https://doi.org/10.1140/epjc/s10052-013-2690-5
Regular Article - Theoretical Physics
Evolution of primordial magnetic fields in mean-field approximation
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
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 
and the magnetic correlation length 
evolve asymptotically with the temperature, 
, as 
and 
. Here, 
, 
, and 
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 
, 
, 
, 
, 
, and 
, depend on the index of the assumed initial power-law magnetic spectrum, 
, and on the particular regime, with the order-one constants and 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 equal to zero.
© The Author(s), 2014
