Interaction of inhomogeneous axions with magnetic fields in the early
  universe
  We study the system of interacting axions and magnetic fields in the early
universe after the quantum chromodynamics phase transition, when axions acquire
masses. Both axions and magnetic fields are supposed to be spatially
inhomogeneous. We derive the equations for the spatial spectra of these fields,
which depend on conformal time. In case of the magnetic field, we deal with the
spectra of the energy density and the magnetic helicity density. The evolution
equations are obtained in the closed form within the mean field approximation.
We choose the parameters of the system and the initial condition which
correspond to realistic primordial magnetic fields and axions. The system of
equations for the spectra is solved numerically. We compare the cases of
inhomogeneous and homogeneous axions. The evolution of the magnetic field in
these cases is different only within small time intervals. Generally, magnetic
fields are driven mainly by the magnetic diffusion. We find that the magnetic
field instability takes place for the amplified initial wavefunction of the
homogeneous axion. This instability is suppressed if we account for the
inhomogeneity of the axion.
