Instability of the Ackerman-Carroll-Wise model, and problems with massive vectors during inflation

We prove that the anisotropic inflationary background of the Ackerman-Carroll-Wise model, characterized by a fixed-norm vector field, is unstable. We found the instability by explicitly solving the linearized equations for the most general set of perturbations around this background, and by noticing that the solutions diverge close to horizon crossing. This happens because one perturbation becomes a ghost at that moment. A simplified computation, with only the perturbations of the vector field included, shows the same instability, clarifying the origin of the problem. We then discuss several other models, with a particular emphasis on the case of a nonminimal coupling to the curvature, in which vector fields are used either to support an anisotropic expansion, or to generate cosmological perturbations on an isotropic background. In many cases, the mass squared of the vector needs to be negative; we show that, as a consequence, the longitudinal vector mode is a ghost (a field with negative kinetic term, and negative energy, and not simply a tachyon). We comment on problems that arise at the quantum level. In particular, the presence of a ghost can be a serious difficulty for the UV completion that such models require in the subhorizon regime.