Vector theories in cosmology

Abstract

This article provides a general study of the Hamiltonian stability and the hyperbolicity of vector field models involving both a general function of the Faraday tensor and its dual, $f({F}^{2},F\stackrel{\texttildelow{}}{F})$, as well as a Proca potential for the vector field, $V({A}^{2})$. In particular it is demonstrated that theories involving only $f({F}^{2})$ do not satisfy the hyperbolicity conditions. It is then shown that in this class of models, the cosmological dynamics always dilutes the vector field. In the case of a nonminimal coupling to gravity, it is established that theories involving $Rf({A}^{2})$ or $Rf({F}^{2})$ are generically pathologic. To finish, we exhibit a model where the vector field is not diluted during the cosmological evolution, because of a nonminimal vector field-curvature coupling which maintains second-order field equations. The relevance of such models for cosmology is discussed.