  The universe cools down monotonically following its expansion.This generates
a sequence of phase transitions. If a second order phase transition happens
during the radiation dominated era with a charged order parameter, spinodal
unstabilities generate large numbers of charged particles. These particles
hence produce magnetic fields.We use out of equilibrium field theory methods to
study the dynamics in a mean field or large N setup.The dynamics after the
transition features two distinct stages: a spinodal regime dominated by linear
long wave length instabilities, and a scaling stage in which the
non-linearities and backreaction of the scalar fields are dominant. This second
stage describes the growth of horizon sized domains. We implement a formulation
based on the non equilibrium Schwinger-Dyson equations to obtain the spectrum
of magnetic fields that includes the dissipative effects of the plasma. We find
that large scale magnetogenesis is efficient during the scaling regime. Charged
scalar field fluctuations with wavelengths of the order of the Hubble radius
induce large scale magnetogenesis via loop effects.The leading processes
are:pair production, pair annihilation and low energy bremsstrahlung, these
processes while forbidden in equilibrium are allowed strongly out of
equilibrium. The ratio between the energy density on scales larger than L and
that in the background radiation r(L,T)= rho_B(L,T)/rho_{cmb}(T) is r(L,T) ~
10^{-34} at the Electroweak scale and r(L,T) ~ 10^{-14} at the QCD scale for L
sim 1 Mpc. The resulting spectrum is insensitive to the magnetic diffusion
length and equipartition between electric and magnetic fields does not hold. We
conjecture that a similar mechanism could be operative after the QCD chiral
phase transition.
