Singular $F(R)$ Cosmology Unifying Early and Late-time Acceleration with
  Matter and Radiation Domination Era
  We present some cosmological models which unify the late and early-time
acceleration eras with the radiation and the matter domination era, and we
realize the cosmological models by using the theoretical framework of $F(R)$
gravity. Particularly, the first model unifies the late and early-time
acceleration with the matter domination era, and the second model unifies all
the evolution eras of our Universe. The two models are described in the same
way at early and late times, and only the intermediate stages of the evolution
have some differences. Each cosmological model contains two Type IV
singularities which are chosen to occur one at the end of the inflationary era
and one at the end of the matter domination era. The cosmological models at
early times are approximately identical to the $R^2$ inflation model, so these
describe a slow-roll inflationary era which ends when the slow-roll parameters
become of order one. The inflationary era is followed by the radiation era and
after that the matter domination era follows, which lasts until the second Type
IV singularity, and then the late-time acceleration era follows. The models
produce a nearly scale invariant power spectrum of primordial curvature
perturbations and a scalar-to-tensor ratio which are compatible with the most
recent observational data and it seems that the deceleration acceleration
transition is crucially affected by the presence of the second Type IV
singularity which occurs at the end of the matter domination era. We perform an
analysis of the Hubble horizon evolution, and in addition, we investigate which
$F(R)$ gravity can successfully realize each of the four cosmological epochs.
