Self-accelerating massive gravity: Time for field fluctuations

Abstract

The ghost-free theory of massive gravity has exact solutions where the effective stress energy generated by the graviton mass term is a cosmological constant for any isotropic metric. Since they are exact, these solutions mimic a cosmological constant in the presence of any matter-induced isotropic metric perturbation. In the St\"uckelberg formulation, this stress energy is carried entirely by the spatial St\"uckelberg field. We show that any stress energy carried by fluctuations in the spatial field away from the exact solution always decays away in an expanding universe. However, the dynamics of the spatial St\"uckelberg field perturbation depend on the background temporal St\"uckelberg field, which is equivalent to the unitary gauge time coordinate. This dependence resolves an apparent conflict in the existing literature by showing that there is a special unitary time choice for which the field dynamics and energy density perturbations vanish identically. In general, the isotropic system has a single dynamical degree of freedom requiring two sets of initial data; however, only one of these initial data choices will affect the observable metric. Finally, we construct cosmological solutions with a well-defined perturbative initial value formulation and comment on alternate solutions that evolve to singularities.