Cosmology in nonrelativistic general covariant theory of gravity

Horava and Melby-Thompson recently proposed a new version of the Horava-Lifshitz theory of gravity, in which the spin-0 graviton is eliminated by introducing a Newtonian prepotential $\phi$ and a local $U(1)$ gauge field $A$. In this paper, we first derive the corresponding Hamiltonian, supermomentum constraints, the dynamical equations, and the equations for $\phi$ and $A$, in the presence of matter fields. Then, we apply the theory to cosmology and obtain the modified Friedmann equation and the conservation law of energy, in addition to the equations for $\phi$ and $A$. When the spatial curvature is different from zero, terms behaving like dark radiation and stiff-fluid exist, from which, among other possibilities, a bouncing universe can be constructed. We also study linear perturbations of the Friedmann-Robertson-Walker universe with any given spatial curvature $k$, and we derive the most general formulas for scalar perturbations. The vector and tensor perturbations are the same as those recently given by one of the present authors [A. Wang, Phys. Rev. D 82, 124063 (2010).] in the setup of Sotiriou, Visser, and Weinfurtner. Applying these formulas to the Minkowski background, we have shown explicitly that the scalar and vector perturbations of the metric indeed vanish, and the only remaining modes are the massless spin-2 gravitons.