-f'(\chi)$, thus obtaining, as in mimetic gravity [@Norbert], one more degree of freedom than in the case of GR. However, as we will show in next Section, when dealing with perturbations in longitudinal gauge, the degrees of freedom are the Newtonian potential $\Phi$ and the perturbation of the scalar field $\delta\phi$ for scalar perturbations and two degrees for the tensor ones, exactly the same as in GR. This is the same as what happens in mimetic gravity [@hap], where the degrees of freedom are the perturbed part of the mimetic field and $\delta\phi$.

Perturbations
=============

In this section we will calculate, for a given $f$-theory, the scalar and tensor perturbations using for scalar perturbations the longitudinal gauge (see for a review of the used setup [@mfb; @cgk]).

Scalar perturbations
--------------------

In Newtonian gauge the the line element is given by [@mukhanovbook] $$\begin{aligned}
ds^2=-(1+2\Phi)dt^2+(1-2\Psi)a^2\delta_{ij}dx^idx^j.\end{aligned}$$ where the potentials $\Phi$ and $\Psi$ coincide with the gauge invariant ones.

A simple calculation leads to $\delta u_0=\Phi$ and $\delta u_k=\frac{\partial_k \delta\phi}{\dot{\bar{\phi}}}$, where $\delta\phi$, in this gauge, coincides with the $\delta\phi^{gi}$ (the gauge invariant perturbation of the scalar field). Then, at linear order, we have $$\begin{aligned}
\chi= 3H-3(\dot{\Psi}+H\Phi)-\frac{1}{a^2\dot{\bar{\phi}}}\Delta\delta\phi.\end{aligned}$$

To obtain the dynamical equations for perturbations, note 