\sigma_\mathrm{int}^ 2}\,.  
\label{chi2_salt2_new}$$ Notice that, in this expression, $\boldsymbol{\eta}$ is not a parameter of the $\chi^2_{\mathrm{SALT2}}$. In order to obtain the best fit values for the parameters, $\boldsymbol{\eta}$ is given initial values and the optimization is performed on $\boldsymbol{\theta}, \boldsymbol{\delta}$ and ${\cal M}$. After this step, $\boldsymbol{\eta}$ is updated with the best fit value of $\boldsymbol{\delta}$ and the optimization is performed again. The process continues until a convergence is achieved, which means that $\boldsymbol{\eta}$ does not change under the required precision.

During this process $\sigma_\mathrm{int}$ is not considered as a free parameter to be optimized, being determined rather by the following procedure: start with a guess value (usually $\sigma_\mathrm{int}=0.15$). Perform the iterative procedure described above. The value of $\sigma_\mathrm{int}$ is then obtained by fine tuning it so that the reduced $\chi^2$ equals unity (with all the other parameters fixed on their best fit values). The iterative procedure is repeated once more with this new value and the final best fit values are obtained. It is important to note that the value of $\sigma_{\rm int}$ affects both the best fit and the confidence levels of the parameters, since it changes the weight given to each supernova in the $\chi^2$ \[cf. Eq. 