The role of the tachyonic instability in Horndeski gravity

TL;DR

This paper investigates the impact of tachyonic instability on the stability of Horndeski gravity models, proposing comprehensive conditions for stability across all cosmological scales and analyzing their effects on model parameter spaces.

Contribution

It introduces a full set of stability conditions combining no-ghost, no-gradient, and no-tachyon criteria, applied to Horndeski and related scalar-tensor theories, enhancing understanding of their viability.

Findings

Tachyonic instability affects the parameter space of scalar-tensor models.

No-tachyon condition excludes models with  in  in generalized Brans-Dicke theories.

In Horndeski gravity, the larger number of free functions reduces the impact of the no-tachyon condition.

Abstract

The tachyonic instability is associated with the unboundedness of the Hamiltonian from below and results in 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 $w$CDM 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.