assumed in our analysis, which is strongly supported by the recent Planck results [@Ma19] and independently supported by the observations of milliarcsecond compact structure of radio quasars at $z\sim 3.0$ [@Cao17; @Cao19]. In a zero-curvature universe filled with ordinary pressureless matter (cold dark matter plus baryons), dark energy, and negligible radiation, the Friedmann equation reads $$H^2(z)=H_0^2[\Omega_m(1+z)^3+(1-\Omega_{m})],$$ where $\Omega_{m}$ is the current density fraction of matter component. In the framework of this standard cosmological model, the angular diameter distance between redshifts $z_1$ and $z_2$ becomes $$\begin{aligned}
D_A^{\Lambda CDM}(z_1,z_2)&=&\frac{c}{H_0 (1+z_2)}\times\nonumber\\
\null&\null&\hskip-0.3in
\int^{z_2}_{z_1}\frac{dz}{\sqrt{\Omega_m(1+z)^3+1-\Omega_m}}.\nonumber\end{aligned}$$ Over the past decades, the importance of modified gravity theories was stressed again. The DGP model [@Dvali00b], which accounts for the cosmic acceleration without dark energy, arises from a class of brane world theories in which gravity leaks out into the bulk at large distances. More specifically, this leaking of gravity takes place only above a certain cosmological scale $r_c$. In the framework of a spatially flat DGP model, the Friedmann equation can be expressed as $$H^2-\frac{H}{r_c}=\frac{8\pi G}{3}\rho_m,$$ where the length at which the leaking occurs can be associated with the density parameter: $\Omega_{r_c}=1/(4r_c^2H_0^2)$. It is also straightforward to check the validity of the relation $\Omega_{r_c}=\frac{1}{4}(1-\Omega_m)^2$ in the flat DGP 