the unperturbed part of $g$.

3.  $g_{\chi}$ means derivative of $g$ with respect to $\chi$ and $g_{\phi}$ derivative of $g$ with respect to $\phi$, for a given function $g$.

A class of modified gravitational theories
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All models to be considered here come from a simple action, which consists in adding a term of the form $f(\chi)$ to the Einstein-Hilbert action. Here $f$ is a given function which, in order to recover GR, vanishes at low energy densities, while $\chi$ is a fully covariant scalar built from the scalar field that fills the whole universe and whose value in any co-moving slicing of the flat FLRW spacetime is proportional to the Hubble parameter. More precisely, we consider $$\begin{aligned}
\label{action}
 S=\int\sqrt{-g}\left(\frac{1}{2}R+f(\chi)    +{\mathcal L}_{matt}\right) d^4x,
 \end{aligned}$$ $R$ being the scalar curvature. We have assumed that the matter sector of the universe is described by a scalar field $\phi$ with a potential, $V(\phi)$ which is minimally coupled to gravity and whose Lagrangian is given by $$\begin{aligned}
  {\mathcal L}_{matt}= \left(-\frac{\phi_{,\mu}\phi^{,\mu}}{2}-V(\phi)\right),
    \end{aligned}$$ and $\chi\equiv -\nabla_{\mu} u^{\mu}$, being the vector $u^{\mu}$ the time-like unitary eigenvector of the stress tensor $$\begin{aligned}
 T_{\mu}^{\nu}=\phi_{,\mu}\phi^{,\nu}-\left(\frac{1}{2}\phi_{,\alpha}\phi^{,\alpha}+V(\phi)   \right)\delta_{\mu}^{\nu}, \end{aligned}$$ that is, $$\begin{aligned}
 u^{\mu}=\frac{\phi^{,\mu}}{\sqrt{-\phi^{,\alpha}\phi_{,\alpha}}}.
 \end{aligned}$$

The dynamical equations are obtained by performing the 