The role of the tachyonic instability in Horndeski gravity
  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.
