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.Comment: 38 pages, 4 figures, 2 tables, references added, paraghraph added, minor changes, results unchanged, to appear in Eur. Phys. J.

On the self-similarity of nonhelical magnetohydrodynamic turbulence

We re-analyze the Olesen arguments on the self-similarity properties of freely evolving, nonhelical magnetohydrodynamic turbulence. We find that a necessary and sufficient condition for the kinetic and magnetic energy spectra to evolve self-similarly is that the initial velocity and magnetic field are not homogeneous functions of space of different degree, to wit, the initial energy spectra are not simple powers of the wavenumber with different slopes. If, instead, they are homogeneous functions of the same degree, the evolution is self-similar, it proceeds through selective decay, and the order of homogeneity fixes the exponents of the power laws according to which the kinetic and magnetic energies and correlation lengths evolve in time. If just one of them is homogeneous, the evolution is self-similar and such exponents are completely determined by the slope of that initial spectrum which is a power law. The latter evolves through selective decay, while the other spectrum may eventually experience an inverse transfer of energy. Finally, if the initial velocity and magnetic field are not homogeneous functions, the evolution of the energy spectra is still self-similar but, this time, the power-law exponents of energies and correlation lengths depend on a single free parameter which cannot be determined by scaling arguments. Also in this case, an inverse transfer of energy may in principle take place during the evolution of the system.Comment: 4 pages, 1 figure, typos correcte

Evolution of Magnetic Fields in Freely Decaying Magnetohydrodynamic Turbulence

We study the evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence. By quasi-linearizing the Navier-Stokes equation, we solve analytically the induction equation in quasi-normal approximation. We find that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time respectively as E_B \propto t^{-2(1+p)/(3+p)} and \xi_B \propto t^{2/(3+p)}, where p is the index of initial power-law spectrum. In the helical case, the magnetic helicity is an almost conserved quantity and forces the magnetic energy and correlation length to scale as E_B \propto (log t)^{1/3} t^{-2/3} and \xi_B \propto (log t)^{-1/3} t^{2/3}.Comment: 4 pages, 2 figures; accepted for publication in PR