Singular Inflationary Universe from $F(R)$ Gravity
  Unlike crushing singularities, the so-called Type IV finite-time singularity
offers the possibility that the Universe passes smoothly through it, without
any catastrophic effects. Then the question is if the effects of a Type IV
singularity can be detected in the process of cosmic evolution. In this paper
we address this question in the context of $F(R)$ gravity. As we demonstrate,
the effects of a Type IV singularity appear in the Hubble flow parameters,
which determine the dynamical evolution of the cosmological system. So we study
various inflation models incorporating a Type IV singularity, with the
singularity occurring at the end of inflation. Particularly we study a toy
model and a singular version of the $R^2$ gravity Hubble rate. As we evince,
some of the Hubble flow parameters become singular at the singularity, an
effect which indicates that at that point a dynamical instability occurs. This
dynamical instability eventually indicates the graceful exit from inflation. We
demonstrate that the toy model has an unstable de Sitter point at the
singularity, so indeed graceful exit could be triggered. In the case of the
singular inflation model, graceful exit proceeds in the standard way. In the
case of the singular inflation model, we found various scenarios for singular
evolution, most of which are compatible with observations, and only one leads
to severe instabilities. We also compare the ordinary Starobinsky with the
singular inflation model, and we point out the qualitative and quantitative
differences. Finally, we study the late-time dynamics of the toy model and of
the singular inflation model and we demonstrate that the unification of early
and late-time acceleration can be achieved. We also show that it is possible to
achieve late-time acceleration similar to the $\Lambda$-Cold Dark Matter model.
