains of interest to
continue searching for FRB-GRB coincidences.

## acknowledgments

We acknowledge the use of public data from the Swift data archive and the
Fermi data archive. We acknowledge that CHIME is located on the traditional,
ancestral, and unceded territory of the Syilx/Okanagan people. We are grateful
to the staff of the Dominion Radio Astrophysical Observatory, which is
operated by the National Research Council of Canada. CHIME is funded by a
grant from the Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund
(Project 31170) and by contributions from the provinces of British Columbia,
Québec and Ontario. The CHIME/FRB Project is funded by a grant from the CFI
2015 Innovation Fund (Project 33213) and by contributions from the provinces
of British Columbia and Québec, and by the Dunlap Institute for Astronomy and
Astrophysics at the University of Toronto. Additional support was provided by
the Canadian Institute for Advanced Research (CIFAR), McGill University and
the McGill Space Institute thanks to the Trottier Family Foundation, and the
University of British Columbia.

A.P.C is a Vanier Canada Graduate Scholar. SPT is a CIFAR Azrieli Global
Scholar in the Gravity and Extreme Universe Program. V.M.K. holds the Lorne
Trottier Chair in Astrophysics & Cosmology, a Distinguished James McGill
Professorship, and receives support from an NSERC Discovery grant (RGPIN
228738-13), from an R. Howard Webster Foundation Fellowship from CIFAR, and
from the FRQNT CRAQ. A.B.P. is a Banting Fellow, a McGill Space Institute
(MSI) Fellow, and a Fonds de Recherche du Quebec – Nature et Technologies
(FRQNT) postdoctoral fellow. M.B. is supported by an FRQNT Doctoral Research
Award. C.L. was supported by the U.S. Department of Defense (DoD) through the
National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.
B.M.G. is supported by an NSERC Discovery Grant (RGPIN-2022-03163), and by the
Canada Research Chairs (CRC) program. A.C. is supported by an Ontario Graduate
Student Award. Z.P. is a Dunlap Fellow. The National Radio Astronomy
Observatory is a facility of the National Science Foundation (NSF) operated
under cooperative agreement by Associated Universities, Inc. S.M.R. is a CIFAR
Fellow and is supported by the NSF Physics Frontiers Center awards 1430284 and
2020265. F.A.D is supported by the U.B.C Four Year Fellowship. E.P.
acknowledges funding from an NWO Veni Fellowship. P.S. is a Dunlap Fellow. FRB
Research at UBC is supported by an NSERC Discovery Grant and by the Canadian
Institute for Advanced Research. The Dunlap Institute is funded through an
endowment established by the David Dunlap family and the University of
Toronto.

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##  Appendix A Upper Limits as a Function of Time for SGRBs and LGRBs

Below, in Tables LABEL:table:_SGRBs_Swift, LABEL:table:_swift_long, and 6, we
present our most constraining radio flux limits and fluence ratios for SGRBs
detected by Swift/BAT, LGRBs detected by Swift/BAT, and LGRBs detected by
Fermi/GBM. For each kilosecond timestamp, we give the range of our most
constraining upper limits calculated considering the entire sample of GRBs
within that category (e.g., for Table LABEL:table:_SGRBs_Swift, all SGRBs
detected by Swift/BAT) are used to determine the limits.

aafootnotemark:

End time of a given ks bin for which the radio flux/fluence limit ranges
apply. Times are relative to the detected high-energy emission, with negative
times indicating emission prior to the high-energy emission. Start time for
the range of fluxes/fluences is 1 ks prior to the end time.

bbfootnotemark:

Range of upper limits on the possible radio flux at the 99$\%$ confidence
level for a 10-ms radio burst for the given ks time bin. The flux range is
calculated using all SGRBs detected by Swift/BAT within our sample.

ccfootnotemark:

Range of upper limits on the possible radio-to-high-energy fluence ratio at
the 99$\%$ confidence level for a 10-ms radio burst for the given ks time bin.
The fluence ratio range is calculated using all SGRBs detected by Swift/BAT
within our sample.

ddfootnotemark:

Range of upper limits on $\eta$ (unitless radio-to-high-energy fluence ratio
assuming a 400-MHz radio emission bandwidth) at the 99$\%$ confidence level
for a 10-ms radio burst for the given ks time bin. The fluence ratio range is
calculated using all SGRBs detected by Swift/BAT within our sample. The
Swift/BAT high-energy band is 15 to 150 keV, while the CHIME/FRB radio band is
400- to 800-MHz.

Table 4: CHIME/FRB Upper Limits on Radio Emission from SGRBs detected by Swift/BAT Time aafootnotemark:  | Flux bbfootnotemark:  | Fluence Ratio ccfootnotemark:  |  $\eta$ddfootnotemark:
---|---|---|---
(ks) | (Jy) | ($10^{8}$ Jy ms | ($10^{-11}$)
|  | erg-1 cm${}^{2})$ |
$-20.6$ |  $<$2000-6000 |  $<$2000-4000 |  $<$70000-200000
$-19.6$ |  $<$300-9000 |  $<$200-6000 |  $<$7000-200000
$-18.6$ |  $<$20-2000 |  $<$13-1300 |  $<$500-50000
$-17.6$ |  $<$2000-9000 |  $<$1400-6000 |  $<$60000-300000
$-16.6$ |  $<$3000-8000 |  $<$2000-6000 |  $<$90000-200000
$-15.6$ |  $<$4000-10000 |  $<$3000-7000 |  $<$110000-300000
$20.4$ |  $<$4000-11000 |  $<$600-2000 |  $<$20000-70000
$21.4$ |  $<$4000-9000 |  $<$600-1400 |  $<$20000-50000
$22.4$ |  $<$700-20000 |  $<$110-4000 |  $<$4000-140000
$23.4$ |  $<$20-1100 |  $<$3-200 |  $<$110-6000
$24.4$ |  $<$400-1200 |  $<$200-500 |  $<$7000-20000
$25.4$ |  $<$400-900 |  $<$200-500 |  $<$9000-20000
$26.4$ |  $<$400-1300 |  $<$200-800 |  $<$9000-30000
$27.4$ |  $<$16-1200 |  $<$10-700 |  $<$400-30000
$28.4$ |  $<$1-30 |  $<$0.9-15 |  $<$40-600
$29.4$ |  $<$20-1500 |  $<$14-900 |  $<$600-30000
$30.4$ |  $<$500-1500 |  $<$300-900 |  $<$11000-40000
$31.4$ |  $<$500-1200 |  $<$300-700 |  $<$11000-30000
$32.4$ |  $<$500-1600 |  $<$300-1000 |  $<$13000-40000
$33.4$ |  $<$1100-1600 |  $<$700-900 |  $<$30000-40000
$35.4$ |  $<$300-700 |  $<$200-400 |  $<$7000-20000
$36.4$ |  $<$200-600 |  $<$160-400 |  $<$6000-16000
$37.4$ |  $<$90-600 |  $<$60-400 |  $<$2000-16000
$38.4$ |  $<$2-90 |  $<$1-60 |  $<$50-2000
$39.4$ |  $<$2-110 |  $<$4-120 |  $<$140-5000
$40.4$ |  $<$3-500 |  $<$5-300 |  $<$200-12000
$41.4$ |  $<$400-900 |  $<$200-600 |  $<$10000-20000
$42.4$ |  $<$500-1400 |  $<$300-2000 |  $<$12000-90000
|  |  |

aafootnotemark:

End time of a given ks bin for which the radio flux/fluence limit ranges
apply. Times are relative to the detected high-energy emission, with negative
times indicating emission prior to the high-energy emission. Start time for
the range of fluxes/fluences is 1 ks prior to the end time.

bbfootnotemark:

Range of upper limits on the possible radio flux at the 99$\%$ confidence
level for a 10-ms radio burst for the given ks time bin. The flux range is
calculated using all LGRBs detected by Swift/BAT within our sample.

ccfootnotemark:

Range of upper limits on the possible radio-to-high-energy fluence ratio at
the 99$\%$ confidence level for a 10-ms radio burst for the given ks time bin.
The fluence ratio range is calculated using all LGRBs detected by Swift/BAT
within our sample.

ddfootnotemark:

Range of upper limits on $\eta$ (unitless radio-to-high-energy fluence ratio
assuming a 400-MHz radio emission bandwidth) at the 99$\%$ confidence level
for a 10-ms radio burst for the given ks time bin. The fluence ratio range is
calculated using all LGRBs detected by Swift/BAT within our sample. The
Swift/BAT high-energy band is 15 to 150 keV, while the CHIME/FRB radio band is
400- to 800-MHz.

Table 5: CHIME/FRB Upper Limits on Radio Emission from LGRBs detected by Swift/BAT Time aafootnotemark:  | Flux bbfootnotemark:  | Fluence Ratio ccfootnotemark:  |  $\eta$ddfootnotemark:
---|---|---|---
(ks) | (Jy) | ($10^{8}$ Jy ms | ($10^{-11}$)
|  | erg-1 cm${}^{2})$ |
$-20.6$ |  $<$300-1000 |  $<$3-8 |  $<$130-300
$-19.6$ |  $<$300-1000 |  $<$3-8 |  $<$120-300
$-18.6$ |  $<$20-800 |  $<$0.4-10 |  $<$20-400
$-17.6$ |  $<$4-70 |  $<$0.02-0.9 |  $<$0.6-40
$-16.6$ |  $<$11-1000 |  $<$1.0-13 |  $<$40-500
$-15.6$ |  $<$2-20 |  $<$0.4-5 |  $<$20-200
$-14.6$ |  $<$3-900 |  $<$0.11-40 |  $<$5-2000
$-13.6$ |  $<$400-1000 |  $<$40-60 |  $<$2000-2000
$-12.6$ |  $<$500-2000 |  $<$30-90 |  $<$1300-4000
$-11.6$ |  $<$120-2000 |  $<$20-80 |  $<$700-3000
$-10.6$ |  $<$10-1000 |  $<$0.8-20 |  $<$30-1000
$-9.6$ |  $<$800-2000 |  $<$30-70 |  $<$1000-3000
$-8.6$ |  $<$900-2000 |  $<$40-90 |  $<$2000-4000
$-7.6$ |  $<$14-2000 |  $<$0.7-100 |  $<$30-4000
$-6.6$ |  $<$3-20 |  $<$0.02-0.7 |  $<$0.7-30
$-5.6$ |  $<$2-10 |  $<$0.08-0.5 |  $<$3-20
$-4.6$ |  $<$2-8 |  $<$0.11-1.0 |  $<$4-40
$-3.6$ |  $<$4-60 |  $<$0.2-5 |  $<$9-200
$-2.6$ |  $<$7-900 |  $<$1-11 |  $<$60-400
$-1.6$ |  $<$500-2000 |  $<$20-100 |  $<$800-4000
$-0.6$ |  $<$400-2000 |  $<$20-90 |  $<$900-3000
$0.4$ |  $<$70-2000 |  $<$10-70 |  $<$400-3000
$1.4$ |  $<$4-70 |  $<$0.6-9 |  $<$20-400
$2.4$ |  $<$6-1000 |  $<$0.8-13 |  $<$30-500
$3.4$ |  $<$5-600 |  $<$0.2-8 |  $<$9-300
$4.4$ |  $<$5-900 |  $<$0.04-40 |  $<$1-1000
$5.4$ |  $<$50-1000 |  $<$0.3-20 |  $<$14-900
$6.4$ |  $<$130-1000 |  $<$2-12 |  $<$70-500
$7.4$ |  $<$12-120 |  $<$0.2-2 |  $<$7-70
$8.4$ |  $<$11-40 |  $<$0.1-0.6 |  $<$6-20
$9.4$ |  $<$40-200 |  $<$0.6-3 |  $<$20-130
$10.4$ |  $<$200-800 |  $<$3-13 |  $<$130-500
$11.4$ |  $<$300-800 |  $<$3-20 |  $<$100-700
$12.4$ |  $<$200-600 |  $<$2-6 |  $<$90-200
$13.4$ |  $<$6-400 |  $<$0.13-5 |  $<$5-200
$14.4$ |  $<$1-10 |  $<$0.015-0.13 |  $<$0.6-5
$15.4$ |  $<$8-400 |  $<$0.07-11 |  $<$3-400
$16.4$ |  $<$12-300 |  $<$0.8-7 |  $<$30-300
$17.4$ |  $<$4-20 |  $<$0.3-2 |  $<$11-60
$18.4$ |  $<$2-12 |  $<$0.1-0.8 |  $<$6-30
$19.4$ |  $<$2-8 |  $<$0.13-0.6 |  $<$5-20
$20.4$ |  $<$3-20 |  $<$0.2-2 |  $<$8-60
$21.4$ |  $<$11-50 |  $<$0.7-4 |  $<$30-100
$22.4$ |  $<$3-20 |  $<$0.2-1 |  $<$7-50
$23.4$ |  $<$20-700 |  $<$0.9-8 |  $<$40-300
$24.4$ |  $<$500-2000 |  $<$2-6 |  $<$100-200
$25.4$ |  $<$500-1000 |  $<$2-6 |  $<$100-200
$26.4$ |  $<$500-1000 |  $<$3-6 |  $<$110-200
$27.4$ |  $<$120-2000 |  $<$1-9 |  $<$50-300
$28.4$ |  $<$3-120 |  $<$0.03-1 |  $<$1-50
$29.4$ |  $<$3-30 |  $<$0.014-0.14 |  $<$0.6-6
$30.4$ |  $<$30-2000 |  $<$0.15-5 |  $<$6-200
$31.4$ |  $<$800-2000 |  $<$4-10 |  $<$150-400
$32.4$ |  $<$700-1000 |  $<$3-7 |  $<$130-300
$33.4$ |  $<$700-1000 |  $<$3-8 |  $<$130-300
$34.4$ |  $<$700-2000 |  $<$4-9 |  $<$150-400
$35.4$ |  $<$900-2000 |  $<$4-30 |  $<$160-1300
$36.4$ |  $<$60-1000 |  $<$3-30 |  $<$110-1300
$37.4$ |  $<$7-1100 |  $<$0.4-20 |  $<$14-800
$38.4$ |  $<$20-1300 |  $<$0.3-20 |  $<$11-800
$39.4$ |  $<$20-1200 |  $<$1-20 |  $<$50-700
$40.4$ |  $<$8-900 |  $<$0.6-20 |  $<$20-800
$41.4$ |  $<$15-2000 |  $<$0.11-30 |  $<$5-1100
$42.4$ |  $<$800-2000 |  $<$11-30 |  $<$400-1100
|  |  |
Table 6: CHIME/FRB Upper Limits on Radio Emission from LGRBs detected by
Fermi/GBM

Time aafootnotemark:  | Flux bbfootnotemark:  | Fluence Ratio ccfootnotemark:  |  $\eta$ddfootnotemark:
---|---|---|---
(ks) | (Jy) | ($10^{8}$ Jy ms | ($10^{-11}$)
|  | erg-1 cm${}^{2})$ |
$-5.882$ |  $<$2000-3000 |  $<$2-7 |  $<$90-300
$-4.882$ |  $<$20-5000 |  $<$0.3-5 |  $<$12-200
$-3.882$ |  $<$20-3000 |  $<$0.3-8 |  $<$11-300
$-2.882$ |  $<$20-1100 |  $<$0.013-0.9 |  $<$0.5-40
$-1.882$ |  $<$1100-5000 |  $<$0.9-13 |  $<$40-500
$-0.882$ |  $<$2000-8000 |  $<$3-7 |  $<$130-300
$0.118$ |  $<$2000-12000 |  $<$3-30 |  $<$130-1200
$1.118$ |  $<$1400-4000 |  $<$20-60 |  $<$900-2000
$2.118$ |  $<$1400-3000 |  $<$20-40 |  $<$900-1500
$40.118$ |  $<$100-1100 |  $<$0.006-0.06 |  $<$0.2-2
$41.118$ |  $<$700-2000 |  $<$0.04-0.1 |  $<$2-4
$42.118$ |  $<$700-2000 |  $<$0.04-0.09 |  $<$2-4
$43.118$ |  $<$200-3000 |  $<$0.011-0.2 |  $<$0.5-6

aafootnotemark:

End time of a given ks bin for which the radio flux/fluence limit ranges
apply. Times are relative to the detected high-energy emission, with negative
times indicating emission prior to the high-energy emission. Start time for
the range of fluxes/fluences is 1 ks prior to the end time.

bbfootnotemark:

Range of upper limits on the possible radio flux at the 99$\%$ confidence
level for a 10-ms radio burst for the given ks time bin. The flux range is
calculated using all LGRBs detected by Fermi/GBM within our sample.

ccfootnotemark:

Range of upper limits on the possible radio-to-high-energy fluence ratio at
the 99$\%$ confidence level for a 10-ms radio burst for the given ks time bin.
The fluence ratio range is calculated using all LGRBs detected by Fermi/GBM
within our sample.

ddfootnotemark:

Range of upper limits on $\eta$ (unitless radio-to-high-energy fluence ratio
assuming a 400-MHz radio emission bandwidth) at the 99$\%$ confidence level
for a 10-ms radio burst for the given ks time bin. The fluence ratio range is
calculated using all LGRBs detected by Fermi/GBM within our sample. The
Fermi/GBM high-energy range is 10 to 1000 keV, while the CHIME/FRB radio band
is 400- to 800-MHz.