$c$) and two new parameters, $\boldsymbol{\delta}:=(\alpha,\beta)$, plus the peak absolute magnitude, in $B$ band, $M_B$. Defining the corrected magnitude as $$m^\mathrm{corr}_B(\boldsymbol{\delta}):=m_B^{\ast}+\alpha x_1 -\beta c
\label{m_corr}$$ we can write $$\mu(\boldsymbol{\delta},M_B) = m^\mathrm{corr}_B(\boldsymbol{\delta}) - M_B$$

Assuming that all SNe Ia events are independent, one can rewrite Eq. (\[chi2\_base\]) as $$\chi^2_{\mathrm{SALT2}}(\boldsymbol{\theta},
\boldsymbol{\delta}, {\cal M}(M_B, h)) = \sum_{i=1}^N
\frac{[\mu_{i}(\boldsymbol{\delta},M_B)-\mu_\mathrm{th}(z_i;
\boldsymbol{\theta}, h)]^2}{\sigma_i^2(\boldsymbol{\delta}) +
\sigma_\mathrm{int}^ 2}
\label{chi2_salt2}$$ where $N$ is the number of SNe Ia in the sample, $\boldsymbol{\theta}$ denotes the cosmological parameters other than $h$, with the present value of the Hubble parameter given by $H_0= 100 h \mbox{ km}\cdot\mbox{s}^{-1}\cdot\mbox{Mpc}^{-1}$. The theoretical distance modulus is given by $$\mu_\mathrm{th}(z; \boldsymbol{\theta},
h)=5\log[\mathcal{D}_L(z; \boldsymbol{\theta})]+\mu_0(h)\,.
\label{m_th_def}$$ with $$\mu_0(h) := 5\log\left( \frac{100c/(\mbox{km/s})}{h} \right)\,.$$ The dimensionless luminosity distance (in units of the Hubble distance today), $\mathcal{D}_L$, for comoving observers in a Robertson-Walker universe, is given by $$\mathcal{D}_L(z; \boldsymbol{\theta})  = \left\{\begin{array}{ll}
 (1+z)\left(\frac{1}{\sqrt{\Omega_{k0}}}\right)\sinh\left(\sqrt{\Omega_{k0}}{
\int_{z'=0}^z\frac{1}{E(z'; \boldsymbol{\theta})}dz'}\right), &\hbox{if}\;
\Omega_{k0}>0,  \\
 (1+z){\int_{z'=0}^z\frac{1}{E(z'; \boldsymbol{\theta})}dz'}, &\hbox{if}\;
\Omega_{k0}=0,  \\
 (1+z)\left(\frac{1}{\sqrt{-\Omega_{k0}}}\right)\sin\left(\sqrt{-\Omega_{k0}}{
\int_{z'=0}^z\frac{1}{E(z'; \boldsymbol{\theta})}dz'}\right), &\hbox{if}\;
\Omega_{k0}<0,
 \end{array} \right.$$ where $\Omega_{k0}$ is the “curvature density parameter” (whatever the underlying dynamical gravitational theory), such that it is proportional to the three-curvature, and $$E(z;
\boldsymbol{\theta}):=H(z;\boldsymbol{\theta},h)/H_0$$ is the dimensionless Hubble parameter. As indicated in Eq. (\[chi2\_salt2\]), the $\chi^2$ function depends on the parameters $M_B$ and 