# A broad survey of spectro-temporal properties from FRB 20121102A

Mohammed A. Chamma,1 Fereshteh Rajabi,1,2 Aishwarya Kumar1 and Martin Houde1
1Department of Physics and Astronomy, The University of Western Ontario, 1151
Richmond Street, London, Ontario N6A 3K7, Canada
2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,
Canada E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We survey the spectro-temporal properties of fast radio bursts from FRB
20121102A observed by earlier studies across a wide range of frequencies. We
investigate 167 bursts from FRB 20121102A spanning frequencies 1–7.5GHz,
durations of less than 1 ms to approximately 10 ms, with low and high
energies, and with different wait-times. We find from this sample of bursts a
strong agreement with the inverse relationship between sub-burst slope and
duration and with other predictions made by the triggered relativistic
dynamical model (TRDM). Earlier results found agreement with those predictions
across three different repeating FRB sources. For this sample of bursts, we
find that the sub-burst slope as well as the ‘sad trombone’ drift rate are
consistent with being in a quadratic relationship with frequency and that both
these quantities are inversely proportional to the duration. We also find that
the duration decreases with increasing frequency as well as a statistically
significant correlation between the sub-burst duration and bandwidth
(proportional to $t^{-1/2}$) that is unexpected. No distinct group of bursts
in this sample deviated from these relationships, however significant scatter
can be seen in measurements. This study demonstrates the consistent existence
of relationships between the spectro-temporal properties of bursts from a
repeating FRB source. A simple explanation for the inverse relation between
the sub-burst slope and duration is an inherently narrowband emission process.
We make all measurements available as well as a graphical user interface
called Frbgui developed and used to perform measurements of burst waterfalls.

###### keywords:

(transients:) fast radio bursts – methods: data analysis – relativistic
processes – radiation: dynamics – radiation mechanisms: non-thermal

††pubyear: 2023††pagerange: A broad survey of spectro-temporal properties from
FRB 20121102A–A broad survey of spectro-temporal properties from FRB 20121102A

##  1 Introduction

Fast radio bursts (FRBs) are short and intense pulses of radiation whose
emission mechanism eludes understanding despite a multitude of theoretical
models and many recent observational efforts. Observationally, the originating
environments, energy distributions, polarization properties, and activity
cycles of FRB sources have been investigated, with each new study often adding
a new unexpected characteristic (see Petroff et al., 2022, for a recent
review). Theoretical explanations for FRBs center around extreme environments
such as magnetars where large numbers of particles in a plasma state emit
coherently (Platts et al., 2019; Lyubarsky, 2021). In recent years, hundreds
of new sources have been discovered with a clear distinction in properties
between repeating and non-repeating sources (CHIME/FRB Collaboration et al.,
2021; Pleunis et al., 2021). This flurry of activity around FRBs has revealed
significant challenges in understanding this phenomena and in constraining the
possible explanations.

Many of the observational characteristics of FRBs vary dramatically from
source to source, making it unclear which features are arising due to the
emission mechanism, the environment, or propagation effects. The spectral
luminosities of FRB sources span several orders of magnitude and the durations
of bursts can range from tens of nanoseconds to tens of milliseconds (Nimmo et
al., 2022). The polarization properties of FRBs vary as well; for example in
FRB 20121102A (Spitler et al., 2014, 2016) bursts have a constant polarization
across their duration (Michilli et al., 2018), whereas in FRB 20180301A the
polarization angle of some bursts show diverse behaviours (Luo et al., 2020).
In FRB 20190520B, the observed rotation measures (RMs) exhibited changes in
sign (Anna-Thomas et al., 2022), while FRB 20121102A has the largest absolute
RM and range of RMs observed (Hilmarsson et al., 2021). These characteristics
of FRB sources complicate our understanding of the processes involved and are
likely due to an overlap of multiple different phenomena.

One avenue for understanding the emission mechanism of FRBs is to study the
spectro-temporal properties of bursts, which has revealed several
relationships that are common from burst to burst and even from source to
source. Among these quantities, which include the bandwidth, duration, and
central frequency, are the drift rate and the similar but distinct sub-burst
slope or intra-burst drift111The ‘sub-burst slope’ terminology was used in
Chamma et al. (2021) while ‘intra-burst drift’ was used in Jahns et al.
(2022). Both terms describe the same measurement and we will use ‘sub-burst
slope’ hereafter.. The drift rate refers to the change in frequency of
multiple resolved sub-bursts within a single waterfall, and the tendency for
later sub-bursts to arrive at lower frequencies is called the ‘sad trombone’
effect. The sub-burst slope on the other hand refers to the change in
frequency with time within a single sub-burst or pulse. Hessels et al. (2019)
studied bursts from FRB 20121102A and the relationship between their frequency
and the drift rate of multiple resolved sub-bursts, finding that the drift
rate increased with frequency. This relationship appeared to be linear
(Josephy et al., 2019).

Modelling and understanding the origins of spectro-temporal relationships is
an informative avenue for understanding FRBs. In the triggered relativistic
dynamical model (TRDM) proposed by Rajabi et al. (2020), an inverse
relationship was predicted between a sub-burst’s slope and its duration from
the assumptions of a narrow-band emission process and up-to-relativistic
motions in the source. This prediction agreed with measurements made for a
sample of bursts from FRB 20121102A. This relationship was further explored in
Chamma et al. (2021) where bursts analysed from three repeater sources (FRB
20180916B (CHIME/FRB, 2020) and FRB 20180814A (CHIME/FRB Collaboration et al.,
2019) in addition to FRB 20121102A) also had sub-burst slopes that varied
inversely with duration, indicating that the same relationship could describe
the features of bursts from different sources. The model in Rajabi et al.
(2020) also predicted a quadratic relationship between the sub-burst slope and
the frequency, and some evidence of this can be seen in various datasets as
plotted in Fig. 5 of Chamma et al. (2021) and Fig. 8 of Wang et al. (2022).
Jahns et al. (2022) studied over 800 bursts from FRB 20121102A in the 1.1-1.7
GHz band and also found the sub-burst slope to be inversely proportional to
duration. In addition they compared the sub-burst slope to a dozen drift rates
and found that the drift rates were larger but seemed to extend the same trend
with the duration as the sub-burst slopes do (Jahns et al., 2022). This
behaviour for drift rates and sub-burst slopes obeying similar or identical
relationships is expected within the TRDM when groups of sub-bursts are
emitted at roughly the same time (Sec. 3.1 of Chamma et al., 2021; Rajabi et
al., 2020). Given these recent discoveries it is fruitful to study the
spectro-temporal features of bursts in order to better characterize these
relationships, to find their limitations, and to find possible commonalities
between sources that will contribute to our understanding of the FRB emission
mechanism.

Studies thus far have focused on samples of bursts that are observed in
relatively narrow frequency ranges (limited by the observing instrument), and
often spectro-temporal measurements are not performed. While studies that
perform such measurements exist using deep samples of bursts (e.g. Aggarwal et
al. 2021; Pleunis et al. 2021; Jahns et al. 2022), there is a dearth of
studies that broadly and comprehensively sample the properties of all of a
source’s bursts. This leaves a potential gap in our knowledge of the spectro-
temporal relationships that exist and their applicability to a broad sample of
a source’s bursts.

In this work we collect a large and diverse sample of bursts from a single
source and search for spectro-temporal relationships, while also investigating
if the same relationships, such as that between the sub-burst slope and the
duration, are verified across the entire sample. This will indicate if bursts
from a single source can be differentiated based on the spectro-temporal
relationships they obey as well as the robustness of these relationships. To
this end we collect bursts from multiple observational studies of the
repeating source FRB 20121102A and measure their spectro-temporal properties.
As one of the best observed repeaters, FRB 20121102A has been localized to a
star-forming dwarf galaxy with a persistent radio counterpart and has bursts
that cover a wide range of frequencies and durations (Bassa et al., 2017;
Tendulkar et al., 2017; Chatterjee et al., 2017; Marcote et al., 2017). Our
measurements include the central frequency, the bandwidth, the sub-burst
slope, the sad-trombone drift rate (when applicable), and the duration of
every burst and sub-burst. We investigate the relationships between these
quantities, and whether a single relationship is sufficient to describe the
data or if deviations exist. Understanding the scope of these relationships
and how well they describe the complete cohort of bursts from a source can
provide valuable information about the emission mechanism and host
environment.

In the following, Section 2 will describe the observations used and the bursts
sampled for this study. Section 3 will describe how data are prepared, how
measurements are obtained using 2D autocorrelations, and how the dispersion
measure (DM) for each burst is handled and optimized within the context of the
burst sample. Section 4 will describe the measurements obtained and explore
different correlations between the spectro-temporal properties of the bursts
while Section 5 will discuss the relationships observed, the predictions of
the TRDM in more detail, and possible interpretations and implications. The
paper is summarized with our conclusions in Section 6.

##  2 Burst Sampling

We describe here the observations used and the properties of the bursts
sampled for our study.

Michilli et al. (2018) observed 16 bursts from FRB 20121102A using the Arecibo
Observatory in a band that spanned 4.1–4.9 GHz. We use all the bursts they
observed and separate the components of three of their bursts (M9, M10, and
M13) for a total of 19 single pulses. The durations of these bursts range from
0.03–1.36 ms, as measured by their full-width-half-maximum (FWHM) and
excluding the bursts whose components we have separated (Michilli et al.,
2018).

Gajjar et al. (2018) observed 21 bursts from FRB 20121102A using the Green
Bank Telescope in a band of 4–8 GHz all of which were 100% linearly polarized.
Of these observations we exclude 5 due to their low SNR (which prevents or
complicates measurements) and split three of their bursts (11A, 12A, and 12B)
for a total of 21 single pulses, with durations that range from 0.18–1.74 ms
(Gajjar et al., 2018).

Oostrum et al. (2020) observed 30 bursts from FRB 20121102A using the
WSRT/Apertif telescope in a band spanning 1250–1450 MHz (and 1220–1520 MHz for
one burst) with much lower levels of linear polarization than observed at
higher frequencies. We use 23 of these bursts with the remaining seven
excluded due to low SNR. All bursts from this dataset were single pulses. The
durations of these bursts, as measured by a top hat pulse with an equivalent
integrated flux density, span 1.6–8.2 ms (Oostrum et al., 2020).

Aggarwal et al. (2021) searched data observed by Gourdji et al. (2019) using
the Arecibo telescope in a band spanning 580 MHz (with a backend spanning 800
MHz) and centered around 1375 MHz. Including the 41 bursts found by Gourdji et
al. (2019), the search yielded a total of 133 bursts in three hours of data.
Notably, almost all of these bursts occur above 1300 MHz. Of these bursts we
exclude almost half due to a low SNR and separate 6 into single pulses for a
total of 63 bursts. The durations of the bursts used span 1.16–17.16 ms based
on the FWHM obtained from the burst autocorrelation (see Section 3).

Li et al. (2021) used the FAST telescope to detect 1652 bursts in about 60
hours of data over 47 days in a band spanning 1000–1500 MHz. These bursts
followed a bimodal energy distribution222For a discussion on differences in
methodology when using the observing bandwidth versus burst bandwidth to
compute burst energies see Aggarwal (2021) and Section 3 of Jahns et al.
(2022). with peaks around $10^{37.8}$ and $10^{38.6}$ erg as well as a bimodal
wait-time distribution with peaks at around 3.4 ms and 70 s. Jahns et al.
(2022) found a weak bimodality in the burst energy distribution in their
sample of 849 bursts and could not conclusively confirm the result of Li et
al. (2021).

In order to investigate if these differing properties of wait-time and energy
might correspond to different spectro-temporal features or relationships we
sampled bursts from both peaks of the bimodal energy and wait-time
distributions reported by Li et al. (2021). For the energy distribution we
sampled 20 bursts from both peaks (10 each) of the energy distribution by
filtering their list of bursts to those with estimated energies between
$10^{37.7}$ and $10^{37.8}$ erg for the first peak and $10^{38.6}$ and
$10^{38.7}$ erg for the second peak. To select bursts with high enough SNR for
a good measurement we additionally filter bursts from the first peak that have
a peak flux density greater than 10 mJy and greater than 100 mJy from the
second peak. Different flux limits are chosen for each peak to ensure a large
enough sample and are somewhat arbitrary. For the wait-time distribution, we
filtered for bursts with wait-times between $\sim$4 to $\sim$6 ms and with a
peak flux density above 40 mJy, and wait-times between $\sim$63 to $\sim$100 s
above 100 mJy. This yielded 11 and 13 bursts from each peak, respectively, for
an additional 24 bursts. Of this sample, several more were excluded due to an
SNR that was still too low to obtain measurements and many of the short wait-
time bursts were split into multiple components, which finally resulted in a
total of 41 single pulses. Of this total, 6 and 12 bursts were from the short
and long wait-time peaks, respectively. Only 2 bursts remained from the low
energy peak and 7 from the high energy peak, 1 burst was included for having a
long duration, and 13 were new pulses that we had separated. Due to challenges
in acquiring the data we decided to proceed with this sample. The bursts used
span a frequency range of 1080–1430 MHz and their FWHM durations span
0.56–15.43 ms.

The sample of bursts analysed in this study total 167 and broadly represent
all the types of bursts that have been observed from FRB 20121102A, spanning
frequencies ranging from 1080 MHz to 7.4 GHz and durations from less than 1 ms
to about and greater than 10 ms. Figure 1 shows a distribution of the
frequencies and durations of the bursts used in our sample.

Figure 1: Histograms of the sub-bursts sampled for this study showing the
frequency (top), sub-burst duration (middle), and sub-burst bandwidth
(bottom).

##  3 Methods and Analysis

This section will describe how the burst waterfalls are loaded and their
spectro-temporal properties measured via 2D autocorrelations of the waterfall
(Hessels et al., 2019; Chamma et al., 2021), as well as the dispersion measure
(DM) ranges used for the measurements, and how measurements are reviewed and
validated. We also describe a graphical user interface (GUI) we developed to
aid in the measurement of burst properties that is easily extensible and
publicly available.

The FRB observations described in Section 2 were obtained from the authors of
their respective publications and we describe in the following how they are
pre-processed in order to reduce computation time, remove radio frequency
interference (RFI), and/or crop burst files before performing measurements.
FRB waterfall data are available in various formats and include filterbank and
PSRFITS files, which were loaded using the
PyPulse333https://github.com/mtlam/PyPulse and
Your444https://github.com/thepetabyteproject/your software packages before
being stored as 2D Python numpy arrays.

The waterfalls of bursts from Michilli et al. (2018) were provided via an
ASCII dump, dedispersed, and at frequency and time resolutions of 1.5625 MHz
and 0.01024 ms, respectively, with 512 frequency channels and 512 time samples
(i.e. with dimensions 512 $\times$ 512).

Burst waterfalls from Gajjar et al. (2018) were available dedispersed in
PSRFITS format at a resolution of 183 kHz and 0.01024 ms with 19456 frequency
channels and 2048 time samples, which we subsampled in frequency by a factor
of 8 to obtain 2432 frequency channels at a resolution of 1.464 MHz. These
were then stored in arrays of size 1690 $\times$ 2048 to exclude masked
channels present at the bottom of the band.

Waterfalls from Oostrum et al. (2020) were obtained un-dedispersed in the
PSRFITS format with resolutions of about 0.195 MHz and 0.04096 ms with size
1024 $\times$ 25000 channels, which we dedispersed to each burst’s reported DM
and downsampled by a factor of 8 to resolutions of 1.5625 MHz and 0.32768 ms.
These waterfalls were then centered and cropped to be stored at a size of 128
$\times$ 200 channels. In addition, before downsampling we applied both a
spectral kurtosis and Savitzky-Golay filter (SK-SG filter; Agarwal et al.,
2020; Nita et al., 2016) that is available via the Your package to mask
channels with high RFI.

The waterfall data from Aggarwal et al. (2021) are provided in long duration
filterbank files with a list of candidate timestamps, and the
Burstfit555https://thepetabyteproject.github.io/burstfit/ package is used to
dedisperse the data to each burst’s DM and select a 0.1 s long waterfall
around the burst. These waterfalls are then saved as arrays of size 64
$\times$ 1220 at resolutions of 12.5 MHz and 0.08192 ms, respectively.

Data from Li et al. (2021) were obtained via correspondence and consisted of
PSRFITS files for each burst. We load these with Your at a size of 4096
$\times$ 131072 channels and resolutions of 122 kHz and 0.098304 ms, and then
filter them with an SK-SG filter. We then dedisperse each waterfwall to the
reported burst DM, subsample to 256 frequency channels, center about the time
channel with the peak frequency-averaged intensity and crop to an array of
size of 256 $\times$ 1000 with resolutions of 1.952 MHz and 0.098304 ms.

All the bursts in this study are preprocessed in the way described in order to
facilitate the measurements of their spectro-temporal features.

Despite the pre-processing of the waterfalls there are still several tasks
that are needed on a burst-by-burst basis, that, with a large number of bursts
and measurements to manage, can quickly become overwhelming and difficult to
review. These tasks include additional noise removal (such as masking channels
or applying an SK-SG filter), additional subsampling to increase the S/N, and
separating the components of bursts with multiple pulses. While these tasks
can be automated to an extent, being able to manage and refine a measurement
allows for more accurate results and the inclusion of strange bursts that
might not fit in an automation pipeline of limited complexity. To this end we
developed an extensible graphical user interface (GUI) called Frbgui that
allows a user to input additional masks, change the subsampling of the data,
and isolate components of a burst, and used it to prepare bursts and obtain
measurements of the spectro-temporal features.

The spectro-temporal features of each burst are obtained via a 2D Gaussian fit
to the 2D autocorrelation of the burst waterfall. An autocorrelation of the
waterfall helps increase the S/N for measurement and limits the effect of
spectral structures, noise, and banding in the burst. The Gaussian model of
the autocorrelation therefore provides an analytical and robust way of
measuring the spectro-temporal features from a small number of parameters.
This technique is detailed in, for example, Appendix A of Chamma et al. (2021)
(see Jahns et al. 2022 for alternative formulations), and for this study, the
Gaussian fit is obtained using the normalized physical coordinates of the
autocorrelation. Thus, the dimensions of the Gaussian model’s parameters are
unitless, and we write with $x=t/(1\,\text{ms})$ and $y=\nu/(1\,\text{MHz})$

$\displaystyle G\left(x,y\right)$
$\displaystyle=C\exp\left\\{-\frac{1}{2}\left[(x-x_{0})^{2}\left(\frac{\cos^{2}\theta}{b^{2}}+\frac{\sin^{2}\theta}{a^{2}}\right)\right.\right.$
$\displaystyle\rule{0.0pt}{17.07164pt}+2(x-x_{0})(y-y_{0})\sin\theta\cos\theta\left(\frac{1}{b^{2}}-\frac{1}{a^{2}}\right)$
$\displaystyle\left.\rule{0.0pt}{17.07164pt}\left.+(y-y_{0})^{2}\left(\frac{\sin^{2}\theta}{b^{2}}+\frac{\cos^{2}\theta}{a^{2}}\right)\right]\right\\},$
(1)

where $C$, $x_{0}$, $y_{0}$, $a$, $b$, and $\theta$ are the model parameters
corresponding to the amplitude, central $x-$ and $y-$ positions, the standard
deviations of the Gaussian, and the orientation of the semi-major axis ($a$)
measured counterclockwise from the positive $y-$axis. The fit is found with
the scipy.optimize.curve_fit package using the normalized physical coordinates
defined above. The sub-burst slope and duration are obtained via equations
A2-A3 of Chamma et al. (2021), with the modification that the unit conversion
becomes unity due to the choice of coordinates666When using coordinates
numbered by channel fits did not always converge or were inaccurate. Using
normalized physical coordinates resolved this issue. when obtaining the
Gaussian fit, so that

$\displaystyle\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}$
$\displaystyle=-\left(1\,\frac{\text{MHz}}{\text{ms}}\right)\,\cot\theta,$ (2)
$\displaystyle t_{\text{w}}$
$\displaystyle=\left(1\,\text{ms}\right)\,\frac{ab}{\sqrt{b^{2}\sin^{2}\theta+a^{2}\cos^{2}\theta}},$
(3)

where $d\nu_{\text{obs}}/dt_{\text{D}}$ and $t_{\text{w}}$ are the sub-burst
slope and sub-burst duration777The duration defined in eq. (3) is the
correlation length of the burst, and can be converted to other definitions of
burst duration with a simple scaling. If the burst is Gaussian with a standard
deviation of $\sigma_{\text{p}}$ and FWHM
$t_{\text{FWHM}}=2\sqrt{2\ln{2}}\sigma_{\text{p}}$, then the correlation
length $t_{\text{w}}$ is related to those durations by
$t_{\text{w}}=\sqrt{2}\,\sigma_{\text{p}}$, and
$t_{\text{w}}=1/\sqrt{4\ln{2}}\,t_{\text{FWHM}}\simeq 0.6\,t_{\text{FWHM}}$.,
respectively, and $\nu_{\text{obs}}$ and $t_{\text{D}}$ are the observed
frequency and delay time (or arrival time) of the burst, written in the
formalism of Rajabi et al. (2020). We also compute the total bandwidth
$B_{\text{tot}}$ according to

$\displaystyle B_{\text{tot}}=(1\,\text{MHz})\,\sqrt{8\ln 2}\,a\cos\theta,$
(4)

which is the semi-major axis of the Gaussian ellipsoid scaled to its FWHM
value and projected on to the frequency axis. Figure 2 shows an example
measurement of the spectro-temporal properties of burst B006 from Aggarwal et
al. (2021). For bursts with multiple components in a single waterfall, the
autocorrelation changes in size to include the multiple components and
equations (2) to (4) are still valid. We can therefore perform the same
analysis to obtain the drift rate if a fit can be found to the larger
autocorrelation. The drift rates we obtain are treated distinctly from the
sub-burst slopes as these potentially arise from different phenomena.

Figure 2: Example measurement of burst B006 from Aggarwal et al. (2021). The
left panel shows the burst waterfall dedispersed to 559.5 pc/cm3 with the
dashed line, horizontal bar, and vertical bar indicating the sub-burst slope,
duration, and burst bandwidth obtained via the 2D Gaussian model of the
autocorrelation, shown on the right. On the right, the image shows the
computed autocorrelation, and the blue outlines are contours of the Gaussian
model at a quarter and 90% of the peak. The dashed green line shows the
corresponding slope.

Because the choice of DM affects the value of the sub-burst slope and, to a
lesser extent, the sub-burst duration, each burst is measured over a range of
trial DMs. We measure each burst over a grid of DMs spanning 555 to 575 pc/cm3
in steps of 0.5 pc/cm3, chosen to account for the historical range of DMs
observed from FRB 20121102A. Each burst is incoherently dedispersed from its
reported DM to the trial DM and measured via autocorrelation, resulting in 42
sets of measurements per burst including the reported burst DM.

The total set of measurements obtained over the DM grid are then filtered to
exclude sub-burst slopes that are positive or with measurement uncertainties
larger than 40%. We exclude all positive sub-burst slopes under the assumption
that they are unphysical and due to over dedispersion (Chamma et al., 2021;
Jahns et al., 2022). Most of the measurement uncertainties larger than 40% are
due to the maximal DM in the range being greater than the published DM of a
burst. Each burst therefore has a sub-burst slope that is nearly vertical at
some point over the DM range where uncertainties can be very large (see eq.
2).

For each dataset, the remaining measurements are then grouped by DM and a fit
is found between the sub-burst slope and duration with the form
$(d\nu_{\text{obs}}/dt_{\text{D}})/\nu_{\text{obs}}=A/t_{\text{w}}$, which is
the relationship predicted for these two properties by the TRDM. Of this set
of fits, we compute a reduced-$\chi^{2}$ to assess the goodness of the fit and
tabulate the remaining number of bursts that passed the filtering process for
each DM, by dataset. For each of the datasets listed in Section 2, the range
of DMs is then limited to only include those DMs that include all the bursts
in the sample, and, within this limited range, the DM with optimal (closest to
unity) reduced-$\chi^{2}$ is chosen as the representative DM for that sample.

The range of values in the remaining measurements in the limited DM range are
then used as an estimate of the uncertainties. For example, the bursts from
Michilli et al. (2018) are measured over the DM range 555–575 pc/cm3, and,
after filtering invalid measurements, the remaining DMs that include all of
the bursts range from 555–560 pc/cm3. Within this limited DM range, the DM
with optimal reduced-$\chi^{2}$ is found to be 558 pc/cm3, and the range of
measurements over the limited DM range are used as the uncertainties of the
measurement at 558 pc/cm3. Treating the measurements of each burst in this
manner helps account for the effect of the DM on the measurement and allows
for an estimate of the uncertainties beyond those from the Gaussian model. To
summarize, this measurement filtering process described in the preceding
paragraphs includes the exclusion of positive sub-burst slopes and slopes with
large uncertainties, the determination of the maximum valid and optimal DMs
using the sub-burst slope law, and the determination of the range of
measurement values over the valid DM range.

The measurements are reviewed visually with the help of the bars and slope
indicators such as those shown on both panels of Figure 2. Plots and tables of
all measurements can be found online at github.com/mef51/SurveyFRB20121102A.

##  4 Results

We describe the results of the autocorrelation analysis, the measurement
filtering process, and the relationships between the spectro-temporal
properties in this section. The sub-burst slope (normalized by the observing
frequency) vs. sub-burst duration is shown in Figure 3. Figures 4 to 6 show
different (potential) correlations between the measured spectro-temporal
properties, including the sub-burst duration and sub-burst slope with
frequency, and correlations between the bandwidth with frequency, duration and
sub-burst slope. Figure 5 examines the small sample of bursts with multiple
components and shows the drift rate vs. duration and drift rate vs. frequency.
In Figure 6 we find an unexpected correlation between the bandwidth and
duration (and thus sub-burst slope). Finally we look in particular at the
spectro-temporal features of bursts from Li et al. (2021) in Figure 7 and the
absence of correlations between bursts from different energy and wait-time
peaks.

As mentioned in the previous section, the measurements are filtered to exclude
invalid values and this process results in some bursts not having a valid
measurement at a particular DM, which results in a limited DM range after
requiring that all bursts are included. After this filtering, the ranges of
DMs for which bursts have valid measurement varied from dataset to dataset;
while all five datasets had valid measurements down to 555 pc/cm3, the highest
DMs at which all bursts in a dataset still had valid measurements differed.
These maximal DMs along with the DM in that range that had the optimal fit
between the sub-burst slope and duration are listed in Table 1. The average
maximal DM (weighted by the number of bursts in each dataset) was 561.5(4)
pc/cm3 and the average optimal DM was 560.1(8) pc/cm3. In the figures to
follow, burst measurements are always displayed at the optimal DM of the
dataset they come from with uncertainties estimated from the range of
measurements found within the limited DM range.

Dataset | Max valid DM | Optimal DM | # of bursts
---|---|---|---
| (pc/cm3) | (pc/cm3) |
Michilli et al. (2018) | 560.0 | 558.0 | 19
Gajjar et al. (2018) | 563.5 | 557.5 | 21
Oostrum et al. (2020) | 563.0 | 563.0 | 23
Aggarwal et al. (2021) | 560.5 | 559.5 | 63
Li et al. (2021) | 562.0 | 562.0 | 41
Weighted Average: | 561.5(4) | 560.1(8) | 167
Table 1: Results from the measurement exclusion and DM optimization process
described in Section 3. The maximum valid DM is the DM beyond which bursts
start being excluded for having invalid sub-burst slope measurements (ie.
being over-dedispersed). The optimal DM is the DM for which the
reduced-$\chi^{2}$ of the fit between the sub-burst slope and duration
measurements at that DM is closest to unity. The DM step size used was 0.5
pc/cm3.

###  4.1 Spectro-temporal Properties

The correlations between the spectro-temporal properties shown in Figures 3 to
6 confirm known relationships across a broad parameter range and reveal a
potentially new relationship between the bandwidth and sub-burst duration (or
normalized sub-burst slope). A summary of the fits found is shown in Table 2.
A discussion of these results will be found in Section 5.

Figure 3: The relationship between the sub-burst slope and sub-burst duration. The sub-burst slope is normalized by the observed sub-burst frequency in order for the different datasets to be plotted together with the same relationship. The black line is a fit of the form $A\,t^{-1}_{\text{w}}$ which is the relationship predicted in the triggered relativistic dynamical model of Rajabi et al. (2020), while the other are more general fits of the form $A\,t^{n}_{\text{w}}$ and $A\,t^{n}_{\text{w}}+B$. The shaded region shows the intersection of the range of fits found over the DM ranges of each of the five datasets. Each measurement is displayed at the optimal DM for its source dataset, listed in Table 1. Uncertainty bars show the range of measurements found over the limited DM range (see Section 3). We see decent agreement between the three fits with the exponent of the expected relation lying between those found for the general fits. # | Form | $A$ | $n$ | $C$ | Figure
---|---|---|---|---|---
1 | $\frac{1}{\nu_{\text{obs}}}\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|=A_{1}t_{\text{w}}^{-1}$ | $0.113\pm 0.003$ |  |  | 3
2 | $\frac{1}{\nu_{\text{obs}}}\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|=A_{2}t_{\text{w}}^{n_{2}}$ | $0.110\pm 0.003$ | $-0.81\pm 0.04$ |  | 3
3 | $\frac{1}{\nu_{\text{obs}}}\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|=A_{3}t_{\text{w}}^{n_{3}}+C_{3}$ | $0.046\pm 0.003$ | $-1.25\pm 0.07$ | $0.015\,\text{ms}^{-1}$ | 3
4 | $t_{\text{w}}=A_{4}\nu_{\text{obs}}^{-1}$ | $1474\pm 48\,\text{ms$\cdot$MHz}$ |  |  | 4
5 | $t_{\text{w}}=A_{5}\nu_{\text{obs}}^{n_{5}}$ | $(8.3\pm 0.5)\times 10^{5}$ | $-1.77\pm 0.07$ |  | 4
5b | $t_{\text{w}}=A_{\text{5b}}\nu_{\text{obs}}^{-2}$ | $(5.9\pm 0.3)\times 10^{6}\,\text{ms$\cdot$MHz}^{2}$ |  |  | 4
6 | $\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|=A_{6}\nu_{\text{obs}}^{2}$ | $(6.1\pm 0.3)\times 10^{-5}\,\text{ms}^{-1}\text{$\cdot$MHz}^{-1}$ |  |  | 4
7 | $\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|=A_{7}\nu_{\text{obs}}^{n_{7}}$ | $(1.4\pm 0.1)\times 10^{-5}$ | $2.18\pm 0.07$ |  | 4
8 | $\frac{1}{\nu_{\text{obs}}}\left|\frac{\Delta\nu_{\text{obs}}}{\Delta t_{\text{D}}}\right|=A_{8}t_{\text{w}}^{-1}$ | $0.145\pm 0.012$ |  |  | 5
9 | $\frac{1}{\nu_{\text{obs}}}\left|\frac{\Delta\nu_{\text{obs}}}{\Delta t_{\text{D}}}\right|=A_{9}t_{\text{w}}^{n_{9}}$ | $0.119\pm 0.012$ | $-0.71\pm 0.1$ |  | 5
10 | $\left|\frac{\Delta\nu_{\text{obs}}}{\Delta t_{\text{D}}}\right|=A_{10}\nu_{\text{obs}}^{2}$ | $(2.1\pm 0.2)\times 10^{-5}\,\text{ms}^{-1}\text{$\cdot$MHz}^{-1}$ |  |  | 5
11 | $\left|\frac{\Delta\nu_{\text{obs}}}{\Delta t_{\text{D}}}\right|=A_{11}\nu_{\text{obs}}^{n_{11}}$ | $(1.3\pm 0.8)\times 10^{-4}$ | $1.77\pm 0.08$ |  | 5
12 | $B_{\nu}=A_{12}\nu_{\text{obs}}$ | $0.14\pm 0.004$ |  |  | 6
13 | $B_{\nu}=A_{13}\nu_{\text{obs}}^{n_{13}}$ | $0.48\pm 0.16$ | $0.85\pm 0.04$ |  | 6
14 | $B_{\nu}=A_{14}t_{\text{w}}^{n_{14}}$ | $272\pm 12$ | $-0.53\pm 0.04$ |  | 6
15 | $B_{\nu}=A_{15}\left(\frac{1}{\nu_{\text{obs}}}\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|\right)^{n_{15}}$ | $1225\pm 93$ | $0.52\pm 0.05$ |  | 6
16 | $B_{\nu}=A_{16}t_{\text{w}}^{-1}$ | $162\pm 6\,\text{ms$\cdot$MHz}$ |  |  | 6
17 | $B_{\nu}=A_{17}\left(\frac{1}{\nu_{\text{obs}}}\left|\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}\right|\right)$ | $2570\pm 108\,\text{ms}$ |  |  | 6
Table 2: A summary of fits attempted with the measurement results. The columns
$A$, $n$, and $C$, respectively, show the amplitude of some kind of power-law,
the index of that power law, and any additive constants when applicable. The
variable $\nu_{\text{obs}}$ denotes the sub-burst frequency, $t_{\text{w}}$
denotes the sub-burst duration, and $(d\nu_{\text{obs}}/dt_{\text{D}})$ and
$(\Delta\nu_{\text{obs}}/\Delta t_{\text{D}})$ denote the sub-burst slope and
drift rate between multiple sub-bursts, respectively.

In Figure 3 we plot the sub-burst slope normalized by the observing frequency
versus the sub-burst duration. We find three fits to the data using the
scipy.odr package, which performs an orthogonal distance regression, a method
that takes into account the uncertainties on both variables. We use the range
of measurements over the DM range as the variable uncertainties when
performing the fits. The first fit is of the form
$(d\nu_{\text{obs}}/dt_{\text{D}})/\nu_{\text{obs}}=A_{1}\,t^{-1}_{\text{w}}$
which is the prediction of the TRDM and we find $A_{1}=0.113\pm 0.003$. We fit
two general power-laws, which are free of assumptions, of the form
$A_{2}\,t^{n_{2}}_{\text{w}}$ and $A_{3}\,t^{n_{3}}_{\text{w}}+C_{3}$ to see
how they compare with the predicted fit. These yield $A_{2}=0.110\pm
0.003,\,n_{2}=-0.81\pm 0.04\,$ and $A_{3}=0.046\pm 0.003,\,n_{3}=-1.25\pm
0.07,\,C_{3}=0.015\text{ms}^{-1}$. For each dataset, a range of fits of the
form $A\,t^{-1}_{\text{w}}$ is found over the DM range and the shaded region
shows the intersection of those ranges across the five datasets. That is, the
shaded region shows the range of fits possible within the limited DM ranges of
all the datasets and is an estimate of the uncertainty on the parameter
$A_{1}$ as a function of the DM. We see that a majority of points fall within
this region and are well described by the fits, with a small population of
outliers located above the rest of the points in the sub-bursts with durations
greater than 1 ms. We also see the measurement ranges systematically tending
to smaller sub-burst slopes with increasing duration.

In Figure 4, we plot the sub-burst duration and sub-burst slope against the
frequency. In those two panels, the black line represents a fit of the form
with the exponent fixed to the prediction of the TRDM and the tan line
represents a fit where the exponent is free to vary. In the sub-burst duration
vs. frequency plot, the black fit is of the form
$A_{4}\,\nu_{\text{obs}}^{-1}$, while the tan fit leaves the index free and is
of the form $A_{5}\,\nu_{\text{obs}}^{n_{5}}$. The free fit finds an exponent
of $n_{5}=-1.77\pm 0.07$, higher than the predicted $n=-1$, with
$A_{4}=1474\pm 48\,\text{ms $\cdot$ MHz}$ and $A_{5}=(8.3\pm 0.5)\times
10^{5}$. The yellow fit is of the form $A_{5\text{b}}\,\nu_{\text{obs}}^{-2}$,
and will be discussed in Section 5.2. In the sub-burst slope vs. frequency
plot we fit to the TRDM form of $A_{6}\,\nu_{\text{obs}}^{2}$ and again to a
free index form of $A_{7}\,\nu_{\text{obs}}^{n_{7}}$, finding $A_{6}=(6.1\pm
0.3)\times 10^{-5}\text{ ms${}^{-1}\cdot$ MHz${}^{-1}$}$, $A_{7}=(1.4\pm
0.1)\times 10^{-5}$ and an index of $n_{7}=2.18\pm 0.07$, slightly higher than
the predicted $n=2$ relationship. Both fits visually describe the data well,
and we see a large spread in the bursts in both plots.

Figure 4: Relationships between the sub-burst duration (top) and the sub-burst
slope (bottom) with burst frequency. Colors for all are shown in the top panel
and are the same as in Figure 3. In the top and bottom panels, the black line
shows a fit to the relationship predicted by the triggered relativistic
dynamical model in Rajabi et al. (2020), and the tan lines are general fits of
the form $A\nu_{\text{obs}}^{n}$. Figure 5: Relationships between the drift
rate of multiple components and the duration (top) and the frequency of
observation (bottom). On the left, fits of the form $A\,t^{-1}_{\text{w}}$ and
$A\,t^{n}_{\text{w}}$ are made for comparison with the similar inverse
relationship between the frequency-normalized sub-burst slope and duration
shown in Figure 3. On the right, we find two fits with one locked to a power
of 2 and the other allowed to change.

The drift rate of a burst with multiple components may be due to a different
phenomenon than that which results in the sub-burst slope, and so we separate
the drift rate measurements from the sub-burst slope measurements and show
their correlations in Figure 5. We find that their relationships are similar
to those found for the sub-burst slopes.

Of our sampled bursts only a handful showed multiple components. Most of these
were measurable via the autocorrelation method but a few were not, such as in
the cases where one of the bursts dominated the autocorrelation or when the
drift rate was very small. The drift rates are put through the same filtering
process as the sub-burst slopes and the top panel of Figure 5 shows the
normalized drift rate vs. the duration of the event, while the bottom panel
shows the drift rate vs. the frequency. As before, for the relationship with
the duration we find one fit with the predicted form
$A_{8}\,t_{\text{w}}^{-1}$ which yielded $A_{8}=0.145\pm 0.012$ and a fit with
a free exponent of the form $A_{9}\,t_{\text{w}}^{n_{9}}$ which yielded
$A_{9}=0.119\pm 0.012$ and $n_{9}=-0.71\pm 0.1$. This first fit has an $A$
value higher than that found for the sub-burst slopes (i.e. $A_{8}>A_{1}$) and
outside the range of fits shown in Figure 3. The second fit found an exponent
$n_{9}$ with a smaller magnitude than $n_{2}$, the corresponding index for the
sub-burst slopes. This suggests a relationship for the drift rates and
duration similar to that found for the sub-burst slopes but with possibly a
slightly different scaling.

For the relationship between the drift rates and the frequency (bottom panel
of Figure 5) we perform two fits, with the first fit of the TRDM form
$A_{10}\,\nu_{\text{obs}}^{2}$ yielding $A_{10}=(2.1\pm 0.2)\times
10^{-5}\text{ ms${}^{-1}\cdot$ MHz${}^{-1}$}$, which is about a third the
value found between the sub-burst slope and frequency ($A_{6}$, and shown in
the bottom panel of Figure 4). A fit of the form
$A_{11}\,\nu_{\text{obs}}^{n_{11}}$ found $A_{11}=(1.3\pm 0.8)\times 10^{-4}$
and $n_{11}=1.77\pm 0.08$, with quite a large uncertainty on $A_{11}$. For the
limited drift rates available, the fits to the drift rates’ relationships are
similar enough to those found for the sub-burst slopes that it is tempting to
conclude that the same phenomena is responsible for both.

Figure 6 shows plots of the sub-burst bandwidth against the frequency, sub-
burst duration, and normalized sub-burst slope, respectively. For the
bandwidth versus frequency plot we find a linear fit of the form
$A_{12}\,\nu_{\text{obs}}$ consistent with Doppler broadening of narrowband
emission and a free index fit of the form $A_{13}\,\nu_{\text{obs}}^{n_{13}}$.
This yields $A_{12}=0.14\pm 0.004$, $A_{13}=0.48\pm 0.16$ and exponent
$n_{13}=0.85\pm 0.04$. For the fits to bandwidth versus the sub-burst duration
and normalized slope (middle and right panels), we leave the exponent free and
use the forms $A_{14}\,t_{\text{w}}^{n_{14}}$ and
$A_{15}\,(\nu_{\text{obs}}^{-1}d\nu_{\text{obs}}/dt_{\text{D}})^{n_{15}}$. We
find $A_{14}=272\pm 12$ and $n_{14}=-0.53\pm 0.04$ for the duration and
$A_{15}=1225\pm 93$ and $n_{15}=0.52\pm 0.05$ for the normalized sub-burst
slope. Plotting against the sub-burst slope instead of the normalized sub-
burst slope has little effect on the exponent $n_{15}$. That the exponents
$n_{14}$ and $n_{15}$ are inverses of each other within the uncertainties is
expected from the inverse trend between the slope and duration shown in Figure
3, in agreement with the prediction of the TRDM.

Figure 6: Relationships between the bandwidth and the burst frequency (left),
sub-burst duration (middle) and sub-burst slope (right). The relationship
between the bandwidth and frequency is linear, while the relationship with the
duration goes, within uncertainties, to the power of $-$1/2. The origin of
this relationship is unclear, and may be due to observational bias as
indicated by the overlayed regions showing the bandwidth limits. The dashed
lines in the middle and right panels show fits of the expected relationship
implied by the left panel, and do not describe the data well. In the right
panel we see that the fit found between the bandwidth and slope is the inverse
of that for the duration, as might be expected based on their inverse
relationship shown in Figure 3.

That there is a relationship between the bandwidth and duration or slope of a
burst can be expected from the other correlations shown but, as will be
discussed in Section 5.2, the exact form of relationship seen here is not, and
is visible in these data because of the large range in frequency of the bursts
sampled. However, as shown by the bandwidth limits plotted as regions in the
figures, the maximum observable bandwidths for the Oostrum et al. (2020),
Aggarwal et al. (2021), and Li et al. (2021) datasets are 200 MHz, 580 MHz,
and 500 MHz, respectively, due to their instrumentation (with a few exceptions
for bursts that had slightly different backends), implying that sub-bursts
with high bandwidths and long durations could be missed. This however does not
explain the observed absence of low bandwidth, short duration bursts.

The nearest and simplest fractional value consistent with the uncertainties of
the exponent $n_{14}=-0.53\pm 0.04$ is $-$1/2 and so the two rightmost panels
of Figure 6, if observationally complete, suggest a proportionality of
$B_{\nu}\propto
t^{-1/2}_{\text{w}}\propto(\nu_{\text{obs}}^{-1}d\nu_{\text{obs}}/dt_{\text{D}})^{1/2}$,
where $B_{\nu}$ is the bandwidth. The colored lines in the two rightmost
panels show fits of the form $A_{16}\,t^{-1}_{\text{w}}$ and
$A_{17}\,(\nu_{\text{obs}}^{-1}d\nu_{\text{obs}}/dt_{\text{D}})$,
respectively, with $A_{16}=162\pm 6$ ms$\cdot$MHz and $A_{17}=2570\pm 108$ ms.
These last two fits are the expected relationships inferred from our results
and will be discussed in Section 5.2 where we also further explore this
unusual dependence.

The set of correlations described above provide evidence that the
relationships predicted by the TRDM between the sub-burst duration, slope and
frequency apply for a majority of bursts from this source and possibly
indicate an additional unknown relationship between the bandwidth and sub-
burst slope/duration.

For the sampled bursts from the bimodal peaks in energy and wait-time
distributions reported by Li et al. (2021), we searched for correlations
between our measured spectro-temporal properties and the burst energy and
wait-time, finding that they were unrelated to each other. Figure 7 shows
plots of the slope, sub-burst duration, bandwidth, and frequency as functions
of the wait-time (top row) and burst energy (bottom row). In the top row we
can see that the bursts cluster around wait-times of $10^{-2}$ and $10^{2}$
seconds, corresponding to the two sampled peaks, and that the spectro-temporal
measurements in all panels vary randomly and with similar ranges. In the
bottom row a large number of bursts cluster around $10^{39}$ erg while the low
energy peak at $<10^{38}$ erg only has two points, due to difficulties we
encountered in measuring these lower S/N bursts. Nonetheless, across the four
panels we again see that the measurements vary widely at the high energy peak
and cover similar values as the measurements from the low energy peak. For
this limited sample of data we therefore conclude that no clear relationship
exists between the wait-time and energy of a burst with any of its spectro-
temporal properties.

Figure 7: Search for correlations between measured spectro-temporal properties
of 28 bursts from Li et al. (2021) and their wait-times (top) and energies
(bottom), as reported by those authors. This figure does not include the 13
sub-bursts that were separated ourselves. The two peaks of the wait-time
distributions can be seen in the top row around 10-2 and 102 seconds. The two
peaks of the energy distribution are less obvious due to difficulties
measuring bursts from the low energy ($<10^{38}$ erg) peak and the small
sample limits the possible conclusions. No obvious dependence of a burst’s
spectro-temporal properties on either its wait-time or energy is seen in these
data.

##  5 Discussion

We discuss here our results in the context of other studies and the TRDM,
including the expected relationships for the drift rates and their connection
to those for the sub-burst slopes. We will also discuss the unexpected
relationship observed between the bandwidth, duration, and normalized sub-
burst slope, the possible implications of the relationships observed (and not
observed, in the case of the spectro-temporal properties with the burst energy
and wait-time), as well as the large spread in values seen in the measurements
of the spectro-temporal properties and what these results mean for
understanding FRBs overall.

Several studies have now closely examined the relationships between spectro-
temporal properties of FRBs and found strong correlations between them,
especially between the normalized sub-burst slope and sub-burst duration. The
first report of the inverse relationship between the slope and duration was in
Rajabi et al. (2020), seen with 25 bursts from FRB 20121102A. Chamma et al.
(2021) reported the same relationship for multiple sources seen with bursts
from FRB20180916B and FRB20180814A as well as FRB 20121102A, while also
reporting that the same value of the fit parameter $A$ characterized the
relationship for the three sources. In earlier work the data for FRB 20121102A
were limited to bursts with short durations and frequencies greater than 4
GHz, and since then large numbers of bursts were made available at longer
durations and lower frequencies, presenting the opportunity to understand the
relationships seen earlier with improved statistics.

In this study the frequency and duration ranges are broad and the number of
bursts studied from FRB 20121102A is large enough to conclude that a majority
of bursts from this source follow a tight relationship between the normalized
sub-burst slope and duration. The optimum fit parameter found here was
$A_{1}=0.113\pm 0.003$, which is higher than the $A=0.078\pm 0.006$ found by
Chamma et al. (2021), along with a narrower range of fits centered around
$A=0.09$. We note the range of fits includes this earlier value.

In Jahns et al. (2022), 849 bursts from FRB 20121102A were analysed on the
lower end of our frequency range (around 1400 MHz) and a strong inverse
correlation between the sub-burst duration and slope can be seen in their
Figure 5. Their formalism finds the inverse of the sub-burst slope relation
used here and their fit parameter of $|b|=0.00862(37)$ MHz-1 can be converted
to our formalism for comparison via $A=1/(\nu_{\text{obs}}b)$, which yields
$A_{\text{Jahns}}=0.0839$ when using $\nu_{\text{obs}}=1400$ MHz, the most
common burst frequency in that sample. This value is comparable to the results
found here, and differences can be due to the burst definition used and
details of the analysis (such as normalizing each point by the burst
frequency).

The TRDM predicts the relationship between the sub-burst duration and
frequency to be

$\displaystyle
t_{\text{w}}=\tau^{\prime}_{\text{w}}\frac{\nu_{0}}{\nu_{\text{obs}}},$ (5)

where $t_{\text{w}}$ is the sub-burst duration, $\tau^{\prime}_{\text{w}}$ is
the sub-burst duration in the rest frame of emission, $\nu_{0}$ is the rest
frame frequency, and $\nu_{\text{obs}}$ is the frequency in the observer
frame. The values of $\tau^{\prime}_{\text{w}}$ and $\nu_{0}$ can be absorbed
into a constant and so an inverse relationship is expected. Other evidence of
this relationship can be found in Figure 7 of Gajjar et al. (2018) and Figure
3 of Hessels et al. (2019). Each ‘clump’ of bursts seen in Figure 4 viewed
independently appears uncorrelated with the observing frequency, but placed
together a downward trend becomes clear.

The relationship between sub-burst slope and frequency is similar in this
regard as well, since over a narrow spectral range the two observables can
appear uncorrelated. Wang et al. (2022) in their Figure 8 using data from
multiple studies and multiple FRB sources and spanning a large frequency range
found that trends proportional to $\nu^{2}$ and $\nu^{2.29}$ fit the data
well. A quadratic relationship between sub-burst slope and frequency was
previously implied by the results of Chamma et al. (2021) in their Figure 5.
Our results here again show that a quadratic relationship fits well, with the
free exponent fit finding a value of $\nu^{2.18\pm 0.07}$.

The relationship between the sub-burst slope and frequency was derived in
Chamma et al. (2021) from the TRDM using equation (5) and the relationship
between the sub-burst slope and duration, which is given by

$\displaystyle\frac{1}{\nu_{\text{obs}}}\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}=-\left(\frac{\tau^{\prime}_{\text{w}}}{\tau^{\prime}_{\text{D}}}\right)\frac{1}{t_{\text{w}}}=-\frac{A}{t_{\text{w}}},$
(6)

where as before $d\nu_{\text{obs}}/dt_{\text{D}}$ is the sub-burst slope,
$t_{\text{D}}$ is the delay time (or arrival time), and
$\tau^{\prime}_{\text{D}}$ is the delay time in the rest frame of the FRB. The
fit parameter $A$ is seen here as the ratio between the rest frame sub-burst
duration and delay time, and this relationship is the one predicted for the
sub-burst slope and duration, shown with the black fit in Figure 3. Combining
equations (5) and (6) we obtain

$\displaystyle\frac{d\nu_{\text{obs}}}{dt_{\text{D}}}=-\frac{A\nu_{\text{obs}}^{2}}{\tau^{\prime}_{\text{w}}\nu_{0}}\equiv-
A_{6}\,\nu_{\text{obs}}^{2},$ (7)

where the new constant
$A_{6}=A/(\tau^{\prime}_{\text{w}}\nu_{0})=1/(\tau^{\prime}_{\text{D}}\nu_{0})$.
This is a quadratic relationship between the sub-burst slope and frequency and
is consistent with our results as well as the results of Chamma et al. (2021)
and Wang et al. (2022).

The TRDM assumes a fundamentally narrow-band emission process at a rest frame
frequency $\nu_{0}$ and if the rest frame sub-burst duration
$\tau^{\prime}_{\text{w}}$ can be estimated with a more detailed model of the
emission process, then the ratio of $A$ and $A_{6}$ would provide a measure of
$\nu_{0}$, through

$\displaystyle\frac{A}{A_{6}}=\nu_{0}\tau^{\prime}_{\text{w}}.$ (8)

For example using our results of $A_{1}=0.113\pm 0.003$ and $A_{6}=(6.1\pm
0.3)\times 10^{-5}\text{ ms${}^{-1}\cdot$ MHz${}^{-1}$}$ and with an order of
magnitude estimate of $\tau^{\prime}_{\text{w}}\approx 1\,\text{ms}$, we can
very roughly estimate $\nu_{0}$ to be on the order 1 GHz.

The TRDM offers a simple explanation for the relationships discussed here
through its assumption of a narrow-band emission process for FRBs at a common
rest frame frequency $\nu_{0}$.

###  5.1 Drift rate vs duration and frequency

As discussed in Section 4, the drift rate relationships with duration and
frequency are plotted in Figure 5 and these relationships appear to be
analogous to those for the sub-burst slope. For the limited data available,
the drift rate relationship with frequency seems to be adequately described by
the fit proportional to $\nu_{\text{obs}}^{2}$, as the free fit is
proportional to $\nu_{\text{obs}}^{1.77}$, which is closer to quadratic than
not. We can derive a relationship between the drift rate and the frequency
using equation 8 of Rajabi et al. (2020), which describes the change in
frequency with arrival time of distinct sub-bursts. Expanding the
$d\nu_{\text{obs}}/d\tau^{\prime}_{\text{D}}$ term in that equation we get

$\displaystyle\frac{\Delta\nu_{\text{obs}}}{\Delta
t_{\text{D}}}=\frac{\nu_{\text{obs}}}{\nu_{0}}\frac{d\nu_{\text{obs}}}{d\tau^{\prime}_{\text{D}}}=\frac{\nu_{\text{obs}}}{\nu_{0}}\frac{d\nu_{\text{obs}}}{d\beta}\frac{d\beta}{d\tau^{\prime}_{\text{D}}},$
(9)

where $\Delta\nu_{\text{obs}}/\Delta t_{\text{D}}$ is the drift rate and
$\beta$ is the fraction of the speed of light the FRB source is moving with
along the line of sight. By using the relativistic Doppler shift formula
$\nu_{\text{obs}}=\nu_{0}\sqrt{(1+\beta)/(1-\beta)}$, we can evaluate
$d\nu_{\text{obs}}/d\beta$ in the above expression to obtain

$\displaystyle\frac{\Delta\nu_{\text{obs}}}{\Delta
t_{\text{D}}}=\frac{\nu_{\text{obs}}^{2}}{\nu_{0}}\gamma^{2}\frac{d\beta}{d\tau^{\prime}_{\text{D}}},$
(10)

where $\gamma=1/\sqrt{1-\beta^{2}}$. Equation (10) shows a
$\nu_{\text{obs}}^{2}$ dependence with the frequency with
$\gamma^{2}d\beta/d\tau^{\prime}_{\text{D}}$ a parameter characterizing the
region under study. More data are required to verify what relationship exists
between the drift rate and frequency, and the result of such an analysis can
be interpreted within the model based on the relations described above. The
data also seem to suggest an inverse relationship analogous to the
relationship between the sub-burst slope and sub-burst duration. As mentioned
earlier, this can be expected in the TRDM when groups of sub-bursts are
emitted at roughly the same time (Sec. 3.1 of Chamma et al., 2021; Rajabi et
al., 2020).

###  5.2 Bandwidth vs. Sub-burst Duration relationship

The correlations of spectro-temporal properties with the bandwidth shown in
Figure 6 reveal known as well as unknown relationships. One known spectro-
temporal correlation seen in FRBs is the scaling of bandwidth with frequency.
For example, in Houde et al. (2019), the linear relationship shown in the left
panel of Figure 6 between bandwidth and frequency is seen in their Figure 5,
and includes points between 2 and 4 GHz. Their slope, once converted for
comparison is $0.156$, close to the value $A_{12}=0.14\pm 0.004$ obtained
here. This linear relationship between the bandwidth and frequency follows
from the non- or weakly-relativistic Doppler effect (Houde et al., 2019).

The relationships found in the middle and right panels of Figure 6 where
$B_{\nu}\propto t^{-1/2}_{\text{w}}$ and
$B_{\nu}\propto(d\nu_{\text{obs}}/dt_{\text{D}})^{1/2}$ are inconsistent with
the TRDM, difficult to verify, and statistically significant enough to be
considered real. Based on our results as well as the predictions of the TRDM,
the expected dependence is $B_{\nu}\propto t^{-1}_{\text{w}}$ consistent with
the dependencies $t_{\text{w}}\propto\nu_{\text{obs}}^{-1}$ (top panel of
Figure 4) and $B_{\nu}\propto\nu_{\text{obs}}$ (left panel of Figure 6).
Accordingly, the expected dependence between $B_{\nu}$ and
$d\nu_{\text{obs}}/dt_{\text{D}}$ is linear. However, fits of the form
$B_{\nu}\propto t^{-1}_{\text{w}}$ and $B_{\nu}\propto
d\nu_{\text{obs}}/dt_{\text{D}}$, shown by the dashed lines in the middle and
right panels of Figure 6, fail to simultaneously cross both the low frequency
and high frequency bursts as well as the $B_{\nu}\propto t^{-1/2}_{\text{w}}$
and $B_{\nu}\propto(d\nu_{\text{obs}}/dt_{\text{D}})^{1/2}$ fits do.

In an effort to explain this relationship outside of the TRDM, we can take
advantage of the fact that the bandwidth appears to be linear with frequency
and perform a test fit on the sub-burst duration and frequency that assumes
the form $t_{\text{w}}\propto\nu_{\text{obs}}^{-2}$ since this would imply a
bandwidth-duration relationship of the form $B_{\nu}\propto
t^{-1/2}_{\text{w}}$ consistent with our observations. Indeed, the
$\nu_{\text{obs}}^{-2}$ fit (shown in the first panel of Figure 4) is adequate
enough of a fit that we cannot be entirely certain whether
$t_{\text{w}}\propto\nu^{-1}_{\text{obs}}$ or
$t_{\text{w}}\propto\nu^{-2}_{\text{obs}}$. However, we note that the
relationship $t_{\text{w}}\propto\nu^{-1}_{\text{obs}}$ is at the basis for
the sub-burst slope law, i.e.,
$(d\nu_{\text{obs}}/dt_{\text{D}})/\nu_{\text{obs}}\propto t^{-1}_{\text{w}}$,
through a simple derivative, and that this relation appears to be securely
confirmed here and elsewhere (Rajabi et al., 2020; Chamma et al., 2021; Jahns
et al., 2022). On the other hand, we note that a combination of the sub-burst
slope law with $t_{\text{w}}\propto\nu^{-2}_{\text{obs}}$ leads to a
relationship $d\nu_{\text{obs}}/dt_{\text{D}}\propto\nu^{3}_{\text{obs}}$,
that is not quite compatible with our observed results pointing to a quadratic
relationship. If we are inclined to favour a
$t_{\text{w}}\propto\nu^{-1}_{\text{obs}}$ relation then this can naturally
explain the sub-burst slope law. In that case, the $B_{\nu}\propto
t^{-1/2}_{\text{w}}$ functionality appears at odds with the TRDM. If instead
we favour the $t_{\text{w}}\propto\nu^{-2}_{\text{obs}}$ relation, then we
lack an explanation for the sub-burst slope law and we are in tension with the
observed evidence for a quadratic relationship between the slope and
frequency.

There is no known relationship within the TRDM that explains the
$B_{\nu}\propto t^{-1/2}_{\text{w}}$ relationship between the bandwidth and
the sub-burst duration, or the
$B_{\nu}\propto\left[(d\nu_{\text{obs}}/dt_{\text{D}})/\nu_{\text{obs}}\right]^{1/2}$
relationship between the bandwidth and sub-burst slope. If the velocity
distribution of the FRB source, which determines the bandwidth in the TRDM, is
an independent variable that helps in determining the sub-burst duration and
sub-burst slope, then it is not clear what physical mechanism enables the
velocity of one sample of material to determine the duration of the pulse
emitted by material moving at a different velocity.

The absence of a readily available explanation for the bandwidth-
duration/slope relationship from our results suggests that the explanation may
be related to the physics of the emission process. However, a resolution to
this inconsistency could involve changing or discarding the TRDM, ruling out
the observed bandwidth-duration relationship, or some mixture of both.

###  5.3 Spectro-temporal properties and their relationship to the emission
process

Across the different relationships explored and the broad sample of bursts
considered from FRB 20121102A, there does not seem to be a large population of
outliers that breaks sharply from the relationships described. This fact, in
light of the absence of correlations between the spectro-temporal properties
and the energies and wait-times, suggests that some of the spectro-temporal
properties of an FRB and its emission mechanism could be independent of one
another. If we were to assume the predictions of the TRDM are consistently
true, then a simple interpretation for the absence of outliers is that the
correlations between properties like the sub-burst slope, duration and
frequency of FRBs (at least from this source) originate from the relativistic
dynamics of the source as explained by the TRDM. Under this assumption, the
correlation between the bandwidth and duration, unexplained in the TRDM as
discussed above, falls into a different category. Of course, another
interpretation is that the TRDM is incomplete.

However, the results presented here are evidence that the FRB emission from
this source comes from material that is moving relativistically, since several
spectro-temporal correlations of FRB 20121102A are explainable through the
dynamics of that material and the TRDM, and not due to the actual emission
process.

This view is supported by the results shown in Figure 7, where several
spectro-temporal properties appear uncorrelated with both the wait-time and
energy of a burst. Further supporting examples are the plots with energy
displayed in Figure 5 of Jahns et al. (2022) for bursts around 1400 MHz, the
relationship between the frequency and flux density shown in Figure 4 of
Gajjar et al. (2018) for bursts around 7 GHz, and plots of different
properties with the fluence shown in Figure 5 of Aggarwal et al. (2021) for
bursts around 1400 MHz, all of which imply no clear relationship between the
spectro-temporal properties and burst energies.

These aforementioned studies are done independently over a narrow range of
frequencies and durations. An analysis with a broad sample such as the one
used for this study that focuses on the energy and/or fluxes of bursts would
be needed to confirm the absence of correlation between burst energies and
spectro-temporal properties such as the sub-burst slope or duration, as the
present results and literature suggest. If so, and assuming the validity of
the TRDM, then such an absence would be evidence that what we observe from
FRBs can be broken into two regimes; properties which are determined from the
motion of the source (TRDM), and properties which are determined from the so
far unknown emission process.

We also note there is considerable statistical spread in the values of sub-
burst slope, duration and bandwidth seen in Figures 3 to 6 from each dataset
that could be related to the emission process. For example, the data for
Michilli et al. (2018) cluster around 4.5 GHz, but values for the sub-burst
slope range from $5\times 10^{2}$ MHz/ms to $4\times 10^{3}$ MHz/ms, and the
sub-burst duration spans nearly an order of magnitude. Similar statements are
true for all the other datasets considered. In the TRDM the sub-burst duration
and sub-burst slope (see equations 5 and 6) are related to the rest frame
frequency $\nu_{0}$, and the rest frame duration and delay times
$\tau^{\prime}_{\text{w}}$ and $\tau^{\prime}_{\text{D}}$. The dependence of
the sub-burst duration and slope on these properties, which in the TRDM are
determined exclusively by the emission process and FRB environment, is
possibly the cause of the scatter seen in the spectro-temporal properties.
Characterizing the scatter of spectro-temporal properties for large samples of
bursts may therefore be an additional avenue for studying the emission
process.

##  6 Conclusions

We have measured the spectro-temporal properties and the relationships between
them of a broad sample of 167 bursts from FRB 20121102A with frequencies
ranging from 1–7.5 GHz and sub-burst durations ranging from less than 1 ms to
about 10 ms from the observational studies of Michilli et al. (2018), Gajjar
et al. (2018), Oostrum et al. (2020), Aggarwal et al. (2021), and Li et al.
(2021). Bursts from the latter study were sampled from the two peaks of the
bimodal energy and bimodal wait-time distributions reported therein in order
to search for relationships between those parameters and the spectro-temporal
properties of the bursts. Our measurements of the spectro-temporal properties
include measurements of the sub-burst slope, sub-burst duration, burst
bandwidth, center frequency, and sad trombone drift rate (when applicable) at
each DM between 555 to 575 pc/cm3 in steps of 0.5 pc/cm3. The complete set of
measurements are available online along with an extensible graphical user
interface called Frbgui that was developed and used to prepare and perform
measurements on the burst waterfalls.

We characterized multiple relationships between the measured spectro-temporal
properties of bursts from FRB 20121102A finding general agreement with
multiple predictions from the TRDM described in Rajabi et al. (2020), which
fundamentally assumes a narrow-band emission process and up-to relativistic
motion. We found, as in Rajabi et al. (2020), Chamma et al. (2021) and Jahns
et al. (2022), an inverse relationship between the sub-burst slope and sub-
burst duration. In addition, we find an inverse relation between the sub-burst
duration and frequency, a quadratic relation between the sub-burst slope and
frequency, and a linear relationship between the sub-burst bandwidth and
frequency.

The 12 drift rates measured were seen to follow relationships with the
duration and frequency that are analogous to the ones obeyed by the sub-burst
slope. However, more drift rates are needed to properly characterize these
trends.

We also found an unexpected correlation with power-law index consistent with
$-$1/2 between the sub-burst bandwidth and sub-burst duration $(B_{\nu}\propto
t_{\text{w}}^{-1/2})$. Consistent with the inverse relationship between the
sub-burst slope and duration, we also found a relationship between the
bandwidth and sub-burst slope with power-law index 1/2 (i.e.
$B_{\nu}\propto\left[(d\nu_{\text{obs}}/dt_{\text{D}})/\nu_{\text{obs}}\right]^{1/2}$).
This relationship is hard to explain within the TRDM and is potentially
uncertain due to the limited observing bands at lower frequencies, however
this would not explain the absence of low bandwidth, short duration sub-bursts
observed at high frequencies.

We found no correlations between the spectro-temporal properties of a burst
and its energy or wait-time.

Except for the bandwidth relationship with the normalized sub-burst slope and
duration, the multiple relationships observed between the spectro-temporal
properties generally agree with the predictions made in Rajabi et al. (2020).
Therefore, more data are needed from FRB 20121102A between 2–4 GHz and high
bandwidth bursts should be searched for at low frequencies (below 2 GHz), as
well as low bandwidth bursts at high frequencies, in order to test the limits
of the bandwidth-duration/slope relationship found. Such observations are also
needed to pin down the relationship between the sub-burst duration and
frequency, and sub-burst slope and frequency.

The general agreement of the spectro-temporal properties with the predictions
of the TRDM from this broad sample of bursts from FRB 20121102A suggests that
several of the spectro-temporal relationships of all bursts from FRB 20121102A
can be explained by dynamical motions of the FRB source, though there still
remain challenges to validating all the predictions of the TRDM. No
significant population of outliers was found with our burst sample and the
correlations studied were sufficiently characterized by a single relationship.
Meanwhile, the scatter of values observed in the spectro-temporal properties
could arise from parameters in the model that are determined exclusively by
the emission mechanism such as the rest frame frequency of emission and the
rest frame timescales. Therefore, characterizing the scatter of spectro-
temporal properties from large samples of bursts may also inform on the
emission mechanism of FRBs.

## Acknowledgements

We are grateful to Di Li for arranging access to and support with the FAST
data. The authors are grateful to Christopher Wyenberg for feedback and
discussions on drafts of this manuscript and to the referee for their careful
reading and suggestions on improving clarity. M.H.’s research is funded
through the Natural Sciences and Engineering Research Council of Canada
Discovery Grant RGPIN-2016-04460. Plots in the paper were prepared using the
Matplotlib package (Hunter, 2007).

## Data Availability

All spectro-temporal measurements are available online at
github.com/mef51/SurveyFRB20121102A. The waterfalls of the FRBs can be
obtained from the authors of their respective publications. The graphical user
interface Frbgui and related scripts used to perform the measurements are
available at github.com/mef51/frbgui.

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