We study two nonlinear extensions of the nonlocal R \Box ^ { -2 } R gravity theory .
We extend this theory in two different ways suggested by conformal symmetry , either replacing \Box ^ { -2 } with ( - \Box + R / 6 ) ^ { -2 } , which is the operator that enters the action for a conformally-coupled scalar field , or replacing \Box ^ { -2 } with the inverse of the Paneitz operator , which is a four-derivative operator that enters in the effective action induced by the conformal anomaly .
We show that the former modification gives an interesting and viable cosmological model , with a dark energy equation of state today w _ { DE } \simeq - 1.01 , which very closely mimics \Lambda CDM and evolves asymptotically into a de Sitter solution .
The model based on the Paneitz operator seems instead excluded by the comparison with observations .
We also review some issues about the causality of nonlocal theories , and we point out that these nonlocal models can be modified so to nicely interpolate between Starobinski inflation in the primordial universe and accelerated expansion in the recent epoch .