Self-accelerating Massive Gravity: Time for Field Fluctuations

TL;DR

This paper investigates the behavior of field fluctuations in ghost-free massive gravity, showing that certain perturbations decay in an expanding universe and identifying conditions under which these fluctuations do not affect observable metrics.

Contribution

It demonstrates that fluctuations in the spatial Stueckelberg field decay and depends on background choices, clarifying the dynamics and initial conditions in massive gravity cosmological solutions.

Findings

Fluctuations decay in an expanding universe.

A special gauge choice eliminates perturbations.

Only one initial data set influences observable metrics.

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 Stueckelberg formulation, this stress energy is carried entirely by the spatial Stueckelberg 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 Stueckelberg field perturbation depend on the background temporal Stueckelberg 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.