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 $F(R)$.
