# Kinetic equation for Lifshitz scalar

## TL;DR

This paper derives kinetic equations for Lifshitz scalar fields on curved spaces, extending to general dispersion relations, with implications for cosmology in Hořava-Lifshitz gravity.

## Contribution

It develops a method to derive kinetic equations for Lifshitz scalars with arbitrary dispersion relations on curved geometries, generalizing previous specific cases.

## Key findings

- Derived kinetic equations for z=2 and z=3 Lifshitz scalars.
- Proposed a conjecture for general dispersion relations in curved spaces.
- Potential applications in cosmology within Hořava-Lifshitz gravity.

## Abstract

Employing the method of Wigner functions on curved spaces, we study classical kinetic (Boltzmann-like) equations of distribution functions for a real scalar field with the Lifshitz scaling. In particular, we derive the kinetic equation for $z=2$ on general curved spaces and for $z=3$ on spatially flat spaces under the projectability condition $N=N(t)$, where $z$ is the dynamical critical exponent and $N$ is the lapse function. We then conjecture a form of the kinetic equation for a real scalar field with a general dispersion relation in general curved geometries satisfying the projectability condition, in which all the information about the non-trivial dispersion relation is included in the group velocity and which correctly reproduces the equations for the $z=2$ and $z=3$ cases as well as the relativistic case. The method and equations developed in the present paper are expected to be useful for developments of cosmology in the context of Ho\v{r}ava-Lifshitz gravity.