the SALT2 data. In order to perform such analysis the dataset was divided in redshift bins and the cosmological parameters were fixed in the best-fit values obtained from the global fit, then releasing $\alpha$, $\beta$, $\mathcal{M}$ and $\sigma_\text{int}$ to be determined in each bin. The results are shown in Fig. \[fig:SDSS\_zevol\]. We found evidence of evolution for the parameter $\beta$, in agreement with [@Kessler2009]; furthermore we also found evidence of evolution for $\sigma_{\text{int}}$, which might support the use of a variable $\sigma_{\text{int}}$ instead of a constant one.

In addition to the SDSS compilation, we also used the Union2 compilation [@Amanullah2010] and the SNLS third year data [@Guy2011]. However these references do not provide the covariances between the SALT2 parameters ($m_B^*$,$x_1$,$c$). In order to make a fair comparison with the SDSS data, we ran again the analyses for all three samples, this time neglecting the cross terms in Eq. (\[sigsalt2\]). Without taking into account the covariances, the differences in the areas for the SDSS compilation are essentially the same. The main difference relies on the fact that, for the F$w$CDM model, the constraints are tighter for the $\chi^2$ approach than for the likelihood one. For the Union 2 compilation, the difference 