Gamma Ray Burst Prompt correlations

The mechanism responsible for the prompt emission of gamma-ray bursts (GRBs) is still a debated issue. The prompt phase-related GRB correlations can allow to discriminate among the most plausible theoretical models explaining this emission. We present an overview of the observational two-parameter correlations, their physical interpretations, their use as redshift estimators and possibly as cosmological tools. The nowadays challenge is to make GRBs, the farthest stellar-scaled objects observed (up to redshift $z=9.4$), standard candles through well established and robust correlations. However, GRBs spanning several orders of magnitude in their energetics are far from being standard candles. We describe the advances in the prompt correlation research in the past decades, with particular focus paid to the discoveries in the last 20 years.


Introduction
Gamma-ray bursts (GRBs) are highly energetic events with the total isotropic energy released of the order of 10 48 − 10 55 erg (for recent reviews, see Nakar 2007;Zhang 2011;Gehrels and Razzaque 2013;Berger 2014;Kumar and Zhang 2015;Mészáros and Rees 2015). GRBs were discovered by military satellites Vela in late 1960's and were recognized early to be of extrasolar origin (Klebesadel et al. 1973). A bimodal structure (reported first by Mazets et al. 1981) in the duration distribution of GRBs detected by the Burst and Transient Source Experiment (BATSE) onboard the Compton Gamma-Ray Observatory (CGRO) (Meegan et al. 1992), based on which GRBs are nowadays commonly classified into short (with durations T 90 < 2 s, SGRBs) and long (with T 90 > 2 s, LGRBs), was found . BATSE observations allowed also to confirm the hypothesis of Klebesadel et al. (1973) that GRBs are of extragalatic origin due to isotropic angular distribution in the sky combined with the fact that they exhibited an intensity distribution that deviated strongly from the −3/2 power law (Paczynski 1991a,b;Meegan et al. 1992;Briggs et al. 1996). This was later corroborated by establishing the first redshift measurement, taken for GRB970508, which with 0.835 < z 2.3 was placed at a cosmological distance of at least 2.9 Gpc (Metzger et al. 1997). Despite initial isotropy, SGRBs were shown to be distributed anisotropically on the sky, while LGRBs are distributed isotropically (Balazs et al. 1998;Mészáros et al. 2000a,b;Mészáros andŠtoček 2003;Magliocchetti et al. 2003;Bernui et al. 2008;Vavrek et al. 2008;Tarnopolski 2015a). Cosmological consequences of the anisotropic celestial distribution of SGRBs were discussed lately by  and Mészáros and Rees (2015). Finally, the progenitors of LGRBs are associated with supernovae (SNe) (Hjorth et al. 2003;Malesani et al. 2004;Woosley and Bloom 2006;Sparre et al. 2011;) related with collapse of massive stars. Progenitors of SGRBs are thought to be neutron star-black hole (NS-BH) or NS-NS mergers (Eichler et al. 1989;Paczynski 1991b;Narayan et al. 1992;Nakar and Piran 2005), and no connection between SGRBs and SNe has been proven ).
While the recent first direct detection of gravitational waves (GW), termed GW150914, by the Laser Interferometer Gravitational Wave Observatory (LIGO) (Abbott et al. 2016), interpreted as a merger of two stellar-mass BHs with masses 36 +5 −4 M ⊙ and 29 +4 −4 M ⊙ , is by itself a discovery of prime importance, it becomes especially interesting in light of the finding of  who reported a weak transient source lasting 1 s and detected by Fermi/GBM (Narayana ) only 0.4 s after the GW150914, termed GW150914-GBM. Its false alarm probability is estimated to be 0.0022. The fluence in the energy band 1 keV − 10 MeV is computed to be 1.8 +1.5 −1.0 × 10 49 erg s −1 . While these GW and GRB events are consistent in direction, its connection is tentative due to relatively large uncertainties in their localization. This association is unexpected as SGRBs have been thought to originate from NS-NS or NS-BH mergers. Moreover, neither INTEGRAL (Savchenko et al. 2016), nor Swift (Evans et al. 2016) detected any signals that could be ascribed to a GRB. Even if it turns out that it is only a chance coincidence (Lyutikov 2016), it has already triggered scenarios explaining how a BH-BH merger can become a GRB, e.g. the nascent BH could generate a GRB via accretion of a mass ≃ 10 −5 M ⊙ , indicating its location in a dense medium (see also Loeb 2016), or two high-mass, low-metallicity stars could undergo an SN explosion, and the matter ejected from the last exploding star can form-after some time-an accretion disk producing an SGRB (Perna et al. 2016). Also the possible detection of an afterglow that can be visible many months after the event (Morsony et al. 2016) could shed light on the nature of the GW and SGRB association.
From a phenomenological point of view, a GRB is composed of the prompt emission, which consists of high-energy photons such as γ-rays and hard X-rays, and the afterglow emission, i.e. a long lasting multi-wavelength emission (X-ray, optical, and sometimes also radio), which follows the prompt phase. The first afterglow observation (for GRB970228) was due to the BeppoSAX satellite van Paradijs et al. 1997). Another class, besides SGRBs and LGRBs, i.e. intermediate in duration, was proposed to be present in univariate duration distributions (Horváth 1998(Horváth , 2002Horváth 2009;Řípa et al. 2009), as well as in higher dimensional parameter spaces (Mukherjee et al. 1998;Horváth et al. 2006;Řípa et al. 2009;Koen and Bere 2012). On the other hand, this elusive intermediate class might be a statistical feature that can be explained by modelling the duration distribution with skewed distributions, instead of the commonly applied standard Gaussians (Zitouni et al. 2015;Tarnopolski 2015bTarnopolski , 2016a. Additionally, GRB classification was shown to be detector dependent (Nakar 2007;Bromberg et al. 2013;Tarnopolski 2015c). Moreover, a subclass classification of LGRBs was proposed (Gao et al. 2010), and Norris and Bonnell (2006) discovered the existence of an intermediate class or SGRBs with extended emission, that show mixed properties between SGRBs and LGRBs. GRBs with very long durations (ultra-long GRBs, with T 90 > 1000 s) are statistically different than regular (i.e., with T 90 < 500 s) LGRBs (Boër et al. 2015), and hence might form a different class (see also Virgili et al. 2013;Levan 2015). Another relevant classification appears related to the spectral features distinguishing normal GRBs from X-ray flashes (XRFs). The XRFs ) are extragalactic transient X-ray sources with spatial distribution, spectral and temporal characteristics similar to LGRBs. The remarkable property that distinguishes XRFs from GRBs is that their νF ν prompt emission spectrum peaks at energies which are observed to be typically an order of magnitude lower than the observed peak energies of GRBs. XRFs are empirically defined by a greater fluence (time integrated flux) in the X-ray band (2 − 30 keV) than in the • T 45 is the time spanned by the brightest 45% of the total counts detected above background (Reichart et al. 2001).
• T peak is the time at which the pulse (i.e., a sharp rise and a slower, smooth decay (Fishman et al. 1994;Norris et al. 1996;Stern and Svensson 1996;Ryde and Svensson 2002)) in the prompt light curve peaks (see Fig. 1). • T break is the time of a power law break in the afterglow light curve , i.e. the time when the afterglow brightness has a power law decline that suddenly steepens due to the slowing down of the jet until the relativistic beaming angle roughly equals the jet opening angle θ jet (Rhoads 1997) • τ lag and τ RT are the difference of arrival times to the observer of the high energy photons and low energy photons defined between 25 − 50 keV and 100 − 300 keV energy band, and the shortest time over which the light curve increases by 50% of the peak flux of the pulse.
• T p is the end time prompt phase at which the exponential decay switches to a power law, which is usually followed by a shallow decay called the plateau phase, and T a is the time at the end of this plateau phase ).
• T f is the pulse width since the burst trigger at the time T ej of the ejecta.
• E peak , E iso , E γ and E prompt are the peak energy, i.e. the energy at the peak of the νF ν spectrum (Mallozzi et al. 1995), the total isotropic energy emitted during the whole burst (e.g., Amati et al. 2002), the total energy corrected for the beaming factor [the latter two are connected via E γ = (1 − cos θ jet )E iso ], and the isotropic energy emitted in the prompt phase, respectively.
• F peak , F tot are the peak and the total fluxes respectively (Lee and Petrosian 1996).
• L a , L X,p and L f are the luminosities respective to T a , T p (specified in the X-ray band) and T f .
• L is the observed luminosity, and specifically L peak and L iso are the peak luminosity (i.e., the luminosity at the pulse peak, Norris et al. 2000) and the total isotropic luminosity, both in a given energy band. More precisely, L peak is defined as follows: with D L (z, Ω M , Ω Λ ) the luminosity distance given by where Ω M and Ω Λ are the matter and dark energy density parameters, H 0 is the present-day Hubble constant, and z is the redshift. Similarly, L iso is given by • S γ , S obs , S tot indicate the prompt fluence in the whole gamma band (i.e., from a few hundred keV to a few MeV), the observed fluence in the range 50 − 300 keV and the total fluence in the 20 keV − 1.5 MeV energy band.
• V is the variability of the GRB's light curve. It is computed by taking the difference between the observed light curve and its smoothed version, squaring this difference, summing these squared differences over time intervals, and appropriately normalizing the resulting sum (Reichart et al. 2001). Different smoothing filters may be applied (see also Li and Paczyński 2006 for a different approach). V f denotes the variability for a certain fraction of the smoothing timescale in the light curve.
Most of the quantities described above are given in the observer frame, except for E iso , E prompt , L peak and L iso , which are already defined in the rest frame. With the upper index " * " we explicitly denote the observables in the GRB rest frame. The rest frame times are the observed times divided by the cosmic time expansion, for example the rest frame time in the prompt phase is denoted with T * p = T p / (1 + z). The energetics are transformed differently, e.g. E * peak = E peak (1 + z). The Band function (Band et al. 1993) is a commonly applied phenomenological spectral profile, such that where A norm is the normalization. Here, α and β are the low-and high-energy indices of the Band function, respectively. N E (E) is in units of photons cm −2 s −1 keV −1 . For the cases β < −2 and α > −2, the E peak can be derived as E peak = (2 + α)E 0 , which corresponds to the energy at the maximum flux in the νF ν spectra (Band et al. 1993;Yonetoku et al. 2004).
The Pearson correlation coefficient (Kendall and Stuart 1973;Bevington and Robinson 2003) is denoted with r, the Spearman correlation coefficient (Spearman 1904) with ρ, and the p-value (a probability that a correlation is drawn by chance) is denoted with P .
Finally, we mostly deal with correlations of the form y = ax + b. However, when the intercept b is neglected in the text, but its value is non-negligible (or not known due to lacking in the original paper), we use the notation y ∼ ax to emphasize the slope.

The Prompt Correlations
Several physical relations between relevant quantities in GRBs were found since the 1990's. In each paragraph below we follow the discovery of the correlation with the definition of the quantities, the discussions presented in literature and their physical interpretation.
3.1. The L peak − τ lag correlation 3.1.1. Literature overview Liang and Kargatis (1996), using 34 bright GRBs detected by BATSE, found that E peak depends linearly on the previous flux emitted by the pulse, i.e. that the rate of change of E peak is proportional to the instantaneous luminosity. Quantitatively, where N is a normalization constant expressing the luminosity for each pulse within a burst, and L peak was calculated from the observed flux via Eq. (1).
The L peak − τ lag correlation was introduced for the first time by Norris et al. (2000) who examined a sample of 174 GRBs detected by BATSE, among which 6 GRBs had an established redshift and those were used to find an anticorrelation between L peak and τ lag in the form of (see the left panel of Fig. 2) log L peak = 55.11 − 1.14 log τ * lag , with L peak , in units of 10 53 erg s −1 , computed in the 50 − 300 keV range, and τ * lag is measured in seconds. A remarkably consistent relation was found by Schaefer et al. (2001), who used a sample of 112 BATSE GRBs and reported that log L peak = 52.46 − (1.14 ± 0.20) log τ lag , being in perfect agreement with the result of Norris et al. (2000). Here, L peak is in units of 10 51 erg s −1 , and τ lag in seconds. This relation has been confirmed by several studies (e.g. Salmonson 2000; Daigne and Mochkovitch 2003;. Schaefer (2004) showed that the L peak −τ lag relation is a consequence of the Liang and Kargatis (1996) empirical relation from Eq. (5), and he derived this dependence to be of the form log L peak ∼ − log τ lag . This correlation was useful in the investigation of Kocevski and Liang (2003), who used a sample of 19 BATSE GRBs and the L peak −τ lag relation from (Schaefer et al. 2001) to infer their pseudo-redshifts. Their approach was to vary the guessed z until it allowed to match the luminosity distance D L measured with the GRB's energy flux and the D L that can be calculated from the guessed redshift within a flat ΛCDM model, until the agreement among the two converged to within 10 −3 . Next, the rate of E peak decay, as in (Liang and Kargatis 1996), was measured. Finally, Kocevski and Liang (2003) showed that the L peak is directly related to the GRB's spectral evolution. However, Hakkila et al. (2008) found a different slope, −0.62 ± 0.04, and argued that the L peak − τ lag relation is a pulse rather than a burst property, i.e. each pulse is characterized by its own τ lag , distinct for various pulses within a GRB. Tsutsui et al. (2008), using pseudo-redshifts estimated via the Yonetoku relation (see Sect. 3.6.2) for 565 BATSE GRBs, found that the L peak − τ lag relation has a ρ of only 0.38 (see the right panel of Fig. 2). However, assuming that the luminosity is a function of both the redshift and the lag, a new redshift-dependent L peak − τ lag relation was found as log L peak = 50.88 + 2.53 log(1 + z) − 0.282 log τ lag , with L peak in units of 10 50 erg s −1 , τ lag in seconds, ρ = 0.77 and P = 7.9×10 −75 . Although the spectral lag is computed from two channels of BATSE, this new L peak − τ lag relation suggests lag distribution for six GRBs with measured redshifts. The dashed line represents the power law fit to the lag times for ranges consisting of count rates larger than 0.1 × peak intensity (squares), yielding log L peak ∼ −1.14 log(τ * lag /0.01 s). The lag time is computed using channel 1 (25 − 50 keV) and channel 3 (100 − 300 keV) of the BATSE instrument. ( that a future lag-luminosity relation defined within the Swift data should also depend on the redshift. Afterwards, Sultana et al. (2012) presented a relation between the z-and k-corrected τ lag for the Swift energy bands 50−100 keV and 100−200 keV, and L peak , for a subset of 12 Swift long GRBs. The z-correction takes into account the time dilatation effect by multiplying the observed lag by (1 + z) −1 to translate it into the rest frame. The k-correction takes into account a similar effect caused by energy bands being different in the observer and rest frames via multiplication by (1 + z) 0.33 . The net corrected τ * lag is thence (1 + z) −0.67 τ lag . In addition, Sultana et al. (2012) demonstrated that this correlation in the prompt phase can be extrapolated into the L a − T * a relation (Dainotti et al. 2008(Dainotti et al. , 2010(Dainotti et al. , 2011a(Dainotti et al. , 2013. Sultana et al. (2012) found 1 : and log L a = (51.57 ± 0.10) − (1.10 ± 0.03) log T * a , with τ lag in ms, T * a in seconds, and L in erg s −1 . The correlation coefficient is significant for these two relations (ρ = −0.65 for the L peak − τ lag and ρ = −0.88 for the L a − T * a relations) and have surprisingly similar best-fit power law indices (−1.19 ± 0.17 and −1.10 ± 0.03, respectively). Although τ lag and T * a represent different GRB time variables, it appears distinctly that the L peak − τ lag relation extrapolates into L a − T * a for timescales τ lag ≃ T * a . A discussion and comparison of this extrapolation with the L f − T f relation is extensively presented in (Dainotti et al. 2015). Ukwatta et al. (2010) confirmed that there is a correlation between L * peak and the z-and k-corrected τ lag among 31 GRBs observed by Swift, with r = −0.68, P = 7 × 10 −2 and the slope equal to −1.4 ± 0.3, hence confirming the L peak − τ lag relation, although with a large scatter. This was followed by another confirmation of this correlation (Ukwatta et al. 2012) with the use of 43 Swift GRBs with known redshift, which yielded r = −0.82, P = 5.5×10 −5 , and a slope of −1.2 ± 0.2, being consistent with the previous results.
Finally, Margutti et al. (2010) established that the X-ray flares obey the same L peak −τ * lag relation (in the rest-frame energy band 0.3 − 10 keV) as GRBs, and proposed that their underlying mechanism is similar.

Physical interpretation of the L peak − τ lag relation
The physical assumption on which the work by Norris et al. (2000) was based is that the initial mechanism for the energy formation affects the development of the pulse much more than dissipation. From the study of several pulses in bright, long BATSE GRBs, it was claimed that for pulses with precisely defined shape, the rise-to-decay ratio is ≤ 1. In addition, when the ratio diminishes, pulses show a tendency to be broader and weaker. Salmonson (2000) proposed that the L peak −τ lag relation arises from an entirely kinematic effect. In this scenario, an emitting region with constant (among the bursts) luminosity is the source of the GRB's radiation. He also claimed that variations in the line-of-sight velocity should affect the observed luminosity proportionally to the Lorentz factor of the jet's expansion, Γ = [1 − (v/c) 2 ] −1/2 (where v is the relative velocity between the inertial reference frames and c is the speed of light), while the apparent τ lag is proportional to 1/Γ. The variations in the velocity among the line-of-sight is a result of the jet's expansion velocity combined with the cosmological expansion. The differences of luminosity and lag between different bursts are due to the different velocities of the individual emitting regions. In this case, the luminosity is expected to be proportional to 1/τ lag , which is consistent with the observed relation. This explanation, however, requires the comoving luminosity to be nearly constant among the bursts, which is a very strong condition to be fulfilled. Moreover, this scenario has several other problems (as pointed out by Schaefer 2004): 1. it requires the Lorentz factor and luminosity to have the same range of variation. However, the observed L peak span more than three orders of magnitude (e.g., Schaefer et al. 2001), while the Lorentz factors span less than one order of magnitude (i.e., a factor of 5) (Panaitescu and Kumar 2002); 2. it follows that the observed luminosity should be linearly dependent on the jet's Lorentz factor, yet this claim is not justified. In fact, a number of corrections is to be taken into account, leading to a significantly nonlinear dependence. The forward motion of the jet introduces by itself an additional quadratic dependence (Fenimore et al. 1996). Ioka and Nakamura (2001) proposed another interpretation for the L peak − τ lag correlation. From their analysis, a model in which the peak luminosity depends on the viewing angle is elaborated: the viewing angle is the off-axis angular position from which the observer examines the emission. Indeed, it is found that a high-luminosity peak in GRBs with brief spectral lag is due to an emitted jet with a smaller viewing angle than a fainter peak with extended lag. It is also claimed that the viewing angle can have implications on other correlations, such as the luminosity-variability relation presented in Sect. 3.2. As an additional result from this study, it was pointed out that XRFs can be seen as GRBs detected from large angles with high spectral lag and small variability.
On the other hand, regarding the jet angle distributions, Liang et al. (2008) found an anticorrelation between the jet opening angle and the isotropic kinetic energy among 179 X-ray GRB light curves and the afterglow data of 57 GRBs. Assuming that the GRB rate follows the star formation rate, and after a careful consideration of selection effects, Lü et al. (2012b) found in a sample of 77 GRBs an anticorrelation between the jet opening angle θ jet and the redshift in the form with ρ = 0.55 and P < 10 −4 . Using a mock sample and bootstrap technique, they showed that the observed θ jet − z relation is most likely due to instrumental selection effects. Moreover, they argued that other types of relation, e.g. τ lag − z (Yi et al. 2008) or the redshift dependence of the shallow decays in X-ray afterglows Stratta et al. (2009), while might have connections with the jet geometry, are also likely to stem from observational biases or sample selection effects. Also, Ryan et al. (2015) investigated the jet opening angle properties using a sample of 226 Swift/XRT GRBs with known redshift. They found that most of the observed afterglows were observed off-axis, hence the expected behaviour of the afterglow light curves can be significantly affected by the viewing angle.  argued, on the basis of the kinematic model, that the origin of the L peak − τ lag relation is due to a more intrinsic L peak − V relation (see Sect. 3.2). They also gave an interpretation of the latter relation within the internal shock model (see Sect. 3.2.2). Recently, Uhm and Zhang (2016) constructed a model based on the synchrotron radiation mechanism that explains the physical origin of the spectral lags and is consistent with observations.
Another explanation for the origin of the L peak − τ lag relation, given by Sultana et al. (2012), involves only kinematic effects. In this case, L peak and τ lag depend on the quantity: depicting the Doppler factor of a jet at a viewing angle θ and with velocity β 0 ≡ v/c at redshift z. In this study there is no reference to the masses and forces involved and, as a consequence of the Doppler effect, the factor D associates the GRB rest frame timescale τ with the observed time t in the following way: Therefore, considering a decay timescale ∆τ in the GRB rest frame, Eq. (13) in the observer frame will give ∆t = ∆τ /D. Furthermore, taking into account a spectrum given by Φ(E) ∝ E −α , the peak luminosity (as already pointed out by Salmonson 2000) can be computed as with α ≈ 1. In such a way, Eqs. (13) and (14) allow to retrieve the observed L peak − τ lag relation. Finally, the analogous correlation coefficients and best-fit slopes of the L peak − τ lag and L a − T * a correlations obtained by Sultana et al. (2012) seem to hint toward a similar origin for these two relations.

The L peak − V correlation
The first correlation between L peak and V was discovered by Fenimore and Ramirez-Ruiz (2000), and was given as log L peak = 56.49 + 3.35 log V, with L peak measured in erg s −1 . Here, the luminosity is per steradian in a specified (rest frame) energy bandpass (50 − 300 keV), averaged over 256 ms. First, seven BATSE GRBs with a measured redshift were used to calibrate the L peak − V relation. Next, the obtained relationship was applied to 220 bright BATSE GRBs in order to obtain the luminosities and distances, and to infer that the GRB formation rate scales as (1 + z) 3.3±0.3 . Finally, the authors emphasized the need of confirmation of the proposed L peak − V relation. Reichart et al. (2001) used a total of 20 GRBs observed by CGRO/BATSE (13 bursts), the KONUS/Wind (5 bursts), the Ulysses/GRB (1 burst), and the NEAR/XGRS (1 burst), finding:

Literature overview
with ρ = 0.8 and P = 1.4 × 10 −4 (see the left panel of Fig. 3); L peak was computed in the 50 − 300 keV observer-frame energy band, which corresponds roughly to the range 100 − 1000 keV in the rest frame for z ≃ 1 − 2, typical for GRBs in the sample examined. The distribution of the sample's bursts in the log L peak − log V f plane appears to be well modeled by the following parameterization: where b = 0.013 +0.075 −0.092 is the intercept of the line, m = 0.302 +0.112 −0.075 is its slope, andV f andL peak are the median values of V f and L peak for the bursts in the sample for which spectroscopic redshifts, peak fluxes, and 64-ms or better resolution light curves are available. Later,  updated the sample to 32 GRBs detected by different satellites, i.e. BeppoSAX, CGRO/BATSE, HETE-2 and KONUS (see the middle panel of Fig. 3). The existence of a correlation was confirmed, but they found a dramatically different relationship with respect to the original one: with ρ = 0.625 and P = 10 −4 , and L peak in units of 10 50 erg s −1 .
However, Reichart and Nysewander (2005) using the same sample claimed that this result was the outcome of an improper statistical methodology, and confirmed the previous work of Reichart et al. (2001). Indeed, they showed that the difference among their results and the ones from  was due to the fact that the variance of the sample in the fit in  was not taken into account. They used an updated data set, finding that the fit was well described by the slope m = 3.4 +0.9 −0.6 , with a sample variance σ V = 0.2 ± 0.04.
Subsequently, Guidorzi et al. (2006) using a sample of 551 BATSE GRBs with pseudoredshifts derived using the L peak − τ lag relation (Guidorzi 2005), tested the L peak − V correlation (see right panel of Fig. 3). They also calculated the slope of the correlation of the samples using the methods implemented by Reichart et al. (2001) andD'Agostini (2005). The former method provided a value of the slope in the L peak − V correlation consistent with respect to the previous works: Instead, the slope for this sample using the latter method is much lower than the value in (Reichart et al. 2001): The latter slope m is consistent with the results obtained by , but inconsistent with the results derived by Reichart and Nysewander (2005).
Afterwards, Rizzuto et al. (2007) tested this correlation with a sample of 36 LGRBs detected by Swift in the 15−350 keV energy range and known redshifts. The sample consisted of bright GRBs with L peak > 5 × 10 50 erg s −1 within a 100 − 1000 keV energy range. In their study, they adopted two definitions of variability, presented by Reichart et al. (2001), called V R , and by Li and Paczyński (2006), hereafter V LP . V R and V LP differ from each other with a different smoothing filter which, in the second case, selects only high-frequency variability. Finally, Rizzuto et al. (2007) confirmed the correlation and its intrinsic dispersion around the best-fitting power law given by with ρ = 0.758 and P = 0.011, and with σ log L = 0.58 +0.15 −0.12 , ρ = 0.115, and P = 0.506. Six low-luminosity GRBs (i.e., GRB050223, GRB050416A, GRB050803, GRB051016B, GRB060614 and GRB060729), out of a total of 36 in the sample, are outliers of the correlation, showing values of V R higher than expected. Thus, the correlation is not valid for low-luminosity GRBs.
As is visible from this discussion, the scatter in this relation is not negligible, thus making it less reliable than the previously discussed ones. However, investigating the physical explanation of this correlation is worth to be depicted for further developments.

Physical interpretation of the L peak − V relation
We here briefly describe the internal and external shock model (Piran 2004;Mészáros 2006a), in which the GRB is caused by emission from a relativistic, expanding baryonic shell with a Lorentz bulk factor Γ. Let there be a spherical section with an opening angle θ jet . In general, θ jet can be greater than Γ −1 , but the observer can detect radiation coming only from the angular region with size ≃ Γ −1 . An external shock is formed when the expanding shell collides with the external medium. In general, there might be more than one shell, and the internal shock takes place when a faster shell reaches a slower one. In both cases one distinguishes an FS, when the shock propagates into the external shell or the external medium, and a reverse shock (RS), when it propagates into the inner shell. Fenimore and Ramirez-Ruiz (2000) pointed out that the underlying cause of the L peak − V relation is unclear. In the context of the internal shock model, larger initial Γ factors tend to produce more efficient collisions. After changing some quantities such as the Γ factors, the ambient density, and/or the initial mass of the shells, the observed variability values are not recovered. Therefore, the central engine seems to play a relevant role in the explanation for the observed L peak − V correlation. In fact, this correlation was also explored within the context of a model in which the GRB variability is due to a change in the jet-opening angles and narrower jets have faster outflows (Salmonson and Galama 2002). As a result, this model predicts bright luminosities, small pulse lags and large variability as well as an early jet break time for on-axis observed bursts. On the other hand, dimmer luminosities, longer pulse lags, flatter bursts and later jet break times will cause larger viewing angles. Guidorzi et al. (2006) gave an interpretation for the smaller value of the correlation in the context of the jet-emission scenario where a stronger dependence of the Γ of the expanding shells on the jet-opening angle is expected. However, Schaefer (2007) attributed the origin of the L peak − V relation to be based on relativistically shocked jets. Indeed, V and L peak are both functions of Γ, where L peak is proportional to a high power of Γ, as was already demonstrated in the context of the L peak − τ lag relation (see Sect. 3.1.2), and hence fast rise times and short pulse durations imply high variability.
3.3. The L iso − τ RT correlation and its physical interpretation  predicted that τ RT should be connected with L iso in a following manner: with the exponent N ≃ 3 (see also Schaefer , 2007. Therefore, fast rises indicate high luminosities and slow rises low luminosities. The τ RT can be directly connected to the physics of the shocked jet. Indeed, for a sudden collision of a material within a jet (with the shock creating an individual pulse in the GRB light curve), τ RT will be determined as the maximum delay between the arrival time of photons from the center of the visible region versus their arrival time from its edge. The angular opening of the emitted jet, usually associated with Γ, could cause this delay leading to a relation τ RT ∝ Γ −2 . The radius at which the material is shocked affects the proportionality constant, and the minimum radius under which the material cannot radiate efficiently anymore should be the same for each GRB (Panaitescu and Kumar 2002). In addition, a large scatter is expected because it will depend on how near this minimum radius the collisions are observed.
With both τ RT and L iso being functions of Γ, Schaefer (2007) confirmed that log L iso should be ∼ −N/2 log τ RT . From 69 GRBs detected by BATSE and Swift, the following relation was obtained: with L iso in erg s −1 and τ * RT measured in seconds. The 1σ uncertainties in the intercept and slope are σ a = 0.06 and σ b = 0.06 (see the left panel of Fig. 4). The uncertainty in the log of the burst luminosity is where Schaefer (2007) takes into account the extra scatter, σ sys . When σ RT,sys = 0.47, the χ 2 of the best fit line is unity. Xiao and Schaefer (2009) explained in details the procedure of how they calculated the τ RT using 107 GRBs with known spectroscopic redshift observed by BATSE, HETE, KONUS and Swift (see the right panel of Fig. 4), taking into account also the Poissonian noise. Their analysis yielded with the same units as in Eq. (24). As a consequence, the flattening of the light curve before computing the rise time is an important step. The problem is that the flattening should be done carefully, in fact if the light curve is flattened too much, a rise time comparable with the smoothing-time bin is obtained, while if it is flattened not enough, the Poissonian noise dominates the apparent fastest rise time, giving a too small rise time. Therefore, for some of the dimmest bursts, the Poissonian-noise dominant region and the smoothing-effect dominant region can coincide, thus not yielding τ RT values for the weakest bursts. Finally, the physical interpretation of this correlation is given by Schaefer (2007). It is shown that the fastest rise in a light curve is related to the Lorentz factor Γ simply due to the geometrical rise time for a region subtending an angle of 1/Γ, assuming that the minimum radius for which the optical depth of the jet material is of order of unity remains constant. The luminosity of the burst is also a power law of Γ, which scales as Γ N for 3 < N < 5. Therefore, the τ RT − Γ and the L iso − Γ relations together yield the observed L iso − τ RT relation.
3.4. The Γ 0 − E prompt and Γ 0 − L iso correlations and their physical interpretation Freedman and Waxman (2001) in their analysis of the GRB emission, considering a relativistic velocity for the fireball, showed that the radiation detected by an observer is within an opening angle ≃ 1/Γ(t). Hence, the total fireball energy E should be interpreted as the energy that the fireball would have carried if this is assumed spherically symmetric. In particular, it was claimed that the afterglow flux measurements in X-rays gave a strong evaluation for the fireball energy per unit solid angle represented by ǫ e = ξ e E/4π, within the observable opening angle 1/Γ(t), where ξ e is the electron energy fraction. It was found that Γ(t) = 10.6 1 + z 2 where E prompt is in units of 10 53 erg, n 0 is the uniform ambient density of the expanding fireball in units of cm −3 , and t is the time of the fireball expansion in days. Finally, it was pointed out that ξ e from the afterglow observations should be close to equipartition, namely ξ e ≃ 1 3 . For example, for GRB970508 it was found that ξ e ≃ 0.2 (Waxman 1997;Wijers and Galama 1999;Granot et al. 1999). A similar conclusion, i.e. that it is also close to equipartition, could be drawn for GRB971214, however Wijers and Galama (1999) proposed another interpretation for this GRB's data, demanding ξ e ≃ 1.  selected from the Swift catalogue 20 optical and 12 X-ray GRBs showing the onset of the afterglow shaped by the deceleration of the fireball due to the circumburst medium. The optically selected GRBs were used to fit a linear relation in the log Γ 0 − log E prompt plane, where Γ 0 is the initial Lorentz factor of the fireball and E prompt is in units of 10 52 erg (see left panel of Fig. 5). The best fit line of the Γ 0 − E prompt relation is given by with ρ = 0.89, P < 10 −4 , and σ = 0.11 which can be measured with the deviation of the ratio Γ 0 /E 0.25 prompt . It was found that most of the GRBs with a lower limit of Γ 0 are enclosed within the 2σ region represented by the dashed lines in the left panel of Fig. 5, and it was pointed out that GRBs with a tentative Γ 0 derived from RS peaks or the afterglow peaks, as well as those which lower limits of Γ 0 were derived from light curves with a single power law, are systematically above the best fit line. The lower values of Γ 0 , obtained from a set of optical afterglow light curves with a decaying trend since the start of the detection, were compatible with this correlation.
Later, this correlation was verified by Ghirlanda et al. (2011) and Lü et al. (2012a). Ghirlanda et al. (2011), studying the spectral evolution of 13 SGRBs detected by Fermi/GBM, investigated spectra resolved in the 8 keV − 35 MeV energy range and confirmed the results of . Lü et al. (2012a) enlarged this sample reaching a total of 51 GRBs with spectroscopically confirmed redshifts, and engaged three methods to constrain Γ 0 : (1) the afterglow onset method  which considers T peak of the early afterglow light curve as the deceleration time of the external FS; (2) the pair opacity constraint method (Lithwick and Sari 2001) which requires that the observed high energy γ-rays (i.e., those in the GeV range) are optically thin to electron-positron pair production, thus leading to a lower limit on Γ 0 of the emitting region; (3) the early external forward emission method (Zou and Piran 2010) where an upper limit of Γ 0 can be derived from the quiescent periods between the prompt emission pulses, in which the signal of external shock has to go down the instrument thresholds. Considering some aspects of the external shock emission, the Γ 0 − E prompt correlation was statistically re-analysed using 38 GRBs with Γ 0 calculated using method (1) (as the other two provide only a range of the Lorentz factors, not a definite value), finding with r = 0.67, and E prompt in units of 10 52 erg. In addition, applying the beaming correction, a relation between Γ 0 and L iso , using the same sample (see right panel of Fig. 5), was found to be with r = 0.79, and L iso in units of 10 52 erg s −1 .
Regarding the physical interpretation,  claimed that this correlation clearly shows the association of E prompt with Γ 0 angular structure, and this result yielded another evidence for the fireball deceleration model. Instead, Lü et al. (2012a) found that this relation is well explained by a neutrino-annihilation-powered jet during the emission, indicating a high accretion rate and not very fast BH spin. Besides, evidence for a jet dominated by a magnetic field have already been presented (Zhang and Pe'er 2009;Fan 2010;Zhang and Yan 2011). From the studies of the BH central engine models it was also indicated that magnetic fields are a fundamental feature (Lei et al. 2009). Nevertheless, the baryon loading mechanism in a strongly magnetized jet is more complex, and it has still to be fully investigated.
3.5. Correlations between the energetics and the peak energy 3.5.1. The E peak − F peak and the E peak − S tot correlations Mallozzi et al. (1995) analysed 399 GRBs observed by BATSE and discovered a correlation between the logarithmic average peak energies E peak and F peak . Choosing as a selection criterion for the bursts F peak ≥ 1 ph cm −2 s −1 , they derived F peak from the count rate data in 256 ms time bins in the energy band 50−300 keV and used the E peak distribution derived from the Comptonized photon model (the differential photon number flux per unit energy): with A the normalization, β S the spectral index, and E piv = 100 keV. Then, they grouped the sample into 5 different width F peak bins of about 80 events each (see Fig. 6). The bursts were ranked such that group 1 had the lowest peak flux values and group 5 had the highest values. They found a correlation with ρ = 0.90 and P = 0.04. Lower intensity GRBs exhibited a lower E peak .
Later, Lloyd et al. (2000a) examined the E peak − S tot correlation with 1000 simulated bursts in the same energy range as Mallozzi et al. (1995), and found a strong correlation between E peak and S tot (see the left panel of Fig. 7). The relation between the two variables was as follows: log E peak ∼ 0.29 log S tot , with the Kendall correlation coefficient (Kendall 1938) τ = 0.80 and P = 10 −13 . In addition, they compared it to the E peak − F peak relation (see right panel of Fig. 7). This relation was for the whole spectral sample, and consistent with earlier results (Mallozzi et al. 1995(Mallozzi et al. , 1998. However, they selected a subsample composed of only the most luminous GRBs, because spectral parameters obtained from bursts near the detector threshold are not robust. Therefore, to better understand the selection effects relevant to E peak and burst strength, they considered the following selection criteria: F peak ≥ 3 ph cm −2 s −1 , S obs ≥ 10 −6 erg cm −2 , and S tot ≥ 5 × 10 −6 erg cm −2 . Due to the sensitivity over a certain energy band of all the detectors, especially BATSE, and to some restrictions to the trigger, the selection effects are inevitable. However, the subsample of the most luminous GRBs presents a weak E peak −F peak correlation. Instead, a tight E peak − S tot correlation was found for the whole sample as well as the subsample of the brightest GRBs. Lloyd et al. (2000a) paid more attention to the E peak −S tot correlation for the brightest GRBs because it is easier to deal with the truncation effects in this case, and the cosmological interpretation is simpler.
This correlation has been the basis for the investigation of the Amati relation (see Sect. 3.5.2), and the Ghirlanda relation (see Sect. 3.5.3). Lloyd et al. (2000a) concluded that "the observed correlation can be explained by cosmological expansion alone if the total radiated energy (in the γ-ray range) is constant". In fact, their finding does not depend on the GRB rate density or on the distribution of other parameters. However, the data from GRBs with known redshift are incompatible with a narrow distribution of radiated energy or luminosity.
Following a different approach,  pointed out that the ratio E peak /S tot can serve as an indicator of the ratio of the energy at which most of the γ-rays are radiated to the total energy, and claimed that the E peak − S tot relation is a significant tool for classifying LGRBs and SGRBs. The fluence indicates the duration of the burst without providing a biased value of T 90 , and E peak /S tot displays, as a spectral hardness ratio, an increased hardness for SGRBs in respect to LGRBs, in agreement with ). This correlation is quite interesting, since the energy ratio, being dependent only on the square of the luminosity distance, gets rid of the cosmological dependence for the considered quantities. Therefore, it was evaluated that the energy ratio could be used as a GRB classifier.
Later, Lu et al. (2012) with the results of time-resolved spectral analysis, computed the E peak − S tot relation for 51 LGRBs and 11 bright SGRBs observed with Fermi/GBM. For each GRB, they fitted a simple power law function. They measured its scatter with the distance of the data points from the best fit line. The measured scatter of the E peak − S tot relation is 0.17 ±0.08. This result was reported for the first time by Golenetskii et al. (1983), and later confirmed by Borgonovo and Ryde (2001)

The E peak − E iso correlation
Evidence for a correlation between E peak and S tot was first found by Lloyd and Petrosian (1999) and Lloyd et al. (2000b) based on 46 BATSE events, but this relation was in the observer frame due to the paucity of the data with precise redshift measurement, as was shown in previous paragraphs. Evidence for a stronger correlation between E peak and E iso , also called the Amati relation, was reported by Amati et al. (2002) based on a limited sample of 12 GRBs with known redshifts (9 with firm redshift and 3 with plausible values) detected by BeppoSAX. They found that with r = 0.949, P = 0.005, and E iso calculated as Regarding the methodology considered, instead of fitting the observed spectra, as done for example by Bloom et al. (2001), the GRB spectra were blue-shifted to the rest frames to obtain their intrinsic form. Then, the total emitted energy is calculated by integrating the Band et al. (1993) Amati et al. (2002) by including 20 GRBs from BeppoSAX with known redshift for which new spectral data (BeppoSAX events) or published best-fitting spectral parameters (BATSE and HETE-2 events) were accessible. The relation was found to be log E peak = (2.07 ± 0.03) with r = 0.92, P = 1.1 × 10 −8 , E peak in keV and E iso in units of 10 52 erg. Therefore, its statistical significance increased, providing a correlation coefficient comparable to that obtained by Amati et al. (2002), but based on a larger set.
Based on HETE-2 measurements, Lamb et al. (2004) and Sakamoto et al. (2004) verified the previous results and considered also XRFs, finding out that the Amati relation remains valid over three orders of magnitude in E peak and five orders of magnitude in E iso . The increasing amount of GRBs with measured redshift allowed to verify this relation and strengthen its validity, as found by Ghirlanda et al. (2004b) with 29 events (r = 0.803 and P = 7.6 × 10 −7 ; see left panel in Fig. 9). Ghirlanda et al. (2005a) verified the E peak − E iso correlation among LGRBs considering a set of 442 BATSE GRBs with measured E peak and with pseudo-redshifts computed via the L peak − τ lag correlation. It was shown that the scatter of the sample around the best fitting line is comparable with that of another set composed of 27 GRBs with measured spectroscopic redshifts. This is because the weights of the outliers were marginal. It was noted that the relation for the 442 BATSE GRBs has a slope slightly smaller (0.47) than the one obtained for the 27 GRBs with measured spectroscopic redshifts (0.56).
Afterwards, Amati (2006) (see the upper left and bottom left panels in Fig. 8) updated the study of the E peak − E iso correlation considering a sample of 41 LGRBs/XRFs with firm values of z and E peak , 12 GRBs with uncertain z and/or E peak , 2 SGRBs with certain values of z and E peak , and the sub-energetic events GRB980425/SN1998bw and GRB031203/SN2003lw. The different sets are displayed in the upper right panel in Fig. 8. Taking into account also the sample variance it was found: with ρ = 0.89, P = 7 × 10 −15 , and units the same as in Eq. (35). Moreover, sub-energetic GRBs (980425 and possibly 031203) and SGRBs were incompatible with the E peak − E iso relation, suggesting that it can be an important tool for distinguishing different classes of GRBs. Indeed, the increasing number of GRBs with measured z and E peak can provide the most reliable evidence for the existence of two or more subclasses of outliers for the E peak − E iso relation. Moreover, the relation is valid also for the particular sub-energetic event GRB060218. Finally, the normalization considered by Amati (2006) is consistent with those obtained by other instruments. Ghirlanda et al. (2008) confirmed the E peak − E iso correlation for softer events (XRFs). The sample consisted of 76 GRBs observed by several satellites, mainly HETE-2, KONUS/Wind, Swift and Fermi/GBM. The most important outcome is a tight correlation with no new outliers (with respect to the classical GRB980425 and GRB031203) in the E peak − E iso plane. The obtained relation was log E peak ∼ (0.54 ± 0.01) log E iso .

Amati et al. (2009) studied 95
Fermi GRBs with measured z and obtained an updated E peak − E iso relation, which read with ρ = 0.88 and P < 10 −3 . In particular, they investigated two GRBs (080916C and 090323) with very energetic prompt emission, and found that they follow the E peak − E iso relation well. On the other hand, an SGRB, 090510, also a very luminous and energetic event, was found not to obey the relation. Hence, Amati et al. (2009) proposed that the correlation might serve as a discriminating factor among high-energetic GRBs. In addition, they claimed that the physics of the radiation process for really luminous and energetic GRBs is identical to that for average-luminous and soft-dim long events (XRFs), because all these groups follow the Amati relation.
Later,  provided an update of the analysis by Amati et al. (2008) with a larger sample of 120 GRBs (see the upper right panel of Fig. 8) finding it to be consistent with the following relation: log E peak = 2 + 0.5 log E iso .
with units the same as in Eqs. (35) and (36). Afterwards, Qin and Chen (2013) analysed a sample of 153 GRBs with measured z, E peak , E iso and T 90 , observed by various instruments up to 2012 May. The distribution of the logarithmic deviation of E peak from the Amati relation displayed a clear bimodality which was well represented by a mixture of two Gaussian distributions. Moreover, it was suggested to use the logarithmic deviation of the E peak value for distinguishing GRBs in the E peak versus E iso plane. This procedure separated GRBs into two classes: the Amati type bursts, which follow the Amati relation, and the non-Amati type bursts, which do not follow it. For the Amati type bursts it was found that log E peak = (2.06 ± 0.16) + (0.51 ± 0.12) log E iso with r = 0.83 and P < 10 −36 , while for non-Amati bursts: with r = 0.91 and P < 10 −7 . In both relations E peak is in keV, and E iso is in units of 10 52 erg.
In addition, it was pointed out that almost all Amati type bursts are LGRBs at higher energies, as opposed to non-Amati type bursts which are mostly SGRBs. An improvement to this classification procedure is that the two types of GRBs are clearly separated, hence different GRBs can be easily classified. Heussaff et al. (2013), applying particular selection criteria for the duration and the spectral indices, obtained a set of Fermi GRBs and analysed their locations in the E peak − E iso plane. The sample, composed of 43 GRBs with known redshifts, yielded the following relation: log E peak = 2.07 + 0.49 log E iso , with ρ = 0.70, P = 1.7 × 10 −7 , and the same units as in previous relations of this type.

Amati and Della Valle (2013) pointed out that an enlarged sample of 156
LGRBs with known z and E peak also follows the Amati relation with a slope ≃ 0.5 (see the bottom right panel of Fig. 8). Additionally, Basak and Rao (2012) showed that a time-resolved Amati relation also holds within each single GRB with normalization and slope consistent with those obtained with time-averaged spectra and energetics/luminosity, and is even better than the time-integrated relation (Basak and Rao 2013). Time-resolved E peak and E iso are obtained at different times during the prompt phase (see also Ghirlanda et al. 2010;Lu et al. 2012;Frontera et al. 2012 and Sect. 3.6).

The E peak − E γ correlation
The E peak − E γ relation (also called the Ghirlanda relation) was first discovered by Ghirlanda et al. (2004b), who used 40 GRBs with z and E peak known at their time of writing. Considering the time T break , its value can be used to deduce E γ from E iso . Indeed, even if only a little less than half of the bursts have observed jet breaks (47%), from  we know that where T break is measured in days, n is the density of the circumburst medium in particles per cm 3 , η γ is the radiative efficiency, and E iso is in units of 10 52 erg. Here, θ jet is in degrees and it is the angular radius (the half opening angle) subtended by the jet. For GRBs with no measured n, the median value n = 3 cm −3 of the distribution of the computed densities, extending between 1 and 10 cm −3 , was considered (Frail et al. 2000;Yost et al. 2002;Panaitescu and Kumar 2002;Schaefer 2003a).
Later, Liang and Zhang (2005) using a sample of 15 GRBs with measured z, E peak and T break , considered a purely phenomenological T * break of the optical afterglow light curves, thus avoiding the assumption of any theoretical model, contrary to what was done by Ghirlanda et al. (2004b). The functional form of this correlation is given by: log E γ = (0.85 ± 0.21) + (1.94 ± 0.17) log E * peak − (1.24 ± 0.23) log T * break , where E γ is in units of 10 52 erg, E * peak in units of 100 keV, T * break is measured in days, and ρ = 0.96 and P < 10 −4 . Nava et al. (2006) found that the Ghirlanda relation, assuming a wind-like circumburst medium, is as strong as the one considering a homogeneous medium. They analysed the discrepancy between the correlations in the observed and in the comoving frame (with Lorentz factor identical to the fireball's one). Since both E peak and E γ transform in the same way, the wind-like Ghirlanda relation remains linear also in the comoving frame, no matter what the Lorentz factor's distribution is. The wind-like relation corresponds to bursts with the same number of photons emitted. Instead, for the homogeneous density medium scenario, it is common to consider a tight relation between the Lorentz factor and the total energy, thus limiting the emission models of the prompt radiation. Using 18 GRBs with firm z, E peak and T break , Nava et al. (2006) with ρ = 0.92 and P = 6.9 × 10 −8 . Ghirlanda et al. (2007) tested the E peak − E γ correlation using 33 GRBs (16 new bursts detected by Swift with firm z and E peak up to December 2006, and 17 pre-Swift GRBs). They claimed that for computing the T break it is required that: 1. the detection of the jet break should be in the optical, 2. the optical light curve should continue up to a time longer than the T break , 3. the host galaxy flux and the flux from a probable SN should be removed, 4. the break should not depend on the frequency in the optical, and a coincident break in the X-ray light curve is not necessary, because the flux in X-rays could be controlled by another feature, 5. the considered T break should be different from the one at the end of the plateau emission (the time T a in , otherwise the feature affecting the X-ray flux is also influencing the optical one. Therefore, considering all these restrictions, the sample was reduced to 16 GRBs, all compatible with the following E peak − E γ relation: log E peak 100 keV = (0.48 ± 0.02) + (0.70 ± 0.04) log E γ 4.4 × 10 50 erg .
No outliers were detected. Therefore, the reduced scatter of the E peak − E γ relation corroborates the use of GRBs as standardizable candles. Lloyd et al. (2000a) investigated the physical explanation of the E peak − S tot correlation assuming the emission process to be a synchrotron radiation from internal and external shocks. Indeed, they claimed that this correlation is easily obtained considering a thin synchrotron radiation by a power law distribution of electrons with Γ larger than some minimum threshold value, Γ m . Moreover, the internal shock model illustrates the tight E peak − S tot relation and the emitted energy better than the external shock model. Lloyd-Ronning and Petrosian (2002) pointed out that the GRB particle acceleration is not a well analysed issue. Generally, the main hyphotesis is that the emitted particles are accelerated via recurrent scatterings through the (internal) shocks. They found that the recurrent crossings of the shock come from a power law distribution of the particles with a precise index, providing a large energy synchrotron photon index. Moreover, the connection between E peak and the photon flux can be justified by the variation of the magnetic field or electron energy in the emission events. Finally, they claimed that in the majority of GRBs, the acceleration of particles is not an isotropic mechanism, but occurs along the magnetic field lines. Amati et al. (2002) confirmed the findings of Lloyd et al. (2000b) that the log E peak ∼ 0.5 log E iso relation is obtained assuming an optically thin synchrotron shock model. This model considers electrons following the N(Γ) = N 0 Γ −p distribution for Γ > Γ m with Γ m , GRB duration, and N 0 constant in each GRB. However, the above assumptions are not fully justifiable. In fact the duration is different in each GRB and E iso might be smaller in the case of beamed emission. Amati (2006) pointed out the impact that the correlation has on the modeling of the prompt emission and on the possible unification of the two classes of GRBs and XRFs. In addition, this correlation is often applied for checking GRB synthesis models (e.g. Zhang and Mészáros 2002;Ghirlanda et al. 2013).

Physical interpretation of the energetics vs. peak energy relations
In every model, E peak and E iso depend on Γ, and the E peak − E iso relation can help to relate the parameters of the synchrotron shock model and inverse Compton model (Zhang and Mészáros 2002;Schaefer 2003a). Specifically, Zhang and Mészáros (2002) and Rees and Mészáros (2005) found that, for an electron distribution given by a power law and produced by an internal shock in a fireball with velocity Γ, the peak energy is given as where L is the total fireball luminosity and t ν the variability timescale. However, to recover the E peak − E iso relation from this relation, Γ and t ν should be similar for each GRB, a condition that cannot be easily supported. A further issue arises when one considers that L ∝ Γ N , with N between 2 and 3 in different models (Zhang and Mészáros 2002;Schaefer 2003a;Ramirez-Ruiz 2005). An explanation could be that direct or Comptonized thermal radiation from the fireball photosphere (Zhang and Mészáros 2002;Ramirez-Ruiz 2005;Ryde 2005;Rees and Mészáros 2005;Beloborodov 2010;Guiriec et al. 2011;Guiriec et al. 2013;Vurm and Beloborodov 2015;Guiriec et al. 2015a,b) can affect significantly the GRB prompt emission. This can be a good interpretation of the really energetic spectra presented for many events (Frontera et al. 2000;Preece et al. 2000;) and the flat shape in GRB average spectra. In such cases, E peak depends on the peak temperature T bb,peak of photons distributed as by a blackbody, and therefore it is associated to the luminosity or emitted energy. For Comptonized radiation from the photosphere the relations are log E peak ∼ log Γ + log T bb,peak ∼ 2 log Γ − 0.25 log L or log E peak ∼ log Γ + log T bb,peak ∼ −0.5 log r 0 + 0.25 log L, where r 0 is a particular distance between the central engine and the energy radiating area, such that the Lorentz factor evolves as Γ ≃ r/r 0 up to some saturation radius r s (Rees and Mészáros 2005). As suggested by Rees and Mészáros (2005), in this scenario the E peak − E iso relation could be recovered for particular physical cases just underneath the photosphere, though it would rely on an undefined number of unknown parameters.
Also for high-energetic GRBs (i.e., E iso ≈ 10 55 erg) the nonthermal synchrotron emission model can explain the E peak − E iso correlation. This can be possible either considering the minimum Lorentz factor and the normalization of the power law distribution of the emitting electrons constant in each GRB, or by constraints on the slope of the relation between Γ and the luminosity (Lloyd et al. 2000b;Zhang and Mészáros 2002). Panaitescu (2009) used 76 GRBs with measured redshifts to analyse the case in which the E peak − E iso relation for LGRBs is due to the external shock generated by a relativistic outflow interacting with the ambient medium. He considered the effect of each parameter defining the E peak − E iso relation on the radial distribution of the external medium density and pointed out that the log E peak ∼ 0.5 log E iso relation is recovered if the external medium is radially stratified. For some combinations of radiative (synchrotron or inverse-Compton) and dissipation (such as RS or FS) mechanisms, it is concluded that the external medium requires a particle density distributed distinctly from R −2 , with R the distance at which the GRB radiation is generated. This tendency should be commonly associated to uniform mass-loss rate and final velocity. Mochkovitch and Nava (2015) checked whether the E peak − E iso relation can be recovered in a case when the prompt emission is due to internal shocks, or alternatively if the correlation can give some limits for the internal shock scenario defined through the impact of only two shells. Simulated GRB samples were obtained considering different model parameter distributions, such as the emitted power in the relativistic emission and Γ. Simulated E peak − E iso distributions were plotted for each case and analysed together with the observed relation (based on 58 GRBs). The sample contained only luminous Swift GRBs with F peak > 2.6 ph cm −2 s −1 in the 15 − 150 keV energy band. In conclusion, a correspondence between the model and data was found, but exclusively if the following restrictions for the dynamics of the emission and for the dispersion of the energy are assumed: 1. the majority of the dispersed energy should be radiated in few electrons; 2. the spread between the highest and the lowest Lorentz factor should be small; 3. if the mean Lorentz factor grows asΓ ∝Ė 1/2 (whereĖ is the rate of injected energy, or mean emitted power, in the relativistic outflow), the E peak − E iso relation is not retrieved and E peak is diminishing with larger E iso . However, the E peak − E iso relation can be regained ifΓ ∝Ė 1/2 is a lower constraint for a particularĖ; 4. when the timescale or the width of the variability of the Lorentz factor is associated withΓ, E peak − E iso relation is recovered.
For the Ghirlanda relation (Ghirlanda et al. 2004b), with the assumption that the line of sight is within the jet angle, the E peak − E γ relation indicates its invariance when moving from the rest frame to the comoving frame. As a result, the number of radiated photons in each GRBs is comparable and should be about 10 57 . The last characteristic could be important for understanding the dynamics of GRBs and the radiative mechanisms (see also right panel of Fig. 9). Collazzi et al. (2011) found that the mean E * peak is near to 511 keV, the electron restmass energy m e c 2 . Therefore, it is claimed that the tight shape of the E peak distribution does not stem only from selection effects. No studied mechanism can drive this effect, however with the E * peak compatible with the effective temperature of the γ-ray radiating area, the almost constant temperature needs some mechanism similar to a thermostat, keeping the temperature at a steady value. It was suggested that such a mechanism could be an electronpositron annihilation. Ghirlanda et al. (2013), using a simulated sample, analysed if different intrinsic distributions of Γ and θ jet can replicate a grid of observational constraints. With the assumption that in the comoving frame each GRB has similar E peak and E γ , it was found that the distributions of Γ and θ jet cannot be power laws. Instead, the highest concordance between simulation and data is given by log-normal distributions, and a connection between their maxima, like θ 2.5 jet,max Γ max = const. In this work θ jet and Γ are important quantities for the calculation of the GRB energetics. Indeed, from a sample of ≈ 30 GRBs with known θ jet or Γ it was found that the E γ distribution is centered at 10 50 − 10 51 erg and it is tightly related to E peak . It was obtained that Present values of Γ and θ jet rely on incomplete data sets and their distributions could be peak − E γ relation for GRBs with known redshift. The filled circles represent E γ for the events where a jet break was detected. Grey symbols indicate lower/upper limits. The solid line represents the best fit, i.e. log E peak ∼ 53.68+0.7 log E γ . Open circles denote E iso for the GRBs. The dashed line represents the best fit to these points and the dash-dotted line is the relation shown by Amati et al. (2002). (Figure after Ghirlanda et al. (2004b); see Fig. 1 therein. @ AAS. Reproduced with permission.) Right panel: rest frame plane of GRB energy. The large black dot indicates that all simulated GRBs were assigned E * peak = 1.5 keV and E * γ = 1.5 × 10 48 erg. Since Γ > 1 but less then 8000, regions (I) are forbidden. Since for all the simulated GRBs θ jet ≤ 90 • , they cannot stay in region (II). When Γ is small, the beaming cone turns out to be larger than the jet. Therefore, the isotropic equivalent energy is given by log E iso = log E γ +log(1+β 0 )+2 log Γ, lower than the energy computed by log E iso = log E γ −log(1−cos θ jet ). This brings in a constraint, log E peak ∼ 1/3 × log E iso , and GRBs cannot lie to the right of this constraint. Hence, region (III) is not allowed. The black dots indicate the actual GRBs of the Swift sample. The fit to the Swift sample is displayed as the dot-dashed line. (Figure after  affected by biases. Neverthless, Ghirlanda et al. (2013) claimed that greater values of Γ are related to smaller θ jet values, i.e. the faster a GRB, the narrower its jet.
Furthermore, GRBs fulfilling the condition sin θ jet < 1/Γ might not display any jet break in the afterglow light curve, and Ghirlanda et al. (2013) predicted that this group should comprise ≈ 6% of the on-axis GRBs. Finally, their work is crucial as it allowed to find that the local rate of GRBs is ≈ 0.3% of the local SNe Ib/c rate, and ≈ 4.3% of the local hypernovae (i.e., SNe Ib/c with wide-lines) rate.
3.6. Correlations between the luminosity and the peak energy The L iso − E peak relation was discovered by Schaefer (2003a) who used 84 GRBs with known E peak from the BATSE catalogue (Schaefer et al. 2001), and 20 GRBs with luminosities based on optically measured redshift (Amati et al. 2002;Schaefer 2003b). It was found that (see Fig. 10) for the 20 GRBs with r = 0.90 and P = 3 × 10 −8 , and among the 84 GRBs the relation was log E peak ∼ (0.36 ± 0.03) log L iso .
The underlying idea is that the L iso varies as a power of Γ, as we have already discussed in Sect. 3.1.2, and the E peak also varies as some other power of Γ, so that E peak and L iso will be correlated to each other through their dependence on Γ. For the general case where the luminosity varies as Γ N and E peak varies as Γ M , and therefore log E peak will vary as (M + 1)/N × log L iso . , using a sample of 9 GRBs detected simultaneously with the Wide Field Camera (WFC) on board the BeppoSAX satellite, and by the BATSE instrument, reported the results of a systematic study of the broadband (2 − 2000 keV) time-resolved prompt emission spectra. However, only 4 of those GRBs (970111,980329,990123,990510) were bright enough to allow a fine time-resolved spectral analysis, resulting in a total of 40 spectra. Finally, the study of the time-resolved dependence (see also the end of Sect. 3.5.2) of E peak on the corresponding L iso was possible for two bursts with known redshift (i.e., 990123 and 990510), and found using the least squares method (see Fig. 11): -39 - Fig. 10.-Direct fit of log L iso − log E peak data. This is shown here for two independent data sets for which the luminosities are derived by two independent methods. The first data set consists of 20 bursts with spectroscopically measured redshifts (the open circles). The second one is for 84 bursts (whose binned values are shown as filled diamonds, and the horizontal bars are the bin widths) whose luminosity (and then redshift) were determined with the spectral lag and variability light curve parameters. Both data sets show a highly significant and similar power law relations. with ρ = 0.94 and P = 1.57 × 10 −13 .
Afterwards, Nava et al. (2012), using a sample of 46 Swift GRBs with measured z and E peak , found a strong L iso − E peak correlation, with a functional form of log E * peak = −(25.33 ± 3.26) + (0.53 ± 0.06) log L iso , with ρ = 0.65 and P = 10 −6 ; E peak is in keV, and L iso is in units of 10 51 erg s −1 . Furthermore, using 12 GRBs with only an upper limit on z (3 events) or no redshift at all (3 events), or with a lower limit on E peak (3 events) or no estimate at all (3 events), they found that these bursts also obey the obtained L iso − E peak relation.

The L peak − E peak correlation
It was also found that the Amati relation holds even if E iso is substituted with L iso and L peak , which is not surprising given that these "energy indicators" are strongly correlated. To this end, the Yonetoku correlation (Yonetoku et al. 2004, see the left panel of Fig. 12), relates E peak with L peak . The relation was obtained employing 11 GRBs with known redshifts detected by BATSE, together with BeppoSAX GRBs from (Amati et al. 2002). This relation uses the L peak of the burst instead of L iso , and it is tighter than previous prompt correlations.
The best-fit line is given by with r = 0.958, P = 5.31 × 10 −9 , and the uncertainties are 1σ error. This relation agrees well with the standard synchrotron model (Zhang and Mészáros 2002;Lloyd et al. 2000b). Finally, it has been used to estimate pseudo-redshifts of 689 BATSE LGRBs with unknown distances and to derive their formation rate as a function of z.  Ghirlanda et al. (2004a) selected 36 bright SGRBs detected by BATSE, with an F peak on the 64 ms timescale in the 50 −300 keV energy range exceeding 10 ph cm −2 s −1 . In 7 cases, the signal-to-noise-ratio was too low to reliably constrain the spectral best-fit parameters. One case yielded missing data. Hence, the sample consisted of 28 events. Due to unknown redshifts, E * peak , E iso and L peak were expressed as functions of the redshift in the range z ∈ [0.001, 10]. It was found that SGRBs are unlikely to obey the Amati relation, E iso −E * peak , but the results were consistent with the L peak − E * peak relation of Yonetoku et al. (2004). Hence, assuming that this relation indeed holds for SGRBs, their pseudo-redshifts were estimated and found to have a similar distribution as LGRBs, with a slightly smaller average redshift.
Afterwards,  investigated the prompt emission of 101 GRBs with measured redshifts and a reported F peak detected until the end of 2009. The sample comes from events detected in a number of independent missions: the satellites used for this purpose are KONUS, Swift, HXD-WAM and RHESSI. Using this data set, the E peak −L peak correlation was revised, and its functional form could be written as log L peak = (52.43 ± 0.037) + (1.60 ± 0.082) log E * peak , with r = 0.889 for 99 degrees of freedom and an associated P = 2.18 × 10 −35 ; L peak is expressed in erg s −1 , and E * peak in units of 355 keV. To provide reference to previous works, the 1 − 10 4 keV energy band in the GRB rest frame was used to calculate the bolometric energy and L peak . Finally, it was demonstrated that this relation is intrinsic to GRBs and affected by the truncation effects imposed by the detector threshold. Lu and Liang (2010), using time-resolved spectral data for a sample of 30 pulses in 27 bright GRBs detected by BATSE, investigated the L peak − E peak relation in the decay phases of these pulses (see right panel of Fig. 12). Quite all of the pulses followed a narrow L peak − E peak relation given by with r = 0.91 and P < 10 −4 , but the power law index varied. The statistical or observational effects could not account for the large scatter of the power law index, and it was suggested to be an intrinsic feature, indicating that no relation common for all GRB pulses L peak − E peak would be expected. However, in the light of Fermi observations that revealed deviations from the Band function (Abdo et al. 2009;Guiriec et al. 2010;Ackermann et al. 2010Ackermann et al. , 2011Ackermann et al. , 2013; see also Lin et al. 2016), it was proposed recently that the GRB spectra should be modelled not with the Band function itself (constituting a non-thermal component), but with additional black-body (BB, thermal) and power law (PL, non-thermal) components (Guiriec et al. 2013(Guiriec et al. , 2015a(Guiriec et al. ,b, 2016. The non-thermal component was well described within the context of synchrotron radiation from particles in the jet, while the thermal component was interpreted by the emission from the jet photosphere. The PL component was claimed to originate most likely from the inverse Compton process. The results point toward a universal relation between L peak and E * peak related to the non-thermal components. Tsutsui et al. (2013) analysed 13 SGRB candidates (i.e., an SGRB with T * 90 < 2 s), from among which they selected 8 events considering them as secure ones. An SGRB candidate is regarded as a misguided SGRB if it is located within the 3σ int dispersion region from the best-fit E * peak − E iso function of the correlation for LGRBs, while the others are regarded as secure SGRBs. The relation obtained with secure GRBs is the following: log L peak = (52.29 ± 0.066) + (1.59 ± 0.11) log E * peak , with r = 0.98 and P = 1.5 × 10 −5 , where E * peak (in units of 774.5 keV) is from the timeintegrated spectrum, while L peak (in erg s −1 ) was taken as the luminosity integrated for 64 ms at the peak considering the shorter duration of SGRBs. Application of this relation to 71 bright BATSE SGRBs resulted in pseudo-redshifts distributed in the range z ∈ [0.097, 2.258], with z = 1.05, which is apparently lower than z = 2.2 for LGRBs. Finally, Yonetoku et al. (2014), using 72 SGRBs with well determined spectral features as observed by BATSE, determined their pseudo-redshifts and luminosities by employing the L peak − E peak correlation for SGRBs found by Tsutsui et al. (2013). It was found that the obtained redshift distribution for z ≤ 1 was in agreement with that of 22 Swift SGRBs, indicating the reliability of the redshift determination via the E * peak − L peak relation.

Physical interpretation of the luminosity vs. peak energy relations
As pointed out by Schaefer et al. (2001) and Schaefer (2003a), E peak and L iso are correlated because of their dependence on Γ. The L iso − E peak relation could shed light on the structure of the ultra relativistic outflow, the shock acceleration and the magnetic field generation (Lloyd-Ronning and Ramirez-Ruiz 2002). However, since only few SGRBs are included in the samples used, the correlations and interpretations are currently only applicable to LGRBs. Schaefer et al. (2001) and Schaefer (2003a) claimed that the values of E peak are approximately constant for all the bursts with z ≥ 5. However, with the launch of the Swift satellite in the end of 2004 the hunt for "standard candles" via a number of GRB correlations is still ongoing. Thus, the great challenge is to find universal constancy in some GRB parameters, despite the substantial diversity exhibited by their light curves. If this goal is achieved, GRBs might prove to be a useful cosmological tool . Liang et al. (2004) defined a parameter ω = (L iso /10 52 erg s −1 ) 0.5 /(E peak /200 keV) and discussed possible implications of the E peak − L iso relation for the fireball models. They found that ω is limited to the range ≃ 0.1 − 1. They constrained some parameters, such as the combined internal shock parameter, ζ i , for the internal as well as external shock models, with an assumption of uncorrelated model parameters. Their distributions suggest that the production of prompt γ-rays within internal shocks dominated by kinetic energy is in agreement with the standard internal shock model. Similarly in case when the γ-rays come from external shocks dominated by magnetic dissipation. These results imply that both models can provide a physical interpretation of the L iso ∝ E 2 peak relation as well as the parameter ω.
To explain the origin of this correlation, Mendoza et al. (2009) considered simple laws of mass and linear momentum conservation on the emission surface to give a full description of the working surface flow parameterized by the initial velocity and mass injection rate. They assumed a source-ejecting matter in a preferred direction x with a velocity v(t) and a mass ejection rateṁ(t), both dependent on time t as measured from the jet's source; i.e., they studied the case of a uniform release of mass and the luminosity was measured considering simple periodic oscillations of the particle velocity, a common assumption in the internal shock model scenario.
Due to the presence of a velocity shear with a considerable variation in Γ at the boundary of the spine and sheath region, a fraction of the injected photons are accelerated via a Fermilike acceleration mechanism such that a high energy power law tail is formed in the resultant spectrum. Ito et al. (2013) showed in particular that if a velocity shear with a considerable variance in Γ is present, the high energy part of the observed GRB photon spectrum can be explained by this photon acceleration mechanism. The accelerated photons may also account for the origin of the extra hard power law component above the bump of the thermal-like peak seen in some peculiar GRBs (090510, 090902B, 090926A). It was demonstrated that time-integrated spectra can also reproduce the low energy spectra of GRBs consistently due to a multi-temperature effect when time evolution of the outflow is considered.
Regarding the Yonetoku relation, its implications are related to the GRB formation rate and the luminosity function of GRBs. In fact, the analysis of Yonetoku et al. (2004) showed that the existence of the luminosity evolution of GRBs, assuming as a function a simple power law dependence on the redshift, such as g(z) = (1 + z) 1.85 , may indicate the evolution of GRB progenitor itself (mass) or the jet evolution. To study the evolution of jet opening angle they considered two assumptions: either the maximum jet opening angle decreases or the total jet energy increases. In the former case, the GRB formation rate obtained may be an underestimation since the chance probability to observe the high redshift GRBs will decrease. If so, the evolution of the ratio of the GRB formation rate to the star formation rate becomes more rapid. On the other hand, in the latter case, GRB formation rate provides a reasonable estimate.
Recently Frontera et al. (2016), building on the spectral model of the prompt emission of Titarchuk et al. (2012), gave a physical interpretation of the origin of the time resolved L iso − E peak relation. The model consists of an expanding plasma shell, result of the star explosion, and a thermal bath of soft photons. Frontera et al. (2016) showed analytically that in the asymtotic case of the optical depth τ ≫ 1 the relation log L iso − log E peak indeed has a slope of 1/2. This, in turn, is evidence for the physical origin of the Amati relation (see Sect. 3.5.2).

3.7.
Comparisons between E peak − E iso and E peak − L peak correlation For a more complete dissertation we compare the E peak − E iso correlation with the E peak − L peak correlation. To this end, Ghirlanda et al. (2005b) derived the E peak − L peak relation with a sample of 22 GRBs with known z and well determined spectral properties. This relation has a slope of 0.51, similar to the one proposed by Yonetoku et al. (2004) with 12 GRBs, although its scatter is much larger than the one originally found. Tsutsui et al. (2009) investigated these two relations using only data from the prompt phase of 33 low-redshift GRBs with z ≤ 1.6. In both cases the correlation coefficient was high, but a significant scatter was also present. Next, a partial linear correlation degree, which is the degree of association between two random variables, was found to be ρ L peak ,E iso ,E peak = 0.38. Here, ρ 1,2,3 means the correlation coefficient between the first and the second parameter after fixing the third parameter. This fact indicates that two distance indicators may be independent from each other even if they are characterized by the same physical quantity, E peak , and similar quantities, L peak and E iso . To correct the large dispersion of the Yonetoku correlation, Tsutsui et al. (2009) introduced a luminosity time constant T L defined by T L = E iso /L peak as a third parameter and a new correlation was established in the form with r = 0.94 and P = 10 −10 . Here, L peak is in units of 10 52 erg s −1 , E peak is in keV, and T L in seconds. In this way the systematic errors were reduced by about 40%, and the plane represented by this correlation might be regarded as a "fundamental plane" of GRBs.
Later,  reconsidered the correlations among E peak , L peak and E iso , using the database constructed by , which consisted of 109 GRBs with known redshifts, and E peak , L peak and E iso well determined. The events are divided into two groups by their data quality. One (gold data set) consisted of GRBs with E peak determined by the Band function with four free parameters. GRBs in the other group (bronze data set) had relatively poor energy spectra so that their E peak were determined by the Band function with three free parameters (i.e., one spectral index was fixed) or by the cut-off power law (CPL) model with three free parameters. Using only the gold data set, the intrinsic dispersion, σ int , in log L peak is 0.13 for the E peak − T L − L peak correlation, and 0.22 for the E peak − L peak correlation. In addition, GRBs in the bronze data set had systematically larger E peak than expected by the correlations constructed with the gold data set. This indicates that the quality of the sample is an important issue for the scatter of correlations among E peak , L peak , and E iso .
The difference between the E peak − L peak correlation for LGRBs from (Ghirlanda et al. 2010) and the one from ) is due to the presence of GRB060218. In the former, it was considered an ordinary LGRB, while in the latter, an outlier by a statistical argument. Because GRB060218 is located far from the L peak − E peak correlation in ) (more than 8σ), it makes the best-fit line much steeper.
Regarding the high-energetic GRBs, Ghirlanda et al. (2010) considered 13 GRBs detected by Fermi up to the end of July 2009, and with known redshift. They found a tight relation: with a scatter of σ = 0.26. A similarly tight relation exists between E * peak and E iso : The time integrated spectra of 8 Fermi GRBs with measured redshift were consistent with both the E peak − E iso and the E peak − L iso correlations defined by 100 pre-Fermi bursts.
Regarding the study of SGRBs within the context of these two correlations, Tsutsui et al. (2013) used 8 SGRBs out of 13 SGRB candidates to check whether the E peak − E iso and E peak − L peak correlations exist for SGRBs as well. It was found that the E peak − E iso correlation seemed to hold in the form log E iso = (51.42 ± 0.15) + (1.58 ± 0.28) log E * peak , with r = 0.91, P = 1.5 × 10 −3 , E iso in erg s −1 and E * peak in units of 774.5 keV. They also found that the E peak − L peak correlation with a functional form as in Eq. (59) is tighter than the E peak − E iso one. Both correlations for SGRBs indicate that they are less luminous than LGRBs, for the same E peak , by factors ≃ 100 (for E peak − E iso ), and ≃ 5 (for E peak − L peak ). It was the first time that the existence of distinct E peak − E iso and E peak − L peak correlations for SGRBs was argued.
3.8. The L X,p − T * p correlation and its physical interpretation Using data gathered by Swift,  proposed a unique phenomenological function to estimate some relevant parameters of both the prompt and afterglow emission. Both components are well fitted by the same functional form: The index i can take the values p or a to indicate the prompt and afterglow, respectively. The complete light curve, f tot (t) = f p (t) + f a (t), is described by two sets of four parameters where α i is the temporal power law decay index, the time t i is the initial rise timescale, F i is the flux and T i is the break time. Fig. 13 schematically illustrates this function.
Following the same approach as adopted in (Dainotti et al. 2008), Qi and Lu (2010) investigated the prompt emission properties of 107 GRB light curves detected by the XRT instrument onboard the Swift satellite in the X-ray energy band (0.3 − 10 keV). They found that there is a correlation between L X,p and T * p . Among the 107 GRBs, they used only 47, because some of the events did not have a firm redshift and some did not present reliable spectral parameters in the prompt decay phase. Among the 47 GRBs, only 37 had T * p > 2 s, and 3 of them had T * p > 100 s. The functional form for this correlation could be written in the following way: where L X,p is in erg s −1 , and T * p is in seconds. The fits were performed via the D'Agostini (2005) fitting method applied to the following data sets: 1. the total sample of 47 GRBs (see the left panel of Fig. 14), 2. 37 GRBs with T * p > 2 s (see the middle panel of Fig. 14), 3. 34 GRBs with 2 s < T * p < 100 s (see the right panel of Fig. 14).
The results of these fittings turned out to give different forms of Eq. (65). In case 1., a = 50.91 ± 0.23 and b = −0.89 ± 0.19 were obtained. The slope b is different in cases 2. and 3., b = −1.73 and b = −0.74, respectively. The best fit with the smallest σ int comes from case 3. Remarkably, in this case the slope b is close to the slope (−0.74 +0.20 −0.19 ) of a similar log L X − log T * a relation (Dainotti et al. 2008). Qi and Lu (2010) noticed a broken linear relation of the L X,p − T * p correlation. More specifically, an evidence of curvature appears in the middle panel of Fig. 14. One can see, from the left panel of Fig. 14, that if the best-fit line is extended to the range of T * p < 2 s, all the GRBs with T * p < 2 s are located below this line. However, the small sample of GRBs used in their analysis is still not sufficient to conclude whether the change in the slope is real or just a selection bias caused by outliers. If there is a change in the slope this may suggest that GRBs could be classified into two groups, long and short, based on their values of T * p instead of T 90 , since T * p is an estimate of the GRB duration based on temporal features of the light curves, and T 90 is a measure based on the energy. This idea has actually been proposed for the first time by O'Brien and . It is worth noting that while T 90 and T p are both estimates of the GRB duration, the correlation does not hold if T p is replaced with T 90 . For an analysis of an extended sample and comparison of T 45 versus T p also, see (Dainotti et al. 2011b).
Regarding the physical interpretation, the change of the slope in the L X,p − T * p relation at different values of T * p in (Qi and Lu 2010) can be due to the presence of few GRBs with a large T * p , but it might also be due to different emission mechanisms. Unfortunately, the paucity of the sample prevents putting forward any conclusion due to the presence of (potential) outliers in the data set. A more detailed analysis is necessary to further validate this correlation and better understand its physical interpretation.

The L f − T f correlation and its physical interpretation
In most GRBs a rapid decay phase (RDP) soon after the prompt emission is observed , and this RDP appears continue smoothly after the prompt, both in terms of temporal and spectral variations . This indicates that the RDP could be the prompt emission's tail and a number of models have been proposed to take it into account (see , in particular the high latitude emission (HLE). This model states that once the prompt emission from a spherical shell turns off at some radius, then the photons reach the observer from angles apparently larger (relative to the line of sight) due to the added path length caused by the curvature of the emitting region. The Doppler factor of these late-arriving photons is smaller.
A successful attempt to individually fit all the distinct pulses in the prompt phase and in the late X-ray flares observed by the complete Swift/BAT+XRT light curves has been performed by  using a physically motivated pulse profile. This fitting is an improved procedure compared to the  one. The pulse profile has the following functional form: where T 0 = T f − T rise (with T rise the rise time of the pulse) is the arrival time of the first photon emitted from the shell. It is assumed here that the emission comes from an ultrarelativistic thin shell spreading over a finite range of radii along the line of sight, in the observer frame measured with respect to the ejection time, T ej . From these assumptions it is possible to model the rise of the pulse through α, T rise and T f (see also Fig. 1).
The combination of the pulse profile function P (t, T f , T rise ) and the blue-shift of the spectral profile B(x) produces the rise and fall of the pulse. B(x) is approximated with the Band function in the form where x = (E/E f ) [(T − T ej ) /T f ] −1 , with E f the energy at the spectral break, and B norm is the normalization.
Using this motivated pulse profile,  found that, within a sample of 12 GRBs observed by Swift in the BAT and XRT energy bands, L f is anti-correlated with T * f in the following way: log L f ∼ −(2.0 ± 0.2) log T * f .
Therefore, high luminosity pulses occur shortly after ejection, while low luminosity pulses appear at later time (see the left panel of Fig. 15). Moreover,  also found a correlation between L f and E peak as shown in the right panel of Fig. 15. This is in agreement with the known correlation between the L peak for the whole burst and the E peak of the spectrum during the time T 90 (Yonetoku et al. 2004;Tsutsui et al. 2013); for comparison with the L peak − E peak correlation, see also Sect. 3.6.2.
In the 12 light curves considered by , 49 pulses were analysed. Although several pulses with a hard peak could not be correctly fitted, the overall fitting to the RDP was satisfactory and the HLE model was shown to be able to take into account phase of the GRB emission. However, it is worth to mention the hard pulse in GRB061121 which requires a spectral index β S = 2.4, larger than the value expected for synchrotron emission, i.e. β S = 1. Lee et al. (2000) and Quilligan et al. (2002) discussed analogous correlations, although these authors considered the width of a pulse rather than T f , which is in fact closely correlated with pulse width. Many authors afterwards (Littlejohns et al. 2013;Bošnjak and Daigne 2014;Evans et al. 2014;Hakkila and Preece 2014;Laskar et al. 2014;Littlejohns and Butler 2014;Roychoudhury et al. 2014;Ceccobello and Kumar 2015;Kazanas et al. 2015;Peng et al. 2015) have used the motivated pulse profile of  for various studies on the prompt emission properties of the pulses.
Regarding the physical interpretation, in ) the flux density of each prompt emission pulse is depicted by an analytical expression derived under the assumption that the radiation comes from a thin shell, as we have already described. The decay after the peak involves the HLE (Genet and Granot 2009) along the considered shell which is delayed and modified with a different Doppler factor due to the curvature of the surface (Ryde and Petrosian 2002;Dermer 2007).

Summary
In this work we have reviewed the bivariate correlations among a number of GRB prompt phase parameters and their characteristics. It is important to mention that several of these correlations have the problem of double truncation which affects the parameters. Some relations have also been tested to prove their intrinsic nature like the E peak − S tot , E peak − E iso and L peak − E peak relations. For the others we are not aware of their intrinsic forms and consequently how far the use of the observed relations can influence the evaluation of the theoretical models and the "best" cosmological settings. Therefore, the evaluation of the intrinsic correlations is crucial for the determination of the most plausible model to explain the prompt emission. In fact, though there are several theoretical interpretations describing each correlation, in many cases more than one is viable, thus showing that the emission processes that rule GRBs still need to be further investigated. These correlations might also serve as discriminating factors among different GRB classes, as several of them hold different forms for SGRBs and LGRBs, hence providing insight into the generating mechanisms. Hopefully those correlations could lead to new standard candles allowing to explore the high-redshift universe.