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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 ($kgtrsim 10^{-3}~{rm Mpc}^{-1}$ 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.
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A phenomenological attempt at alleviating the so-called coincidence problem is to allow the dark matter and dark energy to interact. By assuming a coupled quintessence scenario characterized by an interaction parameter $epsilon$, we investigate the precision in the measurements of the expansion rate $H(z)$ required by future experiments in order to detect a possible deviation from the standard $Lambda$CDM model ($epsilon = 0$). We perform our analyses at two levels, namely: through Monte Carlo simulations based on $epsilon$CDM models, in which $H(z)$ samples with different accuracies are generated and through an analytic method that calculates the error propagation of $epsilon$ as a function of the error in $H(z)$. We show that our analytical approach traces simulations accurately and find that to detect an interaction {using $H(z)$ data only, these must reach an accuracy better than 1%.
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