are the present-day energy densities of baryons and photons, respectively. In this work, we adopt $3\Omega_{{\rm b}}/4\Omega_{{\rm \gamma}}=31500\Omega_{\rm{b}}h^{2}(T_{\rm{cmb}}/2.7{\rm K})^{-4}$ and $T_{\rm{cmb}}=2.7255$ K. $z_{\ast}$ can be calculated by the fitting formula [@Hu:1995en], $$z_{\ast}=1048[1+0.00124(\Omega_{{\rm b}}h^{2})^{-0.738}][1+g_{1}(\Omega_{{\rm m}}h^{2})^{g_{2}}],$$ where $$g_{1}=\frac{0.0783(\Omega_{\rm{b}}h^{2})^{-0.238}}{1+39.5(\Omega_{\rm{b}}h^{2})^{-0.76}}, \;  g_{2}=\frac{0.560}{1+21.1(\Omega_{\rm{b}}h^{2})^{1.81}}.$$

The three values can be obtained from the Planck TT+LowP data [@Ade:2015rim]: $R=1.7488\pm0.0074$, $\ell_{\rm{A}}=301.76\pm0.14$, and $\Omega_{\rm{b}}h^{2}=0.02228\pm0.00023$. The $\chi^{2}$ function for CMB is $$\chi^{2}_{\rm{CMB}}=\Delta p_{i}[{\rm Cov}^{-1}_{\rm{CMB}}(p_{i},p_{j})]\Delta p_{j}, \quad \Delta p_{i}=p_{i}^{\rm{th}}-p_{i}^{\rm{obs}},$$ where $p_{1}=\ell_{\rm{A}}$, $p_{2}=R$, $p_{3}=\omega_{b}$, and ${\rm Cov}^{-1}_{\rm CMB}$ is the inverse covariance matrix and can be found in Ref. [@Ade:2015rim].

**BAO data:** From the baryon acoustic oscillations (BAO) measurements, we can obtain the distance ratio $D_{{\rm V}}(z)/r_{\rm s}(z_{\rm d})$ or $D_{{\rm M}}(z)/r_{\rm s}(z_{\rm d})$. The spherical average gives the expression of $D_{{\rm V}}(z)$, $$D_{\rm{V}}(z)\equiv\left[D^{2}_{\rm{M}}(z)\frac{z}{H(z)}\right]^{1/3},$$ where $D_{\rm{M}}(z)=(1+z)D_{\rm A}(z)$ is the the comoving angular diameter distance [@Alam:2016hwk]. $r_{\rm s}(z_{\rm d})$ is the comoving sound horizon size at the redshift $z_{\rm d}$ of the drag epoch and its calculated value can be given by Eq. (\[rs\]). $z_{\rm{d}}$ is given by the fitting formula [@Hu:1995en], $$z_{\rm{d}}=\frac{1291(\Omega_{\rm{m}}h^2)^{0.251}}{1+0.659(\Omega_{\rm{m}}h^2)^{0.828}}[1+b_1(\Omega_{\rm{b}}h^2)^{b_2}],$$ with $$\begin{gathered}
b_1=0.313(\Omega_{\rm{m}}h^2)^{-0.419}[1+0.607(\Omega_{\rm{m}}h^2)^{0.674}],\\
b_2=0.238(\Omega_{\rm{m}}h^2)^{0.223}.
\end{gathered}$$

We use five BAO data points form the 6dF Galaxy Survey at $z_{\rm eff} = 0.106$ [@Beutler], the SDSS-DR7 at $z_{\rm eff} = 0.15$ [@Ross:2014qpa], and 