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.
