Nonlinear stability of cosmological solutions in massive gravity

TL;DR

The work addresses nonlinear stability of cosmological solutions in dRGT massive gravity by constructing perturbations around axisymmetric Bianchi-I backgrounds and expanding around fixed points for two classes of FLRW-like solutions. The approach reveals that isotropic FLRW self-accelerating solutions exhibit nonlinear ghost instabilities, while certain anisotropic FLRW configurations can be ghost-free over a range of parameters and initial conditions, indicating a healthy region between isotropic and anisotropic branches. By formulating gauge-invariant perturbations and separating odd and even sectors, the paper provides explicit conditions and parameter regimes where all five graviton polarizations can be non-ghost, albeit with strong-coupling caveats exactly at the fixed point. The results highlight how breaking FLRW symmetry lifts degeneracies in the Stückelberg sector and suggest that viable cosmological backgrounds may reside in anisotropic neighborhoods, guiding future explorations of stability in nonlinear massive gravity.

Abstract

We investigate nonlinear stability of two classes of cosmological solutions in massive gravity: isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions and anisotropic FLRW solutions. For this purpose we construct the linear cosmological perturbation theory around axisymmetric Bianchi type--I backgrounds. We then expand the background around the two classes of solutions, which are fixed points of the background evolution equation, and analyze linear perturbations on top of it. This provides a consistent truncation of nonlinear perturbations around these fixed point solutions and allows us to analyze nonlinear stability in a simple way. In particular, it is shown that isotropic FLRW solutions exhibit nonlinear ghost instability. On the other hand, anisotropic FLRW solutions are shown to be ghost-free for a range of parameters and initial conditions.