Effective Field Theory of Broken Spatial Diffeomorphisms

TL;DR

This work develops an effective field theory for gravity with broken spatial diffeomorphisms, where in unitary gauge the graviton becomes a massive spin-2 with 5 healthy degrees of freedom. On FRW backgrounds with enhanced symmetries, the perturbations are governed by a five-parameter set including the usual cosmological parameters and an extra coupling $M_2^2$, with all five helicities acquiring mass and a common IR mass scale $m_g^2 = 4 H^2 \epsilon + 8 M_2^2 / a^4$. In the de Sitter and Minkowski limits, the lowest-derivative kinetic terms can vanish and higher-derivative operators become essential, restoring well-defined dynamics and leaving two propagating tensor modes after integrating out the Goldstones; the framework also encompasses various massive-gravity scenarios and self-accelerating solutions. Overall, the paper provides a systematic EFT description of broken spatial diffeomorphisms, with concrete perturbation spectra and regimes that connect to massive gravity phenomenology and cosmological tensor modes.

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.