Bumblebee Gravity - Lessons from Perturbation Theory

TL;DR

These proceedings analyze Bumblebee gravity, a vector-tensor theory with spontaneous breaking of local Lorentz and diffeomorphism invariance, on a de Sitter background using cosmological perturbation theory. They show that with non-minimal couplings $\xi$ and $\sigma$, the scalar sector generically contains a ghost due to a rank-3 kinetic matrix, unless the degeneracy condition $\sigma = -\tfrac{1}{2}\xi$ is imposed; under this condition the model becomes a subset of generalized Proca with a specific $G_2$-$G_4$ mapping and a background relation $H = H_{\rm dS}/(1 - \xi \tilde{B}_0^2)$, with $V_B(-\tilde{B}_0^2) = \Lambda_B/(1 - \xi \tilde{B}_0^2)$. They further find that the tensor mode speed $c_T$ is tightly constrained by the GW170817 event, yielding bounds on $\xi \tilde{B}_0^2$ and implying strong coupling in the minimal coupling limit $\xi \to 0$. Overall, the degeneracy condition yields a ghost-free, theoretically consistent framework that remains compatible with gravitational-wave speed constraints, provided the background parameters are small.

Abstract

These proceedings summarize some recent efforts in understanding a class of vector-tensor theories known as {\it bumblebee} models, which spontaneously break local Lorentz and diffeomorphism invariance. Using cosmological perturbation theory on an exact dS background, we find that for non-minimal coupling to gravity, the theory contains a ghost mode unless a degeneracy condition is imposed, after which the model becomes a subset of generalized Proca theory. We go further to show that scalar perturbations become strongly coupled in the minimal-coupling limit, which shows the necessity of the non-minimal coupling. Moreover, we find a constraint on the bumblebee field from the speed of tensor modes on the order of $10^{-15}$.