  The evolution of the gauge coupling is modelled using a scalar degree of
freedom whose homogeneous and inhomogeneous modes evolve from an ordinary
inflationary phase to the subsequent epochs. Depending upon various parameters
(scalar mass, curvature scale at the end of inflation and at the onset of the
radiation epoch), the two-point function of the magnetic inhomogeneities grows
during the de Sitter stage of expansion and, consequently, large scale magnetic
fields are generated. The requirements coming from inflationary magnetogenesis
are examined together with the theoretical constraints stemming from the
explicit model of the evolution of the gauge coupling. Galactic magnetogenesis
is possible for a wide range of parameters. Inter-galactic and inter-cluster
magnetogenesis are also discussed.
