Ghost-free non-local $F(R)$ Gravity Cosmology
  In this work we shall study ghost-free non-local $F(R)$ gravity models.
Firstly we shall demonstrate how the ghost degrees of freedom may occur in the
non-local $F(R)$ gravity models, and accordingly we shall modify appropriately
the gravitational action of non-local $F(R)$ gravity models in order to
eliminate the ghosts. Also we shall investigate how the (anti-)de Sitter and
the Minkowski spacetime cosmological solutions may arise in the theory, and we
investigate when these solutions are stable. Moreover, we shall examine the
inflationary phenomenology of the Jordan frame ghost-free non-local $F(R)$
gravity. We shall study two $F(R)$ gravity models, the power law $F(R)$ gravity
model $\sim R^n$ with $1<n<2$, $n\neq 2$ and the $R^2$ model, assuming that the
slow-roll condition holds true for the Hubble rate during the inflationary era
$\dot{H}\ll H^2$ and that the general constant-roll condition
$\ddot{\phi}=3\beta H\dot{\phi}$ holds true for the evolution of the scalar
field, which includes the slow-roll case for $\beta=0$. As we shall
demonstrate, the power-law non-local $F(R)$ gravity case can produce a viable
inflationary era, compatible with observations, in the constant-roll case,
unlike for the $R^2$ model. It is conceivable that the results are model
dependent, as in the ordinary vacuum $F(R)$ gravity.
