models, there is a small difference in the best fit values and confidence contours ($\sim$ 30% difference in area) (cf. Figs. \[fig:SDSS\_LCDM\_cosmopar\] and \[fig:SDSS\_wCDM\_cosmopar\]) in case the MLCS2k2 fitter is adopted. For SALT2 the difference is less significant ($\lesssim 13\%$ difference in areas). In both cases the likelihood approach gives more restrictive constraints. We can understand these results by observing, from Fig. \[fig:SDSS\_sint\_dist\], that the estimated value of $\sigma_\text{int}$ in the traditional approach, is higher than the peak of the $\sigma_\text{int}$ distribution obtained with the likelihood method. In fact, for MLCS2k2 the $\sigma_\text{int}$ value obtained with the traditional $\chi^2$ approach is outside the 99% confidence level of the distribution obtained with the likelihood method. For SALT2 it is of the order of 68% confidence level and this might explain why for SALT2 the differences between these two approaches are less significant. We should also remark that we noticed that the covariance between the SALT2 parameters ($m^*_B,x_1,c$) has an important role in the above result since we obtained, for the 68% confidence level contour, an area ratio of 1.13, when considering the covariance, and of 0.95, when neglecting the covariance.

We also studied the possible evolution of the SALT2 parameters $\alpha$, 