Isotropy theorem for cosmological Yang-Mills theories

TL;DR

This paper demonstrates that rapidly evolving homogeneous non-abelian vector fields in an expanding universe have an average energy-momentum tensor that is isotropic and behaves like a perfect fluid, regardless of potential, polarization, or gauge-fixing.

Contribution

It generalizes the isotropy theorem for Yang-Mills fields, showing isotropic energy-momentum tensors under broad conditions and including gauge-fixing effects and arbitrary backgrounds.

Findings

Average energy-momentum tensor is diagonal and isotropic.

Isotropy holds for arbitrary potentials and polarization patterns.

Results extend to any background geometry and locally inertial observers.

Abstract

We consider homogeneous non-abelian vector fields with general potential terms in an expanding universe. We find a mechanical analogy with a system of N interacting particles (with N the dimension of the gauge group) 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 a generalization of the virial theorem that for arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. We consider also the case in which a gauge-fixing term is introduced in the action and show that the average equation of state does not depend on such a term. Finally, we extend the results to arbitrary background geometries and show that the average energy-momentum tensor of a rapidly evolving Yang-Mills fields is always isotropic and has the perfect fluid form for any locally inertial observer.