Evolution of Primordial Magnetic Fields from Phase Transitions
  We consider the evolution of primordial magnetic fields generated during
cosmological, electroweak or QCD, phase transitions. We assume that the
magnetic field generation can be described as an injection of magnetic energy
to cosmological plasma at a given scale determined by the moment of magnetic
field generation. A high Reynolds number ensures strong coupling between
magnetic field and fluid motions. The subsequent evolution of the magnetic
field is governed by decaying hydromagnetic turbulence. Both our numerical
simulations and a phenomenological description allow us to recover "universal"
laws for the decay of magnetic energy and the growth of magnetic correlation
length in the turbulent (low viscosity) regime. In particular, we show that
during the radiation dominated epoch, energy and correlation length of
non-helical magnetic fields scale as conformal time to the powers -1/2 and
+1/2, respectively. For helical magnetic fields, energy and correlation length
scale as conformal time to the powers -1/3 and +2/3, respectively. The
universal decay law of the magnetic field implies that the strength of magnetic
field generated during the QCD phase transition could reach $\sim 10^{-9}$\,G
with the present day correlation length $\sim 50$ kpc. The fields generated at
the electroweak phase transition could be as strong as $\sim 10^{-10}$ G with
correlation lengths reaching $\sim 0.3$\,kpc. These values of the magnetic
fields are consistent with the lower bounds of the extragalactic magnetic
fields.
