A complete analysis of linear cosmological perturbations in Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper provides a comprehensive analysis of linear cosmological perturbations in Hořava-Lifshitz gravity with a scalar field, revealing the degrees of freedom and gauge invariances involved.

Contribution

It offers a detailed decomposition and Hamiltonian reduction of perturbations, clarifying the degrees of freedom in scalar, vector, and tensor sectors under different conditions.

Findings

Two tensor degrees of freedom identified.

No vector degrees of freedom present.

Scalar degrees depend on the projectable condition.

Abstract

We investigate the linear cosmological perturbations in Ho\v{r}ava-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. For both cases, we lose the time reparametrization symmetry of any kind.