by a cosmological constant, but in the presence of this additional degree of freedom. The assumption of a scalar field ensures that there are no gross violations of isotropy. The limitation to second order is a necessary condition to ensure that the model is not subject to instabilities (see e.g. [@Woodard:2006nt]).[^2] We will therefore assume that the dark energy is governed by the most general Lagrangian which fulfills these requirements: $\frac{1}{2}R+\mathcal{L}_{x}$ will form the Horndeski Lagrangian (HL [@Horndeski:1974; @Deffayet:2011gz]). We dedicate section \[sec:Horn\] to this system.

Background Observables {#sec:Bckg}
======================

From assumptions $(a)$-$(c)$, by varying the action eq.  with respect to the metric, we obtain a Friedmann equation that can be written as $$H^{2}-H_{0}^{2}\Omega_{k0}a^{-2}=\frac{1}{3}(\rho_{x}+\rho_{\text{m}})\,,\label{eq:Friedmann}$$ where $H_{0}$ is the present value of the Hubble parameter, $\Omega_{k0}$ the present curvature density parameter and $\rho_{\text{m}}$ is the matter energy density. From assumption $(b)$, $\rho_{\text{m}}$ evolves as $a^{-3}$, and $\rho_{x}$ is the energy density of the terms coming from $\mathcal{L}_{x}$.

Observations of the cosmic expansion are essentially estimations of distances $D(z)$ (i.e. luminosity or angular-diameter distances) or directly $H(z)$ (e.g. using measurements of longitudinal baryon acoustic oscillations, or real-time redshift-drift observations [@Quercellini:2010zr]) based on the existence of standard candles, rods or clocks. More exactly, 