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 $\Lambda$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 $F(R)$.