in the presence of dark energy.

When focusing on the normal branch and in the quasi-static limit for sub-horizon scales, the perturbations in the conformal Newtonian gauge give the modified Poisson system of equations [@PhysRevD.79.123512]: $$\begin{aligned}\label{eq:poisson_dgp}
\nabla^2 \Phi_N &= 4 \pi G  a^2 \delta \rho_m + \frac{1}{2} \nabla^2 \varphi, \\
\nabla^2 \varphi &= \frac{8 \pi G a^2}{3 \beta} \delta \rho_m - \frac{r_c^2}{3 \beta a^2}\Big[(\nabla^2 \varphi)^2- (\nabla_i \nabla_j \varphi)^2 \Big],
\end{aligned}$$ with the coupling $\beta$ given by $$\begin{aligned}
\label{eq:dgpbeta}
\beta(a) = 1 + 2 H(a) r_c \left(1 + \frac{\dot{H(a)}}{3 H(a)^2} \right).\end{aligned}$$

The $n$DGP model is a typical example of a scalar-tensor theory that realizes the Vainshtein screening mechanism [@VAINSHTEIN1972393; @Babichev:2013usa], in which the modifications to gravity are suppressed in the existence of large second derivatives of the scalar field. In the second equation of (\[eq:poisson\_dgp\]), in particular, it can be seen how, once the second derivatives of the scalar field become large in high densities, the second term on the right hand side (r.h.s.) becomes significant and effectively weakens the source strength, resulting in strong screening of the fifth forces.

We consider the $n$DGP model for two choices of the free parameter $r_c$, corresponding to $n\equiv H_0 r_c=1$ and $n=5$, which 