Ghost instabilities of cosmological models with vector fields
  nonminimally coupled to the curvature
  We prove that many cosmological models characterized by vectors nonminimally
coupled to the curvature (such as the Turner-Widrow mechanism for the
production of magnetic fields during inflation, and models of vector inflation
or vector curvaton) contain ghosts. The ghosts are associated with the
longitudinal vector polarization present in these models, and are found from
studying the sign of the eigenvalues of the kinetic matrix for the physical
perturbations. Ghosts introduce two main problems: (1) they make the theories
ill-defined at the quantum level in the high energy/sub horizon regime (and
create serious problems for finding a well behaved UV completion); (2) they
create an instability already at the linearized level. This happens because the
eigenvalue corresponding to the ghost crosses zero during the cosmological
evolution. At this point the linearized equations for the perturbations become
singular (we show that this happens for all the models mentioned above). We
explicitly solve the equations in the simplest cases of a vector without vev in
a FRW geometry, and of a vector with vev plus a cosmological constant, and we
show that indeed the solutions of the linearized equations diverge when these
equations become singular.
