Inflation and late-time cosmic acceleration in non-minimal   Maxwell-$F(R)$ gravity and the generation of large-scale magnetic fields
  We study inflation and late-time acceleration in the expansion of the universe in non-minimal electromagnetism, in which the electromagnetic field couples to the scalar curvature function. It is shown that power-law inflation can be realized due to the non-minimal gravitational coupling of the electromagnetic field, and that large-scale magnetic fields can be generated due to the breaking of the conformal invariance of the electromagnetic field through its non-minimal gravitational coupling. Furthermore, it is demonstrated that both inflation and the late-time acceleration of the universe can be realized in a modified Maxwell-$F(R)$ gravity which is consistent with solar system tests and cosmological bounds and free of instabilities. At small curvature typical for current universe the standard Maxwell theory is recovered. We also consider classically equivalent form of non-minimal Maxwell-$F(R)$ gravity, and propose the origin of the non-minimal gravitational coupling function based on renormalization-group considerations. 
Kazuharu Bamba and Sergei D. Odintsov