Cosmological magnetic fields: generation during inflation and evolution

This paper concerns the generation and evolution of the cosmological (large-scale $\sim Mpc$) magnetic fields in an inflationary universe. The universe during inflation is represented by de Sitter space-time. We started with the Maxwell equations in spatially flat Friedmann-Robertson-Walker (FRW) Cosmologies. Then we calculated the wave equations of the magnetic field and electric field for the evolution. We consider the input current that was produced from a massless charged scalar complex field. This field minimally coupled to both gravity and the electromagnetic fields. The Lagrangian for massless scalar electrodynamics is then $L=\sqrt{-g}(D_\mu\phi(D^\mu\phi)^*-{1/4}F_{\mu\nu}F^{\mu\nu})$ . The complex scalar field couples to electromagnetism through the usual gauge covariant derivative $D_\mu=\partial_\mu-ieA_\mu$ . After the quantum field theoretical deduction for the current, we put it back into the wave equation of the magnetic field. After solving this wave equation, our result is $a^2B\sim \frac{eH}{\sqrt{2}k^2}\mid\sin\sqrt{2}k\eta\mid$ . At the time $\eta_{RH}$ we have $B_{RH}=\frac{e}{k_{phys}}$. This may imply that the breaking of the conformal invariance due to the minimal coupling of a massless charged scalar complex field to both gravitational and electromagnetic fields is not sufficient for the production of seed galactic magnetic fields during inflation. But since we are interested in the large-scale cosmological magnetic field, this could be still a candidate, because of the $1/k$ factor.