Complete analysis of linear cosmological perturbations in Hořava-Lifshitz gravity

We investigate the linear cosmological perturbations in Hořava-Lifshitz gravity with a scalar field. Starting from the most general expressions of the metric perturbations as well as that of a canonical scalar field, we decompose the scalar, vector, and tensor parts of the perturbed action. By reducing the Hamiltonian, we find that there are two independent degrees of freedom for the tensor perturbations while none for the vector perturbations. For the scalar perturbations, the remaining number of degrees of freedom, which are all gauge invariant, depends on whether the projectable condition is applied or not: two when applied, with one of them being possibly a ghost, and one when not applied. For both cases, we lose the time reparametrization symmetry of any kind.