# Comprehensive Bayesian analysis of FRB-like bursts from SGR 1935+2154
observed by CHIME/FRB

Utkarsh Giri Department of Physics, University of Wisconsin-Madison, 1150
University Ave, Madison, WI 53706, USA  Perimeter Institute for Theoretical
Physics, 31 Caroline Street N, Waterloo, ON N25 2YL, Canada  Bridget C.
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  Pragya Chawla Anton
Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands  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  Emmanuel Fonseca Center for Gravitational Waves and Cosmology, West
Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505,
USA  Department of Physics and Astronomy, West Virginia University, P.O. Box
6315, Morgantown, WV 26506, USA  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  Hsiu-Hsien Lin Institute of Astronomy and Astrophysics, Academia
Sinica, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei
10617, Taiwan  Canadian Institute for Theoretical Astrophysics, 60 St. George
Street, Toronto, ON M5S 3H8, Canada  K. W. Masui 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  Ketan R. Sand 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  Paul Scholz
Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON M5S 3H4, Canada  Department of Physics and
Astronomy, York University, 4700 Keele Street, Toronto, Ontario, ON MJ3 1P3,
Canada  Thomas C. Abbott 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, Carnegie Mellon University, 5000 Forbes Avenue,
Pittsburgh, 15213, PA, USA  Fengqiu Adam Dong Department of Physics and
Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,
BC V6T 1Z1 Canada  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
Present address: Division of Physical and Biological Sciences, University of
California Santa Cruz, Santa Cruz, CA 95064, USA  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  NHFP Einstein Fellow  Daniele Michilli 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  Moritz Münchmeyer Department of Physics, University
of Wisconsin-Madison, 1150 University Ave, Madison, WI 53706, USA  Ayush
Pandhi David A. Dunlap Department of Astronomy & Astrophysics, University of
Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada  Dunlap Institute
for Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  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  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  Alex Reda
Department of Physics, Yale University, New Haven, CT 06520, 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  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  David C.
Stenning Department of Statistics and Actuarial Science, Simon Fraser
University, 8888 University Dr, Burnaby, BC V5A 1S6, 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  CIFAR Azrieli
Global Scholars Program, MaRS Centre, West Tower, 661 University Ave, Suite
505, Toronto, ON, M5G 1M1 Canada Utkarsh Giri<EMAIL_ADDRESS>
###### Abstract

The bright millisecond-duration radio burst from the Galactic magnetar SGR
1935+2154 in 2020 April was a landmark event, demonstrating that at least some
fast radio burst (FRB) sources could be magnetars. The two-component burst was
temporally coincident with peaks observed within a contemporaneous short X-ray
burst envelope, marking the first instance where FRB-like bursts were observed
to coincide with X-ray counterparts. In this study, we detail five new radio
burst detections from SGR 1935+2154, observed by the CHIME/FRB instrument
between October 2020 and December 2022. We develop a fast and efficient
Bayesian inference pipeline that incorporates state-of-the-art Markov chain
Monte Carlo techniques and use it to model the intensity data of these bursts
under a flexible burst model. We revisit the 2020 April burst and corroborate
that both the radio sub-components lead the corresponding peaks in their high-
energy counterparts, respectively. For a burst observed in 2022 October, we
find that our estimated radio pulse arrival time is contemporaneous with a
short X-ray burst detected by GECAM and HEBS, and Konus-Wind and is consistent
with the arrival time of a radio burst detected by GBT. We present flux and
fluence estimates for all five bursts, employing an improved estimator for
bursts detected in the side-lobes. We also present upper limits on radio
emission for X-ray emission sources which were within CHIME’s field-of-view at
trigger time. Finally, we present our exposure and sensitivity analysis and
estimate the Poisson rate for FRB-like events from SGR 1935+2154 to be
$0.005^{+0.082}_{-0.004}$ events/day above a fluence of
$10~{}\mathrm{kJy~{}ms}$ during the interval from 28 August 2018 to 1 December
2022, although we note this was measured during a time of great X-ray activity
from the source. This rate is informative for other future wide-field radio
telescope detections of Galactic magnetars.

Magnetars (992), Radio transient sources (2008)

††facilities: CHIME††software: matplotlib (Hunter, 2007), numpy (Harris et
al., 2020), emcee (Foreman-Mackey et al., 2019), JAX (Bradbury et al., 2018),
NumPyro (Phan et al., 2019), getdist (Lewis, 2019)

##  1 Introduction

Fast radio bursts are extremely bright transient radio pulses of very short
($\sim$ millisecond) duration, with dispersion measures (DMs) suggestive of
extra-galactic origin. Discovered for the first time in 2007 (Lorimer et al.,
2007), FRBs have emerged as one of the most exciting phenomena in the field of
time-domain astronomy (Cordes & Chatterjee, 2019; Petroff et al., 2019).
Explaining their origin has become a central unresolved problem in astronomy,
and numerous theories have been proposed to explain the extraordinary emission
mechanism that powers them and progenitors that host them (Platts et al.,
2019).

FRBs exhibit diverse phenomenology, adding richness and mystery to their
possible origin mechanism. To date, thousands of FRBs have been detected (e.g.
CHIME/FRB Collaboration et al., 2021a) with a high fraction of them being one-
off events where only a single component of emission from the source was
observed. However, a small number of FRB sources have been seen to repeat,
i.e., the same source emits pulses intermittently (e.g. CHIME/FRB
Collaboration et al., 2019a; Fonseca et al., 2020; CHIME/FRB Collaboration et
al., 2023). A handful of these repeaters have also shown periodicity in their
activity periods (CHIME/FRB Collaboration et al., 2020a, 2021b; Cruces et al.,
2021). The repeaters and apparent non-repeaters also show statistically
different morphologies (Pleunis et al., 2021; CHIME/FRB Collaboration et al.,
2019a, 2023) with some repeaters having a puzzling downward-drifting structure
in time-frequency phase space.

The sources of FRBs are still unknown. To understand them better, source
localization and host association are key. However, there is one case where we
have detected FRB-like radio pulses from a previously known source: the
Galactic magnetar SGR 1935+2154 (CHIME/FRB Collaboration et al., 2020b;
Bochenek et al., 2020). While the sources of FRBs are still not definitively
known, the detection of an FRB-like 111We call the radio bursts FRB-like as
their spectral luminosity is several orders of magnitude greater than the
pulsed emission seen from SGR 1935+2154 (Zhu et al., 2023) and above that of a
typical Giant pulse, but still a few orders of magnitude below that of FRBs.
burst FRB 20200428D from SGR 1935+2154 was the first direct evidence that
magnetars could be a source of at least some FRBs. Specifically, a radio burst
with two sub-components was seen on 2020 April 28, with the sub-components
separated by $\sim 30$ ms, following a period of unusually high X-ray
activity. The radio sub-bursts were contemporaneous with hard X-ray emission
peaks observed independently by Konus-Wind, INTEGRAL, Insight-HXMT and AGILE
(Ridnaia et al., 2021; Mereghetti et al., 2020; Borghese et al., 2020; Tavani
et al., 2021).

This landmark event and the associated high X-ray activity prompted several
follow-up observations. On 2020 April 30, during the course of a month-long
monitoring campaign, the Five-hundred-meter Aperture Spherical Telescope
(FAST) observed a faint and highly polarized radio pulse from SGR 1935+2154
which had a fluence of 51 mJy ms (Zhang et al., 2020). Subsequently, Kirsten
et al. (2021) performed hundreds of hours of follow-up observations of SGR
1935+2154 using several telescopes under a coordinated campaign, and detected
two radio bursts on 2020 May 24 within a span of $\sim 1.4$s, with fluences of
$112\pm 22$ Jy ms and $24\pm 5$ Jy ms, respectively. Unlike the April 2020
event however, none of these radio bursts was found to be accompanied with
emission at X-ray wavelengths.

On 2020 October 8, CHIME/FRB detected three more bursts from SGR 1935+2154
within a span of $\sim$3 sec. The detection was rapidly communicated with the
community (Good & CHIME/FRB Collaboration, 2020) and prompted a new follow-up
observation campaign by the Five-hundred-meter Aperture Spherical Telescope
(FAST). The campaign resulted in the detection of hundreds of periodic, sub-
Jansky radio pulses, anti-aligned with the X-ray emission and likely of
pulsar-mode origin, powered by spin-down energy (Zhu et al., 2023). After a
prolonged lull in activity of over 2 years, CHIME/FRB detected another FRB-
like burst from SGR 1935+2154 on 2022 October 14 (Dong & CHIME/FRB
Collaboration, 2022) and another one on 2022 December 1 (Pearlman & CHIME/FRB
Collaboration, 2022). Interestingly, based on preliminary estimates of the
arrival times, the October 14 burst was reported to be coincident with a
short, 250 ms X-ray burst observed by GECAM and HEBS (Wang et al., 2022) and
by Konus-Wind (Frederiks et al., 2022a) and also with a radio bursts detected
by the Green Bank Telescope (GBT) at 5 GHz during a C-Band session (Maan et
al., 2022).

Studies of these events from SGR 1935+2154 offer a unique opportunity to
deepen our understanding of the transient universe. For example, comparing
burst rates of regular FRBs with rates of FRB-like bursts from SGR
1935+2154-like magnetars can tell us what fraction of FRBs might be coming
from magnetars. Identifying coincident radio and X-ray bursts in space and
time or putting upper limits on radio flux is an important step towards
constraining FRB models. Moreover, given their special placement in the
luminosity-duration energetics space, accurate flux measurements for SGR
1935+2154 events and FRBs at large can offer critical insights.

The need for accurate and precise determination of burst parameters goes
further than this. For example, a precise measurement of relative arrival time
between a radio burst and an associated high-energy counterparts could help
distinguish between different models of FRB emission (e.g. Margalit et al.,
2020; Lyutikov & Popov, 2020). In the case of SGR 1935+2154, we already have
two events where such an accurate time of arrival (ToA) estimation is needed.
Estimating radio ToA correctly requires an adequate burst model together with
a robust inference framework to disentangle propagation effects. More
generally, as the FRB field develops, the FRB data quality as well as the
count is expected to grow very rapidly, enabling interesting science analysis
via observables like pulse dispersion and scatter broadening. This
necessitates the development of a fast, accurate and modular analysis
framework that will enable processing large sets of data correctly while
allowing for seamless integration of custom burst models for complex
morphology.

In this paper, we present an analysis of more SGR 1935+2154 radio bursts seen
by CHIME/FRB using a fast and flexible Bayesian inference pipeline. After
briefly describing the CHIME/FRB backend in §2 and our observations in §3, we
describe the burst modeling in §4 . In §5, we describe our Bayesian inference
pipeline and validate it using simulations. We analyze the intensity data of
SGR 1935+2154 events and present results in §6. We reanalyze FRB 20200428D
with a more flexible model and present an updated estimate of its arrival
time. We confirm that both the radio sub-bursts of FRB 20200428D lead their
contemporaneous high-energy counterparts and rule out their simultaneity at a
high significance. In §6.4, we present flux and fluence estimates for all
observed bursts. We employ an improved estimator for new bursts detected in
the side-lobes. In §6.5, based on non-detection of radio activity, we present
upper limits on radio emission for X-ray pulses that were observed while their
sources were within CHIME’s field-of-view. Finally in §6.6, we present our
exposure and sensitivity analysis which is used to obtain the rate of FRB-like
bursts from SGR 1935+2154. Throughout the paper, our unit for DM are
$\mathrm{pc~{}cm^{-3}}$ and time-like parameters are in milliseconds unless
stated otherwise.

##  2 Instrument

The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is an
interferometric telescope at the Dominion Radio Astrophysical Observatory in
British Columbia, Canada. Designed initially with the aim of mapping neutral
hydrogen in the universe (Newburgh et al., 2014; Bandura et al., 2014; CHIME
Collaboration et al., 2022), the instrument was supplemented with a dedicated
backend for real-time detection of FRBs — CHIME/FRB (CHIME/FRB Collaboration
et al., 2018). Here we summarize the CHIME/FRB system.

The CHIME telescope comprises four 20-m $\times$ 100-m cylindrical reflectors
with no moving parts, aligned in the North-South direction and operating
between $400-800$ MHz. Each of these reflectors has 256 dual-polarization
antennas installed along its focal plane axis. The analog signal received by
the feeds gets digitized to a time-series with a 1.25 ns time-resolution
before getting channelized to 1024 frequency channels at a resolution of 2.56
$\mu s$ using customized integrated circuits called Field-Programmable Gate
Arrays (FPGAs). The channelized output from the FPGAs is processed by a series
of pipelines. First among them is L0 pipeline which beamforms using Fast-
Fourier Transforms (Tegmark & Zaldarriaga, 2009; Ng et al., 2017; Masui et
al., 2019). It is built out of 256 GPU nodes that correlate data from all the
feeds with appropriate time delays to constructively interfere them to produce
beams digitally “pointed” to a particular location in the sky. In total, L0
produces 1024 beams on the sky and further up-channelizes the data stream by a
factor of 16 (yielding a total of 16384 radio-frequency channels) at a time
resolution of 0.983 ms.

The data then go to the L1 pipeline, which is responsible for detecting FRBs.
After filtering out terrestrial radio frequency interference (Rafiei-Ravandi &
Smith, 2023), the CPU-based L1 pipeline uses a near optimal tree-based
incoherent dedispersion algorithm bonsai to search for FRBs in real-time. The
pipeline utilizes a ring-buffer to store intensity data, and upon detecting a
candidate event above a nominal signal-to-noise (S/N), it triggers a dump of a
few seconds of data around the event for further offline analysis. Finally,
the pipeline also has a machine-learning based routine for sifting through the
candidate events to identify genuine FRBs. Subsequently, the L2/L3 pipeline
performs another iteration of RFI removal and groups together instances of an
event that are detected in multiple beams. L2/L3 is also responsible for
possible source identification and association. The last in the series is the
L4 pipeline, which is responsible for maintaining a database for candidate
meta-data and carrying out user-defined specific actions dependent on source
identity.

##  3 Observations

CHIME/FRB detected an extremely bright radio burst from the direction of SGR
1935+2154 on 2020 April 28 with two sub-components at approximate UTC
(topocentric, 400MHz) 14:34:33 (Scholz & Chime/Frb Collaboration, 2020;
CHIME/FRB Collaboration et al., 2020b). The burst was detected in the far
side-lobe of the telescope at an hour-angle (HA) of $\sim 22$°. As is typical
of extremely bright far side-lobe events, it was detected in many of the
CHIME/FRB beams.222For CHIME/FRB, events are in far side-lobes if they are
located at least several beam widths (1.3∘$-$2.6∘) away from the meridian. The
signature of such a detection is a dynamic spectrum (i.e., waterfall) with
multiple regular “spikes” in radio frequency, as was seen for these events.
The second of the two sub-components was also detected by STARE-2 (Bochenek et
al., 2020) at 1281-1486 MHz. These radio observations occurred during a period
of intense X-ray activity, in which hundreds of bursts were detected in the
X-ray and $\gamma$-ray range. Several satellites reported X-ray bursts
contemporaneous with the radio pulse (e.g., Mereghetti et al., 2020; Ridnaia
et al., 2021; Tavani et al., 2021). A first analysis of the CHIME/FRB-detected
radio bursts, including flux and fluence estimates, was presented in CHIME/FRB
Collaboration et al. (2020b).

On 2020 October 8, CHIME/FRB detected three more bursts from SGR 1935+2154
within a span of $\sim$3 seconds. All three bursts were detected in the main
lobe of CHIME and the detection was reported via ATel including preliminary
estimates of key burst parameters (Good & CHIME/FRB Collaboration, 2020).
After a gap of over 2 years, CHIME/FRB detected another burst from this source
on 2022 October 14 at UTC 19:21:39.47 (Dong & CHIME/FRB Collaboration, 2022)
and another one on 2022 December 1 at UTC 22:06:59.08 at an HA of $-$99.8° and
$-$11.1° from the CHIME meridian, respectively (Pearlman & CHIME/FRB
Collaboration, 2022). These bursts were detected in four beams and appeared to
have spiky spectra indicative of far side-lobe detections. For all these
events, our automated trigger system saved a few seconds of intensity data in
the events vicinity for offline post-processing and analysis. For the bursts
observed on 2020 October 14 and 2022 December 1 event, we were successful in
saving a block of raw voltage data as well, an analysis of which will be
presented elsewhere. In the next section, we describe the details of the model
that we use to characterize the physical properties of observed bursts in the
intensity data.

##  4 Burst Modeling

In this section, we describe our model for the radio burst intensity
$I(\nu,t)$ that we fit to the dynamic spectrum $\mathcal{D}(\nu,t)$ of
observed radio bursts. The basic pulse model we fit to an observed FRB
accounts for its intrinsic features as well as propagation effects imprinted
on the burst (Masui et al., 2015; CHIME/FRB Collaboration et al., 2021a;
McKinnon, 2014). The intrinsic profile of an FRB is modeled as a Gaussian
pulse in time with a frequency-dependent modulation given by its spectrum. The
Gaussian profile is a two parameter model given by

$I_{Gaussian}=\frac{1}{\sqrt{2\pi
w^{2}}}\exp\Bigg{(}-\frac{(t-t_{0})^{2}}{2w^{2}}\Bigg{)},$

where $t_{0}$ is the time of arrival of the burst at our telescope at a
reference frequency (which we set to $\nu_{ref}=400.195$ MHz internally) and
$w$ is the width of the pulse. We model the spectrum of the burst using a
three-parameter power-law given by

$I_{spectrum}=A_{s}\Bigg{(}\frac{\nu}{\nu_{0}}\Bigg{)}^{n_{s}+\alpha\ln(\frac{\nu}{\nu_{0}})},$

where $A_{s}$ is the amplitude, $n_{s}$ is the spectral index and $\alpha$
models the running of spectral index as a function of frequency. They combine
together to give us the total intrinsic pulse profile

$I_{intrinsic}(\nu,t)=I_{spectrum}*I_{Gaussian}.$ (1)

The resulting profile captures features that are likely intrinsic to the FRB
generation process.

In addition to these intrinsic features, the observed pulse is modified due to
propagation effects. Chief among these are scattering and dispersion.
Scattering, or multi-path propagation of the FRB, causes many copies of the
pulse to be superimposed with different delays. This results in a temporal
broadening of the pulse which can be modeled under a thin screen approximation
where the Gaussian profile gets convolved with a decaying exponential of the
form $e^{-(t-t_{0})/\tau_{s}}$, where the scattering time is given by

$\tau_{s}=\tau\bigg{(}\frac{\nu}{\nu_{s}}\bigg{)}^{SI},$ (2)

where $\tau$ is called the scattering time, is defined at a reference
frequency $\nu_{s}$, and parameterizes the spectral broadening. We choose the
center of the observing bandwidth as the reference frequency for scattering
i.e. $\nu_{s}=600$ MHz. The parameter $SI$ is called the scattering index and
characterizes the frequency dependence of the broadening. The scattering index
depends on the nature of plasma inhomogeneities and in particular their power
spectrum ($\propto\langle n_{e}^{2}\rangle$). It can be a free parameter in
principle but in the literature, its value is fixed to the theoretically
motivated value of $-$4 or $-$4.4 (e.g. Lang, 1971; Rickett, 1977). Thus,

${I_{scattered}(\nu,t)=I_{intrinsic}\otimes\Bigg{[}\frac{\theta(t-t_{0})}{\tau_{s}}e^{-(t-t_{0})/\tau_{s}}\Bigg{]}}$
(3)

where $\theta(t)$ is the Heaviside function and $\otimes$ denotes a
convolution operation.

Finally, the cold plasma dispersion leads to a frequency dependent delay in
the arrival of the FRB pulse with a time shift given by

$\displaystyle\Delta t(\nu)$ $\displaystyle=\frac{1}{K}\frac{DM}{\nu^{DI}}$
(4)

where the dispersion constant $K=241$ GHz-2 cm-3 pc s-1 (Manchester & Taylor,
1972) and $DM=\int n_{e}dl$. The integral over the electron density $n_{e}$
counts the number of electrons encountered by the FRB along its path from
source to observer and is called the dispersion measure ($DM$). The parameter
$DI$ is the dispersion index and captures the frequency dependence of the
dispersion relation. The $DI$ parameter is typically set to $-$2 in most
pulsar and FRB analyses. The leading order correction from this quadratic
behavior is proportional to $({\nu_{plasma}}/{\nu_{emission}})^{4}$ and is
highly suppressed. Incorporating this time shift, we have

$I(\nu,t)_{overhead}=I_{scattered}(\nu,t-\Delta t({\nu})),$ (5)

where $\Delta t({\nu})$ is given by Eq. (4) and is dependent on two new
parameters: $DM$ and $DI$.

Lastly, the intensity profile $I(\nu,t)_{overhead}$ gets attenuated by the
beam response 333https://chime-frb-open-data.github.io/beam-model of the
telescope as the signal is captured at the feed and processed. While fitting
for the FRB parameters, we account for the attenuation by using models of the
primary and the FFT-based synthesized beams. The primary beam is basically the
beam response of a single feed over its cylinder. However, additional complex
features appear on top of it due to instrumental effects like reflections and
cross-talk between feeds. The primary beam for CHIME/FRB is calibrated using
data-driven holography and is accurate to within 10%. A more detailed
discussion of the primary beam can be found in CHIME Collaboration et al.
(2022). The FFT-based synthesized beams, although complex and full of spatial
and spectral features, are fully deterministic (Ng et al., 2017). Their effect
can be accounted for up to the resolution of the beam. Incorporating the
composite beam attenuation, the intensity data can be modeled as

$I(\nu,t)=I_{overhead}*\mathcal{B}(Dec,RA)$ (6)

where $\mathcal{B}(Dec,RA)$ is the composite beam model at the known
declination and right ascension of the SGR 1935+2154. A discretized version of
Eq. (6) gives the final model we use for modeling the 2-d data array
$\mathcal{D}(\nu_{i},t_{j})$ given arbitrary values of the parameters
${\bm{\theta}}=(t_{0},DM,w,A_{s},n_{s},\alpha,\tau,SI,DI)$. In our fitting,
the position ($RA,Dec$) is kept fixed to the known sky position of the SGR
1935+2154 given in degrees by $RA$ $=293.732\degree$ and $Dec$
$=+21.897\degree$ (Israel et al., 2016).

For bursts with multiple components such as FRB 20200428D, we model the
intrinsic profile of each component independently while the propagation
effects and instrument effects are modeled jointly. In particular, we expand
the above prescription to model intrinsic features like width and spectrum
independently for each of the additional components, whereas propagation
effects like scattering and dispersion are fit using a common set of
parameters for all the sub-components.

##  5 Bayesian modeling

Given the _forward model_ of an FRB as in Eq. (6), the process of burst
parameter estimation is the inverse problem of inferring the region in
parameter space that agrees with the observed FRB data.

In order to infer the model parameters, we work under the probabilistic
Bayesian framework which formalizes the inference problem. The Bayesian
approach returns a distribution over the parameter space assigning a higher
probability density to regions in the space which best fit the data. The
approach naturally incorporates our previous beliefs or knowledge about the
parameters expressed in the form of a distribution called the _prior_. One of
the advantages of the Bayesian formulation is the ability to consistently
model and marginalize over parameters that one is not interested in but are
still needed to describe the data.

Formally, the posterior distribution $\mathcal{P({\bm{\theta}}|{\bm{D}})}$
over parameters ${\bm{\theta}}=(t_{0},DM,w,A_{s},n_{s},\alpha,\tau,SI,DI)$ of
our fiducial model described in §4, for a dataset $\mathcal{{\bm{D}}}$ is
given by

$\mathcal{P({{\bm{\theta}}}|{{\bm{D}}})}\propto\mathcal{P({\bm{D}}|{\bm{\theta}})P({\bm{\theta}})}=\mathcal{L_{{\bm{\theta}}}({\bm{D}})P({\bm{\theta}})}\\\
$ (7)

where $\mathcal{P({\bm{\theta}})}$ is the joint prior distribution over the
model parameters and $\mathcal{L_{{\bm{\theta}}}({\bm{D}})}$ is the
likelihood. We assume prior independence and define
$\mathcal{P({\bm{\theta}})}$ as a product of independent distributions; either
uniform or Gaussian distributions. Our likelihood function is a Gaussian noise
model given by

$\mathcal{L_{\theta}}=\exp({-\chi^{2}({\bm{\theta}},\mathcal{{\bm{D}}})}/2)\\\
$ (8)

with

$\displaystyle\ln\mathcal{L_{\theta}}=-\chi^{2}({\bm{\theta}},\mathcal{{\bm{D}}})/2=-\sum_{t,\nu}\frac{(\mathcal{{\bm{D}}}(t,\nu)-I(t,\nu|\theta))^{2}}{2\sigma_{\nu}^{2}}$
(9)

where $\sigma_{\nu}^{2}$ is the per channel variance of our data in the
neighbourhood of the event.

The posterior distribution over the parameters can be explored by obtaining an
approximate sample using Monte Carlo Markov Chain (MCMC).

###  5.1 Implementation

####  5.1.1 Likelihood

The likelihood defined in Eq. (8) is a numerically intensive function. For
data $\mathcal{D}(\nu,t)$ of size
${16k\mbox{-}\mathrm{by}\mbox{-}\mathcal{O}(100)}$, the model ${I(t,\nu)}$,
consists of ${16k\times m\times\mathcal{O}(100)\times n}$ elements, where $m$
and $n$ are factors by which model composition is upsampled in frequency and
time with respect to the native data resolution. This property of the data
lends itself to high throughput, single instruction multiple data (SIMD)
computation. To leverage this property, we implement the model using the
numerical library JAX (Bradbury et al., 2018). JAX is a modern high-
performance library for numerical calculations on GPUs, implemented primarily
for machine learning research and application. JAX uses XLA (Accelerated
Linear Algebra; a domain-specific compiler for linear algebra) to compile and
run programs on GPUs and has an API very similar to the popular numerical
library NumPy. Compilation happens under the hood by default, with library
calls getting just-in-time compiled and executed. But JAX also lets us compile
our own Python functions into XLA-optimized kernels. We make use of this just-
in-time compilation feature as well as efficient vectorization offered in JAX
to optimize our likelihood code. This likelihood model has been used
previously in several of our studies (e.g. CHIME/FRB Collaboration et al.,
2019b, 2020b), but is re-written from scratch in JAX and integrated into the
object-oriented approach of the pipeline.

####  5.1.2 Sampling

In order to further optimize our analysis, we implement a vectorized version
of the popular affine-invariant MCMC sampling algorithm from scratch in JAX.
The parallelized version of this algorithm, first presented in Foreman-Mackey
et al. (2013) along with a pure Python implementation, utilized disjoint sets
of random walkers to sample the posterior distribution, where the walkers in
one set had their position updated based on the walkers in the other set,
making it amenable to SIMD principle. We implement the same basic idea in JAX
and make it publicly available for general
use444https://github.com/utkarshgiri/jaims.

Among the various known sampling approaches, and beyond the basic random-walk
based samplers, there exists a class of samplers which utilize local gradient
information of the distribution to efficiently converge to the target
distribution. With JAX, one can precisely compute the gradient of a Python
function relative to its parameters via the Automatic Differentiation
algorithm. We leverage this feature to integrate the gradient-based No U-Turn
Sampler (NUTS) (Neal et al., 2011; Hoffman & Gelman, 2011) into our repertoire
as well.

For each of our burst analysis, described further in the next sub-section, we
use both the gradient-based NUTS sampler from the numpyro library and the
native JAX-based version of the affine-invariant sampler. We find that both
samplers give consistent results.

####  5.1.3 MCMC pipeline

For a single component event, our _default_ analysis fits progressively
complex burst models to an approximately dedispersed,
$16k\mbox{-}\mathrm{by}\mbox{-}\mathcal{O}(100)$ size dynamic spectrum data
around the burst in an iterative fashion. The analysis begins by MCMC fitting
a Gaussian burst model with parameters ${\bm{\theta}}=(t_{0},DM,w,A_{s})$ to
band-summed intensity time-series. The initial estimates for $t_{0}$ and $DM$
come from the real-time detection pipeline. The width $w$ is initialized to 1
ms and $log_{10}(A_{s})$ is initially set to a reasonable value of ${-4}$ in
native data units. All these parameters get a broad uniform prior around their
initial estimate. We use a uniform prior of 1D radius ($\mathrm{4,8}$) in
units of $\mathrm{ms}$ and $\mathrm{pc\;cm^{-3}}$ around the real-time
detection estimates of $t_{0}$ and $DM$. This is a sufficiently generous prior
given the accuracy of our real-time estimates. The uniform prior range for $w$
in ms is $\mathcal{U}(0,10)$ and for $log_{10}(A_{s})$ it is
$\mathcal{U}(-10,2)$. We fix the $DI$ to $-2$. The posterior is then sampled
using MCMC. This initial iteration refines the location of the burst in time-
DM space and provides a good first estimate of the intrinsic pulse width along
with a rough estimate of the amplitude. Using mean based point-estimates for
$(t_{0},DM,w,A_{s})$ from this first MCMC iteration, we initialize our second
iteration, where we additionally fit for the spectral parameters $n_{s}$ and
$\alpha_{s}$, which have their initial values set to 0.

Basic Model: In our third iteration of the MCMC fitting, we include scatter
broadening by convolving the Gaussian pulse profile with a one-sided
exponential as given by Eq. (6). The additional parameters needed to model
scattering are $\tau$ and $SI$ from Eq. (2), but we fix $SI$ to $-4$ and fit
only for $\tau$. Thus our model now has seven parameters in total;
${\bm{\theta}}=(t_{0},DM,w,A_{s},n_{s},\alpha,\tau)$ with $DI$ still kept
fixed to $-2$ as before. Our initial estimates for
$t_{0},DM,w,A_{s},n_{s},\alpha$ come from the mean of the converged samples
from the last iteration while $\tau$ is initialized to a reasonable value of 1
ms. We define a broad uniform prior around this initial value with a 1D radius
of (5, 2, 4, 4, 500, 500, 4) where time-like values are in ms. For e.g, our
prior on time $t_{0}$ is $\mathcal{U}(\bar{t}_{0}-5,\bar{t}_{0}+5)$ where
$\bar{t}_{0}$ is the mean estimate of $t_{0}$ from the last iteration of the
fit. We call this model the ‘basic model’ henceforth.

Fiducial Model: For the analysis presented in this work, our ‘fiducial model’
is the full nine-parameter model given by Eq. (6) which we fit for in the
fourth and final iteration of the fitting process. Compared to the seven-
parameter basic model, we now additionally fit for the parameters $DI$ and
$SI$. The parameters $t_{0},DM,w,A_{s},n_{s},\alpha$ and $\tau$ are
initialized to the mean value of MCMC samples obtained from the previous step.
The two new free parameters $DI$ and $SI$ are initialized to their fiducial
values of $-2$ and $-4$. We use flat priors for
$t_{0},DM,w,A_{s},n_{s},\alpha$ and $\tau$ with 1d radius given by (5, 2,
3.16, 4, 200, 200, 1) where time-like values are in ms. We use a Gaussian
prior with width of 0.001 for $DI$ and a width of 0.5 for $SI$. The burst
model for this final iteration now also includes a treatment to model intra-
channel dispersion smearing, as has been done in our previous works (CHIME/FRB
Collaboration et al., 2019b). We compose the model at an upsampled factor of
16 and 8 in time and frequency, respectively, which we then boxcar-convolve to
get the predicted pulse at our instrumental resolution. The boxcar convolution
serves to model the impulse response of our instrument. This additional step
in the analysis increases our robustness to often ignored issues like
dispersion smearing and instrument impulse modeling. This four-step iterative
fitting procedure constitutes our default analysis.

The burst models described above are the same as that used in Masui et al.
(2015) and implemented in the fitburst burst fitting library (Fonseca et al.,
2023). An initial version of this MCMC pipeline was used for the Bayesian
modeling of FRBs in CHIME/FRB’s first science paper (CHIME/FRB Collaboration
et al., 2019b). The codebase has since been re-implemented from scratch in JAX
with several optimizations incorporated along the way. The inference pipeline
outlined above, with its relatively uninformative choice of prior has been
successfully applied to hundreds of CHIME/FRB events. However, some atypical
events require special treatment. We describe these choices in detail in §6.

Figure 1: Results from validation study of our MCMC pipeline using simulated
FRBs as described in §5.2. The study uses simupulse to simulate radio pulses
with varying true physical parameters which are then fed to the MCMC pipeline
for modeling and parameter estimation. The top panel shows S/N of the
simulated burst. Remaining panels show difference between the true simulation
values of parameters and their MCMC based median estimates for 80
independently simulated bursts along with the $2\sigma$ uncertainty. We use
broad uniform priors on all the parameters and recover $\sim 95\%$ of the
parameters within the uncertainty of $2\sigma$ and without any systematic
bias. The study was performed on 200 simulated events but for the sake of
clarity we show results for a subset of simulated bursts. The spectral
parameters $n_{s}$ and $\alpha_{s}$ were also found to be consistent with
their true simulation values.

###  5.2 Validation

Since the objective behind performing a comprehensive modeling of SGR
1935+2154, and FRBs in general, is to obtain an accurate estimate of its
parameters and their associated uncertainties, thoroughly testing our analysis
pipeline is of paramount importance. We test the robustness of the MCMC
pipeline by using it to model a suite of radio bursts simulated using high
precision burst simulation library
simpulse555https://github.com/kmsmith137/simpulse/ and with specifications of
CHIME/FRB. For each simulation, we randomly and sparsely sample burst
parameters from a hyper-space of $DM$, width and scattering. For simplicity,
we keep remaining parameters of the bursts fixed to some fiducial value. In
particular, the intrinsic spectrum is assumed to be broadband and flat
($n_{s}=0$, $\alpha_{s}=0$) and the $DI$ and $SI$ are fixed to $-2$ and $-4$,
respectively. We simulate an FRB for the sampled parameter and add Gaussian
noise and normalize it to a randomly chosen S/N in the range of 20 to 50. The
simulated time-frequency data are seamlessly fed to the MCMC pipeline. The
initial guess for time and $DM$ parameters needed by the MCMC pipeline are
obtained by adding Gaussian perturbation to the true, known values of the
parameters used for simulation. We follow this prescription to simulate and
model 200 bursts. In our post-processing, we find that $\sim 95\%$ of each of
the recovered parameters lie within $2\sigma$ of the true values used for
simulation. None of the parameters show any kind of systematic bias. In Figure
1, we show the fidelity of recovered parameters compared to true simulation
parameters. We note that the validation analysis presented here is simple and
idealized where the noise and RFI are not fully representative of what is
encountered in reality.

Figure 2: Triangle plot showing the two-dimensional joint posterior
distributions of parameters as well as marginalized posterior distribution for
a subset of parameters of our fiducial burst model for FRB 20200428D. The
modeling shown here is performed on data for FRB 20200428D. The two sub-bursts
are fit jointly using a common set of dispersion and scattering parameters as
described in §5.1 and §6.1.1. All time measurements are in milliseconds while
DM is in $\mathrm{pc~{}cm^{-3}}$. The values of $t_{1}$ and $t_{2}$ are
standardized with respect to arrival time of the sub-bursts at 400.195 MHz.
The plot is produced using getdist (Lewis, 2019).

##  6 Results

###  6.1 Burst modeling and Inference

####  6.1.1 FRB 20200428D

A first analysis of FRB 20200428D was published in CHIME/FRB Collaboration et
al. (2020b). The arrival times of the two sub-bursts were found to lead the
contemporaneous/corresponding peaks in their high-energy counterpart light-
curves by a few milliseconds. Establishing their simultaneity or their order
of arrival is important for discriminating among magnetar and FRB emission
models. A conclusive result would also guide future model building attempts.
We therefore revisit the analysis using our newly developed MCMC pipeline. For
the multi-component FRB 20200428D, we model the two components of the bursts
jointly, where the intrinsic parameters are fit independently while
propagation effects are fit using a common set of parameters for both the sub-
components. Thus we have a pair of estimate for the arrival time ($t_{1}$ and
$t_{2}$) and pulse width ($w_{1}$ and $w_{2}$) corresponding to the two
components but a single estimate for the propagation effects $DM$, $\tau$,
$SI$ and $DI$.

For multi-beam detections, we have generally analyzed and presented results
for the highest S/N beam data, as was the case in CHIME/FRB Collaboration et
al. (2020b). But for the extremely bright FRB 20200428D, our burst model is
inadequate in characterizing the highest S/N beam data and our Gaussian
likelihood assumption breaks down. We do obtain reasonable results for each of
the first 3 steps of the iterative fitting, including for the basic model
(which is effectively the same as the fiducial model with sharply peaked
priors on $DI$ and $SI$). However, once we fit the full nine-parameter
fiducial model with relatively weak priors on model parameters as described in
§5.1, the inadequacies become apparent. The $DI$ retrieved after fitting the
fiducial model to the data from the highest S/N beam is several sigmas away
from the expected value of $-2$. Although such a discrepancy could point to
interesting physics, we believe the deviation is possibly due to the failure
of the Gaussian likelihood model for this high S/N, non-Gaussian event. The
band-attenuated dynamic spectrum of FRB 20200428D shows a complex structure
near the bottom of the band which is only made worse by the polyphase filter
bank (PFB) leakage (Pleunis, 2021). Such a structure, which is not accurately
described by our burst model, can lead to covariant shifts in the estimated
values of the highly correlated parameter pairs $DM$ and $DI$.

To circumvent this issue, we make the conservative decision to analyze data
and present results from a different beam that has a much lower detection S/N.
To decide which beam to analyze out of the available options, we look for the
beam with highest S/N detection for which the PFB leakage is not visually
apparent in the dynamic spectra plot. We find that beam 2068 satisfies this
criteria and so we re-run our fiducial analysis on its intensity data.

In Figure 2, we present two-dimensional joint posterior distributions of
parameters for FRB 20200428D and report the best-fit parameter values in Table
3. The best-fit parameters are reasonably consistent with those reported in
CHIME/FRB Collaboration et al. (2020b) particularly when we consider that the
data, their modeling as well as the RFI masking strategy employed here are
different compared to those of CHIME/FRB Collaboration et al. (2020b).
Crucially, the qualitative conclusions made about the arrival time of the sub-
bursts remain unchanged. Both the radio sub-bursts temporally lead their
corresponding high-energy counterparts.

Quantifying the arrival time offset between radio sub-bursts and X-ray peaks:
We use the approach and values reported in Ge et al. (2023) to make
quantitative statements about the offset between the arrival time of the two
radio bursts and the corresponding peaks seen in the X-ray light-curve of
several instruments. The _weighted_ X-ray arrival time of the first and second
peaks, based on measurements made by Insight-HXMT High Energy, Insight-HXMT
Medium Energy, Konus-Wind and INTEGRAL in UTC, geocentric is
14:34:24.42942$\pm$0.00043 and UTC 14:34:24.46167$\pm$0.00043, respectively
(Ge et al., 2023). The ToA estimate for the observed peaks in Insight-HXMT
High Energy, Insight-HXMT Medium Energy and INTEGRAL come from a multi-
Gaussian profile fit to the data in Ge et al. (2023), Li et al. (2021), and
Mereghetti et al. (2020). For the ToA from the Konus-$Wind$ light-curve data,
the starting time of the 4-ms bin associated with the two _wide_ peaks are
reported by Ridnaia et al. (2021). To get a refined estimate of the ToA from
the Konus-$Wind$ light-curve data, Gaussian profiles were fit to the publicly
available data in Ge et al. (2023). 666The result from this Gaussian model fit
agrees reasonably well with a FRED model fit performed subsequently by the
Konus-$Wind$ team (Ridnaia et al. (2023, private communication)). Based on
these ToA estimates from Gaussian model fits, the _weighted_ average delay of
X-ray arrival time for the first peak with respect to the arrival time of
first radio sub-burst reported in CHIME/FRB Collaboration et al. (2020b) is
$\tilde{\Delta}_{1}=2.92\pm 0.43$ ms (refer to Table 3 of Ge et al. 2023). The
weighted uncertainty is dominated by the measurement made by Insight-HXMT High
Energy which has the least uncertainty. Our refined estimate for the
geocentric ToA at infinite frequency for the two radio components are UTC
14:34:24.42595$\pm$0.00061 and UTC 14:34:24.45497$\pm$0.00051, respectively.
With our refined estimate, the delay is updated to $\Delta_{1}=3.47\pm 0.73$
ms where the final uncertainty comes from adding radio and X-ray timing
uncertainties in quadrature. Thus, under these modelling choices, we rule out
simultaneity in the arrival times for the first peak of the radio burst and
the contemporaneous high-energy counterparts at $4.7\sigma$. Similarly, for
the second radio sub-burst and the corresponding high-energy counterpart, we
find $\Delta_{2}=6.70\pm 0.67$ ms and thus rule out simultaneity of the two at
$10\sigma$. These offsets are many times the radio burst widths, many times
the widths of the X-ray peaks but less than the overall X-ray burst envelope
(see Fig. 1 in Mereghetti et al. 2020), and much less than the 3.24-s period
of the magnetar. In Figure 3, the arrival time estimates of radio and X-ray
pulses are shown in UTC (geocentric).

We emphasize again that the radio ToA estimates are obtained using a very
conservative choice of data and burst modeling where the choices were made _a-
priori_ and were mainly dictated by the desire to robustly quantify the radio
burst ToA. While we have not shown results for the highest S/N beam data due
to its subpar fit quality, the offset in arrival time between the radio and
X-ray bands is present even in that case, and is notably more significant than
the conservative result we have presented. Finally, we note that our offset
estimate and its significance analysis rely on the estimate of the X-ray
arrival time published by several other groups, utilizing different datasets,
modeling, and systematic mitigation strategies. The time resolution of the
X-ray light-curve data and its binning, as well as the energy window of these
datasets and dead-time correction strategies, are different for each group.
Therefore, systematic biases to the individual timing estimates cannot be
ruled out. More details about the data and modelling can be found in
Mereghetti et al. (2020); Ge et al. (2023); Li et al. (2021); Ridnaia et al.
(2021). We have aggregated these estimates using inverse variance weighting
and do not attempt to analyze them ourselves.

Figure 3: Estimated time of arrival (ToA) in UTC, geocentric for the two sub-
components of FRB 20200428D. The histogram in green is the marginalized
arrival time for the radio pulse under our fiducial model which has scattering
index and dispersion index, among others, as free parameters. The orange
dashed vertical line is the radio arrival time reported by CHIME/FRB
Collaboration et al. (2020b). The blue curve denotes the weighted arrival time
of the near-coincident X-ray peak estimated by Ge et al. (2023). All the
timestamps are at infinite frequency and include a correction for the
dispersion delay for the radio pulse.

Figure 4: Dynamic spectra of all the SGR 1935+2154 radio bursts modeled in
this paper. The top panel of each subplot shows de-dispersed and band-averaged
data (grey) along with the model curve (red). The right panel of each sub-plot
shows the time-averaged spectrum of the burst along with the beam-convolved
model spectrum. The data correspond to the beam with the highest detection S/N
and we use a customized mask for each spectrum. The curves around the brighter
events are characteristic features resulting from spectral leakage in the
polyphase filterbank (Pleunis, 2021)

####  6.1.2 2020 October 08 events

The events FRB 20201008B, FRB 20201008D and FRB 20201008C were detected in the
main lobe of CHIME/FRB on 2020 October 8 within a time window of 3 seconds.
The first and second bursts were separated by 1.949 s while the time gap
between the second and third burst was 0.954 s, indicating that all three
bursts occurred within one 3.24-s SGR 1935+2154 rotation period. The three
detections, along with their preliminary estimates of $DM$ and fluence, were
first reported in an Atel (Good & CHIME/FRB Collaboration, 2020). Although the
sub-leading bursts FRB 20201008D, and FRB 20201008C are orders of magnitude
dimmer than the leading burst FRB 20201008B and typical FRBs in general, the
fact that they are temporally very close to the bright FRB-like FRB 20201008B
and happen within a rotational period of the magnetar, argues in favor of
classifying them as FRB-like. We therefore use names assigned by the Transient
Name Server777https://www.wis-tns.org/ (TNS).

When modeling these bursts, we chose to treat them as independent events and
model them individually. We note however that regarding them as independent
events as opposed to a single event with multiple components is somewhat
ambiguous. As was the case with FRB 20200428D, FRB 20201008B is also an
extremely bright event and our fiducial model is a poor fit to the data
corresponding to the highest S/N detection beam. Therefore, we apply our model
to a more amenable data from a different beam.

For FRB 20201008C, a single-component fit to the data suggested a complex
morphology, possibly indicating the presence of multiple sub-components. The
$DM$ and width retrieved from the fit were considerably different from other
SGR 1935+2154 bursts. Consequently, we chose to apply a two-component model to
FRB 20201008C. For this fitting, we employed a relatively strong yet a
reasonable prior on $DM$ and width. Given its low S/N, we only fit up to the
basic model with fixed $DI$ and $SI$. The results from this fitting are
reported in Table 3.

####  6.1.3 Remaining two bursts from 2022

The remaining two events, FRB 20221014A and FRB 20221201A were observed on
2022 October 14 and 2022 December 01, respectively. These events were detected
in the side-lobes, at hour angles of $-$99.8$\degree$ and $-$11.1$\degree$
respectively with estimated fluence of $\mathcal{O}$(10 kJy ms) as detailed
later in §6.4. Given this high fluence, we register them as FRBs in the
Transient Name Server. We model the highest S/N beam data of both these bursts
under the fiducial model. The best-fit point estimate of parameters, based on
the median of the converged MCMC chain, are reported in Table 3 while the
dynamic spectrum is included in Figure 4.

For the radio burst observed on 2020 October 14, FRB 20221014A, independent
X-ray instruments, GECAM and HEBS, and Konus-Wind (Frederiks et al., 2022a;
Wang et al., 2022) reported a short X-ray burst association. The GECAM
observation consisted of a single pulse in the 20-100 keV band with a duration
of about 250 ms. The Konus-Wind emission was observed in two instrument’s
energy bands: G1(20-80 keV) and G2 (80-320 keV). The Green Bank telescope
(GBT), which was actively monitoring SGR 1935+2154 at that time, observed at
least five bursts with significant signal-to-noise during a C-Band session
(Maan et al., 2022). All five radio bursts were detected within a time span of
1.5 seconds, well within one rotation of the magnetar, but over a range of
phases. We find that our estimated arrival time of 2022-10-14T19:21:39.130
(UTC, topocentric) for FRB 20221014A is contemporaneous with the short X-ray
burst reported by GECAM and HEBS (Wang et al., 2022) and Konus-Wind (Frederiks
et al., 2022a) and remains consistent with two of the brightest bursts seen by
GBT within their 0.1s uncertainty as reported by Maan et al. (2022). A higher
precision estimate of the X-ray arrival time can establish or rule out
simultaneity of the two events. In order to aid such a comparative analysis in
the future, we present parameter covariance result from our MCMC analysis of
FRB 221014A in Figure 5.

Figure 5: A triangle plot showing the covariance between parameters pairs as
well as marginalized posterior distribution for all the parameters of our
fiducial burst model for FRB 20221014A. The parameter $t_{0}$ is with respect
to the arrival time of the burst at $\nu_{ref}=400.195$ MHz. All time
measurements are in millisecond units. $DM$ is in $\mathrm{pc~{}cm^{-3}}$.

###  6.2 Properties & Morphology

The dispersion measures as well as scattering measures of the bursts analyzed
in this paper and reported in Table 3 are close to each other and do not show
any secular evolution, suggesting no drastic evolution in the properties of
the intervening medium over the 2.5-yr period. The dispersion indices of all
the bursts, which are fit using a weak Gaussian prior, are within 2-3$\sigma$
of the theoretically expected value of $-2$. Except for FRB 20221201A,
scattering indices are also found to be consistent with the theoretically
motivated value of close to $\sim-4$, when fit with a Gaussian prior of width
$\sigma=0.5$ centered around the value of $-4$.

None of the bursts show any hint of the ‘sad trombone’ features often seen in
repeating FRB spectra (e.g. Hessels et al., 2019; CHIME/FRB Collaboration et
al., 2019a). Moreover, all these events are quite broadband. This is of note
as statistical studies of FRB population by Pleunis et al. (2021) found
repeater events to be on average broader in temporal width and more narrow-
band than apparent non-repeaters.

###  6.3 Localization

The detected bursts were localized using the intensity localization pipeline
of CHIME/FRB. This pipeline fits a model of CHIME/FRB’s beams and an
underlying spectral model to the spectra from all beams that detected the
burst as well as all beams adjacent to those detection beams. For the events
detected in CHIME’s far side-lobes, FRB 20221014A and FRB 20221201A, we used a
method identical to that presented by Lin et al. (2023a). That is, we use a
power-law model for the underlying burst spectrum and include only the FFT-
formed synthesized beam model. For the events detected in CHIME’s main lobe,
FRB 20201008B, FRB 20201008D, and FRB 20201008C, a power-law model was also
used, but the primary beam model was included. Table 1 shows the resulting
burst localizations and their uncertainties. The positional uncertainties are
derived from the statistical uncertainty summed in quadrature with the
systematic errors estimated by Lin et al. (2023a): 0.07 deg in RA and 0.10 deg
in Dec. These burst positions are all consistent with the known position of
SGR 1935+2154 (Israel et al., 2016).

TNS Name | RA | Dec
---|---|---
| deg | deg
FRB 20221201A | $293.75\pm 0.06$ | $21.86\pm 0.06$
FRB 20221014A | $293.84\pm 0.18$ | $21.84\pm 0.17$
FRB 20201008A | $293.8\pm 0.2$ | $21.86\pm 0.11$
$292.0\pm 0.1$ | $21.85\pm 0.12$
FRB 20201008B | $293.8\pm 0.2$ | $21.88\pm 0.11$
FRB 20201008C | $293.75\pm 0.06$ | $21.87\pm 0.05$
Table 1: The fitted sky positions for the five new SGR 1935+2154 events.

###  6.4 Flux and Fluence estimation

Lower limit fluence and flux values are estimated for each burst using the
automated intensity flux calibration pipeline described by Andersen et al.
(2023). In brief, the spectrum of each burst is calibrated for flux and beam
response using a steady source transit located closest in declination and
time. In this automated analysis, we calculate the burst flux assuming that
each burst was detected at “beam boresight”, which we take to be the transit
location of the burst position along the local meridian. Thus, these flux
measurements are biased low, as bursts off-meridian will experience beam
attenuation that is not accounted for. Note that, although the location of SGR
1935+2154 is precisely-known, we do not use the method in CHIME/FRB
Collaboration et al. (2020a) to scale the fluence by the beam response and
obtain non-lower-limits for the bursts in this sample that were detected in
the main lobe of the primary beam. Each of these bursts, although in the main
lobe of the primary beam, are outside of the $600$ MHz FWHM of the formed
beam, meaning that parts of the bandwidth will be attenuated to the noise
floor by the beam response (for a description of the formed beam response, see
Section 2.2.1 of Andersen et al. 2023).

For FRB 20221014A and FRB 20221201A that were detected in the far side-lobe
region (CHIME/FRB Collaboration et al., 2020b; Lin et al., 2023a, b), we
complete additional analysis to obtain non-lower-limit estimates of the flux.
Lin et al. (2023b) mentioned an approach to calibrate the S/N of the side-lobe
event. Here we use the similar approach to calibrate the flux,

$\mathrm{Flux}=\mathrm{\frac{G}{B}}\mathrm{Flux_{main}},\\\ $ (10)

where G is the geometrical factor in the range of 4-5, B is the beam response
at the given hour-angle (HA) (Lin et al., 2023b), and Fluxmain is the flux
reported by the automated pipeline assuming that the event was detected along
the meridian in the main lobe.

Figure 6: The left panel shows the map of beam response at $400-725$ MHz
obtained using the Crab Nebula as calibrator for HAs ranging from $-$91.5 to
91.5 deg. Channels contaminated with RFI have been masked manually. The right
panel shows the frequency-averaged beam-response of the left panel.

We measure the beam response B using holographic measurements of the Crab
Nebula by following the pre-processing procedures mentioned in Lin et al.
(2023b), in which Figure 6 shows the 2D and 1D beam response with the HA from
$-$91.5 to 91.5 deg. To estimate the uncertainty of the beam response, we use
the standard deviation of the frequency-averaged beam-response values 10 deg
within the given HA. For FRB 20221201A with HA of $-$11.1 deg, the beam-
response is 0.011$\pm$0.006. For FRB 20221014A, the HA of $-$99.8 deg is
outside of the HA of holographic data (up to $-$91.5 deg). We assume the beam
response is constant outside of the boundary, and the beam-response at $-$99.8
deg is the same as the beam response of 0.0015$\pm$0.0001 at $-$91.5 deg. The
result is consistent with the solar beam measurement (Amiri et al., 2022). We
summarize the beam response measurements in Table 2.

The flux and fluence of the two side-lobe events are on the order of $\sim$1
kJy and $\sim$10 kJy ms, respectively, and we summarize the results in Table
3.

TNS Name | HA | B
---|---|---
| deg |
FRB 20221201A |  $-$11.1 | 0.011$\pm$0.006
FRB 20221014A |  $-$99.8 | 0.0015$\pm$0.0001
Table 2: The hour-angle and the beam-response for the far side-lobe events. Table 3: MCMC based estimates of model parameters for the radio bursts from SGR 1935+2154. The estimates correspond to the median of converged MCMC samples. The uncertainty on time, flux, fluence and $DI$ are $1\sigma$ standard deviation of the samples. For DM, width, scattering and $SI$ we report the 68% highest density interval. Properties | FRB 20200428D aaResults are a for 2-component fit to data. For flux and fluence, we simply quote the estimates from CHIME/FRB Collaboration et al. (2020b) which uses a different masking and calibration approach. | FRB 20201008B | FRB 20201008D | FRB 20201008C bbWe fit a 2-component basic model to the burst with DI and SI fixed and only report the values corresponding to the primary peak.  | FRB 20221014A | FRB 20221201A
---|---|---|---|---|---|---
UTC Date | 2020-04-28 | 2020-10-08 | 2020-10-08 | 2020-10-08 | 2022-10-14 | 2022-12-01
Time ccArrival time is in UTC (topocentric) at infinite reference frequency. | 14:34:24.4080(6) 14:34:24.4370(5) | 02:23:33.3578(8) | 02:23:35.307(1) | 02:23:36.261(3) | 19:21:39.130(2) | 22:06:59.0762(6)
DM [pc cm-3] ddThe DM is derived from the dispersion slope using a dispersion constant of $K=241$ GHz-2 cm-3 pc s-1 (Manchester & Taylor, 1972; Kulkarni, 2020) | $332.77^{+0.08}_{-0.08}$ | $332.50^{+0.13}_{-0.11}$ | $332.09^{+0.19}_{-0.20}$ | $332.67^{+0.12}_{-0.13}$ | $332.72^{+0.26}_{-0.30}$ | $333.11^{+0.09}_{-0.08}$
Width [ms] |  $0.66^{+0.05}_{-0.05}$ $0.33^{+0.07}_{-0.08}$ | $0.23^{+0.02}_{-0.04}$ | $0.29^{+0.08}_{-0.11}$ | $1.03^{+0.92}_{-0.97}$ | $1.48^{-0.25}_{-0.25}$ | $2.02^{+0.05}_{-0.04}$
Peak Flux | 110 kJy 150 kJy | 266 $\pm$ 64 Jy | 16.3 $\pm$ 3.9 Jy | 1.8 $\pm$ 0.7 Jy | 2.1 $\pm$ 1.4 kJy | 3.8 $\pm$ 3.0 kJy
Fluence | 420 kJy ms 220 kJy ms | 966 $\pm$ 239 Jy ms | 34.1 $\pm$ 8.1 Jy ms | 5.9 $\pm$ 1.7 Jy ms | 9.7 $\pm$ 6.7 kJy ms | 23.7 $\pm$ 18.0 kJy ms
Hour Angle [deg] | 22° | -0.5° | -0.5° | -0.5° | -99.8° | -11.1°
Beam ID eeCHIME/FRB beam ID of the data analyzed | 2068 | 0062 | 0059 | 0059 | 0181 | 3061
Scattering [ms] ffThe scattering is with respect to pivot frequency of 600 MHz | $0.76^{+0.09}_{-0.10}$ | $0.72^{+0.08}_{-0.09}$ | $0.52^{+0.08}_{-0.09}$ | $1.13^{+0.5}_{-0.8}$ | $0.93^{+0.35}_{-0.45}$ | $0.92^{+0.11}_{-0.12}$
Dispersion index | $-1.9998(2)$ | $-2.0004(3)$ | $-2.0014(5)$ | $-2$ | $-2.000(1)$ | $-1.99934(23)$
Scattering Index | $-3.96^{+0.35}_{-0.37}$ | $-3.70^{+0.46}_{-0.34}$ | $-4.27^{+0.45}_{-0.45}$ | $-4$ | $-3.92^{+0.45}_{-0.54}$ | $-2.62^{+0.32}_{0.30}$

###  6.5 Radio Upper limits on X-ray Bursts

In addition to the radio bursts discussed above, there were over 156 more
X-ray or $\gamma$-ray bursts from SGR 1935+2154 detected by either Swift/BAT,
XMM-Newton, NuStar, Konus-Wind, Fermi/GBM, Agile, or INTEGRAL IBIS/ISGRI
between 2022 May 25 and 2022 December 13 that were published in either public
ATels or GCNs.888We note that some of these 156 bursts may be repeats e.g., a
burst detected both by Swift/BAT and Fermi/GBM will be counted twice.999This
time range was chosen as this was the period in which the authors maintained
notices of all X-ray and $\gamma$-ray bursts from SGR1935+2154. Out of these
156 bursts, seven were within the field of view of CHIME/FRB at the time of
their high-energy emission, where we define the field of view as within $\sim$
17 degrees from the meridian101010This is the region in which the CHIME/FRB
beam is well modeled and understood.. However, of these seven high-energy
bursts, three occurred during times of non-nominal CHIME/FRB sensitivity, and
hence were ignored for the analysis that follows.

For the four bursts which occurred within the FOV of CHIME/FRB and occurred at
times of nominal CHIME/FRB sensitivity (Frederiks et al., 2022b; Palmer &
Swift/BAT Team, 2022; Veres & Fermi-GBM Team, 2022; Veres et al., 2022), we
calculate an upper limit on FRB-like radio emission associated with the burst.
To calculate an upper limit on FRB-like radio emission, we follow the methods
presented in Curtin et al. (2023). In summary, we use an FRB nearby in
declination to act as our S/N-to-flux calibrator. We use models of both the
primary and formed beams at CHIME/FRB to account for any subtle positional
differences between the FRB calibrator and SGR 1935+2154, and use system
sensitivity metrics to account for temporal system sensitivity differences
between the time of the FRB and that of the high-energy burst from SGR
1935+2154. We then account for differences in the sky temperature (and hence
sensitivity) between the FRB location and that of SGR 1935+2154 using the 2014
Haslam all-sky continuum map at 408 MHz (Haslam et al., 1982). Finally, we
scale our results to a detection S/N of 10, an intrinsic radio burst
width111111This only affects our fluence radio limits. of 10 ms, and an
estimated scattering time of 0.41 ms (estimated using NE2001). We also account
for the different time of arrivals between the X-ray burst and the radio burst
assuming a DM of 332.699 pc cm-3. The resulting upper limits for the four
bursts in question are provided in Table 4.

The three bursts from Swift/BAT and Fermi/GBM do not have readily available
X-ray fluxes and fluences. However, for the X-ray burst detected on 2022
October 13 at 02:02:46, Konus-Wind reported a preliminary burst fluence of
3.30 $\pm$ 0.11 $\times 10^{-6}$ erg/cm2 and a burst flux of 16.9 $\pm$ 1.4
$\times 10^{-6}$ erg/cm2/s in the 20-500 keV band. Thus, the upper limits on
the radio-to-high-energy flux and radio-to-high-energy fluence ratios are
$8.5\times 10^{-9}$ and $5.5\times 10^{-11}$, respectively, assuming a 400-MHz
bandwidth for CHIME/FRB. This radio-to-fluence ratio upper limit is five
orders of magnitude lower than the radio-to-fluence ratio of
$4^{+4}_{-2}\times 10^{-6}$ between the radio detection and Konus-Wind
detection of SGR 1935+2154 from 2020 April.

While the burst on 2020 April 20 was particularly hard, with a peak energy in
the Konus-Wind band (20 - 500 keV) of 82${}^{+12}_{-9}$ keV, the burst on 2022
October 13 had a lower peak energy of 33${}^{+3}_{-2}$ keV. The peak energy of
the burst on 2022 October 13 was similar to the accompanying high-energy
counterpart detected by Konus-Wind for the radio burst detected by CHIME/FRB
and presented above on 2022 Oct 14. The peak energy of this high-energy
counterpart was 40${}^{+6}_{-6}$ keV, in agreement within 1$\sigma$ with the
peak energy report for the burst on 2022 October 13. Thus, not all radio
bursts from SGR 1935+2154 require a particularly high peak energy for an
accompanying X-ray burst, yet bursts of similar peak X-ray energy are not
always accompanied by observable radio bursts.

Table 4: Upper Limits on FRB-Like Radio Emission from SGR 1935+2154 at the times of X-ray bursts occuring near CHIME’s meridian  TimeaaDetection time at the respective high-energy instrument. | Instrument bbInstrument for the high-energy detection. | Hour Angle ccHour angle at CHIME/FRB accounting for the DM delay at 400 MHz associated with a DM of 332.699 pc cm-3. | Flux ddUpper limit on the FRB-like radio flux in the 400-to-800 MHz band. | Fluence eeUpper limit on the FRB-like radio fluence in the 400-800 MHz band scaling the burst to a 10-ms width. |  $\eta$ ffUnitless ratio between the radio and high-energy fluences assuming a 400 MHz bandwidth for CHIME/FRB.
---|---|---|---|---|---
(UTC) |  | (degrees) | Jy | Jy ms | (unitless)
10-13-2022 02:02:46 | Konus-Wind |  $-$1.0 |  $<$36 |  $<$234 |  $<$$5.5\times 10^{-11}$
10-13-2022 02:36:00 |  Swift/BAT | 7.2 |  $<$2300 |  $<$14000 | N/A
10-14-2022 02:20:34 |  Fermi/GBM | 4.4 |  $<$4000 |  $<$25000 | N/A
10-15-2022 02:13:55 |  Fermi/GBM | 3.7 |  $<$1800 |  $<$11000 | N/A

###  6.6 Exposure and Sensitivity

Among the six CHIME/FRB detections, three bursts occurred during the transit
of the source in the main lobe of the telescope. The exposure in the main lobe
was calculated following the methodology outlined by CHIME/FRB Collaboration
et al. (2021a). This corresponds to a cumulative duration of 111.14 hours
above a fluence threshold limit of 10.3 Jy ms, with a confidence level of 95%,
from 2018 August 28 to 2022 December 1.

To determine the fluence threshold, we employed FRB
20190107A121212https://www.chime-frb.ca/catalog/FRB20190107A from CHIME/FRB
Collaboration et al. (2021a) as it has a similar declination to that of SGR
1935+2154. Using the CHIME/FRB beam model131313https://chime-frb-open-
data.github.io/beam-model/, we assessed the sensitivity along the transit path
for both the reference FRB and SGR 1935+2154. The sensitivity ratio between
SGR 1935+2154 and the reference FRB was 0.8. Subsequently, we scaled the
fluence threshold of the reference FRB by this ratio to obtain the final value
for SGR 1935+2154.

For events detected in a sidelobe, we present an upper limit on the exposures,
taking into account the operational uptime of the CHIME/FRB system. The system
remained operational for a duration of 1191.45 days, starting from 2018 August
28, until the cutoff date of 2022 December 1, for this publication. On
average, only 95% of the synthesized beams were online, corresponding to 984
beams out of 1024 beams (CHIME/FRB Collaboration et al., 2018). Consequently,
the total operational duration is reduced to 1145 days, denoted as “beam days”
in this analysis. The exposure in days from the source can be estimated as
follows -

$Exposure\,(in\,days)=\frac{\Delta H}{\cos\delta\times 360^{\circ}}\times
beam\;days$ (11)

Here, the first term represents the fraction of a day during which a source at
a given declination $\delta$ is observable by CHIME within a specific hour
angle range $\Delta H$. For SGR 1935+2154, the furthest detection has an hour
angle of $-$100.7 deg. As the beam response cannot be accurately characterized
at this position, we consider the second furthest detection from zenith, FRB
20221014A, corresponding to an hour angle of 91.5 degrees. Thus, the total
hour angle transit range amounts to 183 degrees. By substituting these values,
we obtain an upper limit on exposure of 627 days at a fluence threshold of
10.2 kJy ms. More details on the analysis are presented by Lin et al. (2023a).

The fluence threshold in the side-lobe is three orders of magnitude higher
than that in the main lobe. This disparity arises because side-lobes are
solely sensitive to exceedingly bright detections. To determine the
sensitivity threshold for burst FRB 20221014A, we utilized the following
relation:

$Fluence~{}threshold\,(\mathrm{Jy~{}ms})=f\times\frac{G}{B_{R}}\times\frac{8}{S/N}.$
(12)

Here $f$ is the lower limit on the main lobe fluence of the burst estimated
using the methodology described by Andersen et al. (2023), $G$ is the
geometrical factor derived by Lin et al. (2023b), $B_{R}$ is the beam response
at hour angle 91.5 degrees estimated using Tau A holography, details of which
are discussed by Lin et al. (2023b), $S/N$ is the real-time (bonsai) signal-
to-noise ratio for this event (CHIME/FRB Collaboration et al., 2021a) and 8 is
the cutoff S/N for which we store total intensity data.

We detected a total of three bursts in the side-lobes during the interval from
28 August 2018 to 1 December 2022, so the rate of FRB-like bursts from the
source for fluences $\gtrsim$ 10 kJy is $0.005^{+0.082}_{-0.004}$ burst per
day, where the uncertainty is the 95% confidence interval assuming a
Poissonian process. This suggests that very bright bursts from SGR 1935+2154
are rare occurrences, even during times of great X-ray activity.

##  7 Discussion

In this paper we report on additional radio bursts, following those found in
2020 April (CHIME/FRB Collaboration et al., 2020b), detected by CHIME/FRB from
the Galactic magnetar SGR 1935+2154. In total, we present six bursts observed
on four different occasions between 2020 April and 2023 December, with at
least one burst with fluence $\gtrsim\mathrm{1~{}kJy~{}ms}$ on each of these
occasions. The peak fluxes of these bursts span a remarkable range of five
orders of magnitude. Although the brightest bursts seen thus far from SGR
1935+2154 have luminosities in the range of those the least luminous FRBs
known, as discussed by CHIME/FRB Collaboration et al. (2020a), FRBs as a class
have luminosities over six orders of magnitude yet higher. Thus it is yet
unclear whether most FRBs are magnetars. Still, the large dynamic flux range
reported on in this work makes clear even a single magnetar is capable of a
wide variety of radio burst luminosities.

We present a novel MCMC burst fitting code that has been developed for current
and future precision FRB science needs, and use it to confirm that even when
allowing both $DI$ and $SI$ to vary, the 2020 April FRB-like radio burst from
SGR 1935+2154 had both radio sub-bursts leading their X-ray counterparts by a
few ms, far less than the rotation period of the magnetar, ruling out models
that predict X-ray emission to precede or be coincident with the radio
emission of this event. For example, in a model that invokes synchrotron maser
emission from decelerating relativistic blast waves to produce FRBs, Metzger
et al. (2019) and Margalit et al. (2020) argue that X-ray emission arises from
incoherent synchrotron emission from the same location as the radio burst
(i.e. in the downstream shock; see also Lyubarsky, 2014; Beloborodov, 2020).
They therefore predict near-coincident X-ray and radio bursts, or at most
under a radio burst width apart. However the X-rays in this model are
optically thin at the shock deceleration radius, in contrast to the radio
photons which are optically thick to induced scattering at the same location,
hence escape after the X-rays. This is hard to reconcile with the results we
have obtained in our analysis. On the other hand, Lyutikov & Popov (2020)
argue that FRBs are reconnection events in magnetar magnetospheres (see
Lyutikov, 2002; Wadiasingh & Timokhin, 2019; Lyutikov, 2020) and predict that
any radio burst must lead an associated X-ray burst by at most a few
milliseconds, because it is only early in the reconnection event that the
radio waves can escape. This prediction is consistent with our results.
However, Beloborodov (2021, 2023) argue that such luminous radio bursts are
damped within the magnetosphere and cannot escape; along the magnetic axis,
the radio waves interact with plasma particles and accelerate them to high
Lorenz factor at the expense of the wave energy. If so, FRB emission cannot
originate therein. Meanwhile, Qu et al. (2022) argue that in the open field
line region of magnetar magnetosphere, both the likely high outward speed of
the plasma together with alignment of the field with the radio propagation
direction reduce the expected interaction between the radio waves and the
plasma, mitigating damping. If so, then the close precedence of the radio to
the X-ray bursts may indeed indicate a magnetospheric origin.

From a morphological point of view, the reported bursts from SGR 1935+2154
appear different from typical repeating FRBs (e.g., Hessels et al., 2019;
CHIME/FRB Collaboration et al., 2019a), which tend to have broader bandwidths
than the average for the repeaters (Pleunis et al., 2021). Also, we find no
evidence for downward frequency drifting (the “sad trombone” effect) so common
to repeating FRBs. However, small number statistics might be at play, given
that bursts from repeaters can be broadband and exhibit no drifting. A clear
prediction of the implication that at least some FRBs are magnetars (CHIME/FRB
Collaboration et al., 2020a; Bochenek et al., 2020) is that narrow-band and/or
frequency drifting will one day be seen in radio bursts from SGR 1935+2154
and/or other established magnetars.

In this work we find that CHIME/FRB, by virtue of its wide side-lobes, has
excellent daily exposure to SGR 1935+2154 and other Galactic magnetars for
bursts above a fluence threshold of $\mathrm{10~{}kJy}$ ms when above the
horizon. Our estimated radio burst rate for SGR 1935+2154, is
$0.005^{+0.082}_{-0.004}$ bursts per day for fluences $\gtrsim$ 10 kJy ms in
the CHIME 400-800-MHz band in the period 28 August 2018 to 1 December 2022,
suggesting that only 1–2 such bright bursts on average will be observable by
CHIME/FRB per magnetar each year, and this is likely a strong upper limit
since this rate was measured during a period of intense activity by SGR
1935+2154. Indeed although there are 10 catalogued Galactic magnetars known in
the declination range covered by CHIME/FRB (see Olausen & Kaspi,
2014)141414https://www.physics.mcgill.ca/~pulsar/magnetar/main.html, radio
bursts from only SGR 1935+2154 have been detected thus far by CHIME/FRB. This
suggests that the per source long-term average bright radio burst rate per
Galactic magnetar is far lower than the value reported above. This information
from CHIME/FRB will be helpful for informing on the expected event rates of
Galactic magnetar radio bursts for other planned wide-field radio instruments
such as GReX (Connor et al., 2021) and BURSTT (Lin et al., 2022) whose aim is
to detect rare, high fluence short-duration radio bursts like those from SGR
1935+2154.

We acknowledge that CHIME is located on the traditional, ancestral, and
unceded territory of the Syilx/Okanagan people. We thank the Dominion Radio
Astrophysical Observatory, operated by the National Research Council Canada,
for gracious hospitality and expertise. 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 Trottier Space Institute via the Trottier Family Foundation, and the
University of British Columbia. The CHIME/FRB baseband system was funded in
part by a CFI John R. Evans Leaders Fund grant to IHS. The Dunlap Institute is
funded through an endowment established by the David Dunlap family and the
University of Toronto. Research at Perimeter Institute is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research & Innovation. The National Radio Astronomy
Observatory is a facility of the National Science Foundation (NSF) operated
under cooperative agreement by Associated Universities, Inc. FRB research at
UBC is supported by an NSERC Discovery Grant and by the Canadian Institute for
Advanced Research. U.G. thanks Mingyu Ge and Anna Ridnaia for their assistance
with interpreting their data and answering some basic questions about their
analysis. U.G. also thanks Daniel Foreman-Mackey for his suggestions regarding
sampling-algorithms and Mark Halpern and Antonio Herrera Martin for their
useful feedback. Support for this research was provided by the University of
Wisconsin - Madison Office of the Vice Chancellor for Research and Graduate
Education with funding from the Wisconsin Alumni Research Foundation. B. C. A.
is supported by an FRQNT Doctoral Research Award. A.P.C is a Vanier Canada
Graduate Scholar. 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. K.W.M. holds the
Adam J. Burgasser Chair in Astrophysics and is supported by NSF grants
(2008031, 2018490). K.R.S. acknowledges support from Fonds de Recherche du
Quebec – Nature et Technologies (FRQNT) Doctoral Research Award. M.B. is a
Mcwilliams fellow and an IAU Gruber fellow. F.A.D is supported by the UBC Four
Year Fellowship. B.M.G. is supported by an NSERC Discovery Grant
(RGPIN-2022-03163), and by the Canada Research Chairs (CRC) program. C. L. is
supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51536.001-A
awarded by the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. M.M is supported by NSF grant 2307109 and DOE grant DE-
SC0017647. A.B.P. is a Banting Fellow, a McGill Space Institute (MSI) Fellow,
and a FRQNT postdoctoral fellow. A.P. is funded by the NSERC Canada Graduate
Scholarshops – Doctoral program. Z.P. is a Dunlap Fellow. K.S. is supported by
the NSF Graduate Research Fellowship Program. D.C.S. is supported by an NSERC
Discovery Grant (RGPIN-2021-03985) and by a Canadian Statistical Sciences
Institute (CANSSI) Collaborative Research Team Grant. S.P.T. is a CIFAR
Azrieli Global Scholar in the Gravity and Extreme Universe Program. , astropy
(Astropy Collaboration et al., 2022)

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