The role of the tachyonic instability in Horndeski gravity

The tachyonic instability is commonly associated with mass terms having a negative sign and evolving faster than the Hubble scale, leading to an unstable low-k regime. In the cosmological exploration of modified gravity, it is seldom taken into account, with more focus given to the popular no-ghost and no-gradient conditions. The latter though are intrinsically high-k statements. Here we combine all three conditions into a full set of requirements that we show to guarantee stability on the whole range of cosmological scales. We then explore the impact of the different conditions on the parameter space of scalar-tensor gravity, with particular emphasis on the no-tachyon one. We focus on Horndeski gravity and also consider separately the two subclasses of f(R) and Generalized Brans Dicke theories. We identify several interesting features, for instance in the parameter space of designer f(R) on a wCDM background, shedding light on previous findings. When looking at the phenomenological functions Sigma and mu, associated to the weak lensing and clustering potential respectively, we find that in the case of Generalized Brans Dicke the no-tachyon condition clearly cuts models with mu, Sigma > 1. This effect is less prevalent in the Horndeski case due to the larger amount of free functions in the theory.