11institutetext: School of Physics, Nanjing University, Nanjing 210023,
People’s Republic of China
22institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210023, People’s Republic of China
22email<EMAIL_ADDRESS>33institutetext: School of Astronomy and Space
Science, Nanjing University, Nanjing 210023, People’s Republic of China
44institutetext: Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, Nanjing 210023, People’s Republic of China

# Evolution Patterns of the Peak Energy in the GRB Prompt Emission

Hao-Xuan Gao 11 Jin-Jun Geng 22 Yong-Feng Huang 3344

(Received Month xx, yyyy; accepted Month xx, yyyy)

###### Abstract

Context. There are two different evolution patterns of the peak energy
($E_{\text{p}}$) exhibited during the prompt emission phase of gamma-ray
bursts (GRBs), i.e., hard-to-soft and intensity-tracking, of which the
physical origin remains unknown. Except for low-energy indices of GRB prompt
spectra, the evolution patterns of $E_{\text{p}}$ may be another crucial
indicator to discriminate radiation mechanisms (e.g., synchrotron or
photosphere) for GRBs.

Aims. We explore the parameter space to find conditions that could generate
different evolution patterns of the peak energy in the framework of
synchrotron radiation.

Methods. We have developed a code to calculate the synchrotron emission from a
simplified shell numerically, considering three cooling processes
(synchrotron, synchrotron self-Compton (SSC), and adiabatic) of electrons, the
effect of decaying magnetic field, the effect of the bulk acceleration of the
emitting shell, and the effect of a variable source function that describes
electrons accelerated in the emitting region.

Results. After exploring the parameter space of the GRB synchrotron scenario,
we find that the intensity-tracking pattern of $E_{\text{p}}$ could be
achieved in two situations. One is that the cooling process of electrons is
dominated by adiabatic cooling or SSC+adiabatic cooling at the same time. The
other is that the emitting region is under acceleration in addition to the
cooling process being dominated by SSC cooling. Otherwise, hard-to-soft
patterns of $E_{\text{p}}$ are normally expected. Moreover, a chromatic
intensity-tracking pattern of $E_{\text{p}}$ could be induced by the effect of
a variable source function.

###### Key Words.:

gamma-ray burst: general — radiation mechanisms: non-thermal — relativistic
processes

##  1 Introduction

Gamma-ray bursts (GRBs), the most energetic stellar explosions in the
universe, have been observed and studied for several decades. The prompt
emission of GRBs typically lasts for less than one second to several minutes
and consists of several spikes in the lightcurve, which is thought to come
from the relativistic jet launched from the central compact remnant (Blandford
& Znajek 1977; Eichler et al. 1989; Piran 2004; Kumar & Zhang 2015). There are
still several fundamental issues for GRBs that remain unsolved, e.g., the
composition of GRB jets, the radiation mechanism producing GRBs, and the
distance of the emitting region from the central engine (Zhang 2018).

The spectrum of the prompt emission is often described by the so-called Band
function empirically (Band et al. 1993). Except for the Band component,
evidence has shown that multiple spectral components exist in some GRBs, such
as an additional power-law component (Abdo et al. 2009a; Ackermann et al.
2011) in the high-energy band or the thermal component (Ghirlanda et al. 2003;
Ryde 2005; Ryde & Pe’er 2009; Ryde et al. 2010; Pe’er & Ryde 2011; Guiriec et
al. 2011, 2013; Lazzati et al. 2013; Iyyani et al. 2015; Li 2019; Li et al.
2021). Both synchrotron radiation of electrons (e.g., Mészáros et al. 1994;
Tavani 1996; Zhang & Yan 2011) and photospheric emission (e.g., Rees &
Mészáros 2005; Pe’er et al. 2006; Ryde et al. 2011) have been proposed to
explain the GRB prompt emission. However, it is still under hot debate that
which radiation mechanism is responsible for most GRB spectra.

The Band function is characterized by three parameters, the low-energy and
high-energy photon spectral indices ($\alpha$ and $\beta$) as well as the peak
energy ($E_{\text{p}}$). The “synchrotron line of death problem” (Preece et
al. 1998) involving the statistics of $\alpha$ has been proposed to be against
the synchrotron origin for GRBs, which is later eased by detailed treatment on
cooling processes of electrons (Derishev et al. 2001; Daigne et al. 2011; Uhm
& Zhang 2014; Zhao et al. 2014; Geng et al. 2018b). Meanwhile, some efforts
have been made on the revision of the method that observed spectra can be
modeled (Zheng et al. 2012; Oganesyan et al. 2017, 2018; Ravasio et al. 2018).
It is shown that there is a low-energy break in the prompt spectrum of some
long GRBs. The low-energy break is described by two power-law indices
$\alpha_{1}$ and $\alpha_{2}$, which have distributions centered around -2/3
and -3/2, respectively, and agree with the predictions of the synchrotron
model. Notably, this additional spectral break at low energies was also
confirmed in some short GRBs through a systematic search (Ravasio et al.
2019). Moreover, Burgess et al. (2020) suggest that synchrotron spectra from
electrons in evolving (fast-to-slow) cooling regimes are capable of fitting
$\sim$95% of all time-resolved spectra of the brightest long GRBs observed by
gamma-ray burst monitor (GBM: 8 keV - 40 MeV) on board the NASA Fermi Gamma-
Ray Observatory, by comparing the theoretical results with observed data
directly.

Thus $\alpha$ may not be a good indicator to justify the radiation mechanism
(Burgess et al. 2015; Iyyani et al. 2015; Meng et al. 2018). It was found that
there are two distinct evolution patterns for $E_{\text{p}}$, i.e., hard-to-
soft (Norris et al. 1986) and intensity-tracking(Golenetskii et al. 1983; Bhat
et al. 1994; Kargatis et al. 1994; Ford et al. 1995; Preece et al. 2000; Lu et
al. 2010; Ghirlanda et al. 2011; Lu et al. 2012; Li et al. 2019, 2021). In
addition, Hakkila & Preece (2011) show that all single pulses GRBs could have
the hard-to-soft evolution pattern. Although $E_{\text{p}}$ is also derived
from the Band function, its evolution pattern with a burst may represent an
intrinsic trend rather than an artificial effect like $\alpha$. Revealing the
enigma of these distinct patterns may provide vital clues for understanding
the physical mechanism of the GRB prompt emission.

The evolution patterns of $E_{\text{p}}$ were studied in different
dissipation/radiative processes. Zhang & Yan (2011) proposed that the GRB
prompt emission comes from sudden discharge of magnetic energy through
turbulent magnetic reconnection, in which a hard-to-soft evolution of
$E_{\text{p}}$ throughout a pulse is predicted. In a sophisticated quasi-
thermal GRB photosphere scenario, an intensity-tracking pattern for
$E_{\text{p}}$ could be naturally produced but the hard-to-soft pattern
requires contrived physical condition (Deng & Zhang 2014). Beniamini & Granot
(2016) discussed the temporal/spectral properties of prompt emission from
magnetic reconnection. They found that magnetic reconnections could account
for the hard-to-soft pattern and the intensity-tracking pattern for different
comoving Lorentz factors of the field lines. However, the evolution of the
electron spectra is not incorporated in this work.

In prevailing synchrotron models, the prompt emission comes from a
relativistic jet/outflow launched from the central compact object (e.g.,
Bromberg et al. 2018; Kathirgamaraju et al. 2019; Geng et al. 2019). For a
baryon-dominated jet, internal shocks could efficiently dissipate the relative
kinetic energy of colliding shells into relativistic electrons (Narayan 1992;
Rees & Meszaros 1994; Piran 1999). If the jet is Poynting flux-dominated,
turbulent magnetic reconnections are expected to accelerate electrons in the
jet (Drenkhahn & Spruit 2002; Lyutikov & Blandford 2003; Zhang & Yan 2011).
Uhm et al. (2018) considered a simple physical scenario, in which a thin
relativistic spherical shell expands while emitting photons isotropically in
the shell. Assuming that the shell is under bulk accelerating (Uhm & Zhang
2016; Jia et al. 2016) and that photons emitted obeys an ad-hoc spectral
function without specifying its physical origin, two evolution patterns of
$E_{\text{p}}$ could then be reproduced. However, a robust conclusion could be
achieved only if the modeling of the energy spectrum of electrons is
spontaneously included in the calculations. Here, we will focus on the
synchrotron scenario and numerically calculate the evolution of the electron
distribution, to test the self-consistency of synchrotron origin for the
$E_{\text{p}}$ evolution.

In this study, we investigate the conditions that make the evolution patterns
of $E_{\text{p}}$ in the framework of synchrotron radiation. Three main
cooling processes (i.e., adiabatic, synchrotron, and synchrotron self-Compton
(SSC)) for electrons, and the effect of a decaying magnetic field are
coherently considered in the calculation of electron distributions (Geng et
al. 2018b; Zhang et al. 2019). Also, other possible effects like the dynamics
of the shell (Uhm & Zhang 2016; Li & Zhang 2021), and the evolving pre-
description of accelerated electrons are included in our investigation. A
brief description of our synchrotron scenario is exhibited in Section 2. Our
numerical results are presented in Section 3, which include three distinct
physical effects (3.1-3.3). To test the applicability of our results, two
short GRBs, i.e., GRB 090227B and GRB 110529A are modeled (3.4). Moreover, as
another possible way to distinguish different radiation mechanisms, the
numerical results of the prompt optical emission are presented (3.5). Finally,
in Section 4, we summarize and discuss our results. Some relevant formulations
are given in the appendix.

Table 1: Parameters of numerical calculations.

Model | $\Gamma$ | $\gamma^{\prime}_{m}$ | $B^{\prime}_{0}$ | $N^{\prime}_{\text{inj}}$ | $R_{0}$ | $s$ | $g$
---|---|---|---|---|---|---|---
|  | $(10^{4})$ | ($10^{2}$ G) |  $(10^{47}\text{s}^{-1}$) | (cm) |  |
$\text{C}1\text{R}14\Gamma 240$ | 240 | 2 | 2 | 4 | $10^{14}$ |  |
$\text{C}1\text{R}15\Gamma 120$ | 120 | 10 | 1 | 16 | $10^{15}$ |  |
$\text{C}2\text{R}14\Gamma 240$ | 240 | 2 | 2 | 64 | $10^{14}$ |  |
$\text{C}2\text{R}15\Gamma 460$ | 460 | 5 | 1 | 2 | $10^{15}$ |  |
$\text{C}3\text{R}14\Gamma 1900$ | 1900 | 1.4 | 3 | 1.2 | $10^{14}$ |  |
$\text{C}3\text{R}15\Gamma 1900$ | 1900 | 1.4 | 3 | 1.2 | $10^{15}$ |  |
$\text{C}4\text{R}14\Gamma 460$ | 460 | 5 | 0.8 | 4 | $10^{14}$ |  |
$\text{A}\text{R}14\Gamma 200$ | 200 | 3.5 | 0.6 | 20 | $10^{14}$ | 0.35 |
$\text{A}\text{R}14\Gamma 150$ | 150 | 3.5 | 0.6 | 20 | $10^{14}$ | 0.55 |
$\text{A}\text{R}14\Gamma 80$ | 80 | 3.5 | 1.0 | 80 | $10^{14}$ | 1.0 |
$\text{A}\text{R}15\Gamma 230$ | 230 | 5 | 2 | 8 | $10^{15}$ | 1.0 |
$\text{AG}1\text{R}14\Gamma 300$ | 300 | 4.5 | 0.6 | 32 | $10^{14}$ | 0.0 | 0.7
$\text{AG}2\text{R}14\Gamma 150$ | 150 | 5.0 | 0.6 | 20 | $10^{14}$ | 0.55 | 0.5
$\text{AG}1\text{R}15\Gamma 250$ | 250 | 5 | 2 | 8 | $10^{15}$ | 0.0 | 1.0
$\text{AG}2\text{R}15\Gamma 230$ | 230 | 5 | 2 | 8 | $10^{15}$ | 1.0 | 1.0
111 Three distinct groups of cases are listed. In the fist group, letter C in
the model name denotes that the shell is moving at a constant Lorentz factor,
the number (1-4) means the cooling process is dominated by synchrotron
cooling, SSC cooling, adiabatic cooling and SSC+adiabatic cooling,
respectively. In the second group, letter A denotes that the shell is moving
at an increasing Lorentz factor, and $s$ here describes the index of bulk
acceleration law. In the third group, letter G means that a broken power-law
profile of $\gamma_{\mathrm{m}}^{\prime}(R)$ is considered, while the
acceleration effect is also taken into account (letter A). And the index $g$
describes the evolution of $\gamma_{\mathrm{m}}^{\prime}(R)$, which could
refer to Equation (5).

##  2 Physical scenario

Let’s consider a relativistic jet moving from an initial radius of $R_{0}$
with a bulk Lorentz factor of $\Gamma$ and emitting electrons in it. We trace
the evolution of the distribution of electrons and derive observed synchrotron
flux density $F_{\nu_{\mathrm{obs}}}$ from these electrons. The electrons are
injected into the co-moving background magnetic field of $B^{\prime}$ (the
superscript prime $\prime$ denotes the quantities in the comoving frame
hereafter), and the evolution of electrons could be obtained by solving the
continuity equation in the energy space. The detailed procedure can be found
in Appendix A and the numerical method is the same as that in Geng et al.
(2018b). The distribution of injected electrons is assumed to be a power-law
of $Q(\gamma_{\rm e}^{\prime},t^{\prime})=Q_{0}(t^{\prime})(\gamma_{\rm
e}^{\prime}/\gamma_{\rm m}^{\prime})^{-p}$ for $\gamma_{\rm
e}^{\prime}>\gamma_{\rm m}^{\prime}$, where $Q_{0}$ is related to the
injection rate by $N_{\rm inj}^{\prime}=\int_{\gamma_{\rm
m}^{\prime}}^{\gamma_{\rm max}^{\prime}}Q(\gamma_{\rm
e}^{\prime},t^{\prime})d\gamma_{\rm e}^{\prime}$ 222$\gamma_{\rm
max}^{\prime}$ is the maximum Lorentz factor of electrons and is given by the
approximation $\gamma_{\rm max}^{\prime}\simeq
10^{8}\left(\frac{B^{\prime}}{1\leavevmode\nobreak\ \mathrm{G}}\right)^{-0.5}$
(Huang et al. 2000).. The electron spectral index $p$ in shock theories ranges
from 2.3 to 2.8 (Kirk et al. 2000; Sironi & Spitkovsky 2009). Quite large
values ($p>3$) in some GRBs are obtained through synchrotron fitting to the
GRB spectra, making it a challenge to standard simulations in shocks and
magnetic reconnection (Oganesyan et al. 2019; Ronchi et al. 2020; Burgess et
al. 2020). However, the high-energy part of the spectrum is rarely constrained
because GBM data are missing in most cases. Also, the evolution of the low-
energy electron distribution mainly depends on its cooling process rather than
$p$ when high-energy electrons are continuously injected. Here, an
intermediate value $p=2.7$ is commonly adopted in view of particle simulations
and observations, and this has little influence on our conclusions. The
temporal structure of a GRB pulse consists of a fast rise and a shallow decay.
To mimic such a GRB pulse, the injection of electrons into the shell is ceased
at a turn-off time $t_{\text{off}}$.

The Lorentz factor of the electrons that radiate at the GRB spectral peak
energy is roughly $\gamma_{\rm m}^{\prime}$, then we have

$E_{\rm p}=\frac{1}{1+z}\frac{3hq_{\rm e}B^{\prime}}{4\pi m_{\rm
e}c}\Gamma\gamma_{\rm m}^{\prime 2},$ (1)

where $h$ is the Planck constant, $c$ is the speed of light, $q_{\rm e}$ and
$m_{\rm e}$ are the electron charge, and electron mass respectively, $z$ is
the redshift of the burst. $z=1$ is adopted in this work unless a specific
statement. When the jet encounters the surrounding interstellar medium, its
bulk Lorentz factor $\Gamma$ will decrease. Meanwhile, in a rapidly expanding
jet, the magnetic field strength $B^{\prime}$ would decay with radius by a
general form of $B^{\prime}=B_{0}^{\prime}(R/R_{0})^{-q}$, where $R$ is the
radius of the expanding shell. Since the radial component of the magnetic
field decrease more rapidly than the toroidal component, the magnetic field is
soon toroidally dominated, which decreases as $R^{-1}$ (Lyubarsky 2009). So
$q$ is always set to be 1 in the main text (Uhm & Zhang 2015; Geng et al.
2018a), while the effect of different $q$ (e.g., Ronchini et al. 2021) on our
results is further discussed in Appendix C. Hence, as a function of decreasing
$\Gamma$ and $B^{\prime}$, a hard-to-soft evolution pattern for $E_{\text{p}}$
is naturally expected according to Equation (1). Nevertheless, the real
situation is probably much complicated. The jet may be under acceleration
driven by radiative pressure or the magnetic pressure gradient. Moreover, the
observed flux comes from an equal-arrival-time surface (EATS) rather than a
point-like emission site (Fenimore et al. 1996; Dermer 2004; Huang et al.
2007; Geng et al. 2017), and Equation (1) is invalid for an integrated
spectrum. Furthermore, $\gamma_{\rm m}^{\prime}$ might vary with time due to
detailed particle acceleration processes (e.g., Guo et al. 2014). Therefore,
other evolution patterns of $E_{\text{p}}$ are likely to arise in specific
conditions.

##  3 Numerical calculations

Instead of assuming a detailed model, we try to derive some constraints on
relevant parameters used in our calculations from observations. The pulse
duration $t_{\rm obs}$ could be estimated by the angular timescale of the
emitting shell, i.e.,

$\delta t_{\mathrm{obs}}\simeq\frac{(1+z)R_{0}}{2\Gamma^{2}c},$ (2)

from which the corresponding $\Gamma$ could be obtained when $t_{\rm obs}$ and
the initial emission radius $R_{0}$ are given. On the other hand, the specific
flux at $E_{\text{p}}$ in the observer frame can be estimated as (Sari et al.
1998; Kumar & McMahon 2008)

$F_{\nu_{\mathrm{obs}}}=N_{\mathrm{e}}\frac{\sqrt{3}q_{\mathrm{e}}^{3}B^{\prime}\Gamma}{m_{\mathrm{e}}c^{2}}\frac{1+z}{4\pi
D_{L}^{2}},$ (3)

where $N_{\mathrm{e}}$ is the total (already corrected for $4\pi$ solid angle)
number of electrons with
$\gamma_{\mathrm{e}}^{\prime}>\gamma_{\mathrm{m}}^{\prime}$ and $D_{L}$ is the
luminosity distance of the burst. Taking typical values for a GRB pulse as
$E_{\text{p}}\simeq 1\mathrm{MeV}$, $\delta t_{\text{obs}}\simeq 1\mathrm{s}$,
and $F_{\nu_{\mathrm{obs}}}\simeq 1\mathrm{mJy}$, we could obtain plausible
values of different sets of parameters ($\Gamma$, $\gamma_{\rm m}^{\prime}$,
$B_{0}^{\prime}$, $N_{\rm inj}^{\prime}$, $R_{0}$) from Equations (1-3). More
details could be found in Appendix B. Detailed parameter values are listed in
Table 1, from which we derive evolving $E_{\rm p}$ (corresponding to the peak
point of $\nu_{\rm obs}F_{\nu_{\rm obs}}$) from the observed spectrum
integrated over the EATS for each specific time (Appendix D). The curvature
effect on $E_{\rm p}$ and $F_{\nu_{\rm obs}}$ is thus naturally incorporated.

To explore the conditions that generate distinct evolution patterns of
$E_{\text{p}}$, we will show results by adding physical effects progressively
in the following subsections. A shell moving at a constant Lorenz factor with
two different starting radii where the shell begins to emit photons is modeled
in subsection 3.1. Two typical dissipation radii for GRB prompt emissions,
i.e., $R_{0}=10^{14}$ cm and $10^{15}$ cm, are considered. A shell moving at
an accelerating Lorenz factor is assumed and calculated in subsection 3.2. We
further investigate the case that the parameter $\gamma_{\mathrm{m}}^{\prime}$
is varying in subsection 3.3. Moreover, two short GRBs, i.e., GRB 090227B and
GRB 110529A are modeled (3.4). Finally, the prompt optical emission is
discussed in subsection 3.5.

###  3.1 Constant $\Gamma$

Here, we first assume that the expanding shell is moving at a constant Lorentz
factor of $\Gamma$. In this group of calculations, we perform seven
calculations, which are named in the form of “$\text{CXRY}\Gamma\text{Z}$”
with the letter C representing that $\Gamma$ is constant. X is an integer
ranging from 1 to 4, which denotes cases that the cooling process of electrons
is dominated by synchrotron cooling, synchrotron self-Compton (SSC) cooling,
adiabatic cooling, and SSC+adiabatic cooling at the same time in order. Y
presents the logarithmic index of the starting radius and Z shows the value of
$\Gamma$. The corresponding results, including the evolution of the electron
distribution ($dN_{\rm e}/d\gamma^{\prime}_{\rm e}$), the flux density
($F_{\nu}$), the spectral energy distribution ($\nu F_{\nu}$), lightcurves,
and the evolution of $E_{\rm p}$ for each case are exhibited in Fig. 1.

In the case of $\text{C1R14}\Gamma\text{240}$, the cooling process of
electrons is forced to be dominated by synchrotron cooling by artificially
ignoring SSC cooling. The electron distribution obeys the standard fast-
cooling pattern ($dN_{\rm e}/d\gamma^{\prime}_{\rm
e}\propto{\gamma^{\prime}_{\rm e}}^{-2}$) in the early stage, and its slope
gets flatter soon since adiabatic cooling begins to dominate for low-energy
electrons when $B^{\prime}$ decreases. The corresponding synchrotron spectra
and lightcurves show that $E_{\text{p}}$ always decreases gradually during the
rising and decaying stages of the pulse. Thus a hard-to-soft pattern of
$E_{\text{p}}$ arises in this case. In $\text{C1R15}\Gamma\text{120}$, the
cooling of electrons exactly trace the standard fast-cooling pattern for the
whole duration. This is owing to that adiabatic cooling rate is inversely
proportional to the radius of the shell (Uhm et al. 2012; Geng et al. 2014).
So, synchrotron cooling always dominates through the whole stage. Meanwhile,
$E_{\text{p}}$ also exhibits a hard-to-soft evolution.

SSC cooling dominates in both $\text{C}2\text{R}14\Gamma 240$ and
$\text{C}2\text{R}15\Gamma 460$, characterized by that the slope of the low-
energy electron spectrum approaches -1 (Bošnjak et al. 2009; Wang et al.
2009). Although the electron/photon spectra are harder, $E_{\text{p}}$ still
appears a hard-to-soft evolution in both cases. As shown in Fig. 1, for
$\text{C3R14}\Gamma\text{1900}$, the dominance of adiabatic cooling leads to
an even harder low-energy electron spectrum (Geng et al. 2018b). The evolution
of $E_{\text{p}}$ follows an intensity-tracking pattern. Electrons tend to
gather in the low energy scope due to the inefficient cooling rate. As a
result, the slope of low-energy electron spectrum turns to be positive in one
segment. Correspondingly, the peak intensity as well as the peak frequency in
$F_{\nu}$ spectrum changes slowly and slightly, which is different from the
above cases. Equation (1) is now invalid and $E_{\text{p}}$ is related to the
integral of this segment of electrons. As an integral effect, the upwarp of
the electron spectrum leads to a right shift of the peak point of $\nu
F_{\nu}$ spectrum. The electron spectrum of $dN_{e}/d\gamma^{\prime}_{\rm
e}\propto{\gamma^{\prime}_{\rm e}}^{0}$ would generate a flux density of
$F_{\nu}\propto{\nu}^{1/2}$. If the slope of the low-energy spectra exceeds 0,
the index of corresponding $F_{\nu}$ spectra will be greater than $1/2$. As a
result, the peak point of $\nu F_{\nu}$ spectrum will shift towards high
frequencies.

Unlike $\text{C3R14}\Gamma\text{1900}$, a larger initial radius of the shell
in $\text{C3R15}\Gamma\text{1900}$ results in a slower magnetic decay, and the
cooling process is dominated by synchrotron cooling rather than adiabatic
cooling. In this case, $E_{\text{p}}$ exhibits a hard-to-soft evolution. If
SSC cooling together with adiabatic cooling governs the cooling process, the
electron spectrum will quickly get harden. As a consequence, it also shows an
intensity-tracking evolution for $E_{\text{p}}$. This is presented in the case
of $\text{C}4\text{R}14\Gamma 460$.

According to these results, hard-to-soft evolution for $E_{\text{p}}$ is
normally expected in constant $\Gamma$. The necessary condition that could
generate the intensity-tracking pattern is that the slope of the low-energy
electron spectra at the early stage must be harder than -1.0. This requires
that electrons are dominated by adiabatic cooling or adiabatic+SSC cooling.

###  3.2 Increasing $\Gamma$

A bulk acceleration of $\Gamma$ may lead to the increase of $E_{\rm p}$
according to Equation (1). Both radiative pressure and the magnetic pressure
gradient within the jet could be responsible for the acceleration of the jet
(e.g., Piran et al. 1993; Drenkhahn 2002; Geng et al. 2021). Hence, we
consider that the Lorentz factor of the shell is increasing gradually (Uhm et
al. 2018), i.e.,

$\Gamma(R)=\Gamma_{0}\left(R/R_{0}\right)^{s},$ (4)

where $R$ is the radius of the shell at some time. The acceleration index $s$
ranges from $1/3$ to 1 in different acceleration theories (Drenkhahn 2002;
Komissarov et al. 2009). Three representative values, i.e., 0.35, 0.55, and
1.0 are taken in calculations. Similar to those in the previous section, we
consider two possible initial radiation radiuses of the shell. This group of
numerical results are presented in Fig. 2, which are named in the form of
“$\text{A}\text{RY}\Gamma Z$”. A means that the shell is accelerating, Y is
the logarithmic index of the starting radius, and Z is the initial Lorentz
factor $\Gamma_{0}$ of the shell.

In $\text{A}\text{R}14\Gamma 200$, $s$ is set to be 0.35 and the cooling
process is dominated by SSC cooling. The low-energy electron spectrum gets
hard within a very early period, while $F_{\nu}$ spectrum in the high-
frequency domain does not change much over time. $E_{\text{p}}$ exhibits an
intensity-tracking pattern evolution, which is distinct from case
$\text{C2}\text{R}14\Gamma 240$. On the one hand, the increasing $\Gamma$
profits to the rise of $E_{\text{p}}$ analytically. On the other hand, we find
that the EATS effect is against the occurrence of the intensity-tracking
pattern. The acceleration effect weakens the EATS effect, owing to compel low
latitude photons of the EATS to be more significant. Thus the case of
$\text{C2}\text{R}14\Gamma 240$ does not present an intensity-tracking pattern
of $E_{\text{p}}$. In $\text{A}\text{R}14\Gamma 150$, $s$ is 0.55 and SSC
cooling is also dominant. The slope of the low-energy electron spectrum tends
to be -1 early and exceeds -1 at the late stage. The evolution curve of
$E_{\text{p}}$ also shows an intensity-tracking pattern. In
$\text{A}\text{R}14\Gamma 80$, $s$ is 1.0. The electron distribution and the
flux density are similar to those in $\text{A}\text{R}14\Gamma 150$.
$E_{\text{p}}$ presents an intensity-tracking evolution again.

As for the case of $\text{A}\text{R}15\Gamma 230$, $s$ is 1.0, a hard-to-soft
evolution for $E_{\rm p}$ is shown, which is a natural result from the
dominance of synchrotron cooling and the EATS effect. Owing to a larger
initial radius, the photon energy density is reduced significantly so that it
is hard to make SSC cooling dominate the cooling process at an early time. As
a result, the low-energy electron spectra follow the standard cooling except
for the last stage. Moreover, the moving distance of the shell during $\delta
t_{\rm obs}$ is only the magnitude of $R_{0}$, which implies a strong EATS
effect on $F_{\nu}$.

In this subsection, the significance of SSC cooling, as well as the bulk
acceleration effect, are discussed. SSC cooling could not produce an
intensity-tracking pattern of $E_{\text{p}}$ alone, due to the impediment of
the EATS effect. However, an accelerating shell working together with a
dominant SSC cooling could generate an intensity-tracking pattern of
$E_{\text{p}}$ for small initial radius of $10^{14}$ cm. For a larger initial
radius of $10^{15}$ cm, only a hard-to-soft pattern is resulted even for an
accelerating shell because of the absence of SSC cooling dominance.

###  3.3 A variable source function

The particle-in-cell simulations for relativistic magnetic reconnections
indicate that the accelerated particle spectrum is evolving with time (e.g.,
Guo et al. 2014). Here, we take a variable source function into account.
Specifically, a broken power-law profile of $\gamma_{\mathrm{m}}^{\prime}(R)$
(Uhm et al. 2018) is assumed,

$\gamma_{\mathrm{m}}^{\prime}(R)={\gamma_{\mathrm{m}}^{0}}\times\left\\{\begin{array}[]{ll}\left(R/R_{\rm
a}\right)^{g}&\text{ if }R\leqslant R_{\rm a},\\\ \left(R/R_{\rm
a}\right)^{-g}&\text{ if }R>R_{\rm a},\end{array}\right.$ (5)

where $R_{\rm a}$ is a radius that particle acceleration efficiency begins to
decrease. In the real situation, the turnover time for particle acceleration
efficiency is probably not identical with $t_{\rm off}$. Thus, $R_{\rm a}$ is
set to be smaller than $R_{\rm off}$ which corresponds to $t_{\rm off}$.
Similarly, four cases are included in this group of numerical results (see
Table 1), and the corresponding figures are exhibited in Fig. 4. The cases are
named in the form of “$\text{AGX}\text{RY}\Gamma Z$”, where G means that the
evolution of $\gamma_{\mathrm{m}}^{\prime}(R)$ is taken into account, X is an
integer ranging from 1 to 2, denoting the bulk acceleration effect is
considered or not, respectively. The meanings of Y and Z are similar to those
in subsection 3.2.

In $\text{AG}1\text{R}14\Gamma 300$, SSC cooling is dominant at the early
stage and the slope of the low-energy electron spectra tends to -1, which is
the same as the case of $\text{AG}2\text{R}14\Gamma 150$. In
$\text{AG}1\text{R}14\Gamma 300$, we first consider the evolution of
$\gamma_{\text{m}}^{\prime}$ only but ignore the bulk acceleration effect. It
presents a chromatic intensity-tracking pattern for $E_{\rm p}$, which means
the peak time of $E_{\text{p}}$ and lightcurves is mismatched. When the bulk
acceleration effect is also included in $\text{A}2\text{R}14\Gamma 150$, it
shows a chromatic intensity-tracking too. This chromatic intensity-tracking
pattern for $E_{\rm p}$ has not been discovered from observations. Different
from the above intensity-tracking patterns, the chromatic intensity-tracking
pattern is induced by the effect of a variable source function which reflects
the intrinsic particle acceleration. An intensity-tracking pattern is expected
for $E_{\text{p}}$ when $R_{\rm a}$ is set to be the same value as $R_{\rm
off}$. If this new pattern is confirmed from detailed data analyses in the
future, it would be evidence for the varying $\gamma_{\text{m}}^{\prime}$ from
particle acceleration.

In results of $\text{AG}1\text{R}15\Gamma 250$, owing to the large initial
radius, synchrotron cooling merely dominates the whole cooling process and SSC
cooling emerges at the last stage. Also, a large initial radius makes the EATS
effect strongly affect the observed flux density due to little movement of the
shell relative to the initial radius (also see Appendix B). Consequently, the
role of a varying $\gamma_{\text{m}}^{\prime}$ is suppressed by the
synchrotron cooling dominance and the EATS effect. Hence it is understandable
that $E_{\text{p}}$ shows a hard-to-soft evolution pattern. While for
$\text{AG}2\text{R}15\Gamma 230$, both the bulk acceleration effect and the
varying $\gamma_{\text{m}}^{\prime}$ are considered. The EATS effect is
weakened so that $E_{\text{p}}$ presents a chromatic intensity-tracking
pattern again at the very early stage.

###  3.4 Application to GRB data

To test the applicability of our calculations, we perform theoretical modeling
of two representative short GRBs, i.e., GRB 110529A (intensity-tracking) and
GRB 090227B (hard-to-soft) respectively. The data used from Fermi-
GBM333https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html is analyzed
with the standard Bayesian approach and the time bin selection method to
obtain the time-resolved spectra (see details in Li et al. 2019). Because of
the huge computational cost in each calculation and potential strong
degeneracy between the parameters, a strict fitting procedure with the
Bayesian method is not carried out in the current work. Instead, we change the
physical parameters involved manually through some trials, until a visual good
match between the numerical results and the data is reached. The fitting
parameters are listed in Table 4. As is shown in Fig. 5, the evolution of
$E_{\rm p}$ and flux for these two GRBs could be roughly reproduced.

In GRB 110529A, the shell is under accelerating and SSC cooling is dominant,
making a rapid rise of the lightcurve and an intensity-tracking pattern of
$E_{\rm p}$. In GRB 090227B, the shell is moving at a constant Lorentz factor
and the cooling process is dominated by synchrotron cooling, resulting in a
hard-to-soft pattern of $E_{\rm p}$. Note that there is a distinct sub-pulse
within 0.1-0.15 s as shown in photon counts of GRB 090227B, which leads to the
significant deviation of theoretical $E_{\rm p}$ from the observed ones during
this period.

Table 2: The fitting parameters.

GRB | $\Gamma$ | $\gamma^{\prime}_{m}$ | $B^{\prime}_{0}$ | $N^{\prime}_{\text{inj}}$ | $R_{0}$ | $s$
---|---|---|---|---|---|---
|  | $(10^{4})$ | ($10^{2}$ G) |  $(10^{47}\text{s}^{-1}$) | (cm) |
110529A | 300 | 3.0 | 0.6 | 1.4 | $10^{14}$ | $0.35$
090227B | 240 | 5.0 | 2 | 4.0 | $10^{14}$ |
444We assume a redshift of $z=1$ for both GRBs during fitting.

###  3.5 The prompt optical emission

Relevant researches indicate that we need more probes to distinguish possible
radiation mechanisms, except for the evolution patterns of $E_{\text{p}}$ and
the low-energy spectral indices. Therefore, in this subsection, we calculate
the prompt optical emission of the above fifteen distinct cases. The
representative results are exhibited in Fig. 6, including
$\text{C}1\text{R}15\Gamma 120$, $\text{C}2\text{R}15\Gamma 460$, and
$\text{A}\text{R}14\Gamma 150$.

As shown in Fig. 6, the flux of the prompt optical emission is higher than the
prompt X-ray emission but within only one magnitude. A clear spectral lag and
a shallower decay for the optical emission are present. The detection
sensitivity of the Ground-based Wide Angle Camera (GWAC) and future Wide Field
Survey Telescope (WFST) of China is roughly 15 and 20 apparent magnitudes in
the $R$ band within an exposure time of one second. Accordingly, prompt
synchrotron optical emissions from a burst within $\sim$ 1 Gpc are detectable
by GWAC ($\sim$ 10 Gpc by WFST). Future observations on the prompt optical
emission will help to examine the synchrotron radiation scenarios.

##  4 Conclusions and discussion

In this article, we numerically trace the evolution of electrons and the
corresponding synchrotron emissions in the framework of synchrotron radiation,
to explore possible evolution patterns of $E_{\text{p}}$. The effects of the
bulk acceleration of the emitting shell, the varying
$\gamma_{\text{m}}^{\prime}$, and the EATS are analyzed by performing groups
of calculations. In addition, the prompt optical emission is calculated. The
following conclusions are drawn.

1\. For a shell moving at a constant Lorentz factor, the intensity-tracking
pattern of $E_{\text{p}}$ could emerge when electrons are dominated by
adiabatic cooling or adiabatic+SSC cooling. For the case of pure dominance by
adiabatic cooling, a bulk Lorentz factor $\Gamma$ of $>1000$ for the shell and
a small emitting radius of $R_{0}\leq 10^{14}$ cm is required. Otherwise, a
hard-to-soft pattern is normally expected.

2\. For a shell under bulk acceleration, a dominant SSC cooling could generate
an intensity-tracking pattern of $E_{\text{p}}$ for a small initial radius of
$10^{14}$ cm. However, for a larger initial radius of $10^{15}$ cm, only a
hard-to-soft pattern is resulted even for an accelerating shell owing to the
synchrotron cooling dominance and the EATS effect.

3\. When the role of the intrinsic evolution of $\gamma_{\mathrm{m}}^{\prime}$
is taken into account, a chromatic intensity-tracking for $E_{\text{p}}$ may
occur. Also, the rapid decay of $B^{\prime}$ ($q\geq 1.5$), accompanied with a
dominant adiabatic cooling, would generate a chromatic intensity-tracking for
$E_{\rm p}$ (Appendix C).

4\. The flux density of the prompt optical emission is slightly higher than
the prompt X-ray emission, and is detectable for bursts within $\sim$ 1Gpc by
current wide-field telescopes like GWAC, and $\sim$ 10Gpc by future WFST of
China.

Recently, Ronchini et al. (2021) show a relation between the spectral index
and the flux by investigating the X-ray tails of bright GRB pulses, which is
incompatible with the long-standing scenario which invokes the delayed arrival
of photons from high-latitude parts of the jet. It is suggested that a
dominating adiabatic cooling process, accompanied by the decay of the magnetic
field could explain this relation after testing several possible cooling
mechanisms. This study supports that adiabatic cooling could play a
significant role in the GRB prompt phase.

These distinct evolution patterns, i.e., hard-to-soft, intensity-tracking and
chromatic intensity-tracking, reflect different physical processes in the GRB
prompt emission. In general, the intensity-tracking pattern of $E_{\rm p}$
indicates that the bulk Lorentz factor is initially very large or it is under
acceleration, and the emitting radius should be relatively small. The
evolution of $\gamma_{\mathrm{m}}^{\prime}$ from the particle acceleration
process could generate a chromatic intensity-tracking pattern. Further studies
could help to test our conclusions.

Although only the intensity-tracking pattern of a single short pulse of the
GRB is given in our work, the result is applicable for a long-duration burst
that consists of several pulses. Owing to the proportional relation between
$F_{\nu_{\mathrm{obs}}}$ and $E_{\text{p}}$ in time-resolved pulses
observationally, it is naturally expected that a long duration burst presents
the same evolution pattern of $E_{\text{p}}$ as the single pulse.

In the current work, the source function $Q(\gamma_{\rm
e}^{\prime},t^{\prime})$ is set the form of power-law, i.e., $Q(\gamma_{\rm
e}^{\prime},t^{\prime})=Q_{0}(t^{\prime})(\gamma_{\rm e}^{\prime}/\gamma_{\rm
m}^{\prime})^{-p}$, which should be derived from detailed modeling of the
particle acceleration process in our future works. On the other hand, we have
assumed that the shell is homogeneous in latitude and its thickness could be
ignored. Further considerations on the structure of the shell may be specified
in the future.

Except for studies on the evolution pattern of $E_{\text{p}}$ within the
synchrotron radiation scenarios, its interpretation within the framework of
the photosphere radiation is urgently invoked. Also, better indicators (like
$E_{\rm p}$, the curvature of the spectrum, etc.) to identify the underlying
radiation mechanism are now under discussion. For example, the characteristics
of the low energy band (i.e., less than 1 keV) for GRB prompt emissions may
show distinct properties within different physical scenarios (Oganesyan et al.
2018; Toffano et al. 2021). Therefore, theoretical and observational efforts
are encouraged in the research of the prompt optical emission.

###### Acknowledgements.

We appreciate the anonymous referee for constructive suggestions. We
acknowledge the use of the public data from the Fermi data archives. We also
would like to thank Xue-Feng Wu for the helpful discussion and Liang Li for
providing the GRB data. This work is supported by National SKA Program of
China No. 2020SKA0120300, by the National Natural Science Foundation of China
(Grant Nos. 11903019, 11873030, 11833003, 12041306, U1938201, 11535005), and
by the Strategic Priority Research Program of the Chinese Academy of Sciences
(“multi-waveband Gravitational-Wave Universe”, Grant No. XDB23040000).

(a)  (b)

(c)  (d)

Figure 1: Numerical results for seven cases in the first group shown in Table
1. The diagram of each case consists of 4 panels: the electron distribution
$dN_{e}/d\gamma^{\prime}_{e}$ (upper left), the corresponding synchrotron flux
density $F_{\nu}$ from electrons (upper right), the spectral energy
distribution $\nu F_{\nu}$ (lower left), and lightcurves at three bands $\&$
the evolution of $E_{\text{p}}$ (lower right). Notably, in the electron
distribution, orange dashed lines are the standard fast-cooling pattern, i.e.,
$dN_{\mathrm{e}}/d\gamma_{\mathrm{e}}^{\prime}\propto\gamma_{\mathrm{e}}^{\prime-2}$,
and gray dashed lines present the expected cooling pattern, i.e.,
$dN_{\mathrm{e}}/d\gamma_{\mathrm{e}}^{\prime}\propto\gamma_{\mathrm{e}}^{\prime-1}$.
In the flux density, three vertical lines are given corresponding to three
different frequencies of 300 keV, 10 keV, and 1 keV respectively. The peak
energy $E_{\text{p}}$ is derived from the peak point of $\nu F_{\nu}$.

(a)  (b)

(c)

Figure 2: (continued.)

(a)  (b)

(c)  (d)

Figure 3: Numerical results for four cases in the second group shown in Table
1.

(a)  (b)

(c)  (d)

Figure 4: Numerical results for four cases in the third group shown in Table
1.

(a)  (b)

Figure 5: Theoretical modeling of lightcurves and $E_{\rm p}$ evolutions for
GRB 110529A and GRB 090227B. In each diagram, numerical results are shown with
blue (for flux) and red (for $E_{\rm p}$) lines, while the observational data
are presented by blue (for flux) and red (for $E_{\rm p}$) dots. The grey
histogram shows the photon-count lightcurve in a scaled way.

(a)  (b)  (c)

Figure 6: Representative results of the prompt optical emission from three
cases in Table 1. There are three lightcurves in different energy band in the
diagram of each case, including 2 eV, 1 keV, and 10 keV.

(a)  (b)

(c)

Figure 7:  Numerical results for three cases shown in Table 3.

##  Appendix A Formulations for synchrotron radiation model

There are three main cooling mechanisms for relativistic electrons. An
electron will lose energy when it is moving in the magnetic field with a
Lorentz factor $\gamma_{\mathrm{e}}^{\prime}$ (Rybicki & Lightman 1986),

$\dot{\gamma}_{\text{e,syn}}^{\prime}=-\frac{\sigma_{T}B^{\prime
2}\gamma_{\mathrm{e}}^{\prime 2}}{6\pi m_{\mathrm{e}}c},$ (6)

where $\sigma_{T}$ is the Thomson cross-section. Meanwhile, the electron
undergoes an adiabatic cooling (Uhm et al. 2012; Geng et al. 2014) in the
moving jet,

$\dot{\gamma}_{\mathrm{e},\mathrm{adi}}^{\prime}=\frac{1}{3}\gamma_{\mathrm{e}}^{\prime}\frac{d\ln
n_{\mathrm{e}}^{\prime}}{dt^{\prime}}=-\frac{2}{3}\frac{\gamma_{\mathrm{e}}^{\prime}}{R}\frac{dR}{dt^{\prime}},$
(7)

where $n_{\mathrm{e}}^{\prime}$ is the comoving electron number density.
Moreover, electrons would cool by scattering with self-emitted synchrotron
photons, called the SSC process (Blumenthal & Gould 1970),

$\displaystyle\dot{\gamma}_{\mathrm{e},\mathrm{SSC}}^{\prime}=$
$\displaystyle-\frac{1}{m_{\mathrm{e}}c^{2}}\frac{3\sigma_{T}c}{4\gamma_{\mathrm{e}}^{\prime
2}}\int_{\nu_{\min}^{\prime}}^{\nu_{\max}^{\prime}}\frac{n_{\nu^{\prime}}d\nu^{\prime}}{\nu^{\prime}}$
(8)
$\displaystyle\times\int_{\nu_{\mathrm{ic},\min}^{\prime}}^{\nu_{\mathrm{ic},\max}^{\prime}}h\nu_{\mathrm{ic}}^{\prime}d\nu_{\mathrm{ic}}^{\prime}F(q,g),$

where $n_{\nu^{\prime}}$, $\nu^{\prime}$, and $\nu_{\mathrm{ic}}^{\prime}$ are
the synchrotron seed photon spectrum, the frequency of the photon before
scattering, and the frequency of the photon after scattering, respectively.
Here, $F(q,g)=2q\ln q+(1+2q)(1-q)+\frac{1}{2}\frac{(4qg)^{2}}{1+4qg}(1-q)$,
$g=\frac{\gamma_{\mathrm{e}}^{\prime}h\nu^{\prime}}{m_{\mathrm{e}}c^{2}}$,
$w=\frac{h\nu_{\mathrm{ic}}^{\prime}}{\gamma_{\mathrm{e}}^{\prime}m_{\mathrm{e}}c^{2}}$,
and $q=\frac{w}{4g(1-w)}$, which means that the Klein-Nishina effect was fully
taken into account. The upper limit of the internal integral can be derived as
$h\nu_{\mathrm{ic},\max}^{\prime}=\gamma_{\mathrm{e}}^{\prime}m_{\mathrm{e}}c^{2}\frac{4g}{4g+1}$,
and the lower limit is $\nu_{\text{ic,min}}^{\prime}=\nu^{\prime}$. To sum up
the above, we could obtain the electron distribution by solving the continuity
equation of electrons in energy space (Longair 2011),

$\frac{\partial}{\partial
t^{\prime}}\left(\frac{dN_{\mathrm{e}}}{d\gamma_{\mathrm{e}}^{\prime}}\right)+\frac{\partial}{\partial\gamma_{\mathrm{e}}^{\prime}}\left[\dot{\gamma}_{\mathrm{e},\mathrm{tot}}^{\prime}\left(\frac{dN_{\mathrm{e}}}{d\gamma_{\mathrm{e}}^{\prime}}\right)\right]=Q\left(\gamma_{\mathrm{e}}^{\prime},t^{\prime}\right),$
(9)

where
$\dot{\gamma}_{\mathrm{e},\mathrm{tot}}^{\prime}=\dot{\gamma}_{\mathrm{e},\mathrm{syn}}^{\prime}+\dot{\gamma}_{\mathrm{e},\mathrm{adi}}^{\prime}+\dot{\gamma}_{\mathrm{e},\mathrm{SSC}}^{\prime}$.
More details on the numerical method could be found in Yabe et al. (2001) and
Geng et al. (2018b).

##  Appendix B Constraints of parameters

Here, some constraints on parameters used in our scenario are derived. The
photosphere radius is estimated by (Mészáros & Rees 2000; Rees & Mészáros
2005; Deng & Zhang 2014)

$r_{\text{ph}}=(1-\beta)\frac{L\sigma_{\text{T}}}{4\pi m_{\mathrm{p}}\beta
c^{3}\Gamma},$ (10)

where $L$ is the luminosity of the shell, $m_{\mathrm{p}}$ is the mass of the
proton, and $\beta$ is the dimensionless velocity of the jet. For a range of
the bulk Lorentz factor $\Gamma$ from 50 to 300, the photosphere radius
$r_{\text{ph}}$ varies from $10^{11}$ cm to $10^{13}$ cm, where the radiation
is described by the photosphere emission. Meanwhile, we found that the
scattering between electrons and photons is dramatic below $r_{\text{ph}}$ and
our calculations are invalid under such large optical depth. A more delicate
model containing both the photosphere emission and the synchrotron radiation
should be invoked in the future. On other hand, when the radius $R_{0}$ is
larger than $10^{16}$ cm, synchrotron cooling always dominates the process and
electrons are in the fast cooling regime. Thus we focus on the initial radius
ranging from $10^{14}$ cm to $10^{15}$ cm for the shell in this work.

The radiative cooling time for an electron of $\gamma_{\mathrm{e}}^{\prime}$
in the observer frame is (Sari et al. 1998; Sari & Esin 2001)

$t_{\mathrm{c}}=\frac{6\pi m_{\mathrm{e}}c(1+z)}{\sigma_{T}B^{\prime
2}\gamma_{\mathrm{e}}^{\prime}\Gamma(1+Y)},$ (11)

where $Y$ is the Compton parameter. The fast-cooling condition of
$t_{\mathrm{c}}\left(\gamma_{\mathrm{m}}^{\prime}\right)\leqslant t_{\rm obs}$
requires

$B^{\prime}>400\leavevmode\nobreak\
\Gamma_{2}^{1/2}\gamma_{\mathrm{m},3}^{\prime-1/2}R_{0,14}^{-1/2}\,\mathrm{G},$
(12)

where the convention $Q_{x}=Q/10^{x}$ in cgs units are adopted hereafter.
Since the extremely fast-cooling regime is not favored by observations, we set
the upper limit of $B^{\prime}<400$ G in our calculations. The range of is
$\Gamma$ taken to be within [50,2000] according to constraints from other
observations in the literature (Lithwick & Sari 2001; Abdo et al. 2009b).
Furthermore, $\gamma_{\rm m}^{\prime}$ varies from $10^{3}$ to several
$10^{4}$ in line with particle-in-cell simulations (Sironi & Spitkovsky 2009;
Sironi et al. 2013; Guo et al. 2014).

##  Appendix C Effect of different $q$

The value of the magnetic field decaying index ($q$) varies from 1 to 2 due to
the geometry of the shell and the structure of the magnetic field (Mészáros &
Rees 1999; Spruit et al. 2001). Moreover, it was suggested that $q$ could be
0.6 in recent studies (Ronchini et al. 2021). To investigate the effect of
different $q$ on our results in the main text, we adopt the other three
possible values of $q$ (i.e., 0.5, 1.5, and 2) and perform numerical
calculations for different cases as above. We find that the results under the
cooling process dominated by synchrotron cooling or SSC cooling are weakly
influenced by different $q$ since the absolute value of $B^{\prime}$ is more
crucial than its decaying law. For simplicity, only the results in the
dominance of adiabatic cooling are exhibited here. The model name follows the
previous definition, i.e., “$\text{CXRY}\Gamma\text{Z}$” but with an
additional parameter $q$, and detailed parameters could be found in Table 3.

As shown in Fig. 7, in $\text{C3R14}\Gamma\text{1900}q0.5$, where $q=0.5$,
$E_{\rm p}$ shows a hard-to-soft evolution, owing to the slow decay of the
magnetic field and the adiabatic cooling can not dominate at the early time.
In $\text{C3R14}\Gamma\text{1000}q1.5$, the evolution of the electron
distribution obeys the adiabatic cooling case. However, at an early stage, due
to a large value of $q$, the synchrotron cooling rate decreases rapidly the
adiabatic cooling dominates soon and $E_{\rm p}$ rises. At a late time, the
observed lightcurve may even enter the decay phase before $R_{\rm off}$. Also,
the rapid decay of synchrotron intensity along with $R$ makes the observed
spectrum/$E_{\rm p}$ is dominated by the high latitude emission of the EATS.
Similarly, Ronchini et al. (2021) find that the spectral evolution becomes
dominated by the emission at larger angles for large values of $q$. Therefore,
a chromatic intensity-tracking evolution pattern of $E_{\rm p}$ is produced.
Moreover, a similar result is shown in $\text{C3R14}\Gamma\text{800}q2.0$.
These results support our conclusion on the crucial role of adiabatic cooling
dominance for an intensity-tracking pattern. On the other hand, a large $q$,
accompanying with a dominated adiabatic cooling, would generate a chromatic
intensity-tracking evolution of $E_{\rm p}$.

Table 3: Parameters of numerical calculations for different $q$. Model | $\Gamma$ | $\gamma^{\prime}_{m}$ | $B^{\prime}_{0}$ | $N^{\prime}_{\text{inj}}$ | $R_{0}$ | $q$
---|---|---|---|---|---|---
|  | $(10^{4})$ | ($10^{2}$ G) |  $(10^{47}\text{s}^{-1}$) | (cm) |
$\text{C}3\text{R}14\Gamma 1900$ | 1900 | 1.4 | 3 | 0.1 | $10^{14}$ | $0.5$
$\text{C}3\text{R}14\Gamma 1000$ | 1000 | 1.4 | 3 | 1.2 | $10^{14}$ | $1.5$
$\text{C}3\text{R}14\Gamma 800$ | 800 | 1.4 | 3 | 5.8 | $10^{14}$ | $2.0$

##  Appendix D Concerns on EATS

By ignoring the EATS effect, one could often estimate the observed flux
density by using

$F_{\nu_{\mathrm{obs}}}=\frac{(1+z)\Gamma
P^{\prime}\left(\nu^{\prime}\left(\nu_{\mathrm{obs}}\right)\right)}{4\pi
D_{L}^{2}},$ (13)

where $P^{\prime}$ is the synchrotron radiation power of all electrons at
frequency $\nu^{\prime}=(1+z)\nu_{\mathrm{obs}}/D$,
$D=1/[\Gamma(1-\beta\cos\theta)]$ is the Doppler factor, and $\theta$ is the
angle between the line of sight and the local radial direction. When we take
the EATS effect into account, the observed flux density now writes as (Geng et
al. 2016)

$F_{\nu_{\mathrm{obs}}}=\frac{1+z}{4\pi
D_{L}^{2}}\int_{0}^{\theta_{j}}P^{\prime}\left(\nu^{\prime}\left(\nu_{\mathrm{obs}}\right)\right)D^{3}\frac{\sin\theta}{2}d\theta,$
(14)

where $\theta_{j}$ is the half-opening angle of the jet.

In general, the results from the integral of Equation (14) should degenerate
to results from Equation (13) in practice. However, this is not the case in
scenarios invoking a starting radius $R_{0}$. At the beginning of the
radiation, the shell moves forward a little distance relative to its initial
radius. When we are performing the integral of Equation (14), only a small
region contributes to the observed flux since the high latitude locations of
the full EATS are not filled with photons at all. On the other hand, if we use
Equation (13) and take $\theta$ to be 0, it means that we have supposed all
photons of the shell converge on the beam of light of sight. This treatment
would highly overestimate $F_{\nu_{\rm obs}}$ in comparison with that from
Equation (14). The approximate formulation for $F_{\nu_{\rm obs}}$ in Equation
(13) is therefore invalid in this work, which should also be noted in future
relevant works.

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