# Diagnosing Circumburst Environment with Multiband Gamma-Ray Burst Radio
Afterglows

Bo Zhang National Astronomical Observatories, Chinese Academy of Sciences, 20A
Datun Road, Beijing, 100012, People’s Republic of China  CAS Key Laboratory of
FAST, National Astronomical Observatories, Chinese Academy of Sciences,
People’s Republic of China  Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing, Jiangsu, 210034, People’s Republic of China  Liang-Duan Liu
Institute of Astrophysics, Central China Normal University, Wuhan 430079,
People’s Republic of China  Department of Astronomy, Beijing Normal
University, Beijing, 100875, People’s Republic of China  Tian-Rui Sun  Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing, Jiangsu, 210034,
People’s Republic of China  School of Astronomy and Space Science, University
of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of
China  Fen Lyu  Astronomical Research Center, Shanghai Science and Technology
Museum, Shanghai, 201306, People’s Republic of China  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing, Jiangsu, 210034, People’s
Republic of China  Xue-Feng Wu  Purple Mountain Observatory, Chinese Academy
of Sciences, Nanjing, Jiangsu, 210034, People’s Republic of China

(Received 00 00, 0000; Revised 00 00, 0000; Accepted )

###### Abstract

It has been widely recognized that gamma-ray burst (GRB) afterglows arise from
interactions between the GRB outflow and circumburst medium, while their
evolution follows the behaviors of relativistic shock waves. Assuming the
distribution of circumburst medium follows a general power-law form, that is,
$n=A_{\ast}R^{-k}$, where $R$ denotes the distance from the burst, it is
obvious that the value of the density-distribution index $k$ can affect the
behaviors of the afterglow. In this paper, we analyze the temporal and
spectral behaviors of GRB radio afterglows with arbitrary $k$ values. In the
radio band, a standard GRB afterglow produced by a forward shock exhibits a
late-time flux peak, and the relative peak fluxes as well as peak times at
different frequencies show dependencies on $k$. Thus, with multiband radio-
peak observations, one can determine the density profile of the circumburst
medium by comparing the relations between peak flux/time and frequency at each
observing band. Also, the effects of transrelativistic shock waves, as well as
jets in afterglows are discussed. By analyzing 31 long and 1 short GRB with
multiband data of radio afterglows, we find that nearly half of them can be
explained with uniform interstellar medium ($k=0$), $\sim 1/5$ can be
constrained to exhibiting a stellar-wind environment ($k=2$), while less than
$\sim 1/3$ of the samples show $0<k<2$.
Unified Astronomy Thesaurus concepts: Burst astrophysics (187), Gamma-ray
bursts (629)

††journal: ApJ

##  1 Introduction

Gamma-ray bursts (GRBs) are stellar explosions arising from the collapses of
massive stars (for long bursts, e.g., see Woosley 1993; Paczyński 1998;
MacFadyen & Woosley 1999) or mergers of binary compact stars (for short
bursts, e.g., see Paczynski 1986; Narayan et al. 1992; Gehrels et al. 2009;
Abbott et al. 2017). The afterglows of GRBs are the results of interactions
between relativistic ejecta from central engines and the circumburst medium.
The shock waves produced during such interactions can accelerate the swept-up
electrons, thus giving rise to nonthermal radiation, such as synchrotron
emission (e.g., see Piran 1999; van Paradijs et al. 2000; Mészáros et al.
2002). Our understanding of GRB afterglows has been greatly improved with the
discoveries of GRB multiband afterglows (Costa et al. 1997; van Paradijs et
al. 1997; Frail et al. 1997; Zhang 2007).

Assuming the profile of the circumburst density $n$ follows a power-law form,
that is, $n=A_{\ast}R^{-k}$, where $R$ denotes the distance from GRB central
engine, it is obvious that different values of the power-law index $k$ lead to
different light-curve behaviors (e.g., see Wu et al. 2005). The most widely
discussed circumburst density model is the homogeneous interstellar medium
(ISM) case with $k=0$ (e.g., see Sari et al. 1998; Kobayashi 2000; Panaitescu
& Kumar 2004), while the stellar-wind environment with $k=2$ has also been
proposed (e.g., see Dai & Lu 1998; Mészáros et al. 1998; Chevalier & Li 2000;
Panaitescu & Kumar 2000, 2004; Kobayashi & Zhang 2003; Wu et al. 2003, 2004;
Zou et al. 2005). Previously, it was thought that because the long GRBs
originate from massive stars with significant mass losses via stellar winds
before their demise, such events should preferably occur in stellar-wind
environments. For example, Starling et al. (2008) performed a joint fitting
for X-ray to IR afterglows of 10 GRBs, and put constraints on $k$ for half of
the sample, with 4 windlike cases and 1 ISM one. However, Panaitescu & Kumar
(2002) found that half of their 10 GRBs favor a $k=0$ medium; while Curran et
al. (2009) made $k$ constraints for 6 out of 10 GRBs, with 2 consistent with
the ISM environment, 2 with the wind case, and another 2 samples compatible
with both $k=0$ and $2$ cases. Yi et al. (2013) and van Eerten (2014)
presented various relationships between observables in GRB X-ray and optical
afterglows for arbitrary $k$. However, Yi et al. (2013) have drawn a typical
value of $k\sim 1$ for 19 Swift long bursts by analyzing their early
afterglows that originated from forward-reverse-shock interactions, implying a
mass-loss scheme of the GRB progenitors other than stellar wind.

In order to give a better understanding of circumburst medium distributions,
it is necessary to investigate this issue with methods other than early
X-ray/Optical observations. In this work, we utilize a new way to constrain
the density distribution of the circumburst medium, that is, to put a
constraint on $k$ by comparing multiband GRB radio afterglow peak times/fluxes
with the corresponding frequencies. Observationally, the light curves of GRB
radio afterglows usually show a double-peaked structure. In the rest frame,
the typical peak times are $0.1-0.2$ days and 2 days after triggers of prompt
emissions, respectively (Chandra & Frail, 2012). Generally speaking, the first
peak can be explained with forward–reverse-shock interactions at early times,
while the second peak is due to late-time forward-shock evolution and is
unique for the radio band. In fact, because a strong dependency exists between
GRB afterglow behaviors in the radio band and the underlying synchrotron
radiation, Barniol Duran (2014) and Beniamini & van der Horst (2017) have
already put constraints on the GRB microphysical parameters with GRB radio
peaks, including the fraction of electron and magnetic energy among shock
energy, that is, $\varepsilon_{e}$ and $\varepsilon_{b}$, respectively.

In our circumburst density analysis, we ignore the effects of early radio
peaks due to reverse shocks and only take the second peak into consideration.
Since the peak flux $F_{{\rm peak},\nu}$ and the peak time $t_{{\rm
peak},\nu}$ of this late peak depend on $k$, the value of $k$ can be deduced
with multiband radio-peak observations. Although in theory, the $k$ value can
also change the evolution of radio afterglows, it should be noted that radio
observations are largely constrained by instrumental sensitivities, refractive
interstellar scintillation (Chandra & Frail, 2012), or the intrinsic
properties of GRBs (Hancock et al., 2013). Thus, the precise shapes of the
complete light curves are usually hard to obtain. Hence, the main advantage of
our radio-peak method is that the peak times/fluxes can be determined more
easily and reliably due to the high fluxes of such peaks, as long as the
afterglow light curves are sampled frequently enough. And because radio
afterglows occur at a larger radius than early forward-reverse shocks,
combining density profiles for earlier afterglows using the method proposed by
Yi et al. (2013), and those for later radio peaks as shown in this work, a
more complete circumburst medium profile at various $R$ can be obtained.

This paper is organized as follows. In Section 2, we search for the peak time
$t_{\rm peak,\nu}$ in the analytical light curve of the GRB radio afterglow
with an arbitrary $k$, and calculate the relations between $t_{\rm peak,\nu}$
and $F_{\rm peak,\nu}$ with observing frequencies, with the role played by the
$k$ value clarified. Meanwhile, the effects of transrelativistic shocks, as
well as jets, are discussed. In Section 3, we apply our theoretical
predictions to 31 long GRBs, as well as 1 short burst, with multiband radio
afterglow observations, and draw a conclusion that most of them can be
explained with transrelativistic shock waves under ISM environments, which is
different from Yi et al. (2013). This implies that the circumburst density
profile may change with radius. In Section 4 our results are discussed and
summarized. In our calculations, the $\Lambda$CDM cosmology is adopted, with
cosmological parameters $H_{0}=71{\rm km}$ $\rm s^{-1}$ $\rm Mpc^{-1}$,
$\Omega_{m}=0.3$, and $\Omega_{\Lambda}=0.7$.

##  2 GRB Radio Light Curves and Multifrequency Peak Evolution

Figure 1: The relations between $k$ and $b$. In this figure, solid, dotted,
and dashed lines represent the $\epsilon=0$, $\epsilon=1/3$, and
$\epsilon=1/2$ cases, respectively.

In this section, our goal is to derive the relations between GRB radio
afterglow peak times $t_{{\rm peak},\nu}$ versus $k$, as well as peak fluxes
$F_{{\rm peak},\nu}$ versus $k$, i.e., the $k$-dependent expressions of the
power-law indices $a$ and $b$ for $t_{{\rm peak},\nu}\propto\nu^{-a}$, and
$F_{{\rm peak},\nu}\propto\nu^{b}$, respectively. We investigate the
analytical behaviors of multiband afterglow with arbitrary circumburst
density-distribution index $k$ in order to get the expressions of $a$ and $b$.
Ignoring the early afterglow evolution due to forward-reverse-shock
interactions, we consider the shock with radius $R>R_{{\rm dec}}$ only, in
which $R_{{\rm dec}}$ is the deceleration radius of relativistic shock, with a
corresponding deceleration time $t_{{\rm dec}}=R_{{\rm
dec}}\left(1+z\right)/2\Gamma_{0}^{2}c$, a cosmological redshift of $z$, and
an initial bulk Lorentz factor of $\Gamma_{0}$. By this time, the GRB
afterglow has already entered the self-similar phase. Here we do calculations
for semiradiative shock waves for generality, although by the time of the
emergence of radio peaks, i.e., several days after the initial GRB trigger,
the shock waves are usually adiabatic. Assuming the circumburst density can be
described as $n=A_{\ast}R^{-k}=A\left(R/R_{*}\right)^{-k}$ ($k=0$ for
uniformly distributed interstellar medium, $k=2$ for stellar wind, with a
typical value of the characteristic length $R_{*}\sim 10^{17}$ cm; see Wu et
al. 2005), the hydrodynamical evolution should be $\Gamma^{2}\propto R^{-m}$,
and $R\propto t^{1/\left(m+1\right)}$ (Blandford & McKee 1976), with self-
similar index $m$ as

$m=\frac{3-k}{1-\epsilon},$ (1)

where $\epsilon$ denotes the radiative efficiency of the shock, $R$ the radius
of the shock wave, $\Gamma$ the bulk Lorentz factor of the shock, and $t$ the
observed time after the initial burst (Wu et al., 2005). The radiative
efficiency $\epsilon$ can be considered as a constant at the early fast-
cooling regime, and it evolves slowly after entering the slow-cooling regime
at later times. However, as long as we have power-law-distributed electrons,
$N_{e}\left(\gamma_{e}\right)\propto\gamma_{e}^{-p}$, with $\gamma_{e}$ as the
Lorentz factor of the electrons and $N_{e}$ the electron number, and the
distribution index $p$ does not deviate too much from $2$, the effect of the
evolving $\epsilon$ is not significant (Wu et al., 2005). Because the typical
value of $p$ for relativistic-shock-accelerated electrons is $2.2$ (Achterberg
et al. 2001, also see Curran et al. 2009 for observational constraints of an
average value of $k\sim 2.24$ with six Swift GRBs), which is close to $2$,
here we ignore the evolution of $\epsilon$.

In this work, we focus on radio peaks occurring at $\geqslant T_{90}+0.5$ days
in the rest frame only, thus the effects of the equal-arrival-time surface
(EATS) can be ignored. That is because, for a typical GRB, the radio-peak time
$t_{{\rm peak}}\sim T_{90}+0.5$ days $<\Gamma^{-2}R/2c$, which is the maximum
time delay due to the EATS effects. Also, because the major concern of this
paper is the behavior of GRB radio afterglows, we ignore the inverse Compton
(IC) scattering, for IC is not a dominant factor at lower frequencies (Wu et
al., 2005).

Following the procedures of Wu et al. (2005) and van der Horst (2007), we
first consider the evolution of spherical shock waves with various $k$ values.
For synchrotron radiation caused by power-law-distributed electrons, a key
time in the light curve is $t_{cm}$, which marks the transition point between
the early fast-cooling phase to the late slow-cooling phase, with an electron
minimum Lorentz factor $\gamma_{m}$ equal to the electron cooling Lorentz
factor $\gamma_{c}$, and a minimum frequency $\nu_{m}$ equal to the cooling
frequency $\nu_{c}$ (Sari et al. 1998; Panaitescu & Kumar 2000). We define
$\nu_{cm}=\nu_{m}\left(t_{cm}\right)=\nu_{c}\left(t_{cm}\right)$. The
transition occurs at

$\displaystyle t_{cm}$ $\displaystyle=$ $\displaystyle
A\sigma_{T}^{\left(3-k\right)/2}\frac{\left(1+z\right)\sigma_{T}^{1/2}}{2c}\left[\frac{\left(3-k\right)E_{cm}}{4\pi
m_{p}c^{2}}\right]^{\left(2-k\right)/2}$ (2)
$\displaystyle\left(\frac{2m_{p}}{3m_{e}}\varepsilon_{e}^{3/4}\varepsilon_{B}^{1/4}\zeta_{1/6}^{1/2}\right)^{4-k}.$
$\displaystyle\nu_{cm}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(\frac{3}{2\pi}\right)^{5/4}\frac{m_{e}c^{3}}{e^{2}\left(1+z\right)}\left(\frac{2m_{p}}{3m_{e}}\right)^{\left(3k-7\right)/2}$
(3)
$\displaystyle\times\varepsilon_{e}^{\left(9k-20\right)/8}\varepsilon_{B}^{\left(3k-8\right)/8}\zeta_{1/6}^{\left(3k-4\right)/4}$
$\displaystyle\times\left(A\sigma_{T}^{\left(3-k\right)/2}\right)^{-3/2}\left[\frac{\left(3-k\right)E_{cm}}{4\pi
m_{p}c^{2}}\right]^{\left(3k-4\right)/4},$

where $m_{e}$ is the electron mass, $m_{p}$ the mass of a proton, $e$ the
charge of an electron, $c$ the speed of light in vacuum, $\sigma_{T}$ the
electron’s Thomson scattering cross section, $E_{cm}$ the isotropic energy of
the shock wave at $t_{cm}$, and
$\zeta_{1/6}=6\left(p-2\right)/\left(p-1\right)$.

Following the assumptions of Wu et al. (2005), we take typical values for
parameters as follows: the isotropic energy $E_{cm}=10^{53}$ erg,
$\varepsilon_{e}=1/3$, $\varepsilon_{B}=10^{-2.5}$, and the electron
distribution index $p=2.2$. It should be noted that although some later works,
including Nava et al. (2014), Beniamini & van der Horst (2017), as well as
Santana et al. (2014), hinted that $\varepsilon_{e}\sim 0.15$ and
$\varepsilon_{B}\sim 10^{-8}-10^{-3}$ should be the case for some GRBs, the
exact values of $\varepsilon_{e}$ and $\varepsilon_{B}$ do not change the
overall shape of the afterglow light curves. Thus, our calculations still
apply to their samples.

The characteristic flux density $F_{\nu,max}$ at any given time can be
obtained with the characteristic flux density of synchrotron radiation at a
certain frequency on $t_{cm}$ as $F_{\nu,max}\left(t_{cm}\right)$. (It should
be noted that $F_{\nu,max}$ is different from $F_{{\rm peak},\nu}$ because the
latter denotes the peak flux of the radio light curve at a given frequency
$\nu$, rather than the radiation peak across the radio spectrum).
$F_{\nu,max}$ can be calculated as follows, regardless of fast or slow cooling

$F_{\nu,max}=F_{\nu,max}\left(t_{cm}\right)\left(\frac{t}{t_{cm}}\right)^{\left(6-3k-2m\right)/\left[2\left(m+1\right)\right]}.$
(4)

Also, at the radio band with low-enough frequencies, the synchrotron self-
absorption (SSA) effects cannot be ignored. In this case, we have another
characteristic time $t_{a}$, which denotes the time when $\nu=\nu_{a}$, where
$\nu_{a}$ is the self-absorption frequency. Because in this work the later-
time behavior of the radio light curve is our main focus, we only discuss the
later-time slow-cooling regime, with
$\nu_{a}=\min\left\\{\nu_{as,<},\nu_{as,>}\right\\}$, and

$\displaystyle\nu_{as,<}$ $\displaystyle=$
$\displaystyle\nu_{m}\left[\frac{c_{0}\left(p-1\right)}{3-k}\frac{enR}{B\gamma_{{\rm
min}}^{5}}\right]^{3/5},\nu_{a}<\nu_{m},$ (5) $\displaystyle\nu_{as,>}$
$\displaystyle=$
$\displaystyle\nu_{m}\left[\frac{c_{0}\left(p-1\right)}{3-k}\frac{enR}{B\gamma_{{\rm
min}}^{5}}\right]^{2/\left(p+4\right)},\nu_{m}<\nu_{a}<\nu_{c}.$ (6)

where $c_{0}\approx
10.4\left[\left(p+2\right)/\left(p+2/3\right)\right]\approx 15.24$, and the
strength of the magnetic field $B$ as well as shock radius $R$ can be
calculated with their corresponding values at $t_{cm}$ (e.g., see Equations
(13) and (14) in (Wu et al., 2005)).

In the slow-cooling phase, the transition from $\nu_{as,<}$ to $\nu_{as,>}$
occurs at $t_{am}$, when $\nu_{a}=\nu_{m}$. For various combinations of
physical parameters, typically we have $t_{am}\sim 10^{3}t_{cm}$. For many
cases, the GRB afterglow may have already finished the ultra-relativistic
self-similar evolution and even began the transition to the Newtonian regime
by the time of $t_{am}$, making Blandford & McKee (1976) no longer applicable.
And it is worth noting that in our numerical discussions of late-time
afterglow, including the transrelativistic phase, in Section 2.1, the
evolution of $\nu_{as,>}$ is included in calculations, with corresponding
effects of this factor in this regime considered. Thus, we do not take
$\nu=\nu_{as,>}$ into consideration in analytical calculations.

For typical parameter values, we have $\nu_{cm}\sim 10^{3}-10^{4}$ GHz, which
is much higher than in the radio band; hence in this work, we only take the
$\nu<\nu_{cm}$ band into consideration. Firstly, we calculate the light curves
for radio afterglows. If the flux $F_{\nu}\propto t^{\alpha}$ has a positive
temporal index $\alpha$ before a turning point and then shows a negative
$\alpha$, we define the turning point as a radio peak. We only consider the
$0\leqslant k\leqslant 2$ case. The reason is that the self-similar solution
proposed by Blandford & McKee (1976) only stands as long as $k<4$. And in
reality, it is difficult for the circumburst environment to show density
distributions steeper than $k>2$, since the original medium distribution and
the stellar wind from the progenitor are the only main factors affecting $k$.

Figure 2: The multiband GRB radio afterglow light curves for $k=0$ (upper
panel), $k=1.1$ (middle panel), and $k=2$ (lower panel) calculated with the
numerical model proposed by Huang et al. (1999). Here we take the GRB
isotropic energy $E_{\rm iso}=10^{53}$ erg, characteristic density $A=1$ ${\rm
cm}^{-3}$, and adiabatic shock wave $\epsilon=0$. The y-axis are shown in
arbitrary units. It can be seen that for the $k=0$ case, the low-frequency
afterglows peak during the transrelativistic-to-Newtonian phase, and the
relations between peak times/fluxes and frequencies are different from the
ultrarelativistic regime, while for the $k=1.1$ and $2$ cases, the radio
afterglows never enter the trans-relativistic phase during our calculations.
Figure 3: The upper panel shows the temporal evolution of $\Gamma v/c$ with
various isotropic energies. The transrelativistic evolution featuring
$0.2<\Gamma v/c<2$ is marked with the gray-shaded region bounded by the dotted
lines. Here we take $k=0$ and $n=1$ $\text{cm}^{-3}$. It can be seen that the
lower the $E_{\rm iso}$ is, the earlier the transrelativistic phase begins.
The middle panel shows the temporal evolution of $\Gamma v/c$ for various
circumburst densities, with $k=0$, $E_{\rm iso}=10^{53}$ erg. It can be seen
that the larger the density $n$, the earlier the transition begins. The lower
panel shows the temporal evolution of $\Gamma v/c$ with various $k$, $E_{\rm
iso}=10^{53}$ erg, and characteristic density $A=100$ ${\rm cm}^{-3}$. It can
be seen that the smaller the $k$ is, the earlier the transition time.

Thus, similar to Wu et al. (2005) and van der Horst (2007) by comparing the
values of characteristic times $t_{a}$, $t_{cm}$, $t_{c}$, as well as $t_{m}$,
with the latter two items corresponding to the time points when $\nu_{c}$ and
$\nu_{m}$ equal to the observing frequency $\nu$, respectively, we can get the
overall profile of GRB radio afterglow light curves. Depending on the values
of $k$, three types of light curves can be obtained. For smaller $k$ with
$0<k<\frac{32-12p-8\epsilon}{12-3p+\left(p-4\right)\epsilon}$, when
$\nu_{a}\left(t_{cm}\right)<\nu<\nu_{cm}$ , with $\nu_{a}\left(t_{cm}\right)$
means the SSA frequency at $t_{cm}$, we have $t_{a}<t_{cm}<t_{m}<t_{c}$, and
the peak occurs at $t_{m}$, with flux before and after peak as

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{cm}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t<t_{c},\\\
\end{array}\right.$ (7)

For $\nu<\nu_{a}\left(t_{cm}\right)$, the order changes to
$t_{cm}<t_{a}<t_{m}<t_{c}$, and the flux peaks again at $t_{m}$. The light
curve around peak time can be expressed as

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{a}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t<t_{c},\\\
\end{array}\right.$ (8)

It should be noted that, in this case, the upper limit of $k$, i.e.,
$\frac{32-12p-8\epsilon}{12-3p+\left(p-4\right)\epsilon}$, depends on the
electron distribution index $p$, as well as the shock radiation efficiency
$\epsilon$, and is quite sensitive to both parameters. For $p\sim 2.2$,
$\epsilon\sim 0$, this upper limit is $\sim 1.037$.

When the value of $k$ increases to
$\frac{32-12p-8\epsilon}{12-3p+\left(p-4\right)\epsilon}<k<4/3$, at higher
frequencies, $\nu_{a}\left(t_{cm}\right)<\nu<\nu_{cm}$, we have
$t_{a}<t_{cm}<t_{m}$, with $t_{c}$ disappearing in the radio band. In this
case, the high-frequency light curve still peaks at $t_{m}$, and the radio
flux around this time can be calculated as

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{cm}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t.\\\
\end{array}\right.$ (9)

For $\nu<\nu_{a}\left(t_{cm}\right)$, $t_{cm}<t_{a}<t_{m}$, $t_{m}$ remains as
$t_{\rm peak,\nu}$, and the peak-time light curve is

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{a}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t.\\\
\end{array}\right.$ (10)

And both the high- and low- frequency radio fluxes peak at $t_{m}$ again.

For a larger $k$ ($4/3<k<2$), the SSA frequency at $t_{ac}$ is lower than
$\nu_{cm}$, that is, $\nu_{ac}<\nu_{cm}$, and the radio band should be divided
into three sections. For $\nu_{ac}<\nu<\nu_{cm}$, $t_{a}<t_{c}<t_{cm}<t_{m}$,
while we have $t_{c}<t_{a}<t_{cm}<t_{m}$ for $\nu<\nu_{a}\left(t_{cm}\right)$.
In these two cases, the slope between $t_{cm}$ and $t_{m}$ is slightly
negative as long as $\epsilon>0$, only with an early peak at $t_{a}$. However,
a significant break does exist at $t_{m}$, as can be seen from the equation
below. Considering possible observational errors, such a turnover could still
be taken as peak time:

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{cm}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t.\end{array}\right.$
(11)

While for $\nu<\nu_{a}\left(t_{cm}\right)$, we expect
$t_{c}<t_{cm}<t_{a}<t_{m}$, again with $t_{m}$ serving as peak time. Here the
light curve can be expressed as

$F_{\nu}\propto\left\\{\begin{array}[]{ll}\nu^{1/3}t^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]},&t_{a}<t<t_{m},\\\
\nu^{-\left(p-1\right)/2}t^{\left[12-5k-p\left(4m+k\right)\right]/\left[4\left(m+1\right)\right]},&t_{m}<t.\end{array}\right.$
(12)

It can be seen that the GRB radio afterglow peaks at $t_{{\rm peak}}=t_{m}$,
regardless of $k$ value or observing frequency. Because we have
$t_{m}\propto\nu^{-2\left(m+1\right)/\left(4m+k\right)}$, the peak times and
peak fluxes can be calculated with Eqs. 7-12 as

$\displaystyle t_{{\rm peak},\nu}$ $\displaystyle\propto$
$\displaystyle\nu^{-2\left(m+1\right)/\left(4m+k\right)},$ (13) $\displaystyle
F_{{\rm peak},\nu}$ $\displaystyle\propto$
$\displaystyle\nu^{1/3}t_{m}^{\left(9-4k-m\right)/\left[3\left(m+1\right)\right]}$
(14) $\displaystyle\propto$
$\displaystyle\nu^{\left(-6+3k+2m\right)/\left(4m+k\right)}.$

That is, the $a$ and $b$ indices we seek should be

$\displaystyle a$ $\displaystyle=$
$\displaystyle-2\left(m+1\right)/\left(4m+k\right),$ (15) $\displaystyle b$
$\displaystyle=$ $\displaystyle\left(-6+3k+2m\right)/\left(4m+k\right).$ (16)

For $\epsilon=0$, it can be seen that $t_{{\rm peak},\nu}\propto\nu^{-2/3}$.
Even for $\epsilon>0$ cases, as long as the shock radiation efficiency is low
enough, the $t_{{\rm peak},\nu}\propto\nu^{-a}$ index $a$ should not deviate
too much from $2/3$. For the $F_{{\rm peak},\nu}\propto\nu^{b}$ index $b$ with
$\epsilon=0$, $b$ increases with $k$, that is, $b=0$ when $k=0$, and $b=1/3$
when $k=2$. If $\epsilon>0$, the $b$ value is larger than that in the
$\epsilon=0$ case, although such a difference is not significant for larger
$k$. In Figure 1, the relations between $k$ and $b$ with various $\epsilon$
are shown, according to which the $k$ value can be drawn from $b$. And it
should be noted that for later peaks occurring at $T_{90}+$several days,
$\epsilon=0$ usually applies, and thus a single $k$ value can be constrained
with $b$, as shown in Eq. 16.

Eqs 15 and 16 cannot be applied for high-frequency ($\nu_{ac}<\nu<\nu_{cm}$)
radio afterglows with $k>4/3$, the flux of which peaks at $t_{a}\ll 1$ day.
Because for real GRBs radio follow-up observations usually do not begin at
such early times, and the behaviors of afterglows are often contaminated by
the early reverse shocks, such a high-frequency early peak with a large $k$
have little value for our purpose, and its behavior is beyond the scope of
this work.

###  2.1 Late-time Afterglow Behaviors and Transition from Relativistic to
Newtonian Phase

Figure 4: The temporal evolution of characteristic frequencies for various $k$
(lower panel). Here the dashed, dotted and dash-dotted lines represent the
evolution of cooling frequency $\nu_{c}$, minimum frequency $\nu_{m}$, as well
as SSA frequency $\nu_{a}$. The solid lines show the evolution of multi-band
radio peak frequencies, as well as peak times. It can be seen that considering
trans-relativistic effects and evolution of $\nu_{a}$, the peak time $t_{{\rm
peak},\nu}$ for these cases should be $\approx t_{a}$. However, other
possibilities for peak also exist for various physical parameters.

The GRB afterglow begins the transition from relativistic to Newtonian phase
(the so-called “transrelativistic” stage) when the shock wave expands to a
larger radius. The evolutionary phase with Lorentz factor $0.2\leqslant\Gamma
v/c\leqslant 2$ can be considered as “transrelativistic”, because by this
time, the shock velocity has been significantly decreased and the amount of
swept-up circumburst medium becomes large enough, while the afterglow
evolution in this stage can be described by neither the relativistic self-
similar solution proposed by Blandford & McKee (1976), nor the Newtonian
Sedov-Taylor solution (Taylor 1950; Sedov 1969). Here, numerical calculations
based upon dynamical equations proposed by Huang et al. (1999), which provide
a unified framework from relativistic to deep Newtonian stages of shock-wave
evolution, are required, with examples shown in Figure 2. And because in late
phases GRB afterglows should be adiabatic, it is safe to assume $\epsilon=0$
in our transrelativistic calculations. Figure 3 shows the temporal evolution
of $\Gamma v/c$ for adiabatic shocks exhibiting various physical parameters,
with the trans-relativistic phase occurring during $T_{90}+$dozens to hundreds
of days. And considering the transrelativistic phase only begins when the
swept-up mass is comparable with the mass of the shock wave itself, it is
clearly seen that a larger $E_{\rm iso}$, a smaller $A$, and a larger $k$ can
lead to a later transition.

According to numerical calculations, for spherical ejecta, the radio flux peak
no longer appears at $t_{m}$ during such late times, due to the change of
blast-wave dynamics, as well as the transition from $\nu_{a}=\nu_{as,<}$ to
$\nu_{a}=\nu_{as,>}$, which may occur before or after the beginning of
transrelativistic phase, depending on the circumburst density profiles and
$E_{\rm iso}$. For example, as can be seen from Figure 4, $t_{a}$ acts as the
peak time for our typical parameters, though other possibilities also exist.
Hence, the values of $a$ and $b$ in this phase differ significantly with
predictions made by the self-similar shock-wave theory.

Figure 5: The temporal evolution of jet opening angles $\theta_{j}$ for
sideway-expanding jets under various circumburst medium distribution index
$k$. Here different colors represent various $k$-values, the solid line the
evolution of $\theta_{j}$, and the dashed line the evolution of $1/\Gamma$. It
can be seen that during late times, the condition of $\Gamma>\theta_{j}^{-1}$
can be satisfied once more, making the afterglow flux to rise again, providing
a constant expanding speed.

In late times covering the transrelativistic regime, the analytical
expressions to describe the evolution of the bulk Lorentz factor $\Gamma$ and
other quantities no longer exist. However, the relations between $t_{{\rm
peak},\nu}$, $F_{{\rm peak},\nu}$ and $\nu$ can be analyzed numerically. We
fitted $a$ and $b$ with various parameter values and found that both $a$ and
$b$ show no strong dependence on $E_{\rm iso}$ or $A$. While these indices do
change more significantly with $k$. For typical parameter values, the larger
the $k$ value, the flatter the late time $a$, and the smaller the $b$.
Generally speaking, for shocks with $E_{\rm iso}=10^{53}$ erg, $A=100{\rm
cm}^{-3}$, $\epsilon=0$, $p=2.2$, $\varepsilon_{e}=1/3$, and
$\varepsilon_{B}=10^{-2.5}$, we have $a\sim 1.6$, and $b\sim 0.95$ for the
$k=0$ case. While $a$ decreases from $\sim 1.6$ to $\sim 1.0$, $b$ decreases
from $\sim 1.0$ to $\sim 0.89$ when $k$ increases from $0$ to $2$. By
comparing Fig. 4 with Fig. 3(c), it can be seen that the late peaks due to the
transition of $\nu_{a}$ could appear earlier than the beginning of the
transrelativistic phase for the circumburst environment with $k\geqslant 1$
and small $A$. However, because our numerical calculations cover the complete
from as early as $<T_{90}+0.1$ days, the general trend mentioned above still
applies.

We also investigate the effects on transrelativistic $a$ and $b$ caused by the
electron distribution index $p$, the proportion of electron and magnetic field
energies $\varepsilon_{e}$ and $\varepsilon_{B}$, as well as shock radiation
efficiency $\epsilon$. We find that for electrons with distribution index
$2.0<p<2.5$, the values of $a$ and $b$ depend weakly on $p$, and a larger $p$
can lead up to a steeper $b$, as well as a shallower $a$. However, such a
trend is not as clear as the $k$ dependence mentioned above, and the change in
$a$ is hardly noticeable. For example, $a$ decreases from 1.6 to 1.55 when $p$
increases from $2.1$ to $2.4$. Meanwhile, a larger $p$ can give rise to a $b$
value as large as $\sim 1.1$. Besides, both $a$ and $b$ depend weakly on
$\varepsilon_{B}$, and a smaller $\varepsilon_{B}$ can lead to flatter $a$ and
$b$. For example, $a$ decreases from $\sim 1.6$ to $\sim 1.4$ when
$\varepsilon_{B}$ decreases from $0.01$ to $0.001$, while neither
$\varepsilon_{e}$ nor $\epsilon$ shows a strong influence on $a$ and $b$. In
short, the variation of $b$ within a wide parameter range is relatively small,
while the value of $a$ is largely determined by $k$. With the value of $a$ in
the transrelativistic regime, one can put a stringent constraint on $k$.

For wideband observations covering $\sim 1-10^{2}$ GHz frequency range, the
values of the $t_{\rm peak,\nu}-\nu$ and $F_{\rm peak,\nu}-\nu$ indices $a$
and $b$ fitted from peak data should fall between $a\sim 2/3$, $b\sim 0-1/3$
for the ultrarelativistic regime, and $a\sim 1-0.9$ as well as $b\sim 1.6-1$
for the transrelativistic regimes, depending on $k\sim 0-2$. Similar to Eqs.
15 and 16, such a transrelativistic trend with steeper $a$ and $b$ compared
with the ultrarelativistic case can also be used to put constraints on $k$
qualitatively, using $a$ and $b$ derived with multiband light-curve peaks.

###  2.2 Corrections for the Jet

Figure 6: Theoretical GRB radio afterglow light curves considering forward-
reverse shock interactions with $E_{\rm iso}=10^{52}$ erg(left) and $10^{53}$
(right) erg, respectively. The other parameters are adopted as the typical
values listed in Section 2. It can be seen that such interactions mainly
affect higher frequency light curves at early times ($\leqslant 1$ day), with
a smaller $E_{\rm iso}$ can give rise to an earlier FS/RS peak. Also, For
frequencies $<4.8$ GHz, the FS/RS peak is hardly noticeable, with the only
prominent structure as the later peak shown in Fig. 2.

Many GRBs have already entered postjet break evolution when reaching radio
afterglow peak times, especially for low observing frequencies, that is,
$t_{{\rm peak},\nu}>t_{b}$, where $t_{b}$ denotes the jet break time (Chandra
& Frail, 2012). It is necessary to discuss the effects brought by a GRB jet in
order to get a full understanding of multiband afterglows. Assuming the GRB
jet-opening angle is $\theta_{j}$, with an initial opening angle $\theta_{0}$.
When the bulk Lorentz factor of GRB shock wave decreases to
$\Gamma<\theta_{j}^{-1}$, the jet-opening angle becomes larger than the
relativistic beaming angle $1/\Gamma$, and thus, approximately speaking, the
flux density from the jet is $\sim\Gamma^{2}\theta_{0}^{2}$ times smaller than
that from spherical shock.

Firstly, we consider a jet without sideways expansions, that is,
$\theta_{j}=\theta_{0}$. Here we perform a similar analysis to Mészáros & Rees
(1999). Assuming the flux density from spherical shock evolves as
$F_{\nu,1}\propto t^{a_{1}}$, a jet with the same parameters can give rise to
a flux density $F_{\nu,2}\propto t^{a_{2}}$. Because in the ultrarelativistic
phase the bulk Lorentz factor evolves as $\Gamma\propto
t^{\left(3-k\right)/\left(8-2k\right)}$, and $F_{\nu,1}/F_{\nu,2}\propto
t^{a_{1}-a_{2}}\propto\Gamma^{-2}\theta_{0}^{-2}$, it can be seen that

$a_{2}=a_{1}-\frac{3-k}{4-k},$ (17)

That is, analytically speaking, for a relativistic jet without sideways
expansions, the slope of the afterglow light curves during $t>t_{j}$ is the
same as the slopes presented by Eqs. 7-12 minus a correction factor
$\frac{3-k}{4-k}$. It can be seen that no radio peak exists after $t_{j}$, no
matter the $k$ value. If a jet exists during the radio afterglow evolution,
observationally the radio-peak time $t_{{\rm peak},\nu}$ should equal the jet
break time $t_{j}$, if no peak occurs before $t_{j}$, and $t_{{\rm peak},\nu}$
no longer depends on the observing frequency $\nu$ in this case, making $a=0$.
If we assume that $t_{j}<t_{m}$, and that $t_{j}$ is not far from $t_{m}$, we
can calculate that $b=1/3$. Such a $b$ value cannot be distinguished from
peaks caused by spherical shocks. Although Huang et al. (2000) show that the
shock dynamics can be affected by the jet, according to the numerical
calculation, the afterglow flux decreases smoothly after jet break when
sideways expansions of the jet can be ignored even with evolving temporal
index, and later emission peaks do not appear. Thus, the analytical
conclusions on $a$ and $b$ still apply.

Analytical expressions for jets with sideways expansions only exist when
$k=0$, according to Sari et al. (1999). In this case, we also have
$a_{2}<a_{1}$, and $a_{2}\leqslant 0$. That is, no emission peak exists after
the jet break. Hence, similar to jets without expansions, the possible peak
marks the jet break time $t_{j}$, with $a=0$, $b=1/3$ (providing that $t_{j}$
occurs not too early). For arbitrary $k$ values, a numerical calculation based
upon dynamical equations provided by Huang et al. (2000) is needed. We take
the local speed of sound $c_{s}=c/\sqrt{3}$ as the speed of jet expansion to
calculate the dynamical evolution and find that the existence of a jet has a
greater impact on late-time afterglow behaviors. During this time, the
condition $\Gamma>\theta_{j}^{-1}$ can be satisfied once more, making the
afterglow flux rise again, forming a new emission peak. As early peaks are
produced by spherical shocks, the late peaks due to jet effects appear earlier
at higher frequencies, too. Because for typical parameters the late peak
appears at several hundred days after GRB prompt emission, the analytical
solutions proposed by Sari et al. (1999) or Rhoads (1999) do not apply, even
for the $k=0$ ISM environment. The flux level of such a late peak is much
lower than that of early peaks, for the afterglow has already entered
transrelativistic phase during this time, making the relativistic beaming
effect much less significant. And it should be noted that when $k\sim 2$, we
can almost have $\Gamma>\theta_{j}^{-1}$ satisfied all the way. As a result,
it is possible that no jet break exists in the stellar-wind environment.
Figure 5 shows the temporal evolution of the jet-opening angle for various
$k$, and its relation with $1/\Gamma$.

According to numerical results, the $a$ value of the late emission peaks for
sideways-expanding jets decreases with increasing $k$, which is similar to the
case with spherical shock waves. The value of $a$ decreases from $\sim 1.1$ to
$\sim 0.88$ when $k$ increases from $0$ to $2$. Meanwhile, the value of $b$
for jets increases with larger $k$, which is opposite to the spherical case.
And $b$ is quite small this time, even smaller than the ultrarelativistic
case. For example, our results show that $b\sim 0.15$ for $k=0$, and $b\sim
0.45$ for $k=2$. However, it should be noted that all such results are based
upon one single assumption, that is, the jet expands at a constant speed of
$c/\sqrt{3}$. In reality, this assumption may not be satisfied after the
afterglow enters the transrelativistic regime. Late-time afterglows may expand
quite slowly, making the jet-opening angle $\theta_{j}<1/\Gamma$, thus no late
peak can be formed. Also, earlier numerical simulation shows that jets do not
have significant sideways expansions (Kumar & Granot 2003; Granot & Kumar
2003). In one word, effects caused by jet may be more complex in real GRBs.

##  3 Observational Implications

###  3.1 Sample Selection

We collected data on GRBs with multiband radio light-curve observations (that
is, GRBs with radio light curves in no fewer than two bands) from the
literature, in order to put constraints on the circumburst density profiles
based on Eqs. 13 and 14. Only the observed flux and corresponding epochs,
rather than fitted peak values, such as the ones shown in Table 4 of Chandra &
Frail (2012) or Table 1 of Li et al. (2015), were utilized in this analysis.
Because it can be seen from Eqs. 13 and 14, as well as in Section 2.1 that
theoretically speaking, both relativistic and transrelativistic cases give
rise to nonnegative $a$ and $b$, no matter of the $k$ value, high-frequency
GRB radio afterglow light curves should exhibit a higher $F_{\rm peak}$, as
well as a $t_{\rm peak}$ no later than lower-frequency peaks, a general trend
of later and weaker peaks at lower frequencies is required during our sample
selection, in order to comply with the theoretical framework of this analysis.
For this reason, GRB 980519 (Frail et al. 2000b) was excluded, due to the late
appearance of higher-frequency peaks compared with the lower-frequency ones,
which is hard to explain with the GRB afterglow model based upon synchrotron
radiation, unless we take large observation errors and light-curve modulation
due to interstellar scintillation into consideration. Besides, GRBs 050416A
(Soderberg et al. 2007), 020813, and 050713B (Li et al. 2015) were omitted,
due to their higher peak fluxes compared with their lower-frequency ones.
Although our strategy is somewhat biased, such a bias could be considered as
originating from model constraints, especially for our radio-peak-focused
approach to get the $k$ value.

Table 1: Characteristics of GRB Multi-Band Radio Afterglow Emission Peaks

GRB | Telescopea | Frequency | $F_{\rm peak,\nu}$ | $t_{\rm peak,\nu}$ | Reference
---|---|---|---|---|---
|  | (GHz) | ($T_{90}$ \+ days) | ($\mu$Jy)($T_{90}$ \+ days) |
970508 | VLA | $1.43$ | $81.03^{+21.08}_{-10.49}$ | $485\pm 45$ | Frail et al. (2000a)
| VLA | $4.86$ | $20.2^{+1.84}_{-1.43}$ | $1350\pm 55$ | Frail et al. (1997)
| VLA | $8.46$ | $20.2^{+1.84}_{-1.43}$ | $1200\pm 47$ | Frail et al. (1997)
970828 | VLA | $4.86$ | $5.6^{+20.8}_{-2.1}$ | $99\pm 50$ | Djorgovski et al. (2001)
| VLA | $8.46$ | $3.5^{+2.1}_{-1.0}$ | $147\pm 33$ | Djorgovski et al. (2001)
980329 | VLA | $4.9$ |  $28.86^{+3.08}_{-3.92}$ e | $128\pm 42$ | Taylor et al. (1998)
| VLA | $8.3$ | $12.93^{+6.95}_{-3.03}$ | $274\pm 34$ | Taylor et al. (1998)
980425 | ATCA | $1.38$ | $45.7^{+6.0}_{-5.7}$ | $28900\pm 500$ | Kulkarni et al. (1998)
| ATCA | $2.49$ | $34.7^{+2.1}_{-1.8}$ | $33500\pm 500$ | Kulkarni et al. (1998)
| ATCA | $4.86$ | $11.7^{+2.9}_{-1.8}$ | $44600\pm 300$ | Kulkarni et al. (1998)
| ATCA | $8.46$ | $11.7^{+2.9}_{-1.8}$ | $49400\pm 200$ | Kulkarni et al. (1998)
980703 | VLA | $1.43$ | $49.10^{+0.94}_{-9.99}$ | $148\pm 37$ | Frail et al. (2003)
| VLA | $4.86$ | $14.38^{+2.77}_{-2.15}$ | $1200\pm 48$ | Frail et al. (2003)
| VLA | $8.46$ | $12.93^{+2.15}_{-6.92}$ | $1050\pm 35$ | Frail et al. (2003)
990510 | ATCA | $4.8$ |  $9.22$ $\left(>0.72\right)^{b}$ | $177\pm 36$ | Harrison et al. (1999)
| ATCA | $8.7$ | $3.31^{+1.93}_{-2.59}$ | $227\pm 30$ | Harrison et al. (1999)
991208f | VLA | $4.86$ |  $21.96$ $\left(>12.04\right)^{b}$ | $820\pm 52$ | Galama et al. (2000)
| VLA | $8.46$ | $11.73^{+1.00}_{-1.11}$ | $1990\pm 33$ | Galama et al. (2000)
| Ryle | $15.0$ | $5.36^{+2.06}_{-1.11}$ | $2100\pm 600$ | Galama et al. (2000)
000301C | VLA | $4.86$ | $48.06^{+16.07}_{-6.00}$ | $226\pm 51$ | Berger et al. (2000)
| VLA | $8.46$ | $12.17^{+2.00}_{-7.19}$ | $483\pm 26$ | Berger et al. (2000)
| Ryle | $15.0$ |  $3.57$ $\left(<8.8\right)^{c}$ | $660\pm 160$ | Berger et al. (2000)
000418g | VLA | $4.86$ | $14.63^{+1.93}_{-1.98}$ | $1120\pm 52$ | Berger et al. (2001)
| VLA | $8.46$ | $14.63^{+1.93}_{-1.98}$ | $1240\pm 46$ | Berger et al. (2001)
| Ryle | $15.0$ | $12.32^{+2.20}_{-1.98}$ | $1350\pm 480$ | Berger et al. (2001)
000911 | VLA | $4.86$ | $11.05^{+6.15}_{-4.86}$ | $71\pm 23$ | Price et al. (2002a)
| VLA | $8.46$ | $3.06^{+1.00}_{-0.87}$ | $278\pm 36$ | Price et al. (2002a)
000926 | VLA | $4.86$ | $10.75^{+0.96}_{-3.55}$ | $395\pm 52$ | Harrison et al. (2001)
| VLA | $8.46$ | $8.22^{+2.56}_{-1.03}$ | $566\pm 34^{h}$ | Harrison et al. (2001)
| Ryle | $15.0$ | $4.70^{+4.06}_{-0.78}$ | $820\pm 390$ | Harrison et al. (2001)
| VLA | $22.5$ |  $7.19$ $\left(<8.22\right)^{c}$ | $1415\pm 185$ | Harrison et al. (2001)
010921 | VLA | $4.86$ |  $37.77$ $\left(>34.82\right)^{d}$ | $140\pm 28$ | Price et al. (2002b)
| VLA | $8.46$ | $27.01^{+10.76}_{-1.08}$ | $229\pm 22$ | Price et al. (2002b)
| VLA | $22.5$ |  $25.93$ $\left(<32.80\right)^{c}$ | $330\pm 90$ | Price et al. (2002b)
011121 | ATCA | $4.80$ | $6.86^{+8.16}_{-3.44}$ | $510\pm 38$ | Price et al. (2002c)
| ATCA | $8.70$ | $6.86^{+8.16}_{-3.56}$ | $610\pm 39$ | Price et al. (2002c)
020903 | VLA | $1.5$ | $36.7^{+37.3}_{-8.10}$ | $294\pm 91$ | Soderberg et al. (2004a)
| VLA | $4.9$ | $36.7^{+37.3}_{-8.10}$ | $832\pm 47$ | Soderberg et al. (2004a)
| VLA | $8.5$ |  $23.8$ $\left(<25.7\right)^{c}$ | $1058\pm 19$ | Soderberg et al. (2004a)
021004 | VLA | $4.86$ | $27.2^{+4.5}_{-7.0}$ | $476\pm 38$ | de Ugarte Postigo et al. (2005)
| VLA | $8.46$ | $16.3^{+3.9}_{-3.7}$ | $720\pm 49$ | de Ugarte Postigo et al. (2005)
| VLA | $22.5$ | $6.2^{+4.0}_{-0.8}$ | $1282\pm 85$ | de Ugarte Postigo et al. (2005)
030329 | GMRT | $0.61$ | $334.15^{+43.86}_{-52.66}$ | $1140\pm 250$ | van der Horst et al. (2008)
| WSRT | $0.84$ |  $318.555$ $\left(<379.267\right)^{c}$ | $2332\pm 288$ | van der Horst et al. (2008)
| WSRT | $1.4$ | $170.739^{+25.152}_{-44.507}$ | $2290\pm 190$ | van der Horst et al. (2005)
| WSRT | $2.3$ | $72.356^{+49.741}_{-31.013}$ | $3840\pm 70$ | van der Horst et al. (2005)
| VLA | $4.86$ | $33.58^{+2.94}_{-2.07}$ | $11620\pm 80$ | Berger et al. (2003)
| VLA | $8.46$ | $14.87^{+1.79}_{-2.18}$ | $19150\pm 80$ | Berger et al. (2003)
| VLA | $15.0$ | $11.90^{+0.79}_{-2.41}$ | $31400\pm 250$ | Berger et al. (2003)
| VLA | $22.5$ | $9.49^{+2.41}_{-1.81}$ | $48160\pm 230$ | Berger et al. (2003)
| VLA | $43.3$ | $6.89^{+0.79}_{-2.13}$ | $55330\pm 430$ | Berger et al. (2003)
031203i | VLA | $4.86$ | $65.32^{+7.98}_{-11.90}$ | $751\pm 45^{j}$ | Soderberg et al. (2004b)
| VLA | $8.46$ | $811\pm 40$ | $19.45^{+3.03}_{-2.02}$ | Soderberg et al. (2004b)
050820A | VLA | $4.86$ | $2.19^{+1.96}_{-2.034}$ | $256\pm 78$ | Cenko et al. (2006)
| VLA | $8.46$ | $0.93^{+1.23}_{-0.814}$ | $634\pm 62$ | Cenko et al. (2006)
051022 | WSRT | $4.9$ | $2.22^{+1.08}_{-1.03}$ | $342\pm 34$ | Rol et al. (2007)
| VLA | $8.5$ |  $1.64$ $\left(<13.37\right)$ | $585\pm 49^{c}$ | Cameron & Frail (2005)
060218k | VLA | $1.43$ | $6.97^{+5.11}_{-5.12}$ | $134\pm 145$ | Soderberg et al. (2006)
| VLA | $4.86$ | $4.85^{+2.12}_{-1.02}$ | $328\pm 61$ | Soderberg et al. (2006)
| VLA | $8.46$ |  $1.87$ $\left(<3.00\right)^{c}$ | $453\pm 77$ | Soderberg et al. (2006)
070125 | VLA | $4.86$ | $27.90^{+7.01}_{-5.17}$ | $308\pm 78$ | Chandra et al. (2008)
| VLA | $8.46$ | $18.86^{+1.01}_{-1.01}$ | $660\pm 29$ | Chandra et al. (2008)
| VLA | $14.94$ | $16.06^{+1.83}_{-1.09}$ | $1410\pm 137$ | Chandra et al. (2008)
| VLA | $22.5$ | $12.80^{+1.29}_{-2.07}$ | $1603\pm 235$ | Chandra et al. (2008)
071003 | VLA | $4.86$ |  $3.84$ $\left(<21.68\right)^{c}$ | $220\pm 54$ | Perley et al. (2008)
| VLA | $8.46$ | $3.82^{+4.90}_{-2.06}$ | $430\pm 50^{l}$ | Perley et al. (2008)
100418Am | JVLA | $4.95$ | $72^{+23}_{-16}$ | $513\pm 44$ | Moin et al. (2013)
| JVLA | $8.46$ | $48^{+8}_{-15}$ | $1454\pm 21$ | Moin et al. (2013)
100814An | JVLA | $4.7$ | $11.3^{+12.9}_{-3.9}$ | $461.13$ | De Pasquale et al. (2012)
| JVLA | $7.9$ | $11.3^{+12.9}_{-3.9}$ | $560.43$ | De Pasquale et al. (2012)
111215Ao | WSRT | $1.4$ | $31.01^{+16.99}_{-8.98}$ | $373\pm 170$ | van der Horst et al. (2015)
| WSRT | $4.8$ | $22.03^{+8.98}_{-6.94}$ | $1079\pm 51$ | van der Horst et al. (2015)
| JVLA | $6.7$ | $15.41^{+10.99}_{-1.97}$ | $1180\pm 30$ | Zauderer et al. (2013)
| JVLA | $8.4$ |  $15.41$ $\left(<26.40\right)^{c}$ | $1340\pm 63$ | Zauderer et al. (2013)
| JVLA | $19.1$ | $11.49^{+3.92}_{-8.14}$ | $1670\pm 47$ | Zauderer et al. (2013)
| JVLA | $24.4$ |  $3.35$ $\left(<11.49\right)^{c}$ | $1990\pm 47$ | Zauderer et al. (2013)
130427A | JVLA | $5.1$ | $2.04^{+2.71}_{-1.36}$ | $1760\pm 88$ | Perley et al. (2014)
| JVLA | $6.8$ |  $0.68$ $\left(<2.04\right)^{c}$ | $2570\pm 160$ | Perley et al. (2014)
| AMI | $15.7$ | $0.64^{+0.91}_{-0.28}$ | $4160\pm 220$ | Anderson et al. (2014)
140304A | JVLA | $4.90$ | $8.55^{+9.71}_{-4.00}$ | $112\pm 16$ | Laskar et al. (2018)
| JVLA | $7.00$ | $8.55^{+9.71}_{-4.00}$ | $166\pm 12$ | Laskar et al. (2018)
| JVLA | $8.55$ | $8.55^{+9.70}_{-4.00}$ | $187\pm 16$ | Laskar et al. (2018)
| JVLA | $11.0$ | $4.55^{+4.00}_{-3.01}$ | $241\pm 17$ | Laskar et al. (2018)
| JVLA | $13.5$ | $4.55^{+4.00}_{-3.01}$ | $280\pm 13$ | Laskar et al. (2018)
| JVLA | $16.0$ | $4.55^{+4.00}_{-3.01}$ | $323\pm 13$ | Laskar et al. (2018)
| JVLA | $19.2$ | $4.55^{+4.00}_{-3.00}$ | $333\pm 18$ | Laskar et al. (2018)
| JVLA | $24.5$ |  $1.55$ $\left(<3.00\right)^{c}$ | $307\pm 28$ | Laskar et al. (2018)
| JVLA | $30.0$ |  $1.55$ $\left(<3.00\right)^{c}$ | $384\pm 40$ | Laskar et al. (2018)
| JVLA | $37.0$ |  $1.55$ $\left(<3.00\right)^{c}$ | $469\pm 55$ | Laskar et al. (2018)
141121Ap | JVLA | $3$ | $21.350^{+2.979}_{-4.965}$ | $156\pm 21$ | Cucchiara et al. (2015)
| JVLA | $5$ | $21.350^{+2.979}_{-4.965}$ | $184\pm 15$ | Cucchiara et al. (2015)
| JVLA | $7$ | $16.385^{+4.965}_{-4.983}$ | $270\pm 18$ | Cucchiara et al. (2015)
| AMI | $15$ | $3.037^{+5.315}_{-2.127}$ | $460\pm 50$ | Cucchiara et al. (2015)
160509Aq | JVLA | $1.644$ |  $10.03$ $\left(>6.00\right)^{d}$ | $170.2\pm 102.6$ | Laskar et al. (2016)
| JVLA | $2.679$ | $6.00^{+4.03}_{-0.69}$ | $334.8\pm 40.1$ | Laskar et al. (2016)
| JVLA | $3.523$ | $4.06^{+1.25}_{-4.06}$ | $384.4\pm 27.1$ | Laskar et al. (2016)
| JVLA | $5.0$ | $2.1479^{+0.8458}_{-0.9797}$ | $509.1\pm 18.9$ | Laskar et al. (2016)
| JVLA | $7.4$ | $4.06^{+1.94}_{-1.0663}$ | $1389.1\pm 20.2$ | Laskar et al. (2016)
| JVLA | $8.5$ | $2.9781^{+1.0819}_{-1.8253}$ | $1049.0\pm 17.1$ | Laskar et al. (2016)
| JVLA | $11.0$ | $2.1479^{+1.94}_{-1.0819}$ | $1261.0\pm 17.9$ | Laskar et al. (2016)
| JVLA | $13.5$ |  $1.1369$ $\left(<2.9623\right)^{c}$ | $946.0\pm 27.2$ | Laskar et al. (2016)
| JVLA | $16.0$ |  $1.1369$ $\left(<2.9623\right)^{c}$ | $1025.9\pm 31.7$ | Laskar et al. (2016)
| JVLA | $19.2$ |  $1.1163$ $\left(<2.1259\right)^{c}$ | $1341.0\pm 33.7$ | Laskar et al. (2016)
| JVLA | $24.5$ |  $1.1163$ $\left(<2.1259\right)^{c}$ | $1891.0\pm 56.3$ | Laskar et al. (2016)
| JVLA | $30.0$ |  $1.0911$ $\left(<2.9165\right)^{c}$ | $2117.8\pm 47.9$ | Laskar et al. (2016)
160625Br | JVLA | $1.77$ | $22.52^{+25.86}_{-10.02}$ | $346\pm 62$ | Alexander et al. (2017)
| JVLA | $2.68$ | $12.94^{+10.02}_{-6.19}$ | $621\pm 31$ | Alexander et al. (2017)
| JVLA | $3.52$ | $12.94^{+10.02}_{-6.19}$ | $475\pm 40$ | Alexander et al. (2017)
| JVLA | $5.0$ | $2.50^{+3.79}_{-1.13}$ | $932\pm 24$ | Alexander et al. (2017)
| JVLA | $7.1$ | $2.50^{+3.79}_{-1.13}$ | $1310\pm 20$ | Alexander et al. (2017)
| JVLA | $8.5$ | $2.49^{+3.79}_{-1.14}$ | $1135\pm 28$ | Alexander et al. (2017)
| JVLA | $11.0$ | $2.49^{+3.79}_{-1.14}$ | $946\pm 25$ | Alexander et al. (2017)
170817A | JVLA | $2.6$ | $162.9^{+35.8}_{-57.8}$ | $69.5\pm 16.3$ | Margutti et al. (2018)
| ATCA | $5.5$ | $149.6^{+13.3}_{-24.0}$ | $99.1\pm 8.4$ | Dobie et al. (2018)
| ATCA | $9.0$ | $125.6^{+24.0}_{-10.2}$ | $84.5\pm 10.9$ | Dobie et al. (2018)
171010A | JVLA | $4.5$ | $9.66^{+18.93}_{-7.09}$ | $929\pm 42$ | Bright et al. (2019)
| JVLA | $5.5$ | $9.66^{+18.93}_{-7.09}$ | $978\pm 33$ | Bright et al. (2019)
| JVLA | $6.5$ | $9.66^{+18.93}_{-7.09}$ | $1025\pm 35$ | Bright et al. (2019)
| JVLA | $7.5$ | $9.66^{+18.93}_{-7.09}$ | $1090\pm 35$ | Bright et al. (2019)
| JVLA | $8.5$ |  $2.56$ $\left(<9.27\right)^{c}$ | $1024\pm 40$ | Bright et al. (2019)
| JVLA | $9.5$ |  $2.56$ $\left(<9.27\right)^{c}$ | $1580\pm 54$ | Bright et al. (2019)
| JVLA | $10.5$ |  $2.56$ $\left(<9.27\right)^{c}$ | $1858\pm 64$ | Bright et al. (2019)
| JVLA | $11.5$ |  $2.56$ $\left(<9.27\right)^{c}$ | $2009\pm 71$ | Bright et al. (2019)
| JVLA | $15.5$ | $2.37^{+0.94}_{-1.02}$ | $2520\pm 140$ | Bright et al. (2019)

Note. —

a “VLA” for Very Large Array; “JVLA” for the upgraded Karl G. Jansky Very
Large Array; “ATCA” for Australia Telescope Compact Array; “GMRT” for Giant
Meterwave Radio Telescope; “WSRT” for Westerbork Synthesis Radio Telescope;
“Ryle” for Ryle Telescope; “AMI” for Arcminute Mircrokelvin Imager.

b Error bars of $t_{\rm peak,\nu}$ denotes the times on which the adjacent
measurements are taken around the corresponding peak of the light curve.

c Flux decreases monotonically during the complete observing campaign without
a definite peak; we took the highest $S_{\nu}$ as peak flux.

d Flux increases monotonically during the complete observing campaign without
a definite peak; we took the highest $S_{\nu}$ as peak flux.

e The 4.9 MHz light curve of GRB 980329 exhibits two peaks, according to
Taylor et al. (1998). Since the first peak appears earlier than the 8.3 MHz
peak, which is in contradiction with the standard afterglow model, here we
adopt the later peak for our analysis.

f The 30 GHz data acquired by the Owens Valley Radio Observatory’s 40 m
telescope from Galama et al. (2000) were discarded due to scintillated light
curve without a definite peak, and the later appearance of the highest flux
compared to the 15 GHz data; the 1.43 GHz data acquired by VLA from Galama et
al. (2000) were discarded due to the earlier peak time compared to higher
frequencies.

g The 22.46 GHz data acquired by VLA from Berger et al. (2001) were discarded
due monotonic light curve and later appearance of the highest flux compared to
the 15 GHz data.

h The 8.46 GHz data point with $S_{\nu}=566\pm 34$ $\mu$Jy acquired at 2000
Octpber 5.216 (UTC) by VLA from Harrison et al. (2001) was adopted as the peak
flux; a later rebrightening at October 12.771 with $S_{\nu}=644\pm 126$
$\mu$Jy was discarded due to the later appearance compared to the 4.86 GHz
peak, as well as the larger error bar.

i The 1.43 GHz data acquired by VLA from Berger et al. (2003) were discarded
due to scintillated light curves and earlier appearance of the highest flux
measurement compared to the 4.86 GHz data; the 22.5 GHz data were discarded
due to monotonic light curve and a maximum flux significantly lower than the
8.46 GHz data.

j Two flux measurements with similar flux levels, $S_{\nu}=728\pm 55$ and
$749\pm 63$ $\mu$Jy, were performed at 2004 January 4.33 and January 15.35
(UTC), respectively, as shown in Soderberg et al. (2004b). It seems that the
radio afterglow of this burst suffered significantly from interstellar
scintillation, according to data presented in Soderberg et al. (2004b), thus
making the exact peak time somewhat hard to determine.

k The 15 GHz measurements acquired by the Ryle Telescope from Soderberg et al.
(2006) were discarded due to large measurement errors and lack of data points;
the 22.5 GHz measurements acquired by VLA from Soderberg et al. (2006) were
discarded due to lack of data points.

l An 8.46 GHz measurement performed at 2007 October 12.04 (UTC) shows a flux
level of $S_{\nu}=431\pm 51$ $\mu$Jy (Perley et al. 2008), which is slightly
larger than the peak data listed in this table. The earlier $S_{\nu}=430\pm
50$ $\mu$Jy was chosen as a peak considering the theoretical predictions for
earlier peaks at higher frequencies. However, due to no 8.46 GHz data has been
taken between the two measurements, it is difficult to know the exact peak
time for the GRB 071003 radio afterglow at this frequency.

m ACTA 5.5 GHz data for GRB 100418A listed in Moin et al. (2013) were
discarded, due to the monotonic light-curve behavior, which is inconsistent
with the VLA data.

n No errors in the flux measurements shown in De Pasquale et al. (2012).

o The 15 GHz measurements acquired by AMI from van der Horst et al. (2015)
were discarded due to the lower flux and later peak compared to the 4.8 GHz
data.

p The 13 and 15 GHz measurements acquired by JVLA from Cucchiara et al. (2015)
were discarded due to the lower flux compared to the 7 GHz data.

q The 1.255 GHz measurements acquired by JVLA from Laskar et al. (2016) were
discarded due to a peak flux that appeared earlier than the 1.644 GHz peak,
with a flux higher than the 5.0 GHz peak.

r The 1.45 GHz measurements acquired by JVLA from Alexander et al. (2017) were
discarded due to an earlier peak compared to 1.77 GHz; all data with observed
frequency $>13.5$ GHz were discarded due to monotonic decline with maximum
$S_{\nu}$ lower than 11.0 GHz peak, due to the lack of early observations.

In total, we selected 32 GRBs, including 31 long bursts and 1 short one (GRB
170817A), as our sample. For observations at certain frequencies in our
sample, GRBs do not comply with the theoretical trend of higher $F_{\rm
peak}$, earlier $t_{\rm peak}$ at higher $\nu$ – such data points have been
omitted during our analysis. Meanwhile, we largely ignored the light-curve
peaks occurring earlier than $\sim 1-2$ days in the observer’s frame to avoid
possible contamination from early reverse shocks, because as can be seen in
Fig. 6, the typical peak time due to forward-reverse-shock interactions should
be at $\sim T_{90}+$ several$\times 10^{-1}$ days, which corresponds to $\sim
T_{90}+1-2$ days in the observer’s frame because the typical redshift value of
GRB is $z\sim 3$ (Jakobsson et al. 2005). Also, $\geqslant 50$ GHz data are
not taken into consideration, as in such high frequencies, the $t_{\rm
peak}=t_{m}$ conclusion does not apply (see Section 2). Considering the
sparsely sampled nature of GRB radio observations, as well as the not-so-steep
slopes of temporal evolution for radio afterglow light-curve fluxes (see Fig.
2), we took an approximate approach to make our analysis: The maximum observed
flux is recognized as of the peak flux $F_{\rm peak,\nu}$ at a certain
observing frequency, the corresponding observing time as the estimated peak
time $t_{\rm peak,\nu}$, and the epochs of adjacent observations give the
upper and lower limits of $t_{\rm peak,\nu}$. Besides, in order to make
constraints with as many observing frequencies as possible, we took another
strategy to utilize the monotonically evolved light curves: the reading of the
last data point of a monotonically increasing light curve was taken as the
lower limit of $F_{\rm peak}$, and the second-to-last epoch as the possible
lower range of error for $t_{\rm peak}$. Similarly, for a constantly declining
light curve, the first data point sets the lower limit of peak flux, and the
second observing epoch the upper limit of peak time.

Some GRBs have been observed by various telescopes at similar frequencies. For
example, GRB 030329 has been observed by WSRT at 1.4 GHz (van der Horst et al.
(2005), and GMRT at 1.28 GHz (van der Horst et al. (2008)). Also, GRB 100418A
has been observed by VLA and ACTA at 4.95 and 5.5 GHz, respectively Moin et
al. (2013). Because calibrations between different instruments could lead to
extra complications, in this case, we select peak data for analysis from as
few telescopes as possible. Hence, our main uncertainties arise from errors in
$t_{\rm peak,\nu}$ estimation, especially for GRB light curves with fewer
observed data points. Information on the $t_{\rm peak,\nu}$ and $F_{\rm
peak,\nu}$ for each band of our GRB samples, as well as descriptions of the
discarded outlier data, is listed in Table 1. It is worth noting that as seen
in Fig. 3, considering the long-lasting process of transition between
relativistic to deep Newtonian shock waves, nearly all of our samples should
be still in the relativistic or transrelativistic phase when the light-curve
peaks occur.

###  3.2 Constraints on Circumburst Environment

The values of the $a$ and $b$ indices of 32 sample GRBs are listed in Table 2.
Figure 8 shows the $a$ values, observing frequencies, as well as peak times in
light curves of each sample GRB. Figure 9 shows the corresponding $b$ value,
the observing bands, along with peak fluxes. And a statistic of the $k$ value
in the circumburst environment for our complete sample can be found in Figure
7. We compare these fitted indices with theoretical predictions from Eqs. 15
and 16 and Section 2.1, taking the circumburst density data provided by
Chandra & Frail (2012) as well as possible observational errors into
consideration. It can be seen that more than 40% of our samples can be
explained with uniform ISM distribution, while nearly one-fifth samples can be
described with stellar-wind environment. And due to various observational
complications, both $k=0$ and $k=2$, or any $0<k<2$ circumburst density
distributions work for about 30% of the samples, while two GRBs cannot be
properly constrained with our analysis using multiband radio peaks.

Table 2: Results of Multi-Band Light Curve Peak Fitting

GRB | Band Coverage | $a$ | $b$ | Explanationsa
---|---|---|---|---
| (GHz) |  |  |
970508 | $1.43-8.46$ | $0.84^{+0.09}_{-0.06}$ | $0.56_{-0.06}^{+0.06}$ | (1,2)
970828 | $4.86-8.46$ | $0.84^{+1.99}_{-0.68}$ | $0.71_{-0.80}^{+1.19}$ | (1,2)c
980329 | $4.9-8.3$ | $1.52^{+0.28}_{-0.60}$ | $1.44_{-0.56}^{+0.73}$ | (1)
980425 | $1.38-8.46$ | $0.86^{+0.06}_{-0.08}$ | $0.31_{-0.01}^{+0.01}$ | (2)
980703 | $1.43-8.46$ | $0.86^{+0.19}_{-0.10}$ | $1.21_{-0.14}^{+0.15}$ | (1)
990510 | $4.8-8.7$ | $1.72^{+1.73}_{-1.42}$ | $0.42_{-0.36}^{+0.41}$ | (1,2)
991208 | $4.86-15.0$ | $1.25^{+0.22}_{-0.34}$ | $1.51_{-0.42}^{+0.05}$ | (1,2)
000301C | $4.86-15.0$ | $2.31^{+0.79}_{-0.54}$ | $0.95_{-0.30}^{+0.28}$ | (1)
000418 | $4.86-15.0$ | $0.15^{+0.11}_{-0.11}$ |  $0.17_{-0.37}^{+0.25}$ d | (3,4)
000911 | $4.86-8.46$ | $2.32^{+0.56}_{-0.58}$ | $2.46_{-0.53}^{+0.68}$ | (5)
000926 | $4.86-22.5$ | $0.37^{+0.57}_{-0.17}$ | $0.80_{-0.18}^{+0.16}$ | (1,2)
010921b | $4.86-22.5$ | $0.22^{+0.75}_{-0.09}$ | $0.54_{-0.24}^{+0.20}$ | (1,2)
011121 | $4.80-8.70$ | $0.00^{+0.95}_{-0.85}$ | $0.30_{-0.15}^{+0.15}$ | (2,4)
020903b | $1.5-8.5$ | $0.21^{+0.70}_{-0.11}$ | $0.76_{-0.17}^{+0.20}$ | (1)
021004 | $4.86-22.5$ | $0.97^{+0.07}_{-0.27}$ | $0.64_{-0.07}^{+0.06}$ | (1,2)
030329b | $0.61-43.3$ | $1.00^{+0.03}_{-0.11}$ | $0.96_{-0.02}^{+0.03}$ | (2)
031203 | $4.86-8.46$ | $2.19^{+0.16}_{-0.25}$ | $0.14_{-0.13}^{+0.13}$ | (1)
050820A | $4.86-8.46$ | $1.51^{+1.50}_{-1.72}$ | $1.63_{-0.49}^{+0.62}$ | (1)c
051022 | $4.9-8.5$ | $0.53^{+0.19}_{-3.07}$ | $0.97_{-0.22}^{+0.21}$ | (1)c
060218 | $1.43-8.46$ | $0.67^{+0.50}_{-0.20}$ |  $0.69_{-0.46}^{+0.21}$ e | (1,2)
070125b | $4.86-22.5$ | $0.48^{+0.09}_{-0.07}$ | $1.12_{-0.15}^{+0.19}$ | (1)
071003 | $4.86-8.46$ | $0.01^{+2.08}_{-1.79}$ | $1.21_{-0.42}^{+0.49}$ | (5)
100418A | $4.95-8.46$ | $0.76^{+0.51}_{-0.29}$ | $1.94_{-0.15}^{+0.16}$ | (1)
100814A | $4.7-7.9$ | $0.00^{+0.86}_{-0.80}$ | 0.38 | (3,4)
111215Af | $1.4-24.4$ | $0.64^{+0.26}_{-0.17}$ | $0.55_{-0.12}^{+0.17}$ | (1,2)
130427Ab | $5.1-15.7$ | $0.81^{+0.49}_{-0.64}$ | $0.77_{-0.06}^{+0.06}$ | (1)c
140304A | $4.90-37.0$ | $1.00^{+0.37}_{-0.91}$ | $0.64_{-0.06}^{+0.06}$ | (1,2)
141121A | $3-15$ | $1.26^{+0.37}_{-0.52}$ | $0.70_{-0.09}^{+0.10}$ | (1)c
160509A | $1.644-30.0$ | $0.76^{+0.27}_{-0.16}$ | $0.79_{-0.09}^{+0.15}$ | (1)c
160625B | $1.77-11.0$ | $1.38^{+0.17}_{-0.30}$ | $0.65_{-0.06}^{+0.07}$ | (1)c
170817A | $2.6-9.0$ | $0.20^{+0.09}_{-0.24}$ | $0.18_{-0.21}^{+0.21}$ | (4)
171010A | $4.5-15.5$ | $1.56^{+0.55}_{-0.44}$ | $0.90_{-0.04}^{+0.04}$ | (1)

Note. —

a (1) Transrelativistic afterglows under a $k\sim 0$ ISM environment. (2)
Transrelativistic afterglows under a $k\sim 2$ wind environment. (3)
Ultrarelativistic afterglows with larger $k$. (4) Afterglows produced by
expanding jets with larger $k$. (5) Hard to explain, unless taking larger
observational errors into consideration.

b Show signs of transrelativistic break; both $a$ and $b$ become steeper at
later times.

c Possible reverse-shock contamination due to early peak time and/or large
$E_{\rm iso}$.

d Large fitting errors for $b$ due to large observation errors of 15 GHz data;
fitted as $b=0.18_{-0.10}^{+0.10}$ without 15 GHz flux.

e The 1.43 GHz peak flux as shown in Soderberg et al. (2006) exhibits large
errors comparable with the flux level. The fitting results without the 1.43
GHz data point is $a=1.72^{+2.00}_{-0.55}$, $b=0.58_{-0.43}^{0.42}$, which are
still compatible with the transrelativistic afterglows in the ISM explanation.

f Fitted with the 1.4 GHz data, which exhibits large measured errors
($S_{\nu}=373\pm 170$ $\mu$Jy). The fitting result should be
$a=0.9^{+0.48}_{-0.35}$, $b=0.36_{-0.02}^{+0.02}$ without 1.43 GHz data, which
is compatible with explanations (1, 2, 4).

Figure 7: A statistic on the $k$ distributions for the 32 GRB sample. It can
be seen that the radio peaks of 14 GRBs can be explained by the $k=0$ density
profile (with six samples showing possible contamination by the reverse
shock). Another six samples exhibit behaviors compatible with predictions for
the $k\sim 2$ case. Also, we have 10 GRBs that can be explained by density
distributions between the ISM and stellar-wind environment, along with another
2 bursts occurring in environment hard to constrain, due to observation
errors.

####  3.2.1 GRBs with ISM Density Profiles

According to our method, a total number of 14 GRBs out of the 32 samples can
be solely explained with a $k=0$ ISM environment. As listed in Table 2, we
find that the values of $a$ and $b$ for multiband radio afterglow peaks of
GRBs 980329, 980703, 000301C, 020903, 031203, 070125, 100418A, and 171010A lie
between the ultra- and transrelativistic cases, and thus can be explained with
transrelativistic afterglows under an ISM environment. And as can be seen in
Figs 8 and 9, although due to observational uncertainties, the time-frequency
index $a$ shows large fitting errors for GRBs980329, 020903, and 100418A,
which may lead to multiple possible $k$ values, the flux-$\nu$ index $b$ can
be constrained stringently enough. In fact, as noted in Section 2.1,
considering the late occurrence of the transrelativistic phase for $k=1.1$ or
2 cases, a $b\sim 1$ value alone can still lead to a confident conclusion of
$k=0$, no matter of the $a$ value. Besides, tt is worth noting that the radio
peaks of GRBs 020903 and 070125 show “breaks” of $a$ and $b$ values at a
certain frequency, which could imply a transition from ultra- to
transrelativistic phases, as the high-frequency peaks could appear during the
ultrarelativistic phase with flatter $a$ and $b$ values, while the low-
frequency ones occur later with steeper transrelativistic indices.

Compared with existing works using multiband modeling, it should be noted that
Berger et al. (2000), Soderberg et al. (2004b), and Chandra et al. (2008) have
drawn similar conclusions for GRBs 000301C, 031203, and 070125, respectively,
that is, all these three GRBs occur in ISM environments. And Starling et al.
(2008) found a possible ISM explanation for GRB 980329 and demonstrated that
GRB 980703 can be explained by both $k=0$ and $2$ environments. Frail et al.
(2003) also showed that the wind model cannot be ruled out (although not
favored) to fit the light curve of GRB 980703, which is still compatible with
our constraints. An exception is GRB 171010A, which has been explained by
Bright et al. (2019) using a steep density profile, although some hard-to-
explain features do exist. This conclusion is somewhat inconsistent with our
constraints, which could be due to the relatively large fitting errors for the
$b$ index, thus leaving large room for alternative explanations.

However, although the rest six GRBs in this category, including GRBs 050820A,
051022, 130427A, 141121A, 160509A, and 160625B generally follow predictions of
the $k=0$ ISM environment, Cenko et al. 2006, Rol et al. 2007, Perley et al.
(2014), Cucchiara et al. (2015), along with Alexander et al. (2017) all
pointed out a possible reverse-shock contribution of the radio afterglow peaks
in these bursts. With the existence of reverse-shock contamination, the
multiband peak method of analysis may no longer apply, thus the reliability of
our related results could be compromized.

####  3.2.2 GRBs Compatible with a Stellar-Wind Environment

GRBs occurring in $k=2$ density profiles as implied by multiband radio peaks
include GRBs 980425, 000418, 011121, 030329, and 100814A, along with the only
short GRB in our sample, GRB 170817A. Among these, the behavior of the GRBs
980425 and 030329 light-curve peaks follows predictions for late-time
afterglow ($\nu_{a}=\nu_{a,>}$) in the wind medium. And Berger et al. (2003)
and van der Horst et al. (2005) have fitted the afterglow light curves of GRB
030329 with both ISM and wind models, whose results are compatible with our
conclusions.

Meanwhile, the $b$ indices of GRBs 000418 and 031203 are relatively shallow
and are consistent with expanding jets in the ISM environment. However, their
$a$ indices are too small to be explained with jets. It also should be noted
that for several bursts, the medium- and low-frequency radio peaks may be
explained with jet breaks, although the details depend much on the jet-opening
angles (hence the jet break times). Analysis with multiband radio peaks may
only hint at multiple explanations, and data from other bands (e.g., optical)
should be utilized to clarify the existence of jets. Besides, it is quite
possible that the speed of the jet’s sideways expansion during the
transrelativistic phase should be slowed down, making the results drawn in
Section 2.2 no longer stand.

The value of $a$ index of GRB 011121 is consistent with the existence of a
jet, while the $b$ index falls within predictions for the $k=2$ case.
Considering the simultaneous appearance of the multiband peak and the large
errors in peak-time estimations that can be attributed to observational
effects, a burst occurring in a wind density profile is a more suitable
conclusion, which is supported by Price et al. (2002c). A similar case applies
for GRB 100814A, for which De Pasquale et al. (2012) has pinned down to the
existence of a jet.

For the gravitational-wave-associated GRB 170817A, the only short burst in our
sample, its multiband radio afterglow exhibits a shallow $a$ with a shallow
$b$. Such a case is more consistence with relativistic jet, which is
compatible with the off-axis jet identified by previous works (e.g., see Fong
et al. 2019).

####  3.2.3 GRBs Explained by k between 0 and 2

This category consists of 10 samples, which are GRBs 970508, 970828, 990510,
991208, 000926, 010921, 021004, 060218, 111215, and 140304A. The uncertainties
here mainly arise from the relatively large errors for peak-time/flux
estimates. For example, the peak behavior of GRBs 970508, 990510, and 010921
lies somewhere in between the $k=0$ trans-relativistic and $k=2$ late time
($\nu_{a}=\nu_{a,>}$) cases, when taking fitting errors into consideration,
making any $0\leqslant k\leqslant 2$ possible. This is compatible with the
conclusion drawn by Starling et al. (2008), who found a homogeneous
circumburst medium for GRBs 970508 and 990510 with X-ray, optical, and IR
afterglows. However, the exact $k$ for each sample is hard to pin down
precisely, due to large uncertainties in $a$ and $b$.

For GRBs with high circumburst densities, including GRBs 991208, 000926,
021004, 060218, and 111215A, the indices of $a$ and $b$ can be explained with
later transrelativistic shocks in the $k\sim 0$ environment. However, because
a larger $A$ can lead to a later $t_{cm}$ (thus a later peak time $t_{m}$ in
radio afterglow), the larger $k$ possibility, which can produce a steeper $b$
during the relativistic phase, cannot be excluded for these bursts. When
compared with existing works, Harrison et al. (2001) showed that the temporal
behavior of GRB 000926’s light curve can be better modeled with a Comptonized
uniform medium; Soderberg et al. (2006) fitted GRB 060218 with both $k=2$ and
$k=0$ density profiles, as well as low an $E_{\rm iso}$; and Zauderer et al.
(2013) has pointed out that the circumburst medium of GRB 111215A should be in
the form of stellar wind, which are all consistent with our analysis.

For GRBs 970828 and 140304A, although the $a$ and $b$ indices can be explained
with both $k=0$ and $2$ possibilities when taking fitting errors into
consideration, it should be noted that the afterglows of these bursts were
significantly affected by reverse shock, as shown by Djorgovski et al. (2001)
and Laskar et al. (2018). And the radio light curve of GRB 140304A might even
be contaminated by interstellar scintillation (Laskar et al., 2018). All of
these factors can complicate our analysis, and thus compromise the robustness
of the results.

####  3.2.4 GRBs Hard to Explain with Multiband Light-curve Peaks

Finally, we have two outliers that cannot be well constrained by the multiband
peak method. One of them is GRB 000911, whose $a$ and $b$ values are much
steeper than theoretical predictions for both trans- or ultra- relativistic
shock waves, thus making its afterglow behavior hard to explain, even
considering the fitting errors. Another hard-to-explain case is GRB 071003,
which is located near the Galactic plane, with its radio data severely
affected by scintillation (Perley et al., 2008), thus leaving a large room for
errors in the $a$ and $b$ indices, therefore making multiple explanations,
including both relativistic or transrelativistic ejecta propagating in the
$k=0$ and $k=2$ circumburst environment, possible.

##  4 Summary and Discussion

In this paper, we investigated the radio afterglows of gamma-ray bursts
occurring in a power-law-distributed circumburst medium with $n\propto
R^{-k}$, and an arbitrary $k$. We show that one can use multiband radio
afterglow peak time $t_{{\rm peak},\nu}$ and $F_{{\rm peak},\nu}$ data to put
a constraint on the density-distribution index $k$. We find that in the
relativistic phase of the afterglow evolution, the peak time $t_{{\rm
peak},\nu}$ corresponds to $t_{m}$, that is, the time when the minimum
frequency of electrons equals to the observing frequency. And we have $t_{{\rm
peak},\nu}\propto\nu^{-a}$, $a\sim 2/3$, which is independent of the shock
radiation efficiency, while for $F_{{\rm peak},\nu}\propto\nu^{b}$, the value
of $b$ depends more strongly on $k$. For adiabatic shocks, $b$ increases from
$0$ to $1/3$, if we change $k$ from $0$ to $2$. And in the transrelativistic
phase, similar dependencies between peak time/flux and frequency exist, with
steeper $a$ and $b$ values than in the ultrarelativistic phase.

Figure 8: The relations between radio afterglow peak fluxes $t_{{\rm
peak},\nu}$ and observing frequencies $\nu/\nu_{{\rm min}}$ for our sample
GRBs. The dashed line with data points in each figure shows the fitting
results from observations, the blue dotted lines denote the possible range of
fitting errors, while the black solid lines from top to bottom represent the
theoretical predictions for relativistic jet break, relativistic spherical
shock wave, $k=2$ late time ($\nu_{a}=\nu_{a,>}$), and $k=0$ transrelativistic
cases, respectively.

Figure 8. - Continued Figure 8. - Continued

Figure 9: The relations between radio afterglow peak fluxes $F_{{\rm
peak},\nu}$ and observing frequencies $\nu/\nu_{{\rm min}}$ for our sample
GRBs. The blue dashed line with data points in each figure shows the fitting
results from observations, the blue dotted lines denote the possible range of
fitting errors, while the black solid lines from top to bottom represent the
theoretical predictions for transrelativistic, $k=2$, $\epsilon=0$
relativistic, and $k=0$, $\epsilon=0$ relativistic cases, respectively. It
should be noted that as seen in Section 2.1, the $b$ value for
transrelativistic cases is insensitive to $k$.

Figure 9. - Continued

Figure 9. - Continued

It can be seen that by comparing multiband radio-peak data, one can determine
the value of the circumburst medium distribution index $k$. We carry out an
analysis for 32 GRBs with multiband radio afterglow light-curve data, and find
that half of them can be explained with transrelativistic afterglows under
uniform ISM environments. Only $\sim 20$% of our sample GRBs can be certainly
determined that they occur in wind-like environments with larger $k$, although
it should also be noted that the radio peaks of nearly 30% of our sample GRBs
can be interpreted by a $k$ value between 0 and 2. Most of these results are
compatible with existing analyses of individual bursts. However, Yi et al.
(2013) found that nearly all their 19 sample GRBs show signs of $k\sim
0.4-1.4$, with a typical value of $k=1$, by analyzing early forward-reverse-
shock evolution. Obviously, our conclusion is different from that of Yi et al.
(2013). One of the possible explanations could be that the emission regions of
GRB afterglows become larger after entering the self-similar phase. And
circumburst medium distributions at these regions may be quite different from
the early forward-reverse-shock-dominated areas, which are much closer to the
GRB central engine. That is because the mass-loss processes of GRB progenitors
induced by the stellar wind or other mechanisms have limited influence and may
not change the density distribution farther away. Besides, the sample utilized
by Yi et al. (2013) does not overlap with our radio sample. It is difficult to
judge whether such inconsistency is due to diversity among individual bursts
or changes in density distribution at different distances around a single GRB.
If complete follow-up observations for one burst become available in the
future, from early X-ray and optical detections to late-time radio
observations, we can use these information to fully understand the GRB
circumburst medium distributions at various distances and possible changes
within.

As seen in Section 3, the main uncertainty in our analysis comes from errors
in peak-time/flux estimation. As for each GRB we consider the maximum observed
flux as its approximate peak flux and the corresponding observing time as an
estimated peak time in our index fitting, while the real peak may be situated
at any time between the two observing times adjacent to our estimation, with a
flux no lower than our estimation, if the sampling of the light curve is not
frequent enough during the follow-up observations, especially if the data
points near the peak time are sparsely distributed, the fitted $a$ and $b$
values using our method do have larger errors, and the circumburst medium
distribution cannot be pinned down without doubts. However, it is worth noting
that although such uncertainty does exist, our constraints on $k$ are still
better than those from single-band light-curve slope fitting. On one hand, in
many cases, it is nearly impossible to fit the light-curve slopes reliably in
order to calculate $k$ with only a handful of data points; while our approach
described in Section 3 can show some preliminary results at least. On the
other hand, even if we can make enough observations for one GRB in a single
band, those data far away from the emission peak often exhibit large errors
due to their low fluxes and instrument limitations, and a precise fitting on
$k$ cannot be guaranteed. On the contrary, with more data on hand, the
uncertainties on peak estimation are greatly reduced this time. Hence, with
more high-quality light-curve observations made available in the future, the
parameters on multiband radio peaks can be better determined, and more
stringent constraints on circumburst density distributions can be obtained.

Because multiband afterglow observations covering the low-frequency range may
show signs of transrelativistic transitions, the values of $a$ and $b$ at
various bands could be quite different. Because existing radio observations
usually sample a few frequencies only, we can only see signs of such
transitions in a handful of bursts. Thus, the uncertainties here cannot be
ignored. If one can carry out radio follow-ups at more frequencies in the
future, such transitions should be unveiled. Besides, the new generation of
large radio telescopes, including the Five-hundred-meter Aperture Spherical
radio Telescope (FAST) and the upcoming Square Kilometer Array (SKA), have
much higher sensitivities at lower frequencies, and thus can be used to detect
GRB late-time afterglows (e.g., see Li et al. 2015; Zhang et al. 2015; Ruggeri
& Capozziello 2016), and sample more frequency bands at longer wavelengths in
order to get a more complete picture of transrelativistic shock behaviors.

This work is supported by the National Natural Science Foundation of China
(grant Nos. 11903056, 11725314, 12041306), as well as the Joint Research Fund
in Astronomy (U1731125) under a cooperative agreement between the National
Natural Science Foundation of China and Chinese Academy of Sciences. B.Z. is
also supported by the Open Project Program of the Key Laboratory of FAST,
National Astronomical Observatories, Chinese Academy of Sciences (NAOC), the
Cultivation Project for FAST Scientific Payoff and Research Achievement of
CAMS-CAS, as well as the Young Researcher Grant of NAOC. L.D.L. is also
supported by the National Postdoctoral Program for Innovative Talents (grant
No. BX20190044), China Postdoctoral Science Foundation (grant No.
2019M660515), and “LiYun” Postdoctoral Fellow of Beijing Normal University. FL
is also supported by Shanghai Post-doctoral Excellence Program. The authors
thank Ming Zhu, Cheng Cheng, Heng Yu, and Qing-Zheng Yu for helpful
discussions.

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