7.5.2. Dainotti or *L* <sup>X</sup>–*t*<sup>X</sup> Correlation

The Dainotti correlation, akin to the *L*iso–*τ*lag correlation, may be retrieved from kinematic effects (see Section 3.1.5 and Ref. [148], for details), pointing out to a common origin between the two correlations.

As already discussed above, this correlation is quite scattered and seems to be a selection effect due to the flux detection limit of *Swift*-XRT instrument [130]. Moreover, it is also affected by evolution effects with the redshift [164].

#### 7.5.3. *E* X iso–*E*iso–*E*<sup>p</sup> Correlation

This correlation depends upon two cosmology-dependent quantities, although it is unsuitable to constrain cosmological parameters. Since it holds for both SGRBs and LGRBs, with different progenitor and surrounding medium properties (see Sections 2.4.1 and 2.4.2), its physical interpretation has not been yet established. There is a speculation that it may be connected with the Γ of the outflow, which might regulate the efficiency of conversion from *γ*-rays to X-rays [138].

The *E* X iso–*E*iso–*E*<sup>p</sup> correlation utilizes the prompt emission observables *E*iso and *E*<sup>p</sup> on which the Amati correlation is based. For this reason, it is straightforward to deduce that the biases and selection effects, at work for the Amati correlation, partially affect this hybrid correlation. Moreover, unlike pure afterglow correlations such as the Dainotti one, this correlation is also plagued by double truncation in the flux limit, both in the prompt and X-ray afterglow emissions, making the correction for any selection effect difficult and the use as redshift estimators and cosmological tool (see discussions in Ref. [164]).
