Vacuum self similar anisotropic cosmologies in $F(R)-$gravity

Abstract: The implications from the existence of a proper Homothetic Vector Field (HVF) on the dynamics of vacuum anisotropic models in $F(R)$ gravitational theory are studied. The fact that \emph{every} Spatially Homogeneous vacuum model is equivalent, formally, with a "flux" -free anisotropic fluid model in standard gravity and the induced power-law form of the functional $F(R)$ due to self-similarity enable us to close the system of equations. We found some new exact anisotropic solutions that arise as fixed points in the associated dynamical system. The non-existence of Kasner-like (Bianchi type I) solutions in proper $F(R)-$gravity (i.e. $R\neq 0$) strengthens the belief that curvature corrections will prevent the shear influence into the past thus permitting an isotropic singularity. We also discuss certain issues regarding the lack of vacuum models of type III, IV, VII$_{h}$ in comparison with the corresponding results in standard gravity.