Infrared modified gravity with propagating torsion: instability of torsionfull de Sitter-like solutions

We continue the exploration of the consistency of a modified-gravity theory that generalizes General Relativity by including a dynamical torsion in addition to the dynamical metric. The six-parameter theory we consider was found to be consistent around arbitrary torsionless Einstein backgrounds, in spite of its containing a (notoriously delicate) massive spin-2 excitation. At zero bare cosmological constant, this theory was found to admit a self-accelerating solution whose exponential expansion is sustained by a non-zero torsion background. The scalar-type perturbations of the latter torsionfull self-accelerating solution were recently studied and were found to preserve the number of propagating scalar degrees of freedom, but to exhibit, for some values of the torsion background some exponential instabilities (of a rather mild type). Here, we study the tensor-type and vector-type perturbations of the torsionfull self-accelerating solution, and of its deformation by a non-zero bare cosmological constant. We find strong, "gradient" instabilities in the vector sector. No tuning of the parameters of the theory can kill these instabilities without creating instabilities in the other sectors. Further work is needed to see whether generic torsionfull backgrounds are prone to containing gradient instabilities, or if the instabilities we found are mainly due to the (generalized) self-accelerating nature of the special de Sitter backgrounds we considered.