*9.2. Applications of Statistical Analysis with GRBs*

In this section, we describe a few applications of statistical analysis using GRBs. Clearly, we focus on a particular choice and, in principle, it is possible to work out different fits and/or experimental procedures. In particular, we here propose a calibration at the very beginning, adopting the most consolidated route to handle GRBs. Above, we also described the non-calibration procedure that, for brevity, we do not report here.

A widely consolidated approach is based on sampling the original catalog by means of a Monte Carlo technique, essentially built up using Markov Chain Monte Carlo simulations that are sampled within the widest possible parameter space. Commonly, the most adopted algorithm is the *Metropolis–Hastings* and the standard approach to get limits uses the minimization of the total *χ* 2 function. As we stated above, in this review, the idea of combining more than one sample is essential in order to refine cosmological bounds on the model parameters. Hereafter, we denote with **x** the set of parameters and include in our analysis SNe Ia and BAO data sets together with the calibrated GRB data. The former data have been obtained through calibrating the correlations. For the sake of brevity, as well as above, we only consider Amati, Ghirlanda, Yonetoku, and Combo correlations. In the specific case of our three samples, i.e., GRBs, SNeIa, and BAO, we combine the chi square functions by

$$
\chi^2\_{\text{tot}} = \chi^2\_{\text{GRB}} + \chi^2\_{\text{SN}} + \chi^2\_{\text{BAO}} \tag{67}
$$

with the following recipe:


$$\chi^2\_{\rm GRB} = \sum\_{i=1}^{N\_{\rm GRB}} \left[ \frac{\mu\_{\rm GRB,i}^{\rm obs} - \mu\_{\rm GRB}^{\rm th}(\mathbf{x}\_i \mathbf{z}\_i)}{\sigma\_{\mu\_{\rm GRB,i}}} \right]^2 \tag{68}$$

where *N*GRB and *µ* th GRB are the experimental and theoretical GRB distance moduli.


$$\chi^2\_{\rm SN} = (\Delta\mu\_{\rm SN} - \mathcal{M}\mathbf{1})^{\rm T}\mathbf{C}^{-1}(\Delta\mu\_{\rm SN} - \mathcal{M}\mathbf{1})\tag{69}$$

where ∆*µ*SN ≡ *µ*SN − *µ* th SN(**x**, *zi*) is the module of the vector of residuals, and **C** the covariance matrix.

In particular, we prompt the distance modulus for the most recent SN catalog, named *Pantheon Sample*. This represents the current largest SN sample consisting of 1048 SNe Ia lying on 0.01 < *z* < 2.3 [180]. The corresponding magnitudes read

$$
\mu\_{\rm SN} = m\_{\rm B} - \left(\mathcal{M} - \mathfrak{a}\mathcal{X}\_1 + \beta\mathcal{C} - \Delta\_{\rm M} - \Delta\_{\rm B}\right). \tag{70}
$$

Here, M and *m*<sup>B</sup> are the *B*-band absolute and apparent magnitudes, respectively. The above distance moduli also depend upon other quantities required to standardize/correct the light curves of SNe Ia. The quantities X<sup>1</sup> and C are the light curve shape and color parameters, respectively, whereas *α* and *β* are the coefficients of the

luminosity–stretch and luminosity–color relationships, respectively. ∆<sup>M</sup> is a distance correction determined on host galaxy mass of SNe, while ∆<sup>B</sup> a distance correction that is built up from predicted biases determined by means of simulations.

By marginalizing over M through a flat prior, it is possible to demonstrate that SN uncertainties do not depend on M, and this permits one to simplify the chi square function through

$$
\chi^2\_{\text{SN},\mathcal{M}} = a + \log \frac{e}{2\pi} - \frac{b^2}{e} \tag{71}
$$

where *<sup>a</sup>* ≡ <sup>∆</sup>~**¯** *T* SN**C** <sup>−</sup>1∆~**¯**SN, *<sup>b</sup>* <sup>≡</sup> <sup>∆</sup>~**¯** *T* SN**C** <sup>−</sup>1~**1**, *<sup>e</sup>* <sup>≡</sup>~**<sup>1</sup>** *<sup>T</sup>***C** <sup>−</sup>1~**1**. - **BAO** *χ* 2 . The chi square function for BAO data is given in Equations (27)–(29).

Below, we summarize a couple of statistical methods applied to GRB data to extract cosmological constraints.
