Linear perturbations spectra for dynamical dark energy inspired by
  modified gravity
  In this paper, we study a particular modified gravity Equation of State, the
so-called Jaime-Jaber-Escamilla, that emerges from the first gravity modified
action principle and can reproduce three cosmological viable $f(R)$ theories:
the Starobinsky, Hu-Sawicki, and Exponential models . This EoS is a suitable
candidate to reproduce the dynamical dark energy behaviour already
reconstructed by the current data sets. Based on the joint statistical
analysis, we found that our results are still in good agreement (within
$1\sigma$) with the $\Lambda$CDM, while at perturbative level we notice that
the matter power spectrum normalisation factor $\sigma_8$ shows an agreement
with SDSS and SNeIa+IRAS at 1-$\sigma$ for the Starobinsky model and with
SDSS-vec for the Hu \& Sawicki and Exponential models. Furthermore, we found
that for the $H_0$ values, Starobinsky and Hu \& Sawicki show the least tension
in comparison with PL18 TT. All these aspects cannot be observed
\textit{directly} from other alternatives theories, were a equation of state is
difficult to compute analytically.
