+
    (1-\Omega_{m0}-\Omega_{k0})(1+z)^{3(1+w)}.$$

We chose the $\Lambda$CDM and F$w$CDM models to directly compare the best-fit and the 68% and 95% confidence contours for the parameters $\boldsymbol{\delta}$, $\sigma_{\text{int}}$ and $\boldsymbol{\theta}$, for both SALT2 and MLCS2k2 data. The best fit values were obtained with the MIGRAD minimization of the Minuit [@James1975] implementation in ROOT [@Antcheva2009] and the probability distributions were obtained with Monte Carlo Markov Chains (MCMC). We considered as the probability distributions, in the context of $\chi^2$ approach, the following functions: $$\begin{aligned}
\mathcal{P}_{SALT2}(\boldsymbol{\theta},\boldsymbol{\delta},\mathcal{M})
& = &
N_{SALT2}\exp\left[-\chi^2_{SALT2}(\boldsymbol{\theta},\boldsymbol{\delta},
\mathcal{M},
\sigma_{\text{int}})/2\right],  \\
\mathcal{P}_{MLCS2k2}(\boldsymbol{\theta},h)
& = & N_{MLCS2k2}\exp\left[-\chi^2_{MLCS2k2}(\boldsymbol{\theta},h,
\sigma_{\text{int}})/2\right],\end{aligned}$$ where the normalization factors $N_{SALT2}$ and $N_{MLCS2k2}$ are independent of the parameters to be estimated. Note that, for the traditional $\chi^2$ method, $\sigma_{\text{int}}$ is fixed so the probability distribution does not depend on it.

In Figs. \[fig:SDSS\_LCDM\_cosmopar\] and \[fig:SDSS\_wCDM\_cosmopar\] we show the confidence contours for the parameters $\boldsymbol{\theta}$ for $\Lambda$CDM and F$w$CDM models, respectively. For these models, we note that the differences between the best fit and the area of the contours, for the $\chi^2$ and likelihood approaches, are more significant when the MLCS2k2 fitter is used. In fact, for the SALT2 fitter, the differences are not significant (less than $13\%$) — see also Fig. \[fig:SDSS\_sint\_dist\] 