by Deffayet [@a3], modifies GR at low energies. This scenario produces the self-acceleration of the late-time universe owing to a weakening of gravity at low energies. Similar to the RS scenario, the DGP scenario is also a five-dimensional scenario with infinite extra dimensions. The dynamics of gravity is governed by a competition between a Ricci scalar term in the four-dimensional brane and an Einstein–Hilbert action in the five-dimensional bulk. Then, the first Friedmann equation of the DGP scenario can be expressed as [@60] $$H^2=H_0^2\{\Omega_{k0}(1+z)^2+[\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}(1+z)^3}]^2\}, \label{6}$$ where $H$ denotes the Hubble parameter, $H_0$ is the present value of the Hubble parameter, $\Omega_{m0}$ and $\Omega_{k0}$ are the fractional contributions of matter and curvature, respectively, and $\Omega_{r_c}=1/4r_c^2H_0^2$ is the bulk-induced term with respect to the crossover radius $r_c$. The $z=0$ value of Eq. (\[6\]) leads to the usual normalization condition $\Omega_{k0}+(\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}})^2=1$, and for a flat universe ($\Omega_{k0}=0$), $\Omega_{r_c}=(1-\Omega_{m0})^2/4$. Furthermore, Eq. (\[6\]) can be rewritten as $$E(z)=\{[1-(\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}})^2](1+z)^2+[\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}(1+z)^3}]^2\}^{1/2}, \label{7}$$ where $E(z)$ denotes the normalized Hubble parameter. Furthermore, according to Ref. [@60], the $z$-dependent equation of state (EoS) of the DGP scenario can be expressed as $$\omega_{eff}(z)=\{[\sqrt{4\Omega_{r_c}/\Omega_{m0}(1+z)^3+4}][\sqrt{\Omega_{r_c}/\Omega_{m0}(1+z)^3}+\sqrt{\Omega_{r_c}/\Omega_{m0}(1+z)^3+1}]\}^{-1}-1. \label{8}$$ In what follows, we exhibit our methodology to constrain the DGP scenario by using the current astrophysical observations.

                                  