Conformal symmetry and nonlinear extensions of nonlocal gravity
  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
$w_{\rm DE}\simeq -1.01$, 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.
