coefficients of the relation between luminosity and stretch and of the relation between luminosity and color, respectively, $\Delta_{M}$ is the distance correction from the host-galaxy mass of the SN, and $\Delta_{B}$ is the distance correction from predicted biases of simulations.

The luminosity distance $d_{\rm L}$ to a supernova can be given by $$d_{{\rm L}}(z)=\frac{1+z}{H_{0}} \int_{0}^{z} \frac{dz'}{E(z')},$$ where $E(z)= H(z)/H_{0}$. Note that we consider a flat universe throughout this work. The $\chi^{2}$ function for SN observation is expressed as $$\chi^{2}_{\rm{SN}}=({\mu}-\mu_{\rm{th}})^{\dagger}C_{\rm SN}^{-1}({\mu}-\mu_{\rm{th}}),$$ where $C_{\rm SN}$ is the covariance matrix of the SN observation [@Scolnic:2017caz], and the theoretical distance modulus $\mu_{\rm{th}}$ is given by $$\mu_{\rm{th}}=5\log_{10}\frac{d_{\rm{L}}}{10\rm{pc}}.$$

**CMB data:** For the cosmic microwave background (CMB) anisotropies data, we use the “Planck distance priors” from the Planck 2015 data [@Ade:2015rim]. The distance priors include the shift parameter $R$, the “acoustic scale” $\ell_{\rm{A}}$, and the baryon density $\omega_{b}$, defined by $$R\equiv\sqrt{\Omega_{\rm{m}}H^{2}_{0}}(1+z_{\ast})D_{\rm{A}}(z_{\ast}),$$ $$\ell_{\rm{A}}\equiv(1+z_{\ast})\frac{\pi D_{\rm{A}}(z_{\ast})}{r_{\rm{s}}(z_{\ast})},\label{la}$$ $$\omega_{b}\equiv \Omega_{b}h^{2},$$ where $\Omega_{\rm{m}}$ is the present-day fractional matter density, and $D_{\rm{A}}(z_{\ast})$ denotes the angular diameter distance at $z_{\ast}$ with $z_{\ast}$ being the redshift of the decoupling epoch of photons. In a flat universe, $D_{\rm{A}}$ can be expressed as $$D_{\rm{A}}(z)=\frac{1}{H_{0}(1+z)}\int_{0}^{z}\frac{dz'}{E(z')},\label{DA}$$ and $r_{\rm{s}}(a)$ can be given by $$r_{\rm{s}}(a)=\frac{1}{\sqrt{3}}\int_{0}^{a}\frac{da'}{a'H(a')\sqrt{1+(3\Omega_{{\rm b}}/4\Omega_{{\rm \gamma}})a'}},\label{rs}$$ where $\Omega_{{\rm b}}$ and $\Omega_{{\gamma}}$ 