in [@yoshida]– they can easily lead to Ostrogradski or gradient instabilities, as well as to the appearance of ghost fields. All these problems led specialists to explore other ways, such as modified mimetic gravity [@mukhanov; @mukhanov1; @langlois], so as to find out such scalar.

Following these arguments, guided by Weyl’s principle and taking into account that for co-moving observers in flat FLRW spacetime the divergence of a unitary time-like vector is equal to $-3H$, in the present paper we propose as our fully covariant scalar the divergence of the unitary time-like eigenvector of the stress tensor, which, in the case of a universe filled up with a scalar field $\phi$ minimally coupled to gravity, is equal to $u_{\mu}=\frac{\phi_{,\mu}}{\sqrt{\phi_{, \nu}\phi^{, \nu}}}$ (throughout the work we will use the notation: $\phi_{,\mu}\equiv\partial_{\mu}\phi=\nabla_{\mu}\phi$).

 

With this covariant scalar, we will show how to construct $f$-theories leading to bouncing backgrounds and calculate the perturbed equations for scalar and tensor perturbations for a given $f$-theory. When dealing with perturbations and working in the longitudinal gauge –where the Newtonian potential, namely $\Phi$, and the variation of the scalar field, namely $\delta\phi$, are the dynamical variables– the corresponding dynamical system turns out to be a coupled one. This is 