Phantom and non-phantom dark energy: The cosmological relevance of non-locally corrected gravity

In this paper we have investigated the cosmological dynamics of non-locally corrected gravity involving a function of the inverse d’Alembertian of the Ricci scalar, $f({\mathrm{\square }}^{-1}R))$. Casting the dynamical equations into local form, we derive the fixed points of the dynamics and demonstrate the existence and stability of a one parameter family of dark energy solutions for a simple choice, $f({\mathrm{\square }}^{-1}R)\sim \exp (\alpha {\mathrm{\square }}^{-1}R)$. The effective EoS parameter is given by, $w_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\alpha -1)\mathrm{/}(3\alpha -1)$ and the stability of the solutions is guaranteed provided that $1\mathrm{/}3<\alpha <2\mathrm{/}3$. For $1\mathrm{/}3<\alpha <1\mathrm{/}2$ and $1\mathrm{/}2<\alpha <2\mathrm{/}3$, the underlying system exhibits phantom and non-phantom behavior respectively; the de Sitter solution corresponds to $\alpha =1\mathrm{/}2$. For a wide range of initial conditions, the system mimics dust like behavior before reaching the stable fixed point. The late time phantom phase is achieved without involving negative kinetic energy fields. A brief discussion on the entropy of de Sitter space in non-local model is included.