Diagnostic of Horndeski Theories

We study the effects of Horndeski models of dark energy on the observables of the large-scale structure in the late time universe. A novel classification into {\it Late dark energy}, {\it Early dark energy} and {\it Early modified gravity} scenarios is proposed, according to whether such models predict deviations from the standard paradigm persistent at early time in the matter domination epoch. We discuss the physical imprints left by each specific class of models on the effective Newton constant $\mu$, the gravitational slip parameter $\eta$, the light deflection parameter $\Sigma$ and the growth function $f\sigma_8$ and demonstrate that a convenient way to dress a complete portrait of the viability of the Horndeski accelerating mechanism is via two, redshift-dependent, diagnostics: the $\mu(z)-\Sigma(z)$ and the $f\sigma_8(z)-\Sigma(z)$ planes. If future, model-independent, measurements point to either $\Sigma-1<0$ at redshift zero or $\mu-1<0$ with $\Sigma-1>0$ at high redshifts or $\mu-1>0$ with $\Sigma-1<0$ at high redshifts, Horndeski theories are effectively ruled out. If $f\sigma_8$ is measured to be larger than expected in a $\Lambda$CDM model at $z>1.5$ then Early dark energy models are definitely ruled out. On the opposite case, Late dark energy models are rejected by data if $\Sigma<1$, while, if $\Sigma>1$, only Early modifications of gravity provide a viable framework to interpret data.