Pathological behaviour of the scalar graviton in Ho\v{r}ava-Lifshitz
  gravity
  We confirm the recent claims that, in the infrared limit of
Ho\v{r}ava-Lifshitz gravity, the scalar graviton becomes a ghost if the sound
speed squared is positive on the flat de Sitter and Minkowski background. In
order to avoid the ghost and tame the instability, the sound speed squared
should be negative and very small, which means that the flow parameter
$\lambda$ should be very close to its General Relativity (GR) value. We
calculate the cubic interactions for the scalar graviton which are shown to
have a similar structure with those of the curvature perturbation in
k-inflation models. The higher order interactions become increasing important
for a smaller sound speed squared, that is, when the theory approaches GR. This
invalidates any linearized analysis and any predictability is lost in this
limit as quantum corrections are not controllable. This pathological behaviour
of the scalar graviton casts doubt on the validity of the projectable version
of the theory.
