###### Abstract

The prompt emission of Gamma Ray Bursts (GRBs) is still an outstanding
question in the study of these cataclysmic events. Part of what makes GRBs
difficult to study is how unique each event seems to be. However, aggregating
many GRB observations and analyzing the population allows us to obtain a
better understanding of the emission mechanism that produces the observed
prompt emission. In this review, we outline some of the most prevalent
correlations that have emerged from GRB prompt emission observations and how
these correlations are interpreted in relation to GRB physical properties and
prompt emission models.

###### keywords:

Gamma Ray Bursts; Observational Relationships; Prompt Emission; Radiation
Mechanism;

1 1 0 https://doi.org/ GRB Prompt Emission: Observed Correlations and Their
Interpretations GRB Prompt Emission: Observed Correlations and Their
Interpretations Tyler Parsotan 1,2,3,†,‡ and Hirotaka Ito 4,5,‡ Tyler Parsotan
and Hirotaka Ito Parsotan, T.; Ito, H. Correspondence<EMAIL_ADDRESS>These
authors contributed equally to this work.

##  1 Introduction

Gamma Ray Bursts (GRBs) are the most energetic events in the universe
releasing up to $10^{54}$ ergs over the course of a given event. GRBs were
initially detected in the 1960’s by the Vela satellite as pulses of gamma ray
emission. These pulses were determined to be of extraterrestrial origin
Klebesadel et al. (1973) and eventually of cosmological origin (Reichart,
1998).

###  1.1  Observations

Since the 1960’s a number of satellites have been launched with one of their
main science goals being understanding GRBs. These satellites include:

  * •

BeppoSAX Boella, G. et al. (1997),

  * •

the Compton Gamma Ray Observatory (CGRO), with the Burst and Transient Source
Experiment (BATSE) on board (Fishman et al., 1994),

  * •

the High Energy Transient Explorer (HETE-2) with the Wide-field X-ray Monitor
Kawai, N. et al. (1999),

  * •

the Wind Spacecraft with the Konus Wind GRB Experiment on board Aptekar et al.
(1995),

  * •

the INTEGRAL satellite with the IBIS (Imager on-Board the INTEGRAL Satellite)
instrument (Ubertini, P. et al., 2003),

  * •

the Neil Gehrels Swift Observatory (Gehrels et al., 2004), with the Burst
Alert Telescope instrument (Barthelmy et al., 2005),

  * •

and the Fermi Gamma-ray Space Telescope with the Gamma Ray Burst Monitor (GBM)
instrument onboard Case et al. (2011).

These satellites have detected numerous GRBs over their lifetimes, providing
rich datasets that have contributed to our current state of knowledge
concerning GRBs. Both Konus-Wind and BATSE data showed that GRBs are
isotropically distributed in the sky, supporting the theory that GRBs were
located at cosmological distances (Fishman et al., 1994; Mazets and
Golenetskii, 1988) before the first redshift measurement was obtained.
Additionally, the BATSE dataset hinted at two distinct populations of GRBs,
based on the duration of observed GRBs (Goldstein et al., 2013; Kouveliotou et
al., 1993). This result has been reinforced by modern datasets and through
many other lines of evidence. The two populations of GRBs are known as Short
GRBs (SGRBs), which typically have durations $\lesssim 2$ s, and Long GRBs
(LGRBs) which typically have durations $\gtrsim 2$ s. SGRBs have been
associated with the merger of a Neutron star and a Black hole Abbott et al.
(2017); Goldstein et al. (2017) and are also theorized to be caused by the
merger of a pair of Neutron stars. On the other hand, LGRBs have been
associated with core collapse supernovae from massive stars Bloom et al.
(1999); Hjorth et al. (2003); Woosley and Bloom (2006) 111It is important to
note that this classification based on just the duration of GRBs leaves room
for ambiguity. The distinction between these two classes can be improved by
including more than one classification parameter, for example plotting the
GRBs on a duration-spectral hardness plane (Zhang et al., 2012). The problem
of classifying GRBs properly is still an active topic of research.. There have
been additional subclasses of GRBs that have been proposed recently such as
SGRBs with extended emission (Norris and Bonnell, 2006; Gehrels et al., 2006)
and ultra long GRBs (Boër et al., 2015; Levan et al., 2013), which are still
active topics of research.

Focusing on “traditional” LGRBs and SGRBs, each type exhibits similar phases
of emission. The initial pulse of high energy X-rays and gamma-rays that is
detected within the first few tens of seconds of the event is known as the
prompt emission. This phase of GRBs has also been detected in optical
wavelengths (see Table 1 presented in Parsotan and Lazzati (2021)) and has a
number of polarimetry measurements associated with it (see Table 1 presented
in Gill et al. (2019)). The second phase of typical GRB emission is known as
the afterglow. This phase is detected across the electromagnetic spectrum and
lasts for timescales significantly longer than the prompt emission. As a
result, the followup of GRB prompt emission detections, to conduct
measurements of the afterglow phase, has contributed greatly to our
understanding of GRBs. In this review we focus on the prompt emission phase of
the GRB and correlations that have been discovered with respect to this phase
of emission. Due to the longevity of afterglow emission, some of the
correlations include GRB parameters that are derived from afterglow
observations.

After decades of observing the prompt emission of GRBs, the radiation
mechanism that produces this emission is still not well understood. There are
a number of reviews focused on both the theory and observations of GRB prompt
emission (see Pe’Er (2015); Zhang (2015); Mészáros (2002); Beloborodov and
Mészáros (2017); D’Avanzo (2015); Kumar and Zhang (2015); Mészáros (2019))
which we briefly summarize here. The light curves of GRB prompt emission can
be erratic and variable, with each light curve seeming to be unique (see e.g.
Figure 6 in Yu et al. (2016)). Typically, the pulses in GRB light curves seem
to follow Fast Rise Exponential Decay (FRED) profiles (Norris et al., 1996,
2005). The spectra of GRB are typically non-thermal with typical low energy
photon indices $\alpha\sim-1$, high energy photon indices $\beta\sim-2.4$, and
spectral peak energies for $\nu$Fν spectra of $E_{\mathrm{pk}}\sim 200$ keV
(Yu et al., 2016). There have been a number of detections of thermal
components in GRB spectra (Ryde, 2005; Pe’er et al., 2015; Guiriec et al.,
2011, 2013; Axelsson et al., 2012), however these seem to be in the minority
as of now.

###  1.2  Models

There are two major classes of models used to describe the mechanism behind
GRB prompt emission. The first class of models is based on synchrotron
emission produced by magnetic fields within the GRB jet. These models consider
both ordered and disordered magnetic fields within the GRB jet. These magnetic
fields are typically constrained to shells of the GRB jet that have been
launched by a central engine Rees and Mészáros (1994); Zhang and Yan (2011);
Toma et al. (2009). Each shell moves with a different Lorentz factor with
respect to the other shells which results in collisions between shells at
large distances from the central engine ($\sim 10^{14}-10^{16}$ cm Zhang and
Yan (2011)) where the optical depth is below unity. These collisions excite
electrons within the shells which then emit non-thermal synchrotron emission.
The strength of the magnetic field plays an important role in the synchrotron
emission as well as the time scales over which the electrons cool (Daigne et
al., 2011). The model that describes shells of material colliding within the
jet has historically been known as the internal shock model although that name
is not as widely used since there are a number of models that use the basic
premise outlined previously. The general subclass of internal shock models
were contrasted with the subclass of external shock models, where the GRB was
thought to be produced by the jet colliding with the external medium Mészáros
(2002). The external shock model however, has fallen out of favor due to its
inconsistency with observations Fenimore et al. (1999); Lloyd et al. (2000).

The other major class of models used to describe GRB prompt emission considers
the photospheres of jets, and thus is referred to as the photospheric model.
This model describes thermal radiation that originates deep in the GRB jet at
large optical depths. As the jet propagates it eventually becomes transparent
to this radiation, allowing the radiation to freely travel towards the
observer without being modified significantly by the jet when the optical
depth is $\sim 1$ (Rees and Mészáros, 2005; Lazzati et al., 2013; Thompson,
1994; Mészáros et al., 2002). This model was initially thought to produce only
thermal spectra however, it has been shown to be able to produce non-thermal
spectra due to a number of effects including high latitude emission (Pe’er and
Ryde, 2011; Parsotan et al., 2018), subphotospheric dissipation (Chhotray and
Lazzati, 2015; Bhattacharya and Kumar, 2020), and the property of the
photosphere being a volume of space (instead of a static surface) where
photons can be upscattered through sparse interactions with matter in the
jet222This concept has been called the photospheric region Parsotan and
Lazzati (2018) or the fuzzy photosphere Pe’er (2008); Beloborodov (2010), in
contrast with a photospheric surface. (Ito et al., 2013; Parsotan and Lazzati,
2018; Pe’er, 2008; Beloborodov, 2010).

The correlations that emerge from observations of GRB prompt detections play
an important role in constraining these models. These correlations provide
information on what physical properties GRB jets have and allow us to infer
how these properties may affect the radiation mechanism at play in these
relativistic explosions. While we focus primarily on the physical
interpretations of the correlations presented here in the context of prompt
emission models, these correlations are also important for using GRBs as
cosmological probes. In fact, this motivation led to the discovery of many
correlations discussed in this review. For a comprehensive review in this
context, we refer the reader to Dainotti et al. (2018).

This review is organized as follows: Section 2 provides an overview of various
physical quantities, both observational and theoretical, that are necessary to
understand the various correlations that are presented here. The correlations
and their physical interpretations are presented in Section 3. Finally, in
Section 4 we summarize the current status of prompt emission correlations and
the future of these important tools to understanding the prompt emission of
GRBs.

##  2 Relevant Physical Quantities

###  2.1  Fundamental Quantities of Energy and Spectra

A number of physical quantities are used to understand GRB detections. When a
GRB is detected, the photons interact with the detector which has some
effective area to the photons for a range of energies. These photons are then
tagged with time, energy, and location information which can be used to
construct GRB spectra and gain insight into the energetics of these events.

GRB spectra play an important role in understanding the prompt emission
mechanism. GRB spectra can be fit using a number of different models but here
we highlight two of the most commonly used models. These are:

  * i)

the Band spectrum Band et al. (1993) which is defined as

$N(E)=A\begin{cases}E^{\alpha}e^{-\frac{E}{E_{o}}}&E\leq(\alpha-\beta)E_{o}\\\
[(\alpha-\beta)E_{o}]^{\alpha-\beta}E^{\beta}e^{\beta-\alpha}&E>(\alpha-\beta)E_{o}\end{cases}$
(1)

  * ii)

the Cutoff powerlaw spectrum Yu et al. (2016) (also called the comptonized
spectrum333This name is misleading as Comptonization can produce many types of
spectra while cutoff power laws can be produced for a number of emission
processes.) which is defined as

$N(E)=AE^{\alpha}e^{-\frac{E}{E_{o}}}$ (2)

Here, $A$ is some normalization parameter, $\alpha$ is the low energy photon
index, $\beta$ is the high energy photon index, and $E_{o}$ is the break
energy between the low energy and high energy power law slopes in the case of
the Band spectrum, or the low energy slope and the exponential decay in the
case of the cutoff powerlaw. Both models are in units of photon flux per unit
energy (typically ph s-1 cm-2 keV-1). The spectral peak energy of the
$E^{2}N(E)$ spectrum for both models is located at
$E_{\mathrm{pk}}=(2+\alpha)E_{o}$.

There are different observer energy-related quantities that can be constructed
from a GRB observation. These are:

  * i

Fluence, $S$, which has units of erg cm-2, is the total energy emitted by the
GRB as measured by a detector with some effective area within some energy band

  * ii

Flux, $F$, which has units erg s-1 cm-2, is the energy emitted by the GRB
within some time period, as measured by a detector with some effective area

We obtain the average flux, $F_{\rm tot}$, in the limit that the flux is
calculated over the time period in which we receive 100% of the emission from
a GRB.

Observed GRB spectra can be produced within a given time bin and then
integrated over energy in order to calculate flux within a given time bin.
Alternatively, the spectra can be fit with the aforementioned models and then
the models can be integrated over a larger energy range than what a given
instrument might allow, providing a larger effective bandpass for calculating
the flux and fluence. This is known as the k-correction which additionally
allows spectra to be corrected for cosmological redshifts Bloom et al. (2001).
Time integrated (over the duration of the GRB) spectra are used to calculate
$S$ while time-resolved spectra are used to calculate $F$. Typically, the same
spectrum that is used to calculate $S$ or $F$ is used to determine the
spectral $E_{\mathrm{pk}}$ (see e.g. the Amati relation in Section 3.3,
however also see the Yonetoku relation where the time-integrated spectrum is
used to calculate $E_{\mathrm{pk}}$ but the peak luminosity in a 1 s bin is
also used in the correlation.)

The luminosity, $L$, and emitted energy, $E$, are typically calculated based
on the assumption that the GRB emits its energy isotropically, within a solid
angle of $4\pi$. These quantities are typically calculated in the rest frame
of the GRB. The isotropic energy is calculated as

$E=4\pi D_{L}^{2}S$ (3)

where $D_{L}$ is the luminosity distance which depends on various cosmological
parameters. Similarly, the luminosity is calculated as

$L=4\pi D_{L}^{2}F$ (4)

Different specifications of the energy range and time bins that go into the
calculation of $S$ or $F$ gives rise to the different types of luminosities
and energies that are quoted in the literature:

  * i)

$F_{\rm tot}$ $\xrightarrow{}$ $L_{\rm iso}$

  * ii)

$F_{\rm pk}$ $\xrightarrow{}$ $L_{\rm pk}$

  * iii)

$S_{\rm tot}$ $\xrightarrow{}$ $E_{\rm iso}$

To give some examples, $S_{\rm tot}$ can be the (time-integrated) total energy
emitted by the GRB in the energy range of 1-$10^{4}$ keV (see e.g. the Amati
relation Amati, L. et al. (2002)), the $F_{\rm pk}$ can be the peak flux
measured from 30-$10^{4}$ keV in the time bin of interest (see e.g. the peak 1
s time bin used for the Yonetoku relation Yonetoku et al. (2004)), and $F_{\rm
tot}$ can be the average flux measured in the energy range of 1-$10^{4}$ keV
over the duration of the GRB. The aforementioned energy ranges are typically
in the rest frame of the GRB, which makes $L$ and $E$ rest frame values. These
quantities also include a k-correction since the energy ranges extend much
further than a single instrument’s energy range.

###  2.2  Light Curve Morphology Quantities

The morphology of the light curve and the time scales associated with them are
also important quantities that are measured. The variability of the light
curve, $V$, is typically measured by calculating a smoothed light curve and
taking the difference between the smoothed light curve and the original light
curve. This is done for multiple time intervals and then added over the
duration of the GRB and normalized appropriately Fenimore and Ramirez-Ruiz
(2000); Guidorzi et al. (2005). The light curve can also be characterized by
its rise time, $\tau_{\mathrm{RT}}$, which is the time for the light curve to
reach it’s maximum flux Schaefer (2007). Another timescale that is important
for our discussion of GRB correlations is the lag time, $\tau_{\mathrm{lag}}$,
between photons of various energy ranges. This quantity is calculated by
identifying the peak light curve times in specific instrument energy channels,
which corresponds to specific photon energy ranges, and calculating the
difference between them.

###  2.3  Transformation of Physical Quantities

The spectral energies and the times measured in GRBs are cosmologically
shifted. As a result, if we know the redshift $z$ of the GRB, we can transform
these quantities to the rest frame of the burst (which is the same as the rest
frame of the central engine). An observed energy of $E_{\rm obs}$ has a rest
frame energy $E_{\rm rest}=E_{\rm obs}(1+z)$. Observed times, $T_{\rm obs}$,
are transformed to rest frame times as $T_{\rm rest}=T_{\rm obs}/(1+z)$.

The quantities described here can be defined in the observer frame or the rest
frame of the GRB which can quickly become difficult to track for a newcomer to
the field. We highlight any quantities defined in the observer frame with a
tilde above the quantity, for example a lag measured in the observer frame
will be denoted $\tilde{\tau}$. A lack of a tilde associated with the variable
means that the represented quantity is defined in the rest frame of the GRB.

###  2.4  GRB Jet Model Parameters

GRB outflows are typically thought to be moving relativistically as a
collimated outflow, or jet Lazzati et al. (2018); Gottlieb et al. (2021).
These jets move with Lorentz factors, $\Gamma\gtrsim 100$ Fenimore et al.
(1993), where the bulk Lorentz factor is calculated as
$\Gamma=1/\sqrt{1-\beta^{2}}$. Here $\beta$ is the speed of the jet normalized
by the speed of light. The Lorentz factor plays an important role as it
defines the region of the jet from which the observer receives radiation from,
namely that region is a size that is $\sim 1/\Gamma$ Rybicki and Lightman
(1979).

Another quantity of interest is the half-opening angle of the jet,
$\theta_{\mathrm{j}}$. This quantity can help place constraints on the size
and the launching mechanism of GRB jets. This quantity is difficult to measure
directly but it can be done indirectly by measuring the break time in the
light curve of GRB afterglows Rhoads (1999); Goldstein et al. (2016) and
assuming that the medium that the jet is colliding into is either homogeneous
Piran et al. (1999) or wind-like Chevalier and Li (2000). With
$\theta_{\mathrm{j}}$, we can now calculate the actual energy emitted by a
given GRB without the assumption that it emits isotropically. The non-
isotropic emitted energy is calculated as

$E_{\gamma}=(1-\cos\theta_{\mathrm{j}})E_{\rm iso}$ (5)

GRB jets have traditionally been modelled as a top hat jet, where the Lorentz
factor is a step function out to some $\theta_{\mathrm{j}}$ (Lazzati et al.,
2004; Farinelli et al., 2021). However, theoretical models have advanced to
considering jets with “structure”. This structure is typically a jet with a
fast moving core and a slower moving cocoon Lazzati et al. (2018); Lazzati and
Perna (2019); Salafia et al. (2016); Lundman et al. (2018).

The dynamics of the GRB jet can be understood under the simplistic fireball
model Cavallo and Rees (1978). In this model energy is instantaneously
released into a shell of high temperature plasma. The energy gets converted
into photons at some distance close to the central engine, $r_{0}$. These
photons then accelerate the material in the outflow to relativistic
velocities. The ratio of the released energy to the rest mass energy of the
material in the shell ($\eta=E/Mc^{2}$) is an important factor in the dynamics
of the fireball as it represents the “baryon loading” of the fireball444As the
denominator increases, more baryons are included in the fireball which means
that more energy is necessary to accelerate those baryons to a given bulk
Lorentz factor. This bulk Lorentz factor is equivalent to $\eta$..
Dynamically, this parameter describes different ways that fireballs can be
radiation versus matter dominated. In the case of GRBs, we expect fireballs
that are balanced between the two; the fireball should transfer its initial
radiative energy to accelerating the baryons trapped in the outflow and then,
when the matter is coasting at a relativistic speed, it should become
optically thin to the radiation (Chhotray and Lazzati, 2018). The baryon
loading also defines different regimes in the evolution of a fireball, such as
the radius where the fireball will stop accelerating and coast with a maximal
bulk Lorentz factor equivalent to $\eta$ (see e.g. Pe’Er, 2015, for a
comprehensive overview). Thus, under the interpretation of GRB jets as
fireballs, we are interested in placing constraints on $\eta$ (which is
equivalent to constraining $\Gamma$ and the dynamics of the jet) and $r_{0}$.

There are many other physical quantities of interest in the jet, such as its
composition (see e.g. Beloborodov (2010)), but these other quantities are not
able to be probed using current observed correlations of GRB prompt emission.

##  3 Prompt Observed Correlations

###  3.1 Luminosity-Time Scale Relations

The peak isotropic luminosity-lag relation
($L_{\mathrm{pk}}-\tau_{\mathrm{lag}}$) was first explored by Norris et al.
(2000) who showed that low luminosity GRBs exhibit longer time lags than high
luminosity bursts, meaning that the flux of low energy photons peak later than
the flux of high energy photons for low luminosity bursts. This relation was
found to be

$\frac{L_{\mathrm{pk}}}{10^{53}}\approx
1.3\left(\frac{\tau_{\mathrm{lag}}}{0.01\mathrm{~{}s}}\right)^{-1.14}~{}{\rm
erg~{}s}^{-1}$ (6)

which is in agreement with an independent analysis conducted by Schaefer et
al. (2001). Both analyses calculated the lag between BATSE channels 1 and 3
which correspond to energy ranges of 25-50 keV and 100-300 keV, respectively.
Each analysis also calculated $L_{\rm pk}$ within 50-300 keV and the analysis
conducted by Schaefer et al. (2001) uses the peak flux of the light curve
binned into 256 ms time bins to calculate $L_{\rm pk}$555This analysis has
been expanded more recently to understand how the spectral lag changes as a
function of energy instead of just between 2 instrumental channels (see e.g.
Shao et al. (2017)).. This relationship was analyzed by Ukwatta et al. (2010)
using similar procedures for Swift BAT detected GRBs. They found similar
correlations between $L_{\mathrm{pk}}$ and $\tau_{\mathrm{lag}}$ which was
calculated using various combinations of instrument channels. For each
combination of Swift BAT channels used to calculate the lag, Ukwatta et al.
(2010) provide best fit lines. A similar analysis was done by Kawakubo et al.
(2015) more recently to expand the sample of analyzed GRBs. The relationship
that they obtain,

$L_{\text{pk }}=2.88^{+0.83}_{-0.65}\times
10^{54}\left(\frac{\tilde{\tau}_{\mathrm{lag}}(1+z)^{-1}}{1\mathrm{~{}s}}\right)^{-1.05\pm
0.03}~{}{\rm erg~{}s}^{-1}$ (7)

still has significant scatter. They calculate their $L_{\mathrm{pk}}$ by
considering the 1 s peak photon flux in the 15–150 keV band and they calculate
$\tau_{\mathrm{lag}}$ between energy bands of 15-25 keV and 50-100 keV.

After, Ukwatta et al. (2012) expanded on the study done by Ukwatta et al.
(2010) to provide constraints on the $L_{\rm pk}-\tau_{\mathrm{lag}}$ relation
in the rest frame of GRBs. By considering the redshift of each GRB in their
sample, they converted the observer channels to rest frame energies of 100–150
and 200–250 keV which were then used to calculate the lag. The
$L_{\mathrm{pk}}$ was calculated by considering the 1 s peak photon flux in
the 15–150 keV band. The relationship that Ukwatta et al. (2012) obtained is:

$L_{\text{pk }}=5.01^{+7.58}_{-3.02}\times
10^{54}\left(\frac{\tilde{\tau}_{\mathrm{lag}}(1+z)^{-1}}{0.001\mathrm{~{}s}}\right)^{-1.2\pm
0.2}~{}{\rm erg~{}s}^{-1}$ (8)

which is shown in Figure 1 as the solid black line and the estimated 1$\sigma$
confidence level as the dashed lines.

Figure 1: The $L_{\rm pk}-\tau_{\mathrm{lag}}$ taken from Ukwatta et al.
(2012). The relationship given in Equation 8, is shown as the solid black
line. The 1$\sigma$ confidence level is shown as the dashed lines and the GRBs
used in their study are shown as the red markers with $1\sigma$ errors. The
correlation coefficient for this relationship and the slope of the
relationship are provided. Credit: Ukwatta et al., The lag–luminosity relation
in the GRB source frame: an investigation with Swift BAT bursts. Monthly
Notices of the Royal Astronomical Society 2012, 419, 614–623. By permission of
Oxford University Press on behalf of MNRAS. ©OUP

There is an analogous relationship between the peak isotropic luminosity,
$L_{\mathrm{pk}}$, and the rise time, $\tau_{\mathrm{RT}}$, for the detected
GRB to reach its maximum flux, as shown by Schaefer (2007). This relation is
given as

$L_{\mathrm{pk}}=3.47^{+0.51}_{-0.45}\times
10^{52}\left(\frac{\tilde{\tau}_{\mathrm{RT}}(1+z)^{-1}}{0.1\mathrm{~{}s}}\right)^{-1.21\pm
0.06}~{}{\rm erg~{}s}^{-1}$ (9)

based on 69 BATSE and Swift detected GRBs that were analyzed. The
$L_{\mathrm{pk}}$ is calculated by integrating over the fitted spectrum of
detected photons normalized by the integrated spectrum in the energy range
where the fitting was performed. $\tau_{\mathrm{RT}}$ is calculated by taking
the shortest time over which the GRB light curve rises by 50% of the peak
flux, which involves smoothing the light curve.

A slightly different relationship was also obtained by Xiao and Schaefer
(2009) by analyzing 107 GRBs. The $L_{\mathrm{pk}}-\tau_{\mathrm{RT}}$
relation that they obtain is

$L_{\mathrm{pk}}=6.92^{+0.67}_{-0.61}\times
10^{52}\left(\frac{\tilde{\tau}_{\mathrm{RT}}(1+z)^{-1}}{0.1\mathrm{~{}s}}\right)^{-1.70\pm
0.05}~{}{\rm erg~{}s}^{-1}$ (10)

where the intercept and the slope of the relations are slightly different that
what was found by Schaefer (2007). Xiao and Schaefer (2009) also revisited the
$L_{\mathrm{pk}}-\tau_{\mathrm{lag}}$ relation and found an updated fit of

$L_{\mathrm{pk}}=1.62^{+0.16}_{-0.14}\times
10^{52}\left(\frac{\tilde{\tau}_{\mathrm{lag}}(1+z)^{-1}}{0.1\mathrm{~{}s}}\right)^{-0.98\pm
0.03}~{}{\rm erg~{}s}^{-1}$ (11)

The $L_{\mathrm{pk}}-\tau_{\mathrm{RT}}$ relationship is closely tied to the
relationship between $L_{\mathrm{pk}}$ and light curve variability, $V$, since
the most important factor in determining light curve variability is the rise
time of the light curve Schaefer (2007). This $L_{\mathrm{pk}}-V$ relation was
first introduced by Reichart et al. (2001) who analyzed a sample of 20 GRBs.
They obtained the general scaling

$L_{\mathrm{pk}}\sim V^{3.3^{+1.1}_{-0.9}}$ (12)

Where the peak isotropic luminosity is measured from a 1 s binned light curve
constructed from photons with energies between 50-300 keV. The variability is
calculated by considering the difference between the detected light curve and
a smoothed light curve.

Fenimore and Ramirez-Ruiz (2000) developed a similar relation from their
sample of 220 BATSE GRBs. The relation that they obtain is

$\frac{L_{\mathrm{pk}}}{d\Omega}=3.1\times 10^{56}V^{3.35}~{}{\rm
erg~{}s}^{-1}~{}{\rm sr}^{-1}$ (13)

The variability is calculated in a similar methodology as Reichart et al.
(2001). The luminosity that Fenimore and Ramirez-Ruiz (2000) use here is
calculated from the observed peak photon flux averaged over 256 ms between the
50-300 keV band. Additionally, the relation was constructed such that it can
easily be modified if GRBs were collimated and not isotropic (which was not
well established then), which is why the solid angle factor is explicitly
placed in the relation.

There are many other analyses of the $L_{\mathrm{pk}}-V$ relation, with
differing sample sizes and smoothing methods (see e.g. Guidorzi et al. (2005);
Reichart and Nysewander (2005)). These studies have all recovered a
relationship between $L_{\mathrm{pk}}$ and $V$ however the difference in the
slope of each derived relation is relatively large when compared to other
values. For example, Guidorzi et al. (2006) used two methods to obtain the
slope of the $L_{\mathrm{pk}}-V$ relation and found slopes of
$3.5^{+0.6}_{-0.4}$ and $0.88^{+0.12}_{-0.13}$, one of which is consistent
with $3.3^{+1.1}_{-0.9}$ Reichart et al. (2001) while the other is very
different, and in line with the slope obtained by Guidorzi et al. (2005).
These discrepancies in the slope are shown in Figure 2, where various
$L_{\mathrm{pk}}-V$ relations are plotted. Each relation has been obtained
using different methods of calculating the slope of the relationship and thus
show how unconstrained the relationship is.

Figure 2: The $L_{\mathrm{pk}}-V$ relation taken from Guidorzi et al. (2006)
who used a number of different methods to calculate variability. Shown are the
32 GRBs with known redshifts that were analyzed by Guidorzi et al. (2005). The
relationship that they obtained is plotted as the black dashed line. Also
plotted are the $L_{\mathrm{pk}}-V$ relations obtained with two independent
methods of calculating the slope of the relationship. The method outlined by
Reichart and Nysewander (2005) and Reichart et al. (2001) is shown as the
solid line with its $1\sigma$ confidence interval. The Bayesian method
outlined by D’agostini (2003) is shown as the dashed-dotted line with its
$1\sigma$ confidence interval. Credit: Guidorzi, et al., The slope of the
gamma-ray burst variability/peak luminosity correlation. MNRAS, 2006, 371,
843–851. By permission of Oxford University Press on behalf of MNRAS. ©OUP

####  3.1.1 Physical Interpretation

While the $L_{\mathrm{pk}}-V$ relation is not well constrained, its existence
can provide some insight into the outflow structure of GRB jets and the
radiation processes that may take place within them. Under the internal shock
model, the existence of this correlation that is observed can be contributed
to shells of material colliding with one another in the GRB jet. Shells that
are moving faster, with higher $\Gamma$, are more luminous and can produce
highly variable light curves when they collide. This was shown by Ramirez-Ruiz
and Lloyd-Ronning (2002) for a set of physical parameters. Under the
assumptions that Ramirez-Ruiz and Lloyd-Ronning (2002) use, they find that:

  * •

Inverse Compton must dominate over synchrotron radiation in the emission
region of the internal shock model, to produce spectral energies that are in
agreement with highly variable GRBs Lloyd-Ronning and Ramirez-Ruiz (2002), and

  * •

Larger optical depths, $\tau>1$, to Compton scattering in the shells is
necessary to: (1) produce lower energy photons for collisions that occur close
to the central engine and (2) elongate temporal pulses of radiation that is
produced in these collisions.

The $L_{\mathrm{pk}}-V$ relation has also been interpreted under the
photospheric model. Kobayashi et al. (2002) and Mészáros et al. (2002) who
considered the effect of the location of the e± photosphere with respect to
where the emission was being produced in the GRB jet. Their models still
considered shells of material colliding within the jet as the primary source
of radiation but the location of the photosphere changed the properties of the
emitted radiation. If the shocks occurred below the photosphere, then the
resulting emission was more diffuse and the pulse profiles were less
variable666This type of physical setup can be classified as subphotospheric
dissipation events. We denote these as photospheric models for this reason..
If the shocks occurred above the photosphere, then the typical internal shock
behavior is obtained.

Under this general framework, Kobayashi et al. (2002) found the
$L_{\mathrm{pk}}-V$ relation to be a result of a relationship between the jet
opening angle, $\theta_{\mathrm{j}}$, and the variability of the light curve.
This may be expected as the variability provides constraints on the emitting
region of the GRB due to causality. Smaller emitting regions are a result of
fast moving, more luminous jets with higher variability and smaller
photospheric radii while larger emitting regions imply GRB jets with smaller
$\Gamma$, lower luminosity, lower variability, and larger photospheric radii
Kobayashi et al. (2002). By simulating shells of material within GRB jets
colliding with one another, Kobayashi et al. (2002) showed that differences in
$\theta_{\mathrm{j}}$ from one GRB to the next can produce the
$L_{\mathrm{pk}}-V$ relationship.

The $L_{\mathrm{pk}}-V$ relationship under the photospheric model can also be
due to the observer viewing angle changing with respect to the jet axis for
each GRB, if one assumes a universal jet profile Mészáros et al. (2002); Rossi
et al. (2002); Zhang and Meszaros (2002); Salmonson and Galama (2002).

Regardless of the emission process however, Salafia et al. (2016) show that
viewing angle effects are important to consider. The superposition of
radiation from different regions of the jet combine such that a given light
curve can become smoothed out and intrinsic variability is lost Salafia et al.
(2016).

The $L_{\mathrm{pk}}-\tau_{\mathrm{RT}}$ can be viewed as an extension of the
$L_{\mathrm{pk}}-V$ relation. As Schaefer (2007) point out,
$\tau_{\mathrm{RT}}$ can be interpreted as the elapsed time from which the
observer starts to receive radiation from the center of the emission region to
when the entire emission region. (See Salafia et al. (2016) for details on
this process and its relation to the structure of light curves.) As we can
infer from the $L_{\mathrm{pk}}-V$ relation, the size of the emitting region
is dependent on $1/\Gamma$. Conversely, the luminosity of the emitting region
scales proportionally with $\Gamma$. As a result, these two relationships
combine to produce the $L_{\mathrm{pk}}-\tau_{\mathrm{RT}}$ relation Schaefer
(2007).

The $L_{\mathrm{pk}}-\tau_{\mathrm{lag}}$ relation can be viewed as being
closely related to the $L_{\mathrm{pk}}-V$ relation. As Zhang et al. (2009)
highlight, variability should be inversely proportional to the spectral lag,
with large variability jets exhibiting short spectral lags. Similarly to the
$L_{\mathrm{pk}}-\tau_{\mathrm{RT}}$ relation, this is due to a large $\Gamma$
which produces smaller emission regions that the observer receives radiation
from and thus the spectral lag from seeing photons from the center versus the
edge of the region would be minimal Schaefer (2007). This interpretation is
under the internal shock model.

Using the internal shock model, Ioka and Nakamura (2001) showed that the
$L_{\mathrm{pk}}-\tau_{\mathrm{lag}}$ relation can also be caused by changes
in observer viewing angle with respect to the jet axis. They find that bright
GRBs are expected to have large $\Gamma$ and small observer viewing angles,
meaning that the observer is viewing the GRB jet on axis. Correspondingly, the
spectral lag should be small due to the smaller emitting region. Their
findings are also relevant to the $L_{\mathrm{pk}}-V$ relationship.

Uhm and Zhang (2016) used the framework of a single emitting shell that is
producing synchrotron emission to explain the spectral lags. They showed that
in order for this model to explain the data they need to enforce a number of
physical constraints. Namely,

  * •

The radius of the emission region must be located $\gtrsim 10^{14}~{}{\rm cm}$
away from the central engine

  * •

The magnetic field strength within the emitting region must decrease as it
expands

  * •

There must be a burst of rapid bulk acceleration when the emission is produced

  * •

The photon spectrum must have curvature associated with it (meaning it cannot
be a single powerlaw across typical GRB energies)

  * •

The spectral peak must progress from high to low energies as a function of
time

Under the photospheric model, Jiang et al. (2021) point out that a large
portion of the physical constraints identified by Uhm and Zhang (2016) are
fulfilled. In their study, Jiang et al. (2021) show that the photospheric
model can reproduce a general spectral lag evolution simply due to a change in
optical depth to photons of different energies, as the jet expands. Radiative
transfer simulations also show that structured jets can change the appearance
of photospheric radiation and the time of arrival for photons of different
energies Parsotan and Lazzati (2021, 2021).

###  3.2 Spectral Peak Energy-Emitted Energy Relations

Some of the first relationships that were explored related the spectral peak
energy of observed GRBs with their measured energetics – either the flux or
the fluence of the burst.

Mallozzi et al. (1996) found a positive correlation between the average peak
energy of GRBs, $<\tilde{E}_{\mathrm{pk}}>$, and their peak fluxes,
$\tilde{F}_{\mathrm{pk}}$, both defined in the observer frame. They analyzed
GRBs observed with BATSE and fit the observed GRB spectra with a cut-off power
law in order to obtain the spectral peak energy. To obtain the peak fluxes of
the GRBs, Mallozzi et al. (1996) binned the BATSE count rate data into 256 ms
time bins for photons between 50 and 300 keV. The GRBs that were used in the
correlation were selected such that $\tilde{F}_{\mathrm{pk}}>1$ photon s-1
cm-2. Each GRB that survived the cut, was then sorted into five
$\tilde{F}_{\mathrm{pk}}$ bins of varying widths. The bursts were sorted such
that each bin progress from lowest $\tilde{F}_{\mathrm{pk}}$ values to the
largest and the $\tilde{E}_{\mathrm{pk}}$ within each bin was averaged. This
analysis produced the positive correlation between $<\tilde{E}_{\mathrm{pk}}>$
and $\tilde{F}_{\mathrm{pk}}$.

Subsequently, Lloyd et al. (2000) investigated the relationship between the
average spectral peak energy of GRBs, $<\tilde{E}_{\mathrm{pk}}>$, and their
fluence, $\tilde{S}_{\mathrm{tot}}$, both defined in the observer frame for
the same energy range analyzed by Mallozzi et al. (1996). The
$\tilde{E}_{\mathrm{pk}}$ of each burst was measured by fitting a Band
function to the observed spectrum. The relationship that Lloyd et al. (2000)
discovered is:

$\tilde{E}_{\mathrm{pk}}\propto\tilde{S}_{\mathrm{tot}}^{0.29\pm 0.03}$ (14)

Lloyd et al. (2000) also ensured that their results were consistent against
instrumental biases and analyzed GRBs that were not near the detection
threshold of BATSE. They found consistent results for
$\tilde{E}_{\mathrm{pk}}\propto\tilde{S}_{\mathrm{tot}}$.

Additionally, Lloyd et al. (2000) analyzed the
$\tilde{E}_{\mathrm{pk}}-\tilde{F}_{\mathrm{pk}}$ relation for their
populations of GRBs. Lloyd et al. (2000) correlated the flux with the
$\tilde{E}_{\mathrm{pk}}$ measured in the time bin of
$\tilde{F}_{\mathrm{pk}}$ and found similar results to Mallozzi et al. (1996)
for their entire sample of GRBs. However, when Lloyd et al. (2000) analyzed
the $\tilde{E}_{\mathrm{pk}}-\tilde{F}_{\mathrm{pk}}$ relationship for the
brightest GRBs in their sample, they found a weaker relationship compared to
their whole sample.

When analyzing SGRBs, Ghirlanda et al. (2011) found a similar relationship
between $\tilde{E}_{\mathrm{pk}}$ and $\tilde{F}_{\mathrm{pk}}$.

####  3.2.1 Physical Interpretation

These relations between the spectral peak energy and the emitted energy of the
GRB have led to the establishment of the Amati and Ghirlanda relations (see
sections 3.3 and 3.5). These relations, as we will see, relate the intrinsic
radiated energy of the burst to the rest frame spectral peak energy.

Furthermore, the existence of
$\tilde{E}_{\mathrm{pk}}-\tilde{F}_{\mathrm{pk}}$ relations for both LGRBs and
SGRBs implies that the emission mechanism between both types of GRBs is the
same Ghirlanda et al. (2011).

Brainerd (1997) showed that the relationship discovered by Mallozzi et al.
(1996) could not be the result of cosmological expansion of an otherwise
static source. As a result this relationship should be something that is
intrinsic to GRBs. Lloyd et al. (2000) analyze their obtained relationship in
the context of the prompt emission of the GRB jet being due to internal versus
external shocks that produce synchrotron radiation. They conclude that the
$\tilde{E}_{\mathrm{pk}}-\tilde{S}_{\mathrm{tot}}$ is not compatible with the
external shock model and is in alignment with what is expected from the
internal shock model.

###  3.3 Amati ($E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$) and Yonetoku
($E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$) Relations

One of the most studied empirical relations found in the prompt emission is
the positive correlation between the peak energy of the time-integrated
spectrum, $E_{\mathrm{pk}}$, and the isotropic equivalent energy,
$E_{\mathrm{iso}}$, which are defined in the GRB rest frame. The so-called
Amati relation was initially discovered by Amati, L. et al. (2002) based on
the first sample of 12 LGRBs detected by BeppoSAX with redshift measurement (9
with firm redshift estimates and 3 with plausible values). The subsequent
studies (e.g., Amati, 2006; Nava et al., 2012; Amati and Della Valle, 2013;
Tsvetkova et al., 2017) that explored larger number of samples detected by
various instruments have confirmed and extended the correlation (Fig. 3). The
best fit power-law function to the correlation found by Zaninoni et al. (2016)
is

$E_{\mathrm{pk}}=110^{+150}_{-60}\left(\frac{E_{\mathrm{iso}}}{10^{52}~{}{\rm
erg}}\right)^{0.51\pm 0.04}~{}{\rm keV}.$ (15)

The uncertainties in the above formula indicate 2$\sigma$ confidence level. As
seen in Fig.3, the scatter of the correlation is not small ($\sigma\sim 0.2$
dex in $E_{\mathrm{pk}}$). While slightly different slopes are found for the
correlation curve among the literature, they are consistent within the
dispersion.

Figure 3: The $E_{\mathrm{pk}}-E_{\mathrm{iso}}$ diagram of LGRBs taken from
Amati and Della Valle (2013). The solid and dashed lines show the best fit
function and the 2$\sigma$ systematic errors of the Amati relation,
respectively. Credit: Measuring Cosmological Parameters with Gamma Ray Bursts,
Amati and Della Valle, International Journal of Modern Physics D, 22, 14,
©2013 World Scientific

Soon after the discovery of the Amati relation, Yonetoku et al. (2004) showed
that there is a similar correlation between $E_{\mathrm{pk}}$ and the
isotropic equivalent peak luminosity, $L_{\mathrm{pk}}$, using the BeppoSAX
GRBs reported by Amati, L. et al. (2002) with additional samples detected by
BATSE (16 with firm redshift measurement and 3 with plausible values in
total). Intuitively, it is not surprising to find such a correlation since
$E_{\mathrm{pk}}$ is likely to reflect the instantaneous physical condition of
the emission rather than time-integrated property $E_{\mathrm{iso}}$ (see
Section 3.4). Subsequent studies also confirm the correlation with larger
samples (e.g., Yonetoku et al., 2010; Nava et al., 2012; Tsvetkova et al.,
2017) (Fig. 4). Yonetoku et al. (2010) finds a best-fit power-law function to
the correlation given as

$L_{\mathrm{pk}}=2.7^{+8.5}_{-2.1}\times
10^{52}\left(\frac{E_{\mathrm{pk}}}{355~{}{\rm keV}}\right)^{1.60\pm
0.082}~{}{\rm erg~{}s}^{-1},$ (16)

where the uncertainties indicate 2$\sigma$ errors. The correlation seems to be
slightly tighter than the Amati relation but still has non-negligible
dispersion (According to the analysis of Yonetoku et al. (2010), $\sigma=0.33$
dex is found in $L_{\mathrm{pk}}$ in the Yonetoku relation, while
$\sigma=0.37$ dex is found in $E_{\mathrm{iso}}$ in the Amati relation).
Slightly different slopes found in the literature are consistent within the
dispersion.

Figure 4: The $E_{\mathrm{pk}}-L_{\mathrm{pk}}$ diagram of LGRBs taken from
Yonetoku et al. (2010). The solid and dashed lines show the best fit function
and the 3$\sigma$ systematic errors of the Yonetoku relation, respectively.
Data from Yonetoku et al. (2010).

Several authors have pointed out the effect of observational biases that may
artificially generate these correlations (Nakar and Piran, 2005; Band and
Preece, 2005; Butler et al., 2007; Kocevski, 2012). However, careful analyses
suggest that, while these effects may partially contribute, they cannot fully
be responsible for producing the correlation (Bosnjak et al., 2008; Ghirlanda
et al., 2008; Nava et al., 2011, 2012; Heussaff et al., 2013) (see Dainotti
and Amati (2018) for comprehensive review on the possible selection effects
and systematic). Further evidence for the physical origin of the correlations
comes from the fact that a similar correlation holds between $E_{\mathrm{pk}}$
and luminosity, $L_{\mathrm{iso}}$, measured in the time-resolved analysis of
individual bursts (Golenetskii et al., 1983; Firmani et al., 2009; Ohno et
al., 2009; Ghirlanda et al., 2010, 2011; Frontera et al., 2012) (see Section
3.4). Therefore, it is likely that Amati and Yonetoku relations are intrinsic,
while the scatter may be large.

These correlations span a wide range in $E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$
and $E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$ diagrams. They cover not only the
ordinary LGRBs ($E_{\mathrm{pk}}\gtrsim 100~{}{\rm keV}$) but also the X-ray-
rich GRBs (XRRs) and X-ray flashes (XRFs) which are considered to be a
continuous family (soft version) of LGRBs (Sakamoto et al., 2005; Lamb et al.,
2005; Sakamoto et al., 2008). Interestingly, it is also found that at least
some ULGRBs are consistent with these correlations (Gendre et al., 2013; Peng
et al., 2013; Zaninoni et al., 2016).777For example, the analysis by Zaninoni
et al. (2016) showed GRB 111209A, 101225A, and 130925A agree with the Amati
relation. The analysis of GRB121027A by Peng et al. (2013) finds that, while
both the initial hard emission and later soft emissions agree with the
Yonetoku relation, Amati relation is only satisfied for the initial emission.
On the other hand, it should also be noted that outliers to these correlations
exist (Nava et al., 2008)888Outliers are often defined as a population located
more than $\sim 2$-$3\sigma$ away from the correlation curve. Note that
outliers are discarded in Figs. 3 and 4., although the observed number is
subdominant. Notable examples are so-called low-luminosity GRBs (LLGRBs)
D’Elia et al. (2018)999LLGRBs are likely to be an outliers for both relations.
An exception is GRB060218 which shows an agreement with Amati relation. The
event is, however, a clear outlier of Yonetoku relation Yonetoku et al.
(2010)., a sub-class of LGRBs that exhibit substantially lower luminosities.
They do not follow the extrapolation from the ordinary LGRB luminosity
function and are likely to dominate the local GRB population, suggesting a
distinct origin for these events (Virgili et al., 2009; Sun et al., 2015).

Several authors have tested the existence of
$E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$ and $E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$
correlations in SGRBs (Amati, 2006; Ghirlanda et al., 2009; Zhang et al.,
2012; Tsutsui et al., 2013; D’Avanzo et al., 2014; Zaninoni et al., 2016).
These studies suggest similar correlations also hold in SGRBs, although not as
robust as in the case of LGRBs due to the limited number of events with
redshift measurement. According to the analysis of Tsutsui et al. (2013) based
on a sample of 13 SGRB candidates (out of which 5 are considered to be falsely
classified to SGRBs and therefore discarded), the correlations have a
following form:

$E_{\mathrm{iso}}=2.6^{+1.08}_{-7.68}\times
10^{51}\left(\frac{E_{\mathrm{pk}}}{774.5~{}{\rm keV}}\right)^{1.58\pm
0.28}~{}{\rm erg},$ (17) $L_{\mathrm{pk}}=1.95^{+0.32}_{-0.27}\times
10^{52}\left(\frac{E_{\mathrm{pk}}}{774.5~{}{\rm keV}}\right)^{1.59\pm
0.11}~{}{\rm erg~{}s}^{-1}.$ (18)

The estimated dispersions in $E_{\mathrm{iso}}$ and $L_{\mathrm{pk}}$ for the
above correlations are $\sigma=0.39$ and $0.13$ dex, respectively (Fig. 5). As
seen in the figure, SGRBs constitute a track parallel to LGRBs that are dimmer
by a factor of $\sim 100$ ($\sim 5$) in $E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$
($E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$) diagram. While there are somewhat
difference among the literature, independent studies also seem to suggest that
SGRBs lie on a much dimmer (slightly dimmer or consistent) track in the
$E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$ ($E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$)
diagram.

Figure 5: The $E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$ (left) and
$E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$ (right) diagrams of SGRBs taken from
Tsutsui et al. (2013). According to the analysis of Tsutsui et al. (2013), red
circles represent secure SGRBs, while cyan circles are considered as misguided
SGRBs. The best-fitting function and 3$\sigma$ intervals of the Amati
($E_{\mathrm{pk}}$–$E_{\mathrm{iso}}$) and Yonetoku
($E_{\mathrm{pk}}$–$L_{\mathrm{pk}}$) relations for the secure SGRBs are
indicated with red solid and dashed lines, respectively. For comparison,
observational data of LGRBs taken from Yonetoku et al. (2010) are shown by
gray points. Data from Tsutsui et al. (2013) and Yonetoku et al. (2010).

It should be noted that there is an ambiguity in selecting “true” SGRBs only
from their duration, since LGRBs and SGRBs have an overlap in the duration
distribution. Hence, it is essential to remove the possible contamination from
LGRBs with a short duration to robustly constrain the properties of
SGRBs.101010Tsutsui et al. (2013) attempt to discard ”misguided” SGRBs by
removing data that lies more than 3$\sigma$ away from the Amati relation of
SGRBs. For example, recently found peculiar GRB200826A, which showed a rest
frame duration of $\sim 0.5s$, was confirmed to be association with a
supernova (Ahumada et al., 2021). Therefore, the event should be classified to
the population of LGRBs despite its short duration. Indeed, GRB200826A is
found consistent with the Amati and Yonetoku relations of LGRBs Svinkin et al.
(2020). It should also be noted that GRB170817A, associated with the LIGO-
Virgo GW170817 binary neutron star merger event, is a clear outlier of the
$E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$ and $E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$
correlations (Wang et al., 2017; Pozanenko et al., 2018; Zhang et al., 2018).
This indicates that, while GRB170817A provides the firm evidence for SGRBs to
arise from the compact binary merger, its highly off-axis viewing angle
results in a prompt emission property different from that of the ordinary
SGRBs (see e.g., Nakar, 2020).

####  3.3.1 Physical Interpretation

Despite the considerable amount of theoretical works done up to now on the
nature of GRB emissions, the origin of the prompt GRB correlations remains to
be an open issue. The main difficulty comes from the fact that theoretical
models do not have sufficient predictive power. This is particularly severe
for synchrotron models in which the models themselves do not provide firm
predictions on when and how much kinetic energy should be dissipated in the
outflow. Moreover, there is an additional uncertainty on how the dissipated
energy is transferred to the accelerated particles and magnetic fields. All
these details are treated as model parameters, making it difficult to make
quantitative estimations on $E_{\mathrm{pk}}$, $E_{\mathrm{iso}}$, and
$L_{\mathrm{pk}}$. Therefore, to reproduce the observed correlations due to
the diversity in the intrinsic nature of individual GRBs, one must assume that
there is a self-regulation among imposed parameters (Zhang and Mészáros, 2002;
Zhang and Yan, 2011; Mochkovitch and Nava, 2015). To our knowledge, there is
no convincing explanation to satisfy such constraints.

Apart from the uncertainties on the dissipative processes, several studies
have indicated that the viewing angle effect may be playing important role in
shaping the observed correlations. The most simplified case of a top-hat jet
(a uniform outflow with a constant bulk Lorentz factor, $\Gamma$, and
emissivity within a half-opening angle, $\theta_{\mathrm{j}}$) was explored by
Yamazaki et al. (2004). Assuming that the Amati relation is satisfied for the
on-jet emission (i.e., $\theta_{\mathrm{v}}<\theta_{\mathrm{j}}$, where
$\theta_{\mathrm{v}}$ is the viewing angle of the observer), they demonstrated
that the off-jet emission that shows a rapid decline in $E_{\mathrm{pk}}$ and
$E_{\mathrm{iso}}$ with $\theta_{\mathrm{v}}$ also agrees with the relation up
to $\theta_{\mathrm{v}}\sim 2\theta_{\mathrm{j}}$. Note that this viewing
angle dependence is different from the well-known scaling for the point-source
approximation (infinitesimally narrow jet:
$\theta_{\mathrm{j}}\ll\theta_{\mathrm{v}}$), which exhibits
$E_{\mathrm{pk}}\propto\delta(\theta_{\mathrm{v}})$ and
$E_{\mathrm{iso}}\propto\delta(\theta_{\mathrm{v}})^{3}$, and hence
$E_{\mathrm{pk}}\propto E_{\mathrm{iso}}^{1/3}$. Here
$\delta(\theta_{\mathrm{v}})=[\Gamma(1-\beta{\rm
cos}\theta_{\mathrm{v}})]^{-1}$ is the Doppler factor, where $\beta$ is the
velocity normalized by the speed of light. While the point-source
approximation is valid at $\theta_{\mathrm{v}}\gtrsim 2\theta_{\mathrm{j}}$,
the viewing angle dependence in the range
$\theta_{\mathrm{j}}\lesssim\theta_{\mathrm{v}}\lesssim 2\theta_{\mathrm{j}}$
can be approximated as
$E_{\mathrm{pk}}\propto\hat{\delta}(\theta_{\mathrm{v}})$ and
$E_{\mathrm{iso}}\propto\hat{\delta}(\theta_{\mathrm{v}})^{2}$, where
$\hat{\delta}(\theta_{\mathrm{v}})=[\Gamma\\{1-\beta{\rm
cos}(\theta_{\mathrm{v}}-\theta_{\mathrm{j}})\\}]^{-1}$ Ioka and Nakamura
(2018).111111Note that it is not straightforward to apply a similar estimation
to the isotropic luminosity $L_{\rm iso}$ even for the case of uniform top-hat
jet. This is because the luminosity depends not only on the viewing angle but
also on the time evolution of the emissivity. As a result, the effect of
finite half-opening angle leads to a scaling $E_{\mathrm{pk}}\propto
E_{\mathrm{iso}}^{1/2}$ which agrees with the Amati relation. A similar
analysis was performed by Eichler and Levinson (2004) and Levinson and Eichler
(2005) which showed that an annular-shaped jet can also reproduce Amati
relation by the viewing angle effect. Subsequent studies have expanded the
analysis to explore a more complicated jet structure (Toma et al., 2005;
Graziani et al., 2006; Donaghy, 2006; Salafia et al., 2015). These works have
shown that Amati correlation can be interpreted (at least partially) by the
viewing angle effect within the variety of jet models.121212It should be noted
that the formulations discussed here are only applicable only the optically
thin emissions. Therefore, photospheric emission cannot be modeled.

Compared to the optically thin synchrotron models, photospheric models have a
significant advantage in the predictive power since the emission is
(quasi-)thermal. In this model, $E_{\mathrm{pk}}$ ($L_{\rm iso}$) is
stabilized as it is fixed by the temperature (thermal energy) around the
photosphere regardless of the dissipation mechanism. One of the consensuses in
the recent studies is that the subphotospheric dissipation must take place at
a large distance from the central engine to reproduce the observed
correlations (Rees and Mészáros, 2005; Thompson et al., 2007).131313In the
framework of the original fireball model (Goodman, 1986; Paczynski, 1986), one
naively expects a fireball to be initially formed at the vicinity of GRB
central engine, $r_{0}=10^{6}-10^{7}~{}{\rm cm}$. Hence, in the absence of
subphotospheric dissipation, an adiabatic loss may be severe and reduce the
values of $E_{\mathrm{pk}}$ and $L_{\rm iso}$ from those of the observations
for a plausible baryon loading ($\eta\sim$ a few $100$). Various dissipative
processes have been proposed up to now and explored in detail under the
assumption of simplified spherical geometry (e.g., Pe’er et al., 2006;
Giannios and Spruit, 2007; Beloborodov, 2010; Vurm and Beloborodov, 2016).
These studies have indeed shown that photospheric emission can bring
$E_{\mathrm{pk}}$ and $L_{\rm iso}$ in the range that agrees with the
observations. Some authors (e.g., Giannios, 2012; Fan et al., 2012; Ito et
al., 2013) discuss possible interpretation for the origin of the correlation
without specifying the details of the dissipation process. However, since the
nature of the dissipative process is poorly understood, these arguments are
far from conclusive.

Apart from the 1D models, studies of photospheric emission based on multi-
dimensional hydrodynamical simulations have provided important progress on
this issue. Initial works in 2D (Lazzati et al., 2009, 2011; Mizuta et al.,
2011; Nagakura et al., 2011) have shown that dissipation far above the central
engine, a required condition mentioned above to reproduce typical values of
$E_{\mathrm{pk}}$ and $L_{\rm iso}$, can be achieved by the collimation shock
(see also Gottlieb et al., 2019, for a sophisticated discussion). Soon after,
Lazzati et al. (2013) and López-Cámara et al. (2014) suggested that the
observed correlations can be interpreted as a sequence of viewing angles.
While these simulations have imposed a crude approximation on the emission
process, their claim was verified by the later studies in which radiation
transfer calculations are incorporated into hydrodynamical simulations in 2D
(Lazzati, 2016; Parsotan and Lazzati, 2018; Parsotan et al., 2018; Ito et al.,
2021) and 3D (Ito et al., 2015, 2019).

The $E_{\mathrm{pk}}$-$E_{\mathrm{iso}}$ and
$E_{\mathrm{pk}}$-$L_{\mathrm{pk}}$ correlations found in these simulations
reflect the angular structure of the hydrodynamic jet. At the central part of
the jet up to a certain angle $\theta_{\rm core}$, there is a powerful “core”
in which the power and Lorentz factor are high and close to uniform. Outside
of the core region, an extended “wing” region is formed at the jet–cocoon
interface, in which these quantities rapidly decrease with angle (see e.g.,
Gottlieb et al., 2021, for detail). As a result, for viewing angles that lie
within the core ($\theta_{\mathrm{v}}\lesssim\theta_{\rm core}$), the emission
properties, $E_{\mathrm{pk}}$, $L_{\mathrm{pk}}$, and $E_{\mathrm{iso}}$, do
not vary largely with the viewing angle and exhibit values consistent with
those of the typical GRBs. At larger viewing angles
($\theta_{\mathrm{v}}\gtrsim\theta_{\rm core}$), these quantities show a rapid
decline, reflecting the sharp gradient in the wing region of the jet. This
viewing angle dependence at the wing region reproduces a continuous sequence
of $E_{\mathrm{pk}}$, $L_{\mathrm{pk}}$, and $E_{\mathrm{iso}}$ towards a
dimmer population of GRBs. These results indicate that Amati (Yonetoku)
relation is a robust feature of the photospheric emission arising from a jet
breaking out of a massive star or compact binary merger ejecta. A caveat to
these simulation-based studies is that shock heating is the only dissipative
process playing a role, and the various types of dissipative processes
explored in the 1D models are not considered. Nevertheless, while further
investigation is necessary, these studies provide a natural explanation for
the origin of the observed correlations within a physically motivated setup.

###  3.4 Golenetskii (Hardness-Intensity) Relation

Golenetskii et al. (1983) proposed relating the luminosity (or flux) of a GRB
to the effective temperature of the spectrum in each time bin of the light
curve, thus providing constraints on the emission mechanism as a function of
time. The temperatures of the spectra analyzed by Golenetskii et al. (1983)
were obtained by fitting GRB time resolved spectra with a model consisting of
a powerlaw with an exponential cutoff. That is: $F\propto
E^{-\alpha}\exp(-E/E_{0})$, where $\alpha$ is some powerlaw index and $E_{0}$
is the break energy that can be used to determine the temperature of the GRB
plasma using $E_{0}=kT$. This model was motivated by the physics of thermal
bremsstrahlung radiation being produced by an optically thin emission region
of the jet that radiates at the energy $E_{0}$. In analyzing a number of Konus
observed GRBs, Golenetskii et al. (1983) found a relationship that followed
$L\propto(kT)^{\gamma}$ with $\gamma$ ranging from 1.5-1.7.

While the analysis was criticized by a number of studies (see e.g., Laros et
al., 1985; Norris et al., 1986; Kargatis et al., 1994) the general correlation
was recovered in many subsequent analyses (see e.g., Kargatis et al., 1994;
Borgonovo and Ryde, 2001; Guiriec et al., 2013; Lu et al., 2012; Ghirlanda et
al., 2011, 2011). Ghirlanda et al. (2011) even showed in their analysis of
$\sim 20$ Fermi GRBs that LGRBs and SGRBs follow the same correlation.

Lu et al. (2012) expanded on this finding to construct a universal
relationship between isotropic luminosity, $\mathrm{L}_{\mathrm{iso}}$,
calculated within 1–104 keV, and rest frame time resolved spectral peak energy
$E_{\rm pk}$, which were obtained using Band function fits to the spectra. The
relationship that they obtained is

$\mathrm{L}_{\mathrm{iso}}=1.17^{+2.72}_{-0.82}\times
10^{48}\left(\frac{\mathrm{E}_{\mathrm{pk}}}{1~{}\mathrm{keV}}\right)^{1.61\pm
0.01}~{}{\rm erg~{}s}^{-1}$ (19)

and is shown in Figure 6(a) as the solid black line and it’s $2\sigma$
dispersion as the dotted lines. The obtained relationship was based on Fermi
data of 14 LGRBs and 1 SGRB (GRB090510) with redshift measurements, which are
also shown on the plot. Lu et al. (2012) also point out that their best-fit
relationship has a similar scatter to that of the Yonetoku relationship.

This relation has been expanded upon by Guiriec et al. (2013) who investigate
the Golenetskii relation in observer quantities and in GRB rest-frame
quantities for a small sample consisting of LGRBs and SGRBs. They find that
the time resolved flux-observed spectral peak energy relation seems to be
intrinsic within a given GRB but not among a sample of GRBs. Similar to Lu et
al. (2012), Guiriec et al. (2013) find a relationship between the time
resolved $\mathrm{L}_{\mathrm{iso}}$ and the time resolved rest-frame
$\mathrm{E}_{\mathrm{pk}}$ which they report to be

$\mathrm{L}_{\mathrm{iso}}=(1.59\pm 0.84)\times
10^{50}\left(\frac{\mathrm{E}_{\mathrm{pk}}}{1~{}\mathrm{keV}}\right)^{1.33\pm
0.07}~{}{\rm erg~{}s}^{-1}$ (20)

This relation is shown in Figure 6(b) as the solid black line. The GRBs that
were used to obtain the relationship are shown as different colors and the
similarly colored dashed lines denote the fit to that individual GRB. The
analysis done by Guiriec et al. (2013) uses multispectral component models
with the models denoted in the legend of Figure 6(b). Guiriec et al. (2013)
point out that the multispectral component fits can provide better fits to GRB
spectra than just the Band function and thus help reduce scatter in the
Golenetskii relationship and other correlations presented in this review.

Figure 6: The Golenetskii relations taken from Lu et al. (2012) (panel (a))
and Guiriec et al. (2013) (panel (b)). In panel (a), the best fit given by
Equation 19 is shown as the solid black line and its $2\sigma$ dispersion is
shown as the dotted lines. The 15 GRBs used to obtain the relation are shown
as various marker types and colors. The gray circle markers are the GRBs from
the sample used by Yonetoku et al. (2010) in the location that they would
appear on the Yonetoku relation, thus showing that the dispersion in the
Yonetoku relation is similar to the dispersion in the Golenetskii relation.
Panel (b) shows the bet fit Golenetskii relation given in Equation 20 as the
solid black line. The different color points with errorbars show the location
of various GRBs that were used to determine the best fit Golenetskii relation.
The fits to individual GRBs are shown as dashed lines with identical colors.
The multispectral components that are used to fit each bursts’ spectra are
denoted in the legend, where ‘B’ or ‘Band’ denotes Band, ‘C’ denotes a cutoff
power law, and ‘BB’ denotes a blackbody spectrum. ©AAS. Reproduced with
permission.

The powerlaw slopes for the relationships obtained by Lu et al. (2012) and
Guiriec et al. (2013) are slightly different from one another which shows that
the rest frame correlation is not well constrained. Burgess (2017) used
statistical methods to show that the current formulation of the Golenetskii
relationship is not well constraining, at least in the context of using the
relationship to detemine the redshifts of GRBs. They suggest adding terms to
the relationship that can account for other physical parameters of GRBs such
as the opening angle to make this relationship more universal among all GRBs.

####  3.4.1 Physical Interpretation

The Golenetskii relation Golenetskii et al. (1983) can be viewed as the time
resolved Yonetoku relationship Guiriec et al. (2013), where the relationship
is probing the prompt radiation mechanism in time. This time resolved
relationship places important constraints on GRB models.

The result that both SGRBs and LGRBS seem to follow this same correlation,
implies that the radiation mechanism at play in both of these categories of
GRBs is the same Ghirlanda et al. (2011); Guiriec et al. (2013). This places
another constraint on prompt emission models by requiring the models to be
independent of the progenitor.

Theories predicting and explaining the Golenetskii relation are still in
development Burgess (2017); Ghirlanda et al. (2011) but there are a few
results that have been obtained under each prompt emission model.

Under the internal shock model, Bošnjak and Daigne (2014) found a qualitative
recovery of the Golenetskii relation but the slopes differed from those
obtained from observational studies. Within their model, they find a better
agreement with observations when they allow the fraction of accelerated
electrons (which then cool and radiate photons) to be a function of the shock
Lorentz factor.

On the other hand, the photospheric model has been shown to be successful in
reproducing the Golenetskii relation in LGRBs with little tuning López-Cámara
et al. (2014); Parsotan et al. (2018). These simulations of GRB jets consider
a central engine that is turned on and off periodically. The stellar envelope,
consisting of low energy plasma, is able to contaminate the jet during time
periods when the central engine is turned off. As a result of this
contamination, the average energy of electrons within the jet decreases which
causes the optical depth to increase. As the optical depth increases photons
increasingly scatter with the low energy electrons and gradually lose energy.
Opposing this, portions of the jet that are “turned on” are not contaminated
by the stellar envelope as much and maintain a relatively low optical depth.
The fluid in this portion of the jet is relatively energetic which allows
photons to maintain a large average energy when they scatter with the
energetic electrons. The effect of the optical depth (on the cooling of the
photons) is the main factor at play in these simulations and thus makes the
results of these simulations applicable to situations when an observer is
viewing a GRB far from the jet axis. The concept of the photospheric region,
where the optical depth is different for different regions of the jet, is
important in being able to naturally recover the Golenetskii relation Parsotan
and Lazzati (2018); Parsotan et al. (2018).

###  3.5 Ghirlanda Relation

The Ghirlanda relation relates the source frame collimated corrected energy,
$E_{\gamma}$, released by GRBs to their source frame spectral peak energy,
$E_{\rm pk}$. The collimated corrected energy is calculated from the flux of
the GRB, where the flux is calculated by using the best fit model spectrum to
the time-integrated GRB spectrum and integrating the model spectrum from 1-104
keV. This model spectrum was also used to determine the time-integrated
spectral peak energy of the GRB. The relationship for the 24 GRBs in their
sample that have redshift measurements is found to be

$E_{\rm pk}\sim 480\left(\frac{E_{\gamma}}{10^{51}~{}{\rm
erg}}\right)^{0.7}{~{}\rm keV}$ (21)

which they point out has relatively small logarithmic scatter of $\sim 0.25$
dex. To obtain this result, Ghirlanda et al. (2004) had to make a number of
assumptions including a constant density homogeneous circum-burst medium
(CBM), used to calculate the jet opening angle from the afterglow
measurements, a constant radiative efficiency, and a top-hat jet structure.

The Ghirlanda relation was criticized for its use of these assumptions, with
the greatest critique being based on its assumption of a homogeneous CBM. Nava
et al. (2006) attempted to recover the Ghirlanda relation for a sample of 18
GRBs using two assumptions of the CBM that the afterglow is produced from: one
for a homogeneous medium and another for a wind-like profile with a $r^{-2}$
dependence. The relationship that they find for the homogeneous CBM is:

$E_{\mathrm{pk}}=(279\pm 15)\left(\frac{E_{\gamma}}{2.72\times
10^{50}\mathrm{erg}}\right)^{0.69\pm 0.04}~{}\mathrm{keV}$ (22)

while the relationship that they find under the wind-like medium is:

$E_{\mathrm{pk}}=(300\pm 16)\left(\frac{E_{\gamma}}{2.2\times
10^{50}\mathrm{erg}}\right)^{1.03\pm 0.06}~{}\mathrm{keV}$ (23)

The reduced chi-squared values are 1.13 and 1.4, respectively, showing the
tightness of the two correlations. These relations are shown in Figure 7(a),
under the homogeneous CBM assumption, and in Figure 7(b), under the wind-like
medium assumption.

Figure 7: The Ghirlanda relationship(s) taken from Nava et al. (2006). In the
panel (a) the relationship for the Ghirlanda relationship is shown assuming a
homogeneous circumburst density. The yellow regions denote the 1$\sigma$,
2$\sigma$, and 3$\sigma$ scatter from darkest shading to lightest shading. The
scatter is measured with respect to the best fit line, plotted as the solid
black line (Equation 22), where uncertainties in both quantities were
considered. The dotted line shows a simple regression analysis without
considering the errors of each data point. The inset plot shows the scatter of
the data with respect to the best fit line. The standard deviation of the best
fit gaussian to the scatter distribution is also shown. Panel (b) is identical
except the best fit relation is given by Equation 23, for the assumption of a
wind-like CBM. Credit: Nava et. al., A&A, 450, 471-481, 2006, reproduced with
permission ©ESO.

Campana et al. (2007) then expanded on the study conducted by Nava et al.
(2006) by analyzing a set of GRBs including a number of GRBs observed by Swift
with redshifts and measured afterglow breaks. They found that the relations
are still present, however the dispersion in the relations increase. They
suggested that the increased scatter could be due to the more complex
afterglow behaviors exhibited by Swift afterglows.

Subsequently, Ghirlanda et al. (2007) analyzed a set of 16 GRBs that fulfilled
stringent requirements related to the determination of the jet break time,
which factors into the calculation of the jet opening angle. Among other
things, Ghirlanda et al. (2007) required that the jet breaks be measured from
the optical light curves of the afterglow which prevents unknown physics from
affecting the measurements. The relationship that they obtained under the
homogeneous CBM assumption is:

$E_{\mathrm{pk}}=(302\pm 14)\left(\frac{E_{\gamma}}{4.4\times
10^{50}\mathrm{erg}}\right)^{0.7\pm 0.04}~{}\mathrm{keV}$ (24)

with a reduced chi-squared of 1.08 for 23 degrees of freedom. The relation
under the wind-like medium is:

$E_{\mathrm{pk}}=(334\pm 18)\left(\frac{E_{\gamma}}{2.6\times
10^{50}\mathrm{erg}}\right)^{1.05\pm 0.06}~{}\mathrm{keV}$ (25)

with a reduced chi-squared of 0.89 for 23 degrees of freedom. Ghirlanda et al.
(2007) also highlight that the dispersions about the relationships are found
to be gaussian with relatively small standard deviations ranging from
$0.08-0.1$.

Using a large sample of Swift GRBs, Wang et al. (2018) revisited the Ghirlanda
relation, using stringent selection criteria for the GRBs that they analyzed.
As a result of their analysis they found a Ghirlanda relation similar to
Ghirlanda et al.’s Ghirlanda et al. (2007) homogeneous medium result, however
Wang et al. (2018) find increased scatter in the relation. They attribute this
scatter to the inclusion of GRBs with early time jet breaks in their sample.

####  3.5.1 Physical Interpretation

The Ghirlanda relation, when it was initially proposed, did not have a well
established interpretation as to how a geometric GRB property such as the
collimated corrected energy (which is a function of the opening angle of the
GRB jet) can affect a local emission process in the GRB jet, where the
spectral peak energy is formed Ghirlanda et al. (2004); Liang and Zhang
(2005).

Nava et al. (2006) showed that the Ghirlanda relation is Lorentz invariant,
meaning that it still holds in the comoving frame of the GRB, and thus is
intrinsic to all GRBS. Using this invariance, Nava et al. (2006) highlight
that the number of photons in GRBs should be $\sim 10^{57}$ . This leads to a
relationship where the energy that is injected into the GRB jet and the size
of the injection region stay constant. An issue that arises here is that this
result implies that the radiative efficiency of GRBs is variable, which is
then in conflict with the use of the assumption of constant radiative
efficiency which was necessary to obtain the Ghirlanda relation. This
discrepancy may be resolved by considering that GRBs jets are structured and
that we observe GRB jets at various observer viewing angles.

Ghirlanda et al. (2013) recently showed that the Ghirlanda, Amati, and
Yonetoku relations are natural consequences of the jet opening angle being
inversely related to the bulk Lorentz factor of the outflow. This relationship
was also inferred based on the $L_{\rm pk}-V$ correlation as well and is seen
in simulations of magnetically launched jets Tchekhovskoy et al. (2009);
Komissarov et al. (2010).

To our knowledge, there have been no studies that have attempted to reproduce
or interpret the Ghirlanda relation within the framework of the synchrotron or
photospheric models.

###  3.6 Lorentz Factor ($\Gamma_{0}$) - Prompt Emission ($E_{\mathrm{iso}}$,
$L_{\rm pk}$, and $E_{\mathrm{pk}}$) Relations

In the standard fireball model, the initial light curve of the forward shock
afterglow is predicted to show a rising behavior (or maybe a plateau depending
on the ambient density profile) during which the fireball is coasting with a
constant bulk Lorentz factor, $\Gamma_{0}$ (Rees and Meszaros, 1992; Sari and
Piran, 1999; Kobayashi et al., 1999; Kobayashi and Zhang, 2007). The forward
shock enters the deceleration phase once the energy of the swept-up matter
becomes comparable to the initial energy of the fireball:
$\Gamma_{0}^{2}M_{\rm CBM}c^{2}\sim E_{\mathrm{k}}$, where $M_{\rm CBM}$ and
$E_{\mathrm{k}}$ are the swept-up mass of the CBM and the initial energy of
the fireball141414Assuming the efficiency of the prompt emission, $\epsilon$,
the initial energy of the fireball in the afterglow phase can be estimated
from $E_{\mathrm{iso}}$ as
$E_{\mathrm{k}}=(1-\epsilon)\epsilon^{-1}E_{\mathrm{iso}}$., respectively.
This transition marks a bump in the early afterglow, often termed “the onset
of afterglow”, beyond which the light curve enters a decaying phase. Assuming
such an interpretation, one can deduce the bulk Lorentz factor $\Gamma_{0}$ by
measuring the peak time of early afterglow, $\tilde{t}_{\rm pk}$.
Analytically, for a given energy $E_{\mathrm{k}}$ and a CBM density profile of
$n_{\rm CBM}(r)=n_{0}r^{-s}$, $\Gamma_{0}$ can be expressed as a function of
$\tilde{t}_{\rm pk}$ as

$\Gamma_{0}=K\left(\frac{E_{\mathrm{k}}}{n_{0}m_{\mathrm{p}}c^{5-s}}\right)^{\frac{1}{8-2s}}t_{\rm
pk}^{-\frac{3-s}{8-2s}},$ (26)

where $t_{\rm pk}=\tilde{t}_{\rm pk}/(1+z)$ and $m_{\mathrm{p}}$ are the peak
time in the rest frame and the proton mass, respectively. Here $K$ is a
numerical factor that depends on the power-law index of CBM density profile,
$s$, and the underlying assumptions on the afterglow modeling, which is found
to be within a factor of $\sim 3$ difference among the literature (see
Ghirlanda et al., 2018, for detail). As seen in the above equation, the
dependence of $\Gamma_{0}$ on the parameters, $E_{\mathrm{k}}$ and $n_{0}$, is
quite weak: $\Gamma_{0}\propto(E_{\mathrm{k}}/n_{0})^{1/8}$ in the case of the
homogenous CBM ($s=0$), and $\Gamma_{0}\propto(E_{\mathrm{k}}/n_{0})^{1/4}$ in
the case of the wind CBM ($s=2$).151515In the case of the wind, the afterglow
light curve does not necessarily show a rising phase before the deceleration
(simple model predicts a rather flat behavior $\propto t^{0}$). However, as
discussed in Ghirlanda et al. (2012); Nappo et al. (2014), the initial rising
phase may be realized by considering the pre-acceleration of the CBM. Hence,
the estimation of $\Gamma_{0}$ from $t_{\rm pk}$ is not significantly affected
by uncertainties in the model parameters, making it the more robust than the
other proposed methods (e.g., constraints from the high energy cut-off in the
prompt emission; Fenimore et al., 1993; Woods and Loeb, 1995; Lithwick and
Sari, 2001).161616It should be noted, however, that the method relies on the
correct interpretation of early bump in the afterglow light curve to be
associated with forward shock deceleration. Hence, if the bump has a different
physical origin (e.g., reverse shock; Sari and Piran, 1999; Zhang et al.,
2003), it can lead to an erroneous conclusion. Also, possible ambiguities in
the afterglow modeling (e.g., CBM density profile (Tian et al., 2022),
dynamics of the forward shock deceleration (Fukushima et al., 2017), and/or
jet structure (Racusin et al., 2008)) may also affect the accuracy of the
estimation.

The launch of the Swift satellite, together with the development of efficient
follow-up by the ground-based telescopes, have enabled us to apply this method
to a number of GRB afterglows in NIR, optical, and X-ray bands (Molinari et
al., 2007; Xue et al., 2009; Melandri et al., 2010). After the launch of
Fermi, the detection of an early peak in the GeV light curve, if interpreted
as an afterglow origin, has provided the earliest measurements of $t_{\rm pk}$
(Ghisellini et al., 2010; Kumar and Barniol Duran, 2010; Ghirlanda et al.,
2010). Liang et al. (2010) performed a systematic study based on a sample of
Swift GRBs in which 20 and 12 GRBs showed the onset feature in their optical
and X-ray band, respectively. From the subsample of 17 optically selected GRBs
with firm redshift measurements, they discovered a positive correlation
between the bulk Lorentz factor and the prompt gamma-ray energy in the form
$\Gamma_{0}\propto E_{\mathrm{iso}}^{1/4}$ under the assumption of homogeneous
CBM.

The existence of the $\Gamma_{0}$-$E_{\mathrm{iso}}$ correlation was verified
by later studies, though somewhat difference is found in the best-fit slope
(Lü et al., 2012; Ghirlanda et al., 2012; Liang et al., 2013). Moreover, the
subsequent studies revealed that $L_{\mathrm{pk}}$ and $E_{\mathrm{pk}}$ are
also positively correlated with $\Gamma_{0}$. Recently, Ghirlanda et al.
(2018) analyzed a large sample of GRBs composed of 67 with measured onset time
$t_{\rm pk}$ in the optical or GeV light curves together with 106 with an
upper limit. Of these, 66 with measured $t_{\rm pk}$ plus 85 with upper limits
(and also deduced lower limits based on the assumption of $t_{\rm pk}>t_{90}$,
where $t_{90}$ is the duration of the prompt emission) are used to investigate
the correlations of $\Gamma_{0}$ with $E_{\mathrm{iso}}$, $L_{\mathrm{pk}}$
and $E_{\mathrm{pk}}$.171717The inclusion of the sample with only upper limits
in the analysis is important to avoid slection bias (see Ghirlanda et al.,
2018, for detail). As a result, they derived similar forms for the
$E_{\mathrm{iso}}$-$\Gamma_{0}$ and $L_{\mathrm{pk}}$-$\Gamma_{0}$
correlations:

$\displaystyle E_{\mathrm{iso}}$ $\displaystyle=$
$\displaystyle\begin{cases}0.955^{+0.425}_{-0.294}\times
10^{52}\left(\frac{\Gamma_{0}}{100}\right)^{2.19\pm 0.10}~{}{\rm erg}&{\rm
for}~{}(s=0),\\\ 1.48^{+0.0697}_{-0.0666}\times
10^{51}\left(\frac{\Gamma_{0}}{100}\right)^{2.66\pm 0.05}~{}{\rm erg}&{\rm
for}~{}(s=2),\end{cases}$ (27) $\displaystyle L_{\mathrm{pk}}$
$\displaystyle=$ $\displaystyle\begin{cases}2.14^{+0.317}_{-0.276}\times
10^{51}\left(\frac{\Gamma_{0}}{100}\right)^{2.60\pm 0.16}~{}{\rm
erg~{}s}^{-1}&{\rm for}~{}(s=0),\\\ 5.62^{+0.265}_{-0.253}\times
10^{51}\left(\frac{\Gamma_{0}}{100}\right)^{2.87\pm 0.05}~{}{\rm
erg~{}s}^{-1}&{\rm for}~{}(s=2).\end{cases}$ (28)

Here the upper (lower) part of each equation corresponds to the case of the
homogeneous (wind) CBM. The $L_{\mathrm{pk}}$-$\Gamma_{0}$ correlation (shown
in Fig. 8) is found to be tighter than the $E_{\mathrm{iso}}$-$\Gamma_{0}$
correlation. With a relatively large scatter, they obtained a correlation
between $E_{\mathrm{pk}}$ and $\Gamma_{0}$ expressed as

$\displaystyle
E_{\mathrm{pk}}=\begin{cases}119^{+8.54}_{-7.97}\left(\frac{\Gamma_{0}}{100}\right)^{1.28\pm
0.03}~{}{\rm keV}&{\rm for}~{}(s=0),\\\
352^{+8.21}_{-8.02}\left(\frac{\Gamma_{0}}{100}\right)^{1.43\pm 0.03}~{}{\rm
keV}&{\rm for}~{}(s=2).\end{cases}$ (29)

All correlations show smaller dispersion in the case of wind CBM than in the
case of homogenous CBM.

Figure 8: The $\Gamma_{0}$-$L_{\mathrm{pk}}$ (top) and
$\Gamma_{0}$-$E_{\mathrm{pk}}$ (bottom) correlations taken from Ghirlanda et
al. (2018). The left and right panels display the two cases in which
homogenious and wind profile are assumed for the CBM, respectively. The red
circles (green stars) correspond to the LGRBs with measured $t_{\rm pk}$ in
optical (GeV) band, while the green square corresponds to short GRB090510. The
rightward (leftward) arrows show the deduced values of lower (upper) limits on
$\Gamma_{0}$ for GRBs without detection of $t_{\rm pk}$. Credit: Ghirlanda et.
al., A&A, 609, A112, 2018, reproduced with permission ©ESO.

It is worth noting that, by combining these correlations, they derived
relations $E_{\mathrm{pk}}\propto E_{\mathrm{iso}}^{0.58}$
($E_{\mathrm{pk}}\propto E_{\mathrm{iso}}^{0.53}$) and $E_{\mathrm{pk}}\propto
L_{\mathrm{pk}}^{0.50}$ ($E_{\mathrm{pk}}\propto L_{\mathrm{pk}}^{0.50}$) for
the homogeneous (wind) CBM, which are consistent with the slopes of the Amati
and Yonetoku relations. This result suggests $\Gamma_{0}$ may be a governing
factor that determines the prompt emission properties, providing a possible
hint for the origin of the Amati and Yonetoku relation. Moreover, as mentioned
in section 3.5, $\Gamma_{0}$ may also be related to the origin of the
Ghirlanda relation through the possible anticorrelation between $\Gamma_{0}$
and $\theta_{\rm j}$ (Ghirlanda et al., 2013).

####  3.6.1 Physical Interpretation

The physical origin of the $\Gamma_{0}$-$E_{\mathrm{iso}}$,
$\Gamma_{0}$-$L_{\mathrm{pk}}$, and $\Gamma_{0}$-$E_{\mathrm{pk}}$
correlations remains unclear. As mentioned in section 3.3.1, the lack of
predictive power in many theoretical models prevents us from making a detailed
examination. Some studies attempted to provide possible interpretations
without specifying the details of the emission process. For example, Liang et
al. (2010) suggested that correlations may be related to the angular structure
of energy and Lorentz of the jet. Alternatively, Lü et al. (2012); Lei et al.
(2013) proposed that these correlations may be attributed to the nature of GRB
central engine properties.

In the context of the dissipative photospheric model, Giannios (2012); Fan et
al. (2012) suggested that the correlations can be reproduced within the
reasonable range of model parameters. On the other hand, the study by Lazzati
et al. (2013) demonstrated that the photospheric emission evaluated based on
the hydrodynamical simulations give rise to $\Gamma_{0}$ – $E_{\mathrm{iso}}$
correlation as a result of viewing angle effect. Although quantitative
comparisons are yet to be made, subsequent series of simulation-based studies
(Ito et al., 2015; Lazzati, 2016; Parsotan and Lazzati, 2018; Parsotan et al.,
2018; Ito et al., 2019, 2021) also seem to suggest that, together with the
other prompt correlations (Amati, Yonetoku, and Golenetskii relations; see
sections 3.3.1 and 3.4.1), the correlations between the Lorentz factor
($\Gamma_{0}$) and prompt emission properties ($E_{\mathrm{iso}}$,
$L_{\mathrm{pk}}$, and $E_{\mathrm{pk}}$) are a natural outcome for the
photospheric emission from a hydrodynamic jet breaking out from a massive
envelope.

##  4 Conclusions

We have outlined some of the most prominent prompt emission correlations that
have shed light on the dynamics of GRB jets and the highest energy emission
that is produced in these events. These correlations have probed a number of
physical properties of GRBs such as their energy, timescales for emission, and
spectral peak energies.

The correlations shown here place important constraints on the plausible
emission mechanisms that may produce the prompt emission in GRBs. We have
shown that the correlations can be explained under a number of different
models. However, there still needs to be substantial work in connecting
theoretical models to observational correlations. Conducting global radiative
transfer calculations using GRB jets from hydrodynamic simulations has been a
major step forward in bridging this divide but these types of analysis need to
be conducted on a larger scale to produce predictions that can be compared to
observations for a number of possible jet structures. Additionally, there is a
need for full forward modeling of theoretically produced spectra and light
curves through the responses of various instruments. This is done in other
areas of astrophysics such as galaxy formation research (see e.g. Cochrane et
al., 2019; Parsotan et al., 2021) where these methods have drastically pushed
the field forward.

One important aspect of the observational correlations that we would like to
emphasize is the use of different pieces of information of the GRB phenomenon.
For example, using information from the afterglow can constrain geometric
properties of GRB jets while the variability of the light curves can shed
light on the ‘interior’ structure of shells in the jets. These creative
combinations of parameters have led to significant advances in our
understanding. We expect that the next generation of GRB correlations will
also take into account another highly sought after measurement of GRBs:
polarization – both time resolved and time integrated. With GRB polarization
measurements becoming more precise, the combination of this measurement with
some of the parameters that we have discussed in this review will
significantly advance our understanding of GRB jet geometries and place even
more stringent constraints on the radiation mechanism at play in these events.

The future of GRB research is very bright partially due to their importance in
multi-messenger astronomy and partially due to advanced instrumentation. As
more advanced instrumentation and observational techniques are developed to
measure GRBs, we need to take full advantage of the rich dataset that will be
produced by advancing theoretical predictions of GRB prompt emission. Coupling
global radiative transfer simulations with instrument responses is the primary
way that we will be able to connect future observations to the dynamics and
structure of GRB jets and provide strong predictions that can be tested
against current and future GRB correlations. Multi-messenger detections of
gravitational waves will also push our understanding of GRB prompt emission
forward by providing independent constraints on the structure of GRB jets
(Hayes et al., 2020; Beniamini et al., 2019), GRB jet launching mechanisms
(Zhang, 2019), which can guide prompt emission models, and it will help
constrain the observer viewing angle of SGRB detections that are coincident
with gravitational wave measurements, as in the case of GRB170817A (Goldstein
et al., 2017; Lazzati et al., 2018). Additionally, detections of neutrinos
coincident with GRB prompt emission measurements will place constraints on
dissipation mechanisms that occur in GRB jets and constrain jets that fail to
breakout of the progenitor star, thus providing insight into jet launching
physics (see Kimura, 2022, for a comprehensive review).

writing—original draft preparation, T.P and H.I.; writing—review and editing,
T.P and H.I. All authors have read and agreed to the published version of the
manuscript.

This research received no external funding.

###### Acknowledgements.

The material is based upon work supported by NASA under award number
80GSFC21M0002. H.I. was supported by JSPS KAKENHI grant (No. JP19K03878,
JP19H00693, JP20H04751, and JP17H06362), RIKEN Interdisciplinary Theoretical &
Mathematical Science Program (iTHEMS), and a RIKEN pioneering project
“Evolution of Matter in the Universe (r-EMU)” and “Extreme precisions to
Explore fundamental physics with Exotic particles (E3-Project)”. The authors
declare no conflict of interest. The funders had no role in the writing of the
manuscript or in the decision to publish this review. References yes

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