Evolution of primordial magnetic fields in mean-field approximation

Abstract

We study the evolution of phase-transitiongenerated cosmic magnetic fields coupled to the primeval cosmic plasma in the turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and the correlation length, both in the helical and the non-helical cases, are found by solving the autoinduction and Navier-Stokes equations in the mean-field approximation. Analytical results are derived in Minkowski spacetime and then extended to the case of a Friedmann universe with zero spatial curvature, both in the radiation- and the matterdominated era. The three possible viscous free-streaming phases are characterized by a drag term in the Navier-Stokes equation which depends on the free-streaming properties of neutrinos, photons, or hydrogen atoms, respectively. In the case of non-helical magnetic fields, the magnetic intensity B and themagnetic correlation length ξB evolve asymptotically with the temperature, T, as (Formula presented.) and (Formula presented.). Here, Ti, Ni, and vi are, respectively, the temperature, the number of magnetic domains per horizon length, and the bulk velocity at the onset of the particular regime. The coefficients (Formula presented.), depend on the index of the assumed initial power-law magnetic spectrum, p, and on the particular regime, with the order-one constants KB and Kξ depending also on the cutoff adopted for the initial magnetic spectrum. In the helical case, the quasi-conservation of the magnetic helicity implies, apart from logarithmic corrections and a factor proportional to the initial fractional helicity, power-like evolution laws equal to those in the non-helical case, but with p equal to zero. © 2014 The Author(s).