$h$ only through their combination $${\cal M}(M_B, h) := M_B + \mu_0(h)\,.$$ It may thus be thought of as effectively directly dependent on only the parameters $\boldsymbol{\theta}$, $\boldsymbol{\delta}$ and ${\cal
M}$.

A floating dispersion term, $\sigma_{int}$, “which contains potential sample-dependent systematic errors that have not been accounted for and the observed intrinsic SNe Ia dispersion” [@Amanullah2010], is added in quadrature to the distance modulus dispersion, which is given by $$\sigma_i^2(\boldsymbol{\delta})=\sigma^
2_{m_B^*,i}+\alpha^2\sigma_{x_1,i}^2 + \beta^2\sigma_{c,i}^2 
+2\alpha\sigma_{m_B^*,x_1,i}-2\beta\sigma_{
m_B^*,c,i} -2\alpha\beta\sigma_{x_1,c,i} +\sigma^2_{\mu,z,i}.
\label{sigsalt2}$$ where $\sigma^2_{\mu,z,i}$ is the contribution to the distance modulus dispersion due to redshift uncertainties from peculiar velocities and also from the measurement itself. Following [@Kessler2009], we will model this contribution, for simplicity, using the distance-redshift relation for an empty universe which gives $$\sigma_{\mu,z,i}=\sigma_{z,i}\left(\frac{5}{\ln
10}\right)\frac{1+z_i}{z_i(1+z_i/2)},$$ with $$\sigma_{z,i}^2=\sigma_{spec,i}^2+\sigma_{pec}^2,$$ where $\sigma_{spec,i}$ is the redshift measurement error, and $\sigma_{pec}=0.0012$ is the redshift uncertainty due to peculiar velocity.

As advocated by some groups [@Astier2006], minimizing Eq. (\[chi2\_salt2\]) gives rise to a bias towards increasing values of $\alpha$ and $\beta$. In order to circumvent this feature an iterative method is performed, according to their approach.

In this iterative method, the $\chi^2$ presented in Eq. (\[chi2\_salt2\]) is replaced by $$\chi^2_{\mathrm{SALT2}}(\boldsymbol{\theta}, \boldsymbol{\delta}, {\cal M})
= \sum_{i=1}^N \frac{[\mu_{i}(\boldsymbol{\delta},
M_B)-\mu_\mathrm{th}(z_i;\boldsymbol{\theta},h)]^2}{\sigma_i^2(\boldsymbol{\eta}
) + 