Cosmological Perturbations in Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper analyzes cosmological perturbations within Hořava-Lifshitz gravity, demonstrating the absence of strong coupling issues, deriving the perturbation spectrum, and exploring implications for inflation and the trans-Planckian problem.

Contribution

It provides a complete second-order action for scalar perturbations in Hořava-Lifshitz gravity and clarifies the behavior of the extra degree of freedom.

Findings

The extra degree of freedom is non-dynamical.

The limit λ→1 is well-behaved, avoiding strong coupling.

The perturbation spectrum depends on the potential form and λ.

Abstract

We study cosmological perturbations in Ho\v{r}ava-Lifshitz Gravity. We consider scalar metric fluctuations about a homogeneous and isotropic space-time. Starting from the most general metric, we work out the complete second order action for the perturbations. We then make use of the residual gauge invariance and of the constraint equations to reduce the number of dynamical degrees of freedom. After introducing the Sasaki-Mukhanov variable, the combination of spatial metric fluctuation and matter inhomogeneity for which the action in General Relativity has canonical form, we find that this variable has the standard time derivative term in the second order action, and that the extra degree of freedom is non-dynamical. The limit $\lambda \to 1$ is well-behaved, unlike what is obtained when performing incomplete analyses of cosmological fluctuations. Thus, there is no strong coupling problem for Ho\v{r}ava-Lifshitz gravity when considering cosmological solutions. We also compute the spectrum of cosmological perturbations. If the potential in the action is taken to be of "detailed balance" form, we find a cancelation of the highest derivative terms in the action for the curvature fluctuations. As a consequence, the initial spectrum of perturbations will not be scale-invariant in a general spacetime background. As an application, we consider fluctuations in an inflationary background and draw connections with the "trans-Planckian problem" for cosmological perturbations. In the special case in which the potential term in the action is of detailed balance form and in which $\lambda = 1$, the equation of motion for cosmological perturbations in the far UV takes the same form as in GR. However, in general the equation of motion is characterized by a modified dispersion relation.