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 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 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 wDE, and the dynamics of linear perturbations, in terms of the phenomenological functions μ and Σ, 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 μΣ, and a related tight constraint on the luminal speed of gravitational waves c²T1. 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.