Effective Field Theory of Broken Spatial Diffeomorphisms

TL;DR

This paper develops a low energy effective theory for gravity with broken spatial diffeomorphisms, resulting in a massive graviton with five degrees of freedom and analyzing its cosmological perturbations.

Contribution

It introduces a general framework for broken spatial diffeomorphisms in gravity, including the effects of Goldstone bosons and their impact on cosmological perturbations.

Findings

Graviton becomes a massive spin-2 particle with five degrees of freedom.

Perturbations characterized by five parameters, including cosmological and symmetry-breaking constants.

In certain limits, only two massive tensor modes propagate, simplifying the theory.

Abstract

We study the low energy effective theory describing gravity with broken spatial diffeomorphism invariance. In the unitary gauge, the Goldstone bosons associated with broken diffeomorphisms are eaten and the graviton becomes a massive spin-2 particle with 5 well-behaved degrees of freedom. In this gauge, the most general theory is built with the lowest dimension operators invariant under only temporal diffeomorphisms. Imposing the additional shift and SO(3) internal symmetries, we analyze the perturbations on a FRW background. At linear perturbation level, the observables of this theory are characterized by five parameters, including the usual cosmological parameters and one additional coupling constant for the symmetry-breaking scalars. In the de Sitter and Minkowski limit, the three Goldstone bosons are supermassive and can be integrated out, leaving two massive tensor modes as the only propagating degrees of freedom. We discuss several examples relevant to theories of massive gravity.