The mechanism responsible for the afterglow emission of Gamma Ray Bursts (GRBs) and its connection to the prompt γ-ray emission is still a debated issue. Relations between intrinsic properties of the prompt or afterglow emission can help to discriminate between plausible theoretical models of GRB production. Here we present an overview of the afterglow and prompt-afterglow two parameter relations, their physical interpretations, their use as redshift estimators and as possible cosmological tools. A similar task has already been correctly achieved for Supernovae (SNe) Ia by using the peak magnitude-stretch relation, known in the literature as the Phillips relation (Phillips 1993). The challenge today is to make GRBs, which are amongst the farthest objects ever observed, standardizable candles as the SNe Ia through well established and robust relations. Thus, the study of relations amongst the observable and physical properties of GRBs is highly relevant together with selection biases in their physical quantities. Therefore, we describe the state of the art of the existing GRB relations, their possible and debated interpretations in view of the current theoretical models and how relations are corrected for selection biases. We conclude that only after an appropriate evaluation and correction for selection effects can GRB relations be used to discriminate among the theoretical models responsible for the prompt and afterglow emission and to estimate cosmological parameters.

Gamma Ray Bursts (GRBs) visible up to very high redshift have become attractive targets as potential new distance indicators. It is still not clear whether the relations proposed so far originate from an unknown GRB physics or result from selection effects. We investigate this issue in the case of the L X − T * a correlation (hereafter LT) between the X-ray luminosity L X (T a ) at the end of the plateau phase, T a , and the rest frame time T * a . We devise a general method to build mock data sets starting from a GRB world model and taking into account selection effects on both time and luminosity. This method shows how not knowing the efficiency function could influence the evaluation of the intrinsic slope of any correlation and the GRB density rate. We investigate biases (small offsets in slope or normalization) that would occur in the LT relation as a result of truncations, possibly present in the intrinsic distributions of L X and T * a . We compare these results with the ones in Dainotti et al. (2013) showing that in both cases the intrinsic slope of the LT correlation is ≈ −1.0. This method is general, therefore relevant to investigate if any other GRB correlation is generated by the biases themselves. Moreover, because the farthest GRBs and star-forming galaxies probe the reionization epoch, we evaluate the redshift-dependent ratio Ψ(z) = (1 + z) α of the GRB rate to star formation rate (SFR). We found a modest evolution −0.2 ≤ α ≤ 0.5 consistent with Swift GRB afterglow plateau in the redshift range 0.99 < z < 9.4.

In this work we study the distribution of temporal power-law decay indices, α, in the Gamma Ray Burst (GRB) afterglow phase, fitted for 176 GRBs (139 long GRBs, 12 short GRBs with extended emission and 25 X-Ray Flashes (XRFs)) with known redshifts. These indices are compared with the temporal decay index, α W , derived with the light curve fitting using the Willingale et al. (2007) model. This model fitting yields similar distributions of α W to the fitted α, but for individual bursts a difference can be significant. Analysis of (α, L a ) distribution, where L a is the characteristic luminosity at the end of the plateau, reveals only a weak correlation of these quantities. However, we discovered a significant regular trend when studying GRB α values along the Dainotti et al. (2008) correlation between L a and the end time of the plateau emission in the rest frame, T * a , hereafter LT correlation. We note a systematic variation of the α parameter distribution with luminosity for any selected T * a . We analyze this systematics with respect to the fitted LT correlation line, expecting that the presented trend may allow to constrain the GRB physical models. We also attempted to use the derived correlation of α(T a ) versus L a (T a ) to diminish the luminosity scatter related to the variations of α along the LT distribution, a step forward in the effort of standardizing GRBs. A proposed toy model accounting for this systematics applied to the analyzed GRB distribution results in a slight increase of the LT correlation coefficient.

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