Inflationary magnetogenesis of primordial magnetic fields with multiple vector fields

In this paper, we discussed the multiple vector fields during the inflation era and the inflationary magnetogenesis with multiple vector fields. Instead of a single coupling function in single vector field models, the coupling matrix between vector fields and scalar field which drive the inflation is introduced. The dynamical equations for multiple vector fields are obtained and applied to the inflation era. We discussed three cases for the double-field model. In no mutual-coupling case, one can find that both electric and magnetic spectrum can be scale-invariant at the end of inflation, meanwhile, the strong coupling problem can be avoided. The effect of mutual-coupling between different vector fields is also discussed. We found that weak mutual-coupling can lead to the slightly blue spectrum of the magnetic field. On the other hand, in the strong mutual-coupling case, the scale-invariant magnetic spectrum can also be obtained but the energy density of electromagnetic fields either lead to the backreaction problem or is diluted by inflation.