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 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=\lambda (A^\mu A_\mu)^n, the average equation of state is found to be w=(n-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.Comment: 5 pages, 1 figure. Final version to appear as Rapid Communication in Phys. Rev.

Isotropy theorem for cosmological Yang-Mills theories

“…The different lines correspond to different values of the potential exponent n in (42). We can observe that for large g values, the non-abelian term dominates and the equation of state approaches the radiation behaviour independently of n. On the contrary, for small g, the abelian case is recovered [15].…”

“…1 Now, unlike the abelian case, f dV ef f /df is not proportional to V ef f , except when n = 2, and that is the reason why the virial theorem can not be used to obtain the equation of state analytically as done in [15] (2) non-abelian case with potential (36), for a given energy density and M parameter. The different lines correspond to different values of the potential exponent n in (42).…”

“…We will follow the method presented in [15], in order to compute the average components of the energymomentum tensor. With that purpose, we will generalize the virial theorem in order to apply it to non-abelian fields, thus we define:…”

“…Two possible isotropic configurations discussed in the literature are the presence of a triad of mutually orthogonal vectors [12][13][14] or the existence of a large N number of randomly oriented vectors, so that the average isotropy violation is kept relatively small of order 1/ √ N [9]. However, recently [15] a new possibility was devised in which a general isotropy theorem for vector fields was proved. Provided vector field evolution is bounded and rapid compared to the rate of expansion, it was shown that the average energy-momentum tensor is always diagonal and isotropic for any kind of initial configuration.…”

Perturbations of ultralight vector field dark matter

“…The main problem which arises in the case of vectors or higher spin fields is that coherent homogeneous fields typically break isotropy. However it has been recently shown (see [54,55] for abelian vector fields, [56] for nonabelian theories and [57] for arbitrary spin) that for rapidly oscillating coherent fields, even though the field evolution is generically anisotropic, the average energy-momentum tensor is not. In particular, for massive fields it is straightforward to show that the average energy density scales as a −3 .…”