Quantum approach to F(R) gravity with f-essence

In recent years, due to the rapid development of astrophysics and the capabilities of modern technologies, a huge amount of astronomical data such as type Ia supernova, cosmic microwave background anisotropy (CMB), acoustic baryon oscillations, weak lensing and large-scale structure have been obtained. In this regard, a fundamental discovery was made, which showed that our Universe is expanding with acceleration. To describe this phenomenon, many models have been created. One of the main directions is the modification of the general theory of relativity, in particular, $F(R)$ gravity, which is a kind of functional dependence on the curvature of space $R$ in a scalar. Within this modification, the evolution of various fields, such as scalar or fermion, have been investigated. Then, I talk a modified $F(R) $model of gravity with a fermion field $f$-essence in the quantum approach. The equations of motion of this model are obtained for the homogeneous and isotropic Friedman-Robertson-Walker space-time. To determine the parameters of the connection of matter and space-time, we consider the standard model of the accelerated expansion of the de Sitter Universe.