Primordial Magnetic Fields from Out of Equilibrium Cosmological Phase Transitions

11 pages, no figures. Lecture given at the International Conference Magnetic Fields in the Universe, Angra dos Reis, Brazil, November, 2004The 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