11institutetext: Yun-Wei Yu 22institutetext: Institute of Astrophysics,
Central China Normal University, Wuhan 430079, China
22email<EMAIL_ADDRESS>33institutetext: He Gao 44institutetext: Department
of Astronomy, Beijing Normal University, Beijing 100875, China
44email<EMAIL_ADDRESS>55institutetext: Fa-Yin Wang 66institutetext: Key
Laboratory of Modern Astronomy and Astrophysics, Ministry of Education, China
77institutetext: School of Astronomy and Space Science, Nanjing University,
Nanjing 210093, China
77email<EMAIL_ADDRESS>88institutetext: Bin-Bin Zhang 99institutetext:
Key Laboratory of Modern Astronomy and Astrophysics, Ministry of Education,
China 1010institutetext: School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China
1010email<EMAIL_ADDRESS>
# Gamma-Ray Bursts

Yun-Wei Yu corresponding author He Gao  Fa-Yin Wang  and Bin-Bin Zhang

###### Abstract

Gamma-ray bursts (GRBs) are short and intense bursts of $\sim$100 keV$-$1MeV
photons, usually followed by long-lasting decaying afterglow emission in a
wide range of electromagnetic wavelengths from radio to X-ray and, sometimes,
even to GeV gamma-rays. These emissions are believed to originate from a
relativistic jet, which is driven due to the collapse of special massive stars
and the mergers of compact binaries (i.e., double neutron stars or a neutron
star and a black hole). This chapter first briefly introduces the basic
observational facts of the GRB phenomena, including the prompt emission,
afterglow emission, and host galaxies. Secondly, a general theoretical
understanding of the GRB phenomena is described based on a relativistic jet’s
overall dynamical evolution, including the acceleration, propagation, internal
dissipation, and deceleration phases. Here a long-lasting central engine of
the GRBs can substantially influence the dynamical evolution of the jet. In
addition, a supernova/kilonova emission can appear in the optical afterglow of
some nearby GRBs, which can provide an important probe to the nature of the
GRB progenitors. Finally, as luminous cosmological phenomena, it is expected
to use GRBs to probe the early universe and to constrain the cosmological
parameters.

##  1 Keywords

gamma-ray burst; afterglow; supernova; kilonova; transient; relativistic jet;
shock; radiation mechanism; gravitational wave; cosmology

##  2 Introduction

Gamma-ray bursts (GRBs) are short and intense bursts of $\sim$100 keV$-$1MeV
photons, which were first discovered in the year of 1967 by the Vela
satellites and announced to the public in 1973 as astronomical phenomena
(Klebesadel et al., 1973). The BATSE detector onboard the COMPTON Gamma-Ray
Observatory, which was launched in 1991, revealed the isotropic distribution
of GRBs on the sky Meegan et al. (1992). Such a distribution indicated the
GRBs have an extra-galactic origin at cosmological distances rather than
originate from our Galaxy, which was further confirmed in 1997 by the
measurements of the cosmological redshifts of several GRBs. In that year, the
new-launched Italian/Dutch BeppoSAX satellite captured an X-ray counterpart of
GRB 970228 Costa et al. (1997) and the X-ray emission can exceed the
sensitivity of the detector for about one week. Thanks to the X-ray
observation, the accuracy of the GRB localization was enhanced significantly,
which made it possible to monitor the transient counterparts also in the
optical and radio bands van Paradijs et al. (1997); Frail et al. (1997) and as
well as to identify the host galaxy. Therefore, the redshift of the GRB can be
obtained from the absorption or emission lines. These long-lasting multi-
wavelength transient counterparts of GRBs are now well known as the afterglow
emission, while the sub-MeV gamma-ray emission detected by the GRB triggering
detectors is termed as the prompt emission.

The cosmological origin of GRBs indicates them to be the most explosive
phenomena in the universe after the Big Bang. Meanwhile, the rapid variability
of the GRB prompt emission gives a stringent constraint on the size of the
emission source. Therefore, it is believed that GRBs should be produced by a
relativistic jet launched by a catastrophic event of a stellar system,
including the collapse of a special category of massive stars Woosley (1993)
and the merger of compact binaries Paczynski (1986); Eichler et al. (1989).
The former hypothesis was quickly confirmed by the discovery of a supernova
emission emerging from the optical afterglow of some nearby GRBs, as
represented by the GRB 980425/SN 1998bw, GRB 030329/SN 2003dh, and GRB
060218/SN 2006aj association events. On the contrary, the merger origin
hypothesis is long pending until the observation of GRB 170817A which was
confidently associated with a gravitational wave event GW170817 Abbott et al.
(2017a, b). Then, it becomes the crucial task of the GRB research to answer
how these stellar catastrophic events can generate a relativistic jet and how
such a jet produces the GRB prompt and afterglow emission. Additionally, as
one of the most distant detectable objects, it is also highly concerned
whether and how we can use the GRBs as a tool to probe the universe.

##  3 Observations

###  3.1 Prompt Emission

The prompt emission of GRBs is represented by their complex, highly variable,
and unpredictably shaped pulses, which constitute their light curves in a
typical energy range between the X-ray and $\gamma$-ray bands. The prompt
emission of a GRB may consist of several consecutive episodes separated by
quiet periods in-between. In addition to the main burst, some, if not all,
GRBs are presumed to have a pre-burst phase (“precursor”) and a post-burst
phase (“extended emission”), as shown in Fig. 1. The duration of GRB prompt
emission is usually defined by the so-called $T_{90}$, i.e., the time interval
in which the integrated photon counts increase from 5% to 95% of the total
counts. Based on the bimodal distribution of GRBs in the duration-hardness
diagram (the hardness of a burst is usually denoted as the photon count ratio
in two fixed observational energy bands), GRBs were classified into two
categories: the long-duration, soft-spectrum class (LGRBs) and the short-
duration, hard-spectrum class (SGRBs) (Kouveliotou et al., 1993). Since both
$T_{90}$ and hardness ratio are energy- and sensitivity-dependent, the
separation line between the long and short GRBs remains unclear but is
conventionally set at about 2s. Such an empirical dichotomy could be a natural
result of the two possible different origins mentioned above, i.e., the core
collapse of Wolf$-$Rayet stars for long GRBs and the mergers of two compact
objects for short GRBs.

Figure 1: The light curve of GRB 160625B, which illustrate a precursor, a main
burst and an extended emission phase. Adapted fromZhang et al. (2018).

The temporal properties of prompt emission vary by a burst-by-burst basis.
Their light curve shapes vary from a single smooth pulse to extremely complex
light curves with many erratic pulses with different durations, amplitudes,
and fine structures (Fishman & Meegan, 1995). The power density spectra
analysis shows no indication that the light curve is periodic (Beloborodov et
al., 2000). The light curves may be decomposed as the superposition of an
underlying slow component and a more rapid fast component (Gao et al., 2012).
Moreover, the light curves would vary with energy. The fast component tends to
be more significant in high energies and becomes less significant at lower
frequencies (Vetere et al., 2006). The shape of the slow component is
typically asymmetric, usually with a fast-rising exponential-decay (FRED)
shape (Norris et al., 1996). In more detail, the slow component pulse tends to
be narrower in harder energy bands and wider in softer energy bands (Norris et
al., 2005). The arrival time of a pulse in a softer band is typically delayed
(or “lagged”) with respect to the arrival time in a harder band, which is
called the “spectral lag” effect (Norris et al., 2000).

GRB spectra are typically non-thermal. When the detector’s energy band is wide
enough, the photon number spectrum of prompt emission, both for time-resolved
spectrum and time-integrated spectrum, can usually be fitted with a broken
power law known as the Band function in form of (Band et al., 1993)

$N(E)=\left\\{\begin{array}[]{ll}A\left(\frac{E}{100\rm{keV}}\right)^{\alpha}{\rm
e}^{-E/E_{0}}&,E<(\alpha-\beta)E_{0}\\\
A\left[\frac{(\alpha-\beta)E_{0}}{100\rm{keV}}\right]^{\alpha-\beta}{\rm
e}^{\beta-\alpha}(\frac{E}{100\rm{keV}})^{\beta}&,E\geq(\alpha-\beta)E_{0}\end{array}\right.$
(1)

where $A$ is the normalization factor, $E_{0}$ is the break energy in the
spectrum, $\alpha$ and $\beta$ are the low-energy and high-energy photon
spectral indices with $\alpha$ ranging at $(-2,0)$ and $\beta$ ranging at
$(-4,-1)$ (Preece et al., 2000). The peak energy of the $E^{2}N(E)$ spectrum,
$E_{\rm p}=(2+\alpha)E_{0}$, distributes within several orders of magnitude
but clusters around $200-300\rm{keV}$ (Preece et al., 2000; Goldstein et al.,
2013). When the detector’s energy band is not wide enough or a GRB is not
bright enough, the photon number spectrum of the prompt emission can behave as
a cut-off power-law, which reads

$N(E)=A\left(\frac{E}{100\rm{keV}}\right)^{\alpha}{\rm e}^{-E/E_{\rm c}},$ (2)

where $E_{\rm c}$ is the cutoff energy and the peak energy in the $E^{2}N(E)$
spectrum of this model is $E_{\rm p}=(2+\alpha)E_{c}$. The spectrum might even
behave as a simple power law as

$N(E)=A\left(\frac{E}{100\rm{keV}}\right)^{-\hat{\Gamma}},$ (3)

where $\hat{\Gamma}$ is the photon index, which is positive by definition.

Figure 2: A illustration of the three possible spectral components of GRB
prompt emission. From Zhang et al. (2011)

Although the main component of GRB spectra is non-thermal and “Band”-like, a
few GRBs, however, show that an additional (quasi-)thermal component and/or
high-energy power-law component is required to explain the spectral data
(Zhang et al., 2011). For instance, in the spectra of several Fermi bursts
(e.g., GRBs 100724B, 110721A, and 120323A), a sub-dominant thermal component
appears at the left shoulder of the Band component. Additionally, the
superposition between the Band and high-energy components has been seen in GRB
090926A and other GRBs. In the exceptional case of GRB 090902B, the spectra
only contain the superposition between a thermal component and a high-energy
spectral component. Detailed analysis of the time-resolved spectra of bright
bursts shows that significant spectral evolution is usually observed between
$E_{\rm p}$ and the flux with two typical types of evolution patterns (Lu et
al., 2012), namely the hard-to-soft and intensity-tracking patterns. A sketch
illustrating the combinations of the three elemental spectral components of
prompt emission is shown in Fig. 2.

Finally, it is worth pointing out that broadband emission is naturally
expected and had been indeed observed during the prompt phase. Such
observations include prompt X-ray flares (Hu et al., 2014), prompt optical
flashes (Akerlof et al., 1999), and prompt GeV flashes (Ackermann et al.,
2014). Different patterns are observed for the relationship between the prompt
sub-MeV emission and those in other wavelengths. For instance, both mismatch
and tracking behaviors exist between the optical and gamma-ray peaks, and
bright X-ray emission was observed during the quiescent phase of sub-MeV
emission. Furthermore, delayed onset of the GeV emission is usually observed
with respect to the MeV emission. Therefore, multiple emission sites are
usually required to account for these different behaviors of the prompt
emission components.

###  3.2 Afterglow and Associated Supernova/Kilonova

Because of the limited observational facilities, early studies of GRB
afterglows have relied primarily upon late-time data, which are typically
obtained hours or days after the burst. A power-law decay can be observed at
all wavelengths, as expected by the standard forward shock model. Such a
picture was altered by the launch of NASA Swift mission, which brings
unprecedented early X-ray data of GRB afterglow from its X-Ray Telescope (XRT)
thanks to its rapid slewing and precise localization capability.

As shown in Fig. 3, the X-ray afterglow light curves can be summarized as a
canonical pattern composed of five components, namely a distinct rapidly
decaying component, a shallow decay component, a normal decay component, a
post jet break component, and X-ray flares (Nousek et al., 2006; Zhang et al.,
2006). Each of those components is further itemized below.

Figure 3: A cartoon illustration of the X-ray afterglow light curve based on
the observations of the Swift XRT. From Zhang et al. (2006)

.

  * •

The rapidly decaying component (I) behaves as the “tail” of the prompt
emission, whose decay slope is typically in the range of $\sim 3$ to $\sim 10$
(Tagliaferri et al., 2005). A good fraction of GRBs showed a clear hard-to-
soft evolution during the rapid decay (Zhang et al., 2007).

  * •

The shallow decay component (II) is usually adjacent to the steep decay phase
and is followed by a normal decay. The typical slope of a shallow decay
component is in the range between $0$ and $\sim 0.7$. There is no significant
spectral evolution between the shallow decay segment and its follow-up
segment, and the latter is usually consistent with the external-shock models
(Liang et al., 2007; Zhao et al., 2019). If the slope of the shallow decay
component is close to 0, it is also called a plateau. In rare cases, an X-ray
plateau can be followed by a very steep decay with a decay slope steeper than
$-3$. (Troja et al., 2007), which is called internal plateau because it
probably arises from an internal dissipation process rather than the external
shock.

  * •

A normal decay component (III) usually follows the shallow decay component, or
sometimes directly follows the steep decay component. It has a decay slope of
$\sim 1$, which is the typical value predicted in the standard external
forward shock model.

  * •

A late steeper decay (with a typical slope of approximately $-2$) component
(IV) often follows the normal decay segment, which is also expected in the
external forward shock model due to the so-called jet break effect (Panaitescu
et al., 1998).

  * •

X-ray flares (V) are considered to have different origins from other
components since they are “superposed” on those background power-law decay
components (Burrows et al., 2005; Chincarini et al., 2007). Their light curves
typically show rapid rise and fall with steep indices. Many properties of
X-ray flares in both temporal and spectral domains are analogous to prompt
emission, suggesting that they might be directly powered by the GRB central
engine, similar to prompt emission.

Compared with the X-ray afterglow, the optical afterglow exhibits more
complicated behavior. An optical afterglow light curve can also consist of
shallow decay, normal decay, and jet-break phases. Additionally, optical
flares were also observed in certain instances. The properties of these
components are similar to those observed in the X-ray band. In late stages, a
re-brightening feature is occasionally observed in the optical light curve of
some GRB, which could be related to the jet structure (Nardini et al., 2011).
One point worthy of note is that a large proportion of GRBs have no detectable
optical afterglow. This is primarily due to the heavy dust extinction within
the GRB host galaxies.

In some optical afterglows of long GRBs, a bump feature, usually with a red
color, can appear about a week after the GRB trigger, as shown in Fig. 4. Such
a bump feature is usually interpreted as the signature of an associated
supernova (Hjorth et al., 2003). Comparatively to long GRBs, a fainter-than-
supernova optical/IR transient has been predicted to accompany short GRBs that
originate from neutron star-neutron star (NS-NS) or neutron star-black hole
(NS-BH) mergers (Li & Paczyński, 1998; Metzger et al., 2010). Such a transient
is now commonly referred to as a kilonova. The kilonova prediction had been
first confirmed by the discovery of the optical transient AT2017gfo following
the GW170817/GRB 170817A event Arcavi et al. (2017); Drout et al. (2017).
Additionally, several kilonova candidates were also claimed in several short
GRB events, including GRB 130603B and GRB 050709 (Tanvir et al., 2009; Yang et
al., 2015; Gao et al., 2015).

Figure 4: The $R-$band light curve of GRB 130427A/ SN 2013cq. From Xu et al.
(2013)

The radio and high-energy afterglows, in contrast, exhibit relatively simple
behaviors. According to statistical studies, the radio afterglow typically
rises and reaches a peak around 3-6 days. This is consistent with the standard
external shock model. The detection of afterglow components at high energies
is more challenging than at low energies as the number of photons dramatically
drops at higher energies. The high-energy afterglow typically shows a power-
law decay with time, which could result from external forward shock emission,
either with synchrotron radiation or related to a synchrotron self-Compton
(SSC) spectral component.

###  3.3 Host Galaxy

For long GRBs, most host galaxies are irregular, star-forming galaxies, with a
few being spiral galaxies with active star formation (Fruchter et al., 2006).
Nevertheless, it is suggested that long GRB hosts are relatively metal-poor
compared with field galaxies. Within the host galaxy, most long GRBs reside in
the brightest core regions, where the specific star formation rate is the
highest (Modjaz et al., 2008). These facts appear to support the massive star
origin of long GRBs. Unlike long GRBs, the majority of host galaxies of short
GRBs are elliptical or early type (Gehrels et al., 2005). Some short GRBs are
even host-less, which may be “kicked” away from their host. Moreover, most
short GRBs are found to be far from the bright light of the host galaxies
(Fong et al., 2010). These discoveries indicate that the short GRBs are likely
not directly associated with the deaths of massive stars but are more
consistent with the compact binary mergers.

##  4 Theory

###  4.1 Central Engine and Jet

#### Energy Sources

Assuming the GRB emission originates from a relativistic jet of a Lorentz
factor $\Gamma$, the size of the emission region of GRBs can be constrained at
the scale of

$R_{\rm GRB}=2\Gamma^{2}c\delta t\sim 6\times 10^{12}\Gamma_{2.5}^{2}~{}\delta
t_{-3}{\rm cm},$ (4)

where $\delta t$ is the variability timescale of the GRB light curves.
Hereafter the conventional notation $Q_{x}=Q/10^{x}$ is used in cgs units. Eq.
(4) indicates the progenitors of GRBs can only be in the order of stellar
systems. Meanwhile, for the typical fluence $F\sim(10^{-7}-10^{-4})\rm
erg~{}cm^{-2}$ of GRBs and their cosmological distances, the isotropically-
equivalent energy release of these GRBs can be calculated as

$E_{\rm iso}=4\pi d_{\rm L}^{2}F/(1+z)\sim(1+z)^{-1}10^{50-53}~{}\rm erg,$ (5)

where $d_{\rm L}\sim 10^{28}\rm cm$ is a typical luminosity distance and $z$
is redshift. Even considering that the GRB emission is actually produced by a
highly-beamed jet, the realistic energy release can still be as high as $\sim
10^{50-51}$ erg. For a stellar system, such a huge energy release can only be
supplied by the gravitational energy from the catastrophical contraction of
the system from a large size to several to a few tens of kilometers:

$\Delta U\sim{3\over 5}{GM^{2}\over R}\sim 10^{53}\left({M\over
M_{\odot}}\right)^{2}\left({R\over 10{\rm km}}\right)~{}\rm erg,$ (6)

where $M$ is the mass of the system and $R$ is the radius after the system
contraction. Specifically, such a contracting stellar system could be a
collapsing massive star or a merging compact binary, which corresponds to the
observational long and short GRBs, respectively.

Following the above considerations, the central engine of GRBs should be a
compact object (BH or NS) forming from the stellar collapses and compact
binary mergers. How such a compact object launches relativistic jets and
extracts the released gravitational energy is still an open question. The most
promising mechanism is the hypercritical accretion onto the central engine
during its forming, where the typical accretion rate, i.e.,
$\dot{M}\sim(0.1-1)M_{\odot}~{}s^{-1}$, can in principle be high enough to
explain the GRB luminosities:

$L_{\rm GRB}=\zeta\dot{M}c^{2}=2\times 10^{51}\zeta_{-3}\left({\dot{M}\over
1\rm M_{\odot}~{}s^{-1}}\right),$ (7)

where $\zeta$ is the radiation efficiency of the accretion. About a half of
the released gravitational energy is initially stored as the internal energy
of the accretion disk, which leads the disk to be so hot that a great
abundance of neutrinos/antineutrinos are produced. These
neutrinos/antineutrinos can bring away the internal energy easily because they
can escape from the transparent disk freely. The $\nu\bar{\nu}$ luminosity can
be obtained by calculating the structure and ingredients of the neutrino-
dominated accretion flow (NDAF) in the disk (Qian & Woosley, 1996).

Besides the internal energy, the gravitational energy can also be converted
into the rotational energy of the central engine and the disk. If the central
engine is a BH, its rotational energy can be expressed as

$E_{\rm rot}=2\times 10^{54}f(a_{*}){M_{\rm bh}\over M_{\odot}}\rm erg,$ (8)

where $f(a_{*})=1-\sqrt{(1+q)/2}$, $q=\sqrt{1-a_{*}^{2}}$, and
$a_{*}=Jc/GM_{\rm bh}^{2}$ is the dimensionless spin parameter with $J$ and
$M_{\rm bh}$ being the angular momentum and mass of the BH. For $a_{*}=0.1$,
we have $f(a_{*})\approx 0.001$ and $E_{\rm rot}\approx 10^{51}\rm erg$. When
this rapidly spinning BH is connected with a remote astrophysical load by
magnetic field lines, its rotational energy can be extracted through the
Blandford-Znajek (BZ) mechanism (Blandford & Znajek, 1977). On the other hand,
if the central engine is an NS, then its rotational energy can reach

$E_{\rm rot}={1\over 2}I\Omega^{2}=2\times 10^{52}I_{45}P^{-2}_{-3}\rm erg,$
(9)

where $I$ is the moment of inertial of the NS, $\Omega$ and $P$ are the spin
frequency and period. In this case, if the dipolar magnetic field of the NS
can be as high as $B_{\rm p}\sim 10^{16}$ G, then its rotational energy can be
released in a timescale of $t_{\rm sd}=20I_{45}R_{\rm s,6}^{-6}B_{\rm
p,16}^{-2}P_{-3}^{2}$ s, which could also play a role in driving GRB jet,
where $R$ is the radius of the NS. In these cases, the jet could be dominated
by a Poynting flux.

#### Jet Acceleration

Because of the extremely high density of the released energy, neutrinos and
anti-neutrinos coming from the hypercritical accretion disk can annihilate
above the disk to produce photons and electron-positron pairs, which are
highly coupled with each other to form a fireball (Paczynski, 1986; Meszaros
et al., 1993). For a fireball in thermal equilibrium, its electron-positron
density can be calculated as

$n_{\pm}=\frac{1}{2}\left(\frac{2m_{\rm e}k_{\rm
B}T}{\pi\hbar^{2}}\right)^{3/2}\exp\left(-\frac{m_{\rm e}c^{2}}{k_{\rm
B}T}\right),$ (10)

where $T$ is the temperature of the fireball satisfying ${k}_{\rm B}{T}\ll
m_{\rm e}{c}^{2}$. The fireball can expand drastically due to the high
radiation pressure. Then, a certain mass of baryons can be stripped by the
fireball from the disk to be coupled with the pairs.

The dynamical evolution of the expanding fireball can be described by
relativistic hydrodynamics, which shows the expanding fireball is self-similar
and can be approximated by a thin shell (Meszaros et al., 1993). Then, the
conservation of the mass and energy of the shell can be written as

$\displaystyle\Gamma r^{2}n^{\prime}$ $\displaystyle=$ $\displaystyle{\rm
const.},$ (11) $\displaystyle\Gamma r^{2}{e^{\prime}}^{3/4}$ $\displaystyle=$
$\displaystyle{\rm const.},$ (12)

where $\Gamma$ is the Lorentz factor of the shell, and $r$ is the radius
relative to the center of the system, and $n^{\prime}$ and $e^{\prime}$ are
the densities of the particle number and energy, respectively, which are
measured in the comoving frame of the fireball represented by the prime. In
the radiation-dominated phase ($e^{\prime}\gg n^{\prime}m_{\rm p}c^{2}$), Eq.
(11) gives

$\displaystyle\Gamma\propto r,~{}n^{\prime}\propto r^{-3},~{}e^{\prime}\propto
r^{-4},~{}T\propto\Gamma e^{\prime 1/4}\sim\rm const,$ (13)

where the Lorentz factor increases linearly with the radius. Subsequently,
when the fireball transits into the matter-dominated phase ($e^{\prime}\ll
n^{\prime}m_{p}c^{2}$), we have

$\displaystyle\Gamma\sim{\rm const.},~{}n^{\prime}\propto
r^{-2},~{}e^{\prime}\propto r^{-8/3},~{}T\propto\Gamma e^{\prime 1/4}\propto
r^{-2/3},$ (14)

where the Lorentz factor keeps constant, which is called a coasting phase.

As mentioned above, the outflow driven by the GRB engine can alternatively be
Poynting-flux-dominated (PFD). In this case, the outflow would be accelerated
due to the magnetic reconnection, which converts the magnetic energy to
internal energy and subsequently to kinetic energy. It is not easy to describe
the reconnection processes. An approximative and convenient treatment is to
assume the comoving reconnection speed to be proportional to the Alfven speed
$v^{\prime}_{\rm A}=\sqrt{\sigma/(1+\sigma)}$ (Drenkhahn, 2002), where the
parameter $\sigma$ represents the magnetization of the outflow. Then, for
stripped toroidal magnetic fields of a width $\lambda\sim cP$, the
reconnection timescale can be estimated by $t^{\prime}_{\rm
r}=\lambda^{\prime}/\epsilon v^{\prime}_{\rm A}\propto\Gamma$ for $\sigma\gg
1$. The ratio of $t^{\prime}_{\rm r}$ to the dynamical timescale
$t^{\prime}_{\rm dyn}=r/\Gamma c$ gives the fraction of the magnetic energy
dissipated through the reconnection. Since the dissipated magnetic energy is
finally used to accelerate the outflow, it is considered that

$\displaystyle\Gamma{t^{\prime}_{\rm r}\over t^{\prime}_{\rm dyn}}\propto\rm
const.$ (15)

This yields

$\displaystyle\Gamma\propto r^{1/3},$ (16)

which is much slower than the fireball acceleration as presented in Eq. (13).

#### Jet Propagation

Before the relativistic jet produces the GRB emission, it should first
penetrate the thick progenitor medium, which can be a stellar envelope for
long GRBs and merger ejecta for short GRBs. The collision between the jet and
the progenitor medium can lead to a forward shock sweeping up the medium and a
reverse shock accumulating the jet material. The region between these two
shocks is called the jet head, in which the material is very hot and flows
quickly and laterally to form a cocoon surrounding the jet. The velocity of
the jet head is determined by balancing the ram pressures of the forward-
shocked medium and the reverse-shocked jet (Matzner, 2003):

$\beta_{\rm h}=\frac{\beta_{\rm j}\tilde{L}^{1/2}+\beta_{\rm
e}}{\tilde{L}^{1/2}+1}$ (17)

with $\tilde{L}\simeq{L_{\rm j}}/({\Sigma_{\rm j}\rho_{\rm e}\Gamma_{\rm
e}^{2}c^{3}})$, where $\beta_{\rm j}\simeq 1$, $L_{\rm j}$, and $\Sigma_{\rm
j}$ are the velocity, one-sided luminosity, and cross section of the unshocked
jet, respectively, and $\rho_{\rm e}$, $\beta_{\rm e}$, and $\Gamma_{\rm e}$
are the density, velocity, and Lorentz factor of the circum-material.

The high pressure of the cocoon can drive a collimation shock into the jet
material toward the jet axis. Therefore, the jet can be gradually collimated
and the evolution of the jet cross section is determined by the pressure in
the cocoon, which is given by $P_{\rm c}={E_{\rm c}/(3V_{\rm c}})$. Here the
total energy stored in the cocoon $E_{\rm c}$ and the cocoon’s volume $V_{\rm
c}$ are given by Bromberg et al. (2011)

$\displaystyle E_{\rm c}$ $\displaystyle=$ $\displaystyle\int_{0}^{t}L_{\rm
j}\left(1-\beta_{\rm h}\right)dt^{\prime},$ (18) $\displaystyle V_{\rm c}$
$\displaystyle\sim$ $\displaystyle\pi r_{\rm c}^{2}z_{\rm h},$ (19)

respectively, where $z_{\rm h}=\int_{0}^{t}\beta_{\rm h}cdt^{\prime}$ is the
height of the jet head, $r_{\rm c}=\int_{0}^{t}\beta_{\rm
c,\perp}cdt^{\prime}$, and the lateral expansion velocity reads $\beta_{\rm
c,\perp}=[P_{\rm c}/\bar{\rho}_{\rm e}(z_{\rm h})c^{2}]^{1/2}$ and the average
density of the cocoon $\bar{\rho}_{\rm e}=\int\rho_{\rm e}(z)dV/V_{\rm c}$. By
defining a critical point $\hat{z}\approx(L_{\rm j}/\pi cP_{\rm c})^{1/2}$
where the collimation shock converges, the jet cross section can be estimated
by

$\Sigma_{\rm j}=\left\\{\begin{array}[]{ll}\pi\theta_{\rm j0}^{2}z_{\rm
h}^{2},&{~{}\rm for}~{}z<\hat{z}/2,\\\ \pi\theta_{\rm
j0}^{2}\left(\hat{z}/2\right)^{2},&{~{}\rm
for}~{}z>\hat{z}/2,\end{array}\right.$ (20)

where $\theta_{\rm j0}$ is the initial opening angle of the jet at launching.
Approximately, the shape of the jet gradually transforms from conical to
cylindrical. The opening angles of the jet head and the cocoon relative to the
central engine can be defined as $\theta_{\rm h}=\left({\Sigma_{\rm j}/\pi
z_{\rm h}^{2}}\right)^{1/2}$ and $\theta_{\rm c}={r_{\rm c}/z_{\rm h}}$,
respectively. As the jet is squeezed by the cocoon, the jet material can be
pushed and accelerated.

###  4.2 Prompt Emission

#### Internal Dissipation

The GRB prompt emission can be produced after the jet breaks out from the
progenitor medium. One possibility is that the acceleration of the outflow has
not finished, and a remarkable fraction of internal energy still exists in the
jet after it becomes transparent. In this case, the emission is produced due
to the release of the internal energy, which is called photosphere emission,
characterized by a quasi black-body spectrum. However, in observation,
although such a black-body component has been indeed found in some GRBs, the
typical spectrum of GRB prompt emission is non-thermal as described by the
Band spectrum. Therefore, it is considered that the prompt emission is more
likely to be produced due to the dissipation of the kinetic energy of the jet
during the coasting phase. The photosphere emission, which probably appears
after the internal dissipation, would be usually weak.

It could be reasonable to consider that a GRB jet is not a continuous fluid,
but consists of a series of intermittent shells of very different velocities.
Then, the kinetic energy due to the speed difference can be easily dissipated
when the shells collide with each other. Let us consider two shells launched
by the central engine at an interval of $\delta t$. The collision between them
happens when the later and more rapid one catches up with the previous slower
one at the radius of

$\displaystyle r_{\rm is}={2\Gamma_{\rm s}^{2}c\delta t\over 1-(\Gamma_{\rm
s}/\Gamma_{\rm r})^{2}}\approx 6\times 10^{12}\Gamma_{\rm s,2.5}^{2}\delta
t_{-3}~{}{\rm cm},$ (21)

where $\Gamma_{\rm s}$ and $\Gamma_{\rm r}$ are the Lorentz factors of the
slower and more rapid shells. The above result can be well consistent with the
constraint presented in Eq. (4), as the light curve variability is determined
by the intermittent activity of the central engine. Furthermore, the
relatively small value of the internal shock radius favors the fireball model
more than the PFD model because the magnetic reconnection could be too slow to
complete the outflow acceleration before such a small radius.

Furthermore, by considering of the energy and momentum conservations before
and after the collision, the radiation efficiency of the internal shocks
(i.e., the conversion efficiency of the kinetic energy to the internal energy)
can be constrained to

$\displaystyle\epsilon\leq 1-2\left(\sqrt{\Gamma_{\rm r}\over\Gamma_{\rm
s}}+\sqrt{\Gamma_{\rm s}\over\Gamma_{\rm r}}\right)^{-1}.$ (22)

If $\Gamma_{\rm r}$ is not much higher than $\Gamma_{\rm s}$, the radiation
efficiency should be much smaller than unity. This indicates the majority of
the jet energy should be released in the afterglow phase, which is, however,
not always supported by observations. In comparison, the radiation efficiency
of a PFD outflow could be much higher than the fireball.

An actual GRB jet could be a hybrid of a fireball and a PFD outflow. In this
case, the jet can be accelerated sufficiently quickly and the dissipation
radius is determined by internal collisions. Meanwhile, the majority of the
jet energy is stored in magnetic fields, the reconnection of which can be
triggered by the internal collisions through rapid turbulence. Such a hybrid
model was first proposed by Zhang & Yan (2011), which provides a very
competitive explanation for the internal dissipation of GRB jets (Zhang & Yan,
2011).

#### Shocked Material

As an important mechanism converting kinetic energy into internal energy, a
relativistic shock can store up internal energy behind it of a density of
(Blandford & McKee, 1976)

$\displaystyle e^{\prime}_{\rm sh}=(\Gamma_{\rm sh}-1)(4\Gamma_{\rm
sh}-3)n^{\prime}_{\rm un}m_{\rm p}c^{2},$ (23)

where $\Gamma_{\rm sh}$ is the Lorentz factor of the shock, which corresponds
to the relative velocity between two shells, and $n^{\prime}_{\rm un}$ is the
comoving number density of the unshocked material. Here the shell is assumed
to simply consist of protons and electrons, which share the internal energy by
equipartition factors of $\varepsilon_{\rm p}$ and $\varepsilon_{\rm e}$,
respectively. Furthermore, the numbers of these charge particles would
distribute with their energies as a power law: ${dN_{\rm p/e}/d\gamma_{\rm
p/e}}\propto\gamma_{\rm p/e}^{-p}$, where $\gamma_{\rm p/e}$ is the Lorentz
factor of the random motion of the protons/electrons in the comoving frame of
the shocked region, the index $p$ is a free parameter which could be not much
higher than 2. While the typical Lorentz factor of the protons is around
$\Gamma_{\rm sh}$, the minimum Lorentz factor of the electrons can be
determined at (Sari et al., 1998)

$\displaystyle\gamma_{\rm m}=\varepsilon_{\rm e}\frac{p-2}{p-1}\frac{m_{\rm
p}}{m_{\rm e}}(\Gamma_{\rm sh}-1).$ (24)

Simultaneously, the shock can also effectively amplify the primordial magnetic
field in the shell. By introducing an equipartition parameter of
$\varepsilon_{\rm B}$ for the amplified fields, we can write the magnetic
field strength as

$\displaystyle B^{\prime}=\sqrt{8\pi\varepsilon_{\rm B}e^{\prime}_{\rm
sh}}\approx\sqrt{32\pi\varepsilon_{\rm B}\Gamma_{\rm sh}^{2}n^{\prime}_{\rm
un}m_{\rm p}c^{2}}.$ (25)

This magnetic field can make the electrons very radiative and change the
energy distribution of the electrons. By invoking the total power of the
synchrotron radiation of the electrons, a cooling Lorentz factor can be
defined as (Sari et al., 1998)

$\displaystyle\gamma_{\rm c}=\frac{6\pi m_{\rm e}c}{\sigma_{\rm T}\Gamma_{\rm
sh}B^{\prime 2}t},$ (26)

where $t$ is the dynamical timescale of the shock. The meaning of $\gamma_{\rm
c}$ is that, for $\gamma_{\rm e}>\gamma_{\rm c}$, the electron spectrum has
been changed substantially because of the primary energy of these electrons
has been radiated. On the contrary, the electron spectrum would keep its
original form for $\gamma_{\rm e}<\gamma_{\rm c}$. To be summarized, the final
electron spectrum can be written as:
(i) fast cooling case ($\gamma_{\rm c}<\gamma_{\rm m}$)

$\displaystyle{dN_{\rm e}\over
d\gamma_{e}}\propto\left\\{\begin{array}[]{lll}0,&\gamma_{\rm e}<\gamma_{c}\\\
\gamma_{\rm e}^{-2},&\gamma_{\rm c}<\gamma_{\rm e}<\gamma_{\rm m}\\\
\gamma_{\rm e}^{-p-1},&\gamma_{\rm m}<\gamma_{\rm e},\end{array}\right.$ (30)

(i) slow cooling case ($\gamma_{\rm c}>\gamma_{\rm m}$)

$\displaystyle{dN_{\rm e}\over
d\gamma_{e}}\propto\left\\{\begin{array}[]{lll}0,&\gamma_{\rm e}<\gamma_{\rm
m}\\\ \gamma_{\rm e}^{-p},&\gamma_{\rm m}<\gamma_{\rm e}<\gamma_{\rm c}\\\
\gamma_{\rm e}^{-p-1},&\gamma_{\rm c}<\gamma_{\rm e}.\end{array}\right.$ (34)

#### Synchrotron Emission

The radiation coefficient of the shocked material due to the synchrotron
radiation of electrons is given by

$\displaystyle j^{\prime}_{\nu^{\prime}}=\frac{1}{4\pi}\int_{\gamma_{\rm
m}}{dN_{\rm e}\over d\gamma_{\rm e}}P_{\nu^{\prime}}^{\prime}(\gamma_{\rm
e})d\gamma_{\rm e},$ (35)

where $\nu^{\prime}$ is the emission frequency measured in the comoving frame
and the radiation spectrum of a single electron reads (Rybicki & Lightman,
1979)

$\displaystyle P_{\nu^{\prime}}^{\prime}\left(\gamma_{\rm
e}\right)=\frac{\sqrt{3}q_{\rm e}^{3}B^{\prime}}{m_{\rm
e}c^{2}}\left[\frac{\nu^{\prime}}{\nu_{\rm
c}^{\prime}}\int_{\nu^{\prime}/\nu_{\rm
c}^{\prime}}^{\infty}K_{5/3}(y)dy\right],$ (36)

where $q_{\rm e}$ is the electron’s charge and $\nu^{\prime}_{\rm c}=3q_{\rm
e}B^{\prime}\gamma_{\rm e}^{2}/(4\pi m_{\rm e}c)$.

Because of the quasi-monochromaticity of synchrotron radiation, it can be
found that the radiation spectrum of electrons would basically trace the
energy distribution of them. Therefore, according to Eqs. (30) and (34), the
synchrotron spectrum of the shocked material can be approximated by (Sari et
al., 1998):
(i) Fast cooling case

$\displaystyle
j^{\prime}_{\nu^{\prime}}=j_{\nu^{\prime},\max}\times\left\\{\begin{array}[]{lll}\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
c}}\right)^{1/3},&\nu^{{}^{\prime}}<\nu^{{}^{\prime}}_{\rm c}\\\
\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
c}}\right)^{-1/2},&\nu^{{}^{\prime}}_{\rm
c}<\nu^{{}^{\prime}}<\nu^{{}^{\prime}}_{\rm m}\\\
\left({\nu^{{}^{\prime}}_{\rm m}\over\nu^{{}^{\prime}}_{\rm
c}}\right)^{-1/2}\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
m}}\right)^{-p/2},&\nu^{{}^{\prime}}_{\rm
m}<\nu^{{}^{\prime}},\end{array}\right.$ (40)

(i) Slow cooling case

$\displaystyle
j_{\nu^{\prime}}=j_{\nu^{\prime},\max}\times\left\\{\begin{array}[]{lll}\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
m}}\right)^{1/3},&\nu^{{}^{\prime}}<\nu^{{}^{\prime}}_{\rm m}\\\
\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
m}}\right)^{\left({1-p}\right)/2},&\nu^{{}^{\prime}}_{\rm
m}<\nu^{{}^{\prime}}<\nu^{{}^{\prime}}_{\rm c}\\\
\left({\nu^{{}^{\prime}}_{\rm c}\over\nu^{{}^{\prime}}_{\rm
m}}\right)^{\left({1-p}\right)/2}\left({\nu^{{}^{\prime}}\over\nu^{{}^{\prime}}_{\rm
c}}\right)^{-p/2},&\nu^{{}^{\prime}}_{\rm
c}<\nu^{{}^{\prime}},\end{array}\right.$ (44)

where the peak value of the radiation coefficient and the two characteristic
frequencies are respectively defined as

$\displaystyle j^{\prime}_{\nu^{\prime},\max}=n^{\prime}_{\rm e}\frac{m_{\rm
e}c^{2}\sigma_{\rm T}}{3q_{\rm e}}B^{\prime}$ (45)

and

$\displaystyle{\nu^{\prime}}_{\rm m/c}=\frac{3q_{\rm e}B^{\prime}}{4\pi m_{\rm
e}c}\gamma_{\rm m/c}^{2},$ (46)

where the number density of the electrons is given by $n^{\prime}_{\rm
e}=n^{\prime}_{\rm sh}$.

For an order-of-magnitude analysis of the synchrotron emission of internal
shocks, we can write the comoving particle number density of the unshocked
shells by

$\displaystyle n^{\prime}_{\rm un}={L_{\rm iso}\over 4\pi R_{\rm
is}^{2}\Gamma^{2}m_{\rm p}c^{3}},$ (47)

where $L_{\rm iso}\sim 10^{50}\rm erg~{}s^{-1}$ is the isotropic-equivalent
luminosity of the jet. Supposing the relative Lorentz factor $\Gamma_{\rm
sh}\approx\Gamma_{\rm r}/2\Gamma_{\rm s}$ between different shells is on the
order of a few to a few of tens, we can get

$\displaystyle h{\nu}_{\rm m}=2C_{p}^{2}\delta
t_{-3}^{-1}\Gamma_{2.5}^{-2}\Gamma_{\rm sh,1}^{3}L_{\rm
iso,50}^{1/2}\epsilon_{\rm B,-2}^{1/2}\epsilon_{e,-1}^{2}\rm MeV$ (48)

which is extremely higher than $h{\nu}_{\rm c}$ (fast cooling case) and well
consistent with the peak energy of the prompt GRB emission, where
$C_{p}=(p-2)/(p-1)$. However, the problem of this synchrotron emission model
is that, the spectrum below the peak ($F_{\nu}\propto\nu^{-1/2}$) is too soft
to explain the observed hard spectrum ($F_{\nu}\propto\nu^{0}$). This
indicates that more complex factors should be taken into account for
reproducing the observed GRB spectrum, e.g., the decay of the magnetic field
behind the shock (Pe’er & Zhang, 2006), the Comptonization of the photosphere
emission (Beloborodov, 2010), and the temporal evolution of the emission
spectrum (Deng & Zhang, 2014) etc.

#### High-Energy Photon and Neutrino Emission

Accompanying with the synchrotron radiation, the inverse Compton scattering of
the synchrotron photons off relativistic electrons would lead to high-energy
emissions above hundreds of GeV. Nevertheless, these high-energy photons are
further subjected to severe absorption via two-photon annihilation into
electron-positron pairs. The absorption can happen inside the jet or far away
from it. On the one hand, the optical depth of internal attenuation is highly
dependent on the bulk Lorentz factor $\Gamma$ of the jet. Only for
sufficiently high $\Gamma$, these high-energy photons can escape from the
emitting region, which therefore can be used to constrain the value of
$\Gamma$. On the other hand, those escaping high-energy photons would further
interact with cosmic infrared background photons, leading to electron-positron
pair production. These secondary pairs can further upscatter the cosmic
microwave background, leading to secondary gamma-ray photons, which lag behind
the primary high-energy emission. Moreover, since the secondary pairs can be
deflected by the intergalactic magnetic field, the delayed high-energy
emission should also deviate from the direction of the primary emission and
thus form a diffuse high-energy halo surrounding the GRB.

Because of the intense radiation in the GRB ejecta, the shock-accelerated
protons can lose their energy to produce mesons such as $\pi^{0}$ and
$\pi^{\pm}$ etc, and subsequently generate neutrinos by the decay of
$\pi^{\pm}$, i.e.,

$\displaystyle\pi^{\pm}\rightarrow\mu^{\pm}+\nu_{\mu}(\bar{\nu}_{\mu})\rightarrow
e^{\pm}+\nu_{\rm e}(\bar{\nu}_{\rm e})+\bar{\nu}_{\mu}+\nu_{\mu}.$ (49)

The timescale of the photomeson processes can be calculated by Waxman &
Bahcall (1997)

$\displaystyle{t^{\prime}}_{\pi}^{-1}={c\over 2{\gamma}_{\rm
p}^{2}}\int_{\tilde{E}_{\rm
th}}^{\infty}\sigma_{\pi}(\tilde{E})\xi(\tilde{E})\tilde{E}\left[\int_{\tilde{E}/2\gamma_{\rm
p}}^{\infty}n^{\prime}(E^{\prime}_{\gamma}){E^{\prime}}_{\gamma}^{-2}dE^{\prime}_{\gamma}\right]d\tilde{E},$
(50)

where $\gamma_{\rm p}$ is the proton’s Lorentz factor,
$\sigma_{\pi}(\tilde{E})$ is the cross section of photopion interactions for a
target photon with energy $\tilde{E}$ in the proton’s rest frame, $\xi$ is the
inelasticity defined as the fraction of energy loss of a proton to the
resultant pions, and $\tilde{E}_{\rm th}=0.15\rm GeV$ is the threshold energy
of the interactions. Then, the fraction of the energy loss of the protons to
pions can be written as

$f_{\pi}\approx
1-\exp\left(-{t^{\prime}\over{t^{\prime}}_{\pi}}\right)\approx\min\left[\Gamma
t/{t^{\prime}}_{\pi},1\right],$ (51)

where $t$ is the observer’s time. By taking a constant ratio between different
pions as $\pi^{\pm}:\pi^{0}=2:1$, the time-integrated muon-neutrino spectrum
can be given by

$E_{\nu}^{2}\phi_{\nu}\equiv{1\over 4\pi d_{\rm
L}^{2}}E_{\nu}^{2}{dN_{\nu}\over dE_{\nu}}={1\over 4\pi d_{\rm
L}^{2}}{f_{\pi}\over 3}{E}_{\rm p}^{2}{dN_{\rm p}\over dE_{\rm p}},$ (52)

where $d_{\rm L}$ is the luminosity distance of the burst and the neutrino
energy is related to the primary proton’s energy by $E_{\nu}={1\over 4}\xi
E_{\rm p}$ since the two resultant muon-neutrinos from the decay of a
$\pi^{\pm}$ could inherit half of the pion’s energy roughly evenly. Here, the
energy distribution of the shock-accelerated protons is expected to have a
form of $({dN_{\rm p}/dE_{\rm p}})\propto{E}_{\rm p}^{-2}$, where the
proportional coefficient can be calculated by $\epsilon_{\rm p}E/\ln(E_{\rm
p,\max}/E_{\rm p,\min})$. In addition, due to the presence of the stochastic
magnetic fields, the ultra-high energy pions and muons can also lose their
energy via synchrotron radiation before decay, which therefore steepens the
neutrino spectrum at very high energies by timing an extra power law as
$E_{\nu}^{-2}$.

###  4.3 Multi-Wavelength Afterglows

#### External Reverse Shock

After the internal shock processes, all the jet material would eventually
merge into a whole ejecta of a Lorentz factor $\eta$, which moves into the
circum-burst medium (CBM) persistently. Typically, the CSM can be the
interstellar medium (ISM) or a wind environment produced by the progenitor
star. Then, the interaction between the ejecta and the CBM can drive a forward
shock sweeping up the CBM and a reverse shock crossing the ejecta, as shown in
Fig. 5. This situation is like the jet propagation in the progenitor material,
but the difference is that the shocks here are radiative rather than
adiabatic, which therefore can produce emission directly.

Figure 5: Illustration of the external reverse and forward shocks.

By considering that the GRB ejecta has a very small initial thickness of
$\Delta_{0}\sim 10^{6}$ cm (before spreading), the external reverse shock can
only last a limited period of (Sari & Piran, 1995)

$\displaystyle t_{\rm rs}\sim{\Delta_{0}\over c}\eta f^{1/2},$ (53)

where

$\displaystyle f\equiv{n^{\prime}_{\rm ej}\over
n}={(4\Gamma+3)(\Gamma-1)\over(4\Gamma_{\rm rs}+3)(\Gamma_{\rm rs}-1)},$ (54)

which represents the mechanical equilibrium between the two shocks, where
$n^{\prime}_{\rm ej}$ is the comoving density of the unshocked ejecta, $n$ is
the density of the CBM, $\Gamma$ is the Lorentz factor of the shocked region,
and

$\displaystyle\Gamma_{\rm rs}={1\over
2}\left({\Gamma\over\eta}+{\eta\over\Gamma}\right)$ (55)

is the Lorentz factor of the shocked region measured in the comoving frame of
the ejecta. Solving Eqs. (54) and (55), the Lorentz factor of the shocked
region can be derived to (Sari & Piran, 1995)

$\displaystyle\Gamma=\left\\{\begin{array}[]{ll}{1\over\sqrt{2}}\eta^{1/2}f^{1/4},&{\rm
for~{}}f\ll\eta,~{}~{}({\rm relativistic~{}case})\\\ \eta(1-\sqrt{2\xi}),&{\rm
for~{}}f\gg\eta,~{}~{}({\rm newtonian~{}case})\end{array}\right.$ (58)

where $\xi=4\eta^{2}/(7f)$. For typical parameter values, we can usually
obtain $\Gamma_{\rm rs}\ll\Gamma$, which leads to the reverse shock emission
peaks in the optical bands while the forward shock is in X-rays. The duration
of the optical flash due to the reverse shock is about $t_{\rm rs}\sim 100$ s,
after which the flux decays as $t^{-2}$.

The superposition of the reverse and forward shock emissions would lead the
early optical afterglow to be more complicated than that in the other bands.
When the reverse shock emission is strong and dominated at the very early
stage, a rapid rising and decaying optical flash would show up, which is
gradually followed by a normal decay at a late time or by a rebrightening
signature due to the forward shock emission.

#### External Forward Shock

After the external reverse shock crosses the GRB ejecta, the external forward
shock still exists and moves persistently, which can enter into a self-similar
evolution (Blandford & McKee, 1976). Then, by taking a thin shell
approximation, the dynamical evolution of the forward shock can be easily
determined by the energy conservation law as

$\displaystyle dE_{\rm k}=-\varepsilon\Gamma(\Gamma-1)c^{2}dM_{\rm sw},$ (59)

where $\varepsilon$ represents the radiation efficiency of the shock, $M_{\rm
sw}$ is the mass of the swept-up CBM, and the kinetic energy of the system can
be expressed as

$\displaystyle E_{\rm k}=(\Gamma-1)(M_{\rm ej}+M_{\rm
sw})c^{2}+(1-\varepsilon)\Gamma(\Gamma-1)M_{\rm sw}c^{2},$ (60)

where $M_{\rm ej}$ is the mass of the ejecta. From Eq. (59), a differential
dynamical equation of the shock can be obtained as (Huang et al., 1999, 2000)

$\displaystyle\frac{d\Gamma}{dM_{\mathrm{sw}}}=-\frac{\Gamma^{2}-1}{M_{\mathrm{ej}}+\varepsilon
M_{\mathrm{sw}}+2(1-\varepsilon)\Gamma M_{\mathrm{sw}}}.$ (61)

The deceleration of the forward shock can be significant only when the mass of
the swept-up CBM is much larger than $M_{\rm ej}/\eta$.

For an adiabatic approximation, we can solve from Eq. (6) to $\Gamma\propto
M_{\rm sw}^{-1/2}$ for $M_{\rm ej}/\eta\ll M_{\rm sw}\ll\eta M_{\rm ej}$ and
$\beta\propto M_{\rm sw}^{-1/2}$ for $M_{\rm sw}\gg\eta M_{\rm ej}$, where
$\beta=\sqrt{1-\Gamma^{-2}}$. Here the evolution of the mass of the swept-up
CBM is determined by

$\displaystyle\frac{dM_{\rm{sw}}}{dr}=4\pi r^{2}nm_{p},$ (62)

and

$\displaystyle{dr\over dt}={\beta c\over 1-\beta},$ (63)

where $r$ is the radius of the shock and $t$ is the time in the observer’s
frame. The term $(1-\beta)$ appearing in the above equation is due to the
Doppler effect compressing the local dynamical time of the shock for the
observer. Then, for an ultra-relativistic case, we can approximately take
$r\sim 2\Gamma^{2}ct$.

Figure 6: Dynamical evolution of a GRB forward shock in the ISM (solid) and
wind (dashed) environments for $\varepsilon=0$.

By further taking the CBM density as $n=Ar^{-k}$ ($k=0$ for ISM and $k=2$ for
wind environment), the dynamical evolution of the external forward shock can
be solved to

$\displaystyle\Gamma=\eta\left({t\over t_{\rm dec}}\right)^{-(3-k)/[2(4-k)]}$
(64)

for the relativistic case and

$\displaystyle\beta=\left({t\over t_{\rm nr}}\right)^{-(3-k)/(5-k)}$ (65)

for the non-relativistic case. The deceleration timescale $t_{\rm dec}$ and
the transition time $t_{\rm nr}$ from the relativistic to the non-relativistic
phase are determined by the conditions of $M_{\rm sw}=M_{\rm ej}/\eta$ and
$M_{\rm sw}=\eta M_{\rm ej}$, respectively, which yield

$\displaystyle t_{\rm dec}$ $\displaystyle=$ $\displaystyle\left[{(3-k)E_{\rm
iso}\over 2^{5-k}\pi Am_{\rm p}\eta^{2(4-k)}c^{5-k}}\right]^{1/(3-k)}$ (68)
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}5E_{\rm
iso,52}^{1/3}\eta_{2.5}^{-8/3}n_{0}^{-1/3}{\rm~{}s},~{}~{}{\rm
for}~{}~{}k=0,\\\ 0.003E_{\rm
iso,52}\eta_{2.5}^{-4}A_{35.5}^{-1}{\rm~{}s},~{}~{}{\rm
for}~{}~{}k=2,\end{array}\right.$

and

$\displaystyle t_{\rm nr}$ $\displaystyle=$ $\displaystyle\left[{(3-k)E_{\rm
iso}\over 4\pi Am_{\rm p}c^{5-k}}\right]^{1/(3-k)}$ (71) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}4\times 10^{7}E_{\rm
iso,52}^{1/3}n_{0}^{-1/3}{\rm~{}s},~{}~{}{\rm for}~{}~{}k=0,\\\ 6\times
10^{7}E_{\rm iso,52}A_{35.5}^{-1}{\rm~{}s},~{}~{}{\rm
for}~{}~{}k=2,\end{array}\right.$

where $M_{\rm ej}=E_{\rm iso}/\eta c^{2}$ is used. These characteristic
timescales can be found easily from the numerical calculation results
presented Fig. 6. Based on these dynamical results, the synchrotron emission
of the forward shock can be calculated by using the formulae presented in
previous sections.

The above calculations assume the GRB jet has a top-hat structure, and thus it
can be treated isotropically. However, an actual jet is very likely to have an
angular structure, which could significantly influence the afterglow
observation when the GRB is observed off-axis. In this case, the dynamical
evolution for different directions should be calculated separately. The angle-
dependence of the Doppler effect also needs to be considered as

$\displaystyle{dr_{\theta}\over dt}={\beta_{\theta}c\over
1-\beta_{\theta}\cos\alpha},$ (72)

where the subscript $\theta$ represents the emission element deviates from the
symmetric axis by an angle of $\theta$, and the viewing angle of this element
is given by

$\displaystyle\cos\alpha=\cos\theta\cos\theta_{\rm
obs}+\sin\theta\sin\theta_{\rm obs}\cos\varphi,$ (73)

where $\theta_{\rm obs}$ is the angle between the jet axis and the line of
sight. The total flux of the afterglow emission of a structured jet can be
obtained by

$\displaystyle F_{\rm\nu}(t)={1\over d_{\rm
L}^{2}}\int_{0}^{\pi/2}\int_{0}^{2\pi}\frac{I^{{}^{\prime}}_{\nu^{{}^{\prime}}}(r,\theta,\varphi)}{\Gamma_{\theta}^{3}(1-\beta_{\theta}\cos\alpha)^{3}}d\varphi
d\theta,$ (74)

where $d_{\rm L}$ is the luminosity function of the GRB and the radiation
intensity contributed by the element in the direction $(\theta,\phi)$ can be
written as

$\displaystyle I^{\prime}_{\nu^{\prime}}(r,\theta,\varphi)d\varphi
d\theta={j^{\prime}_{\nu^{\prime}}(r,\theta,\varphi)\over 4\pi}{M_{\rm
sw,\theta}\over 4\pi n^{\prime}_{\rm sh}m_{\rm p}}d\varphi d\theta,$ (75)

where $M_{\rm sw,\theta}$ is an isotropically-equivalent value. Finally, the
calculating result of Eq. (74) can be used to fit the observational afterglow
data, as presented in Fig. 7.

Figure 7: Fitting to the multi-wavelength afterglow light curves of GRB
170817A with an off-axis Gaussian-structured jet.

#### Post-Standard Afterglow Models

The observed GRB afterglows exhibit a large variety and complexity in their
light curves, although the standard external shock model can explain the
general behavior of the light curves. This indicates that some complex factors
need to be invoked in the model, e.g., a complicated angular and radial
structure of the GRB jet and a possible evolution of the microphysical
parameters. Nevertheless, the most critical factor affecting the afterglow
emission comes from the central engine, which could still be very active after
the prompt phase. Such a hypothesis is strongly supported by the observed
X-ray flares and plateaus observed in the afterglow phase. Specifically, a
long-lasting active engine could be an accreting BH or a spinning-down NS.

In the BH case, the rate of the fallback accretion starting from the time of
$\sim 0.1\,\rm s$ would behave as a power law $\dot{M}_{\rm
fb}\propto\,t^{-5/3}$. By assuming a feedback efficiency $\eta_{\rm fb}$, the
outflow luminosity can be written as

$\displaystyle L_{\rm fb}$ $\displaystyle=$ $\displaystyle\eta_{\rm
fb}\dot{M}_{\rm fb}c^{2}$ (76) $\displaystyle\approx$ $\displaystyle 2\times
10^{51}{\rm erg\,s^{-1}}\left({\eta_{\rm fb}\over
0.1}\right)\left({\dot{M}_{\rm fb,i}\over 10^{-3}M_{\odot}\,\rm
s^{-1}}\right)\left(t\over 0.1\rm\,s\right)^{-5/3},$ (77)

where the subscript ‘i’ stands for “initial”. For this function, however, the
primary energy is released at a very early time so that the later afterglow
emission cannot be influenced substantially. On the contrary, in the spinning-
down NS case, the energy release can nearly keep constant for a period of

$t_{\rm
sd}=\frac{3Ic^{3}}{B_{\rm{p}}^{2}R_{\rm{s}}^{6}}\left(\frac{2\pi}{P_{\rm{i}}}\right)^{-2}=2\times
10^{3}I_{45}R_{\rm{s,}6}^{-6}B_{\rm{p,}15}^{-2}P_{\rm i,-3}^{2}\,\rm{s},$ (78)

where $B_{\rm{p}}$, $R_{\rm{s}}$, and $P$ are the polar magnetic field
strength, radius, and spin period of the NS, respectively. Here the NS is
assumed to be a millisecond magnetar. The corresponding luminosity is
determined by the magnetic dipole radiation as

$L_{\rm{md,i}}=\frac{B_{\rm{p}}^{2}R_{\rm{s}}^{6}}{6c^{3}}\left(\frac{2\pi}{P_{\rm
i}}\right)^{4}=9.6\times 10^{42}B_{\rm{p,}12}^{2}R_{\rm{s,}6}^{6}P_{\rm
i,-3}^{-4}\rm{erg\ s}^{-1}.$ (79)

The complete behavior of the spin-down luminosity can be written as

$L_{\rm{md}}(t)=L_{\rm{md,i}}\left(1+\frac{t}{t_{\rm{sd}}}\right)^{-2}.$ (80)

Sometimes, if the magnetic field of the NS is much lower than $10^{15}$ G and
the ellipticity of the NS is high enough, the gravitational radiation of the
NS would also play a role in braking the stellar rotation, which determines a
spin-down timescale of

$t_{\rm{sd}}=\frac{5P_{\rm{i}}^{4}c^{5}}{2048\pi^{4}GI\epsilon^{2}}=9.1\times
10^{5}\epsilon_{-4}^{-2}I_{45}^{-1}P_{\rm i,-3}^{4}\,\rm{\rm s},$ (81)

where $\epsilon$ is the ellipticity. In any case, the constant energy release
of the NS before the spin-down timescale can substantivally slow the
deceleration of the external shock of the GRB ejecta, which provides a natural
explanation of the shallow-decaying light curves of some GRB afterglows Dai &
Lu (1998, 1998); Zhang & Mészáros (2001). The dynamical evolution of the
external shock with an energy injection can be obtained by substituting Eq.
(80) into (59).

In more detail, the energy released from the central engine could be in the
form of a Poynting flux initially and finally evolve into a relativistic wind
consisting of electrons and positrons. When such a relativistic wind collides
with the swept-up medium, a termination shock can be formed and propagate into
the wind. In the NS case, the shocked wind region can contribute an extra
emission component for the GRB afterglow, just like the emission of a pulsar
wind nebulae existing in some supernova remnant Dai (2004); Yu & Dai (2007).
The existence of such an internal-origin afterglow is helpful for
understanding the complexity of GRB afterglow light curves Yu et al. (2010)
including some unusual plateaus followed by an extremely sharp decay.

###  4.4 Supernova and Kilonova

#### Supernova

Since long GRBs originate from the collapse of massive stars, they are
expected to be associated by supernova emission, which reaches a peak at a
time of

$\displaystyle{t}_{\rm d}$ $\displaystyle=$
$\displaystyle\left(\frac{3\kappa{M}_{\rm ej}}{4\pi{cv}}\right)^{1/2}$ (82)
$\displaystyle=$ $\displaystyle 46\left({\kappa\over 0.2\rm
cm^{2}g^{-1}}\right)^{1/2}\left({{M}_{\rm ej}\over
5M_{\odot}}\right)^{1/2}v_{9}^{-1/2}\rm day.$

This peak time is determined by the diffusion timescale of photons in the
supernova ejecta, where $\kappa$, $M_{\rm ej}$, and $v$ are the opacity, mass,
and expanding speed of the supernova ejecta. As usual, the supernova emission
is powered by the radiative decays of 56Ni to 56Co and to 56Fe. The rate of
the energy release of these processes is given by

$\dot{q}_{\rm r}=\left(\epsilon_{\rm Ni}-\epsilon_{\rm Co}\right){\rm
e}^{-t/\tau_{\rm Ni}}+\epsilon_{\rm Co}{\rm e}^{-t/\tau_{\rm Co}},$ (83)

where $\epsilon_{\rm Ni}=3.90\times 10^{10}\rm{erg}~{}{s}^{-1}{~{}g}^{-1}$,
$\epsilon_{\rm Co}=6.78\times 10^{9}\rm{erg}~{}{s}^{-1}{~{}g}^{-1}$,
$\tau_{\rm Ni}=8.76\rm{day}$, and $\tau_{\rm Co}=111.42\rm{day}$. The light
curve of the supernova emission is determined by the radiative transfer of the
thermal energy in the supernova ejecta:

${L}=-4\pi{r}^{2}\frac{{c}}{3\kappa\rho}\frac{\partial{u}}{\partial{r}}.$ (84)

where $r$ is the radius, $u$ and $\rho$ are the energy and mass densities.

For an order-of-magnitude analysis, by taking
$\frac{\partial{u}}{\partial{r}}\sim\frac{{u}}{{R}}\sim\frac{U}{{VR}}$, the
bolometric luminosity of the supernova can be estimated as follows

${L}_{\rm e}=\frac{{Uc}}{{R}{\tau}}\left(1-{e}^{-\tau}\right),$ (85)

where $U$ is the total internal energy, $V$ is the volume, $R$ is the surface
radius, and $\tau=\kappa\rho R$ is the optical depth of the ejecta. The
evolution of the internal energy can be determined by the energy conservation
of the ejecta as (Arnett, 1982; Kasen & Bildsten, 2010)

$\frac{{dU}}{{dt}}=M_{\rm Ni}\dot{q}_{\rm r}-{L}_{\rm
e}-{P}\frac{\partial{V}}{\partial{t}},$ (86)

where $M_{\rm Ni}$ is the total mass of 56Ni and $P=U/3V$ is the pressure. The
term $-PdV$ represents the adiabatic cooling of the ejecta. The combination of
Eq. (85) and (86) can yield a bolometric light curve for the supernova111The
light curve can also derive from the following integral Arnett (1982):
$L_{\rm e}(t)=2M_{\rm Ni}\left(\int_{0}^{\tilde{t}}\dot{q}_{\rm
r}\tilde{t}^{\prime}e^{\tilde{t}^{\prime
2}}d\tilde{t}^{\prime}\right)e^{-\tilde{t}^{2}},$ (87) where
$\tilde{t}=t/t_{\rm d}$.. Sometimes, an extra power could be involved in Eq.
(86), e.g., if the supernova ejecta can also absorbed energy from the central
engine. Finally, by assuming a black-body spectrum for the supernova emission,
an effective temperature of the supernova emission can be defined as $T_{\rm
BB}=({L_{\rm e}/4\pi R_{\rm ph}^{2}\sigma})^{1/4}$, where $R_{\rm ph}$ is the
radius of the photosphere, which corresponds to $\kappa\rho(R-R_{\rm ph})=1$.
Then, the chromatic luminosity for a specific frequency $\nu$ can be given by

$L_{\nu}={8\pi^{2}R_{\rm ph}^{2}\over c^{2}}{h\nu^{3}\over{\rm
e}^{h\nu/kT_{\rm BB}}-1},$ (88)

which can be used to compare with the observational magnitudes of the
supernova.

#### Kilonova/Mergernova

During a merger of double NSs or an NS and a BH, a non-relativistic mass of
$\sim 10^{-4}-10^{-2}M_{\odot}$ can be ejected more widely than the GRB jet
due to the effects of tidal disruption, collision squeeze, and accretion
feedback. It is suggested that nearly half of the elements heavier than iron
in the universe can be synthesized in this neutron-rich merger ejecta, through
the rapid neutron-capture process (r-process) (Lattimer & Schramm, 1974,
1976). Then, like the supernova situation, the radioactive decays of the
$r$-process elements can also lead to a transient thermal emission (i.e.,
kilonova), by heating the merger ejecta with a power of (Lippuner & Roberts,
2015)

$\dot{q}_{\mathrm{r}}=4\times
10^{18}\left[\frac{1}{2}-\frac{1}{\pi}\arctan\left(\frac{t-t_{0}}{\sigma}\right)\right]^{1.3}\rm
erg~{}s^{-1}g^{-1}$ (89)

and a thermalization efficiency of (Barnes et al., 2016)

$\eta_{\mathrm{th}}=0.36\left[\exp\left(-0.56t_{\mathrm{day}}\right)+\frac{\ln\left(1+0.34t_{\mathrm{day}}^{0.74}\right)}{0.34t_{\mathrm{day}}^{0.74}}\right]$
(90)

where $t_{0}=1.3\,{\rm s}$, $\sigma=0.11\,{\rm s}$, and $t_{\rm day}=t/{\rm
day}$.

The kilonova emission calculation is in principle the same as the case of
supernova emission, but with different power and different ejecta. The peak
emission of kilonovae is expected to appear at

$\displaystyle{t}_{\rm d}=5\kappa_{1}^{1/2}M_{\rm ej,-2}^{1/2}v_{10}^{-1/2}\rm
day.$ (91)

Here a high reference value of $\sim 10\rm cm^{2}g^{-1}$ is taken for the
opacity, by considering that the opacity can be significantly increased due to
the formation of a large number of lanthanide elements Barnes & Kasen (2013).
For a reference luminosity $L_{\rm kn}\sim 10^{41}\rm erg~{}s^{-1}$, we can
determine the peak wavelength of the kilonova emission to

$\displaystyle\lambda_{\rm p}={hc\over 5k_{\rm B}}\left[{L_{\rm kn}\over
4\pi(vt_{\rm d})^{2}\sigma}\right]^{-1/4}=1.6\kappa_{1}^{1/4}L_{\rm
kn,41}^{-1/4}M_{\rm ej,-2}^{1/4}v_{10}^{1/4}\rm\mu m,$ (92)

which indicates the kilonova emission is inclined to be red. Nevertheless, the
different components of the merger ejecta could have very different electron
fractions, which can substantially affect the efficiency of the r-processes
and as well as the opacity of the ejecta. Therefore, besides the red emission
component, the kilonova emission can, in principle, also include relatively
blue components. Furthermore, the spectrum and light curve of kilonovae can
also be sensitive to the viewing angle because of the high anisotropic
structure of the merger ejecta (especially for the NS and BH mergers) (Zhu et
al., 2020).

Figure 8: The fitting to the light curves of kilonova AT2017gfo with hybrid
energy sources inlcuding a pinning-down NS and the radioactivity of r-process
elements (solid lines). From Yu et al. (2018)

Finally, for double NS mergers, one of the most concerned topics is the nature
of the merger product. As implied by the afterglows of short GRBs, the merger
product could be a long-lived massive NS (Dai et al., 2006; Giacomazzo &
Perna, 2013). In this case, the kilonova emission can be extra powered by the
relativistic wind from the NS, and thus its luminosity could be significantly
enhancedYu et al. (2013); Metzger & Piro (2014). Therefore, it was suggested
that the transient thermal emission during a merger event could be generally
called a mergernova (Yu et al., 2013), instead of a kilonova. For example, the
relatively high luminosity of kilonova AT 2017gfo is probably powered by
hybrid energy sources Yu et al. (2018), since the pure radioactive scenario
requires too much high ejecta mass to be accounted for by a double NS merger.
Fitting to the observational data, as presented in Fig. 8, give a stringent
constraint on the property of the remnant NS, which indicates the complexity
of the early evolution of such newborn NSs. In addition, the interaction
between the NS wind and merger ejecta can further contribute to a non-thermal
emission component, which is detectable when the merger ejecta becomes
transparent (Gao et al., 2016; Wu et al., 2021).

##  5 Statistics and Cosmological Applications

###  5.1 Luminosity Function

In a statistical view, by denoting the event rate of GRBs and their luminosity
function as $\dot{R}_{\rm GRB}$ and $\Phi(L)$, the detectable numbers of GRBs
in different flux and redshift ranges can be calculated by Tan & Yu (2020)

$\displaystyle N(P_{1},P_{2})$ $\displaystyle=$
$\displaystyle{\Delta\Omega\over
4\pi}T\int_{0}\int^{P_{2}}_{P_{1}}\int_{0}\eta(P)$ (93) $\displaystyle\times$
$\displaystyle\dot{R}_{\rm GRB}(z)\Phi(L)\sin\theta_{\rm v}~{}d\theta_{\rm
v}dP{dV(z)\over 1+z},$

and

$\displaystyle N(z_{1},z_{2})$ $\displaystyle=$
$\displaystyle{\Delta\Omega\over
4\pi}T\int^{z_{2}}_{z_{1}}\int_{0}\int_{0}\eta(P)\vartheta_{z}(z,P)$ (94)
$\displaystyle\times$ $\displaystyle\dot{R}_{\rm GRB}(z)\Phi(L)\sin\theta_{\rm
v}~{}d\theta_{\rm v}dP{dV(z)\over 1+z},$

where $\Delta\Omega$ is the field of view of a telescope, $T$ is the working
time with a duty cycle of $\sim$50%, $\eta(P)$ and $\vartheta(z,P)$ are the
trigger efficiency and the probability of redshift measurement, respectively,
and $dV(z)$ is the comoving cosmological volume element. Here the integral
over the viewing angle $\theta_{\rm v}$ is included, because the GRB
luminosity could be direction-dependent if the jet has a significant angular
structure. By connecting the GRB rates to the cosmic star formation rates
(SFRs), we can use the above equations to model the observational
distributions and constrain the luminosity function. As usual, without
considering the jet structure, the luminosity function is found to have a
broken-power-law form. However, when a Gaussian jet structure is invoked, it
can be found that a single-power-law luminosity function would become a better
choice Tan & Yu (2020). This hints that the usual low-energy power law in the
luminosity function could be a result of off-axis observations.

###  5.2 High-redshift Universe

The association of long GRBs with core-collapse supernovae indicates that the
GRB event rates could trace the cosmic star formation history either
unbiasedly or, more probably, with an additional evolution effect. Therefore,
they can provide a complementary technique to measure the star formation rate
at high redshifts where direct measurement is extremely difficult (Kistler et
al., 2009; Wang & Dai, 2009), as presented in Fig. 9. Here the selection
effect and calibration from the GRB rate to SFR must be properly handled.

Some works find that the rate of long GRBs has an excess compared to SFR at
high redshifts Kistler et al. (2009); Wang (2013). Possible reasons includes
metallicity evolution Li (2018), the evolution of the stellar initial mass
function Wang & Dai (2011), and evolving luminosity function break Virgili et
al. (2011). Theory and observation both support that long GRBs prefer to
explode in a low-metallicity site, leading to more GRBs at a high-redshift
universe with low metals. For evolving initial mass function, “top-heavy”
initial mass function would lead to more massive stars in the early universe
which can form much more GRBs.

Absorption processes imprinted on the spectra of GRBs are the main sources of
information about the chemical properties of the high-redshift universe. The
progenitors of long GRBs are believed to be massive stars, so their number is
abundant in the early universe. The damped Lyman-$\alpha$ and absorption lines
in GRB spectra can be used to probe the metal enrichment history Wang et al.
(2012), and cosmic reionization Totani et al. (2006). The forthcoming James
Webb Space Telescope would detect afterglow spectra of GRBs out to $z>10$,
shielding light on the process of cosmic reionization.

Figure 9: Cosmic star formation history inferred from high-redshift GRBs. From
Wang (2013)

###  5.3 Luminosity Correlations of GRBs

Since GRBs can be observed much more distant than SNe Ia, they can fill the
gap between SNe Ia and cosmic microwave background (CMB) in cosmological
studies. Similar to SNe Ia, it has been proposed to use GRB correlations to
standardize their energies and/or luminosities. Until now, a lot of GRB
correlations have been proposed.

  * •

Amati correlation. The isotropic energy $E_{\rm iso}$ of GRB prompt emission
is associated with the rest-frame peak energy of the prompt spectrum, i.e.,
$E_{\rm p}\propto E_{\rm iso}^{0.52}$ Amati et al. (2002).

  * •

Yonetoku correlation. The correlation $L_{\rm iso}\propto E_{\rm p}^{2}$ is
found with a sample of 16 GRBs (Yonetoku et al., 2004).

  * •

Ghirlanda correlation. A tight correlation between spectral peak energy
$E_{\rm peak}$ and collimated energy $E_{\gamma}$ was discovered (Ghirlanda et
al., 2004; Dai et al., 2004) using 15 GRBs. The intrinsic scatter is up to
$0.1$.

  * •

Dainotti correlation. A tight correlation between the X-ray afterglow
parameters(Dainotti et al., 2008): $L_{X}$ and $T_{a}$, where $L_{X}$ is the
luminosity of an X-ray plateau, and $T_{a}$ is the time at which the X-ray
light curve establishes a normal power-law decay. The intrinsic scatter is
$0.22$.

  * •

Liang-Zhang correlation. Without imposing any theoretical model, an empirical
correlation was found (Liang & Zhang, 2005) with 15 bursts among the isotropic
energy of the prompt gamma-ray emission $E_{\rm iso}$, the rest-frame peak
energy $E_{\rm p}$, and the rest-frame break time in the optical afterglow
light curves $t_{\rm break}$.

According to the properties of the involved parameter, these correlations can
be divided into three categories such as prompt correlations (i.e., the first
three ones), afterglow correlations (i.e., the Dainotti one), and prompt-
afterglow correlations (i.e., the Liang-Zhang one).

Figure 10:  Standardized the light curves of LGRBs. Original and scaled
plateau light curves of 10 LGRBs. Left panel shows the original X-ray (0.3-10
keV) light curves. Right panel shows the scaled light curves of the same GRBs.
The dispersion of the scaled luminosity is 0.5 dex. Adapted from Wang et al.
(2022)

###  5.4 Cosmological Constraints

The most common method to constrain the cosmological parameters is by using
the $\chi^{2}$ statistic. For an instance, in the standard cosmological model,
the corresponding likelihood function can be written as

$\chi^{2}(\Omega_{i})=\sum_{i=1}^{N}\frac{[\mu_{\rm th}(z,\Omega_{i})-\mu_{\rm
obs}]^{2}}{\sigma_{\mu_{\rm obs}}^{2}},$ (95)

where $\Omega_{i}$ represents cosmological parameters, $\mu_{\rm
th}(z,\Omega_{i})$ and $\mu_{\rm obs}$ the theoretical and observed distance
modulus, respectively. While the observed distance modulus can be derived from
GRB correlations, the theoretical one can be calculated by taking the follow
equation

$\mu_{\rm th}=5\log(d_{\rm L})+25,$ (96)

where $d_{\rm L}$ is the luminosity distance. For a flat universe, the
expressed form of luminosity distance can be written as

$\displaystyle d_{\rm
L}=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{\rm
m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$ (97)

where $\Omega_{\rm m}$, $\Omega_{\Lambda}$, $H_{0}$ and $c$ represent the
cosmic matter density, dark energy density, Hubble constant and the speed of
light, respectively. The best-fitting results of $\Omega_{\rm m}$,
$\Omega_{\Lambda}$, $H_{0}$ can be given by minimizing Eq. (95). For different
cosmological models, Eq. (97) needs to take corresponding modifications.

So far, a lot of effort has been made to constrain cosmological parameters
using GRBs since their cosmological origin was confirmed. However, due to the
lack of low-redshift GRBs, the GRB correlations are cosmology-dependent. This
is the so-called “circularity problem”. Many methods have been proposed to
solve this problem, including fitting the cosmological parameters and
luminosity correlation simultaneously, and calibrating GRB relation utilizing
other observations (i.e., SNe Ia). The first method is fitting the
cosmological parameters and luminosity correlation simultaneously (Dai et al.,
2004). The second method is to calibrate GRB correlation with other
observation data at low redshifts. Recently, the light curves of LGRBs showing
plateau phases in X-ray afterglows are standardized Wang et al. (2022). The
standardized result is shown in Fig. 10. The Hubble parameter data is used to
calibrate this correlation. The cosmological constraints are $\Omega_{\rm
m}=0.32^{+0.05}_{-0.10}$ and $\Omega_{\Lambda}$ = $1.10^{+0.12}_{-0.31}$
(1$\sigma$) in the flat universe. Third, the circularity problem could be
partially solved by analyzing a sample of GRBs within a small redshift bin
(Liang & Zhang, 2006). The advent of the multi-messenger era provides more
solutions to the circularity problem. For example, the gravitational wave
events with EM counterparts can be used to calibrate GRB correlations.

###### Acknowledgements.

This chapter is supported by the National Key R&D Program of China (Grant No.
2021YFA0718500 and 2018YFA0404204), the China Manned Spaced Project (CMS-
CSST-2021-A12), and the National Natural Science Foundation of China (Grant
No. 11833003, U1831207, U2038105, and 12121003).

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