in general, four fundamental ingredients are required to form a traversable wormhole:

$\star$ Violate the NEC, i.e., $T_{\mu\nu}k^{\mu}k^{\nu}>0$, where $T_{\mu\nu}$ is the stress–energy tensor and $k^{\mu}$ is any future directed null vector.

$\star$ Satisfy the flare-out conditions, i.e., $b(r_0)=r_0$, $b'(r_0)<1$ and $b(r)<r$ when $r>r_0$.

$\star$ To avoid a horizon, the redshift function $\Phi(r)$ should be finite everywhere.

$\star$ Impose asymptotically flatness conditions, i.e., $b/r\rightarrow0$ and $\Phi\rightarrow0$ when $r\rightarrow\infty$.

Using the Einstein field equation $G_{\mu\nu}=T_{\mu\nu}$ in an orthonormal reference frame, we can obtain the stress–energy scenario as follows: $$\rho=\frac{b'}{r^2}, \label{2}$$ $$p_r=\frac{b}{r^3}-2\frac{\Phi'}{r}\left(1-\frac{b}{r}\right), \label{3}$$ $$p_t=\left(1-\frac{b}{r}\right)\left[\Phi''+(\Phi')^2-\Phi'\frac{b'r-b}{2r^2(1-b/r)}-\frac{b'r-b}{2r^3(1-b/r)}+\frac{\Phi'}{r}\right], \label{4}$$ where $\rho(r)$ is the matter energy density, $p_r(r)$ is the radial pressure, $p_t(r)$ is the lateral pressure orthogonal to the radial direction, and the prime denotes the derivative with respect to $r$. Using the stress–energy conservation equation, $T^{\mu\nu}_{\hspace{3mm};\nu} = 0$, we obtain $$p'_r=\frac{2}{r}(p_t-p_r)-\Phi'(\rho+p_r), \label{5}$$ which can be regarded as the hydrostatic equation of equilibrium for the materials supporting a wormhole or the relativistic Euler equation. In the following section, we review the DGP braneworld scenario briefly to constrain it by cosmic observations and study the corresponding traversable wormhole space–time configurations.

The DGP braneworld scenario
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The DGP braneworld scenario, which was first generalized to the cosmology field 