Anisotropic Weyl symmetry and cosmology

We construct an anisotropic Weyl invariant theory in the ADM formalism and discuss its cosmological consequences. It extends the original anisotropic Weyl invariance of Ho\v{r}ava-Lifshitz gravity using an extra scalar field. The action is invariant under the anisotropic transformations of the space and time metric components with an arbitrary value of the critical exponent $z$. One of the interesting features is that the cosmological constant term maintains the anisotropic symmetry for $z=-3$. We also include the ordinary matter and show that it can preserve the anisotropic Weylinvariance if the equation of state satisfies $P_m= z\rho_m/3$. Then, we study cosmology of the Einstein-Hilbert-anisotropic Weyl (EHaW) action including the ordinary matter both with or without anisotropic Weyl invariance. The correlation of the critical exponent $z$ and the equation of state parameter $\omega_m$ provides a new perspective of the cosmology. It is also shown that for particular value of $z=-3$, the EHaW action admits a late time accelerating universe.