Non-local gravity with a Weyl-square term
  Recent work has shown that modifications of General Relativity based on the
addition to the action of a non-local term $R\,\Box^{-2}R$, or on the addition
to the equations of motion of a term involving $(g_{\mu\nu}\Box^{-1} R)$,
produce dynamical models of dark energy which are 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 to the action 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. Thus,
only non-local terms which depend just on the Ricci scalar $R$ appear to be
cosmologically viable. We discuss how these results can provide a hint for the
mechanism that might generate these effective non-local terms from a
fundamental local theory.
