# Microphysical Parameter Variation in GRB Stratified Afterglows and Closure
Relations: from sub-GeV to TeV Observations

Nissim Fraija,1 Maria G. Dainotti,2,3,4 Boris Betancourt Kamenetskaia,5,6
Antonio Galván-Gámez,1 Edilberto Aguilar-Ruiz1
1Instituto de Astronomía, Universidad Nacional Autónoma de México,
Circuito Exterior, C.U., A. Postal 70-264, 04510 México City, México
2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
3Space Science Institute, Boulder, CO, USA
4The Graduate University for Advanced Studies, SOKENDAI, Shonankokusaimura,
Hayama, Miura District, Kanagawa 240-0193, Japan
5Technical University of Munich, TUM School of Natural Sciences, Physics
Department, James-Franck-Str 1, 85748 Garching, Germany
6Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring
6, 80805 Munich, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Gamma-ray bursts (GRBs) are one of the most exciting sources that offer
valuable opportunities for investigating the evolution of energy fraction
given to magnetic fields and particles through microphysical parameters during
relativistic shocks. The delayed onset of GeV-TeV radiation from bursts
detected by the Fermi Large Area Telescope (Fermi-LAT) and Cherenkov
Telescopes provide crucial information in favor of the external-shock model.
Derivation of the closure relations (CRs) and the light curves in external
shocks requires knowledge of GRB afterglow physics. In this manuscript, we
derive the CRs and light curves in a stratified medium with variations of
microphysical parameters of the synchrotron and SSC afterglow model radiated
by an electron distribution with a hard and soft spectral index. Using Markov
Chain Monte Carlo simulations, we apply the current model to investigate the
evolution of the spectral and temporal indexes of those GRBs reported in the
Second Gamma-ray Burst Catalog (2FLGC), which comprises 29 bursts with photon
energies above 10 GeV and of those bursts (GRB 180720B, 190114C, 190829A and
221009A) with energetic photons above 100 GeV, which can hardly be modeled
with the CRs of the standard synchrotron scenario. The analysis shows that i)
the most likely afterglow model using synchrotron and SSC emission on the
2FLGC corresponds to the constant-density scenario, and ii) variations of
spectral (temporal) index keeping the temporal (spectral) index constant could
be associated with the evolution of microphysical parameters, as exhibited in
GRB 190829A and GRB 221009A.

###### keywords:

Gamma-rays bursts: individual (GRB 160509A, GRB 180720B, GRB 190114C, GRB
190829A and GRB 221009A) — Physical data and processes: acceleration of
particles — Physical data and processes: radiation mechanism: nonthermal —
ISM: general - magnetic fields

††pubyear: 2015††pagerange: Microphysical Parameter Variation in GRB
Stratified Afterglows and Closure Relations: from sub-GeV to TeV
Observations–Microphysical Parameter Variation in GRB Stratified Afterglows
and Closure Relations: from sub-GeV to TeV Observations

##  1 Introduction

Gamma-ray bursts (GRBs), the most extreme phenomena in the universe that emit
non-repeating gamma-ray flashes,111These flashes are usually reported in the
keV-MeV energy range and modeled by the empirical Band function (Band et al.,
1993). originate in the core collapse of massive stars (Woosley, 1993;
Paczyński, 1998; Woosley & Bloom, 2006) or the merger of binary compact
objects, like a black hole (BH) - neutron star (NS) (Narayan et al., 1992), a
BH - BH or a NS - NS (Duncan & Thompson, 1992; Usov, 1992; Thompson, 1994;
Metzger et al., 2011). These scenarios can lead to a hyper-accreting BH or a
millisecond magnetar222A fast-spinning newly born highly-magnetized NS. as a
remnant. Their classification is determined by the duration of the main gamma-
ray episode ($T_{90}$)333$T_{90}$ represents the period during which a GRB
emits between $5\%$ and $95\%$ of the total observed counts from its prompt
emission. as short (sGRBs) and long GRBs (lGRBs). Based on the nature of
progenitors, sGRBs are associated with the merger of binary compact objects
and lGRBs to the core collapse of massive stars. Irrespective of the
progenitor, a late episode, known as afterglow, is observed from radio
wavelengths to TeV gamma-rays. It is usually interpreted in the synchrotron
afterglow scenario, generated by non-thermal electrons accelerated in the
forward shock. This shock originates when the outflow encounters the
circumburst medium and transfers part of its energy to it (Kobayashi et al.,
1999; Granot & Sari, 2002; Sari et al., 1998). Synchrotron photons are
scattered up to energies greater than tens of GeV by the synchrotron self-
Compton (SSC) mechanism (Zhang & Mészáros, 2001; Fraija et al., 2019b), even
if the highest photon energy produced by the synchrotron process is $\sim
5-10~{}{\rm GeV}$ (Piran & Nakar, 2010; Abdo et al., 2009b; Barniol Duran &
Kumar, 2011).

The Major Atmospheric Gamma Imaging Cherenkov (MAGIC; MAGIC Collaboration et
al., 2019a; Acciari et al., 2019, 2021; Fukami et al., 2022), the High Energy
Stereoscopic System (H.E.S.S.; Abdalla et al., 2019; H. E. S. S. Collaboration
et al., 2021) and the Large High Altitude Air Shower Observatory (LHAASO;
LHAASO Collaboration et al., 2023) reported very-high-energy (VHE) emission
with energetic photons above 100 GeV in GRB 160821B, 180720B, 190114C,
190829A, 201216C and 221009A. Ajello et al. (2019) has released the Fermi
Collaboration’s Second Gamma-ray Burst Catalog (2FLGC), which contains data
from the first decade of their program (from 2008 to 2018, August 4). The data
set contains 169 GRBs with high-energy emission $\geq 100$ MeV and a sample of
GRBs with temporarily-extended emission fitted by a power-law (PL) and a
broken power-law (BPL) function. A BPL function with a temporal break at
hundreds of seconds characterizes several light curves. In most cases, the
temporarily-extended emission is explained according to the variation in
temporal and spectral indices called the closure relations (CRs) predicted by
the synchrotron afterglow model. Although it is generally accepted that CRs
will be present if the forward shock dominates the high-energy emission, this
is not the case for all LAT light curves.

Tak et al. (2019) methodically evaluated the CRs in a sample of 59 LAT-
detected bursts using temporal and spectral indices. While they found that the
conventional synchrotron forward-shock emission successfully explained the
spectral ($\beta_{\rm LAT}$) and temporal ($\alpha_{\rm LAT}$) indices in most
cases, they also found that a sizable minority of bursts defied this
explanation. They found that a significant fraction of GRBs satisfies the CRs
when the synchrotron afterglow model: i) is in the slow-cooling regime
($\nu^{\rm syn}_{\rm m}<\nu_{\rm LAT}<\nu^{\rm syn}_{\rm c}$), ii) occurs in
the constant-density medium, and iii) has an abnormally low value of
$\varepsilon_{B}<10^{-7}$. The synchrotron spectral breaks $\nu^{\rm syn}_{\rm
c}$ and $\nu^{\rm syn}_{\rm m}$ are the cooling and characteristic breaks,
respectively. Dainotti et al. (2021b) investigated the PL segment in the X-ray
and optical light curves with a flatter slope than the typical decline, the
so-called “plateau" phase. This phase is seen as being consistent with the
energy injection scenario (Barthelmy et al., 2005; King et al., 2005; Dai et
al., 2006; Perna et al., 2006; Proga & Zhang, 2006; Burrows et al., 2005;
Chincarini et al., 2007; Dall’Osso et al., 2017; Becerra et al., 2019). They
concluded that the most advantageous scenario was one in which the constant-
density environment underwent slow cooling. Fraija et al. (2022) estimated the
CRs for a forward-shock model of the SSC that originates in the stellar-wind
and interstellar medium. The authors demonstrated that the synchrotron
scenario could not explain a sizable proportion of GRBs and that the constant-
density medium is favored without energy injection. In contrast, the stellar-
wind medium is favored with the presence of injection. Several studies have
employed the development of CRs in the interstellar medium ($k=0$) and the
stellar-wind ($k=2$) to verify the development of the LAT light curve.
Nonetheless, a circumburst medium with a density profile between ${\rm k=0}$
and ${\rm k=2}$ is consistent in several GRBs, as revealed by studies of the
progenitor and modeling of multiwavelength data during the afterglow phase. Yi
et al. (2013) looked at the dynamics of forward/reverse shocks as they
traveled through a medium represented by the PL distribution. Based on their
model, the density profile index of 19 GRBs ranged from $0.4\leq{\rm k}\leq
1.4$, with a mean value of ${\rm k\sim 1}$. Similar results were found by
analyzing 146 GRBs instead of only 13, as shown in Liang et al. (2013). Kumar
et al. (2008) derived a density-profile index with $k>2$ after looking at the
possibility that a BH’s accretion of a stellar envelope causes the plateau
phase in X-ray light curves. Multiwavelength observations of GRB 130427A, one
of the most energetic bursts at redshift $z<1$, were modeled by Kouveliotou et
al. (2013) using a synchrotron afterglow model in the adiabatic domain without
energy injection for $p>2$. The authors concluded that the data most aligned
with a density profile value somewhere between ${\rm k}=0$ and ${\rm k}=2$.

Variations in the values of microphysical parameters ($\epsilon_{\rm e}$ and
$\epsilon_{\rm B}$)444The micropysical parameters $\epsilon_{\rm e}$ and
$\epsilon_{\rm B}$ correspond to the fraction of energy given to accelerate
electrons and amplify the magnetic field, respectively. have also been
considered during the afterglow phase (e.g., see Yost et al., 2003; Kumar &
Panaitescu, 2003; Ioka et al., 2006; Fan & Piran, 2006; Panaitescu et al.,
2006; Ioka, 2005; Granot et al., 2006; Fraija et al., 2020). Given the lack of
knowledge about how energy is transferred from protons to electrons and
magnetic fields in relativistic shocks, this is not unrealistic to assume.
Ioka (2005) proposed that the fraction of energy given to accelerate electrons
could evolve during the early afterglow when the synchrotron radiation was in
the fast-cooling regime. Considering that the efficiency of conversion of the
energy carried by the outflow into radiation depends on the microphysical
parameters, Granot et al. (2006) suggested that the plateau phase could be
interpreted by the variation of the microphysical parameters instead of the
energy injection right after the prompt episode. Fan & Piran (2006) reported
that energy injection and the evolution of microphysical parameters are
consistent with the afterglow observations. To mimic the brightenings observed
in some GRB afterglows, Kong et al. (2010) looked into the evolution of the
microphysical parameters under the setting of a wind bubble environment.
Fraija et al. (2020) considered the variation of microphysical parameters to
explain both features simultaneously, the rapid evolution of the synchrotron
cooling break, and the X-ray plateau phase reported in GRB 160509A. Given the
failure of an early effort to implement inverse Compton cooling, Fan & Piran
(2006) argued that it would be worthwhile to search for as-of-yet-unexamined
mechanisms.

We derive the SSC and synchrotron forward-shock model including the
corresponding light curves, spectra and CRs with the evolution of the
microphysical parameters in the stratified environment with a density profile
$\propto r^{-k}$ with ${\rm k}$ in the range of $0\leq k<3$. We apply the
current model to investigate the evolution of the spectral and temporal
indexes of the GRBs reported in 2FLGC, and of the bursts that emitted VHE
emission. We consider an electron distribution with a hard spectral index
($1\leq p\leq 2$) and $2<p$. This manuscript is organized as follows: in
Section §2, we derive the SSC forward-shock model with the evolution of the
microphysical parameters in a stratified environment with $0\leq k<3$. In
Section §3, we introduce Fermi-LAT data and the methodical approach with a
discussion about the variation of magnetic parameter in decaying
microturbulence, the circumburst environment, the maximum synchrotron photons
and the magnetic microphysical parameter. In addition, we derive and plot the
expected light curves and spectral energy distributions (SEDs) and apply the
theoretical model to describe the latest LAT observations of GRB 160509A. In
Section §4, we introduce and analyze the CRs of SSC afterglow model with the
VHE observations of GRB 180720B, 190114C, 190829A and 221009A. In Section §5,
we summarize previous works and point out the differences with current
manuscript. Finally, in Section §5, we conclude by summarizing our findings
and making some general observations. From this point on, we will denote
quantities as “unprimed" and “primed" if they are measured in the observer and
the comoving frame, respectively. We assume for the cosmological constants a
spatially flat universe $\Lambda$CDM model with $H_{0}=69.6\,{\rm
km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{\rm M}=0.286$ and $\Omega_{\Lambda}=0.714$
(Planck Collaboration et al., 2016) and adopt the convention $Q_{\rm
x}=Q/10^{\rm x}$ in c.g.s. units.

##  2 Closure Relations in a stratified environment with variation of
microphysical parameters

Two types of shocks may be generated by the expansion of a relativistic jet
into the circumburst medium: the forward shock, which travels outward, and the
reverse shock, which travels inward, inside the jet. Taking into account the
hydrodynamic of the forward-shock evolution with the Blandford-Mckee solution
(Blandford & McKee, 1976), the bulk Lorentz factor becomes
$\Gamma=137.9\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{2(4-k)}}A_{\rm
k,-1}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}t_{2}^{\frac{k-3}{2(4-k)}}$,
where $E$ is the isotropic-equivalent kinetic energy, and $(1+z)$ is the
correction due to the redshift. We consider a profile of the circumburst
density $n(r)=A_{\rm k_{s}}r^{\rm-k}$, with $0\leq k<3$ and the evolution of
microphysical parameters to accelerate electrons and amplify the magnetic
field as $\epsilon_{\rm e}=\varepsilon_{\rm e}\left(t/t_{0}\right)^{-\rm a}$
and $\epsilon_{\rm B}=\varepsilon_{\rm B}\left(t/t_{0}\right)^{-\rm b}$,
respectively. The term $t_{0}$ corresponds to the onset of the variation of
the microphysical parameters, which we will set to $10^{2}\,{\rm s}$ with
$a=-0.5$ and $b=0.5$, and we will use ${\rm k=2}$ for $A_{\rm k_{s}}=A_{\rm
k}\frac{\dot{M}_{\rm W}}{4\pi m_{p}v_{\rm W}}=A_{\rm k}3.0\times 10^{35}\,{\rm
cm^{-1}}$,555The terms $v_{\rm W}$ and $\dot{M}_{\rm W}$ correspond to the
wind velocity and the mass-loss rate, respectively, and $m_{p}$ is the proton
mass. unless otherwise indicated. Accelerated electrons are distributed in
accordance with their Lorentz factors ($\gamma_{e}$) and are described by the
electron power index ($p$) as
$N(\gamma_{e})\,d\gamma_{e}\propto\gamma_{e}^{-p}\,d\gamma_{e}$ for
$\gamma_{m}\leq\gamma_{\rm e}$, where $\gamma_{m}$ is the minimum electron
Lorentz factor. The comoving magnetic field in the blastwave $B^{\prime
2}/(8\pi)=\varepsilon_{B}\,U_{B}$ can be obtained from the energy density
$U_{B}=[(\hat{\gamma}\Gamma+1)/(\hat{\gamma}-1)](\Gamma-1)n(r)m_{p}c^{2}$
where $\hat{\gamma}$ is the adiabatic index (Huang et al., 1999), and $c$ is
the speed of light. Given the electron energy density, and the synchrotron,
dynamical and the acceleration timescales, the minimum, cooling and the
maximum electron Lorentz factors become

$\displaystyle\gamma_{\rm m}$ $\displaystyle=$ $\displaystyle 2.9\times
10^{3}\,g_{p}\left(\frac{1+z}{1.022}\right)^{\frac{3-k}{2(4-k)}}\,\,\varepsilon_{e,-1}\,A_{\rm
k,-1}^{-\frac{1}{2(4-k)}}\,E_{53}^{\frac{1}{2(4-k)}}\,t_{2}^{-\frac{3+8a-k(1+2a)}{2(4-k)}},$
(1) $\displaystyle\gamma_{\rm c}$ $\displaystyle=$ $\displaystyle 2.1\times
10^{2}\,(1+Y)^{-1}\left(\frac{1+z}{1.022}\right)^{-\frac{1+k}{2(4-k)}}\,\varepsilon^{-1}_{B,-2}A_{\rm
k,-1}^{-\frac{5}{2(4-k)}}E_{53}^{-\frac{3-2k}{2(4-k)}}\,t_{2}^{\frac{1+8b+k(1-2b)}{2(4-k)}},$
(2) $\displaystyle\gamma_{\rm max}$ $\displaystyle=$ $\displaystyle 1.5\times
10^{7}\,\left(\frac{1+z}{1.022}\right)^{-\frac{3}{4(4-k)}}\,\varepsilon^{-\frac{1}{4}}_{B,-2}\,A_{\rm
k,-1}^{-\frac{3}{4(4-k)}}\,E^{-\frac{1-k}{4(4-k)}}_{53}\,t_{2}^{\frac{3+b(4-k)}{4(4-k)}}\,,\hskip
170.71652pt$ (3)

respectively, where $g_{p}=\frac{p-2}{p-1}$ and $Y$ is the Compton parameter
(Sari & Esin, 2001; Wang et al., 2010). Given the comoving magnetic field, the
bulk and electron Lorentz factors, the synchrotron spectral breaks $\nu^{\rm
syn}_{\rm i}=\frac{hq_{e}}{2\pi m_{e}c}(1+z)^{-1}\Gamma\gamma^{2}_{\rm
i}B^{\prime}$ with ${\rm i=m,c}$ become

$\displaystyle h\nu_{\mathrm{m}}^{\rm syn}$ $\displaystyle=$ $\displaystyle
10.9\,{\rm
eV}\,g_{p}^{2}\left(\frac{1+z}{1.022}\right)^{\frac{1}{2}}\varepsilon_{\rm
e,-1}^{2}\varepsilon_{\rm
B,-2}^{\frac{1}{2}}E_{53}^{\frac{1}{2}}t_{2}^{-\frac{3+4a+b}{2}},$ (4)
$\displaystyle h\nu_{\mathrm{c}}^{\rm syn}$ $\displaystyle=$ $\displaystyle
5.5\times 10^{-2}\,{\rm
eV}\,\left(\frac{1+z}{1.022}\right)^{-\frac{k+4}{2(4-k)}}A_{\rm
k,-1}^{-\frac{4}{4-k}}(1+Y)^{-2}\varepsilon_{\rm
B,-2}^{-\frac{3}{2}}E_{53}^{\frac{3k-4}{2(4-k)}}t_{2}^{\frac{3k+3b(4-k)-4}{2(4-k)}}.\hskip
139.41832pt$ (5)

The terms $h$, $m_{e}$ and $q_{e}$ correspond to the Planck constant, electron
mass and the elementary charge, respectively. Using the synchrotron radiation
power per electron
$P_{\nu_{i}}\simeq\frac{\sqrt{3}q_{e}^{3}}{m_{e}c^{2}}(1+z)^{-1}\Gamma
B^{\prime}$ and the total number of emitting electrons
$N_{e}=\frac{4\pi}{3-k}n(r)r^{3}$ with $r\simeq\frac{2\Gamma^{2}c}{(1+z)}t$
the shock radius, the maximum flux of synchrotron radiation $F^{\rm syn}_{\rm
max}=\frac{(1+z)^{2}}{4\pi D_{z}^{2}}N_{e}P_{\nu_{i}}$, becomes

$\displaystyle F_{\mathrm{\rm max}}^{\rm syn}$ $\displaystyle=$ $\displaystyle
86.2\,{\rm mJy}\,\left(\frac{1+z}{1.022}\right)^{\frac{16-3k}{2(4-k)}}A_{\rm
k,-1}^{\frac{2}{4-k}}D_{\rm z,26.5}^{-2}\varepsilon_{\rm
B,-2}^{\frac{1}{2}}E_{53}^{\frac{8-3k}{2(4-k)}}t_{2}^{\frac{b(k-4)-k}{2(4-k)}}\,,\hskip
176.407pt$ (6)

where $d_{\rm
z}=(1+z)\frac{c}{H_{0}}\int^{z}_{0}\,\frac{d\tilde{z}}{\sqrt{\Omega_{\rm
M}(1+\tilde{z})^{3}+\Omega_{\Lambda}}}$ (Weinberg, 1972) is the luminosity
distance. Taking into account the evolution of spectral breaks and maximum
flux of the synchrotron forward-shock model with variation of the
microphysical parameters (Eqs. 4 and 6), we derive the CRs shown in Table 1.
For the case of $a=0$ and $b=0$, the microphysical parameters do not evolve,
and the synchrotron forward-shock model and CRs derived in Sari et al. (1998);
Chevalier & Li (2000); Panaitescu et al. (1998) are recovered for the
constant-density ($k=0$) and stellar-wind ($k=2$) environments.

When the same set of electrons that emits synchrotron radiation also up-
scatters those photons to higher energy, we have the SSC mechanism. Given the
electron Lorentz factors (Eq. 1) and synchrotron spectral breaks (Eq. 4), the
SSC spectral breaks $h\nu^{\rm ssc}_{\rm i}\simeq\gamma^{2}_{\rm i}h\nu^{\rm
syn}_{\rm i}$ can be written as (e.g., see Sari & Esin, 2001)

$\displaystyle h\nu_{\mathrm{m}}^{\rm ssc}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-1}\,{\rm
GeV}\,g_{p}^{4}\left(\frac{1+z}{1.022}\right)^{\frac{10-3k}{2(4-k)}}A_{\rm
k,-1}^{-\frac{1}{4-k}}\varepsilon_{\rm e,-1}^{4}\varepsilon_{\rm
B,-2}^{\frac{1}{2}}E_{53}^{\frac{6-k}{2(4-k)}}t_{2}^{\frac{bk+8ak+5k-4b-32a-18}{2(4-k)}},$
(7) $\displaystyle h\nu_{\mathrm{c}}^{\rm ssc}$ $\displaystyle=$
$\displaystyle 4.8\times 10^{-6}\,{\rm
GeV}\,\left(\frac{1+z}{1.022}\right)^{-\frac{3(k+2)}{2(4-k)}}A_{\rm
k,-1}^{-\frac{9}{4-k}}(1+Y)^{-4}\varepsilon_{\rm
B,-2}^{-\frac{7}{2}}E_{53}^{\frac{7k-10}{2(4-k)}}t_{2}^{\frac{5k+7b(4-k)-2}{2(4-k)}}.\hskip
184.9429pt$ (8)

Given the maximum synchrotron flux (Eq. 6), the maximum flux of the SSC
process $F^{\rm ssc}_{\rm max}\sim\sigma_{T}n(r)\,r\,F^{\rm syn}_{\rm max}$
yields

$\displaystyle F_{\mathrm{\rm max}}^{\rm ssc}$ $\displaystyle=$ $\displaystyle
3.6\times 10^{-4}\,{\rm
mJy}\,g_{p}^{-1}\,\left(\frac{1+z}{1.022}\right)^{\frac{14-k}{2(4-k)}}A_{\rm
k,-1}^{\frac{5}{4-k}}D_{\rm z,26.5}^{-2}\varepsilon_{\rm
B,-2}^{\frac{1}{2}}E_{53}^{\frac{5(2-k)}{2(4-k)}}t_{2}^{\frac{2-b(4-k)-3k}{2(4-k)}}\,.\hskip
176.407pt$ (9)

Taking into account the evolution of the spectral breaks and maximum flux of
the SSC forward-shock model with variation of the microphysical parameters
(Eqs. 7 and 9), we derive the CRs shown in Table 2. For the case of $a=0$ and
$b=0$, the microphysical parameters are constant, and the SSC forward-shock
model including CRs presented in Fraija et al. (2022) is recovered for the
constant-density ($k=0$) and stellar-wind ($k=2$) environments.

###  2.1 Hard spectral index ($1\leq p\leq 2$)

For an electron distribution with a hard spectral index ($1\leq p\leq 2$), the
minimum Lorentz factor is $\gamma_{\rm
m}=\left[\frac{m_{p}}{m_{e}}\tilde{g}_{p}\epsilon_{e}\Gamma\gamma_{\rm
max}^{p-2}\right]^{\frac{1}{p-1}}$ with $\tilde{g}_{p}=\frac{2-p}{p-1}$ (e.g.,
see Dai & Cheng, 2001). Given the bulk Lorentz factor and the maximum energy
that electrons can radiate, the minimum Lorentz factor becomes

$\gamma_{\rm m}=7.4\times
10^{2}\,\tilde{g}_{p}^{\frac{1}{p-1}}\,\left(\frac{1+z}{1.022}\right)^{\frac{3(4-p)-2k}{4(4-k)(p-1)}}\,\varepsilon^{\frac{1}{p-1}}_{e,-1}\varepsilon^{\frac{2-p}{4(p-1)}}_{B,-2}A_{\rm
k,-1}^{\frac{4-3p}{4(4-k)(p-1)}}E_{53}^{\frac{4-p-k(2-p)}{4(4-k)(p-1)}}\,t_{2}^{-\frac{12+4a(4-k)-2k+b(4-k)(2-p)-3p}{4(4-k)(p-1)}}\,.$
(10)

Eq. 10 is different from the minimum electron Lorentz factor (Eq. 1) usually
derived for $p>2$. Consequently, the characteristic spectral break $\nu^{\rm
syn}_{\rm m}=\frac{hq_{e}}{2\pi m_{e}c}(1+z)^{-1}\Gamma\gamma^{2}_{\rm
i}B^{\prime}$ for the synchrotron forward-shock model becomes

$h\nu_{\mathrm{m}}^{\rm syn}=0.67\,{\rm
eV}\,\tilde{g}_{p}^{\frac{2}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{14+k(p-3)-5p}{2(4-k)(p-1)}}A_{\rm
k,-1}^{\frac{2-p}{2(4-k)(p-1)}}\varepsilon_{\rm
e,-1}^{\frac{2}{p-1}}\varepsilon_{\rm
B,-2}^{\frac{1}{2(p-1)}}E_{53}^{\frac{p+2-k}{2(4-k)(p-1)}}t_{2}^{\frac{(4a+b)(k-4)+k-3p+kp-6}{2(4-k)(p-1)}}\,.$
(11)

For $a=0$ and $b=0$, the typical characteristic spectral breaks evolving as
$\nu_{\mathrm{m}}^{\rm syn}\propto t^{-\frac{3(p+2)}{8(p-1)}}$ and $\propto
t^{-\frac{p+4}{4(p-1)}}$ for constant and wind medium, respectively, are
recovered (Dai & Cheng, 2001). In this case, the SSC spectral break $h\nu^{\rm
ssc}_{\rm i}\simeq\gamma^{2}_{\rm i}h\nu^{\rm syn}_{\rm i}$ evolving during
the afterglow becomes

$\displaystyle h\nu_{\mathrm{m}}^{\rm ssc}=7.4\times 10^{-4}\,{\rm
GeV}\,\tilde{g}_{p}^{\frac{4}{p-1}}\left(\frac{1+z}{1.022}\right)^{\frac{26+k(p-5)-8p}{2(4-k)(p-1)}}A_{\rm
k,-1}^{\frac{3-2p}{(4-k)(p-1)}}\varepsilon_{\rm
e,-1}^{\frac{4}{p-1}}\varepsilon_{\rm
B,-2}^{\frac{3-p}{2(p-1)}}E_{53}^{\frac{6+k(p-3)}{2(4-k)(p-1)}}t_{2}^{\frac{8a(k-4)-18+3k-b(k-4)(p-3)+kp}{2(4-k)(p-1)}}.$
(12)

Tables 1 and 2 show the CRs of synchrotron and SSC forward-shock model,
respectively, for a hard spectral index ($1\leq p\leq 2$). For the case of
$a=0$ and $b=0$, the microphysical parameters are constant, and the
synchrotron CRs presented in Dai & Cheng (2001) are recovered for the
constant-density ($k=0$) and stellar-wind ($k=2$) environments.

##  3 Fermi-LAT GRB Catalog and Closure Relations

###  3.1 The methodical approach

We chose a sample of 86 LAT-detected bursts from the extensive list provided
in 2FLGC (Ajello et al., 2019). The duration of the high-energy emission from
these bursts, which was temporally extended, was between 31 seconds (for GRB
141102) and 34,366 seconds (for GRB 130427). In 2FLGC, the temporal emission
is described with a SPL function; $F(t)\propto t^{-\alpha_{\rm LAT}}$, or a
BPL: $F(t)\propto t^{-\alpha_{\rm LAT,1}}$ for $t<t_{\rm br,LAT}$ and $\propto
t^{-\alpha_{LAT,2}}$ for $t_{\rm br,LAT}<t$ with $t_{\rm br,LAT}$ the break
time. Similarly, the spectral emission is reported through the photon index
$\Gamma_{\rm LAT}=\beta_{\rm LAT}+1$. For this analysis, we follow our
previous strategy exhibited in Dainotti et al. (2021a, b); Srinivasaragavan et
al. (2020). The uncertainties displayed in Figures 3 and 4 are in the
1$\sigma$ range, and we take into account the uncertainties among the $\alpha$
and $\beta$ values to be dependent; thus, these figures are displayed with
ellipses. We require the temporal and spectral indices determined by the PL
function $F_{\rm\nu}\propto t^{-\alpha}\nu^{-\beta}$ to examine the CRs. We
use the temporal and spectral indexes from the 2FLGC. Using the CRs of the
synchrotron and SSC afterglow model derived and listed in Tables 1 and 2, we
investigate if these CRs satisfy the spectral and temporal index of those
bursts reported in 2FLGC.

Figure 1 shows the corner plots of Markov Chain Monte Carlo (MCMC) parameter
estimation ($a$ and $b$) in the scenario of the synchrotron CRs. Table 3
displays the results from MCMC simulations of the synchrotron afterglow model
evolving in a stratified circumburst medium. Each column in Figure 1
corresponds to a different circumburst environment, with the first column
taking the value ${\rm k=0}$ and each subsequent column having a $0.5$
increment from the previous one. The rows follow a different cooling
condition. The first row is in the slow cooling regime
(${\rm\nu_{m}^{syn}<\nu_{\rm LAT}<\nu_{c}^{syn}}$), the second is in the fast
cooling regime (${\rm\nu_{c}^{sync}<\nu_{\rm LAT}<\nu_{m}^{sync}}$) and the
third one is for the condition ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$. Each histogram shows the
marginalized posterior densities with the median values in red lines. Focusing
on the first row, we notice that, in general, $a$ takes negative values and
$b$ positive ones. This means that for our sample of GRBs, the electron
microphysical parameter increases in time while the magnetic one decreases. By
comparing each column in this row, we note that as the value of $\rm k$ is
increased, the magnitude of the power indexes decreases; namely, for $\rm
k=0$, we have $(a,b)=(-1.50,2.77)$ and for $\rm k=2.5$ we find
$(a,b)=(-1.08,1.11)$, so as the stratification parameter increases, the time
evolution of the microphysical parameters becomes weaker. Regarding the third
row, we find that the behavior is the opposite, namely that $a$ takes positive
values and $b$ negative ones. We also notice that variation of the
stratification in this cooling condition leads to a negligible variation of
the best-fit values of the power indexes, especially for $a$, as we find that
for $\rm k=0$, we have $(a,b)=(0.34,-4.74)$ and for $\rm k=2.5$ we find
$(a,b)=(0.32,-4.62)$. Finally, the parameter estimation in the second row,
which corresponds to the fast cooling regime, is not valid, as the corner
plots show no clear region where the index $a$ is extreme. In this case, no
further analysis is possible.

Figure 2 displays the corner plots of MCMC parameter estimation ($a$ and $b$)
in the case of the SSC CRs. Table 3 displays the results from MCMC simulations
of the SSC afterglow model evolving in a stratified circumburst medium. Figure
2 follows the same conventions as the ones in Figure 1, with the sole
difference being that the cooling frequencies are the SSC frequencies instead
of the synchrotron ones. In the case of the SSC CRs, we generally find that
the best-fit parameters are substantially different from the ones in the
synchrotron scenario. We also find that the second row of this Figure suffers
the same problem as the second row of Figure 1, so no parameter estimation is
possible in this case. In the case of the first row, $a$ takes negative
values, but as $\rm k$ increases, they become more positive. On the other
hand, $b$ takes positive values for $0\leq k\leq 2.0$, and as $\rm k$
increases, they decrease until a change of sign for $\rm k=2.5$. For the final
row, we notice that similar to Figure 1, variation in $\rm k$ does not
substantially change the best-fit values of $a$ and $b$. However, we notice
that in this cooling condition, both indexes are negative, which is a stark
contrast to the synchrotron case in which the indexes had different signs.

Figure 3 shows the CRs of the synchrotron forward-shock model evolving in a
stratified medium, from top to bottom $k=0$, $0.5$, $1$, $1.5$, $2$ and $2.5$,
respectively, with the best-fit values obtained with our MCMC and listed in
Table 3. The fast and slow regimes for each cooling condition are shown for
$1<p<2$ and $p>2$. Table 4 summarizes the number and percentage of bursts
following each cooling condition for $1<p<2$ and $p>2$ and the profile index
$k=0$, $0.5$, $1$, $1.5$, $2$ and $2.5$. For the cooling condition
${\rm\nu_{m}^{syn}<\nu_{\rm LAT}<\nu_{c}^{syn}}$ the best case is most
consistent with ${\rm k=0,1.0,1.5,2}$ (23 GRBs, 27.06%). For the cooling
condition ${\rm\nu_{c}^{syn}<\nu_{\rm LAT}<\nu_{m}^{syn}}$, all cases of ${\rm
k}$ are equally likely (1 GRB, 1.18%). For the cooling condition ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$, the best case is most
consistent with ${\rm k=1}$ (24 GRBs, 28.24%) and worst scenario is for ${\rm
k=2.5}$ (21 GRBs, 24.71%).

Figure 4 shows the CRs of the SSC forward-shock model evolving in a stratified
medium, from top to bottom $k=0$, $0.5$, $1$, $1.5$, $2$ and $2.5$,
respectively, with the best-fit values obtained with our MCMC and listed in
Table 3. The fast and slow regimes for each cooling condition are shown for
$1<p<2$ and $p>2$. Table 5 summarizes the number and percentage of bursts
following each cooling condition for $1<p<2$ and $p>2$ and the profile index
$k=0$, $0.5$, $1$, $1.5$, $2$ and $2.5$. For the cooling condition
${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$ the best case is most
consistent with ${\rm k=0}$ (23 GRBs, 27.06%) followed by all cases with 22
GRBs (25.88%). For the cooling condition ${\rm\nu_{c}^{ssc}<\nu_{\rm
LAT}<\nu_{m}^{ssc}}$, all cases of ${\rm k}$ are equally likely (1 GRB,
1.18%). For the cooling condition ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$, the best cases are most
consistent with ${\rm k=0}$ and ${\rm k=0.5}$ (24 GRBs, 28.24%).

We have obtained the synchrotron and SSC closure relations in the slow- and
fast-cooling regimes, which provide the function $\alpha(\beta;a;b)$. We have
noticed that not all cooling conditions are a function of $a$ and $b$
simultaneously, which lead to an inability to obtain a relation between both
parameters in such cases. This, in turn, lead to a degeneracy in either $a$ or
$b$, which appeared in our analysis as a huge uncertainty in our MCMC
simulations.

###  3.2 The light curves and Spectral Energy Distributions: Conditions for
Plateaus and Peaks in 2FLGC

Tables 6 and 7 show the evolution of the synchrotron and SSC light curves with
microphysical parameter variations, respectively, from an outflow that
decelerates in a stellar wind ($k=2$) and constant-density ($k=0$) medium, and
an electron distribution with spectral index in the range of $1<p<2$ and
$p>2$. The evolution of temporal indexes is estimated using the synchrotron
and SSC spectra for fast- and slow-cooling regimes together with the
respective spectral breaks (Eqs. 7 and 12) and the maximum flux (Eq. 9).

Figure 5 presents the SSC light curves and spectra obtained from the uniform-
density afterglow model with a variation of microphysical parameters. The
first and second columns show the spectra at 10 and 100 s, respectively, with
the CTA, MAGIC, and Fermi-LAT sensitivities in colored solid lines. The third
and fourth columns display the light curves for 1 and 10 GeV, respectively.
The first two rows have a value of the electron power index of $p=1.7$, while
the last two have $p=2.4$. The first and third rows consider a variation of
the $a$ parameter while keeping $b=0$ fixed. In this case, the colored lines
correspond to $a=0$ (gray), $a=0.25$ (red), $a=0.5$ (blue) and $a=0.75$
(green). On the other hand, the second and fourth rows keep $a=0$ fixed and
vary the value of $b$. For these rows, the colored lines represent $b=0.25$
(gray), $b=0.5$ (red), $b=0.75$ (blue) and $b=1.25$ (green). Regarding the
different spectra, we notice that variations in either $a$ or $b$ would be
observable by MAGIC and CTA, but not by Fermi-LAT as its sensitivity is not
enough to capture a sufficient part of the spectra. In all panels, we can see
that an increase in any of the microphysical parameters will, most of the
time, lead to an increase in the flux density for small energy. However, this
behavior depends on the energy, as we note in the panels in the last row,
where the gray and red curves dominate for considerable energies. We also
observe that differences in the power index of the microphysical parameters
are not very apparent if the flux density is almost constant, as displayed in
the panel in the first-row second column. In the case of the light curves, we
note that variation in $a$ and $b$ will lead to changes in the cooling
regimes, as exemplified by the panels in the second and last rows. Upon
comparing the first and third rows (second and fourth), we also notice that
variations in the $b$ parameter lead to more considerable changes in the light
curves than variations in $a$.

Figure 6 is analogous to Figure 5, with the only difference being that this
Figure displays the SSC light curves and spectra obtained with the stellar-
wind afterglow model. We note that the spectra predicted by the stellar-wind
model are generally flatter than those in the constant-density medium. We also
notice that for $2<p$, an increase in the power index of the microphysical
parameters leads to less flux density, in contrast to the $1<p<2$ case and the
whole constant-density model. In the case of the light curves, in the same
manner as in Figure 5, changes in $a$ and $b$ may lead to changes in the
cooling conditions, an effect that is most apparent when varying $b$. Compared
to the constant-density case, the wind model predicts more significant fluxes
at early times but with steeper decays at later times. Additionally, we show
that a plateau phase or a MeV-GeV peak in the LAT energy range caused by
synchrotron and SSC processes could be expected depending on the values of $a$
and $b$.

####  3.2.1 Parameter conditions for describing high-energy plateaus

Like the X-ray afterglows recorded by the Swift-XRT instrument, many GRBs
display temporally prolonged high-energy emissions as measured by the Fermi-
LAT calorimeter. There is a late-time flattening in certain LAT light curves
that is suggestive of X-ray plateaus. Dainotti et al. (2021b) investigated the
presence of late-time flattening in LAT observations reported in 2FLGC,
offering evidence that plateaus exist in the light curves. They argued that
the identical phenomenological model required to interpret X-ray plateaus
might be used to describe some of the most energetic bursts; GRB 090510, GRB
090902B, and GRB 160509A, which exhibited gamma-ray photons above 10 GeV. The
synchrotron and SSC flux could mimic a late-time flattening in any cooling
condition of the light curves for both a constant-density and stellar-wind
environment. This condition is fulfilled when the light curve’s time
dependence vanishes when the temporal variable’s power index becomes zero. For
the synchrotron forward-shock model with $p>2$, the value of the parameter $b$
as a function of $a$ in this case becomes $b=1$, $b=-\frac{(3+4a)(p-1)}{p+1}$
($=\frac{1-3p-4a(p-1)}{p+1}$) for $k=0$ ($k=2$) and
$b=\frac{2-3p-4a(p-1)}{p-2}$, for the cooling conditions
${\rm\nu_{c}^{syn}<\nu_{\rm LAT}<\nu_{m}^{syn}}$, ${\rm\nu_{m}^{syn}<\nu_{\rm
LAT}<\nu_{c}^{syn}}$ and ${\rm max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm
LAT}}$, respectively. It is worth noting that the conditions for
${\rm\nu_{c}^{sync}<\nu_{\rm LAT}<\nu_{m}^{syn}}$ and ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$ are independent of the
density profile. Similarly, for the SSC forward-shock model, $b=-\frac{1}{10}$
($=0$), $b=\frac{11-9p-16a(p-1)}{2(p+1)}$ ($=-\frac{4p+8a(p-1)}{p+1}$) and
$b=\frac{10-9p-16a(p-1)}{2(6-p)}$ ($=-\frac{4(2a+1)(p-1)}{6-p}$) for $k=0$
($k=2$), for the cooling conditions ${\rm\nu_{c}^{ssc}<\nu_{\rm
LAT}<\nu_{m}^{ssc}}$, ${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$ and
${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$, respectively.
Therefore, with the appropriate values, the late-time flattening could explain
GRB 090510, GRB 090902B, and GRB 160509A.

The parameter conditions for $a$ and $b$ in the synchrotron model and the
best-fit values of the parameter $a$ and $b$ shown in Table 3 indicate that
the cooling conditions for synchrotron afterglow scenario ${\rm\nu_{c}^{\rm
ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ and ${\rm max\\{\nu_{m}^{\rm
syn},\nu_{c}^{\rm syn}\\}<\nu_{\rm LAT}}$ could exhibit plateaus for both a
homogeneous medium and a stellar wind. In this case, the fraction of energy
given to the electrons decreases and to the magnetic field increases, so that
these variation could be associated with shell collisions. This turbulence
distorts the field lines proving additional magnetic reconnections. It leads
to the release of the accumulated magnetic field energy on the runway (an
ICMART666the Internal-Collision-induced MAgnetic Reconnection and Turbulence.
event; Zhang & Yan, 2011). It is worth noting that the no coincidence of
neutrinos with GRBs (Abbasi et al., 2012; Aartsen et al., 2016, 2015) could be
explained in some magnetic dissipation scenarios such as the ICMART model
(Zhang & Kumar, 2013). Similarly, the parameter conditions for $a$ and $b$ in
the SSC model with the best-fit values of the parameter for SSC afterglow
model shown in Table 3 suggest that the cooling conditions ${\rm\nu_{c}^{\rm
ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ and ${\rm max\\{\nu_{m}^{\rm
syn},\nu_{c}^{\rm syn}\\}<\nu_{\rm LAT}}$ could exhibit high-energy plateaus.
For these these cooling conditions and with the values of $a$ and $b$, the
fraction of energy given to accelerate electrons and amplified magnetic field
increases at the same time, so that these variation could be associated with
continuous energy injection with different efficiencies (Ioka et al., 2006;
Granot et al., 2006).

####  3.2.2 Parameter conditions to explain high-energy peaks

The 2FLGC exhibited a subset of GRBs with a short-lasting peak at the end of
prompt episode and the beginning of the long-lasting emission (e.g., see GRB
080916C, GRB 090510A, GRB 090902B, GRB 090926A, GRB 110731A, GRB 130427A and
GRB 160509A Abdo et al., 2009a, b; Ackermann & et al., 2010; Ackermann et al.,
2011; Ackermann & et al., 2013; Ackermann et al., 2014). For instance, GRB
160509A displayed a bright bump peaking at $\sim$ 20 s, followed by the long-
lasting emission. The synchrotron and SSC flux could mimic a short-lasting
bump when the forward shock evolves in a constant-density and stellar-wind
environment.

The increasing flux $t^{-\alpha}$ with $\alpha>-2$. For the synchrotron
forward-shock model with $p>2$, $b<9$, $b>-\frac{5+3p+4a(p-1)}{p+1}$
($>-\frac{7+3p+4a(p-1)}{p+1}$) for $k=0$ ($k=2$) and
$b>-\frac{3(p+2)+4a(p-1)}{p-2}$, for the cooling conditions ${\rm\nu_{c}^{\rm
syn}<\nu_{\rm LAT}<\nu_{m}^{\rm syn}}$, ${\rm\nu_{m}^{\rm syn}<\nu_{\rm
LAT}<\nu_{c}^{\rm syn}}$ and ${\rm max\\{\nu_{m}^{\rm syn},\nu_{c}^{\rm
syn}\\}<\nu_{\rm LAT}}$, respectively. It is worth noting that the conditions
for ${\rm\nu_{c}^{\rm syn}<\nu_{\rm LAT}<\nu_{m}^{\rm syn}}$ and ${\rm
max\\{\nu_{m}^{\rm syn},\nu_{c}^{\rm syn}\\}<\nu_{\rm LAT}}$ are independent
of the density profile. Similarly, for SSC forward-shock model,
$b<\frac{3}{2}$ ($<\frac{8}{5}$), $b>-\frac{5+9p+16a(p-1)}{2(p+1)}$
($>-\frac{4[(p+2)+2a(p-1)]}{p+1}$) and $b>-\frac{3(3p+2)+16a(p-1)}{2(p-6)}$
($-\frac{4[(p+1)+2a(p-1)]}{p-6}$) for $k=0$ ($k=2$), for the cooling
conditions ${\rm\nu_{c}^{ssc}<\nu_{\rm LAT}<\nu_{m}^{ssc}}$,
${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$ and ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$, respectively. Therefore,
the high-energy peaks exhibited in particular bursts could be described by the
variation of the microphysical parameters in the synchrotron and SSC scenario.
The conditions of $a$ and $b$ to emulate a bright peak can be satisfied when
the instantaneous energy injection is supplied for a small time interval; so
that the microphysical parameters only evolve during a short period. It is
important mentioning that a superposition of synchrotron and SSC process could
describe the high-energy peaks.

###  3.3 Variation of Magnetic Parameter in Decaying Microturbulence

In the GRB afterglow model, particle acceleration in a decelerating
relativistic, collisionless shock front is an essential component (Mészáros &
Rees, 1997). The long-standing questions regarding where the magnetic fields
that penetrate the blast waves of the GRB originate and the mechanisms by
which energy is transported from particles (e.g., protons and electrons) and
magnetic fields in relativistic shocks have not been fully answered.
Relativistic collisionless shock front formation in a weakly magnetized
surrounding medium is explained by the self-generation of intense small-scale
electromagnetic fields mediating the transition between the upstream unshocked
and the downstream shocked media. Accelerated particles play a crucial role in
setting off the microinstabilities that eventually create the self-generated
field. The self-generated microturbulence controls these particles’
dispersion, which in turn guides the complexly non-linear acceleration
process. High-performance particle-in-cell (PIC) simulations have so far
confirmed the validity of this entire approach (e.g., Spitkovsky, 2008; Sironi
& Spitkovsky, 2011, 2009; Martins et al., 2009).

The magnetic field amplification may occur in the GRB’s afterglow shocks and
is linked to the generation of turbulence at a small scale. The analysis
listed in Table 3 shows that the magnetic field is amplified only in the case
of slow cooling regime (${\rm\nu_{m}^{\rm j}<\nu<\nu_{c}^{\rm j}}$) for either
synchrotron or SSC model. In this case, the parameters $a$ and $b$ lies in the
range of $-1.50\leq a\leq-1.08$ and $1.11\leq b\leq 2.77$ for synchrotron and
$-1.00\leq a\leq-0.37$ and $0.77\leq b\leq 3.27$ for SSC. This Table also
shows that the magnetic field is not amplified for $k=2.5$ in the SSC
scenario. It may be due to differences between two collisionless populations
of particles (Medvedev & Loeb, 1999). Due to the generation of
microturbulence, the Fermi acceleration mechanism must be strongly influenced
by it. The results of Lemoine et al. (2006) suggest that Fermi acceleration is
efficient when the magnetic field is amplified in scales shorter than the
typical Larmor radius; in this case, turbulence helps particles gain energy by
completing cycles through the shock front. The formation of a microturbulent
layer guarantees the scattering of accelerated particles. This is the case of
the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
LAT}}$ for SSC scenario where the values of $a$ and $b$ lie in the range of
$-0.46\leq a\leq-0.36$ and $-1.28\leq b\leq-1.02$, as shown in Table 3. In
contrast, in the absence of this layer, particles are advected away with a
transverse magnetic field, suppressing the acceleration process. Table 3 shows
that this criterion is not satisfied for any cooling condition. Therefore, to
understand how particle energy increases via Fermi acceleration and how it
decreases via synchrotron radiation, it is crucial to know how quickly and how
far the microturbulent layer decays (Lemoine, 2013). The author considered the
variation of the magnetic parameter $\propto t^{-b}$ and discussed the effects
of the parameter $b$ on the resulting spectrum for synchrotron and dominant
inverse Compton, identifying two scenarios. In the first scenario, turbulence
gradually decays with $-1<b<0$ and $b>-4/(p+1)$ for synchrotron and dominant
inverse Compton, respectively, and in the second scenario, turbulence rapidly
decays with $b<-1$ for synchrotron model and $-3<b<-4/(p+1)$ for dominant
inverse Compton.

The best-fit values of the parameter $b$ shown in Table 3 indicate that only
the cooling conditions ${\rm\nu_{m}^{\rm syn}<\nu_{\rm LAT}<\nu_{c}^{\rm
syn}}$ and ${\rm\nu_{m}^{\rm ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ lie in the
scenario of rapidly decaying microturbulence for both synchrotron and SSC
processes.

###  3.4 Bursts with a hard spectral index ($1\leq p\leq 2$) in 2FLGC

Table 8 shows the selection of bursts from 2FLGC with spectral index $\beta<1$
and temporal index $\alpha>1.3$, which can hardly be modeled with the CRs of
the standard synchrotron forward-shock model. Tak et al. (2019) showed that
the CRs of the slow-cooling regime ${\rm\nu_{m}^{syn}<\nu_{\rm
LAT}<\nu_{c}^{syn}}$ for $2<p$ could satisfy the evolution of temporal and
spectral index of these bursts, but only with the condition of having an
atypical small value of $\epsilon_{B}<10^{-7}$. This condition must be
considered so that the cooling spectral break is larger than the LAT energy
band ($\nu_{\rm LAT}<\nu_{c}^{syn}$). Ghisellini et al. (2010) modeled the
high-energy emission of 11 GRBs (up until October 2009) identified by the
Fermi-LAT. Based on the LAT observations, the authors determined that the
relativistic forward shock was in the utterly radiative regime for a hard
spectral index $p\sim 2$, $\sim t^{-\frac{10}{7}}$ instead of $t^{-1}$, as
anticipated by the adiabatic synchrotron forward-shock model, being a slow-
cooling regime in the constant-density medium the most favorable afterglow
model. The scenario of microphysical parameter variation could explain the
evolution of this GRB sample’s temporal and spectral indexes. For instance, we
can see that the only cooling conditions for the synchrotron model that
satisfy these bursts would be ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$ for $1\leq p\leq 2$.

For completeness, we now include the CRs of the SSC model in our analysis.
Figure 7 shows the PL indices of the microphysical parameters for which the
CRs are fulfilled in the constant-density medium model for a sample of 8 GRBs.
Each row corresponds to a different cooling condition. The first one is
related to the cooling condition ${\rm
max\\{\nu_{m}^{sync},\nu_{c}^{sync}\\}<\nu_{\rm LAT}}$ for the synchrotron
model with $1<p<2$, the second one to ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$ for the SSC model with
$1<p<2$ and the third one to ${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$
for the SSC model with $p>2$. Each column corresponds to the particular GRB in
question. In all panels, the solid line stands for the relation between the
$a$ and $b$ parameters that satisfy the respective CR, while the gray region
represents the propagated uncertainty from the uncertainty in the temporal and
spectral PL indices from Ajello et al. (2019), summarized in Table 8. In the
first row, we bear in mind that the general behavior is maintained in all
panels, namely that we can only find the best-fit $a$ parameter, but we cannot
conclude on the $b$ parameter. This is easily explained by noticing that the
appropriate CR in Table 3 does not depend on $b$; as such, it remains
unconstrained in our analysis. We also note that a positive $a$ is preferred
for this cooling condition. Regarding the second row, we note that the best-
fit value of $a$ is proportional to $b$, and the same relation seems to be
valid for all GRBs except GRB 100116A, which prefers larger values of $a$. In
the case of the third row, we see that the relation between $a$ and $b$ is now
inversely proportional. In an analogous fashion to the second row, the same
relation is approximately valid for all GRBs except GRB 100116A, which, as in
the previous row, prefers larger values of $a$. It is essential to highlight
that the case for $a=0$ and $b=0$ (standard synchrotron model) is obtained
only in a few bursts when the uncertain regions are considered.

Figure 8 is analogous to Figure 7, with the sole difference being that we
consider the stellar-wind model here. The discussion for Figure 7 can be
repeated for this one, as the same behavior is repeated in all panels. In all
cases, all best-fit values are consistent between both Figures, so this type
of analysis does not lead to a way to differentiate between a constant-density
medium and a stellar wind.

###  3.5 Stellar-wind/constant-density medium

For both short and long-GRB progenitors, the external circumburst medium plays
an essential role in the hydrodynamics of the ejected outflows. A constant
low-density medium is expected for sGRBs, which are originated from the
merging of binary compact objects, such as a BH and a NS or NS-NS (Duncan &
Thompson, 1992; Usov, 1992; Thompson, 1994; Metzger et al., 2011). A
stratified environment with different density profiles is identified for long
GRBs, which are associated with the core collapse of massive stars (Woosley,
1993; Paczyński, 1998). In particular, a transition between a stratified and
constant-density environment is expected in the core-collapse scenario when
the wind from its progenitor is strong enough to overcome the deceleration
ratio (e.g., see Fraija et al., 2017, 2019a). The analysis shows that for the
synchrotron afterglow model, the stellar-wind scenario with ${\rm k=2}$ is the
most favorable one, followed by ${\rm k=2.5}$, and for the SSC afterglow
model, the stellar-wind scenario with ${\rm k=2.5}$ is the most favorable one
followed by ${\rm k=2.0}$. The least favorable scenario is for constant-
density medium in both the synchrotron and SSC afterglow models.

Irrespective of the circumburst medium, the synchrotron flux as a function of
the density evolves as $F_{\nu}\propto A_{\rm k}^{0}\,(A_{\rm k}^{0})$ for
${\rm\nu_{c}^{syn}<\nu_{\rm LAT}<\nu_{m}^{syn}}$, $\propto A_{\rm
k}^{\frac{10-p}{4(4-k)}}\,(A_{\rm k}^{\frac{2}{4-k}})$ for
${\rm\nu_{m}^{syn}<\nu_{\rm LAT}<\nu_{c}^{syn}}$ and $\propto A_{\rm
k}^{\frac{2-p}{4(4-k)}}\,(A_{\rm k}^{0})$ for ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$ for $1<p<2$ ($2<p$).
Similarly, the SSC flux as a function of the density varies as $F_{\nu}\propto
A_{\rm k}^{\frac{1}{2(4-k)}}\,(A_{\rm k}^{\frac{1}{2(4-k)}})$ for
${\rm\nu_{c}^{ssc}<\nu_{\rm LAT}<\nu_{m}^{ssc}}$, $\propto A_{\rm
k}^{\frac{13-2p}{2(4-k)}}\,(A_{\rm k}^{\frac{11-p}{2(4-k)}})$ for
${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$ and $\propto A_{\rm
k}^{\frac{2-p}{4-k}}\,(A_{\rm k}^{\frac{2-p}{2(4-k)}})$ for ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$ for $1<p<2$ ($2<p$). It
shows that an afterglow transition is expected in all cooling conditions for
the SSC scenario but only in some cases for the synchrotron model. For
instance, an afterglow transition is not expected when the synchrotron flux
evolves in the condition ${\rm\nu_{c}^{syn}<\nu_{\rm LAT}<\nu_{m}^{syn}}$ for
either $1<p<2$ or $2<p$.

###  3.6 The maximum energy photons of the synchrotron afterglow model

During forward shocks, electrons are accelerated up to relativistic energies
in a stratified environment through the Fermi mechanisms. The time required
for an electron with Lorentz factor $\gamma_{e}$ to go from one side of the
shock front to the other side is roughly equivalent to the Larmor time
$t^{\prime}_{\rm L}=\gamma_{e}m_{e}c/(q_{e}B^{\prime})$. It is possible to
determine the maximum photon energy output by the synchrotron by equaling the
accelerating ($t^{\prime}_{\rm acc}=\xi 2\pi t^{\prime}_{\rm L}$) and cooling
($t^{\prime}_{\rm syn}=6\pi m_{e}c/(\sigma_{T}\gamma_{e}B^{\prime 2})$) time
scales, with $\xi$ the Bohm parameter, and $\sigma_{T}$ the Thomson cross
section. The limit for synchrotron photons ($\nu^{\rm syn}_{\rm
max}=\frac{q_{e}}{2\pi m_{e}c}(1+z)^{-1}\Gamma\gamma^{2}_{\rm max}B^{\prime}$)
for a uniform magnetic field becomes (Kumar et al., 2012)

$h\nu^{\rm syn}_{\rm max}\approx 2.7\times 10^{2}\,{\rm
MeV}\left(\frac{1+z}{1.022}\right)^{\frac{k-5}{2(4-k)}}\,A_{\rm
k,-1}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}\,t_{2}^{-\frac{3-k}{2(4-k)}}\,.$
(13)

Within the 2FLGC, several bursts represent the extreme cases of Equation 13.
For instance, in the case of redshift variation, we have GRB 080916C and GRB
130702A with the redshifts $z=4.35$ and $z=0.14$, respectively. For a
constant-density medium, and keeping all other parameters in Eq. 13 fixed, the
maximum synchrotron energies would be MeV $h\nu^{\rm syn}_{\rm
max,080916C}\approx 96\ \rm MeV$ and $h\nu^{\rm syn}_{\rm max,130702A}\approx
252\ \rm MeV$, while in a stellar-wind medium we would have $h\nu^{\rm
syn}_{\rm max,080916C}\approx 78$ and $h\nu^{\rm syn}_{\rm max,130702A}\approx
249\ \rm MeV$. Therefore, the larger the redshift, the smaller the synchrotron
energy and this effect is amplified more significantly with the value of
stratification $k$. Similarly, in the case of LAT energy, the most significant
variation in the catalog is between GRB 080916C ($E_{\rm\gamma,iso}=2.3\times
10^{54}\ \rm erg$) and GRB 091127 ($E_{\rm\gamma,iso}=3\times 10^{50}\ \rm
erg$). In an analogous manner as before, in a constant-density (stellar-wind)
medium we would have the maximum synchrotron energy $h\nu^{\rm syn}_{\rm
max,080916C}\approx 400\ \rm MeV$ ($590\ \rm MeV$) and $h\nu^{\rm syn}_{\rm
max,091127}\approx 130\ \rm MeV$ ($63\ \rm MeV$). As such, the synchrotron
energy grows as the GRB’s energy increases, as expected. Finally, in the case
of time, we consider GRB 130427A with $t=3.44\times 10^{4}\ \rm s$. In a
constant-density (stellar-wind) medium, we obtain the maximum synchrotron
energies $h\nu^{\rm syn}_{\rm max,130427A}\approx 30\ \rm MeV$ ($63\ \rm
MeV$), so we notice that at later times the maximum energy decreases.
Considering this scenario and taking into account that the majority of bursts
reported in the 2FLGC have redshifts with $z>1$, isotropic energies
$E_{\rm\gamma,iso}\sim 10^{52}\,{\rm erg}$, circumburst densities around
$n(A_{\rm W})\simeq 1\,cm^{-3}\,(1)$ and durations larger than 100 s, the
limit for synchrotron photons is close or even less than 300 MeV. In this
case, the LAT energy band could be almost described by the SSC process.

On the other hand, due to a lack of knowledge on how long it takes electrons
to move from one side of the shock front to the other, this estimate might be
off by a factor of 5-10 (for a discussion, see Achterberg et al., 2001;
Lemoine & Revenu, 2006; Uchiyama et al., 2007; Sagi & Nakar, 2012).

Synchrotron photons with energies significantly higher than that shown in Eq.
(13) are possible under certain situations. This requires the magnetic field
to decrease behind the shock front on a scale far smaller than the path taken
by electrons with the highest energies before they lose 50 percent of their
original energy. In this case the magnetic field decays as $B(x)=B_{\rm
w}(x/L_{p})^{-\eta}+B_{0}$, were $B_{\rm 0}$ and $B_{\rm w}$ are the weakest
and strongest strengths of magnetic fields, respectively, and $L_{p}$ refers
to the length-scale in which the field gradually loses its strength with
$L_{p}\leq x$ (Kumar et al., 2012). The rate at which an electron loses energy
to synchrotron radiation as it travels behind the shock front is

$\frac{d(\gamma_{e}m_{e}c^{2})}{dt}=-\frac{\gamma_{e}\sigma_{T}c}{6\pi}\left[B_{\rm
w}\left(\frac{x}{L_{p}}\right)^{-\eta}+B_{0}\right]^{2}\,.$ (14)

The situation that attracts our attention is when electrons experience energy
loss when they pass through an area of weak magnetic field. In this case, the
limit for synchrotron photons becomes (Kumar et al., 2012)

$h\nu^{\rm syn}_{\rm max}\approx 2.7\times 10^{2}\,{\rm
MeV}\left(\frac{1+z}{1.022}\right)^{\frac{k-5}{2(4-k)}}\,A_{\rm
k,-1}^{-\frac{1}{2(4-k)}}E_{53}^{\frac{1}{2(4-k)}}\,t_{2}^{-\frac{3-k}{2(4-k)}}\left(\frac{B_{\rm
w}}{B_{\rm 0}}\right)\,,$ (15)

which exceeds the value predicted by Eq. (13) by a factor of $B_{\rm w}/B_{\rm
0}$. Therefore, considering the magnetic field of the interstellar medium with
$B_{\rm 0}=B_{\rm ism,-5}$ and $B_{\rm w}\simeq 8.3\,{\rm
G}\,\left(\frac{1+z}{1.022}\right)^{\frac{3}{2(4-k)}}\epsilon^{\frac{1}{2}}_{\rm
B}A_{\rm
k,-1}^{\frac{3}{2(4-k)}}E^{\frac{1-k}{2(4-k)}}_{53}t^{-\frac{3}{2(4-k)}}$, the
maximum energy of synchrotron radiation becomes $h\nu^{\rm syn}_{\rm max}\sim
10^{2}\,{\rm TeV}$. In this scenario, the synchrotron afterglow model would
almost describe the LAT energy band.

The synchrotron forward-shock model can only explain gamma-ray photons below
the synchrotron limit as reported in Table 7 in the 2FLGC, which listed 29
bursts with photons exceeding energies $>10\,{\rm GeV}$. On the other hand,
since the IceCube team has reported no coincidences of neutrinos with GRBs
(Abbasi et al., 2012; Aartsen et al., 2016, 2015), this data severely limits
the number of hadrons involved in photo-hadronic interactions, which might
yield photons with energy exceeding the synchrotron limit. Given that the
maximum photon energy generated by the SSC forward-shock model is above the
LAT energy range ($h\nu^{\rm ssc}_{\rm max}=\gamma^{2}_{\rm max}h\nu^{\rm
syn}_{\rm max}\gg 300\,{\rm GeV}$), the SSC mechanism is the most natural
process to interpret the photons observed by the Fermi-LAT instrument which is
beyond the synchrotron limit. It is worth noting that although the maximum
synchrotron energy for protons becomes $\sim 400\,{\rm GeV}$, this process is
very ineffective, and it is pretty unlikely that it will play an essential
role in GRBs.

###  3.7 Evolution of the magnetic microphysical parameter

The value of the magnetic microphysical parameter can be estimated through Eq.
7 and equaling the synchrotron cooling break with the Fermi-LAT band
($\nu^{\rm ssc}_{\rm c}=\nu_{\rm LAT}$). This parameter corresponds to

$\varepsilon_{B}\lesssim 3.4\times
10^{-5}\left(\frac{1+z}{1.022}\right)^{-\frac{3(k+2)}{7(4-k)}}\,(1+Y(\gamma_{c}))^{-\frac{8}{7}}A_{\rm
k,-1}^{-\frac{18}{7(4-k)}}E_{53}^{\frac{7k-10}{7(4-k)}}t_{2}^{\frac{5k+7b(4-k)-2}{7(4-k)}}\left(\frac{h\nu^{\rm
ssc}_{c}}{100\,{\rm MeV}}\right)^{-\frac{2}{7}}\,,$ (16)

which is in the usual range of those values ($10^{-5}\leq\varepsilon_{B}$)
reported by modelling the multiwavelength GRB observations (Wijers & Galama,
1999; Panaitescu & Kumar, 2002; Yost et al., 2003; Panaitescu, 2005; Santana
et al., 2014).

In the SSC processes, scattering photons by the relativistic electron can
enter the Klein - Nishina (KN) scattering regime. Above this critical energy,
the SSC flux begins to be attenuated due to the quantum effects. Since the KN
effects are not negligible, the value of the Compton parameter varies, and
therefore it can be estimated as
$\frac{Y(\gamma_{c})[Y(\gamma_{c})+1]}{Y_{N}}=\left(\frac{\nu^{\rm syn}_{\rm
m}}{\nu^{\rm syn}_{\rm c}}\right)^{-\frac{p-3}{2}}\,\left(\frac{\nu^{\rm
syn}_{\rm KN}(\gamma_{\rm c})}{\nu^{\rm syn}_{\rm
m}}\right)^{\frac{4}{3}}\,\,{\rm for}\,\,{\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})<\nu^{\rm syn}_{\rm m}}$, $\left(\frac{\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})}{\nu^{\rm syn}_{c}}\right)^{-\frac{p-3}{2}}\,\,{\rm for}\,\,\nu^{\rm
syn}_{\rm m}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})<\nu^{\rm syn}_{\rm c}$,
$1\,\,{\rm for}\,\,\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm
c})$ where
$Y_{N}=\frac{\varepsilon_{e}}{\varepsilon_{B}}\left(\frac{\gamma_{\rm
c}}{\gamma_{\rm m}}\right)^{2-p}$ where $h\nu^{\rm syn}_{\rm
KN}(\gamma_{c})\simeq\frac{2\Gamma}{(1+z)}\,\frac{m_{e}c^{2}}{\gamma_{c}}$ for
$\nu^{\rm syn}_{\rm m}<\nu^{\rm syn}_{\rm c}$ (see Nakar et al., 2009; Wang et
al., 2010). In addition, it is imperative to estimate the Lorentz factor of
those electrons ($\gamma_{*}$) that could emit gamma-ray photons in the Fermi-
LAT energy range via the synchrotron process. In this case, a new spectral
break $h\nu^{\rm syn}_{\rm KN}(\gamma_{*})$ must be included in the SSC
spectrum with a new Compton parameter $Y(\gamma_{*})$ given by
$Y(\gamma_{*})=Y(\gamma_{c})\left(\frac{\nu_{*}}{\nu_{c}}\right)^{\frac{p-3}{4}}\left(\frac{\nu^{\rm
syn}_{\rm KN}(\gamma_{\rm c})}{\nu^{\rm syn}_{c}}\right)^{-\frac{p-3}{2}}$ for
$\nu^{\rm syn}_{\rm m}<h\nu^{\rm syn}_{\rm KN}(\gamma_{*})=100\,{\rm
MeV}<\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})$ (for
details, see Wang et al., 2010). We can notice that for $Y(\gamma_{c})\ll 1$,
Eq. 16 is not modified, and for the case of $Y(\gamma_{c})\gg 1$, the
parameter $\epsilon_{B}$ increases. As a particular case, given the spectral
breaks in the hierachy $\nu^{\rm syn}_{\rm m}<h\nu^{\rm syn}_{\rm
KN}(\gamma_{*})<\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm KN}(\gamma_{\rm c})$
and with $Y(\gamma_{c})\gg 1$, the new Compton parameter becomes
$Y(\gamma_{*})\simeq 1$ which is ample used for modeling multiwavelength
afterglow observations (see Wang et al., 2010). In each case, the parameter
$\epsilon_{B}$ ranges in those values considered in Santana et al. (2014).

Using the temporal and spectral indices, Tak et al. (2019) methodically
examined the CRs of 59 LAT-detected bursts. The authors showed that these
bursts satisfy the CRs of the slow-cooling regime, but only with the condition
of having an exceptionally small value of $\epsilon_{B}<10^{-7}$. We found
that in the SSC forward-shock scenario, the magnetic parameter ranges of
$10^{-5}\lesssim\epsilon_{B}\lesssim 10^{-1}$, similar to those found after
modeling multiwavelength afterglow observations (e.g., see Santana et al.,
2014).

###  3.8 Particular Burst: GRB 160509A

####  3.8.1 Multiwavelength Observations

Both instruments of the Fermi satellite detected GRB 160509A, GBM (Gamma Burst
Monitor) and LAT, at 8:59:04.36 UTC on 2016-05-09 (Roberts et al., 2016; Longo
et al., 2016). A radius of error of $0.50^{\circ}$ (90 percent containment,
the systematic error alone) was used to pinpoint this burst, which was found
at coordinates of ${\rm R.A.}=310.1$ and ${\rm Dec}=76.0$ (J2000). At about
$\approx 77$ s s after the trigger, the LAT sensor observed an extremely
intense photon with an energy of 52 GeV. Multiple peaks lasting $T_{90}=371$ s
were observed in the GBM light curve in the 50-300 keV energy range. On May 10
at 13:15 U.T., the GMOS-N and NIRI instruments at the Gemini North telescope
detected optical spectroscopy and near-IR imaging, respectively, of this
burst. The spectroscopic study revealed a redshift of z=1.17 (Tanvir et al.,
2016) based on a single, well-defined emission line ([OII] 3727 Å). The VLA
(Very Large Array) radio telescope picked up on this burst 0.36 days after the
trigger time, in frequencies between 1.3 and 37 GHz (Laskar et al., 2016).

Ajello et al. (2019) derived the Fermi-LAT light curve between 100 MeV and 100
GeV band of GRB 160905A, exhibiting two components a short-lasting peak at the
end of the prompt episode and a long-lasting emission with a temporal break at
300 s. In addition, Tam et al. (2017) derived and analyzed the Fermi-LAT
spectrum, photon flux, and the evolution of the photon index between 0.1 and 1
GeV and 1 - 100 GeV bands of GRB 160905A. In particular, they reported photon
emission in the 1 - 100 GeV band and no emission in the lower band (0.1 and 1
GeV) when the photon index was very hard ($\Gamma_{\rm LAT}=1.4\pm 0.3$).
Fraija et al. (2020) modelled the Fermi-LAT and Swift-XRT observations of GRB
160509A. The GeV peak was described with the SSC emission from the reverse
shock in the thick-shell regime, in a uniform-density medium, and the long-
lasting component with synchrotron forward-shock emission. They showed that
the passage of the forward shock synchrotron cooling break through the LAT
band could explain the break in the long-lasting emission, and a fast
evolution of cooling frequency could also explain the Swift-XRT observations.
The authors proposed that one of the possible scenarios with the potential to
describe the fast evolution of the synchrotron cooling break and the plateau
in the X-ray band is the variation of the microphysical parameters with
$\epsilon_{\rm e}\propto t^{\rm 0.319^{+0.366}_{-0.375}}$ and $\epsilon_{\rm
B}\propto t^{-\rm 1.401^{+0.136}_{-0.142}}$ in a very low circumburst density
($4.554^{+1.128}_{-1.121}\times 10^{-4}\,{\rm cm^{-3}}$).

Here, we estimate SSC flux in the scenario of the microphysical parameter
variations using the best-fit values found by Fraija et al. (2020) in order to
describe the latest LAT data point at $4.2\times 10^{4}\,{\rm s}$ reported
from Tam et al. (2017). Figure 9 shows the Fermi-LAT and Swift-XRT
observations of GRB 160509A with the best-fit curves generated by a
synchrotron and SSC from forward and reverse shock, respectively, adapted from
Fraija et al. (2020). The maximum photons energy radiated by synchrotron
afterglow model is $h\nu^{\rm syn}_{\rm max}=95.2\,{\rm MeV}$ at $10^{4}\,{\rm
s}$. While synchrotron flux cannot describe the latest LAT data point, SSC
could marginally describe it. We conclude that a forward-shock scenario with
microphysical parameter variations is needed for describing the early multi-
wavelength observation of GRB 160509A.

Although Laskar et al. (2016) favored a uniform-density environment for this
burst on physical grounds (based on the inferred initial Lorentz factor from
the RS emission), they could not conclusively distinguish between a wind and a
uniform-density medium based on multiwavelength observations alone. Although
Fraija et al. (2020) reported that a uniform-density medium is favored on the
stellar-wind, as shown here, the analysis of the latest LAT observations
confirmed the evolution of the forward shocks in a uniform-density
environment.

##  4 Closure relations of bursts with very-high-energy emission

###  4.1 A GRB sample detected in VHEs with known CRs

####  4.1.1 GRB 180720B

GRB 180720B was detected by Fermi-GBM on July 20th 2018 at $T_{0}=$14:21:39.65
Universal Time (UT) (Roberts & Meegan, 2018) and five seconds after by Swift-
BAT (Siegel et al., 2018). Based on the duration of the prompt emission
$T_{90}\sim 48.9\pm 0.4$ sec, this burst was categorized as a long GRB. In
addition, the isotropic energy released in the energy band of 50-300 keV was
$E_{\rm\gamma,iso}=(6.0\pm 0.1)\times 10^{53}\,\rm erg$. Furthermore, this GRB
was also detected by Fermi-LAT between $T_{0}$ and $T_{0}$+700 sec, with the
most energetic photon of 5 GeV at $T_{0}$ \+ 137 sec (Bissaldi & Racusin,
2018). The prompt emission detected by GBM was identified as extremely bright
and in the top seven brightest. Regarding the afterglow, the observations
performed by the H.E.S.S. collaboration of VHE gamma-ray emission began 10.1
hours after $T_{0}$. It lasted for two hours (Abdalla et al., 2019) revealing
a gamma-ray excess associated with a point-like source centered at $RA=00^{\rm
h}0.2^{\rm min}0.7.6^{\rm sec}$ and
$Dec=0.2^{\circ}56^{\prime}06^{\prime\prime}$ (J2000). Similarly, Swift-XRT
observed a bright afterglow 11 hours after $T_{0}$ in the 0.3-10 keV band,
which was detectable even 30 days after $T_{0}$. The redshift was estimated to
be $z=0.654$ by identifying several absorption features, such as Fe II, Mg II,
Mg I, and Ca II (Vreeswijk et al., 2018).

####  4.1.2 GRB 190114C

GRB 190114C was first detected by Swift-BAT and Fermi-GBM on 2019 January 14,
$T_{0}=$20:57:03 UT (Gropp et al., 2019; Hamburg et al., 2018), categorising
it as a long-GRB type with a $T_{90}$ of $\sim 116$ sec and $\sim 362$ sec by
Swift-BAT and Fermi-GBM, respectively. The redshift estimation was
$z=0.4245\pm 0.0005$ (Castro-Tirado et al., 2019). The isotropic equivalent
energy released in the 10-1000 keV energy band during the prompt episode was
$E_{\rm\gamma,iso}=2.4\pm 0.05\times 10^{53}\,\rm erg$. Approximately 57
seconds after the trigger alert, the MAGIC telescopes observed this burst
until 15,912 seconds after the alert ($\sim 4.42$ hours), detecting VHE gamma-
rays in the energy range of 0.2-1 TeV with more than 50 standard deviations in
the first 20 minutes (MAGIC Collaboration et al., 2019a). The afterglow multi-
frequency observations from $5\times 10^{-6}$ to $10^{12}$ eV were reported by
MAGIC Collaboration et al. (2019b), who accounted a double-peaked feature in
the broadband SED.

####  4.1.3 GRB 190829A

The Fermi-GBM detected GRB 190829A on 29 August 2019 at $T_{0}=$19:55:53 UT,
which was categorized as a long GRB class (The Fermi GBM team, 2019). After
51.6 sec, Swift-BAT triggered and located this burst, and 97.3 sec after the
BAT trigger, Swift-XRT began its observation. The redshift was $z=0.0785$,
confirmed by optical afterglow spectroscopy observations (10.4m GTC telescope
Team, 2019), turning this burst into one of the closest detected. H.E.S.S.
telescopes reported the detection of VHE gamma-rays with statistical
significance of 21.7$\sigma$ and 5.5$\sigma$ during the first two nights.
Their observations started 4.3 hours after the trigger time and ended 55.9
hours after the trigger, reporting the detection of VHE gamma-rays (from 0.18
to 3.3 TeV) during each night (H. E. S. S. Collaboration et al., 2021). The
Fermi-LAT instrument observed the position of GRB 190829A (Piron & et al.,
2019), but no high-energy photons above 100 MeV were detected. The afterglow
observations by the Swift Ultraviolet/optical telescope began 158 seconds
after the trigger time.

####  4.1.4 GRB 221009A

The Fermi-GBM detected GRB 220019A on 2022 October 9 at $T_{0}=$13:16:59:99
UTC. It was categorized as a long GRB with a prompt emission that lasted more
than 600 s (Lesage et al., 2023). This GRB presented the highest total
isotropic equivalent energy $E_{\gamma,\rm iso}=1.0\times 10^{53}\,\rm erg$.
The estimation of the redshift found a value of $z=0.151$. Swift-BAT began the
observations of this burst 53 minutes and 17 seconds after the trigger time,
yielding the best location at $RA=19^{\rm h}13^{\rm min}3.48^{\rm sec}$ and
$Dec=19^{\circ}46^{\prime}24.6^{\prime\prime}$ (J2000). The Swift - XRT
instrument began the observations 143 seconds after the trigger time (Dichiara
et al., 2022). The most energetic Fermi-LAT photon ever observed, with energy
99.3 GeV, arrived 240 seconds after the trigger time (Pillera et al., 2022).
This burst was observed by LHAASO within 2000 seconds after the trigger time,
detecting more than 5000 photons at VHE energies from 0.1 TeV extending up to
18 TeV (Huang et al., 2022).

###  4.2 Analysis and Discussion

Table 9 shows the evolution of the temporal and spectral indexes reported by
Fermi-LAT (Acciari et al., 2019; Fraija et al., 2019b; Fraija et al., 2019c,
2021; Pillera et al., 2022) and TeV gamma-ray telescopes/observatories of a
sample of 4 bursts (GRB 180720B, 190114C, 190829A and 221009A) with energetic
photons above 100 GeV. These luminous bursts were detected by MAGIC (Acciari
et al., 2019, 2021), H.E.S.S. (Abdalla et al., 2019; H. E. S. S. Collaboration
et al., 2021) and LHAASO (Huang et al., 2022; LHAASO Collaboration et al.,
2023).

Figure 10 presents the PL indices of the microphysical parameters for which
the CRs are fulfilled for our GRB sample assuming a constant-density model.
Each column corresponds to a different cooling condition; from left to right
we have the cooling condition ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$,
${\rm\nu_{c}^{ssc}<\nu_{\rm LAT}<\nu_{m}^{ssc}}$ and
${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$, respectively. The first row
refers to GRB 180720B, the second to the Fermi-LAT observations of GRB
190114C, the third to the VHE observations of GRB 190114C, the fourth and
fifth rows correspond to the first and second night of VHE observations of GRB
190829A, respectively. Finally, the sixth row pertains to the Fermi-LAT
observations of GRB 221009A, and the seventh row to the VHE observations of
this GRB. In general, we notice that for all GRBs, the relation between the
power indices of the microphysical parameters depends on the cooling
condition. However, each cooling condition leads to the same relation, as we
notice by comparing each column. The condition ${\rm\nu_{m}^{ssc}<\nu_{\rm
LAT}<\nu_{c}^{ssc}}$ shows an inversely proportional relation between $a$ and
$b$, ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$ presents a
proportional relation and the middle column, corresponding to
${\rm\nu_{c}^{ssc}<\nu_{\rm LAT}<\nu_{m}^{ssc}}$ predicts a negative value of
$b$, but leaves $a$ as a free parameter. Comparing the relations due to the
Fermi-LAT observations with those from the VHE observations, we notice that
the lower-energy regime prefers slightly larger values of $a$. For instance,
in the case of GRB 221009A, we see that the relations obtained for Fermi-LAT
are inconsistent with those for VHEs.

Figure 11 presents the PL indices of the microphysical parameters for which
the CRs are fulfilled for the four GRBs in this section, assuming a stellar-
wind density model. The layout is identical to that of Figure 10. The same
conclusions with the previous Figure apply to this one. The main difference
between the two is found in the first column. We see that for the $k=2$ model,
smaller values of $a$ are predicted. For the second column, we also note that
the wind model predicts slightly smaller values of $b$. On the other hand, the
third column is the same between both Figures, as the cooling condition ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$ does not depend on $k$.

We want to emphasize that the no association of neutrinos with these powerful
GRBs suggests that the ranges of values of $a>0$ and $b<0$ in Figures 10 and
11 are favored. It occurs when the stored magnetic field energy is released by
additional magnetic reconnections, as predicted in some dissipation models,
such as the ICMART scenario (Zhang & Yan, 2011).

####  4.2.1 GRB 180720B

Table 9 shows that the H.E.S.S. collaboration did not report the temporal
index of the VHE gamma-ray emission (Abdalla et al., 2019), so we only analyze
the CRs reported by Fermi-LAT. The spectral analysis performed by LAT
instrument led to an spectral index of $\beta_{\rm LAT}=1.15\pm 0.10$. Taking
into account the CRs of the standard SSC afterglow model ($a=0$ and $b=0$) in
the cooling condition (${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
LAT}}$), the theoretical values of temporal indexes are $\alpha=1.34\pm 0.23$
and $1.30\pm 0.20$ for a homogeneous and stellar-wind environment,
respectively, which are inconsistent with the value of temporal index reported
by MAGIC ($\alpha_{\rm LAT}=1.90\pm 0.10$). It is worth nothing that other
cooling condition would lead to atypical values of the spectral index of
electrons (e.g. $p>3$). A possible explanation is to consider the variation of
the microphysical parameters during the deceleration phase. In this case, the
microphysical parameters would vary following the relation $5.2a-1.85b=1.125$
for ${\rm k=0}$, and $5.2a-1.85b=1.2$ for ${\rm k=2}$.

On the other hand, we discuss the top panels in Figures 10 and 11 around the
scenario of the decaying microturbulence. For the cooling condition ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$, the parameter $a$ lies
in range of $-0.08\leq a\leq-0.39$ for the scenario of gradually decaying
microturbulence and $a\leq-0.39$ for fast decaying microturbulence in both
stellar-wind and ISM. For the cooling condition ${\rm\nu_{m}^{ssc}<\nu_{\rm
LAT}<\nu_{c}^{ssc}}$, the parameter $a$ lies in the range of $0.15\leq a\leq
0.82$ ($0.12\leq a\leq 0.76$) for gradually decaying microturbulence and
$a\leq 0.15$ ($a\leq 0.12$) for fast decaying microturbulence in constant-
density medium (wind).

####  4.2.2 GRB 190114C

Misra et al. (2021) analyzed the radio and optical afterglow observations of
TeV-bright GRB 190114C in the range of $\sim 1$ to $\sim 140$ days. They found
that these observations did not follow the CRs of the standard synchrotron
forward-shock model. Therefore, they required that the microphysical
parameters evolved in time, as $\epsilon_{\rm e}\propto t^{-0.4}$ and
$\epsilon_{\rm B}\propto t^{0.1}$ for ISM, and $\epsilon_{\rm e}\propto
t^{-0.4}$ and $\epsilon_{\rm B}\propto t^{0.76}$ for stellar-wind. To compare
which afterglow model was consistent with MAGIC observation’s temporal and
spectral index, we include these values with red stars in Figures 10 and 11
for constant-density and stellar-wind environments, respectively. These
Figures show that a constant-density medium is self-consistent and not a
stellar-wind medium. In this case, the SSC afterglow model with variation of
microphysical parameters and evolving in constant-density medium is the
afterglow scenario for describing GRB 190114C.

##### LAT CRs.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
LAT}}$, the parameter $a$ lies in range of $-0.65\leq a\leq-0.28$ for the
scenario of gradually decaying microturbulence and $a\leq-0.65$ for fast
decaying microturbulence in both stellar-wind and ISM. For the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$, the parameter $a$
lies in range of $-0.05\leq a\leq 0.32$ ($-0.02\leq a\leq 0.30$) for gradually
decaying microturbulence and $a\leq-0.05$ ($a\leq 0.02$) for fast decaying
microturbulence in ISM (wind).

##### MAGIC CRs.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
MAGIC}}$, the parameter $a$ lies in range of $-0.51\leq a\leq-0.19$ for the
scenario of gradually decaying microturbulence and $a\leq-0.19$ for fast
decaying microturbulence in both stellar-wind and constant-density medium. For
the cooling condition ${\rm\nu_{m}^{ssc}<\nu_{\rm MAGIC}<\nu_{c}^{ssc}}$, the
parameter $a$ lies in range of $-0.02\leq a\leq 0.2$ ($-0.01\leq a\leq 0.18$)
for gradually decaying microturbulence and $a\leq-0.02$ ($a\leq-0.01$) for
fast decaying microturbulence in constant-density medium (wind).

####  4.2.3 GRB 190829A

The Fermi-LAT instrument did not detect photons with energies greater than 100
MeV. Still, the H.E.S.S. collaboration reported the detection of VHE gamma-
rays during the first two nights of observation, as shown in Table 8.
Therefore, we only analyze the CRs reported by H.E.S.S. around the first (*)
and second (**) night.

The H.E.S.S. team reported the detection of VHE gamma-rays with statistical
significances of 21.7$\sigma$ and 5.5$\sigma$, during two consecutive nights
(H. E. S. S. Collaboration et al., 2021). The spectral analysis in the energy
range of 0.1 - 3.3 TeV resulted in photon indices of $\beta_{\rm HESS}=1.06\pm
0.10$ and $0.86\pm 0.26$ for the first and second nights of observations. A
temporal analysis in the 0.2 - 3.0 TeV energy range with a PL function
exhibited a temporal index of $\alpha_{\rm HESS}=1.09\pm 0.05$. A potential
explanation for this behavior where the spectral index varies, and the
temporal index remains constant could be given in terms of the SSC afterglow
scenario with variation of the microphysical parameters. With the values of
spectral indexes, it can be observed that the spectral index of the second
night is a little hard concerning the first one, whereas the temporal index
remains constant. Given that the temporal index is the same for both nights,
it is reasonable to assume that the SSC flux lies in the same cooling
condition (${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm HESS}}$);
otherwise, unreasonable values are obtained. Using the CRs of the standard SSC
afterglow model ($a=0$ and $b=0$), the theoretical temporal indexes when the
first and second night is considered are $\alpha=1.14\pm 0.20$ and $0.69\pm
0.52$ for a homogeneous medium, and $\alpha=1.12\pm 0.20$ and $0.72\pm 0.52$
for stellar-wind environment, respectively. The observed temporal index
($\alpha_{\rm HESS}=1.09\pm 0.05$) strongly agrees with the standard SSC CRs
for a homogeneous medium and stellar wind during the first night, but it does
not with the ones of the second night. This could suggest that between both
nights, the microphysical parameters vary following the relation
$1.44a-1.07b=0.455$ for $k=0$ and by $1.44a-1.07b=0.4$ for $k=2$. We could
conclude that variations of spectral index keeping the temporal index constant
are associated with the evolution of microphysical parameters.

##### First night of HESS observations.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
HESS}}$, the parameter $a$ lies in range $-0.68\leq a\leq-0.36$ for the
scenario of gradually decaying microturbulence, and $a\leq-0.68$ for fast
decaying microturbulence in both stellar-wind and ISM. For the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm HESS}<\nu_{c}^{ssc}}$, the parameter $a$
lies in the range $-0.02\leq a\leq 0.48$ ($-0.01\leq a\leq 0.46$) for
gradually decaying microturbulence and $a\leq-0.02$ ($a\leq-0.01$) for fast
decaying microturbulence in constant-density medium (wind).

##### Second night of HESS observations.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
HESS}}$, the parameter $a$ lies in range $-0.76\leq a\leq-0.39$ for the
scenario of gradually decaying microturbulence, and $a\leq-0.76$ for fast
decaying microturbulence in both stellar-wind and ISM. For the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm HESS}<\nu_{c}^{ssc}}$, the parameter $a$
lies in the range $-0.05\leq a\leq 0.52$ ($-0.04\leq a\leq 0.56$) for
gradually decaying microturbulence and $a\leq-0.05$ ($a\leq-0.04$) for fast
decaying microturbulence in constant-density medium (wind).

####  4.2.4 GRB 221009A

The Water Cherenkov Detector Array (WCDA) instrument of LHAASO detected GRB
221009A with a statistical significance $>250\sigma$. During the first $\sim
3000\,{\rm s}$ after the trigger time, WCDA reported $6.4\times 10^{4}$
photons in the energy range $\sim 0.2-7\,{\rm TeV}$. This observatory reported
three temporal indexes: at the earliest times $\alpha_{\rm
LHAASO,1}=1.82^{+0.21}_{-0.18}$, at intermediate times $\alpha_{\rm
LHAASO,2}=-1.115^{+0.012}_{-0.012}$, and finally, at late times $\alpha_{\rm
LHAASO,3}=-2.21^{+0.03}_{-0.83}$. On the other hand, the collaboration
reported a single spectral index in the small range $2.324\pm 0.065<\beta_{\rm
LHAASO}<2.429\pm 0.062$ during the entire temporal coverage (LHAASO
Collaboration et al., 2023). One possible rationale for the observed
phenomenon, in which the temporal index exhibits variations but the spectral
index remains constant, may be attributed to the scenario of SSC afterglow,
where evolution in microphysical parameters occurs. While $\alpha_{\rm
LHAASO,2}$ and $\alpha_{\rm LHAASO,3}$ can be associated to the deceleration
and the post-jet break phases, respectively, of the standard SSC model
evolving during the cooling condition (${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm HESS}}$) in a constant-density
medium or a stellar-wind (see LHAASO Collaboration et al., 2023), the initial
temporal index of $\alpha_{\rm LHAASO,1}=1.82^{+0.21}_{-0.18}$ could be
consistent with the SSC afterglow scenario during the deceleration phase with
evolution of microphysical parameters in both a constant-density medium or in
a stellar-wind. For instance, in the homogeneous medium, the relation as a
function of $a$ and $b$ is $4.8a-1.9b+6.09=0$, and in the stellar wind, the
relation is $4.8a-1.9b+6.04=0$. Considering that a very early jet break has
never been observed, the last temporal index of $\alpha_{\rm
LHAASO,3}=-2.21^{+0.03}_{-0.83}$ could also be explained in the SSC scenario
evolving in a constant-density medium with the variation of microphysical
parameters. In this case, the relation as a function of $a$ and $b$ is
$4.8a-1.9b-1.97=0$. We could conclude, in general, that variations of the
temporal index keeping the spectral index constant are associated with the
evolution of microphysical parameters.

##### LAT CRs.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
LAT}}$, the parameter $a$ lies in the range $-0.79\leq a\leq-0.81$ for the
scenario of gradually decaying microturbulence and $a\leq-0.81$ for fast
decaying microturbulence in both stellar-wind and ISM. For the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$, the parameter $a$
lies in the range $0.15\leq a\leq 1.21$ ($0.12\leq a\leq 1.13$) for gradually
decaying microturbulence and $a\leq 0.15$ ($a\leq 0.12$) for fast decaying
microturbulence in homogeneous medium (wind).

##### LHAASO CRs.

For the cooling condition ${\rm max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm
LHAASO}}$, the parameter $a$ lies in range of $-0.79\leq a\leq-0.49$ for the
scenario of gradually decaying microturbulence and $a\leq-0.79$ for fast
decaying microturbulence in both stellar-wind and ISM. For the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm LHAASO}<\nu_{c}^{ssc}}$, the parameter
$a$ lies in range of $-0.20\leq a\leq 0.14$ ($-0.19\leq a\leq 0.15$) for
gradually decaying microturbulence and $a\leq-0.20$ ($a\leq-0.19$) for fast
decaying microturbulence in homogeneous medium (wind).

##  5 Discussion: Previous works and our contribution

It is assumed that the microphysical parameters, the fraction of energy that
goes into electron $\epsilon_{e}$ and magnetic fields $\epsilon_{B}$, do not
vary in time. This stationary assumption leads to good consistency between
models and observations of late afterglows. However, this assumption has been
challenged due to different reasons, including the description of the plateau
phase in the X-ray observations (Ioka et al., 2006), the modeling of afterglow
observations (Yost et al., 2003; Panaitescu et al., 2006), the interpretation
of fast evolution of the spectral breaks (Fraija et al., 2020), the
explanation of magnetic field amplification and microturbulence, which in turn
guides the complexly non-linear acceleration process, among others (Rossi &
Rees, 2003; Ioka, 2005; Lemoine et al., 2013; Lemoine, 2013).

Ioka et al. (2006) explored the time variation of the parameters to explain
the plateau phase of many X-ray light curves and found that this behavior
could be explained if $\epsilon_{e}\propto t^{-a}$ with $a=1/2$. Modeling of
radio afterglow observations of GRB 970508 by Wijers & Galama (1999) at 12
days and by Waxman et al. (1998) at $\sim$ 1 year led to different values of
microphysical parameters; the former reported $\epsilon_{e}=0.12$ and
$\epsilon_{B}=0.089$, and the later $\epsilon_{e}\simeq\epsilon_{B}=0.5$.
Huang et al. (2018) modeled the early-time emission of GRB 120729A by
considering a variation of the magnetic parameter given by a BPL of the form
$\epsilon_{B}\propto t^{-\rm b}$ with $\mathrm{b}_{1}=-(0.18\pm 0.04)$ and
$\mathrm{b}_{2}=-(0.84\pm 0.04)$. Yost et al. (2003) performed a study of four
well-observed bursts: GRB 970508, GRB 980329, GRB 980703, and GRB 000926. The
authors first fit the observations with a model with constant microphysical
parameters and noted diversity in the values of $\epsilon_{\rm B}$, namely
that $0.002<\epsilon_{\rm B}<0.25$. Afterward, they proposed the relation
$\epsilon_{\rm B}\propto\Gamma^{\alpha_{x}}$ with $-2\leq\alpha_{x}\leq 1$,
which led to a difference by up to an order of magnitude from the previous
model’s parameters. However, the authors noted that the results were not
unique, so conclusions on the values could not be made. In a similar fashion,
Kong et al. (2010) analyzed GRB 060206, GRB 070311 and GRB 071010A with a two-
region model ($1,2$) in which the microphysical parameters varied as
$\epsilon_{j}=\epsilon_{j,0}\Gamma^{-\alpha_{i}}$, where $j\in[\rm e,B]$ and
$i\in[1,2]$. The authors were able to reproduce the observed R-band and X-ray
light curves with their model and found that $\epsilon_{\rm e,0}=0.3$ and
$\epsilon_{\rm B}=0.03$ were sufficient to fit all GRBs, but different values
of $\alpha_{i}$ were required. Panaitescu et al. (2006) required the evolution
of microphysical parameters as $\epsilon_{\rm B}\propto\Gamma^{-\alpha_{b}}$
and $\epsilon_{\rm e}\propto\Gamma^{-\alpha_{e}}$ to model the X-ray and
optical afterglow observations of GRB 050319, GRB 050401, GRB 050607, GRB
050713A, GRB 050802 and GRB 050922C which exhibited a steepening at $\sim$ 1-4
h after the prompt episode without displaying a break in the optical emission.
They found that indexes $\alpha_{b}$ and $\alpha_{e}$ were in the range of
$4.2\leq\alpha_{b}\leq 7.6$ and $-4.1\leq\alpha_{e}\leq 1.7$. Yost et al.
(2003), Fan & Piran (2006) considered, among other options, a model for the
variation of microphysical parameters of the form $\epsilon_{\rm
e}\propto\Gamma^{-a}$ and $\epsilon_{\rm B}\propto\Gamma^{-b}$. They found
that the X-ray afterglow observations of GRB 050319, GRB 050401, and GRB
050315 could be well explained with $0.6\leq a\leq 0.7$, and $0.45\leq b\leq
1.2$, which implied an increase in time of the microphysical parameters. The
authors noted, however, that while the X-ray afterglow was well-fitted, this
model could not properly account for the R-band afterglow. Granot et al.
(2006) briefly considered a model where $\epsilon_{\rm e}\propto
t^{\alpha_{\rm e}}$ and $\epsilon_{\rm B}\propto t^{\alpha_{\rm B}}$ to
explain an observed stage of flattish decay in Swift X-ray afterglow light
curves. The authors found that the relation between the power indices
$\alpha_{\rm e}+\alpha_{\rm B}\sim 1-2$, was required to reproduce these
observations, which led them to consider linear growth in time of either or
both of the microphysical parameters as a possible explanation.

Relativistic collisionless shock front formation in a weakly magnetized
surrounding medium is explained by the self-generation of intense small-scale
electromagnetic fields mediating the transition between the upstream unshocked
and the downstream shocked media. This scenario was considered by Rossi & Rees
(2003), Ioka (2005) and Lemoine (2013) that showed how the spectrum is
modified if a short magnetic length-scale is considered. Ioka (2005) suggested
that coherent radiation could happen in relativistic collisionless shocks via
two-stream Weibel instabilities. They argued that the coherence amplifies the
radiation power, and electrons/protons cool quickly before being randomized.
Lemoine (2013) considered the variation of magnetic parameter
$\epsilon_{B}\propto t^{-b}$ and discussed the effect of the parameter $b$ on
the resulting spectrum for synchrotron and dominant inverse Compton,
identifying two scenarios. In the first scenario, turbulence gradually decays
with $-1<b<0$ and $b>-4/(p+1)$ for synchrotron and dominant inverse Compton,
respectively, and in the second scenario, turbulence rapidly decays with
$b<-1$ for synchrotron model and $-3<b<-4/(p+1)$ for dominant inverse Compton.
Lemoine et al. (2013) described the multiwavelength observations of GRB
090902B, GRB 090323, GRB 090328 and GRB 110731A, using a synchrotron model
with variation only of the magnetic microphysical parameter
$\epsilon_{B}\propto t^{-b}$ with $b$ in range of $0.4\lesssim b\lesssim 0.5$.
They associated this variation with the PL-decaying micro-turbulence and the
magnetic field amplification, which could occur in the relativistic
collisionless shock of GRB afterglows.

Fraija et al. (2020) modelled the Fermi-LAT and Swift-XRT observations of GRB
160509A. They showed that the fast evolution of cooling frequency exhibited in
the LAT and XRT instruments and the plateau in the X-ray band could be
interpreted as the variation of the microphysical parameters $\epsilon_{\rm
e}\propto t^{-\rm a}$ and $\epsilon_{\rm B}\propto t^{-b}$ with $a=0.319$ and
$b=1.401$. In a more recent article, Misra et al. (2021) analyzed the TeV-
bright long gamma-ray burst 190114C and found that the microphysical
parameters of the forward shock evolve in time. The authors considered two
possibilities for the circumburst density profile ($k=0,2$) and concluded that
time-dependence of both $\epsilon_{e}$ and $\epsilon_{B}$ is necessary to
reproduce the observations in both scenarios. Furthermore, motivated by the
analysis of the afterglow’s SED in several energy bands, the researchers were
able to fix the value of the electron spectral index to $p=2.01$ and the power
of the $\varepsilon_{e}$ variation to $\mathrm{a}=0.4$. This, in turn, allowed
them to analyze the $\mathrm{b}-k$ parameter space that was consistent with
the observations, which they showed in Figure 9. In summary, they found that
for $k=0$, $\mathrm{b}\in[-0.2,0.5]$ was allowed, while for $k=2$,
$\mathrm{b}\in[-0.8,-0.6]$ was permitted.

Previous studies that considered the evolution of microphysical parameters
have focused on the role of this variation as a means to describe particular
segments of GRB light curves and, in a few cases, to explain magnetic field
amplification and microturbulence during the shocks. Given the multiwavelength
observations, they have also only addressed synchrotron afterglow models and
considered, in all cases, an electron population with a PL index $2<p$, but
not with a hard spectral index ($1\leq p\leq 2$). Authors have also considered
differences between afterglow models with constant circumburst medium ($k=0$)
and stellar-wind ($k=2$) but have not tested different types of stratification
$n\propto r^{-k}$ with a general $k$. In the current work, we have derived and
shown the SSC and synchrotron forward-shock model including the light curves
and CRs with the variation of the microphysical parameters ($\epsilon_{\rm
e}\propto t^{-\rm a}$ and $\epsilon_{\rm B}\propto t^{-\rm b}$) evolving in
the stratified environment with a density profile $\propto r^{-k}$ with ${\rm
k}$ in the range of $0\leq k<3$. In addition, we have considered the electron
distribution in the entire range of spectral index $1\leq p\leq 2$ and $2<p$.
We have applied the current model to investigate the evolution of the spectral
and temporal indexes of the GRB sample reported in 2FLGC. Moreover, we have
also considered a sub-sample of bursts from 2FLGC with spectral index
$\beta<1$ and temporal index $\alpha>1.3$, which can hardly be modeled with
the CRs of the standard synchrotron forward-shock model. In this case, a hard
spectral index is favored. We have explicitly shown the evolution of the
synchrotron and SSC light curves with microphysical parameter variations and
demonstrated that a plateau phase or a MeV-GeV peak could be expected in the
LAT energy range depending on the parameter values of $a$ and $b$. This is the
case of GRB 090510, GRB 090902B, and GRB 160509A, which exhibited a plateau
phase, and GRB 080916C, GRB 090510A, GRB 090902B, GRB 090926A, GRB 110731A,
GRB 130427A and GRB 160509A that showed a MeV - GeV peak. In addition, we
found that the cooling conditions for synchrotron afterglow scenario
${\rm\nu_{c}^{\rm ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ and ${\rm
max\\{\nu_{m}^{\rm syn},\nu_{c}^{\rm syn}\\}<\nu_{\rm LAT}}$ could exhibit
plateaus for both a homogeneous medium and a stellar wind. It can occur when
the stored magnetic field energy is released by additional magnetic
reconnections, as predicted in some dissipation models, such as the ICMART
scenario. Similarly, the cooling conditions for the SSC model
${\rm\nu_{c}^{\rm ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ and ${\rm
max\\{\nu_{m}^{\rm syn},\nu_{c}^{\rm syn}\\}<\nu_{\rm LAT}}$ could exhibit
high-energy plateaus. In this case, the variation of microphysical parameters
could be associated with the continuous energy injection with different
efficiencies. We have plotted the SSC light curves and spectra with the CTA
(Southern array), MAGIC, and Fermi-LAT sensitivities and show that using the
exact quantities of kinetic energies, circumburst densities or spectral
indexes, but different evolution of the microphysical parameters, this
emission could be observed in CTA (Southern array), MAGIC and Fermi-LAT. We
have applied the CRs of SSC and synchrotron afterglow model in each cooling
condition to the entire sample of GRBs reported in the 2FLGC using MCMC
simulations and a selection of 4 bursts (GRB 180720B, 190114C, 190829A and
221009A) with energetic photons above 100 GeV, which MAGIC, H.E.S.S. and
LHAASO gamma-ray observatories detected. We have found that the most likely
afterglow model using synchrotron and SSC emission on the 2FLGC corresponds to
the constant-density scenario. Given the best-fit values of the parameter $b$
from the 2FLGC, we have found that only the cooling conditions
${\rm\nu_{m}^{\rm syn}<\nu_{\rm LAT}<\nu_{c}^{\rm syn}}$ and ${\rm\nu_{m}^{\rm
ssc}<\nu_{\rm LAT}<\nu_{c}^{\rm ssc}}$ lie in the scenario of rapidly decaying
microturbulence for both synchrotron and SSC processes. Finally, we have found
that variations of the spectral (temporal) index keeping the temporal
(spectral) index constant are associated with the evolution of microphysical
parameters.

##  6 Conclusion

We have derived the synchrotron and SSC forward-shock scenario, including the
light curves and CR, with variations in the microphysical parameters
($\epsilon_{\rm e}\propto\varepsilon_{\rm e}t^{-\rm a}$ and $\epsilon_{\rm
B}\propto\varepsilon_{\rm B}t^{-\rm b}$). We have considered the forward shock
evolves in a stratified environment with density-profile described by
$n(r)\propto r^{\rm-k}$ with $k=0.0$, $0.5$ $1.0$, $1.5$, $2.0$ and $2.5$ and
relativistic electron population with a spectral index of $1<p<2$ and $p>2$.
We have performed an analysis between these CRs and the spectral and temporal
indices of bursts that were reported in 2FLGC. Considering the synchrotron and
SSC the light curves, we have shown that a plateau phase or a MeV-GeV peak
could be expected in the LAT energy range depending on the parameter
evolution. Potential candidates for a plateau phase are GRB 090510, GRB
090902B, and GRB 160509A, and for a MeV - GeV peak, are GRB 080916C, GRB
090510A, GRB 090902B, GRB 090926A, GRB 110731A, GRB 130427A and GRB 160509A.
We have plotted the SSC light curves and spectra with the CTA (Southern
array), MAGIC, and Fermi-LAT sensitivities and show that using the exact
quantities of kinetic energies, circumburst densities or spectral indexes, but
different evolution of the microphysical parameters, this emission could be
observed in CTA (Southern array), MAGIC and Fermi-LAT.

We have considered LAT-detected bursts reported in 2FLGC and applied the MCMC
method to find the best-fit values that satisfy the CRs in each cooling
condition for the fast- and slow-cooling regimes. The number and percentage of
bursts following each cooling condition for the synchrotron and SSC afterglow
model evolving in a stratified medium are as follows. In the synchrotron
afterglow scenario, for the cooling condition ${\rm\nu_{m}^{syn}<\nu_{\rm
LAT}<\nu_{c}^{syn}}$ the best case is most consistent with ${\rm
k=0,1.0,1.5,2}$ (23 GRBs, 27.06%). For the cooling condition
${\rm\nu_{c}^{syn}<\nu_{\rm LAT}<\nu_{m}^{syn}}$, all cases of ${\rm k}$ are
equally likely (1 GRB, 1.18%). For the cooling condition ${\rm
max\\{\nu_{m}^{syn},\nu_{c}^{syn}\\}<\nu_{\rm LAT}}$, the best case is most
consistent with ${\rm k=1}$ (24 GRBs, 28.24%) and worst scenario is for ${\rm
k=2.5}$ (21 GRBs, 24.71%). In SSC afterglow scenario, for the cooling
condition ${\rm\nu_{m}^{ssc}<\nu_{\rm LAT}<\nu_{c}^{ssc}}$ the best case is
most consistent with ${\rm k=0}$ (23 GRBs, 27.06%) followed by all cases with
22 GRBs (25.88%). For the cooling condition ${\rm\nu_{c}^{ssc}<\nu_{\rm
LAT}<\nu_{m}^{ssc}}$, all cases of ${\rm k}$ are equally likely (1 GRB,
1.18%). For the cooling condition ${\rm
max\\{\nu_{m}^{ssc},\nu_{c}^{ssc}\\}<\nu_{\rm LAT}}$, the best cases are most
consistent with ${\rm k=0}$ and ${\rm k=0.5}$ (24 GRBs, 28.24%). In
particular, we have also considered a sub-sample of bursts from 2FLGC with
spectral index $\beta<1$ and temporal index $\alpha>1.3$, which can hardly be
modeled with the CRs of the standard synchrotron forward-shock model. In this
case, a hard spectral index is favored. Tak et al. (2019) meticulously
analyzed the CRs of 59 LAT-detected bursts by looking at temporal and spectral
indexes. For instance, they discovered that a sizable percentage of bursts
could only be inadequately characterized using the standard synchrotron
emission model, even though this model accurately describes the spectrum and
temporal indices in most cases. Nonetheless, they demonstrated that many GRBs
fulfill the CRs of the slow-cooling regime, but only when the magnetic
microphysical parameter has a value of $\epsilon_{B}<10^{-7}$ or smaller.
Here, we show that a value of $10^{-5}\lesssim\epsilon_{B}\lesssim 10^{-1}$
(e.g., see Santana et al., 2014) is generated by the SSC afterglow model,
regardless of whether or not energy injection is included. The CRs of SSC
afterglow models are required to account for the bursts in the 2FLGC that
cannot be explained in the synchrotron scenario, together with the study
supplied by Tak et al. (2019) (e.g., those with photon energies above 10 GeV).

The afterglow of a GRB and the values of its spectral and temporal indices
could be well explained by the standard synchrotron forward-shock model with
the appropriate values of variation of the microphysical parameters, which is
valid up to the synchrotron limit. In contrast to the synchrotron model, which
predicts that the maximum photon energy emitted during the deceleration phase
will be around a few GeV, we argue that the SSC afterglow model is more suited
to understanding this sample of bursts. Neutrino non-coincidences with GRBs
have been detected by the IceCube Team (Abbasi et al., 2012; Aartsen et al.,
2016, 2015), ruling out photo-hadronic interactions as a possible explanation.
Although the CRs of the synchrotron afterglow model might, in some
circumstances, characterize this sample of 29 bursts, the SSC process may be
more appropriate.

As a particular case, we have applied the SSC scenario of the microphysical
parameter variations evolving in a constant-density medium to describe the
latest Fermi-LAT observations of GRB 160509A. We have shown that these
observations cannot be described with synchrotron flux but with SSC emission
from the forward shock, where it evolves in a uniform-density environment.

We have applied the CRs of the SSC and synchrotron afterglow models in each
cooling condition to the entire sample of GRBs reported in the 2FLGC using
MCMC simulations. We have also applied this procedure to a selection of 4
bursts (GRB 180720B, 190114C, 190829A, and 221009A) that presented photons
with energies above 100 GeV, which the MAGIC, H.E.S.S., and LHAASO gamma-ray
observatories detected. From this analysis, we have concluded that variations
of the spectral (temporal) index keeping the temporal (spectral) index
constant are associated with the evolution of microphysical parameters.

## Acknowledgements

We appreciate the useful discussions with Tanmoy Laskar, Rodolofo Barniol-
Duran, and Peter Veres. NF is grateful to UNAM-DGAPA-PAPIIT for the funding
provided by grant IN106521.

## Data Availability

No new data were generated or analysed in support of this research.

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Table 1: CRs of the synchrotron afterglow scenario with variation of microphysical parameters in a stratified environment. |  |  Synchrotron |  Synchrotron
---|---|---|---
|  |  $1<p<2$ |  $2<p$
|  $\beta$ |  $\alpha$ |  $\alpha$
Fast cooling $(\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm m})$ |  |  |
$\nu^{\rm syn}_{\rm c}<\nu<\nu^{\rm syn}_{\rm m}$ |  $\frac{1}{2}$ |  $\frac{(1-b)\beta}{2}$ |  $\frac{(1-b)\beta}{2}$
$\nu^{\rm syn}_{\rm m}<\nu$ |  $\frac{p}{2}$ |  $\frac{5-2a(k-4)+3\beta-k(\beta+1)}{2(4-k)}$ |  $\frac{3\beta-1+b(\beta-1)+2a(2\beta-1)}{2}$
Slow cooling $(\nu^{\rm syn}_{\rm m}<\nu^{\rm syn}_{\rm c})$ |  |  |
$\nu^{\rm syn}_{\rm m}<\nu<\nu^{\rm syn}_{\rm c}$ |  $\frac{p-1}{2}$ |  $\frac{9-(4a+3b)(k-4)+2\beta(3-k)}{4(4-k)}$ |  $\frac{12\beta+k(1-3\beta)+4a\beta(4-k)+b(4-k)(\beta+1)}{2(4-k)}$
$\nu^{\rm syn}_{\rm c}<\nu$ |  $\frac{p}{2}$ |  $\frac{5-2a(k-4)+3\beta-k(\beta+1)}{2(4-k)}$ |  $\frac{3\beta-1+b(\beta-1)+2a(2\beta-1)}{2}$
Table 2: CRs of the SSC afterglow scenario with variation of microphysical parameters in a stratified environment. |  |  SSC |  SSC
---|---|---|---
|  |  $1<p<2$ |  $2<p$
|  $\beta$ |  $\alpha$ |  $\alpha$
Fast cooling $(\nu^{\rm ssc}_{\rm c}<\nu^{\rm ssc}_{\rm m})$ |  |  |
$\nu^{\rm ssc}_{\rm c}<\nu<\nu^{\rm ssc}_{\rm m}$ |  $\frac{1}{2}$ |  $\frac{[k-5b(4-k)-2]\beta}{2(4-k)}$ |  $\frac{[k-5b(4-k)-2]\beta}{2(4-k)}$
$\nu^{\rm ssc}_{\rm m}<\nu$ |  $\frac{p}{2}$ |  $\frac{8-4a(k-4)-k(\beta+1)+b(k-4)(\beta+1)}{2(4-k)}$ |  $\frac{2(9\beta-5)+k(3-5\beta)+b(4-k)(\beta-3)+4a(4-k)(2\beta-1)}{2(4-k)}$
Slow cooling $(\nu^{\rm ssc}_{\rm m}<\nu^{\rm ssc}_{\rm c})$ |  |  |
$\nu^{\rm ssc}_{\rm m}<\nu<\nu^{\rm ssc}_{\rm c}$ |  $\frac{p-1}{2}$ |  $\frac{7-4a(k-4)+k(1-\beta)+b(k-4)(\beta-2)}{2(4-k)}$ |  $\frac{2(9\beta-1)+k(3-5\beta)+8a\beta(4-k)+b(4-k)(\beta+1)}{2(4-k)}$
$\nu^{\rm ssc}_{\rm c}<\nu$ |  $\frac{p}{2}$ |  $\frac{8-4a(k-4)-k(\beta+1)+b(k-4)(\beta+1)}{2(4-k)}$ |  $\frac{2(9\beta-5)+k(3-5\beta)+b(4-k)(\beta-3)+4a(4-k)(2\beta-1)}{2(4-k)}$
Table 3: Results from MCMC simulations of SSC and synchrotron afterglow model evolving in a stratified circumburst medium. |  | Synchrotron |  |  | SSC |
---|---|---|---|---|---|---
$\nu$ Range | $k$ | a | b |  | a | b
${\rm\nu_{m}^{\rm j}<\nu<\nu_{c}^{\rm j}}$ | $0.0$ | $-1.50\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $2.77\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-1.00\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $3.27\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
| 0.5 | $-1.46\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $2.64\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-0.94\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $2.91\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
| 1.0 | $-1.42\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $2.45\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-0.87\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $2.44\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
| 1.5 | $-1.35\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $2.18\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-0.77\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $1.77\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
| 2.0 | $-1.25\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $1.78\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-0.62\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $0.77\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
| 2.5 | $-1.08\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $1.11\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ |  | $-0.37\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-0.89\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$
${\rm\nu_{c}^{\rm j}<\nu<\nu_{m}^{\rm j}}$ | 0.0 | $0.03\begin{subarray}{c}+10.17\\\ -10.25\end{subarray}$ | $-3.98\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $-0.01\begin{subarray}{c}+3.41\\\ -3.41\end{subarray}$ | $-1.08\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
| 0.5 | $0.15\begin{subarray}{c}+10.15\\\ -10.26\end{subarray}$ | $-3.89\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $-0.01\begin{subarray}{c}+3.42\\\ -3.41\end{subarray}$ | $-1.06\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
| 1.0 | $0.04\begin{subarray}{c}+10.21\\\ -10.24\end{subarray}$ | $-3.89\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $0.00\begin{subarray}{c}+3.40\\\ -3.40\end{subarray}$ | $-1.05\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
| 1.5 | $0.17\begin{subarray}{c}+10.11\\\ -10.29\end{subarray}$ | $-3.89\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $-0.02\begin{subarray}{c}+3.41\\\ -3.39\end{subarray}$ | $-1.02\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
| 2.0 | $0.10\begin{subarray}{c}+10.23\\\ -10.15\end{subarray}$ | $-3.89\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $-0.01\begin{subarray}{c}+3.42\\\ -3.39\end{subarray}$ | $-0.98\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
| 2.5 | $0.03\begin{subarray}{c}+10.25\\\ -10.19\end{subarray}$ | $-3.89\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  | $0.02\begin{subarray}{c}+3.39\\\ -3.41\end{subarray}$ | $-0.91\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$
${\rm max\\{\nu_{m}^{\rm j},\nu_{c}^{\rm j}\\}<\nu}$ | 0.0 | $0.34\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | $-4.74\begin{subarray}{c}+0.10\\\ -0.11\end{subarray}$ |  | $-0.46\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.28\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
| 0.5 | $0.34\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-4.73\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ |  | $-0.45\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.26\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
| 1.0 | $0.34\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-4.71\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ |  | $-0.44\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.23\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
| 1.5 | $0.33\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-4.69\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ |  | $-0.42\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.18\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
| 2.0 | $0.33\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-4.67\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ |  | $-0.40\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.12\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
| 2.5 | $0.32\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-4.62\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ |  | $-0.36\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | $-1.02\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$
Table 4: Number and percentage of bursts satisfying each CR, summarized for the synchrotron forward-shock model with $1<p<2$ and $2<p$. We consider the best-fit values of $a$ and $b$ obtained with our MCMC and reported in Table 3. $\nu$ Range | $k$ | CR: 1 $<$ p $<$ 2 | CR: 2 $<$ p | GRBs Satisfying Relation | Proportion Satisfying Relation
---|---|---|---|---|---
|  |  |  |  | (%)
${\rm\nu_{m}^{\rm syn}<\nu<\nu_{c}^{\rm syn}}$ | $0.0$ | $\frac{16a+3(3+4b+2\beta)}{16}$ | $\frac{b+(3+4a+b)\beta}{2}$ | 23 | 27.06
| 0.5 | $\frac{18+28a+21b+10\beta}{28}$ | $\frac{1+7b+7(3+4a+b)\beta}{14}$ | 22 | 25.88
| 1.0 | $\frac{9+12a+9b+4\beta}{12}$ | $\frac{1+3b+3(3+4a+b)\beta}{6}$ | 23 | 27.06
| 1.5 | $\frac{18+20a+15b+6\beta}{20}$ | $\frac{3+5b+5(3+4a+b)\beta}{10}$ | 23 | 27.06
| 2.0 | $\frac{9+8a+6b+2\beta}{8}$ | $\frac{1+b+(3+4a+b)\beta}{2}$ | 23 | 27.06
| 2.5 | $\frac{18+12a+9b+2\beta}{12}$ | $\frac{5+3b+3(3+4a+b)\beta}{6}$ | 22 | 25.88
${\rm\nu_{c}^{\rm syn}<\nu<\nu_{m}^{\rm syn}}$ | 0.0 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
| 0.5 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
| 1.0 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
| 1.5 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
| 2.0 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
| 2.5 | $\frac{(1-b)\beta}{2}$ | $\frac{(1-b)\beta}{2}$ | 1 | 1.18
${\rm max\\{\nu_{m}^{\rm syn},\nu_{c}^{\rm syn}\\}<\nu}$ | 0.0 | $\frac{5+8a+3\beta}{8}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 23 | 27.06
| 0.5 | $\frac{9+14a+5\beta}{14}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 23 | 27.06
| 1.0 | $\frac{2+3a+\beta}{3}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 23 | 27.06
| 1.5 | $\frac{7+10a+3\beta}{10}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 23 | 27.06
| 2.0 | $\frac{3+4a+\beta}{4}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 22 | 25.88
| 2.5 | $\frac{5+6a+\beta}{6}$ | $-\frac{1+2a+b-(3+4a+b)\beta}{2}$ | 22 | 25.88
Table 5: Number and percentage of bursts satisfying each CR, summarized for the SSC forward-shock model with $1<p<2$ and $2<p$. We consider the best-fit values of $a$ and $b$ obtained with our MCMC and reported in Table 3. $\nu$ Range | $k$ | CR: 1 $<$ p $<$ 2 | CR: 2 $<$ p | GRBs Satisfying Relation | Proportion Satisfying Relation
---|---|---|---|---|---
|  |  |  |  | (%)
${\rm\nu_{m}^{\rm ssc}<\nu<\nu_{c}^{\rm ssc}}$ | $0.0$ | $\frac{7+16a+4b(2-\beta)}{8}$ | $-\frac{1-(9+16a)\beta-2b(1+\beta)}{4}$ | 23 | 27.06
| 0.5 | $\frac{15+28a+7b(2-\beta)-\beta}{14}$ | $-\frac{1-(31+56a)\beta-7b(1+\beta)}{14}$ | 22 | 25.88
| 1.0 | $\frac{4(2+3a)+3b(2-\beta)-\beta}{6}$ | $\frac{1+(13+24a)\beta+3b(1+\beta)}{6}$ | 22 | 25.88
| 1.5 | $\frac{17+20a+5b(2-\beta)-3\beta}{10}$ | $\frac{5+(21+40a)\beta+5b(1+\beta)}{10}$ | 22 | 25.88
| 2.0 | $\frac{9+8a+2b(2-\beta)-2\beta}{4}$ | $\frac{2+b+(4+8a+b)\beta}{2}$ | 22 | 25.88
| 2.5 | $\frac{19+12a+3b(2-\beta)-5\beta}{6}$ | $\frac{11+(11+24a)\beta+3b(1+\beta)}{6}$ | 22 | 25.88
${\rm\nu_{c}^{\rm ssc}<\nu<\nu_{m}^{\rm ssc}}$ | 0.0 | $-\frac{(1+10b)\beta}{4}$ | $-\frac{(1+10b)\beta}{4}$ | 1 | 1.18
| 0.5 | $-\frac{(3+35b)\beta}{14}$ | $-\frac{(3+35b)\beta}{14}$ | 1 | 1.18
| 1.0 | $-\frac{(1+15b)\beta}{6}$ | $-\frac{(1+15b)\beta}{6}$ | 1 | 1.18
| 1.5 | $-\frac{(1+25b)\beta}{10}$ | $-\frac{(1+25b)\beta}{10}$ | 1 | 1.18
| 2.0 | $-\frac{5b\beta}{2}$ | $-\frac{5b\beta}{2}$ | 1 | 1.18
| 2.5 | $\frac{(1-15b)\beta}{6}$ | $\frac{(1-15b)\beta}{6}$ | 1 | 1.18
${\rm max\\{\nu_{m}^{\rm ssc},\nu_{c}^{\rm ssc}\\}<\nu}$ | 0.0 | $\frac{2(1+2a)-b(1+\beta)}{2}$ | $-\frac{5+2b(3-\beta)-9\beta+8a(1-2\beta)}{4}$ | 24 | 28.24
| 0.5 | $\frac{15+28a-7b(1+\beta)-\beta}{14}$ | $-\frac{17+7b(3-\beta)-31\beta+28a(1-2\beta)}{14}$ | 24 | 28.24
| 1.0 | $\frac{7+12a-3b(1+\beta)-\beta}{6}$ | $-\frac{7+3b(3-\beta)-13\beta+12a(1-2\beta)}{6}$ | 23 | 27.06
| 1.5 | $\frac{13+20a-5b(1+\beta)-3\beta}{10}$ | $-\frac{11+5b(3-\beta)-21\beta+20a(1-2\beta)}{10}$ | 23 | 27.06
| 2.0 | $\frac{3+4a-b(1+\beta)-\beta}{2}$ | $-\frac{2+4a+3b-(4+8a+b)\beta}{2}$ | 21 | 24.71
| 2.5 | $\frac{11+12a-3b(1+\beta)-5\beta}{6}$ | $-\frac{5+3b(3-\beta)-11\beta+12a(1-2\beta)}{6}$ | 21 | 24.71
Table 6: Evolution of the synchrotron light curves ($F_{\nu}\propto t^{-\alpha}\nu^{-\beta}$) from an outflow decelerated in a stratified environment with microphysical parameter evolution |  | $k=0$ | $k=2$
---|---|---|---
|  | $1<p<2$ | $2<p$ | $1<p<2$ | $2<p$
| $\beta$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
Fast cooling $(\nu^{\rm syn}_{\rm c}<\nu^{\rm syn}_{\rm m})$ |  |  |  |  |
$\nu<\nu^{\rm syn}_{\rm c}$ | $-\frac{1}{3}$ | $\frac{6b-1}{6}$ | $\frac{6b-1}{6}$ | $\frac{3b+2}{3}$ | $\frac{3b+2}{3}$
$\nu^{\rm syn}_{\rm c}<\nu<\nu^{\rm syn}_{\rm m}$ | $\frac{1}{2}$ | $\frac{1-b}{4}$ | $\frac{1-b}{4}$ | $\frac{1-b}{4}$ | $\frac{1-b}{4}$
$\nu^{\rm syn}_{\rm m}<\nu$ | $\frac{p}{2}$ | $\frac{10+16a+3p}{16}$ | $-\frac{2(1+2a+b)-p(3+4a+b)}{4}$ | $\frac{6+8a+p}{8}$ | $-\frac{2(1+2a+b)-p(3+4a+b)}{4}$
Slow cooling $(\nu^{\rm syn}_{\rm m}<\nu^{\rm syn}_{\rm c})$ |  |  |  |  |
$\nu<\nu^{\rm syn}_{\rm m}$ | $-\frac{1}{3}$ | $-\frac{3(2+p)+16a-4b(3p-4)}{24(p-1)}$ | $-\frac{3-2(b-2a)}{6}$ | $-\frac{5(2-p)+8a-2b(3p-4)}{12(p-1)}$ | $-\frac{2a-b}{3}$
$\nu^{\rm syn}_{\rm m}<\nu<\nu^{\rm syn}_{\rm c}$ | $\frac{p-1}{2}$ | $\frac{3(2+p)+4(4a+3b)}{16}$ | $-\frac{3+4a-b-p(3+4a+b)}{4}$ | $\frac{8+p+2(4a+3b)}{8}$ | $-\frac{1+4a-b-p(3+4a+b)}{4}$
$\nu^{\rm syn}_{\rm c}<\nu$ | $\frac{p}{2}$ | $\frac{10+16a+3p}{16}$ | $-\frac{2(1+2a+b)-p(3+4a+b)}{4}$ | $\frac{6+8a+p}{8}$ | $-\frac{2(1+2a+b)-p(3+4a+b)}{4}$
Table 7: Evolution of the SSC light curves ($F_{\nu}\propto t^{-\alpha}\nu^{-\beta}$) from an outflow decelerated in a stratified environment with microphysical parameter evolution |  | $k=0$ | $k=2$
---|---|---|---
|  | $1<p<2$ | $2<p$ | $1<p<2$ | $2<p$
| $\beta$ | $\alpha$ | $\alpha$ | $\alpha$ | $\alpha$
Fast cooling $(\nu^{\rm ssc}_{\rm c}<\nu^{\rm ssc}_{\rm m})$ |  |  |  |  |
$\nu^{\rm ssc}_{\rm c}<\nu<\nu^{\rm ssc}_{\rm m}$ | $\frac{1}{2}$ | $-\frac{10b+1}{8}$ | $-\frac{10b+1}{8}$ | $-\frac{5b}{4}$ | $-\frac{5b}{4}$
$\nu^{\rm ssc}_{\rm m}<\nu$ | $\frac{p}{2}$ | $\frac{4(1+2a)-b(2+p)}{4}$ | $-\frac{2(5+8a+6b)-p(9+16a+2b)}{8}$ | $\frac{2(3+4a)-p-b(2+p)}{4}$ | $-\frac{2(2+4a+3b)-p(4+8a+b)}{4}$
Slow cooling $(\nu^{\rm ssc}_{\rm m}<\nu^{\rm ssc}_{\rm c})$ |  |  |  |  |
$\nu^{\rm ssc}_{\rm m}<\nu<\nu^{\rm ssc}_{\rm c}$ | $\frac{p-1}{2}$ | $\frac{7+16a+b(10-2p)}{8}$ | $-\frac{11+16a-2b-p(9+16a+2b)}{8}$ | $\frac{2(5+4a)-p+b(5-p)}{4}$ | $-\frac{8a-b-p(4+8a+b)}{4}$
$\nu^{\rm ssc}_{\rm c}<\nu$ | $\frac{p}{2}$ | $\frac{4(1+2a)-b(2+p)}{4}$ | $-\frac{2(5+8a+6b)-p(9+16a+2b)}{8}$ | $\frac{2(3+4a)-p-b(2+p)}{4}$ | $-\frac{2(2+4a+3b)-p(4+8a+b)}{4}$