8.2.1. SN Calibration

A widely-used method to calibrate GRB correlations is through the use of SNe Ia that span within *z* . 2.3. In such a way, assuming this could work for any LGRBs, the GRB data points are mixed with SNe in order to build up a whole, quite large, Hubble diagram, where in the small redshift domain one has the majority of SNe, while, at large *z*, GRBs are the most. Here, the simplest error bars on distance modulus are [168,169]

$$
\sigma\_{\mu} = ([(z\_{i+1} - z) / (z\_{i+1} - z\_i)]^2 \mathfrak{e}\_{\mu, i}^2 + [(z - z\_i) / (z\_{i+1} - z\_i)]^2 \mathfrak{e}\_{\mu, i+1}^2)^{1/2},\tag{43}
$$

where *eµ*,*<sup>i</sup>* and *eµ*,*i*+<sup>1</sup> and *µ<sup>i</sup>* and *µi*+<sup>1</sup> are the errors and distance moduli of the SNe Ia at *z<sup>i</sup>* and *zi*+<sup>1</sup> , respectively.

For each SN catalog, we could find different interpolating functions to model the SN distribution. Thus, calibrating GRBs with SNe would seriously depend on the choice of these expressions for each catalog. Hence, GRB calibrations may turn out to be extremely sensitive to SNe Ia and the approach should be carefully handled since GRB luminosity correlations may no longer be fully independent from SN data points.
