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.