Gamma Ray Burst afterglow and prompt-afterglow relations: an overview
DDraft version April 12, 2017
Preprint typeset using L A TEX style AASTeX6 v. 1.0
GAMMA RAY BURST AFTERGLOW AND PROMPT-AFTERGLOWRELATIONS: AN OVERVIEW
Dainotti M. G. , Del Vecchio R. Physics Department, Stanford University, Via Pueblo Mall 382, Stanford, CA, USA, E-mail: [email protected] INAF-Istituto di Astrofisica Spaziale e Fisica cosmica, Via Gobetti 101, 40129, Bologna, Italy Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Krak´ow, Poland E-mails: [email protected], [email protected]
ABSTRACTThe mechanism responsible for the afterglow emission of Gamma Ray Bursts (GRBs) and its con-nection to the prompt γ -ray emission is still a debated issue. Relations between intrinsic propertiesof the prompt or afterglow emission can help to discriminate between plausible theoretical models ofGRB production. Here we present an overview of the afterglow and prompt-afterglow two parameterrelations, their physical interpretations, their use as redshift estimators and as possible cosmologicaltools. A similar task has already been correctly achieved for Supernovae (SNe) Ia by using the peakmagnitude-stretch relation, known in the literature as the Phillips relation (Phillips 1993). The chal-lenge today is to make GRBs, which are amongst the farthest objects ever observed, standardizablecandles as the SNe Ia through well established and robust relations. Thus, the study of relationsamongst the observable and physical properties of GRBs is highly relevant together with selectionbiases in their physical quantities.Therefore, we describe the state of the art of the existing GRB relations, their possible and debatedinterpretations in view of the current theoretical models and how relations are corrected for selectionbiases. We conclude that only after an appropriate evaluation and correction for selection effects canGRB relations be used to discriminate among the theoretical models responsible for the prompt andafterglow emission and to estimate cosmological parameters. Keywords: gamma rays bursts, accretion model, LT relation. a r X i v : . [ a s t r o - ph . H E ] A p r Contents1. Introduction 32. Notations 53. The Afterglow Relations 73.1. The Dainotti relation ( L X ( T a ) - T ∗ X,a ) 73.1.1. Physical interpretation of the Dainotti relation ( L X ( T a ) - T ∗ X,a ) 103.2. The unified L X ( T a )- T ∗ X,a and L O,a - T ∗ O,a relations 143.2.1. Physical interpretation of the unified L X ( T a )- T ∗ X,a and L O,a - T ∗ O,a relations 153.3. The L O, - α O,> relation and its physical interpretation 164. The Prompt-Afterglow Relations 184.1. The E γ,afterglow − E X,prompt relation and its physical interpretation 184.2. The L X,afterglow − E γ,prompt relation and its physical interpretation 204.3. The L X,a − L O,a relation and its physical interpretation 254.4. The L X ( T a ) − L γ,iso relation 274.5. The L X,peak − L X ( T a ) relation 284.5.1. Physical interpretation of the L X ( T a ) − L γ,iso and the L X,peak − L X ( T a ) relations 304.6. The L FO,peak − T ∗ FO,peak relation and its physical interpretation 315. Selection Effects 335.1. Redshift induced relations 335.2. Redshift induced relations through Efron and Petrosian method 355.2.1. Luminosity evolution 355.2.2. Time Evolution 365.3. Evaluation of the intrinsic slope 385.4. Selection effects for the optical and X-ray luminosities 385.5. Selection effects in the L O, − α O,> relation 396. Redshift Estimator 397. Cosmology 417.1. The problem of the calibration 417.2. Applications of GRB afterglow relations 438. Summary and discussion 479. Conclusions 4810. Acknowledgments 48 INTRODUCTIONGRBs, amongst the farthest and the most powerful objects ever observed in the Universe, are still a mystery after50 years from their discovery time by the Vela Satellites (Klebesadel et al. 1973). Phenomenologically, GRBs aretraditionally classified in short SGRBs ( T < T > T is the time in which the 90% (between 5% and 95%) of radiation isemitted in the prompt emission. However, Norris and Bonnell (2006) discovered the existence of an intermediateclass (IC), or SGRBs with Extended Emission (SGRBsEE), that shows mixed properties between SGRBs and LGRBs.Another relevant classification related to the spectral features distinguishing normal GRBs from X-ray Flashes (XRFs)appears. The XRFs (Heise et al. 2001; Kippen et al. 2001) are extra-galactic transient X-ray sources with spatialdistribution, spectral and temporal characteristics similar to LGRBs. The remarkable property that distinguishesXRFs from GRBs is that their νF ν prompt emission spectrum peaks at energies typically one order of magnitudelower than the observed peak energies of GRBs. XRFs are empirically defined by a greater fluence (time-integratedflux) in the X-ray band (2 −
30 keV) than in the gamma-ray band (30 −
400 keV). This classification is also relevantfor the investigation of GRB relations since some of them become stronger or weaker by introducing different GRBcategories, see sec. 3.1.One of the historical models used to explain the GRB phenomenon is the “fireball” model (Wijers et al. 1997; M´esz´aros1998, 2006) in which a compact central engine (either the collapsed core of a massive star or the merger product of aneutron star binary) launches a highly relativistic, and jetted electron/positron/baryon plasma. Interactions of blobswithin the jet are believed to produce the prompt emission, which consists of high photon energies such as gamma raysand hard X-rays. Instead, the interaction of the jet with the ambient material causes the afterglow phase, namely along lasting multi-wavelength emission (X-ray, optical and sometimes also radio), which follows the prompt. However,problems in explaining the light curves within this model have been shown by Willingale et al. (2007), hereafter W07.More specifically, for ∼
50% of GRBs, the observed afterglow is in agreement with the model, but for the rest, thetemporal and spectral indices do not conform and are suggestive of continued late energy injection. The difficulty ofthe standard fireball models appeared when Swift observations had revealed a more complex behaviour of the lightcurves (O’Brien et al. 2006; Sakamoto et al. 2007; Zhang et al. 2007b) than in the past and pointed out that GRBsoften follow “canonical” light curves (Nousek et al. 2006). In fact, the light curves can be divided into two, three andeven more segments. The second segment, when it is flat, is called plateau emission. X-ray plateaus can be interpretedas occurring due to an accreting black hole (BH) (Cannizzo and Gehrels 2009; Cannizzo et al. 2011; Kumar et al. 2008)or a top-heavy jet evolution (Duffell and MacFadyen 2015). In addition, the fact that a newly born magnetar could beformed either via the collapse of a massive star or during the merger of two neutron stars motivated the interpretationof the X-ray plateaus as resulting from the delayed injection of rotational energy ( ˙ E rot ∼ − erg s − ) from afast spinning magnetar (Usov 1992; Zhang and M´esz´aros 2001; Dall’Osso et al. 2011; Metzger et al. 2011; Rowlinsonand O’Brien 2012; Rowlinson et al. 2014; Rea et al. 2015). These models are summarized in sec. 3.1.1.Therefore, in this context, the discovery of relations amongst relevant physical parameters between prompt andplateau phases is very important so as to use them as possible model discriminators. In fact, many theoretical modelshave been presented in the literature to explain the wide variety of observations, but each model has some advantagesand drawbacks. The use of the phenomenological relations corrected for selection biases can boost the understandingof the mechanism responsible for such emissions. Moreover, being observed at much larger redshift range than the SNe,it has long been tempting to consider GRBs as useful cosmological probes, extending the redshift range by almost anorder of a magnitude further than the available SNe Ia, observed up to z = 2 .
26 (Rodney et al. 2015). Indeed, GRBsare observed up to redshift z = 9 . w , at very high z . So far, the most robust standard candlesare the SNe Ia which, by being excellent distance indicators, provide a unique probe for measuring the expansionhistory of the Universe whose discovery has been awarded the Nobel Prize in 2011 (Riess et al. 1998; Perlmutter et al.1998). Up-to-date, w has been measured to be − Λ , the purevacuum energy. Measurement of the Hubble constant, H , provides another constraint on w when combined withCosmic Microwave Background Radiation (CMBR) and Baryon Acoustic Oscillation (BAO) measurements (Weinberget al. 2013). Therefore, the use of other estimates provided by GRBs would be helpful to confirm further and/or The Swift satellite was launched in 2004. With the instruments on board, the Burst Alert Telescope (BAT, divided in four standardchannels 15-25; 25-50; 50-100; 100-150 keV), the X-Ray Telescope (XRT, 0.3-10 keV), and the Ultra-Violet/Optical Telescope (UVOT,170-650 nm), Swift provides a rapid follow-up of the afterglows in several wavelengths with better coverage than previous missions. constrain the ranges of values of H . However, different from the SNe Ia, which originate from white dwarves reachingthe Chandrasekhar limit and always releasing the same amount of energy, GRBs cannot yet be considered standardcandles with their isotropic energies spanning over 8 orders of magnitude. Therefore, finding out universal relationsamong observable properties can help to standardize their energetics and/or luminosities. It is for this reason that thestudy of GRB relations is relevant for both understanding the GRB emission mechanism, for finding a good distanceindicator and for estimating the cosmological parameters at high z .Until now, for cosmological purposes, the most used relations are the prompt emission relations: Amati (Amatiet al. 2002) and Ghirlanda relations (Ghirlanda et al. 2004). The scatter of these relations is significantly reducedproviding constraints on the cosmological parameters, see Ghirlanda et al. (2006) and Ghirlanda (2009) for details.By adopting a maximum likelihood approach which allows for correct quantification of the extrinsic scatter of therelation, Amati et al. (2008) constrained the matter density Ω M (for a flat Universe) to 0.04-0.40 (68% confidencelevel, CL), with a best-fit value of Ω M ∼ .
15, and exclude Ω M = 1 at > .
9% CL. Releasing the assumption ofa flat Universe, they found evidence for a low value of Ω M (0.04-0.50 at 68% CL) as well as a weak dependence ofthe dispersion of the relation between the prompt peak energy in the νF ν spectrum and the total gamma isotropicenergy, log E γ,peak − log E γ,iso , on Ω Λ (with an upper limit of Ω Λ ∼ .
15 at 90% CL). This approach makes noassumptions about the log E γ,peak − log E γ,iso relation and it does not use other calibrators to set the normalizationof the relation. Therefore, the treatment of the data is not affected by the so-called circularity problem (to calibratethe GRB luminosity relations for constraining cosmological models a particular cosmological model has to be assumeda priori) and the results are independent of those derived via SNe Ia (or other cosmological probes). Nowadays, thevalues of the cosmological parameters confirmed by measurements from the Planck Collaboration for the ΛCDM modelare Ω M = 0 . ± . Λ = 0 . ± . H = 67 . ± .
46 Km s − Mpc − . For the investigation ofthe properties of DE, Amati and Della Valle (2013) showed the 68% CL contours in the Ω M − Ω Λ plane obtained byassuming a sample of 250 GRBs expected shortly compared to those from other cosmological probes such as SNe Ia,CMB and Galaxy Clusters.They obtained the simulated data sets via Monte Carlo techniques by taking into account the slope, normalization, anddispersion of the observed log E γ,peak − log E γ,iso relation, the observed z distribution of GRBs and the distributionof the uncertainties in the measured values of log E γ,peak and log E γ,iso . These simulations indicated that with asample of 250 GRBs, the accuracy in measuring Ω M would be comparable to that currently provided by SNe data.In addition, they reported the estimates of Ω M and the parameter of the DE EoS, w , derived from the present andexpected future samples. They assumed that the log E γ,peak − log E γ,iso relation is calibrated with a 10% accuracy byusing, e.g., the luminosity distances provided by SNe Ia and the self-calibration of the relation with a large enoughnumber of GRBs lying within a narrow range of z (∆ z ∼ . − . M obtained in Amati and Della Valle (2013) may lead to GRBs as promising standardcandles, because they are almost comparable with SNe (0.06 for GRBs versus 0.04 for SNe, as provided for the SNesample by Betoule et al. 2014 and Calcino and Davis 2017), these results show that Ω M has an error which is 20 timeslarger then the value obtained by Planck. Thus, GRBs in a near future can be comparable with SNe Ia, but not likelywith Planck. On the other hand, there is discrepancy among the values of H computed by CMB and SNe (PlanckCollaboration et al. 2016) and thus adding a new effective cosmological probe as GRBs can help to cast light on thisdiscrepancy and break the degeneracy among several cosmological parameters.It is clear from this context that selection biases play a major and crucial role even for the close-by probes suchas SNe Ia in determining the correct cosmological parameters. This problem is more relevant for GRBs, which areparticularly affected by the Malmquist bias effect (Malmquist 1920, Eddington 1940) that favours the brightest objectsagainst faint ones at large distances. Therefore, it is necessary to investigate carefully the problem of selection effectsand how to overcome them before using GRB relations as distance estimators, as cosmological probes, and as modeldiscriminators. This is indeed the major aim of this review. Besides, this work is useful, especially for those embarkingon the study of GRB relations, because it aims at constituting a brief, but a complete compendium of afterglow andprompt-afterglow relations.The review is organized as follows: in section 2, we explain the nomenclature and definitions in all review, in sections3 and 4, we analyze the relations between the afterglow parameters and between parameters of both the prompt andafterglow phases. In section 5, we describe how these relations can be affected by selection biases. In section 6, wepresent how to obtain a redshift estimator and in section 7, we report the use of the Dainotti relation as an example ofGRB application as a cosmological tool. Finally, in section 8, we briefly summarize some findings about the physicalmodels and the cosmological usage of the analyzed relations, while in the last section we draw our conclusions. NOTATIONSFor clarity, we report a summary of the nomenclature adopted in the review. • L , E , F , S , and T indicate the luminosity, the energy, the flux, the fluence and the time which can be observedin several wavelengths, denoted with the first subscript, and at different times or part of the light curve, denotedinstead with the second subscript. In addition, with α , β and ν , we represent the temporal and spectral decayindices and the frequencies.More specifically: • T X,a and T O,a denote the time in the X-ray at the end of the plateau and the same time, but in the opticalwavelength respectively. F X,a are F O,a are their respective fluxes, while L X,a and L O,a are their respectiveluminosities. An approximation of the energy of the plateau is E X,plateau = ( L X,a × T ∗ X,a ), see the left panel ofFig. 1. • T O,peak and T X,f are the peak time in the optical and the time since ejection of the pulse. L O,peak and L X,f aretheir respective luminosities. F O,peak is the respective flux of T O,peak . • T X,peak is the peak time in the X-ray and F X,peak and L X,peak are its flux and luminosity respectively. • T X,p and T X,t are the time at the end of the prompt emission within the W07 model and the time at which theflat and the step decay behaviours of the light curves join respectively. • T and T are the times in which the 90% (between 5% and 95%) and 45% (between 5%-50%) of radiation isemitted in the prompt emission respectively. • τ lag and τ RT are the differences in arrival time to the observer of the high energy photons and low energy photonsand the shortest time over which the light curve increases by the 50% of the peak flux of the pulse. • L X, , L X, , L X, , L X, , L X, and L O, , L O, , L O, , L O, , L O, are the X-ray and optical luminositiesat 200 s, at 10, 11, 12 hours and at 1 day respectively; L O, s , L O, s , L O, s , L O, are the optical luminosityat 100 s, 1000 s, 10000 s and 7 hours; L γ,iso and L L ( ν, T X,a ) are the isotropic prompt emission mean luminosityand the optical or X-ray luminosity of the late prompt emission at the time T X,a . • F X, , F X, and F O, , F O, are the X-ray and optical fluxes at 11 hours and at 1 day respectively; F γ,prompt , F X,afterglow are the gamma-ray flux in the prompt and the X-ray flux in the afterglow respectively. E γ,prompt and E X,afterglow are their respective isotropic energies and L γ,prompt and L X,afterglow are the respective luminosities. S γ,prompt indicates the prompt fluence in the gamma band correspondent to the rest frame isotropic promptenergy E γ,prompt . • E O,afterglow , E γ,iso and E X,f are the optical isotropic energy in the afterglow phase, the total gamma isotropicenergy and the prompt emission energy of the pulse. • E k,aft , E γ,peak and E γ,cor are the isotropic kinetic afterglow energy in X-ray, the prompt peak energy in the νF ν spectrum and the isotropic energy corrected for the beaming factor. • α X,a , α O,> , α X,> , α ν,fl and α ν,st are the X-ray temporal decay index in the afterglow phase, in the opticalafter 200 s, in the X-ray after 200 s and the optical or X-ray flat and steep temporal decay indices respectively. • β X,a , β OX,a and β O,> are the spectral index of the plateau emission in X-ray, the optical-to-X-ray spectralindex for the end time of the plateau and the optical spectral index after 200 s. • ν X , ν O , ν c , ν m are the X-ray and optical frequencies, and the cooling and the peak frequencies of the synchrotronradiation.All the time quantities described above are given in the observer frame, while with the upper index ∗ we denote inthe text the observables in the GRB rest frame. The rest frame times are the observed times divided by the cosmictime expansion, for example, T ∗ X,a = T X,a / (1 + z ) denotes the rest frame time at the end of the plateau emission. Figure 1 . Left panel: the functional form of the fitting model from Willingale et al. (2007). Right panel: the observed light curve forGRB 061121 with the best-fit W07 model superimposed from Dainotti et al. (2016a). The red dot marks the end of the flat plateau phasein the X-ray afterglow ( T X,a , F X,a ). A similar configuration appears in the optical range.
In the following table we will give a list of the abbreviations/acronyms used through the text:
Abbreviation MeaningDE Dark EnergyEoS Equation of StateCL Confidence LevelIC Intermediate Class GRBSGRB Short GRBLGRB Long GRBsSGRBsEE Short GRBs with extended emissionXRFs X-ray FlashesSNe SupernovaeBH Black Holez redshiftFS Forward ShockRS Reverse Shock H Hubble constantΩ M Matter density in ΛCDM modelΩ Λ Dark Energy density in ΛCDM modelΩ k curvature in ΛCDM model σ log L X,a error on the luminosity σ log T ∗ X,a error on the timeE4 sample with σ E = ( σ L X,a + σ T ∗ X,a ) / < σ E = ( σ L X,a + σ T ∗ X,a ) / < . V Variability of the GRB light curve h Hubble constant divided by 100 w , w a coefficients of the DE EoS w ( z ) = w + w a z (1 + z ) − HD Hubble Diagrama normalization of the relationb slope of the relation σ int intrinsic scatter of the relation b int intrinsic slope of the relation Table 1 . Table with abbreviations. THE AFTERGLOW RELATIONSSeveral relations appeared in literature relating only parameters in the afterglow, such as the L X ( T a ) − T ∗ X,a relation(Dainotti et al. 2008) and similar ones in the optical and X-ray bands such as the unified L X ( T a )- T ∗ X,a and L O,a - T ∗ O,a (Ghisellini et al. 2009) and the L O, - α O,> relations (Oates et al. 2012).3.1.
The Dainotti relation ( L X ( T a ) - T ∗ X,a ) The first relation to shed light on the plateau properties has been the L X ( T a ) - T ∗ X,a one, hereafter also referred asLT. The phenomenon is an anti-relation between the X-ray luminosity at the end of the plateau, L X ( T a ), and the timein the X-ray at the end of the plateau, T ∗ X,a , for simplicity of notation we will refer to L X ( T a ) as L X,a .It was discovered by Dainotti et al. (2008) using 33 LGRBs detected by the Swift satellite in the X-ray energy bandobserved by XRT. Among the 107 GRBs fitted by W07 phenomenological model, shown in the left panel of Fig. 1,only the GRBs that have a good spectral fitting of the plateau and firm determination of z have been chosen. Thefunctional form of the LT relation obtained is the following:log L X,a = a + b × log T ∗ X,a , (1)with a normalization a = 48 .
54, a slope b = − . +0 . − . , an intrinsic scatter, σ int = 0 .
43 and a Spearman correlationcoefficient ρ = − . L X,a in the Swift XRT passband, ( E min , E max ) = (0 . ,
10) keV, has been computed from thefollowing equation: L X,a ( z ) = 4 πD L ( z, Ω M , h ) F X,a × K (2)where D L ( z, Ω M , h ) represents the GRB luminosity distance for a given z , F X,a indicates the flux in the X-ray atthe end of the plateau, and K = z ) (1 − βX,a ) denotes the K-correction for cosmic expansion (Bloom et al. 2001). Thisanti-relation shows that the shorter the plateau duration, the more luminous the plateau. Since the ratio between theerrors on both variables is close to unity, it means that both errors need to be considered and the Marquardt Levenbergalgorithm is not the best fitting method to be applied in this circumstance. Therefore, a Bayesian approach (D’Agostini2005) needs to be considered. This method takes into account the errors of both variables and an intrinsic scatter, σ int , of unknown nature. However, the results of both the D’Agostini method and the Marquardt Levenberg algorithmare comparable. Due to the higher accuracy of the first method from now on the authors prefer this technique in theirpapers. Evidently, the tighter the relation, the better the chances to constrain the cosmological parameters. With thisspecific challenge in mind, a subsample of bursts has been chosen with particular selection criteria both on luminosityand time, namely log L X,a >
45 and 1 ≤ log T ∗ X,a ≤
5. After this selection has been applied, a subsample of 28 LGRBswas obtained with ( a, b, σ int ) = (48 . , − . ± . , . b = − . +0 . − . , obtained by Dainotti et al. (2008) when the time is limited between1 ≤ log T ∗ X,a ≤ b = − . +0 . − . , while for the 8 IC GRBs pointed outa much steeper relation ( b = − . +0 . − . ). Finally, taking into account the errors on luminosity ( σ log L X,a ) and time( σ log T ∗ X,a ), the 8 GRBs with the smallest errors were defined as the ones with σ E = ( σ L X,a + σ T ∗ X,a ) / < . − . +0 . − . , see Fig. 2, the right panel of Fig. 3 and Table 2.Similar to Dainotti et al. (2010), also Bernardini et al. (2012a) and Sultana et al. (2012), with a sample of 64 and 14LGRBs respectively, found a slope b ≈ −
1, for details see Table 2.Expanding the sample again to 77 LGRBs, Dainotti et al. (2011a) discovered a relation with b = − . +0 . − . . Later,Mangano et al. (2012), considering in their sample of 50 LGRBs those GRBs with no visible plateau phase andemploying a broken power law as a fitting model, found a steeper slope ( b = − . +0 . − . ). Thus, from all these analysesit is clear that a steepening of the slope has been observed when the sample size is increased.Therefore, before going further with additional analysis, Dainotti et al. (2013a) decided to show how selection biasescan influence the slope of the relation. They showed that the steepening of the relation results from selection biases,while the intrinsic slope of the relation is b = − . +0 . − . , see section 5. Summarizing, Dainotti et al. (2013a) with a A computation of statistical dependence between two variables stating how good the relation between these variables can be representedemploying a monotonic function. It assumes a value between − l og L x * log Ta* 434445464748495051 1 1.5 2 2.5 3 3.5 4 4.5 5434445464748495051 1 1.5 2 2.5 3 3.5 4 4.5 5 ∆∆ ∆∆ ∆∆ ∆∆∆∆∆ ∆∆∆∆∆ ∆∆∆ T * a ( s ) l og L X ( e r g / s ) Figure 2 . Left panel: log L X,a (equivalent to log L ∗ X in this plot) vs. log T ∗ X,a for 62 long afterglows with the error energy parameter σ E <
4, and the best fitted relation line in black, from Dainotti et al. (2010). The red line fitted to the 8 lowest error (red) points producesan upper envelope of the full data set. The upper envelope points with the best fitted line are separately presented in an inset panel. Rightpanel: LONG-NO-SNe 128 GRBs (blue points fitted with a solid blue line) and the 19 events from LONG-SNe (red empty triangles) fittedwith a red dashed line from Dainotti et al. (2016c) sample of 101 GRBs, confirmed the previous results from Dainotti et al. (2010), as well as Rowlinson et al. (2014),with a data set of 159 GRBs.
Figure 3 . Left panel: the plateau flux versus the plateau duration for a sample of 22 SGRBs from Rowlinson et al. (2013). Blue stars areGRBs with two or more breaks in their light curves, while green circles have one break. Right panel: log L X,a versus log F X,a for the fullGRB sample from Dainotti et al. (2010). The 8 upper envelope points are shown as red squares, while the IC GRBs are represented bygreen triangles.
Dainotti et al. (2015b) also confirmed previous results of Dainotti et al. (2013a) but with a larger sample of 123LGRBs. All the samples discussed are observed by SWIFT/XRT.In the context of reducing the scatter of the LT relation, Del Vecchio et al. (2016) investigated the temporal decayindices α X,a after the plateau phase for a sample of 176 GRBs detected by Swift within two different models: a simplepower law, considering the decaying phase after the plateau phase, and the W07 one. It is pointed out that the resultsare independent of the chosen model. It was checked if there are some common characteristics in GRBs phenomenathat can allow them to be used as standardizable candles like SNe Ia and to obtain some constraints revealing which isthe best physical interpretation describing the plateau emission. The interesting result is that the LT relation for thelow and high luminosity GRBs seems to depend differently on the α X,a parameter, thus possibly implying a diversedensity medium.
Author N Type Slope Norm Corr.coeff. PDainotti et al. (2008) 28 1 < T ∗ X,a < − . +0 . − . . × − Dainotti et al. (2008) 33 All GRBs − . +0 . − . − Cardone et al. (2009) 28 L − . +0 . − . − Ghisellini et al. (2009) 33 L − . +0 . − . − Cardone et al. (2010) 66 L − . +0 . − . . +0 . − . -0.68 7 . × − Dainotti et al. (2010) 62 L − . +0 . − . . +1 . − . -0.76 1 . × − Dainotti et al. (2010) 8 high luminosity − . +0 . − . . +0 . − . -0.93 1 . × − Dainotti et al. (2010) 8 IC − . +0 . − . . +1 . − . -0.66 7 . × − Dainotti et al. (2011a) 77 L − . +0 . − . . +0 . − . -0.69 7 . × − Sultana et al. (2012) 14 L − . +0 . − . . +0 . − . -0.88 10 − Bernardini et al. (2012) 64 L − . +0 . − . . × − Mangano et al. (2012) 50 L − . +0 . − . . +0 . − . -0.81 2 . × − Dainotti et al. (2013a) 101 ALL intrinsic − . +0 . − . − Dainotti et al. (2013b) 101 All GRBs − . +0 . − . . +0 . − . -0.74 10 − Dainotti et al. (2013b) 101 without short − . +0 . − . − Dainotti et al. (2013b) 101 simulated − . +0 . − . . +0 . − . -0.74 10 − Postnikov et al. (2014) 101 L ( z < . − . +0 . − . . +0 . − . -0.74 10 − Rowlinson et al. (2014) 159 intrinsic − . +0 . − . − Rowlinson et al. (2014) 159 observed − . +0 . − . . +0 . − . -0.74 10 − Rowlinson et al. (2014) 159 simulated − . +0 . − . . +0 . − . -0.74 10 − Dainotti et al (2015) 123 L − . +0 . − . . +0 . − . -0.74 10 − Dainotti et al. (2016c) 19 L-SNe − . +0 . − . . +0 . − . -0.83 5 × − Table 2 . Summary of the LT relation. All the measurements are performed by the Swift XRT Telescope. The first column represents theauthors, the second one the number of GRBs in the used sample, the third one the GRB type (S=Short, L=Long, IC=Intermediate), thefourth and the fifth ones are the slope and normalization of the relation and the last two columns are the correlation coefficient and thechance probability, P.
Continuing the search for a standard set of GRBs, Dainotti et al. (2016c) analyzed 176 GRB afterglow plateausobserved by Swift with known redshifts which revealed that the subsample of LGRBs associated with SNe (LONG-SNe) presents a very high correlation coefficient for the LT relation. They investigated the category of LONG GRBsassociated spectroscopically with SNe in order to compare the LT correlation for this sample with the one for LGRBsfor which no associated SN has been observed (hereafter LONG-NO-SNe, 128 GRBs). They checked if there is adifference among these subsamples. They adopted first a non-parametric statistical method, the Efron and Petrosian(1992) one, to take into account redshift evolution and check if and how this effect may steepen the slope for theLONG-NO-SNe sample. This procedure is necessary due to the fact that this sample is observed at much higherredshift than the GRB-SNe sample. Therefore, removing selection bias is the first step before any comparison amongsamples observed at different redshifts could be properly performed. They have demonstrated that there is no evolutionfor the slope of the LONG-NO-SNe sample and no evolution is expected for the LONG-SNe sample. The differenceamong the slopes is statistically significant with the probability P=0.005 for LONG-SNe. This possibly suggests thatthe LONG-SNe with firm spectroscopic features of the SNe associated might not require a standard energy reservoir inthe plateau phase unlike the LONG-NO-SNe. Therefore, this analysis may open new perspectives in future theoreticalinvestigations of the GRBs with plateau emission and associated with SNe. They also discuss how much this differencecan be due to the jet opening angle effect. The difference between the SNe-LONG (A+B) and LONG-NO-SNe sampleis only statistically significant at the 10% level when we consider the beaming correction. Thus, one can argue that thedifference in slopes can be partially due to the effect of the presence of low luminosity GRBs in the LONG-SNe samplethat are not corrected for beaming. However, the beaming corrections are not very accurate due to indeterminate jetopening angles, so the debate remains open and it can only be resolved when we will gather more data.In Table 2, we report a summary of the parameters a and b with ρ and P for the LT relation. In conclusion, the mostreliable parameters for this relation are those from Dainotti et al. (2013a), because they have demonstrated that theintrinsic slope not affected by selection biases is determined to be − Physical interpretation of the Dainotti relation ( L X ( T a ) - T ∗ X,a ) Here, we revise the theoretical interpretation of the LT relation, which is based mainly on the accretion (Cannizzoand Gehrels 2009; Cannizzo et al. 2011) and the magnetar models (Zhang and M´esz´aros 2001; Dall’Osso et al. 2011;Rowlinson and O’Brien 2012; Rowlinson et al. 2013, 2014).The first one states that an accretion disc is created from the motion of the material around the GRB progenitorstar collapsing towards its progenitor core. After it is compressed by the gravitational forces, the GRB emission takesplace. For LGRBs, the early rate of decline in the initial steep decay phase of the light curve may provide informationabout the radial density distribution within the progenitor (Kumar et al. 2008).Cannizzo and Gehrels (2009) predicted a steeper relation slope (-3/2) than the observed one ( ∼ − − cm can give anestimate for the plateau duration of around 10 s for LGRBs maintaining the initial fall back mass at 10 − solar masses( M (cid:12) ), see the left panel of Fig. 4. For SGRBs the radius is estimated to be 10 cm. The LT relation provides alower limit for the accreting mass estimates ∆ M ≈ − to 10 − M (cid:12) . From their results, it was claimed that the LTrelation could be obtained if a typical energy reservoir in the fall-back mass is assumed, see the right panel of Fig. 4.However, in their analysis the very steep initial decay following the prompt emission, which have been modelled byLindner et al. (2010) as fall-back of the progenitor core, is not considered. Figure 4 . Left panel: model light curves for LGRB parameters from Cannizzo et al. (2011), keeping the starting fall-back disk massconstant at 10 − M (cid:12) but changing the initial radius and normalization. Right panel: total accretion mass for the plateau + later decayphases of GRBs from Cannizzo et al. (2011), considering 62 LGRBs from Dainotti et al. (2010). The region in red represents a limitingXRT detection flux level f II = 10 − erg cm − s − (assuming a plateau duration t II = 10 s) in order to study a plateau to sufficientaccuracy. A beaming factor f = 1 /
300 and a net efficiency for powering the X-ray flux (cid:15) net = (cid:15) acc (cid:15) X = 0 .
03 were assumed.
Regarding the magnetar model, Zhang and M´esz´aros (2001) studied the effects of an injecting central engine on theGRB afterglow radiation, concentrating on a strongly magnetized millisecond pulsar. For specific starting values ofrotation period and magnetic field of the pulsar, the afterglow light curves should exhibit an achromatic bump lastingfrom minutes to months, and the observation of such characteristics could set some limits on the progenitor models.More recently, Dall’Osso et al. (2011) investigated the energy evolution in a relativistic shock from a spinning downmagnetar in spherical symmetry. With their fit of few observed Swift XRT light curves and the parameters of thismodel, namely a spin period of (1 − B ∼ − G), they managed towell reproduce the properties of the shallow decay phase and the LT relation, see the left panel of Fig. 5.Afterward, Bernardini et al. (2012a) with a sample of 64 LGRBs confirmed, as previously founded by Dall’Osso et al.(2011), that the shallow decay phase of the GRB light curves and the LT relation can be well explained.Then, Rowlinson and O’Brien (2012) and Rowlinson et al. (2013) pointed out that energy injection is a fundamentalmechanism for describing the plateau emission of both LGRBs and SGRBs. In fact, the remnant of NS-NS mergers This value can be derived considering the total inferred accretion mass ∆
M/M = ∆ E X /f − ∗ (cid:15) acc ∗ c where c is the light speed, f isthe X-ray afterglow beaming factor, (cid:15) acc is the efficiency of the accretion onto the BH and E X is the observed total energy of the plateau+ later decay phases (the integral over time between T X,t and the end of afterglow, see Eq. 2 of W07).
100 1000 10 − − − . . L u m i no s it y ( e r g s − ) Observer’s Time (s)E imp,50 =100 B =10 P i,ms =1E imp,50 = 10 B =10 P i,ms =1E imp,50 = 10 B = 3 P i,ms =1E imp,50 = 10 B = 3 P i,ms =3E imp,50 = 1 B ,14 =3 P i,ms =3 Figure 5 . Left panel: five theoretical light curves obtained by Dall’Osso et al. (2011), changing the initial energy of the afterglow, thedipole magnetic field, B, and the initial spin period of the NS, P. Right panel: the grey shaded areas are the homogeneous distribution ofB and P employed to simulate the observable magnetar plateaus from Rowlinson et al. (2014). The upper and lower limits on B and theupper limit on P are computed considering the sample of GRBs fitted with the magnetar model (Lyons et al. 2010; Dall’Osso et al. 2011;Bernardini et al. 2012a; Gompertz et al. 2013; Rowlinson et al. 2013; Yi et al. 2014; L¨u and Zhang 2014). The dashed black vertical line (1)at 0.66 ms is the minimum P allowed. The dotted black line (2) indicates a limit on P and B strengths imposed by the fastest slew timeof XRT in their sample in the rest frame of the highest z GRB, as plateaus with durations shorter than the slew time are unobservable.The black dash-dot lines (3-6) are the observational constraints for the dimmest XRT plateau observable assuming the lowest z in theGRB sample. These limits vary as a function of the beaming and efficiency of the magnetar emission: (3) Minimum beaming angle andefficiency (1 degree and 1% respectively), (4) Minimum efficiency (1%) and maximum beaming angle (isotropic), (5) Maximum efficiency(100%) and minimum beaming angle, (6) Maximum efficiency and beaming angle. The observed distributions indicate that the sampleshave low efficiencies and small beaming angles. Later, Rowlinson et al. (2014), using 159 GRBs from Swift catalogue, analytically demonstrated that the centralengine model accounts for the LT relation assuming that the compact object is injecting energy into the forward shock(FS), a shock driven out into the surrounding circumstellar medium. The luminosity and plateau duration can becomputed as follows: log L X,a ∼ log( B p P − R ) (3)and log T ∗ X,a = log(2 . × IB − p P R − ) , (4)where T ∗ X,a is in units of 10 s, L X,a is in units of 10 erg s − , I is the moment of inertia in units of 10 g cm , B p is the magnetic field strength at the poles in units of 10 G, R is the radius of the NS in units of 10 cm and P is the initial period of the compact object in milliseconds. Then, substituting the radius from eq. 4 into eq. 3, it wasderived that: log ( L X,a ) ∼ log (10 I − P − ) − log ( T ∗ X,a ) . (5)Therefore, an intrinsic relation log L X,a ∼ − log T ∗ X,a is confirmed directly from this formulation. Although somemagnetar plateaus are inconsistent with energy injection into the FS, Rowlinson et al. (2014) showed that this emissionis narrowly beamed and has ≤
20% efficiency in conversion of rotational energy from the compact object into theobserved plateau luminosity. In addition, the intrinsic LT relation slope, namely the one where the selection biasesare appropriately removed, is explained within the spin-down of a newly formed magnetar at 1 σ level, see right panelof Fig. 5. The scatter in the relation is mainly due to the range of the initial spin periods.After several papers discussing the origin of the LT relation within the context of the magnetar model, very recently adebate has been opened by Rea et al. (2015) on the reliability of this model as the correct interpretation for the X-rayplateaus. Using GRBs with known z detected by Swift from its launch to August 2014, Rea et al. (2015) concludedthat the initial magnetic field distribution, used to interpret the GRB X-ray plateaus within the magnetar model does2not match the features of GRB-magnetars with the Galactic magnetar population. Therefore, even though there arelarge uncertainties in these estimates due to GRB rates, metallicity and star formation, the GRB-magnetar model in itspresent form is safe only if two kinds of magnetar progenitors are allowed. Namely, the GRB should be different fromGalactic magnetar ones (for example for different metallicities) and should be considered supermagnetars (magnetarswith an initial magnetic field significantly large). Finally, they set a limit of about ≤
16 on the number of stablemagnetars produced in the Milky Way via a GRB in the past Myr. However, it can be argued that since the rates ofGalactic magnetars and GRBs are really different, the number of Galactic magnetars cannot fully describe the originof GRBs. In fact the Galactic magnetar rate is likely to be greater than 10% than the core collapse SNe rate, whileGRB rate is much lower than that. In addition, the number of magnetars in the Milky Way may not be used as aconstraint on the GRB rate because the spin-down rates of GRB magnetars should be very rapid. Due to the lowGRB rate it would not be easy to detect these supermagnetars. Thus, it can be claimed that no conflict stands amongthis paper and the previous ones.
Figure 6 . Optical and X-ray light curves for wind (left panel) and ISM (right plot) scenario’s from van Eerten (2014a). Thick light greycurves represent the analytical solutions for prolonged and impulsive energy injection. Thick dashed light grey and the thick dotted lightgrey curves indicate the forward shock region emission only and the reverse shock region only respectively. The grey vertical lines show(1) the arrival time of emission from the jet back and (2) the arrival time of emission from the jet front. The solid vertical lines indicatearrival times of emission along the jet axis for these two events; the dashed vertical lines express the arrival times of emission from an angle θ = 1 /γ . Still in the context of the energy injection models, van Eerten (2014a) found a relation between the optical fluxat the end of the plateau and the time at the end of the plateau itself F O,a ∼ T − . ± . O,a (Panaitescu and Vestrand2011; Li et al. 2012) for which observed frame variables were considered. The range of all parameters describing theemission ( E γ,iso , the fraction of the magnetic energy, (cid:15) B , the initial density, n ) is the principal cause of the scatterin the relation, but it does not affect the slope. Finally, it was claimed that both the wind ( ∝ A/r , where A is aconstant) and the interstellar medium can reproduce the observed relation within both the reverse shock (RS, a shockdriven back into the expanding bubble of the ejecta) and FS scenarios, see Fig. 6.Considering alternative models explaining the LT relation, Sultana et al. (2013) studied the evolution of the Lorentzgamma factor, Γ = 1 / (cid:112) − v /c (where v is the relative velocity between the inertial reference frames and c is thelight speed), during the whole duration of the light curves within the context of the Supercritical Pile Model. Thismodel provides an explanation for both the steep-decline and the plateau or the steep-decline and the power law decayphase of the GRB afterglow, as observed in a large number of light curves, and for the LT relation. One of their mostimportant results is that the duration of the plateau in the evolution of Γ becomes shorter with a decreasing value of M c , where M is the initial rest mass of the flow. This occurrence means that the more luminous the plateau, theshorter its duration and the smaller the M c , namely the energy.Instead, in the context of the RS and FS emissions, Leventis et al. (2014), investigating the synchrotron radiation inthe thick shell scenario (i.e. when the RS is relativistic), found that this radiation is compatible with the presence ofthe plateau phase, see the left panel of Fig. 7. In addition, analyzing the log F X,a - log T X,a relation in the frameworkof this model, they arrived at the conclusion that smooth energy injection through the RS is favoured respect to theFS, see the right panel of Fig. 7.3
Figure 7 . Left panel: optical and X-ray light curves before and after the injection break from Leventis et al. (2014). The contributions ofthe FS (dotted line) and RS (dashed line) are shown for both. The considered parameters are E = 10 erg, n = 50 cm − , ∆ t = 5 × s, η = 600, q=0, (cid:15) e = (cid:15) B = 0 .
1, p=2.3, θ j = 90 o , d = 10 cm and z = 0 .
56. Right panel: index of the F X,a − T X,a relation as a functionof the electron distribution index, p, for the FS and the RS from Leventis et al. (2014). The lightly shaded region includes values allowedby the scaling from Panaitescu and Vestrand (2011), while the darker region indicates the scaling from Li et al. (2012). The five dashedlines show the five possible indices for the FS, while the three solid lines display the three possible (independent of p) indices for the RS.
Figure 8 . Comparison of the slopes for 1000 thin shell data set runs and slopes of the observed LT relation in optical (horizontal direction)and the LT relation in X-ray (vertical direction) from van Eerten (2014b) for the FS (left panel) and the RS (right panel) cases. Grey bandexpresses 1 σ errors on the relations, while green dots represent runs consistent at 1 σ error bars for both, orange dots are compatible at3 σ , but not at 1 σ and red dots pass neither test. Vertical grey lines show more scattered LT in optical error bars from Panaitescu andVestrand (2011). van Eerten (2014b), with a simulated sample of GRBs, found out that the observed LT relation rules out basic thinshell models, but not basic thick ones. In fact, in the thick model, the plateau phase comes from the late centralsource activity or from additional kinetic energy transfer from slower ejecta which catches up with the blast wave. Asa drawback, in this context, it is difficult to distinguish between FS and RS emissions, or homogeneous and stellarwind-type environments.In the thin shell case, the plateau phase is given by the pre-deceleration emission from a slower component in a two-component or jet-type model, but this scenario is not in agreement with the observed LT relation, see Fig. 8. This,however, does not imply that acceptable fits using a thin shell model are not possible, but further analysis is neededto exclude without any doubts thin shell models. Another model which has not been tested yet on this correlation isthe photospheric emission model from stratified jets (Ito et al. 2014).4 3.2. The unified L X ( T a ) - T ∗ X,a and L O,a - T ∗ O,a relations
In order to describe the unified picture of the X-ray and optical afterglow, it is necessary to introduce relevantfeatures regarding optical luminosities. To this end, Bo¨er and Gendre (2000) studied the afterglow decay index in 8GRBs in both X-ray and optical wavelengths. In the X-ray, the brightest GRBs had decay indices around 1 . .
11. Instead, they didn’t observe this trend for the optical light curves,probably due to the host galaxy absorption.Later, Nardini et al. (2006) discovered that the monochromatic optical luminosities at 12 hours, L O, , of 24 LGRBscluster at log L O, = 30 .
65 erg s − Hz − , with σ int = 0 .
28. The distribution of L O, is less scattered than the one of L X, , the luminosity at 12 hours in the X-ray, and the one of the ratio L O, /E γ,prompt , where E γ,prompt is the restframe isotropic prompt energy. Three bursts are outliers because they have luminosity which is smaller by a factor ∼
15. This result suggests the existence of a family of intrinsically optically underluminous dark GRBs, namely GRBswhere the optical-to-X-ray spectral index, β OX,a , is shallower than the X-ray spectral index minus 0.5, β X,a − . L O, using 44 GRBs. Nardiniet al. (2008a) also confirmed these findings. They analyzed selection effects present in their observations extendingthe sample to 55 LGRBs with known z and rest-frame optical extinction detected by the Swift satellite.In contrast, Melandri et al. (2008), Oates et al. (2009), Zaninoni et al. (2013) and Melandri et al. (2014) found nobimodality in the distributions of L O, , L O, and L O, , investigating samples of 44, 24, 40 and 47 GRBs respectively.Instead, with the aim of finding a unifying representation of the GRB afterglow phase, Ghisellini et al. (2009) fittedthe light curves assuming this functional form: L L ( ν, t ) = L L ( ν, T X,a ) ( t/T
X,t ) − α ν,fl t/T X,t ) α ν,st − α ν,fl . (6)They used a data sample of 33 LGRBs detected by Swift in X-ray (0.3-10 keV) and optical R bands (see the left andmiddle panels of Fig. 9). Within this approximation, the agreement with data is reasonably good, and they confirmedthe X-ray LT relation. Figure 9 . The light curves of the full sample from Ghisellini et al. (2009) in the X-rays (left panel) and optical (middle panel). Thevertical lines represent log L X, and log L O, in the rest frame time respectively. Instead, the dashed lines indicate the log t − / (blue)and the log t − / (red) behaviours. Right panel: relation between L O,peak (equivalent to L R,p in the picture) and T ∗ O,peak of the data setfrom Liang et al. (2010). Line represents the best fit.
Through their analysis using a data sample of 32 Swift GRBs, Liang et al. (2010) found that the optical peakluminosity, L O,peak , in the R band in units of 10 erg s − and the optical peak time, T ∗ O,peak , are anti-correlated, seethe right panel of Fig. 9, with a slope b = − . ± .
39 and ρ = − .
90. They deduced that a fainter bump has itsmaximum later than brighter ones and it also presents a longer duration.Panaitescu and Vestrand (2011) showed a similar relation to the one presented in Liang et al. (2010). They found alog F O,a ∼ log T − O,a anti-relation using 37 Swift GRBs. This result may indicate a unique mechanism for the opticalafterglow even though the optical energy has a quite large scatter.Afterwards, Li et al. (2012) found a relation (see the left panel of Fig. 10) similar to the LT relation, but in the Rband. They used 39 GRBs with optical data available in the literature. This relation is between the optical luminosityat the end of the plateau, L O,a , in units of 10 erg s − and the optical end of the plateau time, log T ∗ O,a , in the shallow5
Figure 10 . Left panel: L SO,a (equivalent to L SR,p in the picture) as a function of T S, ∗ O,a (equivalent to t b in the picture) from Li et al.(2012). The grey circles represent the X-ray data from Dainotti et al. (2010). Lines correspond to the best fit lines. Middle and Rightpanels: L SO,a and T SO,a distributions for the full GRB data set from Li et al. (2012). decay phase of the GRB light curves, denoted with the index S. They found a slope b = − . ± . ρ = 0 .
86 and
P < − . Correlations Author N Slope Corr.coeff. P L O,peak − T O,peak
Liang et al. (2010) 32 − . ± .
39 -0.90 L O,a − T O,a
Panaitescu & Vestrand (2011) 37 − L SO,a - T SO,a
Li et al. (2012) 39 − . ± .
08 0 . < − Table 3 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, andthe third one the number of GRBs in the used sample. Afterwards, the fourth column is the slope of the relation and the last two columnsare the correlation coefficient and the chance probability, P. L SO,a varies from 10 to 10 erg s − , and in some GRBs with an early break reaches ∼ erg s − , see the middlepanel of Fig. 10. T SO,a spans from tens of seconds to several days after the GRB trigger, with a typical shallow peaktime T SO,a of ∼ seconds, see the right panel of Fig. 10. By plotting L O,a in units of 10 erg s − as a function of T ∗ O,a in the burst frame, they observed that optical data have a similar trend to the X-ray data. In fact, this powerlaw relation, presented in the left panel of Fig. 10, with an index of − . ± .
08 is similar to that derived for theX-ray flares (see sec. 4.6). XRF phenomena are described in sec. 1. As a consequence, they recovered the LT relation.In Table 3 a summary of the relations described in this section is displayed.3.2.1.
Physical interpretation of the unified L X ( T a ) - T ∗ X,a and L O,a - T ∗ O,a relations
In the unified L X ( T a )- T ∗ X,a and L O,a - T ∗ O,a relations Ghisellini et al. (2009) considered the flux as the sum ofsynchrotron radiation caused by the standard FS due to the fireball impacting the circumburst medium and of anothercomponent may be produced by a long-lived central engine, which resembles mechanisms attributed to a “late prompt”.Even if based in part on a phenomenological model, Ghisellini et al. (2009) explained situations in which achromaticand chromatic jet break are either present or not in the observed light curves.In addition, from their analysis, the decay slope of the late prompt emission results to be α X,a = − / ∼ log t − / , see red dashed line for X-rayand for optical emission in the left and middle panels of Fig. 9 respectively). This explains the activity of the centralengine for such a long duration. For a similar interpretation within the context of the accretion onto the BH relatedto LT relation see sec. 3.1.1.Liang et al. (2010) claimed that the external shock model explains well the anti-relation between L O,peak and T O,peak ,because later deceleration time is equivalent to slower ejecta and thus to a less luminous emission.Furthermore, Panaitescu and Vestrand (2008) from the analysis of the log L O,a − log T ∗ O,a relation explained the peakyafterglows (those with L O,a ∝ T − O,a ) as being a bit outside the cone of view, while the plateau as off-axis events anddue to the angular structure of the jet. Later, Panaitescu and Vestrand (2011) asserted that the double-jet structureof the ejecta is problematic. To overcome this issue, they suggested a model in which both the peaky and plateauafterglows depend on how much time the central engine allows for the energy injection. More specifically, impulsiveejecta with a narrow range of Γ are responsible for the peaky afterglows, while the plateau afterglows are produced bya distribution of initial Γ which keeps the energy injection till 10 s.6Later, Li et al. (2012) pointed out that late GRB central engine activities can account for both optical flares and theoptical shallow-decay segments. These activities can be either erratic (for flares) or steady (for internal plateaus). Anormal decay follows the external plateaus with α X,a typically around −
1, thus possibly originated by an externalshock with the shallow decay segment caused by continuous energy injection into the blast wave (Rees and M´esz´aros1998; Dai and Lu 1998; Sari and M´esz´aros 2000; Zhang and M´esz´aros 2001). Instead, the internal plateaus, found firstby Troja et al. (2007) in GRB 070110 and later studied statistically by Liang et al. (2007), are followed by a muchsteeper decay ( α X,a steeper than -3), which needs to be powered by internal dissipation of a late outflow. In summary,the afterglow can be interpreted as a mix of internal and external components.3.3.
The L O, - α O,> relation and its physical interpretation
Oates et al. (2012) discovered a relation between the optical luminosity at 200 s, log L O, , and the optical temporaldecay index after 200 s, α O,> , see the right panel of Fig. 11. They used a sample of 48 LGRB afterglow lightcurves at 1600 ˚A detected by UVOT on board of the Swift satellite, see the left panel of Fig. 11. The best fit line forthis relation is given by: log L O, = (28 . ± . − (3 . ± . × α O,> , (7)with ρ = − .
58 and a significance of 99.998% (4.2 σ ). This relation means that the brightest GRBs decay fasterthan the dimmest ones. To obtain the light curves employed for building the relation, they used the criteria fromOates et al. (2009) in order to guarantee that the entire UVOT light curve is not noisy, namely with a high signalto noise (S/N) ratio: the optical/UV light curves must be observed in the V filter of the UVOT with a magnitude ≤ .
8, UVOT observations must have begun within the first 400 s after the BAT trigger and the afterglow must havebeen observed until at least 10 s after the trigger. Their results pointed out the dependence of this relation is on thedifferences in the observing angle and on the rate of the energy release from the central engine.As a further step, Oates et al. (2015), using the same data set, investigated the same relation both in optical andin X-ray wavelengths in order to make a comparison, and they confirmed previous optical results finding a similarslope for both relations. In addition, they analyzed the connection between the temporal decay indices after 200 s(in X-ray and optical) obtaining as best fit relation α X,> = α O,> − .
25, see the left panel of Fig. 12. Theyyielded some similarities between optical and X-ray components of GRBs from these studies. Their results were indisagreement with those previously found by Urata et al. (2007), who investigated the relation between the opticaland X-ray temporal decay indices in the normal decay phase derived from the external shock model. In fact, a goodfraction of outliers was found in this previous work.Racusin et al. (2016) studied a similar relation using 237 Swift LGRBs, but in X-ray. For this relation, it was found thatslope b = − . ± .
04 and solid evidence for a strong connection between optical and X-ray components of GRBs wasdiscovered as well. In conclusion, the Monte Carlo simulations and the statistical tests validated the relation betweenlog L O, and α O,> by Oates et al. (2012). In addition, it shows a possible connection with its equivalent, the LTrelation in X-ray, implying a common physical mechanism. In Table 4 a summary of the relations described in thissection is reported.Regarding the physical interpretation of the log L O, - α O,> relation, Oates et al. (2012) explored severalscenarios. The first one implies that the relation can be due to the interaction of the jet with the external medium. Ina straightforward scenario α O,> is not a fixed value and all optical afterglows stem from only one closure relationwhere α O,> and β O,> are related linearly. Thus a relation between log L O, and β O,> should naturallyappear. Contrary to this expectation, α O,> and β O,> are poorly correlated, see the right panel of Fig. 12, andthere is no evidence for the existence of a relation between β O,> and log L O, . Therefore, this scenario cannotbe ascribed as the cause of the log L O, - α O,> relation.In the second scenario, they assumed that the log L O, - α O,> relation is produced by few closure relationsindicated by lines in the right panel of Fig. 12. However, from this picture, the α O,> and β O,> values withsimilar luminosities do not gather around a particular closure relation, thus also the basic standard model is not agood explanation of the log L O, − α O,> relation. As a conclusion, the afterglow model is more complex than itwas considered in the past. It is highly likely that there are physical properties that control the emission mechanismand the decay rate in the afterglow that still need to be investigated.Therefore, Oates et al. (2012) proposed two additional alternatives. The first is related to some properties of thecentral engine influencing the rate of energy release so that for fainter afterglows, the energy is released more slowly.Otherwise, the relation can be due to different observing angles where observers at smaller viewing angles see brighter7
Figure 11 . Left panel: “optical light curves of 56 GRBs from Oates et al. (2012)”. Right panel: “log L O, vs. α O,> from Oateset al. (2012). The red solid line indicates the best fit line and the blue dashed lines show the 3 σ variance”. Figure 12 . Left panel: “ α O,> and α X,> from Oates et al. (2015). The red solid line represents the best fit regression and theblue dashed lines represent 3 times the root mean square (RMS) deviation. The relationships expected between the optical/UV X-ray lightcurves from the GRB closure relations are also shown. The pink dotted line represents α O,> = α X,> . The light blue dotted-dashedlines represent α X,> = α O,> ± .
25. In the top right corner it is given the coefficient ρ with P , and it is provided the best fit slopeand constant determined by linear regression”. Right panel: “ α O,> and β O,> for the sample of 48 LGRBs from Oates et al. (2012).The lines represent 3 closure relations and a colour scale is used to display the range in log L O, ”. and faster decaying light curves. Correlations Author N Slope Norm Corr.coeff. P L O, - α O,>
Oates et al. (2012) 48 − . ± .
004 28 . ± . − .
58 2 × − Oates et al. (2015) 48 − . ± .
004 28 . ± . − .
58 2 × − L X, - α X,>
Racusin et al. (2016) 237 − . ± . − . ± .
11 0 .
59 10 − Table 4 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, andthe third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are the slope and normalization of therelation and the last two columns are the correlation coefficient and the chance probability, P.
As pointed out by Dainotti et al. (2013a), the log L O, - α O,> relation is related to the LT one since both showan anti-relation between luminosity and decay rate of the light curve or time. The key point would be to understandhow they relate to each other and the possible common physics that eventually drives both of them. To this end, Oateset al. (2015) compared the observed relations with the ones obtained with the simulated sample. The luminosity-decay8relationship in the optical/UV is in agreement with that in the X-ray, inferring a common mechanism. THE PROMPT-AFTERGLOW RELATIONSAs we have discussed in the previous paragraphs, the nature of the plateau and the relations (e.g. the opticalone) based on similar physics and directly related to the plateau are still under investigation. For this reason, severalmodels have been proposed. To further enhance its theoretical understanding, it is necessary to evaluate the connectionbetween plateaus and prompt phases. To this end, we hereby review the prompt-afterglow relations, thus helping toestablish a more complete picture of the plateau GRB phenomenon.4.1.
The E γ,afterglow − E X,prompt relation and its physical interpretation
W07 analyzed the relation between the gamma flux in the prompt phase, F γ,prompt , and the X-ray flux in theafterglow, F X,afterglow using 107 Swift GRBs, see the upper left panel of Fig. 13. They calculated F X,afterglow in theXRT band (0.3-10 keV), while F γ,prompt in the BAT (15-150 keV) plus the XRT (0.3-10 keV) energy band. For GRBswith known redshift, as shown in the upper right panel of Fig. 13, they investigated the prompt isotropic energy, E γ,prompt , and the afterglow isotropic energy, E X,afterglow , assuming a cosmology with H = 71 km s − Mpc − ,Ω Λ = 0 .
73 and Ω M = 0 . E γ,prompt and E X,afterglow using a sample of53 LGRBs. They pointed out a good relation with b = 1 ± .
16, see the bottom left panel of Fig. 13.In agreement with these results, Liang et al. (2010) and Panaitescu and Vestrand (2011) analyzed this relation, usingrespectively 32 and 37 GRBs, but considering energy bands different from that used in Liang et al. (2007); theyobtained the slopes b = 0 . ± .
14 and b = 1 .
18 respectively (see the left and middle panels of Fig. 14).Rowlinson et al. (2013) and Grupe et al. (2013) confirmed these results, see the left and middle panels of Fig. 15.In fact, they obtained a E γ,prompt − E X,afterglow relation with slope b ∼ E X,afterglow to E γ,prompt ,considering a sample of 123 LGRBs, see the right panel of Fig. 15.Instead, Ghisellini et al. (2009), with a sample of 33 LGRBs, considered a similar relation, but assuming the X-rayplateau energy, E X,plateau , as an estimation of E X,afterglow , see the bottom right panel of Fig. 13; they found a slope b = 0 . E γ,prompt and the kinetic isotropic energy inthe afterglow, E k,aft , with the same sample, finding a relation with b = 0 .
42. Similarly, Racusin et al. (2011) studiedthe same relation, using 69 GRBs and assuming different efficiencies to find some limits between E k,aft and E γ,prompt ,see the right panel of Fig. 14.This relation was most likely used to study the differences in detection of several instruments and to analyze thetransferring process of kinetic energy into the prompt emission in GRBs, making the relation by Racusin et al. (2011)the most reliable one.To summarize, for comparing the energies in the prompt and the afterglow phases, a E γ,prompt − E X,afterglow relationwas studied by Liang et al. (2007) and confirmed by Rowlinson et al. (2013), Grupe et al. (2013) and Dainotti et al.(2015b). The last study found also some limitations on the ratio among the prompt and the afterglow energies.Furthermore, instead of E X,afterglow , E X,plateau was considered for the investigation, although this quantity providedsimilar results to the previous ones (Ghisellini et al. 2009). Finally, the relation between E γ,prompt and E k,aft wasstudied by Ghisellini et al. (2009) and confirmed by Racusin et al. (2011), who examined the energy transfer in theprompt phase. These relations are relevant because of their usefulness for investigating the efficiency of the emissionprocesses during the transition from the prompt phase to the afterglow one, and for explaining which the connectionbetween these two emission phases is. As a main result, Ghisellini et al. (2009) and Racusin et al. (2011) claimed thatthe fraction of kinetic energy transferred from the prompt phase to the afterglow one is around 10%. In particular,Racusin et al. (2011) yielded that this value of the transferred kinetic energy, for BAT-detected GRBs, is in agreementwith the analysis by Zhang et al. (2007a) for which the internal shock model well describes this value in the case of alate energy transfer from the fireball to the surrounding medium (Zhang and Kobayashi 2005).In Table 5, a summary of the relations described in this section is presented.As regards the physical interpretation of the E X,afterglow − E γ,prompt relation, Racusin et al. (2011), estimating theefficiency parameter η for the BAT sample, confirmed the Zhang et al. (2007a) result for which ∼
57% of BAT burstshave η <
Figure 13 . Upper left panel: the F X,afterglow in the XRT band (0.3-10 keV) vs. F γ,prompt computed from the BAT T flux (15-150keV) plus the XRT flux (0.3-10 keV) from Willingale et al. (2007). The dotted line represents where F X,afterglow and F γ,prompt areidentical. Upper right panel: log E γ,prompt vs. log E X,afterglow from Willingale et al. (2007). Symbols show the position of the afterglowin the β X,a - α X,a plane. GRBs that fall in the pre-jet-break region are plotted as dots, those that fall above this in the post-jet-breakregion are plotted as stars, and those below the pre-jet-break band are plotted as squares. The dotted line represents equality betweenlog E γ,prompt and log E X,afterglow . Bottom left panel: the log E γ,prompt − log E X,afterglow relation ( E γ,iso and E X,iso respectively in thepicture) from Liang et al. (2007). The solid line is the best fit. The dashed line indicates the 2 σ area. Bottom right panel: log E X,plateau vs. log E γ,prompt from Ghisellini et al. (2009). The dashed line represents the least square fit with log E X,plateau ( T a L T a in the picture) ∼ . × log E γ,prompt ( E γ,iso in the picture) ( P = 2 × − , without the outlier GRB 070125). (LAT), on board the Fermi satellite , they found that only 25% of the GBM bursts and none of the LAT bursts have η < The Fermi Gamma ray Space Telescope (FGST), launched in 2008 and still running, is a space observatory being used to performgamma ray astronomy observations from low Earth orbit. Its main instrument is the Large Area Telescope (LAT), an imaging gamma raydetector, (a pair-conversion instrument) which detects photons with energy from about 20 MeV to 300 GeV with a field of view of about20% of the sky; it is a sequel to the EGRET instrument on the Compton gamma ray observatory (CGRO). Another instrument aboardFermi is the Gamma Ray Burst Monitor (GBM), which is used to study prompt GRBs from 8 keV to 30 MeV. Figure 14 . Left panel: “relation between E γ,prompt and E O,afterglow ( E γ,iso and E R,iso respectively in the picture), for the opticallyselected sample, from Liang et al. (2010). Line is the best fit”. Middle panel: “relation between log E γ,prompt and log E O,afterglow ( E γ,iso and L p × t p respectively in the picture) from Panaitescu and Vestrand (2011). Black symbols are for afterglows with optical peaks, redsymbols for optical plateaus, open circles for afterglows of uncertain type. r (log E X,afterglow , log E γ,prompt ) = 0 .
66 for all 37 afterglows.This linear correlation coefficients correspond to a probability P = 10 − . ”. Right panel: “ E k,aft as a function of E γ,prompt from Racusinet al. (2011). The dashed lines indicated different values of η . The bursts detected by LAT on board of Fermi tend to have high E γ,prompt ( E γ,iso in the picture), but average E k,aft , and therefore higher values of η than the samples from BAT on board of Swift or GBM onboard of Fermi”.
50 51 52 53 54 554950515253
5) bursts with circles. Right panel: < log E γ,prompt > vs. log E X,afterglow relation from Dainottiet al. (2015b) for 123 LGRBs. The solid line for equal log E γ,prompt and log E X,afterglow is given for reference.Correlations Author N Slope Norm Corr.coeff. P E X,afterglow − E γ,prompt Liang et al. (2007) 53 1 . +0 . − . − . +8 . − . < − E O,afterglow − E γ,prompt Liang et al. (2010) 32 0 . +0 . − . . +0 . − . < − Panaitescu & Vestrand (2011) 37 1.18 0.66 10 − . E X,plateau − E γ,prompt Ghisellini et al. (2009) 33 0.86 2 × − E k,aft − E γ,prompt Ghisellini et al. (2009) 33 0.42 10 − Table 5 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, andthe third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are the slope and normalization of therelation and the last two columns are the correlation coefficient and the chance probability, P.
The L X,afterglow − E γ,prompt relation and its physical interpretation Berger (2007) investigated the prompt and afterglow energies in the observed frame of 16 SGRBs. A large fractionof them (80%) follows a linear relation between the prompt fluence in the gamma band, S γ,prompt , in the BAT rangeand the X-ray flux at 1 day, F X, , in the XRT band given by:log F X, ∼ (1 . ± . × log S γ,prompt , (8)with ρ = 0 .
86 and P = 5 . × − . Gehrels et al. (2008) confirmed his results investigating the same relation, butwith X-ray fluxes at 11 hours, F X, , see Fig. 16.1 Figure 16 . F X, - S γ,prompt ( F X − ray and S γ − ray respectively in the picture) relation for Swift SGRBs (in red) and LGRBs (in blue) fromGehrels et al. (2008) . The XRT F X, are computed at 3 keV and the BAT S γ,prompt are detected between 15 and 150 keV (Sakamotoet al. 2008). Later, Nysewander et al. (2009) considered the relation between F X, or the optical flux at 11 hours, F O, , and E γ,prompt , finding an almost linear relation, see Fig. 17. They used a data set of 37 SGRBs and 421 LGRBs detectedby Swift. Panaitescu and Vestrand (2011) confirmed, in part, these results. They found a similar relation between E γ,prompt and F O,a using 37 GRBs, but with a higher slope ( b = 1 . L X, ∝ E γ,prompt , where L X, is the X-ray luminosity at 10 hourscalculated in the 2-10 keV energy range, while E γ,prompt in the 20-2000 keV energy range, see the left panel of Fig.19. This relation compares four long events spectroscopically associated with SNe with “regular” energetic LGRBs( E γ,prompt ∼ − erg). The results possibly indicate a common efficiency η for transforming kinetic energy intogamma rays in the prompt phase for both these four events and for “regular” energetic LGRBs.The same relation has been studied in the context of the low luminosity versus normal luminosity GRBs. Indeed,Amati et al. (2007) found that the relation between L X, , in the 2-10 keV band, and E γ,prompt , in the 1-10000 keVband, becomes stronger ( P ∼ − ) including sub-energetic GRBs as GRB 060218, GRB 980425 and GRB 031203, seethe middle panel of Fig. 19. Therefore, it is claimed that sub-energetic GRBs are intrinsically faint and are consideredto some extent normal cosmological GRBs.Finally, Berger (2007) also analyzed the relation between the X-ray luminosity at one day, L X, , and E γ,prompt , using13 SGRBs with measured z . They found a slope b = 1 . ± .
16 (see the right panel of Fig. 18).Liang et al. (2010) confirmed his results in the optical range using a sample of 32 Swift GRBs ( E γ,prompt − L O,peak with b = 1 . ± .
08, see the right panel of Fig. 19). In addition, Kann et al. (2010) also confirmed his results with a2
Figure 17 . Upper left panel: “a plot of F O, (corrected for Galactic extinction) vs. 15-150 keV S γ,prompt for both LGRBs (grey) andSGRBs (red) from Nysewander et al. (2009). Note that below a fluence of 10 − erg cm − , no optical afterglow of an SGRB has beendiscovered, while above 10 − , all reasonably deep observing campaigns, but one (GRB 061210) have detected an optical afterglow”. Upperright panel: “a plot of F X, vs. 15-150 keV S γ,prompt ( E γ,iso in the picture) for both LGRBs (grey) and SGRBs (red) from Nysewanderet al. (2009)”. Bottom left panel: “a plot of L O, (corrected for Galactic extinction) vs. E γ,prompt from Nysewander et al. (2009). Dashedupper limits represent SGRBs with a host galaxy determined by XRT error circle only. The classification of GRB 060614 and GRB 060505is uncertain, therefore, they are labelled as “possibly short” ”. Bottom right panel: “a plot of L X, vs. E γ,prompt from Nysewander et al.(2009). The open circles represent SGRBs with a host galaxy determined by XRT error circle only. The classification of GRB 060614 andGRB 060505 is uncertain, therefore, they are labelled as “possibly short” ”. sample of 76 LGRBs ( E γ,prompt − L O, with b = 0 .
36, see the left panel of Fig. 20).Similarly, Dainotti et al. (2011b) analyzed the relation between log L X,a and log E γ,prompt − analyser/. Their sample has been divided into two subsamples: E4 formed of 62 LGRBs and E0095consisting of 8 LGRBs, assuming σ E as a parameter representing the goodness of the fit. For the E4 subsample it was3 Figure 18 . Left panel: E γ,prompt - F O,a ( E γ,iso and F p respectively in the picture) relation from Panaitescu and Vestrand (2011). Blacksymbols are for afterglows with optical peaks, red symbols for optical plateaus, open circles for afterglows of unknown kind. Right panel: L X, vs. E γ,prompt ( E γ,iso in the picture) for the SGRBs with a known z (solid black circles), redshift constraints (open black circles)and without any redshift information (grey symbols connected by dotted lines) from Berger (2007). Figure 19 . Left panel: “ L X , of SN-GRBs (source frame: 2 −
10 keV) as a function of their E γ,prompt , E γ,iso in the picture, (20 − z for each event is also shown in colour”. Middle panel: “ L X, (in 2-10 keV range) vs. E γ,prompt ( E iso in the picture) for the events included in the sample of Nousek et al. (2006) (triangles) plus the 3 sub-energetic GRB 980425, GRB 031203,GRB 060218, the other GRB/SN event GRB 030329 (circles), and 3 GRBs with known z and deep limits to the peak magnitude of associatedSN, XRF 040701, GRB 060505 and GRB 060614 (diamonds) from Amati et al. (2007). Empty triangles indicate those GRBs for which the1-10000 keV E γ,prompt was computed based on the 100-500 keV E γ,prompt reported by Nousek et al. (2006) by assuming an average spectralindex. The plotted lines are the best-fit power laws obtained without (dotted) and with (dashed) sub-energetic GRBs and GRB 030329”.Right panel: “relation between E γ,prompt and L O,peak ( E γ,iso and L R,p respectively in the picture) for the optically selected sample fromLiang et al. (2010). Line is the best fit”. found: log L X,a = 28 . +2 . − . + 0 . +0 . − . × log E γ,prompt , (9)with ρ = 0 .
43 and P = 1 . × − , while for the E0095 subsamplelog L X,a = 29 . +7 . − . + 0 . +0 . − . × log E γ,prompt , (10)with ρ = 0 .
83 and P = 3 . × − . Thus, it was concluded that the small error energy sample led to a higher relationand to the existence of a subset of GRBs which can yield a “standardizable candle”. Furthermore, since log L X,a andlog T ∗ X,a are strongly correlated, and the slope is roughly -1, the energy reservoir of the plateau is roughly constant.Since log E γ,peak and log E γ,prompt are both linked with log L X,a , then the log E γ,peak − log E γ,prompt − log E X,plateau relation is straightforward. For its modification taking into account log E γ,iso of the whole X-ray light curves seeBernardini et al. (2012b). As further confirmations of the L X,a − E γ,prompt relation, D’Avanzo et al. (2012) andMargutti et al. (2013) found a relation between log L X,a and E γ,prompt with slope b ∼ ρ ≈ .
70, using 58 and297 Swift LGRBs respectively.Furthermore, Berger (2014) studied the relation between the X-ray luminosities at 11 hours, L X, , and E γ,peak , and4 Figure 20 . Left panel: “ F O, in the R band plotted against the bolometric E γ,prompt ( E iso,bol in the picture) for all GRBs in theoptically selected sample from Kann et al. (2010) (except GRB 991208, which was only discovered after several days, and GRBs 060210,060607A, 060906 and 080319C, where the follow-up does not extend to one day). While no tight relation is visible, there is a trend ofincreasing optical luminosity with increasing prompt energy release. This is confirmed by a linear fit (in log-log space), using a MonteCarlo analysis to account for the asymmetric errors. The dashed line shows the best fit, while the dotted line marks the 3 σ error region.Several special GRBs are marked”. Middle panel: “ L X, vs. E γ,prompt for SGRBs (blue) and LGRBs (grey) from Berger (2014). Opensymbols for SGRBs indicate events without a known z , for which a fiducial value of z = 0 .
75 is assumed. The dashed blue and red linesare the best-fit power law relations to the trends for SGRBs and LGRBs, respectively, while the dotted black line is the expected relationbased on the afterglow synchrotron model with ν X > ν c and p = 2 . L X, ∼ . × log E γ,prompt ). The inset shows the distributionof the ratio log( L X, × (11 hr) . /E . γ,prompt ), for the full samples (thick lines) and for the region where SGRBs and LGRBs have equal E γ,prompt values (thin lines). The lower level of L X, relative to E γ,prompt for SGRBs is evident from these various comparisons”. Rightpanel: same as in the middle panel, “but for the isotropic-equivalent afterglow optical luminosity at a rest frame time of 7 hours ( L O, ),still from Berger (2014). The dotted black line is the expected relation based on the afterglow model for ν m < ν O < ν c and p = 2 . L O, ∼ . × log E γ,prompt ). The inset shows the distribution of the ratio log( L O, × (7 hr) . /E . γ,prompt ), for the full samples (thicklines) and for the region where SGRBs and LGRBs have equal E γ,prompt values (thin lines). The lower level of L O, relative to E γ,prompt for SGRBs is evident from these various comparisons”. Figure 21 . Left panel: log L O, -log E γ,prompt ( E iso in the picture) relation from Oates et al. (2015). Right panel: log L X, -log E γ,prompt ( E iso in the picture) relation from Oates et al. (2015). the relation between the optical luminosity at 7 hours L O, and E γ,peak for a sample of 70 SGRBs and 73 LGRBsdetected mostly by Swift. He found that the observed relations are flatter than the ones simulated by Kann et al.(2010), see the middle and right panels of Fig. 20.Regarding the relation between E γ,prompt and the optical luminosities, Oates et al. (2015) analyzed the relation between L O, or L X, and log E γ,prompt with a sample of 48 LGRBs. They claimed a strong connection between promptand afterglow phases, see Fig. 21 and Table 6. This relation permits to study some important spectral characteristicsof GRBs, the optical and X-ray components of the radiation process and the standard afterglow model. In Table 6, asummary of the relations described in this section is shown.Regarding the physical interpretation of the L X,afterglow − E γ,prompt relation, Gehrels et al. (2008) underlined that theoptical and X-ray radiation are characterized by β OX,a ≈ .
75. This value matches the slow cooling case, importantat 11 hours, when the electron distribution power law index is p = 2 . ν m < ν O < ν X < ν c .Oates et al. (2015) pointed out that within the standard afterglow model, the log E γ,prompt − (log L O, , log L X, )relations are expected. However, the slopes of the simulated and observed relations are inconsistent at > σ due to5values set for the η parameter. If the distribution of the efficiencies is not sufficiently narrow the relation will be moredisperse. Thus, the simulations repeated with η = 0 . η = 0 . > σ . Correlations Author N Slope Norm Corr.coeff. P F X, − S γ,prompt Berger (2007) 16 1 . +0 . − . . × − F X, − S γ,prompt Gehrels et al. (2008) 111 0 . +0 . − . . +0 . − . × − Gehrels et al. (2008) 10 0 . +0 . − . . +1 . − . F O, − E γ,prompt Nysewander et al. (2009) 421 ∼ F O, − E γ,prompt Nysewander et al. (2009) 37 ∼ F X, − E γ,prompt Nysewander et al. (2009) 421 ∼ F X, − E γ,prompt Nysewander et al. (2009) 37 ∼ F O,a − E γ,prompt Panaitescu&Vestrand (2011) 37 1.67 0.75 10 − . L X, − E γ,prompt Berger (2007) 13 1 . +0 . − . . +0 . − . . × − L O,peak − E γ,prompt Liang et al. (2010) 32 1 . +0 . − . . +0 . − . − L O, − E γ,prompt Kann et al. (2010) 76 0.36 L X,a − E γ,prompt Dainotti et al. (2011b) 62 0 . +0 . − . . +2 . − . . × − Dainotti et al. (2011b) 8 0 . +0 . − . . +7 . − . . × − D’Avanzo et al. (2012) 58 ∼ ≈ . ∼ ≈ . L X, − E γ,prompt Berger (2014) 73 0.72 44.75Berger (2014) 70 0.83 43.93 L O, − E γ,prompt Berger (2014) 73 0.73 43.70Berger (2014) 70 0.74 42.84 L X, − E γ,prompt Oates et al. (2015) 48 1 . +0 . − . − . +7 . − . . × − L O, − E γ,prompt Oates et al. (2015) 48 1 . +0 . − . − . +6 . − . . × − Table 6 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, andthe third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are the slope and normalization of therelation and the last two columns are the correlation coefficient and the chance probability, P.
The L X,a − L O,a relation and its physical interpretation
In the observed frame, Jakobsson et al. (2004) studied the log F O, versus log F X, distribution, in the optical Rband and in the 2 −
10 keV band respectively, using all known GRBs with a detected X-ray afterglow, see the left panelof Fig. 22. Different from the previous definition of dark bursts (where dark bursts were simply defined as those burstsin which the optical transient is not observed), they defined these bursts as GRBs where the optical-to-X-ray spectralindex, β OX,a , is shallower than the X-ray spectral index minus 0.5, β X,a − .
5. They found out 5 dark bursts among52 observed by Beppo-SAX . This analysis aimed at distinguishing dark GRBs through Swift. Gehrels (2007) andGehrels et al. (2008) confirmed the results using a data sample of 19 SGRBs and 37 LGRBs+6 SGRBs respectively,see the middle and right panels of Fig. 22. In particular, Gehrels et al. (2008) obtained a slope b = 0 . ± .
03 forLGRBs and b = 0 . ± .
45 for SGRBs.Instead in the rest-rest frame, Berger (2014) studied the relation between L O, and L X, on 70 SGRBs and 73LGRBs, finding some similarities between SGRBs and LGRBs and a central value < L O, /L X, > ≈ .
08, see the leftpanel of Fig. 23.Oates et al. (2015) improved their study. They analyzed a similar relation with a sample of 48 LGRBs, but using L O, and L X, , see the right panel of Fig. 23. The slope obtained has a value b = 0 . ± . Beppo-SAX, (1996-2003), was an Italian-Dutch satellite capable of simultaneously observing targets over more than 3 decades of energy,from 0 . . −
30 keV and from 100 −
600 keV). The first four instruments point to the same directionallowing observations in the broad energy range (0.1-300 keV). With the WFC it was possible to model the afterglow as a simple powerlaw, mainly due to the lack of observations during a certain period in the GRB light curve. Figure 22 . Left panel: log F O, -log F X, ( F opt and F X respectively in the plot) distribution for the data set from Jakobsson et al.(2004). Filled symbols show optical detections while open symbols represent upper limits. Lines of constant β OX,a are displayed with thecorresponding value. Dark bursts are those which have β OX,a < .
5. Middle panel: F X, - F O, relation for Swift SGRBs and LGRBsfrom Gehrels (2007). Comparison is made to pre-Swift GRBs and to lines of optical to X-ray spectral index from Jakobsson et al. (2004).The grey points indicate LGRBs, the black points represent SGRBs and the small black points without error bars are the pre-Swift GRBs.Right panel: F O, - F X, relation for Swift SGRBs (shown in red) and LGRBs (shown in blue) from Gehrels et al. (2008). The threecircled bursts are those for which z > .
9. The pre-Swift GRBs taken from Jakobsson et al. (2004) are presented in green. Also the darkburst separation line β OX,a = 0 . β OX,a = 1 . Figure 23 . Left panel: “ L O, vs. L X, from Berger (2014). The dotted black line marks a linear relation, expected for ν X ∼ ν c . Theinset shows the distribution of the ratio L O, /L X, , indicating that both SGRBs and LGRBs exhibit a similar ratio, and that in general L O, /L X, ∼
1, indicative of ν X ∼ ν c for SGRBs”. Right panel: “log L O, vs. log L X, from Oates et al. (2015). The red solid linerepresents the best fit regression and the blue dashed line represents 3 times the RMS deviation. In the top right corner, it is given ρ and P and it is provided the best-fit slope and constant determined by linear regression”.Correlations Author N Slope Norm Corr.coeff. P F X, − F O, Gehrels et al. (2008) 6 0 . ± .
45 0 . ± .
94 0 .
06 0 . . ± .
03 1 . ± .
04 0 .
44 0 . L X, − L O, Berger (2014) 70 0.0873 0.08 L X, − L O, Oates et al. (2015) 48 0 . ± .
22 1 . ± .
94 0.81 5 . × − Table 7 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, andthe third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are the slope and normalization of therelation and the last two columns are the correlation coefficient and the chance probability, P. L X,a − L O,a relation, Berger (2014) showed that, in the context of thesynchrotron model, the comparison of L O, and L X, indicated that usually ν c is near or higher than the X-ray band.Indeed, LGRBs have often greater circumburst medium densities (about 50 times greater than SGRBs) and therefore ν c ∼ ν X . 4.4. The L X ( T a ) − L γ,iso relation In Dainotti et al. (2011b) the connections between the physical properties of the prompt emission and log L X,a wereanalyzed using a sample of 62 Swift LGRBs. A relation was found between log L X,a in the XRT band and the isotropicprompt luminosity, log < L γ,iso > ≡ log( E γ,prompt /T ), in the BAT energy band, see the left panel of Fig. 24. Thisrelationship can be fitted with the following equation:log L X,a = 20 . +6 . − . + 0 . +0 . − . × log < L γ,iso > , (11)obtaining ρ = 0 .
59 and P = 7 . × − . In this paper log L X,a was related to several prompt luminosities definedusing different time scales, such as T , T (the time in which the 45% between 5%-50% of radiation is emitted inthe prompt emission), and T X,p (the time at the end of the prompt emission within the W07 model). Furthermore,the E4 (defined in Table 1) subsample of 62 LGRBs with known z from a sample of 77 Swift LGRBs and the E0095subsample of 8 GRBs with smooth light curves were used, see black and red points in the left panel of Fig. 24.Therefore, it has been shown that the GRB subsample with the strongest correlation coefficient for the LT relationalso implies the tightest prompt-afterglow relations. This subsample can be used as a standard one for astrophysicaland cosmological studies.In the middle panel of Fig. 24, the correlation coefficients ρ are shown for the following distributions: (log < L γ,iso > , log < L γ,iso > , log < L γ,iso > T X,p , log E γ,prompt ) − log L X,a , represented by different colours, namely red, black,green and blue respectively.No significant relations for the IC bursts have been found out. However, the paucity of the data does not allow adefinitive statement. From this analysis, it is clear that the inclusion of the IC GRB class does not strengthen theexisting relations.
E4 E0095Correlations ρ (b, a) ρ (b, a)P P L X,a − < L γ,iso > . +0 . − . , . +6 . − . ) 0.95 (0 . +0 . − . , . +5 . − . )0.62 7 . × − . × − L X,a − < L γ,iso > . +0 . − . , . +7 . − . ) 0.93 (0 . +0 . − . , . +3 . − . )0.62 7 . × − . × − L X,a − < L γ,iso > T X,p . +0 . − . , . +4 . − . ) 0.95 (0 . +0 . − . , . +3 . − . )0.56 2 . × − . × − L X,a − E γ,prompt . +0 . − . , . +2 . − . ) 0.83 (0 . +0 . − . , . +7 . − . )0.52 1 . × − . × − T ∗ X,a − E γ,prompt -0.19 ( − . +0 . − . , . +0 . − . ) -0.81 ( − . +0 . − . , . +0 . − . )-0.21 1 . × − -0.69 5 . × − L X,a − E γ,peak . +0 . − . , . +0 . − . ) 0 .
74 (1 . +0 . − . , . +2 . − . )0.51 2 . × − . × − T ∗ X,a − E γ,peak -0.36 ( − . +0 . − . , . +0 . − . ) -0.74 ( − . +0 . − . , . +1 . − . )-0.35 5 . × − -0.77 2 . × − < L γ,iso > − E γ,peak . +0 . − . , . +0 . − . ) 0.76 (1 . +0 . − . , . +1 . − . )0.67 2 . × − . × − Table 8 . Correlation coefficients ρ , the respective relation fit line parameters (a, b), and the correlation coefficient r with the respectiverandom occurrence probability P , for the considered prompt-afterglow and prompt-prompt distributions in log scale from Dainotti et al.(2011b). In general, this study pointed out that the plateau phase results connected to the inner engine. In addition, alsorelations between log L X,a and several other prompt emission parameters were analyzed, including log E γ,peak and8the variability, log V . As a result, relevant relations are found between these quantities, except for the variabilityparameter, see Table 8. ρ u ρ L* a ,
5) burstsby circles.
Finally, as shown in Table 8, only a very small relationship exists between log T ∗ X,a − log E γ,prompt with ρ = − . < L γ,iso > and log L X,a (see the right panelof Fig. 24) and between log < L γ,iso > and log T ∗ X,a using a sample of 232 GRBs. The latter can be derivedstraightforwardly from the log T ∗ X,a − log E γ,prompt relation, being log < L γ,iso > computed as log( E γ,prompt /T ).4.5. The L X,peak − L X ( T a ) relation Dainotti et al. (2015b) further investigated the prompt-afterglow relations presenting an updated analysis of 123Swift BAT+XRT light curves of LGRBs with known z and afterglow plateau phase. The relation between the peakluminosity of the prompt phase in the X-ray, log L X,peak , and log L X,a can be written as follows:log L X,a = A + B × log L X,peak , (12)with A = − . ± . B = 1 . +0 . − . and with ρ = 0 .
79 and
P < .
05, see the left panel of Fig. 25. In theliterature L X,peak is denoted as: L X,peak = 4 π × D L ( z, Ω M , h ) × F X,peak × K. (13)The relation < log L γ,iso > − log L X,a (Dainotti et al. 2011b) for the same GRB sample presented a correlationcoefficient, ρ = 0 .
60, smaller than the one of the log L X,peak − log L X,a relation, see sec. 4.4. This implied that abetter definition of the luminosity or energy parameters improves ρ by 24%. In the left panel of Fig. 25 log L X,peak is calculated directly from the peak flux in X-ray, F X,peak , considering a broken or a simple power law as the bestfit of the spectral model. Thus, the error propagation due to time and energy is not involved, differently fromthe earlier considered luminosities. In addition, Dainotti et al. (2015b) preferred the log L X,peak − log L X,a to therelations presented in Dainotti et al. (2011b), namely the (log E γ,prompt , log E γ,peak ) − log L X,a , due to the fact thatlog E γ,prompt and log E γ,peak can undergo double bias truncation due to high and low energy detector threshold.Instead, this problem does not appear for log L X,peak (Lloyd and Petrosian 1999). Furthermore, to show that thelog L X,peak − log L X,a relation is robust, the redshift dependence induced by the distance luminosity was eliminatedemploying fluxes rather than luminosities. A relation between log F X,a and log F X,peak was obtained with ρ = 0 . σ , from the one of the9 Figure 25 . Left panel: “GRB distributions in redshift bins on the log L X,a -log L X,peak plane from Dainotti et al. (2015b), wherelog L X,peak is computed using the approach used in the Second BAT Catalogue. The sample is split into 4 different equipopulated redshiftbins: z ≤ .
84 (blue), 0 . ≤ z < . . ≤ z < . z ≥ . F X,a − log F X,peak plane from Dainotti et al. (2015b), where log F X,peak is computedfollowing the approach used in the Second BAT Catalogue. The sample is split into 4 different equipopulated redshift bins: z ≤ .
84 (blue),0 . ≤ z < . . ≤ z < . z ≥ . Figure 26 . Left panel: log L -log T ∗ relation for all the pulses in the prompt (black symbols) and in the afterglow (red symbols) emissionsfrom Dainotti et al. (2015b). log L is the same as log L X,f for the prompt emission pulses, while indicates log L X,a for the afterglowemission pulses, and, the time log T ∗ indicates log T ∗ X,f for the prompt emission pulses and log T ∗ X,a for the afterglow phase. The greenpoints show the maximum luminosity prompt emission pulses (log T Lmax , log L max ). Right panel: log E vs. log T ∗ relation for all thepulses in the prompt (black symbols) and in the afterglow (red symbols) emissions from Dainotti et al. (2015b). log E represents log E X,f for the prompt emission pulses, while it represents log E X,plateau for the afterglow emission pulses, and the time log T ∗ indicates log T ∗ X,f for the prompt emission pulses and log T ∗ X,a for the afterglow phase. The yellow points display the maximum energy prompt emissionpulses (log T Emax , log E max ). prompt phase relation between the time since ejection of the pulse and the respective luminosity, log L X,f − log T ∗ X,f (Willingale et al. 2010), see the left panel of Fig. 26. This difference also implied a discrepancy in the distributions ofthe energy and time, see the right panel of Fig. 26. The interpretation of this discrepancy between the slopes opens anew perspective in the theoretical understanding of these observational facts, see the next section for details.As a further step, Dainotti et al. (2016a) analyzed this relation adding as a third parameter T X,a with a sample of 122LGRBs (without XRFs and GRBs associated to SNe). They found a tight relation:log L X,a = (15 . ± .
8) + (0 . ± . × log L X,peak − (0 . ± .
07) log T X,a , (14)with ρ = 0 . P ≤ . × − , and σ int = 0 . ± .
03. Additionally, the scatter could be further reduced consideringthe subsample of 40 LGRBs having light curves with good data coverage and flat plateaus:log L X,a = (15 . ± .
3) + (0 . ± . × log L X,peak − (0 . ± .
1) log T X,a , (15)with ρ = 0 . P = 4 . × − , and σ int = 0 . ± .
04. These results may suggest the use of this plane as a0“fundamental” plane for GRBs and for further cosmological studies.4.5.1.
Physical interpretation of the L X ( T a ) − L γ,iso and the L X,peak − L X ( T a ) relations In Dainotti et al. (2015b), the two distinct slopes of the luminosity-duration and the energy-duration distributions ofprompt and plateau pulses could reveal that these two are different characteristics of the radiation: the former may begenerated by internal shocks and the latter by the external shocks. Indeed, if the plateau is produced by synchrotronradiation from the external shock, then all the pulses have analogous physical conditions (e.g. the power law index ofthe electron distribution). In addition, the prompt-afterglow connections were analyzed in order to better explain theexisting physical models of GRB emission predicting the log L X,a − log L γ,iso and the log L X,peak − log L X,a relationstogether with the LT one in the prompt and afterglow phases. They claimed that the model better explaining theserelationships is the one by Hasco¨et et al. (2014). In this work they considered two scenarios: one in the standard FSmodel assuming a modification of the microphysics parameters to decrease the efficiency at initial stages of the GRBevolution; in the latter the early afterglow stems from a long-lived RS in the FS scenario. In the FS scenario a windexternal medium is assumed together with a microphysics parameter (cid:15) e ∝ n − ν , the amount of the internal energygoing into electrons (or positrons), where n is the density medium. In the case of ν ≈ L X,a − log L X,iso and the log L X,peak − log L X,a relations. Alternatively, in the RS scenario, in order to obtain theobserved prompt-afterglow relationships, the typical Γ of the ejecta should rise with the burst energy.
Figure 27 . “The histogram of the BAT to XRT flux ratio for a number of Swift GRB from Kazanas et al. (2015). The distribution showsclearly a preferred value for this ratio of order ∼ − . The vertical line shows also the proton to electron mass ratio m p /m e ”. In addition, Ruffini et al. (2014) pointed out that the induced gravitational collapse paradigm can recover thelog L X,a − log L γ,iso and the log L X,peak − log L X,a relations. This model considers the very energetic (10 − erg)1LGRBs for which the SNe has been seen. The light curves were built assuming for the external medium either a radialstructure for the wind (Guida et al. 2008; Bernardini et al. 2006, 2007; Caito et al. 2009) or a partition of the shell(Dainotti et al. 2007), therefore well matching the afterglow plateau and the prompt emission.Recently, Kazanas et al. (2015) within the context of the Supercritical Pile GRB Model claimed that they can reproducethe log L X,a − log L γ,iso and the log L X,peak − log L X,a relations, because the ratio, R, between the luminosities appearsconsistent with the one between the mean prompt energy flux from BAT and the afterglow plateau fluxes detected byXRT. In particular, R ≈ The L FO,peak − T ∗ FO,peak relation and its physical interpretation
Liang et al. (2010) studied the relation between the width of the light curve flares, w , and T O,peak of the flares,denoted with the index F, using a sample of 32 Swift GRBs, see the left panel of Fig. 28. This relation reads asfollows: log w F = (0 . ± .
27) + (1 . ± . × log T FO,peak , (16)with ρ = 0 . Figure 28 . Left panel: log w F -log T FO,peak distribution from Liang et al. (2010). Right panel: log w F -log T FO,peak distribution from Liet al. (2012). In both panels lines represent the best fit.
Later, Li et al. (2012) found the same relation as Liang et al. (2010), but with smaller values of normalizationand slope, using 24 flares from 19 single-pulse GRBs observed with CGRO/BATSE , see the right panel of Fig. 28.However, for these 19 GRBs only in 14 GRBs a flare activity is distinctly visible. The relationship was given by:log w F = − .
32 + 1 . × log T FO,peak . (17)They claimed that earlier flares are brighter and narrower than later ones. They compared the w F − T FO,peak distribution for the X-ray flares detected by Swift/XRT with the one for the optical flares in the R band. As aconclusion, they seemed to have a similar behaviour (Chincarini et al. 2007; Margutti et al. 2010), see the right panelof Fig. 28.Furthermore, in the rest frame band, they found a relation between the L O,peak of the flares in the R energy in unitsof 10 erg s − and T ∗ O,peak of the flares using 19 GRBs, see Fig. 29. Both prompt pulses and X-ray and optical flares Among the instruments of the Compton Gamma Ray Observatory (CGRO) satellite, running from 1991 to 2001, the Burst andTransient Source Experiment (BATSE) played a fundamental role in the measurements of GRB spectral features in the range from 20 keVto 8 MeV. Bursts were typically detected at rates of roughly one per day over the 9-year CGRO mission within a time interval rangingfrom ∼ . L FO,peak = (1 . ± . − (1 . ± . × log T ∗ FO,peak , (18)with ρ = 0 .
85 and
P < − . T FO,peak spans from ∼ tens of seconds to ∼ seconds, instead the L FO,peak variesfrom 10 to 10 erg s − , with an average value of 10 erg s − . In addition, considering only the most luminousGRBs, they found that T ∗ FO,peak was strongly anti-correlated to E γ,prompt in the 1 − keV energy band:log T ∗ FO,peak = (5 . ± . − (0 . ± . × log E γ,prompt / , (19)with ρ = 0 .
92. These outcomes revealed that the GRB flares in the optical wavelength with higher E γ,prompt peakearlier and are much more luminous. In Table 9 a summary of the relations described in this section is displayed. Figure 29 . log L FO,peak -log T ∗ FO,peak relation from Li et al. (2012). Lines represent the best fits, black dots indicate optical flares, and thegrey circles with errors show X-ray flares associated with the optical flares.
As regards the physical interpretation of the L FO,peak − T ∗ FO,peak relationship, Li et al. (2012) found that the flares areseparated components superimposed to the afterglow phase. The coupling between L FO,peak and T ∗ FO,peak suggested thatthe prompt γ -ray and late optical flare emission may arise from the same mechanism, namely from a central enginethat can periodically eject a number of shells during the emission. Impacts of these shells could create internal shocksor magnetic turbulent reconnections, which would emerge from the variability (Kobayashi et al. 1997; Zhang and Yan2011). Fenimore et al. (1995) revealed no relevant pattern in the width and intensity distributions using gamma raydata only. In addition, the usual tendency of the w F − T FO,peak relation cannot be due to hydrodynamical diffusionof the shells emitted at recent times, but it is necessary that the central engine radiates thicker and fainter shells atlate stages (Maxham and Zhang 2009). This could be explained as flares generated by clumps, such that the diffusion3
Correlations Author N Slope Norm Corr.coeff. P w F − T FO,peak
Liang et al. (2010) 32 1 . +0 . − . . +0 . − . < − Li et al. (2012) 19 1.01 -0.32 L FO,peak − T ∗ FO,peak
Li et al. (2012) 19 − . +0 . − . . +0 . − . < − T ∗ FO,peak − E γ,prompt Li et al. (2012) 19 − . +0 . − . . +0 . − . < − Table 9 . Summary of the relations in this section. The first column represents the relation in log scale, the second one the authors, thethird one the number of GRBs in the used sample, and the fourth and the fifth columns are the slope and normalization of the relation.The last two columns are the correlation coefficient and the chance probability, P. during the accretion mechanism would extend the accretion duration onto the BH (Perna et al. 2006; Proga and Zhang2006). SELECTION EFFECTSSelection effects are distortions or biases that usually occur when the sample observed is not representative of the“true” population itself. This kind of biases usually affects GRB relations. Efron and Petrosian (1992), Lloyd andPetrosian (1999), Dainotti et al. (2013a, 2015a) and Petrosian et al. (2013) emphasized that when dealing with amultivariate data set, it is imperative to determine first the true relations among the variables, not those introducedby the observational selection effects, before obtaining the individual distributions of the variables themselves. Thisstudy is needed for claiming the existence of the intrinsic relations. A relation can be called intrinsic only if it iscarefully tested and corrected for these biases.The selection effects present in the relations discussed above are mostly due to the dependence of the parameters onthe redshift, like in the case of the time and the luminosity evolution, or due to the threshold of the detector used.In this section, we describe several different methods to deal with selection biases.In paragraph 5.1, we discuss the redshift induced relation through a qualitative method, while in 5.2 we present a morequantitative approach through the EP method. In 5.3, we describe how to obtain the intrinsic relations corrected byselection biases, and in 5.4 we report the selection effects for the optical and X-ray luminosities. Lastly, in 5.5 we showthe evaluation of the intrinsic relation through Monte Carlo simulations.5.1.
Redshift induced relations
An important source of possible selection effects is the dependence of the variables on the redshift. To this end,Dainotti et al. (2011a) investigated the redshift evolution of the parameters of the LT relation, because a change ofthe relation slope has been observed when comparing several analyses (Dainotti et al. 2008, 2010). Namely, in the firstpaper, it was found b = − . +0 . − . and in the latter b = − . +0 . − . . Therefore, it became crucial to understand thereason of this change, even if the two slopes are still comparable at the 1- σ level. The distribution of the 62 LGRBs inthe sample is not uniform within the range ( z min , z max ) = (0 . , .
26) with few data points at large redshifts. Even ifthis sample is sparse, it was important to investigate whether the calibration coefficients ( a, b, σ int ) were in agreementwithin the error bars over this large redshift interval, see the left panel of Fig. 30.For this reason, the data set was separated in three redshift bins with the same number of elements, Z . , . Z . , .
08) and Z . , .
26) presented as blue, green and red points respectively in the left panel of Fig.30. The results are presented in Table 10. Id ρ ( b, a, σ int ) bf b median ( σ int ) median Z1 -0.69 (-1.20, 51.04, 0.98) − . +0 . − . . +0 . − . Z2 -0.83 (-0.90, 50.82, 0.43) − . +0 . − . . +0 . − . Z3 -0.63 (-0.61, 50.14, 0.26) − . +0 . − . . +0 . − . Table 10 . Results of the calibration procedure for GRBs divided in three equally populated redshift bins with ( z min , z max ) = (0 . , . . , . . , .
26) for bins Z1, Z2, Z3 from Dainotti et al. (2011a). The subscript bf displays the best fit values, while the median subscript shows the median values. The correlation coefficient ρ was found quite high in each redshift bins, supporting the independence of the LTrelation on z . The slopes b for bins Z Z Z Z a is comparable in all the bins. Fromthis analysis, it is not possible to confirm that the LT relation is shallower for larger z GRBs, due to the low number4
Figure 30 . Left panel: “log L X,a − log T ∗ X,a relation divided in the three redshift bins Z . , . Z . , .
08) and Z . , .
26) from Dainotti et al. (2011a). With the blue points it is represented the Z Z Z L X,a − log T ∗ X,a distribution fromDainotti et al. (2013a) for the sample of 101 GRB afterglows divided in 5 equipopulated redshift bins shown by different colours: blackfor z < .
89, magenta for 0 . ≤ z ≤ .
68, blue for 1 . < z ≤ .
45, green 2 . < z ≤ .
45, red for z ≥ .
45. Solid lines show the fittedrelations”. of data points and the presence of high σ E GRBs. Finally, bigger samples with small σ E values and a more uniform z binning are required to overcome this problem.For this reason, Dainotti et al. (2013a) performed a similar analysis, but with a larger sample consisting of 101 GRBs.Specifically, this updated sample was split in 5 redshift ranges with the same number of elements, thus having 20GRBs in each subgroup, represented in the right panel of Fig. 30 by different colours: black for z < .
89, magenta for0 . ≤ z ≤ .
68, blue for 1 . < z ≤ .
45, green for 2 . < z ≤ .
45 and red for z ≥ .
45. The fitted lines for eachredshift bin are also shown in the same colour code. The distribution of the subsamples presented different power lawslopes when the whole sample was divided into bins. The objects in the different bins exhibited some separation intodifferent regions of the LT plane. Moreover, the slope of the relation for each redshift bin versus the averaged redshiftrange has also been presented, see the left panel of Fig. 31.In addition, in Dainotti et al. (2015a), the updated sample of 176 GRBs was divided into 5 redshift bins consisting ofabout 35 GRBs for each group, as shown in the right panel of Fig. 31. A small evolution in z has been confirmed withthe following linear function b ( z ) = 0 . z − . Figure 31 . Left panel: “the variation of b (and its error range) with the mean value of the redshift bins from Dainotti et al. (2013a)”.Right panel: “ α τ , which is equivalent to the slope b , vs. z using a linear function α τ = 0 . z − .
38 from Dainotti et al. (2015a)”.
Regarding the log L X,peak − log L X,a relation, Dainotti et al. (2015b) showed that it is not produced by the dependenceon the redshift of its variables. To estimate the redshift evolution, the sample was separated into 4 redshift bins asshown in the left panel of Fig. 25. The GRB distribution in each bin is not grouped or constrained within a specificregion, therefore indicating no strong redshift evolution. For log L X,a it was found that there was negligible redshift5evolution of the afterglow X-ray luminosity (Dainotti et al. 2013a), while for log L X,peak has been demonstrated thatthere is significant redshift evolution (Yonetoku et al. 2004; Petrosian et al. 2013; Dainotti et al. 2015b). For moredetails, see sec. 5.2.1 and 5.2.2.5.2.
Redshift induced relations through Efron and Petrosian method
For a quantitative study of the redshift evolution, which is the dependence of the variables on the redshift, we hererefer to the EP method which is specifically designed to overcome the biases resulting from incomplete data. TheEfron & Petrosian technique, applied to GRBs (Petrosian et al. 2009; Lloyd and Petrosian 1999; Lloyd et al. 2000),allows to compute the intrinsic slope of the relation by creating new bias-free observables, called local variables anddenoted with the symbol (cid:48) . For these quantities, the redshift evolution and the selection effects due to instrumentalthresholds are removed. The EP method uses a modification of the Kendall tau test τ to compute the best fit valuesof the parameters which represent the luminosity and time evolutionary functions. For details about the definition of τ see Efron and Petrosian (1992). 5.2.1. Luminosity evolution
The relation between luminosity and z is called luminosity evolution. We discuss the luminosity evolution forboth prompt and plateau phases. Before applying the EP method to the plateau phase, the limiting plateau flux, F lim , which gives the minimum observed luminosity for a given z needs to be parameterized. The XRT sensitivity, F lim,XRT = 10 − erg cm − s − , is not high enough to represent the truncation of the data set. Hence, as claimedby Cannizzo et al. (2011), a better choice for the flux threshold is 10 − erg cm − s − . Several threshold fluxes wereanalyzed (Dainotti et al. 2013a), finally F lim,XRT = 1 . × − erg cm − s − , which leaves 90 out of 101 GRBs, wasselected (see the left panel of Fig. 32). Regarding instead the prompt limiting flux, Dainotti et al. (2015b) chose aBAT flux limit F lim,BAT = 4 × − erg cm − s − , which also allows 90% of GRBs in the sample, see the right panelof Fig. 32. Figure 32 . Left panel: “the bivariate distribution of log L X,a and z with two different flux limits from Dainotti et al. (2013a). Theinstrumental XRT flux limit, 1 . × − erg cm − s − (dashed green line), is too low to be representative of the flux limit, 1 . × − erg cm − s − (solid red line) represents better the limit of the sample”. Right panel: “the bivariate distribution of log L X,peak and z withthe flux limit assuming the K correction K = 1 from Dainotti et al. (2015b). The BAT flux limit, 4 . × − erg cm − s − (solid red line),better represents the limit of the sample”. In Dainotti et al. (2013a), the relation function, g(z), is defined when determining the evolution of L X,a so that thelocal variable L (cid:48) X,a ≡ L X,a /g ( z ) is not dependent anymore from z . The evolutionary function is parameterized by asimple relation function: g ( z ) = (1 + z ) k LX,a . (20)More complex evolution functions lead to comparable results, see Dainotti et al. (2013a, 2015b). The Kendall τ is a non-parametric statistical test used to measure the association between two measured quantities. It is a measureof rank relation: the similarity of the orderings of the data when ranked by each of the quantities. Figure 33 . Left panel: τ vs. k L X,a from Dainotti et al. (2013a). The red line indicates the full sample, while the green dotted lineindicates the sample of 47 GRBs in common with the 77 LGRBs in Dainotti et al. (2011a). Middle panel: τ vs. k L X,peak , using the eq.20, from Dainotti et al. (2015b). Right panel: τ vs. k L X,peak , using the eq. 21, from Dainotti et al. (2015b).
With this modified version of τ , the value of k L X,a for which τ L X,a = 0 is the one that best represents the luminosityevolution at the 1 σ level. k L X,a = − . +0 . − . means that this evolution is negligible, see the left panel of Fig. 33. Inthe same panel, this distribution is also plotted for a smaller sample of 47 GRBs (green dotted line) in common withthe previous one of 77 LGRBs presented in Dainotti et al. (2011a).The results of the afterglow luminosity evolution among the two samples are compatible at 2 σ . Instead, regarding thestudy of the evolution of L X,peak , the simple relation function (see eq. 20) was compared to a more complex function(Dainotti et al. 2015b) given by: g ( z ) = Z k L (1 + Z k L cr ) Z k L + Z k L cr , (21)where Z = 1 + z and Z cr = 3 .
5. A relevant luminosity evolution was obtained in the prompt, k L X,peak = 2 . +0 . − . ,using the simple relation, while k L X,peak = 3 . +0 . − . for the more complex function, see the middle and right panels ofFig. 33 respectively. The results of the prompt luminosity evolution among the two different functions are compatibleat 2 σ . 5.2.2. Time Evolution
Similarly to the treatment of the luminosity evolution, one has also to determine the limit of the plateau end time, T ∗ X,a,lim = 242 / (1 + z ) s (Dainotti et al. 2013a), and of the prompt peak time T ∗ X,prompt,lim = 1 . / (1 + z ) s (Dainottiet al. 2015b), see the left and right panels of Fig. 34 and Fig. 35 respectively.To determine the evolution of T ∗ X,a , so that the de-evolved variable T (cid:48) X,a ≡ T ∗ X,a /f ( z ) is not correlated with z, therelation function f ( z ) (Dainotti et al. 2013a) was specified: f ( z ) = (1 + z ) k T ∗ X,a . (22)The values of k T ∗ X,a for which τ T ∗ X,a = 0 is the one that best matches the plateau end time evolution at the 1 σ uncertainty. τ T ∗ X,a versus k T ∗ X,a distribution shows a consistent evolution in T ∗ X,a , as seen in the left panel of Fig. 36,namely k T ∗ X,a = − . +0 . − . . In the same panel this distribution is also displayed for a smaller sample of 47 GRBs(green dotted line) in common with the previous one of 77 GRBs presented in Dainotti et al. (2011a). The results ofthe afterglow time evolution among the two samples are compatible at 1.5 σ .Regarding the prompt time evolution, a more complex function was also used in addition to the simple relation function(Dainotti et al. 2015b): f ( z ) = Z k ∗ T (1 + Z k ∗ T cr ) Z k ∗ T + Z k ∗ T cr , (23)where Z = 1 + z and Z cr = 3 . k T ∗ X,prompt = − . +0 . − . , and for the more complex one k T ∗ X,prompt = − . +0 . − . , see the middle and right panels of Fig. 36respectively. The results of the prompt time evolution among the two different functions are compatible at 1 σ .7 Figure 34 . Left panel: “the bivariate distribution of the rest frame time log T ∗ X,a and z from Dainotti et al. (2013a). The red line is thelimiting rest frame time, log( T X,a,lim / (1 + z )) where the chosen limiting value of the observed end-time of the plateau in the sample is T X,a,lim = 242 s”. Right panel: “the bivariate distribution of the rest frame time log T ∗ X,prompt and z from Dainotti et al. (2015b), wherewith log T ∗ X,prompt they denoted the sum of the peak pulses width of each single pulse in each GRB. The chosen limiting value of theobserved pulse width in the sample is log T X,prompt,lim = 0 .
24 s. The red line is the limiting rest frame time, log( T X,prompt,lim / (1 + z ))”. Figure 35 . Distributions between redshift and the observed (left panel) and rest-frame (right panel) T in the BAT energy range fromGrupe et al. (2013). - - - - - - - k T * prompt τ T * p r o m p t - - - - - - - k T * prompt τ T * p r o m p t Figure 36 . Left panel: τ vs. k T ∗ X,a from Dainotti et al. (2013a). The red line indicates the full sample, while the green dotted lineindicates the 47 GRBs in common with the sample presented in Dainotti et al. (2011a). Middle panel: τ vs. k T ∗ X,prompt , using the eq. 22,from Dainotti et al. (2015b). Right panel: τ vs. k T ∗ X,prompt , using the eq. 23, from Dainotti et al. (2015b).
Evaluation of the intrinsic slope
The last step to determine if a relation is intrinsic is to evaluate its “true” slope. To this end, the EP method wasused in the local time ( T (cid:48) X,a ) and luminosity ( L (cid:48) X,a ) space obtaining an intrinsic slope for the LT relation b int = 1 /α = − . +0 . − . . The significance of this relation is at 12 σ level. It can be derived directly from the left panel of Fig. 37(Dainotti et al. 2013a), because if there was no relation it would have been that τ = 0 for b int = 0 at 1 σ .Instead, regarding the evaluation of the intrinsic slope in the log L X,peak − log L X,a relation, Dainotti et al. (2015b)used a different method, namely the partial correlation coefficient. This is the degree of association between tworandom variables calculated as a function of b int in the following way: r L (cid:48) X,peak L (cid:48) X,a ,D L = r L (cid:48) X,peak ,L (cid:48) X,a − r L (cid:48) X,peak ,D L ∗ r L (cid:48) X,a ,D L (1 − r L (cid:48) X,peak ,D L ) ∗ (1 − r L (cid:48) X,a ,D L ) , (24)where log L (cid:48) X,a = L (cid:48) X,a and log L (cid:48) X,peak = L (cid:48) X,peak . Figure 37 . Left panel: τ vs. b int (indicated with α in the picture) from Dainotti et al. (2013a). Right panel: r vs. b int from Dainottiet al. (2015b) with the best value where log L X,peak and log L X,a are strongly correlated (the central thick line). The two thinner linesindicate the 0 .
05% probability that the sample is drawn by chance.
As displayed in the right panel of Fig. 37, the relation is highly significant when b int = 1 . +0 . − . , which is at 1 σ ofthe observed slope.In addition, following an analysis similar to the one of Butler et al. (2010), Dainotti et al. (2015a) simulated a samplefor which biases on both time and luminosity are considered. Particularly, they assumed the biases to be roughly thesame whichever monotonic efficiency function for the luminosity detection is taken. This method presented how anunknown efficiency function could affect the slope of any relation and the GRB density rate. Then, biases in slope ornormalization caused by the truncations were analyzed. This gave distinct fit values that allow for studying the scatterof the relation and its selection effects. This analysis has shown, together with the one in Dainotti et al. (2013a), thatthe LT relation can be corrected by selection effects and therefore can be used in principle as redshift estimator (seesec. 6) and as a valuable cosmological tool (see sec. 7). As regards other relations, D’Avanzo et al. (2012) for the L X,a − E γ,prompt relation, Oates et al. (2015) for the L O, − α O,> relation, and Racusin et al. (2016) for the L X, - α X,> relation, also used the partial correlation coefficient method to show that the redshift dependencedoes not induce these relations.5.4.
Selection effects for the optical and X-ray luminosities
In this section we discuss the selection effects due to the limiting optical and X-ray luminosities relevant for therelations mentioned above. Nardini et al. (2008b) investigated if the observed luminosity distribution can be theresult of selection effects by studying the optically dark afterglows. By simulating the log L O, , z , host galaxy dustabsorption, A hostV , and telescope limiting magnitude for each of the 30000 GRBs, the observed optical luminositydistribution was contrasted to the simulated one. From this simulated distribution regarding the intrinsic one, it is9necessary to take only GRBs with a flux which is larger than the threshold flux of the associated detector. Thiscorresponds with a lower luminosity truncation, which is around log L O, ≈ . − Hz − ). Therefore, the factthat we do not observe GRBs with such a luminosity puts a limit to the luminosity function.They also checked statistically the presence of a low luminosity category of events which are at 3 . σ off the centralvalue of the distribution. They pointed out that if the absorption is chromatic, the observed luminosity distributiondoes not match with any unimodal one. If many GRBs are absorbed by “grey” achromatic dust, then a unimodalluminosity distribution can be obtained. In summary, dark bursts could belong to an optically subluminous group orto a category of bursts for which a high achromatic absorption is present.As regards the evaluation of the selection effects of L O,peak , the biases in the detection of F O,peak need to be considered.As found from Panaitescu and Vestrand (2008), for a typical optical afterglow spectrum ( F O,a ∝ T − O,a ), variations inthe observer offset angle induce a log F O,peak − log T O,peak anti-relation that is flatter than what is measured. In fact,an observational selection effect could steepen the slope of the anti-relation between log F O,peak and log T O,peak .In addition, SGRBs observed by Swift seem to be fluence-limited, while LGRBs detected with the same telescope areflux-limited (Gehrels et al. 2008) due to the instrument trigger.Nysewander et al. (2009) pointed out that the ratio F O, /F X, may be influenced by absorption of photons in the hostgalaxy. Furthermore, they showed that F X, should be precise, because the LGRBs observed in the XRT passbanddo not present X-ray column absorptions, differently from the majority of LGRBs. The computed optical absorptionof LGRB afterglows indicates smaller column densities ( N H ) than in the X-ray, with optical absorptions ( A V ) aboutone-tenth to one magnitude (Schady et al. 2007; Cenko et al. 2009). Regarding the SGRBs, they have more luminousoptical emission relative to the X-ray than what is assumed by the standard model. Later, Kann et al. (2010) claimedthat the grouping of the optical luminosity at the time of 1 day, L O, , is less remarkable than the one described byLiang and Zhang (2006b) and Nardini et al. (2006) for GRBs observed by Swift. This suggested that the groupingpointed out in pre-Swift data can be due to selection effects only. Finally, Berger (2014) claimed that the opticalafterglow detection can influence the luminosity distribution towards places with larger densities medium.5.5. Selection effects in the L O, − α O,> relation
Oates et al. (2012) ensured that a high S/N light curve, covering both early and late times, can be constructed fromthe UVOT multi-filter observations using the criteria from Oates et al. (2009). If the faintest optical/UV afterglowsdecay more slowly than the brightest ones, then at late time the luminosity distribution is less dispersed and thecorrelation coefficient of the log L O, − α O,> relation must become smaller and/or negligible. Indeed, both ofthese effects were observed in their sample. Furthermore, the log L O, − α O,> relation may arise, by chance, fromthe way in which the sample is chosen. Thus, to verify if this is not the case, they computed Monte Carlo simulations.Among the 10 trials, 34 have a correlation coefficient indicating a more significant relation than the original one. Thispoints out that, at 4.2 σ confidence, the log L O, − α O,> relation is not caused by the selection criteria nor doesit happens by chance, and thus it is intrinsic. REDSHIFT ESTIMATORAs we have pointed out in the introduction, the study of GRBs as possible distance estimators is relevant, since formany of them z is unknown. Therefore, having a relation which is able to infer the distance from known quantitiesobserved independently of z would allow a better investigation of the GRB population. Moreover, in the cases inwhich z is uncertain, the estimator can give hints on the upper and lower limits of the distance at which the GRBis placed. Some examples of redshift estimators for the prompt relations (Atteia 2003; Yonetoku et al. 2004; Tsutsuiet al. 2013) have been reported. In these papers, a method is developed for inverting GRB luminosity relations inrespect to the redshift to have an expression of the distance as a function of z. The methodology used for the promptemission relations can be then applied also to the afterglow or prompt-afterglow phase relations.In this respect, Dainotti et al. (2011a) investigated the LT relation as a redshift estimator. From this relation, thebest fit parameters of the slope and normalization are derived, while parameters such as log F X,a , log T X,a and β X,a are known, because they are measured. Therefore, the LT relation can be inverted to obtain an estimate of z as it hasbeen done for the prompt relations by Yonetoku et al. (2004). With this intention, let us return to the eq. 2 and writeit in another form:0 log L X,a = log (4 πF X,a ) + 2 log D L ( z, Ω M , h ) − (1 − β X,a ) log (1 + z )= log (4 πF X,a ) + (1 + β X,a ) log (1 + z ) + 2 log r ( z ) + 2 log ( c/H )= a log (cid:18) T X,a z (cid:19) + b (25)where r ( z ) = D L ( z, Ω M , h ) × ( H /c ). Solving respect to z , it was obtained:(1 + β X,a + a ) log (1 + z ) + 2 log r ( z ) = a log T X,a + b − log (4 πF X,a ) − c/H ) . (26)The numerical solution of this equation may encounter some problems that must be taken into account: (log T X,a , log F X,a , β
X,a ) and the LT calibration parameters ( a, b ) are influenced by their own errors. Furthermore, the errors on( a, b ) are not symmetric and σ int is summed to the total error in a nonlinear way. For details about possible solutionson how to consider the errors see Dainotti et al. (2011a). The above solution was employed for the E4 and the E0095samples, pointing out that the LT relation can still not be considered as a precise redshift estimator, see Fig. 38. z obs z e s t Figure 38 . z obs - z est distribution for the 62 LGRBs divided in three σ E ranges from Dainotti et al. (2011a): σ E ≤ .
095 is represented byred points, 0 . ≤ σ E ≤ . . ≤ σ E ≤ Assuming ∆ z = z obs − z est , where z obs and z est are the observed and the estimated redshifts respectively, it hasbeen shown that ∼
20% of GRBs in the E4 sample (black, 0 . ≤ σ E ≤
4, and blue, 0 . ≤ σ E ≤ .
3, points inFig. 38) has | ∆ z/σ ( z est ) | ≤
1. While for the E0095 subsample 28% has | ∆ z/σ ( z est ) | ≤
1, red dots in Fig. 38. Thepercentage of successful solutions rises to ∼
53% ( ∼ | ∆ z/σ ( z est ) | ≤ σ E has no strong influence on theredshift estimate. The reason why the redshift indicator has not yet given successful results depends on the intrinsicscatter of the LT relation. Thus, it is useful to check whether better results can be achieved by increasing the datasample size. For this reason, an E0095 subsample was simulated creating (log T X,a , β
X,a , z ) values from a distributionsimilar to the observed one for the E4 sample. Then, log L X,a was selected from a Gaussian distribution with mean1value obtained by the LT relation and with σ Gauss = σ int . These values were employed to compute log F X,a and toreproduce the noise for all the quantities so that the relative errors resembled the observations. Then, using Markovchains as input to the redshift estimate formula, it is concluded that only enlarging the sample is not an appropriatemethodology to increase the success of the LT relation as a redshift estimator.In fact, with
N (cid:39)
50, the number of GRBs with | ∆ z/σ ( z est ) | ≤ ∼
34% and then diminish to ∼ N (cid:39) (cid:104) ∆ z/z obs (cid:105) (cid:39) −
17% for both
N (cid:39)
50 and
N (cid:39) a, b, σ int ) values,but does not affect σ int which is the principal cause of inconsistencies between the observed and the estimated z .Therefore, an alternative way was explored: σ int was decreased and the best fit ( a, b ) parameters of the E0095subsample were chosen. In fact, fixing σ int = 0 .
10 gives f ( | ∆ z/z obs | ≤ (cid:39) σ int = 0 . − .
20. If such a sample is achievable is not clear yet due to the paucity of the E0095 subsample. Infact, it is difficult to find out some useful indicators that can help to define GRBs close to the best fit line of the LTrelation. To obtain ∼
50 GRBs to calibrate the LT relation with σ int ∼ .
20 it has been estimated that a sample of ∼
600 GRBs with observed (log T X,a , log F X,a , β
X,a , z ) values is needed. However, even if this is a challenging goal, itmay be possible to find out properties of GRB afterglows which enable us to reduce the σ int of the LT relation witha much smaller sample. Finally, an interesting feature would be to correct for the selection effects all the physicalquantities of the relations mentioned above. In this manner, it would be possible to average them in order to create amore precise redshift estimator. COSMOLOGYThe study of the Hubble Diagram (HD), namely the distribution of the distance modulus µ ( z ) versus z of SNe Ia,opened the way to the investigation of the nature of DE. As it is known from the literature, µ ( z ) is proportional tothe logarithm of the luminosity distance D L ( z, Ω M , h ) through the following equation: µ ( z ) = 25 + 5 × log D L ( z, Ω M , h ) . (27)In addition, D L ( z, Ω M , h ) is related to different DE EoSs.7.1. The problem of the calibration
One of the most important issues presented in the use of GRB relations for cosmological studies is the so-calledcircularity problem. Namely, a cosmological model needs to be assumed to compute D L ( z, Ω M , h ). This is due tothe fact that local GRBs are not available apart from the case of GRB 980425. Indeed, this kind of GRBs would beobserved at z < .
01 and their measure would be independent of a particular cosmological setting. This issue couldbe overcome in three ways: a) through the calibration of these relations by several low z GRBs (in fact, at z ≤ . M and Ω Λ for a given H , where H is between 65 and 72);b) through a solid theoretical model in order to explain the observed 2D relations. Namely, this would fix their slopesand normalizations independently of cosmology, but this task still has to be achieved; c) through the calibration ofthe standard candles using GRBs in a narrow redshift range (∆ z ) near a fiducial redshift, z c . We here describe someexamples on how to overcome the problem of circularity using prompt relations.The treatment of this problem will be the same once we consider afterglow or prompt-afterglow relations. Liangand Zhang (2006b) suggested a new GRB luminosity indicator, E γ,iso = aE b γ,peak T b O,a , different from the previousGRB luminosity indicators that are generally written in the form of L = a (cid:81) x b i i , where a is the normalization, x i is the i-th observable, and b i is its corresponding power law index. It was demonstrated that while a relies on thecosmological parameters, this is not the case for b and b until ∆ z is sufficiently little, see Fig. 39. The choice of ∆ z for a given GRB sample could be evaluated depending on its dimension and the errors on the variables. The mostsuitable approach would be to assemble GRBs within a small redshift range around a central z c ( z c ∼ z c ∼ z distribution peaks in this interval (see also Wang et al. 2011 and Wang et al. 2015).In addition, also Ghirlanda et al. (2006) defined the luminosity indicator E γ,peak = a × E bγ,cor using the log E γ,peak − log E γ,cor relation (Ghirlanda et al. 2004), where E γ,cor = (1 − cos θ jet ) × π × D L ( z, Ω M , h ) × S γ,prompt / (1 + z ) (28) The difference between the apparent magnitude m, ideally corrected from the effects of interstellar absorption, and the absolutemagnitude M of an astronomical object. -4.0-4.0 -8.0-16 -30 -30 -4.0 -4.0-16 -30-8.0
20 30 40 50 60 70 80 90 1000.20.40.60.81.0 Δ z N
20 30 40 50 60 70 80 90 1000.20.40.60.81.0 Δ z N Figure 39 . “Distribution of log P in the (N, ∆ z ) plane from Liang and Zhang (2006b). The grey contours mark the areas where thedependencies of b and b on Ω M are statistically significant (P < − ). The white region is suitable for the calibration purpose”. is the energy corrected for the beaming factor and θ jet is the opening angle of the jet. They calculated the minimumnumber of GRBs (N), within ∆ z around a certain z c , needed to calibrate the relation, considering a sample of 19GRBs detected mostly by Beppo-SAX and Swift. Particularly, they fitted the relation for each value of Ω M and Ω Λ using a set of N GRBs distributed in the interval ∆ z (centered around z c ). If the variation of the slope, b, is lessthan 1% the relation is assumed calibrated. N , ∆ z and z c are free parameters. They checked several z c and distinct z dispersions ∆ z ∈ (0 . , .
5) by Monte Carlo simulations. At every z the smaller the N the bigger the variation ofthe slope, ∆ b (for the same ∆ z ), because the relation is more scattered. On the other hand, for greater z c a tinier ∆ z is necessary to maintain ∆ b in its little state. Finally, they found that 12 GRBs with z ∈ (0 . , .
1) can be sufficientto calibrate the slope of the log E γ,peak − log E γ,cor relation. Instead, at z c = 2 a narrower redshift bin is needed, forexample z ∈ (1 . , . µ ( z ) of aSNe Ia at the same redshift. In this way, GRBs should be considered as complementary to SNe Ia at very high z,thus allowing for the construction of a very long distance ladder. Therefore, interpolating the SNe Ia HD provides thevalue of µ ( z ) for a subsample of GRBs with z ≤ .
4, which can be employed for the calibration of the 2D relations(Kodama et al. 2008; Liang et al. 2008; Wei and Zhang 2009). This value is given by the formula: µ ( z ) = 25 + (5 / y − k )= 25 + (5 / a + b log x − k ) , (29)where y = kD L ( z, Ω M , h ) is a given quantity with k a redshift independent constant, and a and b are the relationparameters. Presuming that this calibration is redshift independent, the HD at higher z can be constructed using thecalibrated relations for the other GRBs in the data set.Finally, Li and Hjorth (2014) analyzed the light curves of 8 LGRBs associated with SNe finding a relation betweenthe peak magnitude and the decline rate at 5, 10 and 15 days as in SNe Ia. However, from the comparison with thewell-known relation for SNe Ia (Phillips 1993), it was pointed out that these two objects have two different progenitors.More importantly, this discovery allowed to use GRBs associated with SNe as possible standard candles. In addition,Cano (2014) investigated the optical light curves of 8 LGRBs associated with SNe discovering evidence of a relationbetween their luminosity and the width of the GRB light curves relative to the template of the well-known SN 1998bw.This result also confirmed the possibility of using GRBs associated with SNe as standard candles.37.2. Applications of GRB afterglow relations
In this section, we describe some applications to cosmology only for the LT relation, because this is the only afterglowrelation that has been used so far as a cosmological probe. However, the method is very general and it can be employedfor all the other relations presented in the review. The idea to use afterglow GRBs phase as cosmological rulers wasproposed for the first time in 2009, when the LT relation was used to derive a new HD (Cardone et al. 2009, 2010).More specifically, Cardone et al. (2009) revised the data set used in Schaefer (2007) appending the LT relation. Theyused a Bayesian fitting method, similar to that used in Firmani et al. (2006) for the log E γ,peak -log E γ,cor relation, tocalibrate the different GRB relations known at that time assuming a fiducial ΛCDM model compatible with the dataprovided by the Wilkinson Microwave Anisotropy Probe, WMAP5.A new HD including 83 objects was obtained (69 from Schaefer (2007) plus 14 new GRBs obtained by the LT relation)computing the mean performed over six relations (log E γ,cor − log E γ,peak , log L γ,iso − log V , with V the variability whichmeasures the difference between the observed light curve and a smoothed version of that light curve, log L X,a − log T ∗ X,a ,log L γ,iso − log τ lag , with τ lag the difference in arrival time to the observer of the high energy photons and low energyphotons, log L γ,iso − log τ RT , with τ RT the shortest time over which the light curve increases by the 50% of the peakflux of the pulse, and log L γ,iso − log E γ,peak ).To elude the circularity problem, local regression was run to calculate µ ( z ) from the newest SNe Ia sample containing307 SNe Ia in the range 0 . ≤ z ≤ .
55. Indeed, the GRB relations mentioned before were calibrated whileconsidering only GRBs with z ≤ . µ ( z ).The basic idea of the local regression analysis consists of several stages described in Cardone et al. (2009). To find outwhich are the optimal parameters of this procedure, a large sample of simulations was carried out. They set the value ofthe model parameters (Ω M , w , w a , h ), with w and w a given by the coefficient of the DE EoS w ( z ) = w + w a z (1+ z ) − (Schaefer 2007), in the ranges 0 . ≤ Ω M ≤ . − . ≤ w ≤ − . − . ≤ w a ≤ . . ≤ h ≤ .
80. Foreach z value, µ ( z i ) was selected from a Gaussian distribution centered on the predicted value and with σ int = 0 . σ int of the SNe Ia absolute magnitude. This way, a mock catalogue with the same z and errordistribution of the SNe sample was built. Each µ ( z ) value derived from this procedure is compared to the input one.The local regression method correctly produces the underlying µ ( z ) at each z from the SNe Ia sample, whichever isthe cosmological model.Furthermore, comparing their HD to the one derived by Schaefer (2007), referred as the Schaefer HD, they have updatedthe Schaefer HD in three ways, namely updating the ΛCDM model parameters, using a Bayesian fitting procedureand adding the LT relation. To analyze the influence of these changes, the sample of 69 GRBs adopted by Schaefer(2007) was also used and the distance moduli were computed with the new calibration, but without considering theLT relation. It was found that µ new /µ old is close to 1 within 5%. Thus, this calibration procedure has not modifiedthe results.In conclusion, it was pointed out that the µ ( z ) for each of the GRBs in common to Schaefer (2007) and Dainottiet al. (2008) samples is compatible with the one computed using the set of Schaefer (2007) relations. Therefore, nosystematic bias is added by also considering the LT relation. On the other hand, the addition of the LT relation to thepre-existing ones not only decreases the errors on µ ( z ) by ∼ z . The GRBs were divided in E0095 and E4 samples,indicating that the introduction of the LT relation alone also provides constraints compatible with previous outcomes,since the HD spans over a large redshift range (0 . , . w a to 0. This result indicates that the consideration of a big sample of E0095 GRBs may lead to aconstant EoS DE model.In addition, we may note that, different from what was done in the literature at the time of their publication, the HDfor the E0095 and E4 samples is the only GRB HD built with a single relation in the afterglow containing a statisticallysignificant sample.Furthermore, the LT relation does not request the mix of several relations to rise the number of GRBs with a known4 µ ( z ). In fact, each relation is influenced by its own biases and intrinsic scatter; therefore, using all of them in thesame HD can affect the evaluation of the cosmological parameters. The σ int of the LT relation may be considerablydecreased if only the E0095 subsample is analyzed. However, considering the whole sample of 66 LGRBs, Cardoneet al. (2010) constrained Ω M and H obtaining values compatible with the ones presented in the literature.This analysis clearly claimed that the LT relation can be considered for building a GRB HD without adding any biasin the study of the cosmological parameters. Equivalent findings were achieved considering E0095 GRBs even if theyare just 12% of the whole sample. Therefore, a further investigation of E0095 GRBs can boost their use as standardsample for studying the DE mystery.As a further development, Dainotti et al. (2013b) pointed out to what extent a separation of 5 σ above and below theintrinsic value, b int = − . +0 . − . , of the slope of the LT relation can influence the cosmological results.For this study, a simulated data set of 101 GRBs obtained through a Monte Carlo simulation was collected assuming b = − . σ int = 0 .
93 (larger than the scatter computed from the original data set, namely σ int = 0 . M = 0 .
291 and H = 71 Km s − Mpc − . They investigated how muchthe scatter in the cosmological parameters can be diminished if, instead of the total sample (hereafter Full), a highlyluminous subsample (hereafter High Luminosity) is considered, constrained by the condition that log L X,a ≥ .
7. Thechoice of this selection cut at a given luminosity is explained in Dainotti et al. (2013a), who showed that the localluminosity function is similar to the observed luminosity one for log L X,a ≥ E γ,peak − log E γ,iso relation,namely the fit has been performed varying simultaneously both the calibration parameters, p GRB = ( a, b, σ int ), andthe cosmological parameters, p c = (Ω M , Ω Λ , w , w a , h ), each time for a given model in order to correctly take this issueinto account.In order to have stronger limits on the cosmological parameters two samples were added to the data set, the H ( z )sample ( H ( z ) = H × (cid:112) Ω M (1 + z ) + Ω k (1 + z ) + Ω Λ ) over the redshift range 0 . ≤ z ≤ .
75 (Stern et al. 2010)and the Union 2 . . ≤ z ≤ .
414 (Suzuki et al. 2012).A Markov Chain Monte Carlo (MCMC) method was used, running three parallel chains and applying the Gelman-Rubin test in order to analyze the convergence for an assumed cosmological model characterized by a given set ofcosmological parameters p c to be determined.From this statistical analysis results regarding the Full GRB sample, b , a and σ int of the LT relation are independentof the chosen cosmological model and the presence of the SNe Ia and H ( z ) data in the sample. In addition, even if a5 σ scatter in b int is assumed, the results for the Full sample are in agreement with earlier outcomes (Dainotti et al.2008, 2011a) where exclusively flat models were assumed.On the other hand, due to the wide errors on the simulated data, the cosmological parameters are not emerging in thecalibration procedure. However, the signature of the cosmology will appear considering a greater data set with lowerrors on (log T ∗ X,a , log L X,a ).Furthermore, for the Full sample, it was studied how much the deviation from the b int of the LT relation influencesthe cosmological parameters. To analyze this problem, a model parameterized in terms of the present day values ofΩ M , Ω Λ and H was considered.Although h is comparable with the values from both the local distance estimators (Riess et al. 2009) and CMBRdata (Komatsu et al. 2011), the median values for (Ω M , Ω Λ ) are broader if compared to a fiducial Ω M ∼ .
27 recoveredin earlier works (Davis et al. 2007). For this reason, considering for the Full sample, a distinct b int will lead to adisagreement of 13% with the best value of the Ω M parameter (see the upper panels of Fig. 40). Even if the medianvalues of the fit for the sample that also has SNe Ia and H ( z ) data do not conduct towards flat models, a spatially flatUniverse accords with, for example, the WMAP7 cosmological parameters within 95% giving Ω k = − . +0 . − . . Thisdifference can be deduced, because in this case it is not possible to distinguish among flat and not flat models and thisdistinction is still not possible when SNe Ia data are present in the fit. Thus, constraining the model to be spatiallyflat, but shaping the DE EoS with w ( z ), leads to a couple ( w , w a ) completely different irrespective of whether SNe Iaand H ( z ) data are considered or not in the sample. Regarding instead the High Luminosity subsample, the limits onthe calibration parameters mostly do not depend on either the used cosmological model or if SNe Ia and H ( z ) dataare considered in the sample. Furthermore, for the High Luminosity subsample it is shown that adding the SNe Iaand H ( z ) data does not ameliorate the constraints on the calibration parameters.Finally, the Full sample outcomes are comparable to those of the flat cosmological model for the SNe Ia sample, while The Gelman-Rubin diagnostics relies on parallel chains to test whether they all converge to the same posterior distribution. - - - - - - Σ i n t W m h Figure 40 . Upper left panel: “regions of confidence for the marginalized likelihood function L ( b, σ ) from Dainotti et al. (2013b), obtainedmarginalizing over a and the cosmological parameters using the Full sample. The bright brown regions indicate the 1 σ (full zone) and2 σ (bright grey) regions of confidence respectively. On the axes are plotted the box-and-whisker diagrams relatively to the b and σ int parameters: the bottom and top of the diagrams are the 25th and 75th percentile (the lower and upper quartiles, respectively), and theband near the middle of the box is the 50th percentile (the median)”. Upper right panel: “regions of confidence for the marginalizedlikelihood function L (Ω M , h ), obtained using the Full sample, from Dainotti et al. (2013b)”. Bottom left panel: “regions of confidence forthe marginalized likelihood function L ( b, σ ) from Dainotti et al. (2013b), obtained marginalizing over a and the cosmological parameters forthe High Luminosity sample. The bright brown regions indicate the 1 σ (full zone) and 2 σ (bright grey) regions of confidence respectively.On the axes are plotted the box-and-whisker diagrams relatively to the b and σ int parameters: the bottom and top of the diagrams arethe 25th and 75th percentile (the lower and upper quartiles, respectively), and the band near the middle of the box is the 50th percentile(the median)”. Bottom right panel: “regions of confidence for the marginalized likelihood function L (Ω M , h ), obtained using the HighLuminosity sample, from Dainotti et al. (2013b)”. the High Luminosity subsample diverges by 5% in the value of H as computed in Petersen et al. (2010), and thescatter in Ω M is underestimated by 13%, see the bottom panels of Fig. 40. In conclusion, an optimal procedure is toconsider a High luminosity subsample provided by a cut exactly at log L X,a = 48; otherwise, the luminosity and timeevolutions should be added in the computation of the cosmological parameters.Later, another application of GRBs to cosmology is presented in Postnikov et al. (2014) where the DE EoS was6analyzed as a function of z without assuming any a priori w ( z ) functional form.To build a GRB ( µ, z ) diagram, 580 SNe Ia from the Union 2.1 compendium (Suzuki et al. 2012) were used togetherwith 54 LGRBs in the overlapping redshift ( z ≤ . w = − Figure 41 . Left panel: “( z j , µ j ± ∆ µ j ) for SNe Ia from Postnikov et al. (2014). GRBs are inferred from the relation assuming a flat w = − w = − z . The SNe Ia data were taken from the Union 2.1 compendium (Suzuki et al. 2012)”. Right panel:“distance ladder from Postnikov et al. (2014). GRBs in the SNe Ia overlap redshift range, where cosmology is well constrained, are used tocalculate the GRB intrinsic correlation coefficient. This relation is then used to calculate D L ( z, Ω M , h ) for high z GRBs from their X-rayafterglow luminosity curves. Standard constant w solutions are shown for reference. Vertical dashed line marks farthest SNe Ia event. Insetto the right shows a histogram of the GRB sample distribution in z . Inset to the left shows resulting most probable EoS, together with asmall sample of models probed, confidence intervals are so large, that only extreme variations with respect to w = − Figure 42 . “Tree of w ( z >
1) curves inferred from synthetic GRB samples constructed for w ( z ) = − z ( z > One order of magnitude expansion in redshift interval is supplied by the GRB data set considering the correlationcoefficients obtained for the SNe Ia. This detail allows for the enlargement of the cosmological model out to z = 8 . L X,a = 53 . +0 . − . − . +0 . − . × log T ∗ X,a , (30)with ρ = − .
74 and P = 10 − .Postnikov et al. (2014) used a Bayesian statistical analysis, similarly to Firmani et al. (2006) and Cardone et al. (2010),in which the hypothesis is related to a particular w ( z ) function with the selection of H and the present DE densityparameter, Ω Λ0 . The assumption of isotropy for the cosmological model, reliable limits on the EoS and also a fixed7value for w ( z ) in the z ≤ .
01 redshift interval were employed. In addition, a huge number of randomly chosen w ( z )models were used.To test the procedure, their pattern is verified through the simulated data sets obtained from several input cosmologicalmodels with relative errors and z distribution equal to the real data. Through this procedure, employing the LT relation,a data set of GRBs detected by the Swift satellite, with z from 0 .
033 to 9 .
44, was adopted (see inset in the right panelof Fig. 41). Thus, it is possible to investigate the history of the Universe out to z ≈
10. (However, an additionalanalysis would be beneficial if we would consider the sample without the GRB at z = 9 .
4. We note that indeed inCardone et al. (2010) a sample of canonical GRBs was used in which this burst has not been included).In order to do that, they simulated 2000 constant EoSs uniformly spaced between − ≤ w Λ ≤
2, with w Λ the DEEoS. Beginning from SNe Ia data sample, a precise solution was found to be in agreement with the cosmologicalconstant and a small confidence interval, w = − . ± .
2, see the right panel of Fig. 41. Furthermore, it is shownthat assuming also that the BAO limits do not differ from the solution of the EoS, but it considerably decreases theconfidence interval ( w = − . ± . Λ0 = 0 . ± . w ( z ) model which leads to the best evaluation of D L ( z, Ω M , h ), z of the SNe Ia sample and theBAO constraints needs to be selected. The confidence region of the allowed w ( z ) curves is significantly constrainedtaking into account also the BAO data.Afterwards, also considering that GRB data should constrain the cosmological parameters, apart from obtaining oneorder of magnitude expansion in the redshift range, it was extremely difficult to constrain the high z w ( z ) functionalform, considered the paucity of points over a broad redshift interval and the error bars related to these data. This isvisible in the left panel of Fig. 42, where a simulated GRB data set having the same z distribution and error bars asthe real data, but with assumed w = − w ( z ) fluctuations are notallowed. Then, decreasing the errors by a factor of 4 led to more intriguing high z DE constraints, see the right panelof Fig. 42.In addition, the small number of elements in the SNe Ia overlapping region indicated broad error bars on the GRBcorrelation coefficients. Meanwhile, the broad error bars for high z GRBs generated a very flat probability distribution(represented by the uniform black shading area in the left panel of Fig. 42) for the several EoSs checked. Therefore,there will be great interest for the 1 < z < SUMMARY AND DISCUSSIONFrom the analysis of the relations mentioned in previous sections, it is visible that:1. The accretion model (Cannizzo and Gehrels 2009; Cannizzo et al. 2011) and the magnetar model (Usov 1992;Dall’Osso et al. 2011; Rowlinson and O’Brien 2012) seem to give the best explanation of the Dainotti relation(giving best fit slopes -3/2 and -1 respectively). The magnetar model seems to be favoured compared to theaccretion one, because the intrinsic slope computed in Dainotti et al. (2013a) is exactly − . +0 . − . .2. A more complex jet structure is needed for interpreting the log L O, - α O,> relation (Oates et al. 2012).Indeed, Oates et al. (2012) showed that the standard afterglow model cannot explain this relation, especiallytaking into account the closure relations (Sari et al. 1998), which relate temporal decay and spectral indices.Therefore, in order to interpret their results, they claimed either the presence of some features of the centralengine which dominate the energy release or that the observations were made by observers at different angulardistances from the source’s axis. Dainotti et al. (2013a) pointed out a similarity between the log L O, - α O,> relation and the L X − T ∗ a relation, making worthy of investigating the possibility of a single physical mechanisminducing both of them.3. In the external shock model the L X ( T a ) − L γ,iso and the L X,peak − L X ( T a ) relations cannot lead to a netdistinction among constant or wind type density media, but they are able to exclude so far the thin shell modelsand to favour the thick shell ones. Among the models that very well describe the L X ( T a ) − L γ,iso and the L X,peak − L X ( T a ) relations there is the one by Hasco¨et et al. (2014). They investigated the standard FS modelwith a wind external medium and a microphysics parameter (cid:15) e ∝ n − ν , and they found out that for values ν ≈ L FO,peak − T ∗ FO,peak relation sheds light on the nature of the flares in the GRB light curves.From the analysis carried on by Li et al. (2012), it was found out that the flares are additional and distinctcomponents of the afterglow phase. They also claimed that a periodically-emitting energy central engine canexplain the optical and γ -ray flares in the afterglow phase.6. One of the greatest issues that may undermine the GRB relations as model discriminators and as cosmologicaltools are selection bias and the evolution with the redshift of the physical quantities involved in these relations.An example of selection biases is given by Dainotti et al. (2013a), who used the Efron and Petrosian (1992)method to deal with the redshift evolution of the X-ray luminosity and the time, to evaluate the intrinsic L X − T ∗ a relation. Furthermore, Dainotti et al. (2015a) assumed an unknown efficiency function for the detectorand investigated the biases due to the detector’s threshold and how they affect the X-ray luminosity and thetime measurements. The methods described can be also useful to deal with the selection effects for the opticalluminosity and in the log L O, - α O,> relation and any other relation.7. Regarding the use of correlations as cosmological tools, we still have to further reduce the scatter of the GRBmeasurements and the dispersion of the relations themselves to allow GRBs to be complementary with themeasurement of SNe Ia. Indeed, the redshift evolution effect and the threshold of the detector can generaterelevant selection biases on the physical quantities which however we know how to treat analytically with robuststatistical techniques as we have shown in several sections. Nevertheless, more precise calibration methods, withthe help of other cosmological objects, and more space missions dedicated to detect faint GRBs and GRBs at highredshift (for example the future SVOM mission) can shed new light on the use of GRBs as cosmological tools.Lastly, other open questions are concerned with how much cosmological parameters can reduce their degeneracyadding GRBs into the set of cosmological standard candles. For example, different results of the value of w canlead to scenarios which can be compatible with a non-flat cosmological model. CONCLUSIONSIn this work, we have summarized the bivariate relations among the GRB afterglow parameters and their character-istics in order to discuss their intrinsic nature and the possibility to use them as standardizable candles. It has beenshown with different methodologies that some of the relations presented are intrinsic. However, the intrinsic slopehas been determined only for a few relations. For the other relations, we are not aware of their intrinsic slopes andconsequently how far the use of the observed relations can influence the evaluation of the theoretical models and the“best” cosmological settings (Dainotti et al. 2013b). Therefore, the estimate of the intrinsic relations is crucial for thedetermination of the most plausible model that can explain the plateau phase and the afterglow emission.In fact, though there are several theoretical interpretations describing each relation, as we have shown, in many cases,more than one is viable. This result indicates that the emission processes that rule the GRBs still have to be furtherinvestigated. To this end, it is necessary to use the intrinsic relations and not the observed ones affected by selectionbiases to test the theoretical models. Moreover, the pure afterglow relations have the advantage of not presentingthe double truncation in the flux limit, thus facilitating the correction for selection effects and their use as redshiftestimators and cosmological tools.A very challenging future step would be to use the corrected relations as a reliable redshift estimator and to determinea further estimate of H , Ω Λ and w . In particular, it is advisable to use all the afterglow relations which are not yetemployed for cosmological studies as new probes, after they are corrected for selection biases, in order to reduce theintrinsic scatter as it has been done in Schaefer (2007) for the prompt relations. ACKNOWLEDGMENTSThis work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. Wethank S. Capozziello for fruitful comments. M.G.D is grateful to the Marie Curie Program, because the research9leading to these results has received funding from the European Union Seventh FrameWork Program (FP7-2007/2013)under grant agreement N 626267. R.D.V. is grateful to the Polish National Science Centre through the grant DEC-2012/04/A/ST9/00083. REFERENCES
L. Amati and M. Della Valle. Measuring CosmologicalParameters with Gamma Ray Bursts.
International Journalof Modern Physics D , 22:1330028, December 2013.doi:10.1142/S0218271813300280.L. Amati, F. Frontera, M. Tavani, J. J. M. in’t Zand,A. Antonelli, E. Costa, M. Feroci, C. Guidorzi, J. Heise,N. Masetti, E. Montanari, L. Nicastro, E. Palazzi, E. Pian,L. Piro, and P. Soffitta. Intrinsic spectra and energetics ofBeppoSAX Gamma-Ray Bursts with known redshifts.
A&A ,390:81–89, July 2002. doi:10.1051/0004-6361:20020722.L. Amati, M. Della Valle, F. Frontera, D. Malesani, C. Guidorzi,E. Montanari, and E. Pian. On the consistency of peculiarGRBs 060218 and 060614 with the E p,i - E iso correlation.
A&A , 463:913–919, March 2007.doi:10.1051/0004-6361:20065994.L. Amati, C. Guidorzi, F. Frontera, M. Della Valle, F. Finelli,R. Landi, and E. Montanari. Measuring the cosmologicalparameters with the E p,i -E iso correlation of gamma-raybursts. MNRAS , 391:577–584, December 2008.doi:10.1111/j.1365-2966.2008.13943.x.J.-L. Atteia. A simple empirical redshift indicator forgamma-ray bursts.
A&A , 407:L1–L4, August 2003.doi:10.1051/0004-6361:20030958.E. Berger. The Prompt Gamma-Ray and Afterglow Energies ofShort-Duration Gamma-Ray Bursts.
ApJ , 670:1254–1259,December 2007. doi:10.1086/522195.E. Berger. Short-Duration Gamma-Ray Bursts.
ARA&A , 52:43–105, August 2014.doi:10.1146/annurev-astro-081913-035926.M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet,A. Corsi, M. G. Dainotti, F. Fraschetti, R. Guida, R. Ruffini,and S. S. Xue. GRB970228 as a prototype for short GRBswith afterglow.
Nuovo Cimento B Serie , 121:1439–1440,December 2006. doi:10.1393/ncb/i2007-10283-0.M. G. Bernardini, C. L. Bianco, L. Caito, M. G. Dainotti,R. Guida, and R. Ruffini. GRB 970228 and a class of GRBswith an initial spikelike emission.
A&A , 474:L13–L16,October 2007. doi:10.1051/0004-6361:20078300.M. G. Bernardini, R. Margutti, J. Mao, E. Zaninoni, andG. Chincarini. The X-ray light curve of gamma-ray bursts:clues to the central engine.
A&A , 539:A3, March 2012a.doi:10.1051/0004-6361/201117895.M. G. Bernardini, R. Margutti, E. Zaninoni, and G. Chincarini.A universal scaling for short and long gamma-ray bursts:E
X,iso - E ,iso - E pk . MNRAS , 425:1199–1204, September2012b. doi:10.1111/j.1365-2966.2012.21487.x. M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas,P. Astier, P. El-Hage, M. Konig, S. Kuhlmann, J. Marriner,R. Pain, N. Regnault, C. Balland, B. A. Bassett, P. J. Brown,H. Campbell, R. G. Carlberg, F. Cellier-Holzem, D. Cinabro,A. Conley, C. B. D’Andrea, D. L. DePoy, M. Doi, R. S. Ellis,S. Fabbro, A. V. Filippenko, R. J. Foley, J. A. Frieman,D. Fouchez, L. Galbany, A. Goobar, R. R. Gupta, G. J. Hill,R. Hlozek, C. J. Hogan, I. M. Hook, D. A. Howell, S. W. Jha,L. Le Guillou, G. Leloudas, C. Lidman, J. L. Marshall,A. M¨oller, A. M. Mour˜ao, J. Neveu, R. Nichol, M. D.Olmstead, N. Palanque-Delabrouille, S. Perlmutter, J. L.Prieto, C. J. Pritchet, M. Richmond, A. G. Riess,V. Ruhlmann-Kleider, M. Sako, K. Schahmaneche, D. P.Schneider, M. Smith, J. Sollerman, M. Sullivan, N. A. Walton,and C. J. Wheeler. Improved cosmological constraints from ajoint analysis of the SDSS-II and SNLS supernova samples.
A&A , 568:A22, August 2014.doi:10.1051/0004-6361/201423413.J. S. Bloom, D. A. Frail, and R. Sari. The Prompt EnergyRelease of Gamma-Ray Bursts using a Cosmologicalk-Correction. AJ , 121:2879–2888, June 2001.doi:10.1086/321093.M. Bo¨er and B. Gendre. Evidences for two Gamma-Ray Burstafterglow emission regimes. A&A , 361:L21–L24, September2000.N. R. Butler, J. S. Bloom, and D. Poznanski. The Cosmic Rate,Luminosity Function, and Intrinsic Correlations of LongGamma-Ray Bursts.
ApJ , 711:495–516, March 2010.doi:10.1088/0004-637X/711/1/495.L. Caito, M. G. Bernardini, C. L. Bianco, M. G. Dainotti,R. Guida, and R. Ruffini. GRB060614: a “fake” short GRBfrom a merging binary system.
A&A , 498:501–507, May 2009.doi:10.1051/0004-6361/200810676.J. Calcino and T. Davis. The need for accurate redshifts insupernova cosmology.
Journal of Cosmology andAstroparticle Physics , 1:038, January 2017.doi:10.1088/1475-7516/2017/01/038.J. K. Cannizzo and N. Gehrels. A New Paradigm forGamma-ray Bursts: Long-term Accretion Rate Modulation byan External Accretion Disk.
ApJ , 700:1047–1058, August2009. doi:10.1088/0004-637X/700/2/1047.J. K. Cannizzo, E. Troja, and N. Gehrels. Fall-back Disks inLong and Short Gamma-Ray Bursts.
ApJ , 734:35, June 2011.doi:10.1088/0004-637X/734/1/35.Z. Cano. Gamma-Ray Burst Supernovae as StandardizableCandles.
ApJ , 794:121, October 2014.doi:10.1088/0004-637X/794/2/121.V. F. Cardone, S. Capozziello, and M. G. Dainotti. An updatedgamma-ray bursts Hubble diagram.
MNRAS , 400:775–790,December 2009. doi:10.1111/j.1365-2966.2009.15456.x.V. F. Cardone, M. G. Dainotti, S. Capozziello, and R. Willingale.Constraining cosmological parameters by gamma-ray burstX-ray afterglow light curves.
MNRAS , 408:1181–1186,October 2010. doi:10.1111/j.1365-2966.2010.17197.x. S. B. Cenko, J. Kelemen, F. A. Harrison, D. B. Fox, S. R.Kulkarni, M. M. Kasliwal, E. O. Ofek, A. Rau, A. Gal-Yam,D. A. Frail, and D.-S. Moon. Dark Bursts in the Swift Era:The Palomar 60 Inch-Swift Early Optical Afterglow Catalog.
ApJ , 693:1484–1493, March 2009.doi:10.1088/0004-637X/693/2/1484.M. Chevallier and D. Polarski. Accelerating Universes withScaling Dark Matter.
International Journal of ModernPhysics D , 10:213–223, 2001. doi:10.1142/S0218271801000822.G. Chincarini, A. Moretti, P. Romano, A. D. Falcone, D. Morris,J. Racusin, S. Campana, S. Covino, C. Guidorzi,G. Tagliaferri, D. N. Burrows, C. Pagani, M. Stroh, D. Grupe,M. Capalbi, G. Cusumano, N. Gehrels, P. Giommi, V. LaParola, V. Mangano, T. Mineo, J. A. Nousek, P. T. O’Brien,K. L. Page, M. Perri, E. Troja, R. Willingale, and B. Zhang.The First Survey of X-Ray Flares from Gamma-Ray BurstsObserved by Swift: Temporal Properties and Morphology.
ApJ , 671:1903–1920, December 2007. doi:10.1086/521591.A. Cucchiara, A. J. Levan, D. B. Fox, N. R. Tanvir, T. N.Ukwatta, E. Berger, T. Kr¨uhler, A. K¨upc¨u Yolda¸s, X. F. Wu,K. Toma, J. Greiner, F. E. Olivares, A. Rowlinson, L. Amati,T. Sakamoto, K. Roth, A. Stephens, A. Fritz, J. P. U. Fynbo,J. Hjorth, D. Malesani, P. Jakobsson, K. Wiersema, P. T.O’Brien, A. M. Soderberg, R. J. Foley, A. S. Fruchter,J. Rhoads, R. E. Rutledge, B. P. Schmidt, M. A. Dopita,P. Podsiadlowski, R. Willingale, C. Wolf, S. R. Kulkarni, andP. D’Avanzo. A Photometric Redshift of z ∼ . ApJ , 736:7, July 2011.doi:10.1088/0004-637X/736/1/7.G. D’Agostini. Fits, and especially linear fits, with errors onboth axes, extra variance of the data points and othercomplications.
ArXiv Physics e-prints , November 2005.Z. G. Dai and T. Lu. Gamma-ray burst afterglows and evolutionof postburst fireballs with energy injection from stronglymagnetic millisecond pulsars.
A&A , 333:L87–L90, May 1998.M. Dainotti, V. Petrosian, R. Willingale, P. O’Brien,M. Ostrowski, and S. Nagataki. Luminosity-time andluminosity-luminosity correlations for GRB prompt andafterglow plateau emissions.
MNRAS , 451:3898–3908, August2015a. doi:10.1093/mnras/stv1229.M. Dainotti, R. Del Vecchio, and M. Tarnopolski. Gamma RayBurst Prompt correlations.
ArXiv e-prints , December 2016a.M. G. Dainotti, M. G. Bernardini, C. L. Bianco, L. Caito,R. Guida, and R. Ruffini. GRB 060218 and GRBs associatedwith supernovae Ib/c.
A&A , 471:L29–L32, August 2007.doi:10.1051/0004-6361:20078068.M. G. Dainotti, V. F. Cardone, and S. Capozziello. Atime-luminosity correlation for γ -ray bursts in the X-rays. MNRAS , 391:L79–L83, November 2008.doi:10.1111/j.1745-3933.2008.00560.x.M. G. Dainotti, R. Willingale, S. Capozziello, V. FabrizioCardone, and M. Ostrowski. Discovery of a Tight Correlationfor Gamma-ray Burst Afterglows with ”Canonical” LightCurves.
ApJL , 722:L215–L219, October 2010.doi:10.1088/2041-8205/722/2/L215.M. G. Dainotti, V. Fabrizio Cardone, S. Capozziello,M. Ostrowski, and R. Willingale. Study of PossibleSystematics in the L* X -T* a Correlation of Gamma-ray Bursts.
ApJ , 730:135, April 2011a. doi:10.1088/0004-637X/730/2/135.M. G. Dainotti, M. Ostrowski, and R. Willingale. Towards astandard gamma-ray burst: tight correlations between theprompt and the afterglow plateau phase emission.
MNRAS ,418:2202–2206, December 2011b.doi:10.1111/j.1365-2966.2011.19433.x. M. G. Dainotti, V. F. Cardone, E. Piedipalumbo, andS. Capozziello. Slope evolution of GRB correlations andcosmology.
MNRAS , 436:82–88, November 2013a.doi:10.1093/mnras/stt1516.M. G. Dainotti, V. Petrosian, J. Singal, and M. Ostrowski.Determination of the Intrinsic Luminosity Time Correlation inthe X-Ray Afterglows of Gamma-Ray Bursts.
ApJ , 774:157,September 2013b. doi:10.1088/0004-637X/774/2/157.M. G. Dainotti, R. Del Vecchio, S. Nagataki, and S. Capozziello.Selection Effects in Gamma-Ray Burst Correlations:Consequences on the Ratio between Gamma-Ray Burst andStar Formation Rates.
ApJ , 800:31, February 2015b.doi:10.1088/0004-637X/800/1/31.M. G. Dainotti, S. Nagataki, K. Maeda, S. Postnikov, andE. Pian. A study of gamma ray bursts with afterglow plateauphases associated with supernovae.
ArXiv e-prints , December2016b.M. G. Dainotti, S. Postnikov, X. Hernandez, and M. Ostrowski.A Fundamental Plane for Long Gamma-Ray Bursts withX-Ray Plateaus.
ApJL , 825:L20, July 2016c.doi:10.3847/2041-8205/825/2/L20.S. Dall’Osso, G. Stratta, D. Guetta, S. Covino, G. De Cesare,and L. Stella. Gamma-ray bursts afterglows with energyinjection from a spinning down neutron star.
A&A , 526:A121,February 2011. doi:10.1051/0004-6361/201014168.P. D’Avanzo, R. Salvaterra, B. Sbarufatti, L. Nava, A. Melandri,M. G. Bernardini, S. Campana, S. Covino, D. Fugazza,G. Ghirlanda, G. Ghisellini, V. L. Parola, M. Perri, S. D.Vergani, and G. Tagliaferri. A complete sample of brightSwift Gamma-ray bursts: X-ray afterglow luminosity and itscorrelation with the prompt emission.
MNRAS , 425:506–513,September 2012. doi:10.1111/j.1365-2966.2012.21489.x.T. M. Davis, E. M¨ortsell, J. Sollerman, A. C. Becker, S. Blondin,P. Challis, A. Clocchiatti, A. V. Filippenko, R. J. Foley, P. M.Garnavich, S. Jha, K. Krisciunas, R. P. Kirshner,B. Leibundgut, W. Li, T. Matheson, G. Miknaitis, G. Pignata,A. Rest, A. G. Riess, B. P. Schmidt, R. C. Smith,J. Spyromilio, C. W. Stubbs, N. B. Suntzeff, J. L. Tonry,W. M. Wood-Vasey, and A. Zenteno. Scrutinizing ExoticCosmological Models Using ESSENCE Supernova DataCombined with Other Cosmological Probes.
ApJ , 666:716–725, September 2007. doi:10.1086/519988.R. Del Vecchio, M. Giovanna Dainotti, and M. Ostrowski. Studyof GRB Light-curve Decay Indices in the Afterglow Phase.
ApJ , 828:36, September 2016.doi:10.3847/0004-637X/828/1/36.P. C. Duffell and A. I. MacFadyen. From Engine to Afterglow:Collapsars Naturally Produce Top-heavy Jets and Early-timePlateaus in Gamma-Ray Burst Afterglows.
ApJ , 806:205,June 2015. doi:10.1088/0004-637X/806/2/205.B. Efron and V. Petrosian. A simple test of independence fortruncated data with applications to redshift surveys.
ApJ ,399:345–352, November 1992. doi:10.1086/171931.E. E. Fenimore, J. J. M. in ’t Zand, J. P. Norris, J. T. Bonnell,and R. J. Nemiroff. Gamma-Ray Burst Peak Duration as aFunction of Energy.
ApJL , 448:L101, August 1995.doi:10.1086/309603.C. Firmani, G. Ghisellini, V. Avila-Reese, and G. Ghirlanda.Discovery of a tight correlation among the prompt emissionproperties of long gamma-ray bursts.
MNRAS , 370:185–197,July 2006. doi:10.1111/j.1365-2966.2006.10445.x. N. Gehrels, S. D. Barthelmy, D. N. Burrows, J. K. Cannizzo,G. Chincarini, E. Fenimore, C. Kouveliotou, P. O’Brien, D. M.Palmer, J. Racusin, P. W. A. Roming, T. Sakamoto,J. Tueller, R. A. M. J. Wijers, and B. Zhang. Correlations ofPrompt and Afterglow Emission in Swift Long and ShortGamma-Ray Bursts.
ApJ , 689:1161–1172, December 2008.doi:10.1086/592766.Neil Gehrels. Short GRB Prompt and Afterglow Correlations.2007.G. Ghirlanda. Advances on GRB as cosmological tools. InG. Giobbi, A. Tornambe, G. Raimondo, M. Limongi, L. A.Antonelli, N. Menci, and E. Brocato, editors,
AmericanInstitute of Physics Conference Series , volume 1111 of
American Institute of Physics Conference Series , pages579–586, May 2009. doi:10.1063/1.3141613.G. Ghirlanda, G. Ghisellini, and D. Lazzati. TheCollimation-corrected Gamma-Ray Burst Energies Correlatewith the Peak Energy of Their ν F nu Spectrum.
ApJ , 616:331–338, November 2004. doi:10.1086/424913.G. Ghirlanda, G. Ghisellini, and C. Firmani. Gamma-ray burstsas standard candles to constrain the cosmological parameters.
New Journal of Physics , 8:123, July 2006.doi:10.1088/1367-2630/8/7/123.G. Ghisellini, M. Nardini, G. Ghirlanda, and A. Celotti. Aunifying view of gamma-ray burst afterglows.
MNRAS , 393:253–271, February 2009.doi:10.1111/j.1365-2966.2008.14214.x.B. P. Gompertz, P. T. O’Brien, G. A. Wynn, and A. Rowlinson.Can magnetar spin-down power extended emission in someshort GRBs?
MNRAS , 431:1745–1751, May 2013.doi:10.1093/mnras/stt293.D. Grupe, J. A. Nousek, P. Veres, B.-B. Zhang, and N. Gehrels.Evidence for New Relations between Gamma-Ray BurstPrompt and X-Ray Afterglow Emission from 9 Years of Swift.
ApJS , 209:20, December 2013.doi:10.1088/0067-0049/209/2/20.R. Guida, M. G. Bernardini, C. L. Bianco, L. Caito, M. G.Dainotti, and R. Ruffini. The Amati relation in the “fireshell”model.
A&A , 487:L37–L40, August 2008.doi:10.1051/0004-6361:200810338.R. Hasco¨et, F. Daigne, and R. Mochkovitch. The prompt-earlyafterglow connection in gamma-ray bursts: implications for theearly afterglow physics.
MNRAS , 442:20–27, July 2014.doi:10.1093/mnras/stu750.J. Heise, J. I. Zand, R. M. Kippen, and P. M. Woods. X-RayFlashes and X-Ray Rich Gamma Ray Bursts. In E. Costa,F. Frontera, and J. Hjorth, editors,
Gamma-ray Bursts in theAfterglow Era , page 16, 2001. doi:10.1007/10853853˙4.H. Ito, S. Nagataki, J. Matsumoto, S.-H. Lee, A. Tolstov, J. Mao,M. Dainotti, and A. Mizuta. Spectral and PolarizationProperties of Photospheric Emission from Stratified Jets.
ApJ , 789:159, July 2014. doi:10.1088/0004-637X/789/2/159.P. Jakobsson, J. Hjorth, J. P. U. Fynbo, D. Watson,K. Pedersen, G. Bj¨ornsson, and J. Gorosabel. SwiftIdentification of Dark Gamma-Ray Bursts.
ApJL , 617:L21–L24, December 2004. doi:10.1086/427089.Y. Kaneko, E. Ramirez-Ruiz, J. Granot, C. Kouveliotou, S. E.Woosley, S. K. Patel, E. Rol, J. J. M. in ’t Zand, A. J. van derHorst, R. A. M. J. Wijers, and R. Strom. Prompt andAfterglow Emission Properties of Gamma-Ray Bursts withSpectroscopically Identified Supernovae.
ApJ , 654:385–402,January 2007. doi:10.1086/508324. D. A. Kann, S. Klose, B. Zhang, D. Malesani, E. Nakar,A. Pozanenko, A. C. Wilson, N. R. Butler, P. Jakobsson,S. Schulze, M. Andreev, L. A. Antonelli, I. F. Bikmaev,V. Biryukov, M. B¨ottcher, R. A. Burenin, J. M. Castro Cer´on,A. J. Castro-Tirado, G. Chincarini, B. E. Cobb, S. Covino,P. D’Avanzo, V. D’Elia, M. Della Valle, A. de Ugarte Postigo,Y. Efimov, P. Ferrero, D. Fugazza, J. P. U. Fynbo, M. G˚alfalk,F. Grundahl, J. Gorosabel, S. Gupta, S. Guziy, B. Hafizov,J. Hjorth, K. Holhjem, M. Ibrahimov, M. Im, G. L. Israel,M. Je´linek, B. L. Jensen, R. Karimov, I. M. Khamitov,¨U. Kiziloˇglu, E. Klunko, P. Kub´anek, A. S. Kutyrev,P. Laursen, A. J. Levan, F. Mannucci, C. M. Martin,A. Mescheryakov, N. Mirabal, J. P. Norris, J.-E. Ovaldsen,D. Paraficz, E. Pavlenko, S. Piranomonte, A. Rossi,V. Rumyantsev, R. Salinas, A. Sergeev, D. Sharapov,J. Sollerman, B. Stecklum, L. Stella, G. Tagliaferri, N. R.Tanvir, J. Telting, V. Testa, A. C. Updike, A. Volnova,D. Watson, K. Wiersema, and D. Xu. The Afterglows ofSwift-era Gamma-ray Bursts. I. Comparing pre-Swift andSwift-era Long/Soft (Type II) GRB Optical Afterglows.
ApJ ,720:1513–1558, September 2010.doi:10.1088/0004-637X/720/2/1513.D. Kazanas, J. L. Racusin, J. Sultana, and A. Mastichiadis. TheStatistics of the Prompt-to-Afterglow GRB Flux Ratios andthe Supercritical Pile GRB Model.
ArXiv e-prints , January2015.R. M. Kippen, P. M. Woods, J. Heise, J. I. Zand, R. D. Preece,and M. S. Briggs. BATSE Observations of Fast X-RayTransients Detected by BeppoSAX-WFC. In E. Costa,F. Frontera, and J. Hjorth, editors,
Gamma-ray Bursts in theAfterglow Era , page 22, 2001. doi:10.1007/10853853˙5.R. W. Klebesadel, I. B. Strong, and R. A. Olson. Observationsof Gamma-Ray Bursts of Cosmic Origin.
ApJL , 182:L85, June1973. doi:10.1086/181225.S. Kobayashi, T. Piran, and R. Sari. Can Internal ShocksProduce the Variability in Gamma-Ray Bursts?
ApJ , 490:92,November 1997. doi:10.1086/512791.Y. Kodama, D. Yonetoku, T. Murakami, S. Tanabe, R. Tsutsui,and T. Nakamura. Gamma-ray bursts in 1.8 ¡ z ¡ 5.6 suggestthat the time variation of the dark energy is small.
MNRAS ,391:L1–L4, November 2008.doi:10.1111/j.1745-3933.2008.00508.x.E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold,G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page,D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, M. Limon,S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland,E. Wollack, and E. L. Wright. Seven-year WilkinsonMicrowave Anisotropy Probe (WMAP) Observations:Cosmological Interpretation.
ApJS , 192:18, February 2011.doi:10.1088/0067-0049/192/2/18.C. Kouveliotou, C. A. Meegan, G. J. Fishman, N. P. Bhat, M. S.Briggs, T. M. Koshut, W. S. Paciesas, and G. N. Pendleton.Identification of two classes of gamma-ray bursts.
ApJL , 413:L101–L104, August 1993. doi:10.1086/186969.P. Kumar, R. Narayan, and J. L. Johnson. Properties ofGamma-Ray Burst Progenitor Stars.
Science , 321:376–, July2008. doi:10.1126/science.1159003.K. Leventis, R. A. M. J. Wijers, and A. J. van der Horst. Theplateau phase of gamma-ray burst afterglows in the thick-shellscenario.
MNRAS , 437:2448–2460, January 2014.doi:10.1093/mnras/stt2055.L. Li, E.-W. Liang, Q.-W. Tang, J.-M. Chen, S.-Q. Xi, H.-J. L¨u,H. Gao, B. Zhang, J. Zhang, S.-X. Yi, R.-J. Lu, L.-Z. L¨u, andJ.-Y. Wei. A Comprehensive Study of Gamma-Ray BurstOptical Emission. I. Flares and Early Shallow-decayComponent.
ApJ , 758:27, October 2012.doi:10.1088/0004-637X/758/1/27. X. Li and J. Hjorth. Light Curve Properties of SupernovaeAssociated With Gamma-ray Bursts.
ArXiv e-prints , July2014.E. Liang and B. Zhang. Identification of Two Categories ofOptically Bright Gamma-Ray Bursts.
ApJL , 638:L67–L70,February 2006a. doi:10.1086/501049.E. Liang and B. Zhang. Calibration of gamma-ray burstluminosity indicators.
MNRAS , 369:L37–L41, June 2006b.doi:10.1111/j.1745-3933.2006.00169.x.E.-W. Liang, B.-B. Zhang, and B. Zhang. A ComprehensiveAnalysis of Swift XRT Data. II. Diverse Physical Origins ofthe Shallow Decay Segment.
ApJ , 670:565–583, November2007. doi:10.1086/521870.E.-W. Liang, S.-X. Yi, J. Zhang, H.-J. L¨u, B.-B. Zhang, andB. Zhang. Constraining Gamma-ray Burst Initial LorentzFactor with the Afterglow Onset Feature and Discovery of aTight Γ -E gamma,iso Correlation.
ApJ , 725:2209–2224,December 2010. doi:10.1088/0004-637X/725/2/2209.N. Liang, W. K. Xiao, Y. Liu, and S. N. Zhang. ACosmology-Independent Calibration of Gamma-Ray BurstLuminosity Relations and the Hubble Diagram.
ApJ , 685:354–360, September 2008. doi:10.1086/590903.C. C. Lindner, M. Milosavljevi´c, S. M. Couch, and P. Kumar.Collapsar Accretion and the Gamma-Ray Burst X-Ray LightCurve.
ApJ , 713:800–815, April 2010.doi:10.1088/0004-637X/713/2/800.N. M. Lloyd and V. Petrosian. Distribution of SpectralCharacteristics and the Cosmological Evolution ofGamma-Ray Bursts.
ApJ , 511:550–561, February 1999.doi:10.1086/306719.N. M. Lloyd, V. Petrosian, and R. D. Preece. Synchrotronemission as the source of GRB spectra, Part II: Observations.In R. M. Kippen, R. S. Mallozzi, and G. J. Fishman, editors,
Gamma-ray Bursts, 5th Huntsville Symposium , volume 526 of
American Institute of Physics Conference Series , pages155–159, September 2000. doi:10.1063/1.1361525.H.-J. L¨u and B. Zhang. A Test of the Millisecond MagnetarCentral Engine Model of Gamma-Ray Bursts with Swift Data.
ApJ , 785:74, April 2014. doi:10.1088/0004-637X/785/1/74.N. Lyons, P. T. O’Brien, B. Zhang, R. Willingale, E. Troja, andR. L. C. Starling. Can X-ray emission powered by aspinning-down magnetar explain some gamma-ray burstlight-curve features?
MNRAS , 402:705–712, February 2010.doi:10.1111/j.1365-2966.2009.15538.x.V. Mangano, B. Sbarufatti, and G. Stratta. Extending theplateau luminosity-duration anticorrelation.
Memorie dellaSocieta Astronomica Italiana Supplementi , 21:143, 2012.R. Margutti, C. Guidorzi, G. Chincarini, M. G. Bernardini,F. Genet, J. Mao, and F. Pasotti. Lag-luminosity relation in γ -ray burst X-ray flares: a direct link to the prompt emission. MNRAS , 406:2149–2167, August 2010.doi:10.1111/j.1365-2966.2010.16824.x.R. Margutti, E. Zaninoni, M. G. Bernardini, G. Chincarini,F. Pasotti, C. Guidorzi, L. Angelini, D. N. Burrows,M. Capalbi, P. A. Evans, N. Gehrels, J. Kennea, V. Mangano,A. Moretti, J. Nousek, J. P. Osborne, K. L. Page, M. Perri,J. Racusin, P. Romano, B. Sbarufatti, S. Stafford, andM. Stamatikos. The prompt-afterglow connection ingamma-ray bursts: a comprehensive statistical analysis ofSwift X-ray light curves.
MNRAS , 428:729–742, January2013. doi:10.1093/mnras/sts066.A. Maxham and B. Zhang. Modeling Gamma-Ray Burst X-RayFlares Within the Internal Shock Model.
ApJ , 707:1623–1633,December 2009. doi:10.1088/0004-637X/707/2/1623. E. P. Mazets, S. V. Golenetskii, V. N. Ilyinskii, V. N. Panov,R. L. Aptekar, Y. A. Guryan, M. P. Proskura, I. A. Sokolov,Z. Y. Sokolova, T. V. Kharitonova, A. V. Dyatchkov, andN. G. Khavenson. Catalog of cosmic gamma-ray bursts fromthe KONUS experiment data.
Ap&SS , 80:85–117, November1981. doi:10.1007/BF00649141.A. Melandri, C. G. Mundell, S. Kobayashi, C. Guidorzi,A. Gomboc, I. A. Steele, R. J. Smith, D. Bersier, C. J.Mottram, D. Carter, M. F. Bode, P. T. O’Brien, N. R. Tanvir,E. Rol, and R. Chapman. The Early-Time Optical Propertiesof Gamma-Ray Burst Afterglows.
ApJ , 686:1209-1230,October 2008. doi:10.1086/591243.A. Melandri, S. Covino, D. Rogantini, R. Salvaterra,B. Sbarufatti, M. G. Bernardini, S. Campana, P. D’Avanzo,V. D’Elia, D. Fugazza, G. Ghirlanda, G. Ghisellini, L. Nava,S. D. Vergani, and G. Tagliaferri. Optical and X-rayrest-frame light curves of the BAT6 sample.
A&A , 565:A72,May 2014. doi:10.1051/0004-6361/201323361.P. M´esz´aros. Theoretical models of gamma-ray bursts. In C. A.Meegan, R. D. Preece, and T. M. Koshut, editors,
Gamma-Ray Bursts, 4th Hunstville Symposium , volume 428 of
American Institute of Physics Conference Series , pages647–656, May 1998. doi:10.1063/1.55394.P. M´esz´aros. Gamma-ray bursts.
Reports on Progress inPhysics , 69:2259–2321, August 2006.doi:10.1088/0034-4885/69/8/R01.B. D. Metzger, D. Giannios, T. A. Thompson, N. Bucciantini,and E. Quataert. The protomagnetar model for gamma-raybursts.
MNRAS , 413:2031–2056, May 2011.doi:10.1111/j.1365-2966.2011.18280.x.M. Nardini, G. Ghisellini, G. Ghirlanda, F. Tavecchio,C. Firmani, and D. Lazzati. Clustering of the optical-afterglowluminosities of long gamma-ray bursts.
A&A , 451:821–833,June 2006. doi:10.1051/0004-6361:20054085.M. Nardini, G. Ghisellini, and G. Ghirlanda. Optical afterglowluminosities in the Swift epoch: confirming clustering andbimodality.
MNRAS , 386:L87–L91, May 2008a.doi:10.1111/j.1745-3933.2008.00467.x.M. Nardini, G. Ghisellini, and G. Ghirlanda. Optical afterglowsof gamma-ray bursts: a bimodal distribution?
MNRAS , 383:1049–1057, January 2008b.doi:10.1111/j.1365-2966.2007.12588.x.J. P. Norris and J. T. Bonnell. Short Gamma-Ray Bursts withExtended Emission.
ApJ , 643:266–275, May 2006.doi:10.1086/502796.J. A. Nousek, C. Kouveliotou, D. Grupe, K. L. Page, J. Granot,E. Ramirez-Ruiz, S. K. Patel, D. N. Burrows, V. Mangano,S. Barthelmy, A. P. Beardmore, S. Campana, M. Capalbi,G. Chincarini, G. Cusumano, A. D. Falcone, N. Gehrels,P. Giommi, M. R. Goad, O. Godet, C. P. Hurkett, J. A.Kennea, A. Moretti, P. T. O’Brien, J. P. Osborne, P. Romano,G. Tagliaferri, and A. A. Wells. Evidence for a CanonicalGamma-Ray Burst Afterglow Light Curve in the Swift XRTData.
ApJ , 642:389–400, May 2006. doi:10.1086/500724.M. Nysewander, A. S. Fruchter, and A. Pe’er. A Comparison ofthe Afterglows of Short- and Long-duration Gamma-rayBursts.
ApJ , 701:824–836, August 2009.doi:10.1088/0004-637X/701/1/824.S. R. Oates, M. J. Page, P. Schady, M. de Pasquale, T. S. Koch,A. A. Breeveld, P. J. Brown, M. M. Chester, S. T. Holland,E. A. Hoversten, N. P. M. Kuin, F. E. Marshall, P. W. A.Roming, M. Still, D. E. vanden Berk, S. Zane, and J. A.Nousek. A statistical study of gamma-ray burst afterglowsmeasured by the Swift Ultraviolet Optical Telescope.
MNRAS , 395:490–503, May 2009.doi:10.1111/j.1365-2966.2009.14544.x. S. R. Oates, M. J. Page, M. De Pasquale, P. Schady, A. A.Breeveld, S. T. Holland, N. P. M. Kuin, and F. E. Marshall.A correlation between the intrinsic brightness and averagedecay rate of Swift/UVOT gamma-ray burstoptical/ultraviolet light curves.
MNRAS , 426:L86–L90,October 2012. doi:10.1111/j.1745-3933.2012.01331.x.S. R. Oates, J. L. Racusin, M. De Pasquale, M. J. Page, A. J.Castro-Tirado, J. Gorosabel, P. J. Smith, A. A. Breeveld, andN. P. M. Kuin. Exploring the canonical behaviour of longgamma-ray bursts using an intrinsic multiwavelength afterglowcorrelation.
MNRAS , 453:4121–4135, November 2015.doi:10.1093/mnras/stv1956.P. T. O’Brien, R. Willingale, J. Osborne, M. R. Goad, K. L.Page, S. Vaughan, E. Rol, A. Beardmore, O. Godet, C. P.Hurkett, A. Wells, B. Zhang, S. Kobayashi, D. N. Burrows,J. A. Nousek, J. A. Kennea, A. Falcone, D. Grupe, N. Gehrels,S. Barthelmy, J. Cannizzo, J. Cummings, J. E. Hill,H. Krimm, G. Chincarini, G. Tagliaferri, S. Campana,A. Moretti, P. Giommi, M. Perri, V. Mangano, andV. LaParola. The Early X-Ray Emission from GRBs.
ApJ ,647:1213–1237, August 2006. doi:10.1086/505457.A. Panaitescu and W. T. Vestrand. Taxonomy of gamma-rayburst optical light curves: identification of a salient class ofearly afterglows.
MNRAS , 387:497–504, June 2008.doi:10.1111/j.1365-2966.2008.13231.x.A. Panaitescu and W. T. Vestrand. Optical afterglows ofgamma-ray bursts: peaks, plateaus and possibilities.
MNRAS ,414:3537–3546, July 2011.doi:10.1111/j.1365-2966.2011.18653.x.S. Perlmutter, G. Aldering, M. della Valle, S. Deustua, R. S.Ellis, S. Fabbro, A. Fruchter, G. Goldhaber, D. E. Groom,I. M. Hook, A. G. Kim, M. Y. Kim, R. A. Knop, C. Lidman,R. G. McMahon, P. Nugent, R. Pain, N. Panagia, C. R.Pennypacker, P. Ruiz-Lapuente, B. Schaefer, and N. Walton.Discovery of a supernova explosion at half the age of theuniverse.
Nature , 391:51, January 1998. doi:10.1038/34124.R. Perna, P. J. Armitage, and B. Zhang. Flares in Long andShort Gamma-Ray Bursts: A Common Origin in aHyperaccreting Accretion Disk.
ApJL , 636:L29–L32, January2006. doi:10.1086/499775.J. H. Petersen, K. K. Holst, and E. Budtz-Jørgensen. Correctinga Statistical Artifact in the Estimation of the Hubble ConstantBased on Type Ia Supernovae Results in a Change in Estimateof 1.2%.
ApJ , 723:966–968, November 2010.doi:10.1088/0004-637X/723/1/966.V. Petrosian, A. Bouvier, and F. Ryde. Gamma-Ray Bursts asCosmological Tools.
ArXiv e-prints , September 2009.Vahe Petrosian, Jack Singal, and Lukasz Stawarz. Luminositycorrelations, luminosity evolutions, and radio loudness of agnsfrom multiwavelength observations. In
Multiwavelength AGNSurveys and Studies , volume 9 of
Proceedings of theInternational Astronomical Union , pages 172–172, 10 2013.doi:10.1017/S174392131400369X. URL http://journals.cambridge.org/article_S174392131400369X .M. M. Phillips. The absolute magnitudes of Type IA supernovae.
ApJL , 413:L105–L108, August 1993. doi:10.1086/186970.Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud,M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B.Barreiro, J. G. Bartlett, and et al. Planck 2015 results. XIII.Cosmological parameters.
A&A , 594:A13, September 2016.doi:10.1051/0004-6361/201525830.S. Postnikov, M. G. Dainotti, X. Hernandez, and S. Capozziello.Nonparametric Study of the Evolution of the CosmologicalEquation of State with SNeIa, BAO, and High-redshift GRBs.
ApJ , 783:126, March 2014. doi:10.1088/0004-637X/783/2/126. D. Proga and B. Zhang. The late time evolution of gamma-raybursts: ending hyperaccretion and producing flares.
MNRAS ,370:L61–L65, July 2006.doi:10.1111/j.1745-3933.2006.00189.x.J. L. Racusin, S. R. Oates, P. Schady, D. N. Burrows, M. dePasquale, D. Donato, N. Gehrels, S. Koch, J. McEnery,T. Piran, P. Roming, T. Sakamoto, C. Swenson, E. Troja,V. Vasileiou, F. Virgili, D. Wanderman, and B. Zhang. Fermiand Swift Gamma-ray Burst Afterglow Population Studies.
ApJ , 738:138, September 2011.doi:10.1088/0004-637X/738/2/138.J. L. Racusin, S. R. Oates, M. de Pasquale, and D. Kocevski. ACorrelation between the Intrinsic Brightness and AverageDecay Rate of Gamma-Ray Burst X-Ray Afterglow LightCurves.
ApJ , 826:45, July 2016.doi:10.3847/0004-637X/826/1/45.N. Rea, M. Gullon, J. A. Pons, R. Perna, M. G. Dainotti, J. A.Miralles, and D. F. Torres. Constraining the GRB-magnetarmodel by means of the Galactic pulsar population.
ArXive-prints , October 2015.M. J. Rees and P. M´esz´aros. Refreshed Shocks and AfterglowLongevity in Gamma-Ray Bursts.
ApJL , 496:L1–L4, March1998. doi:10.1086/311244.A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti,A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan,S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips,D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith,J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry.Observational Evidence from Supernovae for an AcceleratingUniverse and a Cosmological Constant. AJ , 116:1009–1038,September 1998. doi:10.1086/300499.A. G. Riess, L. Macri, S. Casertano, M. Sosey, H. Lampeitl,H. C. Ferguson, A. V. Filippenko, S. W. Jha, W. Li,R. Chornock, and D. Sarkar. A Redetermination of theHubble Constant with the Hubble Space Telescope from aDifferential Distance Ladder. ApJ , 699:539–563, July 2009.doi:10.1088/0004-637X/699/1/539.S. A. Rodney, A. G. Riess, D. M. Scolnic, D. O. Jones,S. Hemmati, A. Molino, C. McCully, B. Mobasher, L.-G.Strolger, O. Graur, B. Hayden, and S. Casertano. Two SNe Iaat Redshift 2: Improved Classification and RedshiftDetermination with Medium-band Infrared Imaging. AJ , 150:156, November 2015. doi:10.1088/0004-6256/150/5/156.A. Rowlinson and P. O’Brien. Energy injection in short GRBsand the role of magnetars. In Gamma-Ray Bursts 2012Conference (GRB 2012) , 2012.A. Rowlinson, P. T. O’Brien, B. D. Metzger, N. R. Tanvir, andA. J. Levan. Signatures of magnetar central engines in shortGRB light curves.
MNRAS , 430:1061–1087, April 2013.doi:10.1093/mnras/sts683.A. Rowlinson, B. P. Gompertz, M. Dainotti, P. T. O’Brien,R. A. M. J. Wijers, and A. J. van der Horst. Constrainingproperties of GRB magnetar central engines using the observedplateau luminosity and duration correlation.
MNRAS , 443:1779–1787, September 2014. doi:10.1093/mnras/stu1277.R. Ruffini, M. Muccino, C. L. Bianco, M. Enderli, L. Izzo,M. Kovacevic, A. V. Penacchioni, G. B. Pisani, J. A. Rueda,and Y. Wang. On binary-driven hypernovae and their nestedlate X-ray emission.
A&A , 565:L10, May 2014.doi:10.1051/0004-6361/201423812.T. Sakamoto, J. E. Hill, R. Yamazaki, L. Angelini, H. A. Krimm,G. Sato, S. Swindell, K. Takami, and J. P. Osborne. Evidenceof Exponential Decay Emission in the Swift Gamma-RayBursts.
ApJ , 669:1115–1129, November 2007.doi:10.1086/521640. T. Sakamoto, S. D. Barthelmy, L. Barbier, J. R. Cummings,E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B.Markwardt, D. M. Palmer, A. M. Parsons, G. Sato,M. Stamatikos, J. Tueller, T. N. Ukwatta, and B. Zhang. TheFirst Swift BAT Gamma-Ray Burst Catalog.
ApJS , 175:179–190, March 2008. doi:10.1086/523646.R. Sari and P. M´esz´aros. Impulsive and Varying Injection inGamma-Ray Burst Afterglows.
ApJL , 535:L33–L37, May2000. doi:10.1086/312689.R. Sari, T. Piran, and R. Narayan. Spectra and Light Curves ofGamma-Ray Burst Afterglows.
ApJL , 497:L17–L20, April1998. doi:10.1086/311269.P. Schady, K. O. Mason, M. J. Page, M. de Pasquale, D. C.Morris, P. Romano, P. W. A. Roming, S. Immler, and D. E.vanden Berk. Dust and gas in the local environments ofgamma-ray bursts.
MNRAS , 377:273–284, May 2007.doi:10.1111/j.1365-2966.2007.11592.x.B. E. Schaefer. The Hubble Diagram to Redshift > ApJ , 660:16–46, May 2007.doi:10.1086/511742.D. Stern, R. Jimenez, L. Verde, S. A. Stanford, andM. Kamionkowski. Cosmic Chronometers: Constraining theEquation of State of Dark Energy. II. A Spectroscopic Catalogof Red Galaxies in Galaxy Clusters.
ApJS , 188:280–289, May2010. doi:10.1088/0067-0049/188/1/280.J. Sultana, D. Kazanas, and K. Fukumura. LuminosityCorrelations for Gamma-Ray Bursts and Implications forTheir Prompt and Afterglow Emission Mechanisms.
ApJ , 758:32, October 2012. doi:10.1088/0004-637X/758/1/32.J. Sultana, D. Kazanas, and A. Mastichiadis. The SupercriticalPile Gamma-Ray Burst Model: The GRB Afterglow SteepDecline and Plateau Phase.
ApJ , 779:16, December 2013.doi:10.1088/0004-637X/779/1/16.N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah,K. Barbary, L. F. Barrientos, J. Botyanszki, M. Brodwin,N. Connolly, K. S. Dawson, A. Dey, M. Doi, M. Donahue,S. Deustua, P. Eisenhardt, E. Ellingson, L. Faccioli,V. Fadeyev, H. K. Fakhouri, A. S. Fruchter, D. G. Gilbank,M. D. Gladders, G. Goldhaber, A. H. Gonzalez, A. Goobar,A. Gude, T. Hattori, H. Hoekstra, E. Hsiao, X. Huang,Y. Ihara, M. J. Jee, D. Johnston, N. Kashikawa, B. Koester,K. Konishi, M. Kowalski, E. V. Linder, L. Lubin,J. Melbourne, J. Meyers, T. Morokuma, F. Munshi, C. Mullis,T. Oda, N. Panagia, S. Perlmutter, M. Postman, T. Pritchard,J. Rhodes, P. Ripoche, P. Rosati, D. J. Schlegel, A. Spadafora,S. A. Stanford, V. Stanishev, D. Stern, M. Strovink,N. Takanashi, K. Tokita, M. Wagner, L. Wang, N. Yasuda,H. K. C. Yee, and T. Supernova Cosmology Project. TheHubble Space Telescope Cluster Supernova Survey. V.Improving the Dark-energy Constraints above z ¿ 1 andBuilding an Early-type-hosted Supernova Sample.
ApJ , 746:85, February 2012. doi:10.1088/0004-637X/746/1/85.E. Troja, G. Cusumano, P. T. O’Brien, B. Zhang, B. Sbarufatti,V. Mangano, R. Willingale, G. Chincarini, J. P. Osborne,F. E. Marshall, D. N. Burrows, S. Campana, N. Gehrels,C. Guidorzi, H. A. Krimm, V. La Parola, E. W. Liang,T. Mineo, A. Moretti, K. L. Page, P. Romano, G. Tagliaferri,B. B. Zhang, M. J. Page, and P. Schady. Swift Observationsof GRB 070110: An Extraordinary X-Ray Afterglow Poweredby the Central Engine.
ApJ , 665:599–607, August 2007.doi:10.1086/519450.R. Tsutsui, D. Yonetoku, T. Nakamura, K. Takahashi, andY. Morihara. Possible existence of the E p -L p and E p -E iso correlations for short gamma-ray bursts with a factor 5-100dimmer than those for long gamma-ray bursts. MNRAS , 431:1398–1404, May 2013. doi:10.1093/mnras/stt262. Y. Urata, R. Yamazaki, T. Sakamoto, K. Huang, W. Zheng,G. Sato, T. Aoki, J. Deng, K. Ioka, W. Ip, K. S. Kawabata,Y. Lee, X. Liping, H. Mito, T. Miyata, Y. Nakada, T. Ohsugi,Y. Qiu, T. Soyano, K. Tarusawa, M. Tashiro, M. Uemura,J. Wei, and T. Yamashita. Testing the External-Shock Modelof Gamma-Ray Bursts Using the Late-Time SimultaneousOptical and X-Ray Afterglows.
ApJL , 668:L95–L98, October2007. doi:10.1086/522930.V. V. Usov. Millisecond pulsars with extremely strong magneticfields as a cosmological source of gamma-ray bursts.
Nature ,357:472–474, June 1992. doi:10.1038/357472a0.H. van Eerten. Self-similar relativistic blast waves with energyinjection.
MNRAS , 442:3495–3510, August 2014a.doi:10.1093/mnras/stu1025.H. J. van Eerten. Gamma-ray burst afterglow plateau breaktime-luminosity correlations favour thick shell models overthin shell models.
MNRAS , 445:2414–2423, December 2014b.doi:10.1093/mnras/stu1921.F.-Y. Wang, S. Qi, and Z.-G. Dai. The updated luminositycorrelations of gamma-ray bursts and cosmologicalimplications.
MNRAS , 415:3423–3433, August 2011.doi:10.1111/j.1365-2966.2011.18961.x.F. Y. Wang, Z. G. Dai, and E. W. Liang. Gamma-ray burstcosmology.
New Astronomy Reviews , 67:1–17, August 2015.doi:10.1016/j.newar.2015.03.001.H. Wei and S.-N. Zhang. Reconstructing the cosmic expansionhistory up to redshift z=6.29 with the calibrated gamma-raybursts.
European Physical Journal C , 63:139–147, September2009. doi:10.1140/epjc/s10052-009-1086-z.D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata,A. G. Riess, and E. Rozo. Observational probes of cosmicacceleration.
PhR , 530:87–255, September 2013.doi:10.1016/j.physrep.2013.05.001.R. A. M. J. Wijers, M. J. Rees, and P. Meszaros. Shocked byGRB 970228: the afterglow of a cosmological fireball.
MNRAS , 288:L51–L56, July 1997.R. Willingale, P. T. O’Brien, J. P. Osborne, O. Godet, K. L.Page, M. R. Goad, D. N. Burrows, B. Zhang, E. Rol,N. Gehrels, and G. Chincarini. Testing the Standard FireballModel of Gamma-Ray Bursts Using Late X-Ray AfterglowsMeasured by Swift.
ApJ , 662:1093–1110, June 2007.doi:10.1086/517989.R. Willingale, F. Genet, J. Granot, and P. T. O’Brien. Thespectral-temporal properties of the prompt pulses and rapiddecay phase of gamma-ray bursts.
MNRAS , 403:1296–1316,April 2010. doi:10.1111/j.1365-2966.2009.16187.x.R. Yamazaki. Prior Emission Model for X-ray Plateau Phase ofGamma-Ray Burst Afterglows.
ApJL , 690:L118–L121,January 2009. doi:10.1088/0004-637X/690/2/L118.S. X. Yi, Z. G. Dai, X. F. Wu, and F. Y. Wang. X-Ray AfterglowPlateaus of Long Gamma-Ray Bursts: Further Evidence forMillisecond Magnetars.
ArXiv e-prints , January 2014.D. Yonetoku, T. Murakami, T. Nakamura, R. Yamazaki, A. K.Inoue, and K. Ioka. Gamma-Ray Burst Formation RateInferred from the Spectral Peak Energy-Peak LuminosityRelation.
ApJ , 609:935–951, July 2004. doi:10.1086/421285.E. Zaninoni, M. G. Bernardini, R. Margutti, S. Oates, andG. Chincarini. Gamma-ray burst optical light-curve zoo:comparison with X-ray observations.
A&A , 557:A12,September 2013. doi:10.1051/0004-6361/201321221.B. Zhang and S. Kobayashi. Gamma-Ray Burst EarlyAfterglows: Reverse Shock Emission from an ArbitrarilyMagnetized Ejecta.
ApJ , 628:315–334, July 2005.doi:10.1086/429787. B. Zhang and P. M´esz´aros. Gamma-Ray Burst Afterglow withContinuous Energy Injection: Signature of a HighlyMagnetized Millisecond Pulsar.
ApJL , 552:L35–L38, May2001. doi:10.1086/320255.B. Zhang and H. Yan. The Internal-collision-induced MagneticReconnection and Turbulence (ICMART) Model ofGamma-ray Bursts.
ApJ , 726:90, January 2011.doi:10.1088/0004-637X/726/2/90. B. Zhang, E. Liang, K. L. Page, D. Grupe, B.-B. Zhang, S. D.Barthelmy, D. N. Burrows, S. Campana, G. Chincarini,N. Gehrels, S. Kobayashi, P. M´esz´aros, A. Moretti, J. A.Nousek, P. T. O’Brien, J. P. Osborne, P. W. A. Roming,T. Sakamoto, P. Schady, and R. Willingale. GRB RadiativeEfficiencies Derived from the Swift Data: GRBs versus XRFs,Long versus Short.
ApJ , 655:989–1001, February 2007a.doi:10.1086/510110.B.-B. Zhang, E.-W. Liang, and B. Zhang. A ComprehensiveAnalysis of Swift XRT Data. I. Apparent Spectral Evolution ofGamma-Ray Burst X-Ray Tails.