Cosmology in $(1+1)$-dimensional Ho\v{r}ava-Lifshitz theory of gravity

The $(1+1)$-dimensional Friedmann-Robertson-Walker (FRW) universe filled with a perfect fluid with equation of state $p=\omega \rho$ is analyzed through the view of Ho\v rava-Lifshitz (HL) theory of gravity. In this theory, the anisotropic scaling of space and time breaks Lorentz invariance of General Relativity (GR) in such a way that the gravitational action is no longer a topological invariant and the theory becomes dynamical. With the introduction of a perfect fluid through Schutz formalism, it is shown that the resulting dynamical theory is very similar to the two-dimensional Jackiw-Teitelboim (JT) model, where a dilatonic degree of freedom is introduced to force a dynamical theory. However, in HL theory, the introduction of a dilaton field is not necessary.