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We summarise the effective field theory of dark energy construction toexplore observable predictions of linear Horndeski theories. Based on\cite{Perenon:2016blf}, we review the diagnostic of these theories on thecorrelation of the large-scale structure phenomenological functions: theeffective Newton constant, the light deflection parameter and the growthfunction of matter perturbations. We take this opportunity to discuss theevolution of the bounds the propagation speed of gravitational waves hasundergone and use the most restrictive one to update the diagnostic.

The effective field theory of dark energy diagnostic of linear Horndeski theories after GW170817 and GRB170817A