11institutetext: Department of Physics and Earth Science, University of
Ferrara, Via Saragat 1, I-44122 Ferrara, Italy 22institutetext: INFN – Sezione
di Ferrara, Via Saragat 1, 44122 Ferrara, Italy 33institutetext: INAF –
Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero
Gobetti 101, 40129 Bologna, Italy 44institutetext: Department of Physics,
University of Cagliari, SP Monserrato-Sestu, km $0.7$, 09042 Monserrato, Italy
55institutetext: Ioffe Institute, Politekhnicheskaya 26, 194021 St.
Petersburg, Russia 66institutetext: INAF, Osservatorio Astronomico d’Abruzzo,
via Mentore Maggini snc, 64100 Teramo, Italy 77institutetext: Key Laboratory
of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, People’s Republic of China 88institutetext:
University of Chinese Academy of Sciences, Beijing 100049, People’s Republic
of China

# Distribution of number of peaks within a long gamma–ray burst

C. Guidorzi<EMAIL_ADDRESS>of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray
burstDistribution of number of peaks within a long gamma–ray burstDistribution
of number of peaks within a long gamma–ray burstDistribution of number of
peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burst 0000-0001-6869-0835 M. Sartori Distribution of number of
peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burst R. Maccary Distribution of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray burst
0000-0002-8799-2510 A. Tsvetkova Distribution of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray
burstDistribution of number of peaks within a long gamma–ray burstDistribution
of number of peaks within a long gamma–ray burstDistribution of number of
peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burst 0000-0003-0292-6221 L. Amati Distribution of number of
peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burst 0000-0001-5355-7388 L. Bazzanini Distribution of number
of peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burstDistribution of number of peaks within a long gamma–ray
burstDistribution of number of peaks within a long gamma–ray burst
0000-0003-0727-0137 M. Bulla Distribution of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray
burstDistribution of number of peaks within a long gamma–ray burstDistribution
of number of peaks within a long gamma–ray burstDistribution of number of
peaks within a long gamma–ray burstDistribution of number of peaks within a
long gamma–ray burst 0000-0002-8255-5127 A. E. Camisasca Distribution of
number of peaks within a long gamma–ray burstDistribution of number of peaks
within a long gamma–ray burst 0000-0002-4200-1947 L. Ferro Distribution of
number of peaks within a long gamma–ray burstDistribution of number of peaks
within a long gamma–ray burstDistribution of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray burst
0009-0006-1140-6913 F. Frontera Distribution of number of peaks within a long
gamma–ray burstDistribution of number of peaks within a long gamma–ray
burstDistribution of number of peaks within a long gamma–ray burstDistribution
of number of peaks within a long gamma–ray burst 0000-0003-2284-571X C. K. Li
Distribution of number of peaks within a long gamma–ray burstDistribution of
number of peaks within a long gamma–ray burst 0000-0001-5798-4491 S. L. Xiong
Distribution of number of peaks within a long gamma–ray burstDistribution of
number of peaks within a long gamma–ray burst 0000-0002-4771-7653 S. N. Zhang
Distribution of number of peaks within a long gamma–ray burstDistribution of
number of peaks within a long gamma–ray burstDistribution of number of peaks
within a long gamma–ray burstDistribution of number of peaks within a long
gamma–ray burst 0000-0001-5586-1017

###### Abstract

Context. The variety and complexity of long duration gamma–ray burst (LGRB)
light curves (LCs) encode a wealth of information on the way LGRB engines
release their energy following the collapse of the progenitor massive star.
Thus far, attempts to characterise GRB LCs focused on a number of properties,
such as the minimum variability timescale, power density spectra (both
ensemble average and individual), or through different definitions of
variability. In parallel, a characterisation as a stochastic process was
pursued by studying the distributions of waiting times, peak flux, fluence of
individual peaks that can be identified within GRB time profiles. Yet, the
question remains as to whether the diversity of GRB profiles can be described
in terms of a common stochastic process.

Aims. Here we address this issue by extracting and modelling for the first
time the distribution of the number of peaks within a GRB profile.

Methods. We analysed four different GRB catalogues: CGRO/BATSE, Swift/BAT,
BeppoSAX/GRBM, and Insight-HXMT. The statistically significant peaks were
identified by means of well tested and calibrated algorithm mepsa and further
selected by applying a set of thresholds on signal-to-noise ratio. We then
extracted the corresponding distributions of number of peaks per GRB.

Results. Among the different models considered (power-law, simple or stretched
exponential) only a mixture of two exponentials turned out to model all the
observed distributions, suggesting the existence of two distinct behaviours:
(i) an average number of $2.1\pm 0.1$ peaks per GRB (“peak poor”) and
accounting for about 80% of the observed population of GRBs; (ii) an average
number of $8.3\pm 1.0$ peaks per GRB (“peak rich”) and accounting for the
remaining 20% of the observed population.

Conclusions. We associate the class of peak-rich GRBs with the presence of
sub-second variability, which instead appears to be absent among peak-poor
GRBs. The two classes could result from two different regimes through which
GRB inner engines release energy or through which energy is dissipated into
gamma-rays.

###### Key Words.:

(Stars:) Gamma-ray burst: general – Methods: statistical

##  1 Introduction

Gamma–ray burst (GRB) prompt emission is observed in at least two kinds of
explosive transients: (i) the merger of a compact object binary system
(Eichler et al., 1989; Paczynski, 1991; Narayan et al., 1992; Abbott et al.,
2017), (ii) the core collapse of some kinds of massive stars also known as
“collapsar” (Woosley, 1993; Paczyński, 1998; MacFadyen & Woosley, 1999; Yoon &
Langer, 2005). While the prompt emission duration remains the strongest hint
on the nature of the progenitor, in a few cases it turned out to be deceitful.
To avoid confusion, the two progenitor families (i) and (ii) are often
referred to as Type-I and Type-II GRBs, respectively (Zhang, 2006). Hereafter,
we will consider only the latter class.

Modelling of thousands of GRB prompt emission time-resolved spectra proved
that the observed variety is likely the result of different radiative
processes in different GRBs, given the variety of spectral components that are
observed: the non-thermal Band function, an occasional quasi-thermal
component, and even a broad-band power-law (see Kumar & Zhang 2015 for a
review). These processes are possibly related to different dissipation
mechanisms taking place at different distances from the progenitor and driven
by key properties, such as the ejecta composition of the relativistic jet that
is launched by the GRB inner engine.

In parallel, while numerous studies that focused on the temporal properties of
GRB time profiles have contributed to characterise their variability, the
large variety and highly erratic nature of GRB prompt emission remains mostly
unexplained and apparently disconnected from the properties that are inferred
from the afterglow modelling, apart from the energetics. Despite being
affected by a large scatter, a positive correlation between peak luminosity
and variability was found as soon as the number of bursts with measured
redshift increased to a dozen or more (Reichart et al., 2001; Fenimore &
Ramirez-Ruiz, 2000). Variability was defined as the net (that is, removed of
the counting statistics noise contribution) variance of the light curve (LC)
with respect to a smoothed version of the same. The correlation was later
confirmed over larger data sets, along with its large scatter (Guidorzi et
al., 2005; Guidorzi, 2005; Guidorzi et al., 2006), with a forthcoming analysis
of Swift and Fermi GRBs (Guidorzi et al. in prep.). Peak luminosity was also
found to correlate with so-called minimum variability timescale (MVT), defined
as the shortest timescale over which an uncorrelated flux change is observed
significantly in excess of statistical fluctuations (see Camisasca et al. 2023
and references therein). A possible interpretation explains more variable GRBs
as the result of narrower jets with higher Lorentz factors and shorter
spreading timescales than wider jets (Kobayashi et al., 2002).

Most GRB LCs can be described as a sequence of peaks of different shapes,
duration, and intensity with no clear rule. While different techniques have
revealed that temporal power is distributed in general over a broad range of
timescales (from $\sim 10$ ms to several $10$ s), two distinct components were
also identified in a number of GRBs: slow ($\gtrsim 1$ s), and fast (down to
few $10$ ms; Vetere et al. 2006; Gao et al. 2012; Guidorzi et al. 2016). Since
the first catalogues of the Burst And Transient Source Experiment (BATSE;
Paciesas et al. 1999) that flew aboard the Compton Gamma–Ray Observatory
(CGRO; 1991–2000), distributions of properties of the peaks, seen as building
blocks of the prompt emission, were studied in detail (e.g., see Ramirez-Ruiz
& Merloni 2001; Quilligan et al. 2002; Nakar & Piran 2002). Yet, despite the
enormous progress in the knowledge of GRB progenitors and their environments,
a precise characterisation of the stochastic process(es) that drive(s) GRB
engines in releasing their energy as a function of time is missing. Hence, the
degree of complexity exhibited by a given GRB LC, especially if it is seen as
a point process given by the sequence of peaks with different intensities and
durations, can hardly be translated into clues on the nature of its inner
engine or on the nature of the powering mechanism that rules its evolution.

In this respect, while the distribution of the number of peaks per GRB was
obtained in past investigations (e.g., Quilligan et al. 2002; Guidorzi et al.
2016), to our knowledge no attempt to model and interpret it has been done to
date. Following previous analogous definitions (Li & Fenimore, 1996; Nakar &
Piran, 2002), a “peak” is defined as a statistically significant increase of
the count rate with respect to the neighbouring time bins, using the
terminology introduced by Li & Fenimore (1996). In this work we addressed this
by applying a well tested peak identification code, mepsa (Guidorzi, 2015), to
four independent catalogues among the different past and present missions
aimed at GRB studies (Tsvetkova et al., 2022): CGRO/BATSE, Swift/BAT,
BeppoSAX/GRBM, and Insight-HXMT. A forthcoming paper will report the still on-
going analysis of Fermi/GBM GRBs, some preliminary results of which will be
mentioned in what follows. We point out that our goal deliberately ignores the
temporal structures of peaks, which then might differ significantly from one
another, or how much they overlap in time, provided that they can be
identified as distinct peaks. Section 2 reports the data sets used, the LC
extraction along with the peak identification. Section 3 reports the results
of the analysis, which are then discussed in Section 4.

##  2 Data sets

We obtained the background-subtracted LC of GRBs with a bintime of 64 ms
(except for BeppoSAX/GRBM for which we made use of both 1-s and $62.5$-ms bin
times) and extracted a list of peaks detected with mepsa, a code specifically
devised and optimised to identify peaks in uniformly sampled and background-
subtracted GRB LCs (Guidorzi, 2015), discarding candidates with a signal-to-
noise ratio S/N $<5$.

From the BATSE 4B catalogue (Paciesas et al., 1999) we took the 64-ms time
profiles of LGRBs ($T_{90}>2$ s) made available by the BATSE
team111https://heasarc.gsfc.nasa.gov/FTP/compton/data/batse/ascii_data/64ms/.
Taken with the eight BATSE Large Area Detectors (LADs), these time profiles
resulted from the concatenation of three standard BATSE types, DISCLA, PREB,
and DISCSC. They are available in four energy channels, 25–55, 55–110,
110–320, and $>320$ keV: we used the summed counts over the four channels. The
background was interpolated with polynomials of up to fourth degree as
prescribed by the BATSE team. These background-subtracted profiles were then
processed with mepsa and we selected the GRBs with at least one significant
peak. We ended up with a sample of 1457 GRBs, which hereafter are referred to
as the BATSE sample.

We started from all the GRBs detected by Swift/BAT in burst mode from January
2005 to July 2023 and rejected those with $T_{90}\leq 2$ s as well as the
long-lasting Type-I candidates, as they were classified by the Swift team:
they include all the so-called short extended-emission GRBs (Norris & Bonnell,
2006), in addition to peculiar events like 060614, 211211A (Gehrels et al.,
2006; Yang et al., 2022). The information on the $T_{90}$ duration and on the
LC characterisation was taken either from the BAT3 catalogue (Lien et al.,
2016) when available, or from the Swift/BAT team circulars. For each burst we
extracted the mask-weighted, 64-ms bin time profile in the 15–150 keV passband
following the standard procedure recommended by the BAT
team222https://swift.gsfc.nasa.gov/analysis/threads/bat_threads.html. and
applied mepsa. Imposing the above mentioned S/N$>5$ threshold on the peak
candidates, we selected all the Type-II GRB candidates with at least one peak
and ended up with 1277 bursts. Hereafter, this will be referred to as the BAT
sample.

The LCs available in the 40–700 keV passband for the GRBs from the
BeppoSAX/GRBM catalogue (Frontera et al., 2009) come after two fashions: for
the 2/3 that triggered the on-board logic, a $7.8125$-ms profile covering the
first 106 s for each of the four independent units is available. For the
remaining 1/3 only 1-s LCs are available. For the former we obtained $62.5$-ms
profiles. The GRBs observed before November 1996 were ignored due to the
presence of extra-Poissonian noise in the background rate of the GRBM, that
was removed after raising the LLT energy threshold (Frontera et al., 2009). We
then discarded short GRBs ($T_{90}<2$ s, as reported by Frontera et al. 2009)
and ran mepsa on the LC of the most illuminated detector (or on the sum of the
two most illuminated ones, depending on which LC had the higher S/N)
independently on both $62.5$-ms and 1-s profiles and merged the results,
making sure to count once the peaks that triggered both timescales. The reason
of this strategy is not to lose peaks that occurred after the first 106 s
acquired with high temporal resolution. We applied the same $S/N>5$ threshold
as for the other catalogues and ended up with 820 GRBs with at least one
significant peak, which hereafter are referred to as the BeppoSAX sample.

From the first Insight-HXMT/HE GRB catalogue (Song et al., 2022) we discarded
all the GRBs with $T_{90}<2$ s as well as those belonging to the iron sample,
for which the electronics saturated due to excessively high count rates. For
the remaining GRBs we extracted the 64-ms, dead-time corrected LCs obtained by
summing the counts of all the 18 HE detector units, selecting only the CsI
events (for which the HE works as an open-sky monitor) following the procedure
described in Song et al. (2022). The total energy passband depends on the HE
operation mode: either 80–800 or 200–3000 keV for the normal and GRB (or low-
gain) mode, respectively. The background interpolation and subtraction was
carried out as in Camisasca et al. (2023), to which the reader is referred. We
finally ran mepsa and after applying the same S/N$>5$ selection we ended up
with 202 GRBs hosting at least one significant peak. Hereafter these GRBs are
referred to as the Insight-HXMT sample.

##  3 Results

For each data set we obtained the distribution of the number of peaks per GRB,
expressed as number of GRBs having $n$ peaks, as a function of $n$ for all
integer values ($n\geq 1$). This is clearly a discrete distribution. For each
of the models considered below, we first determined the best fit parameters by
maximising the corresponding likelihood function. We then estimated the
parameters’ marginalised distributions by sampling the posterior distribution
using Markov Chain Monte Carlo (MCMC) sampling with emcee (Foreman-Mackey et
al., 2013): starting from the maximum likelihood solution, we run 32 walkers
for 5000 steps and discard the ﬁrst 100 steps as burn-in. We tested the
goodness of the best solution by grouping the expected number of GRBs so as to
have at least 5 per class and carried out a $\chi^{2}$ test.

We first tried to model the distributions with a discrete simple exponential,
which is mostly determined by the majority of GRBs, which have 1 or 2 peaks.
However, all data sets showed an excess of GRBs with many ($\gtrsim 10$)
peaks. Motivated by the possible presence of a hard tail, we then considered a
simple power-law model, but the result was poor. A broken power-law did not
provide satisfactory fits for all data sets either.

As a further attempt, we considered a stretched exponential (also known as the
empirical Kohlrausch-Williams-Watts, or KWW, function), which is often
encountered in several contexts in physics for modelling relaxation processes
of systems that dissipate energy and are initially out of equilibrium (e.g.
see Lukichev 2019 and references therein). It can also result from the
combination of numerous (ideally continuous) exponentially distributed
processes (Johnston, 2006). Moreover, a stretched exponential was found to
describe the average temporal decay of peak-aligned and peak-normalised
profiles of BATSE GRBs (Stern & Svensson, 1996). As a result, only the
distribution of the BATSE sample, which is the most numerous one, could not be
modelled satisfactorily with a stretched exponential ($\chi^{2}$ test p-value
of $10^{-6}$).

Table 1: Best fit values and 90% confidence intervals of the parameters of the
mixture model of two exponentials (Eq. 1) applied to the different data sets.

Catalog | $N^{\rm{(a)}}$ | $N_{p}^{\rm{(b)}}$ | $k$ | $n_{1}$ | $\langle n^{(1)}\rangle^{\rm{(c)}}$ | $n_{2}$ | $\langle n^{(2)}\rangle^{\rm{(c)}}$ | $\xi$ | $\bar{w}_{2}^{\rm{(d)}}$ |  $\chi^{2}$ test
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  | p-value
BATSE | 1457 | 8112 | $0.33_{-0.03}^{+0.02}$ | $2.89_{-0.26}^{+0.29}$ | $3.42_{-0.26}^{+0.29}$ | $13.4_{-2.1}^{+3.0}$ | $13.9_{-2.1}^{+3.0}$ | $0.049_{-0.018}^{+0.027}$ | $0.21_{-0.05}^{+0.06}$ | $0.164$
BAT | 1277 | 3817 | $0.78\pm 0.08$ | $1.47\pm 0.17$ | $2.03\pm 0.17$ | $6.4_{-1.2}^{+1.5}$ | $6.9_{-1.2}^{+1.5}$ | $0.043_{-0.017}^{+0.029}$ | $0.20_{-0.06}^{+0.08}$ | $0.845$
BeppoSAX | 820 | 2151 | $1.02_{-0.12}^{+0.13}$ | $1.29_{-0.14}^{+0.15}$ | $1.85_{-0.13}^{+0.15}$ | $7.6_{-1.9}^{+3.4}$ | $8.1_{-1.9}^{+3.4}$ | $0.019_{-0.011}^{+0.015}$ | $0.13_{-0.05}^{+0.06}$ | $0.440$
HXMT | 202 | 616 | $0.99_{-0.46}^{+0.71}$ | $0.85_{-0.23}^{+1.02}$ | $1.45_{-0.20}^{+0.96}$ | $4.3_{-1.2}^{+3.8}$ | $4.8_{-1.2}^{+3.8}$ | $0.107_{-0.076}^{+0.219}$ | $0.43_{-0.26}^{+0.23}$ | $0.600$
All(e) | 3756 | 14696 | $0.59\pm 0.03$ | $1.81_{-0.10}^{+0.11}$ | $2.36\pm 0.10$ | $9.6_{-0.9}^{+1.2}$ | $10.1_{-0.9}^{+1.2}$ | $0.037_{-0.008}^{+0.010}$ | $0.20\pm 0.03$ | $0.033$
W. Aver.(f) | – | – | – | – | $2.1\pm 0.1$ | – | $8.3\pm 0.1$ | – | $0.19\pm 0.03$ | –

  * (a)

Number of GRBs.

  * (b)

Total number of peaks.

  * (c)

Expected number of peaks per GRB of the corresponding exponential component,
calculated as in Eq. (2).

  * (d)

Defined in Eq. (3), it is the total fraction of GRBs contributed by the second
exponential, characterised by a higher average number of peaks per GRB.

  * (e)

All data sets have been merged together and treated as one sample.

  * (f)

Weighted average value using the different estimates of the four catalogues.

---
Figure 1: Distributions of number of GRBs per number of peaks for different
catalogues: CGRO/BATSE (top left), Swift/BAT (top right), BeppoSAX/GRBM
(bottom left), Insight-HXMT (bottom right). Observed data and expected counts
are shown in orange and blue, respectively. All histograms have been grouped
so as to ensure a minimum number of counts per bin. Dashed lines show the best
mixture model of two exponentials that fit each distribution. Grey models are
the result of a random sampling of the posterior distribution of the
parameters as determined via MCMC.

We finally considered the superposition of two simple exponentials with a
relative weight treated as a free parameter. Specifically, the fraction $f(n)$
of GRBs hosting $n$ peaks is described by Eq. (1):

$f(n)\ =\ k\ ({e^{-n/n_{1}}\ +\ \xi\,e^{-n/n_{2}}})\;,$ (1)

where $k$ is a normalisation constant, such that $\sum_{n=1}^{+\infty}f(n)=1$,
$n_{i}$ ($i=1,2$) is the characteristic number of peaks per GRB of the $i$-th
component ($n_{1}\leq n_{2}$ by convention) and $\xi$ is a relative
normalisation parameter, thus totalling 3 free parameters. Due to the discrete
nature of the distribution, the corresponding expected number of peaks per GRB
of the $i$-th exponential component, $\langle n^{(i)}\rangle$, is not simply
$n_{i}$ as in the case of a continuous distribution, but the following:

$\langle n^{(i)}\rangle\ =\
\frac{\sum_{n=1}^{+\infty}ne^{-n/n_{i}}}{\sum_{n=1}^{+\infty}e^{-n/n_{i}}}\
=\frac{1}{1-e^{-1/n_{i}}}\;.$ (2)

Hereafter, Equation (1) is referred to as the mixture model of two
exponentials (M2E).

The resulting distributions along with the best fit M2Es are shown in Figure
1, while Table 1 reports the best fit values of the model parameters and the
corresponding p-value of the $\chi^{2}$ test. All of the four data sets can be
satisfactorily modelled and, what is more, the parameters values are similar
to each other, despite the different effective areas and energy passbands. We
also performed a joint fit of all the four distributions with a common set of
parameters in two ways: (i) by merging all GRBs and treating it as a single
data set, thus dominated by the richest data sets; (ii) by keeping the four
distributions separate and by assigning equal weights in the total likelihood.
Only for (i) we obtained a marginally acceptable solution (p-value of 3%, see
Table 1).

We also calculated the fraction of GRBs $\bar{w}_{2}$ that are contributed by
the second exponential component, according to each best fit solution: this is
calculated simply as

$\bar{w}_{2}\ =\ k\ \xi\sum_{n=1}^{+\infty}e^{-n/n_{2}}\;.$ (3)

Interestingly, all data sets have compatible values, with a weighted average
of $0.19\pm 0.03$ (90% confidence). Analogous results are found for the
expected numbers of peaks of either component: while the first component
presents a range of values for $\langle n^{(1)}\rangle$ only marginally
compatible with each other (weighted average of $2.1\pm 0.1$, p-value of
$6\times 10^{-6}$), the expected value for the second component, $\langle
n^{(2)}\rangle$, is compatible among all data sets, with a weighted average of
$8.3\pm 1.0$ (p-value of $0.06$). Hereafter, the first ($i=1$) and the second
($i=2$) components are referred to as the peak-poor and the peak-rich ones,
respectively.

Figure 2: $T_{90}$ vs. number of peaks per GRBs for the different data sets.
Also shown are the $T_{90}$ median (black) and the geometric mean (olive) as a
function of number of peaks, grouped with at least 9 GRBs each. The dashed-
dotted line (magenta) shows a broken power-law fit of the median behaviour.

Similar results were obtained on a preliminary sample of 393 Fermi Gamma–ray
Burst Monitor (GBM; Meegan et al. 2009) GRBs, 361 of which are shared with the
BAT sample (Maccary et al. in prep.): in particular, we found a peak-rich
fraction $\bar{w}_{2}=0.19_{-0.09}^{+0.13}$, in line with the other four
samples.

We investigated the relation between number of peaks per GRB $n$ and burst
duration as expressed in terms of $T_{90}$, whose values were taken from the
official corresponding catalogue papers. Figure 2 shows the results for all
catalogues. We studied how the median and the logarithmic mean of $T_{90}$ of
all catalogues merged together vary as a function of $n$. To avoid low-count
fluctuations in the median and mean estimates, we grouped bins of $n$ so as to
have at least 9 GRBs per class. The two quantities clearly track each other
very closely (Fig. 2). Interestingly, the dependence of $T_{90}$ on $n$
becomes shallower for $n\gtrsim n_{2}$: modelling the median value of
$T_{90}$, $\tilde{T}_{90}$, with a power-law, $\tilde{T}_{90}/{\rm
s}\simeq\kappa\,n^{a}$, we found $a=0.58$ ($\kappa=18.8$) for $n<n_{2}$, while
$a=0.39$ ($\kappa=23.0$) for $n>n_{2}$ (dash-dotted magenta line in Fig. 2).
The fact that $a$ significantly differs from 1 suggests that the abundance of
peaks within a GRB is not compatible with a common probability per unit time
of emitting a peak, which would predict $a\sim 1$. Moreover, the dynamic range
of $\tilde{T}_{90}$ (19 s for $n=1$, 57 s for $50<n<85$) is significantly
narrower than the scatter of $T_{90}$ for any $n$: this indicates that
$T_{90}$ is not a main driver of $n$. In addition, the fact that the
dependence of $\tilde{T}_{90}$ becomes even shallower for large values $n$,
where most GRBs are contributed by the peak-rich component of Eq. 1, further
supports the possible existence of two families of GRBs or, at least, two
different behaviours of GRB engines.

###  3.1 Instrumental selection effects

The number of peaks clearly depends on the detector sensitivity and on the
threshold on S/N. Secondly, it also depends on the energy passband, given that
harder channels exhibit narrower and spikier temporal structures than softer
channels. This is clearly shown by the higher number of peaks in the BATSE
sample compared with the other samples and is due to the better sensitivity of
the former instrument.

We investigated the impact of S/N by repeating the analysis assuming two
additional threshold values: S/N$>7$ and S/N$>9$. The total number of
identified peaks decreased to a fraction between 70 and 85% for the different
catalogues, when moving from S/N$>5$ to S/N$>9$. Only the average number of
peaks per GRB of the BATSE sample decreased from $5.6$ to $4.9$, whereas for
all the other samples the same quantity remained in the range $2.8$–$3.0$
peaks per GRB: the reason is that for the other three samples the number of
GRBs featuring at least one significant peak decreased proportionally, due to
lower sensitivity with respect to BATSE. Because of this, the weighted average
number of peaks per GRB of both peak-poor and peak-rich components did not
change significantly: $\langle n^{(1)}\rangle$ passed from $2.1\pm 0.1$
(S/N$>5$) to $2.0\pm 0.1$ (S/N$>9$). $\langle n^{(2)}\rangle$ changed from
$8.3\pm 1.0$ (S/N$>5$) to $8.1\pm 1.0$ (S/N$>9$). Finally, the fraction of
peak-rich GRBs moved from $0.19\pm 0.03$ (S/N$>5$) to $0.22\pm 0.03$
(S/N$>9$). Overall, while the number of peaks per GRB is inevitably expected
to decrease by keeping raising the S/N threshold (although in a limited way,
as we found passing from 5 to 9), the fractions of peak-poor and peak-rich
GRBs do not seem to depend sensitively on the S/N threshold.

We identified 160 GRBs in common between the BATSE and the BeppoSAX samples.
For each of these GRBs we calculated the difference between the number of
peaks detected from the two profiles, $\Delta n=n^{\rm(BATSE)}-n^{\rm(SAX)}$.
We divided this common sample into two subsamples: the GRBs for which there
are high-resolution (HR; $62.5$-ms) data, and the GRBs with only 1-s data from
BeppoSAX. Figure 3 shows the distributions of $\Delta n$ for different S/N
thresholds.

Figure 3: Distributions of $\Delta n$, defined as the difference between the
number of peaks detected with BATSE and the one detected with BeppoSAX for a
common sample of 160 GRBs, for three different thresholds on S/N. Blue and
orange show the subsamples of GRBs with/without BeppoSAX high-resolution data,
respectively.

First, the distributions do not depend significantly on the S/N threshold.
Secondly, on average the presence of HR data from BeppoSAX does not impact
appreciably the number of detected peaks. Thirdly, the median values in all
cases is $+2$, which reflects the better sensitivity of BATSE. In addition,
$\sim 60$% ($\sim 70$%) of all common GRBs have $|\Delta n|\leq 2$ ($|\Delta
n|\leq 3$). A discrepancy of $\sim 2$–$3$ in estimating $n$ is small compared
with $\langle n^{(2)}\rangle-\langle n^{(1)}\rangle\simeq 6.2\pm 1.0$. We
derived the analogous distribution for the preliminary common sample of 361
GRBs shared by BAT and Fermi/GBM, confirming that for the majority of bursts
the difference in the number of peaks is small: 86% and 92% GRBs with $|\Delta
n|\leq 2$ and $|\Delta n|\leq 3$, respectively (Maccary et al. in prep.).
Consequently, for most GRBs the probability of belonging to either the peak-
poor or the peak-rich family does not crucially depend on the detector used,
at least as long as these two catalogues are concerned.

A more thorough characterisation through mapping with other GRB observed
properties goes beyond the scope of the present analysis and would deserve a
separate investigation.

###  3.2 Number of peaks vs. peak luminosity

According to the variability-luminosity relation, more luminous GRBs are, on
average, more variable. We therefore studied the relation between number of
peaks and peak luminosity. To this aim, we selected 17 GRBs from the BeppoSAX
and 131 from the BAT samples with spectroscopically measured redshift and for
which the isotropic-equivalent peak luminosity $L_{\rm p,iso}$ in the
comoving-frame $1$–$10^{4}$ keV passband had been estimated with broadband
experiments, such as BeppoSAX and Konus/WIND. These values were taken from
Ghirlanda et al. (2005); Amati et al. (2008); Yonetoku et al. (2010); Dichiara
et al. (2016); Tsvetkova et al. (2017, 2021).

Figure 4: Number of peaks vs. isotropic-equivalent peak luminosity $L_{\rm
p,iso}$ for a sample of 131 (17) GRBs with known $z$ from the BAT (BeppoSAX)
samples. The dashed line marks the median number of peaks of the whole sample.

Figure 4 shows the result. Non-parametric Spearman’s and Kendall’s correlation
tests over the joint sample of 148 GRBs yielded a null hypothesis probability
of no correlation as small as $1.1\times 10^{-7}$ and $2.0\times 10^{-7}$,
respectively. Hence, even though it is highly dispersed, the correlation is
statistically significant, with peak-rich GRBs being, on average, more
luminous.

As a further check, we also split the sample based on the median number of
peaks of $n=3$ of the joint sample, ending up with 79 (69) GRBs with $n\leq 3$
($n>3$) peaks. The two groups can be roughly considered as peak-poor and peak-
rich samples. Median peak luminosities are $1.5\times 10^{52}$ and $6.8\times
10^{52}$ erg s-1 for the peak-poor and peak-rich samples, respectively. The
probability of a common parent population for the two distributions of $L_{\rm
p,iso}$ is $9\times 10^{-5}$ and $<10^{-3}$ according to the Kolmogorov-
Smirnov and Anderson-Darling tests,333These tests were done using
scipy.stats.ks_2samp and scipy.stats.anderson_ksamp, respectively.
respectively. In conclusion, robust evidence is found that peak-rich GRBs are,
on average, more luminous than peak-poor ones, although the two peak
luminosity distributions are strongly overlapped.

##  4 Discussion and conclusions

The analysis of a former Swift/BAT sample of GRB LCs showed that the power
density spectra of peak-rich GRBs tend to be preferentially described by a
shallow PL, which is the result of the superposition of peaks covering a broad
range of timescales, such that no dominant timescale emerges (see Figure 7 of
Guidorzi et al. 2016). In particular, peak-rich GRBs tend to have relatively
more temporal power on shorter ($\lesssim 1$ s) timescales than peak-poor GRBs
due to the presence of a number of narrow peaks which are mostly absent in
peak-poor GRBs. An illustrative example of this difference is shown in Figure
5: two bursts from the BeppoSAX/GRBM catalogue having comparable fluence and
peak flux, one with just 2 peaks and a long smooth decaying tail ($T_{90}\sim
140$ s), and the other featuring 24 peaks within a $T_{90}=60$ s, most of
which have subsecond durations.

Figure 5: Examples of two GRBs having only 2 (left, 981203A) and 24 peaks
(right, 010412), respectively, as observed in the 40–700 keV passband by
BeppoSAX/GRBM. The red points show the identified peaks, while dashed lines
show the interpolated background. The fluence and peak flux of the two GRBs
are very similar (Frontera et al., 2009).

In light of the results here reported on the possible existence of two
distinct dynamics, as revealed by the distribution of number of peaks per GRB,
the emerging picture is suggestive of two kinds of regimes through which Type-
II GRB engines release their energy: (i) one emitting a few ($\sim 2.1$ on
average), typically broad peaks ($\gtrsim 1$ s), makes up $\sim 80$% of the
observed population of GRBs of past and present catalogues; (ii) the other one
emitting several peaks ($\sim 8.3$ on average), covering a broader range of
timescales, including subsecond ones, which makes up the remaining 20% of the
observed population. The coexistence of two kinds of timescales in some GRBs,
the so-called slow and fast variability components, was already suggested in
the literature (Vetere et al., 2006; Margutti, 2009; Gao et al., 2012;
Guidorzi et al., 2016): our results suggest a further characterisation, which
points to the existence of two distinct ways GRB engines may work.

One can look at the class of peak-rich GRBs as being characterised by the
presence of a fast component that manifests itself through a number of narrow
peaks, which are instead mostly missing in the other class of peak-poor GRBs.
Peak richness is also preferentially accompanied, on average, by shorter MVT
and higher peak luminosity, although the latter alone cannot be used to
discriminate between the two classes, given the strong overlap shown in Fig.
4. The origin of the fast component in the literature has been investigated
within the context of different progenitor models and can be summarised in two
main groups, depending on whether the fast component (a) reflects the inner
engine activity or (b) is mainly driven by other factors. Within the context
of the internal shock (IS) model (Kobayashi et al., 1997; Daigne &
Mochkovitch, 1998; Maxham & Zhang, 2009), the number of peaks and the sequence
of time intervals simply reflect the emission time history of the inner
engine. Unlike the slow one, the fast component imprinted by the inner engine
would not be quenched, as a purely hydrodynamic jet propagates through the
stellar envelope (Morsony et al., 2010). When the possibility of a magnetised
outflow is considered, a weakly magnetised and intermittent jet, which is less
subjected to mixing with the surrounding progenitor gas than a pure
hydrodynamic jet, appears more promising to explain the observed variability,
since the predicted GRB emission would be less inhibited (Gottlieb et al.,
2020b, 2021).

Within the (a) scenario, our results suggest that either inner engines of
peak-poor GRBs are quieter, especially at short timescales, or their
progenitor structure is able to suppress the subsecond variability. This could
be due to the different degree of magnetisation of the jet, which can affect
the mixing and consequently the jet baryon load (Gottlieb et al., 2020a).
There are a number of interpretations of how a GRB engine like an accreting
black hole (BH) could explain the fast component: for example, through
fluctuations of the viscosity parameter and how they consequently propagate
through the disk in a similar way to what is believed in the case of BH
binaries and active galactic nuclei (Lin et al., 2016). Alternatively, they
could be caused by the fastest modes of magneto-rotational instabilities that
drive and launch a magnetised jet through the Blandford–Znajek process (Janiuk
et al., 2021). Within these contexts, peak-poor GRBs could be indicative of a
regime in which either viscosity fluctuations or fast instabilities are
strongly inhibited.

Within the alternative (b) scenario, the fast component that characterises
peak-rich GRBs originates at further distance from the inner engine, up to the
GRB emission site. In the Internal-Collision induced MAgnetic Reconnection and
Turbulence (ICMART) model (Zhang & Yan, 2011), a wind of moderately magnetised
shells (with magnetisation $1\lesssim\sigma\lesssim 100$ at the GRB site)
collide and produce several runaway cascades of magnetic reconnection events.
The fast component would be the result of turbulent, Doppler-boosted emission
of mini-jets within the comoving frame of the colliding shells, caused by
particles accelerated by magnetic reconnections. Simulations carried out by
Zhang & Zhang (2014) show that spikier GRB LCs with many distinguishable peaks
are, in particular, favoured by the following factors: (1) large emission site
($R\gtrsim 10^{15}$ cm) from the progenitor giving a longer curvature decay
tail, which thus does not wash out the rapid variability due to the mini-jets;
(2) a more magnetised outflow, which corresponds to a higher contrast between
the Lorentz factors of the mini-jets and that of the corresponding shell. Our
results could suggest that the turbulent magnetic reconnection regime either
sets in efficiently (peak-rich GRB), or it does not (peak-poor GRB).

Ultimately, the possibility remains that the two classes reflect two different
kinds of inner engines, such as ms-magnetars and accreting BHs, for which
other potentially discriminating signatures in the prompt emission are yet to
be identified.

A final note concerns the long-lasting GRBs that are Type-I candidates, such
as 060614, 211211A, and 230307A (Gehrels et al., 2006; Yang et al., 2022;
Troja et al., 2022; Gompertz et al., 2023; Dichiara et al., 2023; Levan et
al., 2023), which were ignored in the present analysis: all their LCs exhibit
numerous peaks (several dozens). While peak-richness alone cannot be a
distinctive feature of this kind of merger candidates, given that there are
several Type-II GRBs with an associated SN that are also peak-rich (e.g.,
080319B, Racusin et al. 2008), nevertheless it could be interpreted as a clue
on the nature of the inner engine that powers these mergers.

###### Acknowledgements.

We acknowledge the anonymous reviewer for a constructive report which improved
the manuscript. C.G. acknowledges the Dept. of Physics and Earth Science of
the University of Ferrara for the financial support through the “FIRD 2022”
grant. M.B. is supported by the Dept. of Physics and Earth Science at the
University of Ferrara through the FIRD 2022 grant. A.T. acknowledges financial
support from “ASI-INAF Accordo Attuativo HERMES Pathinder operazioni n.
2022-25-HH.0”.

## References

  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, ApJ, 848, L13
  * Amati et al. (2008) Amati, L., Guidorzi, C., Frontera, F., et al. 2008, MNRAS, 391, 577
  * Camisasca et al. (2023) Camisasca, A. E., Guidorzi, C., Amati, L., et al. 2023, A&A, 671, A112
  * Daigne & Mochkovitch (1998) Daigne, F. & Mochkovitch, R. 1998, MNRAS, 296, 275
  * Dichiara et al. (2016) Dichiara, S., Guidorzi, C., Amati, L., Frontera, F., & Margutti, R. 2016, A&A, 589, A97
  * Dichiara et al. (2023) Dichiara, S., Tsang, D., Troja, E., et al. 2023, ApJ, 954, L29
  * Eichler et al. (1989) Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989, Nature, 340, 126
  * Fenimore & Ramirez-Ruiz (2000) Fenimore, E. E. & Ramirez-Ruiz, E. 2000, ArXiv Astrophysics e-prints [astro-ph/0004176]
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Frontera et al. (2009) Frontera, F., Guidorzi, C., Montanari, E., et al. 2009, ApJS, 180, 192
  * Gao et al. (2012) Gao, H., Zhang, B.-B., & Zhang, B. 2012, ApJ, 748, 134
  * Gehrels et al. (2006) Gehrels, N., Norris, J. P., Barthelmy, S. D., et al. 2006, Nature, 444, 1044
  * Ghirlanda et al. (2005) Ghirlanda, G., Ghisellini, G., & Firmani, C. 2005, MNRAS, 361, L10
  * Gompertz et al. (2023) Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2023, Nature Astronomy, 7, 67
  * Gottlieb et al. (2021) Gottlieb, O., Bromberg, O., Levinson, A., & Nakar, E. 2021, MNRAS, 504, 3947
  * Gottlieb et al. (2020a) Gottlieb, O., Bromberg, O., Singh, C. B., & Nakar, E. 2020a, MNRAS, 498, 3320
  * Gottlieb et al. (2020b) Gottlieb, O., Levinson, A., & Nakar, E. 2020b, MNRAS, 495, 570
  * Guidorzi (2005) Guidorzi, C. 2005, MNRAS, 364, 163
  * Guidorzi (2015) Guidorzi, C. 2015, Astronomy and Computing, 10, 54
  * Guidorzi et al. (2016) Guidorzi, C., Dichiara, S., & Amati, L. 2016, A&A, 589, A98
  * Guidorzi et al. (2005) Guidorzi, C., Frontera, F., Montanari, E., et al. 2005, MNRAS, 363, 315
  * Guidorzi et al. (2006) Guidorzi, C., Frontera, F., Montanari, E., et al. 2006, MNRAS, 371, 843
  * Janiuk et al. (2021) Janiuk, A., James, B., & Palit, I. 2021, ApJ, 917, 102
  * Johnston (2006) Johnston, D. C. 2006, Phys. Rev. B, 74, 184430
  * Kobayashi et al. (1997) Kobayashi, S., Piran, T., & Sari, R. 1997, ApJ, 490, 92
  * Kobayashi et al. (2002) Kobayashi, S., Ryde, F., & MacFadyen, A. 2002, ApJ, 577, 302
  * Kumar & Zhang (2015) Kumar, P. & Zhang, B. 2015, Phys. Rep., 561, 1
  * Levan et al. (2023) Levan, A. J., Malesani, D. B., Gompertz, B. P., et al. 2023, Nature Astronomy, 7, 976
  * Li & Fenimore (1996) Li, H. & Fenimore, E. E. 1996, ApJ, 469, L115
  * Lien et al. (2016) Lien, A., Sakamoto, T., Barthelmy, S. D., et al. 2016, ApJ, 829, 7
  * Lin et al. (2016) Lin, D.-B., Lu, Z.-J., Mu, H.-J., et al. 2016, MNRAS, 463, 245
  * Lukichev (2019) Lukichev, A. 2019, Physics Letters A, 383, 2983
  * MacFadyen & Woosley (1999) MacFadyen, A. I. & Woosley, S. E. 1999, ApJ, 524, 262
  * Margutti (2009) Margutti, R. 2009, Toward new insights on the gamma-ray burst physics: from X-ray spectroscopy to the identification of characteristic time scales (Milan: Università degli Studi Milano-Bicocca, http://boa.unimib.it/handle/10281/7465)
  * Maxham & Zhang (2009) Maxham, A. & Zhang, B. 2009, ApJ, 707, 1623
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ, 702, 791
  * Morsony et al. (2010) Morsony, B. J., Lazzati, D., & Begelman, M. C. 2010, ApJ, 723, 267
  * Nakar & Piran (2002) Nakar, E. & Piran, T. 2002, MNRAS, 331, 40
  * Narayan et al. (1992) Narayan, R., Paczynski, B., & Piran, T. 1992, ApJ, 395, L83
  * Norris & Bonnell (2006) Norris, J. P. & Bonnell, J. T. 2006, ApJ, 643, 266
  * Paciesas et al. (1999) Paciesas, W. S., Meegan, C. A., Pendleton, G. N., et al. 1999, ApJS, 122, 465
  * Paczynski (1991) Paczynski, B. 1991, Acta Astron., 41, 257
  * Paczyński (1998) Paczyński, B. 1998, ApJ, 494, L45
  * Quilligan et al. (2002) Quilligan, F., McBreen, B., Hanlon, L., et al. 2002, A&A, 385, 377
  * Racusin et al. (2008) Racusin, J. L., Karpov, S. V., Sokolowski, M., et al. 2008, Nature, 455, 183
  * Ramirez-Ruiz & Merloni (2001) Ramirez-Ruiz, E. & Merloni, A. 2001, MNRAS, 320, L25
  * Reichart et al. (2001) Reichart, D. E., Lamb, D. Q., Fenimore, E. E., et al. 2001, ApJ, 552, 57
  * Song et al. (2022) Song, X.-Y., Xiong, S.-L., Zhang, S.-N., et al. 2022, ApJS, 259, 46
  * Stern & Svensson (1996) Stern, B. E. & Svensson, R. 1996, ApJ, 469, L109
  * Troja et al. (2022) Troja, E., Fryer, C. L., O’Connor, B., et al. 2022, Nature, 612, 228
  * Tsvetkova et al. (2017) Tsvetkova, A., Frederiks, D., Golenetskii, S., et al. 2017, ApJ, 850, 161
  * Tsvetkova et al. (2021) Tsvetkova, A., Frederiks, D., Svinkin, D., et al. 2021, ApJ, 908, 83
  * Tsvetkova et al. (2022) Tsvetkova, A., Svinkin, D., Karpov, S., & Frederiks, D. 2022, Universe, 8, 373
  * Vetere et al. (2006) Vetere, L., Massaro, E., Costa, E., Soffitta, P., & Ventura, G. 2006, A&A, 447, 499
  * Woosley (1993) Woosley, S. E. 1993, ApJ, 405, 273
  * Yang et al. (2022) Yang, J., Ai, S., Zhang, B. B., et al. 2022, Nature, 612, 232
  * Yonetoku et al. (2010) Yonetoku, D., Murakami, T., Tsutsui, R., et al. 2010, PASJ, 62, 1495
  * Yoon & Langer (2005) Yoon, S. C. & Langer, N. 2005, A&A, 443, 643
  * Zhang (2006) Zhang, B. 2006, Nature, 444, 1010
  * Zhang & Yan (2011) Zhang, B. & Yan, H. 2011, ApJ, 726, 90
  * Zhang & Zhang (2014) Zhang, B. & Zhang, B. 2014, ApJ, 782, 92