Mimetic F(R) gravity: Inflation, dark energy and bounce

We propose the mimetic F(R) theory and investigate the early-time and late-time acceleration in such theory. It is demonstrated that inflation consistent with observable data may be realized in such theory. The reconstruction of realistic.CDM era is also possible as well as the unification of early-time inflation with late-time acceleration or bounce universe. It is stressed that specific universe evolution is governed by mimetic F(R) theory which is different from convenient F(R) gravity. The corresponding examples are presented. Mimetic F(R) gravity is generalized by the addition of the scalar potential in the formulation of convenient F(R) gravity with specific Lagrange multiplier constraint. It is demonstrated that such theory may admit the arbitrary universe evolution via the corresponding choice of the scalar potential and/or function F(R).