De Sitter vacua in ghost-free massive gravity theory

We present a simple procedure to obtain a large class of different versions of the de Sitter solution in the ghost-free massive gravity theory via applying the Gordon ansatz. For these solutions the physical metric describes a hyperboloid in 5D Minkowski space, while the flat reference metric depends on the Stuckelberg field T(t,r)T(t,r) subject to (∂tT)2−(∂rT)2=1(∂tT)2−(∂rT)2=1. This equation admits infinitely many solutions, hence there are infinitely many de Sitter vacua with different physical properties. Only the simplest solution with T=tT=t has previously been studied, as it is manifestly homogeneous and isotropic, but this solution turns out to be unstable. However, other solutions could be stable. We require the timelike isometry to be common for both metrics and this gives physically distinguished solutions since only for them the canonical Killing energy is time-independent. We conjecture that these solutions minimize the energy and are therefore stable. We also show that in some cases solutions can be homogeneous and isotropic in a non-manifest way such that their symmetries are not obvious. All of this suggests that the theory may admit physically interesting cosmologies.