Accelerating cosmology in modified gravity: from convenient $F(R)$ or
  string-inspired theory to bimetric $F(R)$ gravity
  We consider modified gravity which may describe the early-time inflation
and/or late-time cosmic acceleration of the universe. In particular, we discuss
the properties of $F(R)$, $F(G)$, string-inspired and
scalar-Einstein-Gauss-Bonnet gravities, including their FRW equations and fluid
or scalar-tensor description. Simplest accelerating cosmologies are
investigated and possibility of unified description of the inflation with dark
energy is described. The cosmological reconstruction program which permits to
get the requested universe evolution from modified gravity is developed. As
some extension, massive $F(R)$ bigravity which is ghost-free theory is
presented. Its scalar-tensor form turns out to be the easiest formulation. The
cosmological reconstruction method for such bigravity is presented. The unified
description of inflation with dark energy in $F(R)$ bigravity turns out to be
possible.
