Evolution of primordial magnetic fields in mean-field approximation

Leonardo Campanelli

doi:10.1140/epjc/s10052-013-2690-5

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2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2014) 74: 2690
https://doi.org/10.1140/epjc/s10052-013-2690-5

Regular Article - Theoretical Physics

Evolution of primordial magnetic fields in mean-field approximation

Leonardo Campanelli^*

Dipartimento di Fisica, Università di Bari, 70126, Bari, Italy

^* e-mail: leonardo.campanelli@ba.infn.it

Received: 26 August 2013
Accepted: 5 December 2013
Published online: 23 January 2014

Abstract

We study the evolution of phase-transition-generated 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 matter-dominated 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 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 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.