Anisotropic Weyl symmetry and cosmology

Abstract

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.