Extended Effective Field Theory of Dark Energy: Ghost Condensate Dark Energy with Sextic Dispersion Relation in de Sitter Spacetime

We continue our studies of the ghost condensate (GC) with sixth-order dispersion relation. Contrary to the GC with quartic dispersion relation, we find that the correction to the Newtonian potential explicitly depends on the space and time dependence of matter density. At late times when the Newtonian potential becomes time-independent, one obtains similar oscillatory behavior at the distance $\frac{M_{\text{Pl}}}{M^2}$, but this time at the time scale $\frac{M^4}{M_{\text{Pl}}^3}$, where $M^2$ is the ghost field velocity. We also show that the speed of gravitational wave is modified in a frequency dependent manner at momenta close to $\frac{M_{\text{Pl}}}{\sqrt{\mathrm{\mid }{\sigma }_1\mathrm{\mid }}}$, where ${\sigma }_1$ is the coefficient of ${\gamma }^{ij}{\mathrm{\nabla }}_i{K}_{lr}{\mathrm{\nabla }}_j{K}^{lr}$ operator in the unitary gauge action.