Abstract
We study the accelerating cosmology in massive F ( R ) bigravity via the reconstruction scheme. The consistent solution of the FRW equations is presented: it includes Big and Little Rip, quintessence, de Sitter and decelerating universes described by the physical g metric while the corresponding solution of the universe described by the reference f metric is also found. It is demonstrated that in general the cosmological singularities of g metric are not always manifested as cosmological singularities of the reference f metric. We give two consistent ways to describe the Big and Little Rip, quintessence, de Sitter and decelerating universes. In one of the consistent solutions, the two metrics g and f coincide with each other, which may indicate the connection with the convenient single metric background formulation. For another solution, where two metrics g and f do not coincide with each other, there always appears super-luminal mode.