Anisotropic inflation with a non-minimally coupled electromagnetic field
  to gravity
  We consider the non-minimal model of gravity in $Y(R) F^2$-form. We
investigate a particular case of the model, for which the higher order
derivatives are eliminated but the scalar curvature $R$ is kept to be dynamical
via the constraint $Y_RF_{mn}F^{mn} =-\frac{2}{\kappa^2}$. The effective fluid
obtained can be represented by interacting electromagnetic field and vacuum
depending on $Y(R)$, namely, the energy density of the vacuum tracks $R$ while
energy density of the conventional electromagnetic field is dynamically scaled
with the factor $\frac{Y(R)}{2}$. We give exact solutions for anisotropic
inflation by assuming the volume scale factor of the Universe exhibits a
power-law expansion. The directional scale factors do not necessarily exhibit
power-law expansion, which would give rise to a constant expansion anisotropy,
but expand non-trivially and give rise to a non-monotonically evolving
expansion anisotropy that eventually converges to a non-zero constant. Relying
on this fact, we discuss the anisotropic e-fold during the inflation by
considering observed scale invariance in CMB and demanding the Universe to
undergo the same amount of e-folds in all directions. We calculate the residual
expansion anisotropy at the end of inflation, though as a result of
non-monotonic behaviour of expansion anisotropy all the axes of the Universe
undergo the same of amount of e-folds by the end of inflation. We also discuss
the generation of the modified electromagnetic field during the first few
e-folds of the inflation and its persistence against to the vacuum till end of
inflation.
