Narrow Results

We consider the generation and evolution of magnetic field in a primordial plasma at temperature $T\le 1$MeV in the presence of asymmetric neutrino background, i.e. the number densities of right-handed and left-handed neutrinos are not the same. Semi-classical equations of motion of a charged fermion are derived using the effective low-energy Lagrangian. We show that the spin degree of freedom of the charged fermion couples with the neutrino background. Using this kinetic equation, we study the collective modes of the plasma. We find that there exist an unstable mode. This instability is closely related with the instability induced by chiral-anomaly in high temperature $T\ge 80$TeV plasma where right and left-handed electrons are out of equilibrium. We find that at the temperatures below the neutrino decoupling this instability can produce magnetic field in the universe. We discuss cosmological implications of the results.