\boldsymbol{\Theta}))$, $\boldsymbol{\mu}$ is the set of distance moduli derived from the light curve fitting procedure for each SN Ia event, at redshifts given by $\boldsymbol{z}$, $\boldsymbol{\mu}_{th}(\boldsymbol{z},
\boldsymbol{\Theta})$ is the theoretical prediction for them, given in terms of a vector $\boldsymbol{\Theta}$ of parameters and $\boldsymbol{\Sigma}$ is the covariance matrix of the events.

As discussed in the previous section, each light curve fitter gives a different set of output or processed data, which are not related to the distance modulus in the same way. In this work, to construct the traditional $\chi^2$ function of this section or the proposed likelihood function of the next section, we consider as “data” the distance modulus estimation obtained from MLCS2k2 and SALT2 processing.

The $\chi^2$ approach from SALT2 output
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The SALT2 light curve fitter gives three quantities, with corresponding errors, to be used in the analysis of cosmology:\
$$m^{\ast}_B:=-2.5\log \left[x_0\int d\lambda'
M_0(p=0,\lambda')T^B(\lambda')\right]\,,$$ to be interpreted as the peak de-redshifted (rest-frame) magnitude in the $B$ band, $x_1$, a parameter related to the stretch of the light-curve and $c$, related to the color of the supernova alongside the redshift $z$ of the supernova. The distance modulus is modelled, in this context, as a function of ($m_B^{\ast}$, $x_1$, 