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 .