number of additional assumptions on the source physics. No current or foreseeable method to estimate the peculiar velocity field (let alone its derivative) has been shown to be reliable beyond a few hundred megaparsecs (see e.g. Ref. [@Macaulay:2011av]), so we will not pursue this possibility any further in this paper.

Our first result is that linear cosmological observations can at best determine only $E\equiv H/H_{0}$ as function of time (as well as $\Omega_{k0}$ but *not* $\Omega_{\text{m}0}$) and the observable combinations $P_{1-3}$ as functions of time and space, within the range of the observations themselves. To achieve this, we need to combine galaxy clustering and weak lensing in the same redshift range. It is possible that one can determine other combinations from non-linear effects, but this will certainly introduce new uncertainties (e.g. a non-linear bias).

The Horndeski Lagrangian {#sec:Horn}
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It is now time to use also assumption $(e)$ regarding the explicit form of the dark energy Lagrangian. In the choice of the action , we have assumed that all matter components (i.e. dark matter and baryons) feel the same gravitational force and propagate on geodesics of the metric $g_{\mu\nu}$. In addition, we now explicitly assume that the DE is modeled 