Non-local gravity with a Weyl-square term

Recent work has shown that modifications of General Relativity based on the addition of a non-local term $R\,\Box^{-2}R$ produce a dynamical model of dark energy, which is cosmologically viable both at the background level and at the level of cosmological perturbations. We explore a more general class of models based on the addition of terms proportional to $R_{\mu\nu}\,\Box^{-2}R^{\mu\nu}$ and $C_{\mu\nu\rho\sigma}\, \Box^{-2}C^{\mu\nu\rho\sigma}$, where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. We find that the term $R_{\mu\nu}\,\Box^{-2}R^{\mu\nu}$ does not give a viable background evolution. The non-local Weyl-square term, in contrast, does not contribute to the background evolution, but we find that, at the level of cosmological perturbations, it gives instabilities in the tensor sector. This seems to select $R\,\Box^{-2}R$ as the only cosmologically viable quadratic non-local term in the action. We discuss how these results can provide a hint for the mechanism that generates effective non-local terms from a fundamental local theory.