Evolution of primordial magnetic fields in mean-field approximation

We study the evolution of phase-transition-generated cosmic magnetic fields coupled to the primeval cosmic plasma in turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and correlation length, both in helical and non-helical cases, are found by solving the autoinduction and Navier-Stokes equations in 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 radiation and matter dominated eras. 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 the magnetic correlation length $\xi_B$ evolve asymptotically with the temperature $T$ as $B(T) \simeq \kappa_B (N_i v_i)^{\varrho_1} (T/T_i)^{\varrho_2}$ and $\xi_B(T) \simeq \kappa_\xi (N_i v_i)^{\varrho_3} (T/T_i)^{\varrho_4}$. Here, $T_i$, $N_i$, and $v_i$ 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 $\kappa_B$, $\kappa_\xi$, $\varrho_1$, $\varrho_2$, $\varrho_3$, and $\varrho_4$, depend on the index of the assumed initial power-law magnetic spectrum, $p$, and on the particular regime, with the order-one constants $\kappa_B$ and $\kappa_\xi$ depending also on the cut-off 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.