# Statistical analysis of long GRBs’ prompt emission and X-ray flares:
multivariate clustering and correlations

Joseph Saji Indian Institute of Science Education and Research,
Thiruvananthapuram, Kerala, India, 695551  Shabnam Iyyani Indian Institute of
Science Education and Research, Thiruvananthapuram, Kerala, India, 695551
Kratika Mazde  Indian Institute of Science Education and Research,
Thiruvananthapuram, Kerala, India, 695551

###### Abstract

The extensive observations done by the X-ray telescope onboard Neil Gehrels
Swift observatory has revealed the presence of late time flares concurrent
with the decaying afterglow emission. However, the origin of these flares are
elusive. In this work, we made use of the large database of Swift observations
(2005 - 2020) of long GRBs to conduct a systematic statistical study between
the prompt gamma ray emission and X-ray flares by characterising their
temporal and spectral properties in terms of duration, quiescent period, peak
flux, fluence, minimum variability timescale and spectral power-law index. The
multi-dimensional database of parameters, thereby, generated was investigated
by the principal component analysis which revealed there is no evident
correlation between the different parameters of the prompt emission and X-ray
flares. Furthermore, the correlation studies reveal that while there is a
trend of positive correlation between the minimum variability timescale of
flare and its duration, and a strong negative correlation with its peak flux,
there are no such correlations observed in the prompt emission. Similarly, we
find a positive correlation between the quiescent period and flare duration,
and a negative correlation with the flare peak flux, while no such
correlations are observed for the prompt emission of GRBs. Finally, among the
X-ray flares, we find two dominant classes whose variations are driven by the
minimum variability timescale, peak flux and fluences of the flares. A catalog
of these different parameters characterising the prompt and flare emissions is
presented.

Gamma-ray bursts(629) — X-ray flares — Multivariate analysis(1913) —
Astrostatistics techniques(1886) — Catalogs(205)

##  1 Introduction

Gamma ray bursts (GRBs) are characterised by two main events: firstly, the
prompt emission which composes of the immediate gamma rays and secondly, the
afterglow emission which composes of emission ranging from radio till gamma
rays (Mészáros, 2006; Kumar & Zhang, 2015; Iyyani, 2018). The quick autonomous
re-pointing capability of Neil Gehrels Swift (Burrows et al., 2005a)
spacecraft to the location of the burst, in particular, has enhanced the
observations linking the prompt and X-ray afterglow emissions of GRBs. This is
made possible by the onboard detectors, Burst Alert Telescope (BAT; Barthelmy
et al. 2005 ) and X-ray Telescope (XRT; Burrows et al. 2005b) which make
observations in the energy ranges $15-150\,\rm keV$ and $0.3-10\,\rm keV$,
respectively.

One of the intriguing features of the gamma-ray bursts revealed by Neil
Gehrels Swift observations is the presence of flares in the X-ray afterglow
lightcurves. Systematic survey studies regarding the X-ray flares observed in
Swift XRT were initially carried out by Chincarini et al. 2007a and Falcone et
al. 2007. X-ray flares are found to occur in both long and short GRBs. This
makes the flares a common feature in GRBs despite the difference in
progenitors. Majority of the flares happen before a few $1000$ s (Chincarini
et al., 2007a), however, in a few of them, they are found to occur as late as
$\geq 10^{6}$ s (Swenson et al., 2010; Kumar et al., 2022).

Spectral studies such as Morris 2008 using wider energy range combining data
from BAT, XRT and UVOT111UltraViolet Optical Telescope (UVOT) onboard Neil
Gehrels Swift. simultaneously available in a few cases have shown that the
flare spectrum requires spectral models more complex than a simple power-law
model of the afterglow radiation. This affirmed that the flares are distinctly
different from the observed underlying afterglow emission. Several studies
have been done to analyse various spectral and temporal properties of these
flares observed in X-ray and optical in comparison with the prompt emission
properties (Sonbas et al., 2013a; Peng et al., 2015; Yi et al., 2016, 2017;
Liu & Mao, 2019; Chang et al., 2021; Lü et al., 2022; Shi et al., 2022). The
spectral studies have noted that the flares possess energy fluence comparable
to that of the prompt emission (Burrows et al., 2005c; Falcone et al., 2007).
In addition, multiple flaring episodes are also found in a certain fraction of
GRBs (Chincarini et al., 2007a). As the flare durations tend to increases with
time, the intensity of the flares are found to decrease with time.

Despite several studies, a self-consistent model explaining the origin,
energetics and the evolution of the X-ray flares is yet to be developed. The
various proposed models consider the following scenarios: (a) the late time
flares are due to delayed central engine activity (MacFadyen et al., 2001;
King et al., 2005; Margutti et al., 2011; Dall’Osso et al., 2017; Gibson et
al., 2018); or if it is an extended phase of the prompt emission episode
(Beniamini & Kumar, 2016; Mu et al., 2016); or if it is the late time energy
injection into the blast wave by refreshed shocks (Rees & Mészáros, 1998;
Laskar et al., 2015); or if it is the externals shocks propagating into a
dense and uneven circumburst medium (Hascoët et al., 2017; Ayache et al.,
2020).

In this work, we present a detailed multivariate statistical study of the
temporal and spectral properties of the prompt and X-ray flare behaviours.
Furthermore, a catalog of the studied properties are presented. Firstly, a
reasonably large dataset based on the Swift observations is selected.
Following which the parameters chosen to characterise the prompt and X-ray
flare emissions are defined, and the methodology used to extract those
features are described in the section 2. In section 3, the multivariate
analysis including the distributions of the various parameters, principal
component analysis, detailed correlation analysis cum modelling and clustering
of X-ray flares are presented. Following, in section 4, we discuss and
summarise the main results of the analysis and finally, present our
conclusions.

##  2 Sample selection and Methodology

The primary objective of the statistical analysis of different parameters of
the prompt emission and X-ray flares of the GRBs requires a reasonably large
dataset. The Swift observatory has been discovering approximately $100$ bursts
per year since 2004 and thereby, the large database including the prompt
emission and late time X-ray flares observed by BAT and XRT respectively,
provides the appropriate database for conducting the statistical studies. The
afterglow lightcurves of all the long bursts (lGRBs)222$T_{90}$ represents the
time duration within which $90\%$ of the burst emission counts are detected by
BAT. Long GRBs are those with $T_{90}\geq 2\,s$. detected by Swift XRT until
May 2020 were manually examined to look for the presence of flares, which were
also confirmed by the flare detection algorithm employed by Swift XRT team
(Evans et al., 2009). After the primary inspection, $340$ lGRBs (about $30\%$
of the total Swift lGRBs observed during this period) were found to posses
flaring activity during which the flux evolved in a manner differently from
the otherwise continuously decaying afterglow behaviour. Among such identified
flares, we further sorted and removed certain GRBs with the following light
curve behaviours:

  * •

incomplete flaring episodes such as either the rising or decaying phases of
the flare light curve is completely absent,

  * •

not enough observational data points available in the flare light curve such
that it is insufficient for light curve characterisation,

  * •

the immediate background emission in the regions pre or post the flare, that
is the underlying afterglow emission, is absent or not enough observational
data is available,

  * •

small peaks in the lightcurve that are of very low signal-to-noise
ratio333Refer section 2.1 for definition. ($\ll 0.1$) which are consistent
with the statistical variation in the afterglow observations were strictly
avoided444Note: A very small number of bursts (less than 10) with $0.1<$ SN
ratio $<1$ are included in the sample. In these cases, the parameters were
measurable and we have included them in the database, so as to minimise
selection biases..

The above sorting resulted in the removal of about $35\%$ of the initial
sample and the final sample consisted of $220$ lGRBs. Examples of the
afterglow lightcurves of a few excluded GRBs using the above criteria, and
selected GRBs with single, double and triple flaring episodes are shown in
Appendix A.

###  2.1 Parameters of study

A schematic representation of the prompt emission and afterglow ligthcurves
consisting of late time X-ray flares is shown in the Figure 1a. The following
parameters characterising the prompt and X-ray flare emissions of lGRBs are
chosen for the study and reported in the catalog:

  * •

TP,100 $\&$ TF,100 : The total duration of the gamma-ray burst prompt emission
and X-ray flare emission respectively.

  * •

FluenceP $\&$ FluenceF: The total energy per cm2 obtained after integrating
the energy flux over the duration of the burst and flare respectively.

  * •

Peak Flux, FP,peak $\&$ FF,peak : The maximum energy flux that is recorded
during the burst duration of prompt emission and X-ray flare (subtracting the
underlying afterglow flux) respectively.

  * •

Peak count time, T${}_{P,peak}\,\&\,$ TF,peak: It marks the time when maximum
count rate is observed in the prompt emission and X-ray flare light curves
respectively.

  * •

$\alpha_{P}$ $\&$ $\alpha_{F}$: The low energy spectral index of the power-law
spectra of the prompt emission and X-ray flare, respectively.

  * •

Minimum variability timescale, TP,var $\&$ TF,var: It describes the shortest
time scale over which a significant change in the count rate of prompt
emission and X-ray flare happens, respectively.

  * •

Signal-to-Noise ratio of flare: Considering Poisson statistics, the signal-to-
noise ratio of the flare is estimated as the ratio of the flare peak count
rate (given as the total count rate - afterglow count rate at the peak time)
to the square root of the total count rate at the peak time. This measure
helps to check how distinct and statistically significant the flare is from
the underlying afterglow emission. In addition, the distribution of
statistical significance allows to verify if the sample is biased or not.

  * •

Quiescent period, Tq: It is the difference between the start time of the
TF,100 of the X-ray flare and the end time of the TP,100 of the prompt
emission. During this period, there are no source photon counts other than
background noises in BAT and the underlying afterglow emission in the XRT.

Figure 1: (a) A schematic representation of the GRB prompt emission and
afterglow lightcurves including the late time X-ray flares. The temporal
parameters, TP,100,TF,100 and quiescent period [Tq] are marked. (b) Residue
plot of GRB170705A showing polynomial fit in the rise and decay phases of
X-ray flare is shown. The points at which the fit coincide with residue = 0
line are marked in blue squares. The afterglow(black) and its corresponding
model(blue solid line) of the same GRB is shown in the embedded plot.

###  2.2 Methodology of Extracting the Parameters

In the following subsections, the methodology of how the above defined
parameters are systematically extracted from the data is presented.

####  2.2.1  Prompt Emission Data Analysis

The prompt BAT light curves were analysed using the Heasarc script of Bayesian
Block binning named as battblocks with ncp$\\_$prior = 6 and uniform time
binning of $0.01$s to estimate the total duration, $T_{P,100}$, and the peak
time, $T_{P,peak}$ of the prompt emission of the burst. The time integrated
and peak count BAT spectra of the burst were created for the duration of
$T_{P,100}$ and for 1 second duration around $T_{P,peak}$ of the prompt
emission in the energy range $15-150$ keV, respectively. The spectral data
were then analysed using the Multi-Mission Maximum Likelihood Framework (3ML,
V.2.2.1, Vianello et al. 2015) software package. Using 3ML, the spectra were
analysed using different phenomenological spectral functions such as simple
power-law, cutoff power-law and Band function (Poolakkil et al., 2021). By
comparing the Akaike Information Criterion (AIC)555AIC is the statistic
estimator which is calculated using the formula: $2k-2ln(\hat{L})$ where $k$
is number of estimated parameters and $\hat{L}$ is the maximum likelihood
function., the spectral function fit with the minimum AIC was considered as
the best fit spectral model. From the best fit model, the low energy power-law
index, $\alpha$ of the time integrated spectrum and the total energy fluence
were noted. The peak energy flux was estimated from the analysis of the peak
spectrum generated around the peak time of the prompt emission.

####  2.2.2  X-ray Flare Data Analysis

The pre-processed XRT count rate light curves for the afterglow analysis were
obtained from Swift XRT repository666Data products are available in the link:
https://www.swift.ac.uk/xrt$\\_$live$\\_$cat/ in the energy range, $0.3-10$
keV. The binned GRB XRT afterglow light curves are ensured to contain at least
15 counts and the errors are calculated using the Gaussian statistics (Evans
et al., 2009). The temporal structure of the afterglow emission consists of
several power-laws of decaying count rate which is entirely different from the
highly varying erratic pulses of emission in the flares (Figure 1a).
Therefore, the first step in processing the data was to separate the two
emissions. This was achieved by removing the rough data region that included
the flares from the entire afterglow light curve and modelling the remaining
smoothly decaying afterglow emission using a power-law or a combination of
power-laws (Evans et al., 2007, 2009). The best-fit models were determined
using the 3ML and AIC statistics. A multi-broken power-law model was found to
best represent the underlying afterglow emission. The resulting best-fit model
of the afterglow light curve was subtracted from the total afterglow emission,
which produced a residual light curve including the propagated errors (Figure
1b). The peak count time of the flare was used to identify the rise and decay
phase of the flare. Using the technique of fitting a polynomial function to
the residual counts light curve, the respective last and first intersection
points with the background curve in the rise and decay phases are selected as
the start and the stop times of the flare episode respectively. The best fit
polynomial for the rise and decay phases of the flare was selected after an
iteration of polynomial fits of different orders and the one which gave the
reduced chi-square closest to 1.

To ascertain the variation in the estimated start and stop times of the flare
episode, 5000 MonteCarlo simulated residual light curves were generated by
assuming a Gaussian distribution777 The assumption of Gaussian distribution is
applied to each data point of count rate in the residual light curve, by
taking into account that if in case the count rate takes a negative value
during the simulation, it would still have a physical meaning implying that
the afterglow emission is more dominant at that instance. In addition, count
rate allows for fractional values. to each data point where the original
residual count is the mean and its associated error as the standard deviation.
The rise and decay phases in each residual light curve were analyzed using
polynomial fits with the same best-fit order as the original residual light
curve. The mean and standard deviation of the distributions of the start and
stop times of the flare episode is referred to as TF,start and TF,stop, and
their associated errors respectively. The distribution of the duration of the
flare episode was generated by taking the difference between the start and
stop times of the flares of each simulated residual count curve. The mean and
standard deviation of the distribution of the duration of the flare is
reported as the TF,100 and its associated error.

The X-ray flare spectrum was generated for the whole duration, TF,100 of the
flare in the energy range $0.3-10$ keV using the Swift-XRT repository and
analysed in 3ML. In order to separate the underlying afterglow emission, the
X-ray spectra was firstly generated in regions selected pre and post of the
X-ray flare or either one of them, depending on if enough data were available.
The underlying afterglow emission spectra was first analysed using a power-law
function multiplied by two absorption terms: galactic, $n_{H}$ and source,
$n_{H}$. The galactic, $n_{H}$, was chosen based on the right-ascension (RA)
and declination (Dec) of each burst whereas, the source, $n_{H}$ was left as a
free parameter in the fit. The Galactic $n_{H}$ values were obtained from the
Galactic $n_{H}$ calculator available at Swift science data
centre888https://www.swift.ac.uk/analysis/nhtot/, which is based on Willingale
et al. (2013). Thereafter, the X-ray spectra generated for the flare duration
is analysed using two power-law functions multiplied by the absorption
coefficients (source, $n_{H}$ was fixed at the value obtained during the
afteglow spectral fit). One of the power-law functions was frozen to the
afterglow spectral fit values and the other power-law function was left free
to model the flare spectrum. Thereby, using the flare power-law model, the
respective energy flux and fluence in the energy range, $0.3-10$ keV, were
estimated. In addition, the spectral index, $\alpha_{F}$, along with the above
mentioned parameters were added to the database. The peak flux of flare
corresponding to the peak time is calculated from the flux lightcurve obtained
from the Swift-XRT repository after subtracting the average afterglow flux

In the current sample, there are GRBs with multiple episodes of X-ray flare
emissions. The end of a flaring episode is defined when the decay phase of the
light curve touches the background. The further rise after this instance or
after certain period of quiescence is considered as the onset of the
consecutive flaring episode (Figure 1a). Using this criteria, we identify
multiple episodes of the flare. Out of the $220$ GRBs in the sample, there are
$44$ GRBs which possess two flaring episodes and $4$ GRBs with three flaring
episodes. In $96$ GRBs, late time X-ray flares were observed concurrently by
the BAT and XRT detectors. In such cases, the emission observed in BAT
exempting the X-ray flare detected in XRT, is considered as the prompt
emission in our study. Therefore, the $T_{P,100}$ estimate in such cases
considers only the burst duration of the first episode of radiation that is
observed in BAT. The quiescent period, $T_{q}$ for the various flaring
episodes are respectively measured as the difference between the onset time of
the flaring episode and the end time of the $T_{P,100}$ of the burst (Figure
1a).

####  2.2.3 Variability Measurement

One of the defining characteristics of gamma ray burst emission is its highly
variable light curves observed during the prompt emission (MacLachlan et al.,
2013). Interestingly, the X-ray flares also exhibit strong variability and
erratic light curves (Sonbas et al., 2013b). The minimum timescale of
variability is the minimum timescale over which there is a significant or
reasonable change in count rate in the observed light curve. The study of the
variability pattern can provide us the information regarding the size and
location of the source region (Kobayashi et al., 1997).

In this work, the Bayesian block (BB) technique (Scargle et al., 2013) of
binning the counts light curve to identify the minimum variability timescale
of both the prompt as well as the X-ray flare emissions was employed. BB bins
the light curve such that each time interval possess a constant Poisson rate.
This methodology divides the light curve into various time intervals of
dynamical time widths and signal-to-noise ratio and closely follows the true
underlying variation in the emission (Burgess, 2014; Vianello et al., 2018).
The time interval with the minimum timescale is considered as the minimum
variability ($T_{var}$) of the light curve. The light curves of prompt and
flare emissions are analysed using the Bayesian Block binning method available
in Astropy package (Astropy Collaboration et al., 2022) in python. In order to
compare the estimated variability timescales of both the prompt and flares
emissions of a GRB as well as between different GRBs in the sample, a common
false alarm probability of $p_{0}=0.01$ is adopted throughout the analysis.
The $p_{0}=0.01$ is found to be an optimal choice allowing to effectively
capture the variabilities across different emissions and provides a confidence
level of $99\%$. This approach, thereby, ensures a systematic analysis,
avoiding the use of multiple configurations specific to different GRBs.

##  3 Multivariate Analysis

In this section, we analyse the multi-dimensional database that is generated
via the methodology described in the previous section to assess the
probability distributions, correlation and clustering of the various studied
parameters.

###  3.1 Distributions of prompt and X-ray flare parameters

The histogrammed probability density plots of the temporal and spectral
related parameters of both the prompt and X-ray flare emissions studied for
the GRBs in this work are presented in the Figure 2. In addition, the smoothed
version of the distribution obtained via the kernel density estimation (KDE)
using the Gaussian kernels are also shown.

Figure 2: The distributions obtained for the prompt (red) and flare (orange
for first flare episode and blue for second flare episode) temporal parameters
are shown. The vertical bars represent the histograms and the respective KDE
curves are shown in color filled shaded curves. The distributions of a)
TP,100, b) TF,100, c) Tq, d) TP,var, e) TF,var, f) $\alpha_{P}$, g)
$\alpha_{F}$, h) $Fluence_{P}$, i) $Fluence_{F}$, j) $F_{P,peak}$, and k)
$F_{F,peak}$ are shown.

The X-ray flare episodes have durations on average greater than that of the
prompt emission and the consecutive flaring episodes tend to be even longer.
The mean (median) of the prompt duration is around $51^{+117}_{-35}\,\rm s$
($59\,\rm s$), whereas the first and second X-ray flaring episodes, on average
(median), have longer durations of $188^{+449}_{-133}\,\rm s$ ($169\,\rm s$)
and $344^{+1182}_{-267}\,\rm s$ ($230\,\rm s$), respectively (Figure 2a and
2b). The average (median) quiescent periods of the first and second flaring
episodes are found to be around $69^{+368}_{-58}\,\rm s$ ($91\,\rm s$) and
$303^{+593}_{-200}\,\rm s$ ($224\,\rm s$), respectively (Figure 2c). One of
the immediate similarities that can be noted between the prompt emission and
the X-ray flares via a visual inspection is the erratic nature of both the
light curves which is quantified using variability timescale measurements. The
prompt emission light curves exhibit minimum variabilitPrompt emission and
X-ray flaresies over time scales on average (median) $2.7^{+13.4}_{-2.2}\,\rm
s$ ($2.4\,\rm s$), see Figure 2d, while the minimum variability timescales of
first and second X-ray flare episodes are relatively of longer durations on
average (median) $15^{+56}_{-12}\,\rm s$ ($11\,\rm s$) and
$25^{+144}_{-22}\,\rm s$ ($16\,\rm s$), respectively (Figure 2e). We note that
X-flare light curves depending on the exposure times, for majority of first
episodes an average $1\,s$ time resolutions were available, but, in some
cases, the time resolutions were longer.

The low energy power-law spectral indices, $\alpha_{P}$ for the time
integrated spectrum of the total prompt emission duration (energy band: 15 -
150 keV) are found to be on average (median) around $-1.5\pm 0.40$ (-1.5), see
Figure 2f. However, the lower energy power-law spectral indices, $\alpha_{F1}$
and $\alpha_{F2}$ of the time integrated spectrum of the total first and
second X-ray flare episodes in energy range 0.3 - 10 keV are on average
(median) much softer, around $-2.1\pm 0.83$ ($-2.0$) and $-2.25\pm 0.65$
($-2.17$), respectively (Figure 2g).

The fluence and the peak flux of the prompt episode is on average (median)
around $1.6^{+3.8}_{-1.1}\times 10^{-6}$ ($1.7\times 10^{-6}$) $\rm
erg/cm^{2}$ and $1.2^{+2.1}_{-0.78}\times 10^{-7}$ ($1.1\times 10^{-7}$) $\rm
erg/cm^{2}/s$, respectively in the energy range $15$ \- $150$ keV (Figures 2h
and 2j). The average (median) of the fluence of the first and second X-ray
flaring episodes in 0.3 - 10 keV is around $1.5^{+5.2}_{-1.1}\times 10^{-7}$
($1.5\times 10^{-7}$) $\rm erg/cm^{2}/s$ and $8.7^{+24.7}_{-6.5}\times
10^{-8}$ ($7.2\times 10^{-8}$) $\rm erg/cm^{2}$, respectively (Figure 2i). The
first and second episodes of the X-ray flare episodes have peak flux on
average (median) around $2.9^{+14}_{-2.4}\times 10^{-9}$ ($3.2\times 10^{-9}$)
$\rm erg/cm^{2}/s$ and $9.3^{+47}_{-7.8}\times 10^{-10}$ ($1.2\times 10^{-9}$)
$\rm erg/cm^{2}$, respectively (Figure 2k).

###  3.2 Principal component analysis

Principle Component Analysis (PCA) allows to search for trends or correlations
between parameters and sort them based on their contribution to the observed
variance in the data. The PCA of the entire multi-dimensional dataset
including the parameters of both prompt and X-ray flares, thereby, gives us
some insight into the relation between different parameters and also shows the
parameters that mainly drive the variability in clustering. PCA reduces the
multi-dimensionality of the dataset into a few principal components which is a
linear combination of the original parameters. We find that principal
component 1 (PC1) and principal component 2 (PC2) together account for $46\%$
of the total variability in the dataset. Using PC1 and PC2, the PCA circle of
correlation is created as shown in Figure 3a. Here the correlation refers to
the Pearson correlation coefficient between the parameter and the principal
components. Thereby, the parameters that are positively correlated among each
other are grouped together within the same quadrant and if they are negatively
correlated the parameters would be positioned in opposite quadrants. The
parameters that are less correlated with each other would be positioned in
adjacent quadrants and would be at nearly right angles to each other. It is
also key to note that the length of each vector variable from the origin
(centre of the circle) in the plot represents the quality of representation of
the variable. In other words, it represents the percentage of the contribution
of that parameter in defining the PCA axes 1 and 2 combined, which is provided
in the contribution color bar shown in the Figure 3a.

With the above understanding, from the PCA analysis999For the PC analysis, the
R package FACTOMINER (Lê et al., 2008) was used. of the dataset, we find there
exists no explicit correlation between the temporal and spectral parameters of
prompt emission and X-ray flares, as evident from Figure 3a . In case of
$\alpha$ and $T_{var}$, it is important to note that the quality of
representation of these parameters on PC1 and PC2 are lowest and therefore,
their degree of correlations may not be strong (see section 3.3 for more
details). Using this initial results of PC analysis as a guide, we look into
the correlations between the various parameters and clustering in the
subsequent sections in detail.

Figure 3: (a) PCA circle of correlation showing the relationship between all
the prompt and X-ray flare parameters is shown. The percentage of the total
contribution of each parameter to the PCA axes is represented by the color
coding. (b) The Pearson Correlation Matrix of the various parameters of the
prompt emission and the first (blue), and second (green) X-ray flare episodes
are shown. Markers representing the p-values of the obtained correlations are
shown in each cell representing the statistical significance of the
correlation.

###  3.3 Correlation Analysis

The correlation analysis explores and identifies trends, patterns or
relationships between different parameters under consideration, including
positive, negative or no correlation. This can be quickly assessed by
visualizing data in a scatter plot. Most importantly, correlation does not
imply direct causation, as the relationship between parameters may be complex.
Nonetheless, a correlation does indicate interdependence, allowing for further
exploration of the underlying physical scenario.

A Pearson Correlation Matrix of the various parameters of the prompt emission
and the first and second X-ray flare episode is shown in Figure 3b. The
correlation matrix features markers indicating the various levels of
statistical significance, including p-value $<0.001$, $p<0.01$, $p<0.05$, and
$p>0.05$. The p-value represents the probability of the null hypothesis which
considers there is no correlation between the variables and thereby serves as
the measure of the statistical significance of accepting the alternate model
i.e there exists a correlation between the two variables. The parameters of
the third flare episodes are not included in the correlation matrix because
only a small fraction of the sample consisted of such multiple flaring
episodes. However, they are included in the detailed correlation plots and
modelling. In this work, we have modelled only those correlations between
parameters that exhibited a positive or negative Pearson correlation
coefficient $>0.5$, both in first and second flaring episodes. All the
modelled correlations in our investigation demonstrate strong statistical
significance as indicated by the low p-values, less than 0.001.

In Figure 4a, the duration of the X-ray flare, $T_{F,100}$, is plotted against
the duration of the respective prompt emission, $T_{P,100}$, of the GRB. No
particular correlation can be observed. For the purpose of comparison, the
identity line representing when $T_{P,100}=T_{F,100}$ is marked. The duration
of the flares including both first, second and third episodes of emission
largely lie above this identity line suggesting that the flaring episodes tend
to have durations greater than their respective prompt emissions. In addition,
in case of multiple episodes of flaring, the consecutive episode is always
longer than the preceding episode of flaring.

As evident from the Figure 4b, the flare minimum variability does not show any
explicit correlation between the temporal properties of the prompt emission
properties such as its minimum variability and duration (see Figure 3b). In
comparison to the quiescent period, there is only a trend of positive
correlation between $T_{F,var}$ and $T_{q}$, with a weak Pearson correlation
coefficient of $+0.4$ (Figure 3b). The minimum variability timescale of the
flares is compared with the total duration of the flare as shown in Figure 4c.
We note there is a positive correlation between $T_{F,var}$ and $T_{F,100}$ as
follows

$log_{10}T_{F,var}=(+0.71\pm 0.06)\,log_{10}(T_{F,100})-0.42\pm 0.15$ (1)

We note that this positive correlation is a weak trend as we are limited by
the small number of flares with very long durations ($>1000\,\rm s$). Thus,
this correlation can be affirmed in the future with increased detections of
flares with longer durations and lightcurves with smaller time resolutions.
Interestingly, in contrast to this behaviour of flares, we note that there
exists no significant correlation between the prompt emission minimum
variability and its duration (Figure 4d) as evident from the Pearson
correlation matrix shown in Figure 3b. For a small number of GRBs, the BB
binning of prompt emission resulted in minimum timescale equivalent to the
whole burst duration (Figure 4d).

A positive correlation is found between the quiescent period, $T_{q}$, and the
X-ray flare duration, $T_{F,100}$ (Figure 4e). The positive correlation is
modelled using a linear fit in the log-space as follows

$log_{10}(T_{q}+1)=(+0.97\pm 0.05)\,log_{10}(T_{F,100})-0.12\pm 0.10$ (2)

and is shown in solid red line in Figure 4e. In comparison, there is only a
weak trend of negative correlation observed between the quiescent period and
the prompt emission duration of the bursts as shown in Figure 4f.

Figure 4:  (a) TF,100 is plotted against $T_{P,100}$. The black circles, blue
squares and red triangles represent the first, second and third X-ray flaring
episodes, respectively. The same color code and markers are used hereafter.The
green dashed line represents when TP,100 = TF,100. (b) The $T_{F,var}$ is
plotted against the corresponding $T_{P,var}$. (c)The $T_{F,var}$ is plotted
against $T_{F,100}$. The linear fit in the log space is shown by red solid
line. The shaded red and blue regions correspond to 1$\sigma$ and 2$\sigma$
deviations of data from the fit. The color code represents the same contours
hereafter. d) TP,100 is plotted against $T_{P,var}$ (e) Tq is plotted against
TF,100 in the left hand panel and (f) against TP,100 in the right hand panel
respectively. Note that the grey diamonds represent the GRBs with quiescent
period equal to zero, are not included in the modelling of the correlation.
The parameters of flares that occur during the plateau phase of the afterglow
is marked in green unfilled circle and square markers for the first and second
episodes of flare respectively

The prompt and flare emission spectra were modelled using the BAT and XRT
data, obtained in the energy ranges 15-150 keV and 0.3-10 keV respectively.
For the purpose of comparison of spectral behaviours, we assumed that the
obtained prompt spectral model remains the same in the extrapolated lower
energy range of 0.3 - 10 keV and thereby, studied the correlations between the
spectral properties of the prompt and flare emissions.

Firstly, we studied the correlation between the low energy power-law index
($\alpha_{P}$) of either the power-law or the cutoff power-law model used to
best model the time integrated prompt spectra in the energy range $15-150$ keV
and the low energy power-law index ($\alpha_{F}$) of the power-law model that
was used to model the time integrated flare spectra in the energy range
$0.3-10$ keV, as shown in the Figure 5a. There is no explicit correlation
observed between the $\alpha_{F}$ and $\alpha_{P}$. For majority of the sample
the $\alpha_{F}$ tends to be much softer than $\alpha_{P}$.

The high energy ($>10keV$) part of the flare spectrum is not available for
analysis and therefore, the peak of the flare spectrum is not known. It has
been observed in the prompt emission that the spectral peaks are typically
around a few hundred keVs. Therefore, for the purpose of comparing the energy
fluxes between the prompt and flare, we estimate the prompt flux expected in
the energy range 0.3 -10 keV by extrapolating the spectrum obtained from the
analysis in the 15 -150 keV to those energies. In Figure 5b, we have compared
the peak fluxes of both prompt and X-ray flare estimated in the energy range
for 0.3-10 keV. Again, no significant correlation between these parameters are
found. The flares are found to be less brighter than the prompt emission,
Figure 5c. We further note that among the multiple flaring episodes of the
GRB, the subsequent flaring episodes are found to be less intense as evident
in Figure 5d. This is consistent with the findings by Chincarini et al.
(2007b) and Falcone et al. (2007).
The prompt emission do not show any correlation between its peak flux and its
duration (Figure 5e). However, it is interesting to note that the flare peak
flux shows a weak trend of negative correlation (Pearson correlation
coefficient = -0.42) with flare duration (Figure 5f). Similarly prompt
emission does not show any significant correlation between it’s peak flux and
minimum variability and the quiescent period (Figure 5g and 5i). However, the
peak flux of X-ray flare shows significant negative correlation between its
minimum variability and quiescent period (Figure 5h and 5j) where modelling is
given as follows:

$log_{10}(F_{F,peak})=(-0.89\pm 0.05)\,log_{10}(T_{F,var})-7.44\pm 0.06$ (3)

$log_{10}(F_{F,peak})=(-1.05\pm 0.07)\,log_{10}(T_{q}+1)-6.39\pm 0.17$ (4)

Certain GRBs among the sample show the occurance of flaring partly or fully
during the plateau phase of the afterglow emission (Appendix A). The plateau
phase is considered to be the region where the power law slope of the
afterglow ranges between +/- 0.3. We have identified 13 GRBs wherein 12 of
them have either first or all episodes of flares occurring partly or fully
during the plateau phase while only one case (GRB110414A) has its second
episode of flaring alone, occuring during the plateau phase. These GRBs are
particularly highlighted in all the correlation plots. We note that these GRBs
do not exhibit any unusual trend in the correlation studies in comparison to
the other GRBs in the sample.

Figure 5: (a) $\alpha_{P}$ is plotted against $\alpha_{F}$. (b) FP,peak is
plotted against FF,peak. (c) The ratio of FP,peak to FF1,peak and FF2,peak is
plotted against the TP,100 in black circles and blue squares respectively. (d)
The ratio of FF1,peak to FF2,peak is plotted against TP,100 in blue circles
and the ratio of FF2,peak to FF3,peak in red circles. (e) FP,peak is plotted
against TP,100. (f) FF,peak is compared against TF,100. (g) FP,peak is plotted
against TP,var. (h) FF,peak is compared against TF,var. (i) FP,peak is plotted
against Tq. (j) FF,peak is compared against Tq. The linear fit modelling the
correlations in (h) and (j) are shown in red solid lines. The parameters of
flares occurring during the plateau phase of the afterglow are highlighted in
green unfilled circle and square markers for the first and second episodes of
flare respectively.

###  3.4 Clustering

It is observed that among the GRB population only certain GRBs possess X-ray
flaring episodes and among them certain GRBs are found to have multiple
episodes of flaring, some with and without quiescent periods etc. Such
variability among the different GRBs suggest the possibility of the existence
of subgroups among the GRBs with X-ray flaring.

No major dissimilarity is observable between the distribution of the
properties of the prompt emission of GRBs with and without flares, see
Appendix B. In addition, the Kolmogorov–Smirnov (KS) test shows that it is
highly likely that GRBs with and without X-ray flares belong to the same
population. The Principle component analysis done in section 3.2 shows that
there is no explicit correlation between the studied prompt and flare
properties. Therefore, we incorporated only the X-ray flare properties to
investigate the possibility of clusters among the flaring GRBs. This included
the following X-ray flare parameters: TF,100, TF,var, Tq, FluenceF, FF,Peak
and $\alpha_{F}$.

Unsupervised clustering was carried out using Gaussian Mixture Model (GMM),
Hierarchical and K-means algorithms101010The clustering packages available in
scikit$-$learn (Pedregosa et al., 2011) were used.. To account for the large
difference between numerical values of each parameters in the dataset, the
data was standardised to a fixed order of range. All the chosen parameters
except $\alpha_{F}$ is converted to log scale and later the complete dataset
is scaled and centered by subtracting each values by their mean and dividing
them by their standard deviation. The standardised dataset was then fed into
the different clustering algorithms.

Based on the various evaluation metrics such as Silhouette score, Davies-
Bouldin Index and Calinski-Harabasz Index, the K-means was found to be the
best clustering algorithm for the dataset. The number of clusters in K-means
were determined by two methods: Silhouette score (Rousseeuw, 1987) and the
Elbow method. The elbow was found to be at 4, signifying 4 possible subgroups
within the GRBs with X-ray flares and the Silhouette score also peaked at 4
number of clusters. The obtained clusters are visualised in the principal
component space as shown in Figure 6a. For understanding the origin of the
clusters, we have marked the cases that had $T_{q}=0$ in squares, GRBs with
multiple flaring episodes with star markers and GRBs with flare in the plateau
region of afterglow with yellow triangle marker. Since the $T_{q}$’s
contribution in the PCA analysis is the largest in the fourth principal
component, for the purpose of better visualising of the clusters, we chose to
plot the clusters using PC4 versus PC1 in Figure 6a. The average values of the
parameters of each of the cluster is reported in the Table 1. The GRBs with
flares during the plateau are found across the clusters and does not display
any discernible peculiar behaviours.

##  4 Discussions and Conclusions

The origin of the late time X-ray flares is largely a mystery. Owing to the
similar erratic behaviour of the lightcurves, the prompt emission and X-ray
flares, they are believed to be powered by the central engine. Therefore, in
this work using the extensive database of Swift observations of both prompt
and X-ray flares via BAT and XRT instruments onboard respectively, we
performed a systematic statistical study of the various temporal and spectral
parameters characterising these two emissions. The sample consisted of 220
long GRBs. Among them 172 GRBs possessed single episodes of flare emission,
whereas, 44 and 4 GRBs had double and triple flaring episodes respectively.
Among all the GRBs in the sample, $96$ GRBs had their flaring episode observed
synchronously in BAT as well. The temporal properties of the prompt and X-ray
flare emissions were characterised by parameters such as total duration,
minimum variability and the quiescent period between the episodes; and their
spectral properties were characterised by parameters such as the low energy
spectral power-law index, peak energy flux and energy fluences.

Figure 6: (a) The K-means clustering results are visualized in a PC plot. Black$-$cross and red$-$square markers represent the GRBs which posses multiple flares and GRBs with zero quiescent period respectively. The yellow triangle markers represent GRBs with flares in the plateau region of afterglow. (b) The KDE distributions of the various flare properties of the cluster B and D are shown. Table 1: Cluster Statistics | Cluster A | Cluster B | Cluster C | Cluster D
---|---|---|---|---
log${}_{10}(T_{F,100}$) | 2.08 $\pm$ 0.46 | 2.26 $\pm$ 0.32 | 4.26 $\pm$ 0.49 | 2.18 $\pm$ 0.35
log${}_{10}(T_{F,var}$) | 0.97 $\pm$ 0.42 | 0.76 $\pm$ 0.31 | 3.14 $\pm$ 0.91 | 1.49 $\pm$ 0.49
$\alpha_{F}$ | -1.87 $\pm$ 0.59 | -1.81 $\pm$ 0.56 | -1.64 $\pm$ 0.56 | -2.54 $\pm$ 0.95
log10(FF,peak) | -8.18 $\pm$ 0.43 | -7.97 $\pm$ 0.44 | -10.39 $\pm$ 0.84 | -9.05 $\pm$ 0.44
log10(FluenceF) | -6.59 $\pm$ 0.49 | -6.33 $\pm$ 0.41 | -6.70 $\pm$ 0.82 | -7.42 $\pm$ -0.39
log${}_{10}(T_{q}$) | 0.07 $\pm$ 0.26 | 1.95 $\pm$ 0.35 | 3.54 $\pm$ 0.70 | 2.11 $\pm$ 0.31
Cluster size (% of population) | 12.3 % | 42.3 % | 3.6 % | 41.8 %

The distributions of the total duration of prompt and flaring episodes show
that the flares largely have longer durations than the prompt with the median
of $T_{F,100}$ distribution being nearly 3 times the median of $T_{P,100}$
distribution. The consecutive flaring episodes are found to be longer than the
preceding one. The median of distribution of the duration of the second
flaring episode is around 1.3 times the first episode of flare. This is
consistent with the earlier studies done by Chincarini et al. (2007b).

The distribution of the minimum variability timescales of the prompt emission
has a median around $2$ s. On the contrary, the X-ray flare emissions tend to
have larger variability timescales with a median of 12 s. The quiescent period
between the prompt emission and the flaring episodes are found to range from
none to around a day. The average of quiescence for the first flaring episode
is around 100 s. The spectral parameters such as the low energy power-law
index in the flares is found to be much softer than the prompt emission. The
study of the energy fluence and peak energy flux show that the prompt emission
is more intense and brighter than the X-ray flaring episodes when compared in
the same energy range of $0.3-10$ keV.

The principal component analysis and the correlation studies carried out on
the dataset reveals that there is no explicit correlation between the prompt
and X-ray flare properties in general. The duration of the prompt and flare
episodes are uncorrelated. Interestingly, a positive correlation is observed
between the quiescent period and the flare duration, while there is no
significant correlation with the prompt emission duration. This implies that
the property which drives the quiescent period and the onset time of the flare
emission after the prompt has ended, is intrinsically connected with the
origin of the flare, while it is less dependent on the prompt emission.

The minimum variability timescale observed in GRBs can point towards the size
of the emitting region as well as allows one to understand the central engine
activity, the impact of the circumburst medium as the jet propagates through
the stellar core etc. There is no explicit correlation observed between the
minimum timescale of variabilities of flares and that of prompt emission of
the GRB. While there is no correlation between the prompt emission minimum
variability and its duration, we find that there is a trend of positive
correlation between the flare variability and its duration.

We compared the spectral properties of prompt and flare in the energy range
0.3 - 10 keV. This was done by extrapolating the spectrum obtained for prompt
in the energy range 15-150 keV into the X-ray flare energy range of 0.3-10
keV. We find that the low energy power-law spectral index, $\alpha$ obtained
for the spectrum of the total duration of prompt and flare respectively are
found to be largely uncorrelated. The flare largely tends to have relatively
much softer spectra in comparison to prompt. This may be attributed to the
relatively longer duration of flares and may also point towards more rapidly
evolving spectral behaviour in flares in comparison to prompt. We also find
that the peak fluxes of prompt and flare emissions in the energy range 0.3-10
keV are uncorrelated. Consistent with previous studies (Falcone et al., 2007),
the peak flux of the flare emission are found to be less brighter than the
prompt and also decreases with consecutive flaring episodes. There is no
explicit correlations observed between the peak fluxes of either of the prompt
or flare and its durations. Interestingly, we also note that though there is
no explicit correlation observed between prompt emission’s minimum variability
timescale and its peak flux, however, there exists a strong negative
correlation between the flare’s minimum variability timescale and its peak
flux.

Further, using the flare properties alone, we explored the variety of classes
among the current dataset of flares defined in this paper. Using K-means
clustering algorithm, four clusters are found to exist. Among the clusters, we
note that there is no preferential clustering for GRBs with multiple episodes
of flares and are found to be spread across clusters. Among them we note that
cluster A represents all $25$ cases of flares which have zero quiescent period
(Figure 6a). Thus, we conclude that the segregation of these X-ray flare
events from the rest of the flare sample point out that these observed flaring
episodes are actually the continuing prompt emission. Among the remaining
classes, we note that cluster C contains only $3.6\%$ of the total sample and
represents largely the flares with extreme values of duration, minimum
variability timescale etc. This reduces the clustering to largely two main
clusters B and D, each consisting $\sim 42\%$ of the sample. While inspecting
the distributions of the various properties of these two clusters, we find
that the segregation is largely driven by minimum variability timescale, peak
flux and fluence as evident from the Figure 6b. Certain GRBs in the sample
exhibit flaring during the plateau phase of the afterglow, but they do not
display any unusual trends in correlation studies and are distributed across
clusters without any discernible peculiar behaviours.

Thus, to summarise, in order to understand the origin of the prompt and X-ray
flares, we find the following results of the correlation study: (a) no
explicit correlation between the temporal and the spectral properties of the
prompt and flare emissions; (b) there exists a trend of positive correlation
between minimum variability timescale of flare and its duration, while there
is a strong negative correlation between the minimum variability timescale of
flare and it peak flux. However, there are no such trends to be found in the
prompt emission properties; and, (c) there also exists a positive correlation
between quiescent period and flare duration, and a negative correlation
between the quiescent period and the flare peak flux, however, no such
correlations are observed for the prompt emission of GRBs.
Therefore, using the current parameter characterisation of the prompt and
X-ray flares adapted in this work, we do not find any significant evidence for
the common origin for prompt emission of GRBs and the X-ray flares observed
during the afterglow emission. In addition, we find the X-ray flares
dominantly possess two main classes whose differences are primarily driven by
the variability timescale, peak flux and fluences. Finally, we present the
catalog of the estimated parameters to the community for further use in
Appendix C.

One of the major future prospects of this work is to develop an understanding
of the physical scenario leading to the observed correlations between late-
time X-ray flares and the prompt emission. Furthermore, for a better
comprehensive picture of the origin of flares, an extensive comparative study
need to be carried out between the flares observed in X-rays and optical as
well as high energy emission ($>100$ MeV). The upcoming Cherenkov Telescope
Array (CTA, Kakuwa et al. 2012) holds promise for further insights. We further
note that a significant fraction of the initial sample had to be excluded due
to incomplete data. An enhanced sample can be studied by adopting a light
curve reconstruction method, similar to the one proposed in Dainotti et al.
(2023) as increased sample size can lead to more confident assessments of the
correlations presented in this study.

We thank the anonymous referee for valuable comments and suggestions. J.S. and
S.I. are supported by DST INSPIRE Faculty Scheme (IFA19-PH245). S.I. is also
supported by SERB SRG Grant (SRG/2022/000211). This research has made use of
data obtained from the High Energy Astrophysics Science Archive Research
Center (HEASARC), provided by NASA’s Goddard Space Flight Center and from the
UK Swift Science Data Centre at the University of Leicester.

##  Appendix A Additional Plots

Figure 7: The afterglow lightcurves of (a) GRB 060223A and (b) GRB 180602A
represent cases with no data during the decay and rise phases of the flaring
episode respectively. (c) GRB 100805A and (d) GRB 130803A represent cases that
have less observational points during flare emission. (e) GRB 160408A and (f)
GRB 160624A represent cases where the underlying afterglow emission is absent
or not enough data points are available for proper modelling. (g) GRB 130427A
and (h) GRB 090404 represent cases that are avoided due to their very low
signal-to-noise ratio. The red shaded portion represents the flare duration as
identified by the Swift XRT team. The afterglow lightcurves of (i) GRB 170705A
and (j) GRB 150626B possessing a single flare are shown. (k) GRB 140206A and
(l) GRB 150911A represents afterglows with double flaring episodes, while, (m)
and (n) show lightcurves of GRB 190926A and GRB 050822, which have three
flaring episodes each. (o) and (p) represent lightcurves of GRB 161219B and
GRB 091029 with a full and partial flare, respectively, within the plateau
region of their afterglow. The blue-shaded region represents the flare
duration as identified in this study. The red dotted line shown in all the
plots represents the afterglow model obtained after excluding the flare
region.

##  Appendix B Comparing the prompt emission of GRBs with and without X-ray
flares

Not all GRBs detected by Swift possess X-ray flares in their afterglow
emission. Thus, in order to understand if there exists any difference in the
type of GRBs giving rise to flares, we made comparison plots of the
distributions of different prompt emission properties such as isotropic burst
energy, $E_{iso}$ estimated for known redshift cases, $T_{90}$, low energy
power-law spectral index, $\alpha$ and energy fluence of GRBs with and without
X-ray flares as shown in Figure 8. Furthermore, to assess if both the sample
distributions originate from the same population, we conducted a
Kolmogorov–Smirnov(KS) test111111The scipy package ks$\\_$2samp was used. on
the distributions of the different parameters. The null hypothesis assumes
both the flare and non-flare GRB’s properties are sampled from the same
underlying population and the alternate is they are not identical. The
resultant p-value if lower than the probability $p=0.01$ (chosen threshold),
allows us to reject the null hypothesis and adopt the alternate hypothesis at
a statistical significance greater than $99\%$. However, for the distributions
of $E_{iso}$, $T_{90}$, $\alpha$ and energy fluence in $15-150\rm\,keV$, we
find the p-values from the KS test to be $0.17$, $0.07$, $0.69$ and $0.03$
respectively. Since the p-values lie above the threshold, we conclude that
there is no particular distinction between the population of GRBs that produce
and that does not produce late time X-ray flares.

Figure 8: Comparison of the distribution of the prompt emission properties
such as $E_{iso}$, $T_{90}$, $\alpha$ and energy fluence of GRBs with and
without X-ray flares are shown.

##  Appendix C Table of Prompt $\&$ X-ray Flare properties

The complete catalog of the studied parameters of the sample is made available
online in the machine-readable format. A sample of the catalog is shown in
Table 2 for guidance regarding its form and content.

Table 2: catalog

GRB | Flare | TP,100 | TP,Var | TP,Peak | BFM [I] |  $\alpha_{P}$ [I] | FluenceP | FluenceP | BFM [P] |  $\alpha_{P}$ [P] | FP,Peak | FP,Peak | TF,100 | TF,Var | TF,Peak | $T_{q}$ | $S/N$ | $\alpha_{F}$ | FF,Peak | FluenceF
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
NAME | Episode | (s) | (s) | (s) |  |  | [15-150keV] | [0.3-10keV] |  |  | [0.3-10keV] | [15-150keV] | (s) | (s) | (s) | (s) |  |  | [0.3-10keV] | [0.3-10keV]
|  |  |  |  |  |  | ($\times{10^{-6}}$$\rm erg/cm^{2}$) | ($\times{10^{-6}}$$\rm erg/cm^{2}$) |  |  | ($\times{10^{-8}}$$\rm erg/cm^{2}/s$) | ($\times{10^{-8}}$$\rm erg/cm^{2}/s$) |  |  |  |  |  |  | ($\times{10^{-10}}$$\rm erg/cm^{2}$) | ($\times{10^{-8}}$$\rm erg/cm^{2}$)
170208B† | 1 | 32.4$\pm$2.4 | 0.31 | 1.0 | PL | -1.50${}^{+0.01}_{-0.01}$ | 4.84$\pm$0.37 | 1.52$\pm$0.13 | PL | -1.08${}^{+0.03}_{-0.02}$ | 6.76${}^{+0.47}_{-0.44}$ | 74.27${}^{+1.40}_{-1.39}$ | 172.2$\pm$31.4 | 6.37 | 98.1 | 70.5$\pm$3.37 | 28.9 | -2.88${}^{+0.03}_{-0.03}$ | 338.86$\pm$44.40 | 46.55$\pm$8.53
140419A | 1 | 124.6$\pm$9.3 | 0.36 | 51.4 | PL | -1.02${}^{+0.10}_{-0.10}$ | 15.6$\pm$1.2 | 1.2$\pm$0.1 | PL | -1.02${}^{+0.04}_{-0.04}$ | 3.24${}^{+0.38}_{-0.32}$ | 42.89${}^{+1.25}_{-1.30}$ | 106.3$\pm$9.9 | 10.84 | 209.9 | 60.1$\pm$10.6 | 6.0 | -1.42${}^{+0.16}_{-0.16}$ | 22.45$\pm$5.00 | 4.50$\pm$0.96
121125A | 1 | 78.2$\pm$7.9 | 1.07 | 36.6 | PL | -1.54${}^{+0.03}_{-0.03}$ | 4.37$\pm$0.44 | 1.52$\pm$0.18 | PL | -1.37${}^{+0.07}_{-0.07}$ | 4.35${}^{+0.88}_{-0.73}$ | 21.43${}^{+1.02}_{-1.08}$ | 44.5$\pm$7.4 | 18.9 | 92.0 | 15.7$\pm$8.0 | 5.3 | -1.76${}^{+0.13}_{-0.14}$ | 22.07$\pm$4.42 | 2.73$\pm$0.56
190926A† | 1 | 130.1$\pm$12.0 | 130.10 | 53.0 | PL | -2.06${}^{+0.11}_{-0.10}$ | 1.46$\pm$0.17 | 2.74$\pm$0.89 | PL | -2.03${}^{+0.40}_{-0.40}$ | 5.86${}^{+15.57}_{-3.89}$ | 3.56${}^{+0.78}_{-1.15}$ | 85.5$\pm$6.6 | 14.36 | 242.1 | 78.5$\pm$12.58 | 11.5 | -1.07${}^{+0.07}_{-0.07}$ | 79.21$\pm$13.81 | 16.51$\pm$1.61
190926A | 2 |  |  |  |  |  |  |  |  |  |  |  | 145.2$\pm$6.3 | 9.19 | 361.4 | 194.1$\pm$12.5 | 12.3 | -1.35${}^{+0.04}_{-0.04}$ | 86.43$\pm$13.01 | 34.59$\pm$1.87
190926A | 3 |  |  |  |  |  |  |  |  |  |  |  | 178.9$\pm$18.9 | 81.75 | 584.5 | 370.4$\pm$15.3 | 5.0 | -2.03${}^{+0.06}_{-0.06}$ | 22.74$\pm$4.83 | 11.15$\pm$1.33
121217A† | 1 | 101.4$\pm$16.7 | 3.51 | 0.0 | PL | -1.54${}^{+0.04}_{-0.05}$ | 2.01$\pm$0.34 | 0.71$\pm$0.15 | PL | -1.30${}^{+0.12}_{-0.11}$ | 1.75${}^{+0.65}_{-0.46}$ | 10.39${}^{+0.77}_{-0.83}$ | 764.3$\pm$2.2 | 6.34 | 737.7 | 237.5$\pm$16.83 | 18.3 | -1.41${}^{+0.01}_{-0.01}$ | 146.36$\pm$18.90 | 189.24$\pm$2.15
080906 | 1 | 180.0$\pm$12.4 | 4.37 | 11.0 | PL | -1.61${}^{+0.04}_{-0.04}$ | 3.65$\pm$0.27 | 1.61$\pm$0.22 | PL | -1.60${}^{+0.15}_{-0.15}$ | 3.61${}^{+1.87}_{-1.21}$ | 8.29${}^{+0.80}_{-0.89}$ | 103.2$\pm$14.1 | 45.02 | 186.4 | 60.2$\pm$12.95 | 6.0 | -2.09${}^{+0.05}_{-0.06}$ | 21.40$\pm$3.70 | 6.52$\pm$0.94
080906 | 2 |  |  |  |  |  |  |  |  |  |  |  | 125.1$\pm$34.9 | 115.54 | 596.2 | 447.6$\pm$24.4 | 1.8 | -3.30${}^{+0.41}_{-0.41}$ | 2.30$\pm$0.58 | 0.33$\pm$0.17
200509A† | 1 | $40.0\pm 3.5$ | 0.07 | 1.8 | PL | -1.08${}^{+0.10}_{-0.10}$ | $1.46\pm 0.15$ | $0.13\pm 0.04$ | PL | -0.58${}^{+0.27}_{-0.28}$ | 0.25${}^{+0.31}_{-0.13}$ | 11.65${}^{+1.37}_{-1.67}$ | $810.2\pm 20.1$ | 2.65 | 824.5 | $220.1\pm 18.35$ | 26.1 | -1.80${}^{+0.01}_{-0.01}$ | $293.63\pm 35.30$ | $231.45\pm 5.98$
Notes. BFM - Best Fit Mode; [I] - Integrated; [P] - Peak; $S/N$ \- signal to noise ratio; †- GRBs that had their flaring episodes observed synchronously in BAT |  |  |  |

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# Bernoulli convolutions-2023

Nikita Sidorov  University of Great Northern Tower, 1 Watson Street,
Manchester M3 4EE, United Kingdom  nikita.a.sidorov at gmail.com To Erdős Pál,
the greatest mathematician of the last century

###### Abstract.

Let $\theta\in(1,2)$, and $\mu_{\theta}$ be the Bernoulli convolution
parametrized by $\theta$, that is, the measure corresponding to the
distribution of the random variable $\sum_{n=1}^{\infty}a_{n}\theta^{-n}$,
where the $a_{n}$ are i.i.d. with probability of $a_{n}=0$ equal to
$\frac{1}{2}$.

As is well known, $\mu_{\theta}$ is either equivalent to the Lebesgue measure
on $\text{supp}(\mu_{\theta})$, or singular.

Recall that an algebraic integer $>1$ is called Pisot if all its other Galois
conjugates are smaller than 1 in modulus. It is known that $\mu_{\theta}$ is
singular with $\dim\mu_{\theta}<1$ if $\theta$ is Pisot.

An algebraic integer $\theta$ greater than 1 is called a Salem number if all
its other Galois conjugates are of modulus 1, except $\theta^{-1}$.

I shall prove that

(1) $\dim\mu_{\theta}=1$ if $\theta$ is an algebraic non-Pisot number.

(2) if $\theta$ is Salem, then $\mu_{\theta}$ is equivalent to the Lebesgue
measure on $\text{supp}(\mu_{\theta})$, with an unbounded density in
$L^{p}(\text{supp}(\mu_{\theta}))$ for all $p<\infty$.

(3) Define

$\beta_{\theta,x,n}=\\#\left\\{a_{1}\dots a_{n}:\exists a_{n+1}\dots\text{such
that\ }x=\sum_{k=1}^{\infty}a_{n}\theta^{-k}\right\\}.$

Then

$\lim_{n\to\infty}\sqrt[n]{\beta_{\theta,x,n}}=\theta^{\dim\mu_{\theta}}\text{\
for}\ \mu_{\theta}-\text{a.e.}x.$

(4) Put

$\bigcup_{n=1}^{\infty}\left\\{\sum_{k=1}^{n}a_{k}\theta^{k}\mid
a_{k}\in\\{-1,0,1\\}\right\\}=\\{y_{0}(\theta)<y_{1}(\theta)<\cdots\\},$

and

$\ell(\theta)=\liminf_{n\to\infty}(y_{n+1}(\theta)-y_{n}(\theta)).$

I shall present a short proof of De-Jun Feng’s famous theorem stating that
$\ell(\theta)=0$ for all non-Pisot $\theta$.

###### Key words and phrases:

Bernoulli convolution, Perron number, Pisot number, Salem number.

###### 2010 Mathematics Subject Classification:

28D20; 11R06.

If $\theta$ is transcendental, then $\dim\mu_{\theta}=1$ [9]. I shall assume
henceforth $\theta$ to be an algebraic number.

Let $\theta_{1}=\theta,\dots,\theta_{d}$ be the Galois conjugates of $\theta$
and $\mathcal{M}(\theta)=\prod_{j:|\theta_{j}|>1}|\theta_{j}|$, the Mahler
measure of $\theta$ if $\theta$ is an algebraic integer.

Put

$D_{n}(\theta)=\left\\{\sum_{k=1}^{n}a_{k}\theta^{k}\mid
a_{k}\in\\{0,1\\}\right\\},$

and $d_{n}(\theta)=\\#D_{n}(\theta)$. Put

$L(\theta)=\limsup_{n\to\infty}(y_{n+1}(\theta)-y_{n}(\theta)).$

Recall that $\theta$ is called Perron if all other Galois conjugates of
$\theta$ are smaller than $\theta$ in modulus. By [5, Theorem 2.1],
$L(\theta)=0$ if $\theta<\sqrt{2}$ and $\theta$ is not Perron. 111Condition
(1) in [5, Theorem 2.1] comes with an extra assumption that $-\theta$ is not a
Galois conjugate of $\theta$ which means that $\theta$ is not a root of a
polynomial with no odd powers. I shall deal with this case in the proof of
Theorem 1. This implies

(1) $d_{n}(\theta)\geq c_{n}(\theta)\cdot\theta^{n}\text{\ for some unbounded\
}c_{n}(\theta)>0.$

Let $\theta_{2},\dots,\theta_{s+1}$ denote the non-real conjugates of
$\theta=\theta_{1}$, with $s=0$ if there are none.

By the above and [3, Lemma 1.51], there exists $C_{\theta}>0$ such that for
any interval $J$ with adjacent endpoints in $D_{n}(\theta)$,

(2) $C(\theta)\cdot n^{-s}\mathcal{M}(\theta)^{-n}\leq\mu_{\theta}(J)\leq
d_{n}(\theta)^{-1}\leq c_{n}(\theta)^{-1}\theta^{-n}.$

######  Theorem 1.

  1. (1)

$\dim\mu_{\theta}=1$ iff $\theta$ is not Pisot.

  2. (2)

$\mu_{\theta}$ is equivalent to the Lebesgue measure on $\text{supp}\
\mu_{\theta}$ if $\theta$ is Salem, with density in
$L^{p}(\text{supp}(\mu_{\theta}))$ for all $p<\infty$.

###### Proof.

(1) It suffices to show that for any interval $J$ with adjacent endpoints in
$D_{n}(\theta)$,

(3) $\frac{\log\mu_{\theta}(J)}{\log|J|}\geq 1-\varepsilon$

for any $\varepsilon>0$.

Let us perform some reductions.

We may assume $-\theta$ is not a Galois conjugate of $\theta$, otherwise it is
a root of a polynomial with no odd powers, so we keep extracting square roots
of $\theta$ until we get $\alpha$ that is rid of this constraint. Since
$\theta$ is a positive power of $\alpha$, $\ \alpha$ being fully-dimensional
implies $\theta$ being such.

It suffices to prove the claim for $\theta<\sqrt{2}$, in view of
$\mu_{\theta}=\mu_{\theta^{2}}*\theta\mu_{\theta^{2}}$.

Thus, if $\theta$ is not Perron, then (2) holds. What is left is to obtain a
lower bound for $|J|$ from [3, Lemma 1.51], namely, a constant times the
inverse of the product of the moduli of the other Galois conjugates of
$\theta$. Therefore, $|J|\gg(\mathcal{M}(\theta)-\varepsilon))^{-n}$. Since
$|J|\ll\theta^{-n}$ for all $\theta$ [7], this proves (3).

Assume now $\theta$ to be Perron and neither Pisot nor Salem. Then
$\mathcal{M}(\theta)>\theta$, and the same argument applies.

Finally, assume that $\theta$ is not an algebraic integer. Then all sums in
$D_{n}(\theta)$ are distinct, whence the Garsia entropy
$H_{\theta}=\log_{\theta}2>1$ so $\theta$ is fully-dimensional by Michael
Hochman’s celebrated formula $\dim\mu_{\theta}=\min\\{H_{\theta},1\\}$ [4].

It suffices to recall that $\dim\mu_{\theta}<1$ for a Pisot $\theta$ by [3]
and [4].

(2). Assume $\theta$ to be Salem. Here $\mathcal{M}(\theta)=\theta$, and
$s=d-2$, and $d/2\in\mathbb{Z}$, so (2) is

$n^{2-d}\theta^{-n}\ll\mu_{\theta}(J)\ll\theta^{-n}.$

I shall prove a stronger claim:

(4) $\mu_{\theta}(J)\gg n^{1-d/2}\theta^{-n}.$

We have

(5)
$\sum_{k=1}^{n}\text{Re}(\theta_{j}^{k})=\sum_{k=1}^{n}T_{j}(\cos\text{arg}\theta_{j})\asymp
n,$

where $T_{j}(t)=\cos\arccos jt$ is the $j$th Chebyshëv polynomial. The lower
bound comes from $\alpha$ being badly approximable by the Roth theorem.

If $p$ is a polynomial in $\mathbb{Z}[x]$ with its coefficients bounded by 1
such that $p(\theta)\neq 0$, then $p(\theta)p(\theta_{2})\ldots
p(\theta_{d})\in\mathbb{Z}\setminus\\{0\\}$, whence

$|p(\theta)|\geq\left|\prod_{j=2}^{d}p(\theta_{j})\right|^{-1}\gg
n^{1-d/2}\theta^{n}.$

(4) is proved. Hence

(6) $\frac{\mu_{\theta}(J)}{|J|}\asymp\log|J|,$

whence $\mu_{\theta}$ is equivalent to the Lebesgue measure, with the density
$\delta_{\theta}\in L^{p}(\text{supp}\ (\mu_{\theta}))$ for any finite $p$. ∎

Remark. $\delta_{\theta}$ is 0 on half the points of
$\bigcup_{n=1}^{\infty}D_{n}(\theta)$ and $\infty$ on the rest. Therefore,
$\delta_{\theta}$ is unbounded and discontinuous everywhere. I leave the
details to the interested reader.

The set $\\{\|\theta^{n}\|\\}_{n\geq 1}$ is known to be dense but not
equidistributed in $[0,1]$. Theorem 1 gives a more detailed view.

As a consequence of 5, follows

######  Corollary 2.

For any algebraic integer $\theta$,

$\text{tr}(\theta^{n})=\sum_{j:|\theta_{j}|>1}|\theta_{j}|^{n}+O(n^{s/2}),\quad
n\to\infty.$

This is obvious for the hyperbolic $\theta$ (with $s=0$) but novel – to my
best knowledge – for the non-hyperbolic $\theta$.

It follows from (4) that $d_{n}(\theta)\gg n^{d/2-1}\theta^{n}$ for a Salem
$\theta$. An upper bound is not needed for Theorem 1, yet I believe it may be
of interest.

Towards absolute continuity.

  * •

Every absolutely continuous measure is fully-dimensional but not necessarily
the other way round. The difference is crucial: “dropping the logs” – compare
(3) with (6).

  * •

(2) is crude, yet sufficient for our purposes. The general question is to
study the quantity

$d_{\theta}=\lim_{n\to\infty}\sqrt[n]{d_{n}(\theta)}.$

(The limit exists because $d_{n+k}(\theta)\leq d_{n}(\theta)d_{k}(\theta)$ for
all $n,k\geq 1$.) I believe knowing the exact value of $d_{\theta}$ should
build a such-needed bridge from dimension one to absolute continuity of
Bernoulli convolutions – about which we know very little, generic results
notwithstanding.

Namely, we only know that $\mu_{\theta}$ is absolutely continuous with respect
– and hence equivalent by [6] – to the Lebesgue measure on the support of
$\mu_{\theta}$ if $\theta$ is a Garsia number, that is, an algebraic integer
with all Galois conjugates of modulus greater than 1 and their product being
$\pm 2$ as well as some algebraic $\theta$ very close to 1 recently discovered
by Péter Varjú [8].

Notice that in the former case, $\theta<d_{\theta}=2<\mathcal{M}(\theta)$
unless $\theta=\sqrt{2}$. This gives us hope for the case of algebraic
$\theta$.

  * •

We know from (2) only that

$\theta\leq d_{\theta}\leq\mathcal{M}(\theta),$

which proves sufficient when $\theta$ is Salem since in this case,
$\mathcal{M}(\theta)=\theta$. 222In fact, (2) only works because either
$\mathcal{M}(\theta)>\theta$, or $\theta$ is Salem, in which case $s>0$.

  * •

If $\theta$ is superexponentially well approximable, Michael Hochman’s
techniques from [4] may help proving absolute continuity of $\mu_{\theta}$.

(3) As a direct corollary of Theorem 1, we obtain a result that significantly
extends [2, Theorem 1.1].

######  Theorem 3.

$\lim_{n\to\infty}\sqrt[n]{\beta_{\theta,x,n}}=\theta$

if $\theta$ is not Pisot and $\theta^{\dim\mu_{\theta}}$ otherwise.

(4)

######  Theorem 4.

[1] $\ell(\theta)>0$ iff $\theta$ is Pisot.

###### Proof.

If $\theta$ is not height one, the claim follows from the pigeonhole
principle.

Let now $\theta$ be height one – and therefore, an algebraic unit. Then
$\ell(\theta)=0$ – if

  * •

$\theta<\sqrt{2}$;

  * •

not Perron;

  * •

$-\theta$ is not a Galois conjugate of $\theta$ which means that $\theta$ is
not a root of a polynomial with no odd powers;

I shall now remove these constraints and complete the proof.

If $\theta<\sqrt{2}$ and a root of a polynomial with no odd powers, we keep
extracting square roots of $\theta$ until we get $\nu$ that is rid of this
constraint. Then $\ell(\nu)=0$ implies $\ell(\theta)=0$.

If $\theta=\sqrt{2}$, then $\ell(\theta)=0$ by the pigeonhole principle.

If $\theta>\sqrt{2}$, then $\ell(\sqrt{\theta})=0$ implies $\ell(\theta)=0$.

If $\theta<\sqrt{2}$ and a non-Pisot Perron, its Mahler measure is
$\geq\theta$ and $>\theta$ unless $\theta$ is Salem. It suffices to recall
that $d_{n}(\theta)\geq c_{n}\theta^{n}$ with an unbounded $c_{n}$ and apply
the pigeonhole principle. ∎

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