Cuscuton and low energy limit of Horava-Lifshitz gravity

TL;DR

This paper shows that the low energy limit of Horava-Lifshitz gravity is a covariant scalar field theory called the quadratic cuscuton, which constrains the extra scalar degree of freedom and links cosmological observations to quantum gravity parameters.

Contribution

It establishes the equivalence between low energy Horava-Lifshitz gravity and the quadratic cuscuton model, revealing the non-dynamical nature of the scalar degree of freedom and providing observational constraints.

Findings

The low energy limit is uniquely given by the quadratic cuscuton model.

Current cosmological data constrains the Lorentz breaking parameter |λ-1| < 0.014.

The theory predicts constant mean curvature hypersurfaces at low energies.

Abstract

A proposal for a power-counting renormalizable theory of quantum gravity at a Lifshitz point was recently put forth by Horava (arXiv:0901.3775), and has been since dubbed as Horava-Lifshitz gravity. The theory explicitly breaks Lorentz invariance, which introduces an apparent extra scalar degree of freedom. In this note, we show that the low energy limit of (non-projectible) Horava-Lifshitz gravity is uniquely given by the quadratic cuscuton model: a covariant scalar field theory with an infinite speed of sound and a quadratic potential, which is minimally coupled to Einstein gravity. This implies that the extra scalar is non-dynamical to all orders in perturbation theory. Using current cosmological constraints on the quadratic cuscuton model, we can constrain the low energy Lorentz breaking parameter of Horava-Lifshitz theory (which leads to a running of Planck mass on the Hubble scale) to |lambda-1| < 0.014 (at 95% confidence level). We also point out that, with reasonable boundary conditions, the spatial hypersurfaces in this theory are constant mean curvature (CMC) or uniform expansion surfaces at low energies, and introduce geometrical symmetries that can protect the non-dynamical nature of these theories from quantum corrections. We also notice that the theory with lambda < 1 might suffer from a non-perturbative UV instability.