Phenomenology of Horndeski Gravity under Positivity Bounds

A set of conditions that any effective field theory needs to satisfy in order to allow for the existence of a viable UV completion has recently gained attention in the cosmological context under the name of $\textit{positivity bounds}$. In this paper we revisit the derivation of such bounds for Horndeski gravity and translate them into a complete set of viability conditions in the language of effective field theory of dark energy. We implement the latter into $\texttt{EFTCAMB}$ and explore the large scale structure phenomenology of Horndeski gravity under positivity bounds. We build a statistically significant sample of viable Horndeski models, and derive the corresponding predictions for the background evolution, in terms of $w_{\rm DE}$, and the dynamics of linear perturbations, in terms of the phenomenological functions $\mu$ and $\Sigma$, associated to clustering and weak lensing, respectively. We find that the addition of positivity bounds to the traditional no-ghost and no-gradient conditions considerably tightens the theoretical constraints on all these functions. The most significant feature is a strengthening of the correlation $\mu\simeq\Sigma$, and a related tight constraint on the luminal speed of gravitational waves $c^2_T\simeq1$. In anticipation of a more complete formulation of positivity conditions in cosmology, this work demonstrates the strong potential of such bounds in shaping the viable parameter space of scalar-tensor theories.