Isotropy theorem for cosmological vector fields

Abstract

We consider homogeneous Abelian vector fields in an expanding universe. We find a mechanical analogy in which the system behaves as a particle moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show---by making use of the virial theorem---that for an arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. For simple power-law potentials of the form $V=\ensuremath{\lambda}({A}^{\ensuremath{\mu}}{A}_{\ensuremath{\mu}}{)}^{n}$, the average equation of state is found to be $w=(n\ensuremath{-}1)/(n+1)$. This implies that vector coherent oscillations could act as natural dark matter or dark energy candidates. Finally, we show that under very general conditions, the average energy-momentum tensor of a rapidly evolving bounded vector field in any background geometry is always isotropic and has the perfect fluid form for any locally inertial observer.