\[sec:LPTDM\].

### The $n$DGP model {#nDGPsec}

The second MG model under consideration comes from the realm of higher-dimensional braneworld cosmology. The simplest example of such a configuration is the so-called Dvali-Gabadadze-Porrati (DGP) model [@Dvali:2000hr], which is described by an action of the following form: $$\label{actDGP}
S=\int d^4x \sqrt{-g} \left[\frac{R}{16 \pi G} + \mathcal{L}_m \right] + \int d^5x \sqrt{-g_5} \left(\frac{R_5}{16 \pi G r_c}\right).$$ In this model, the spacetime consists of 5 dimensions, rather than the usual 4, but the standard model fields are restricted to a 4-dimensional (4D) brane and the free parameter, $r_c$, denotes the length-scale below which gravity becomes 4D. $R_5$ and $g_5$ denote the corresponding 5D versions of the Ricci scalar and metric tensor determinant, respectively. The resulting Friedman equation from (\[actDGP\]) is: $$\label{FridDGP}
\epsilon \frac{H}{r_c}=H^2 - \frac{\rho_m}{24 \pi G},$$ where $\epsilon=\pm 1$. Each of the two values of $\epsilon$ represents a particular branch of the model, with $\epsilon=+1$ producing the self-accelerating solution, which, however, suffers from undesirable “ghost” instabilities [@Koyama:2007zz] and is thus an unphysical model to consider. The value of $\epsilon=-1$ corresponds to the so-called normal branch, hereafter called $n$DGP, which is well behaved, but does not self-accelerate and can match a $\Lambda$CDM expansion history only 