Dark sector evolution in Horndeski models
  We use the Equation of State (EoS) approach to study the evolution of the
dark sector in Horndeski models, the most general scalar-tensor theories with
second order equations of motion. By including the effects of the dark sector
into our code EoS\_class, we demonstrate the numerical stability of the
formalism and excellent agreement with results from other publicly available
codes for a range of parameters describing the evolution of the function
characterising the perturbations for Horndeski models, $\alpha_{\rm x}$, with
${\rm x}=\{{\rm K}, {\rm B}, {\rm M}, {\rm T}\}$. After demonstrating that on
sub-horizon scales ($k\gtrsim 10^{-3}~{\rm Mpc}^{-1}$ at $z=0$) velocity
perturbations in both the matter and the dark sector are typically subdominant
with respect to density perturbations in the equation of state for
perturbations, we find an attractor solution for the dark sector
gauge-invariant density perturbation $\Delta_{\rm ds}$ by neglecting its time
derivatives in the equation describing its time evolution, as commonly done in
the well-known quasi-static approximation. Using this result, we provide
simplified expressions for the equation-of-state functions: the dark sector
entropy perturbations $w_{\rm ds}\Gamma_{\rm ds}$ and anisotropic stress
$w_{\rm ds}\Pi_{\rm ds}$. From this we derive a growth factor-like equation for
both matter and dark sector and are able to capture the relevant physics for
several observables with great accuracy. We finally present new analytical
expressions for the well-known modified gravity phenomenological functions
$\mu$, $\eta$ and $\Sigma$ for a generic Horndeski model as functions of
$\alpha_{\rm x}$. We show that on small scales they reproduce expressions
presented in previous works, but on large scales, we find differences with
respect to other works.
