Vector theories in cosmology

TL;DR

This paper analyzes the stability and hyperbolicity of vector field models in cosmology, revealing limitations of certain models and proposing a nonminimal coupling that maintains vector fields during cosmic evolution.

Contribution

It provides a comprehensive stability analysis of vector field theories with various couplings and introduces a novel model with nonminimal coupling that avoids dilution.

Findings

Models with only $f(F^2)$ are not hyperbolic.

Cosmological dynamics tend to dilute vector fields in these models.

Nonminimal couplings like $A^2$ or $F^2$ with gravity are generally problematic.

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, $f(F^2,F\tilde F)$, 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 $R f(A^2)$ 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.