Non-local gravity with a Weyl-square term

TL;DR

This paper investigates non-local modifications of gravity involving the Weyl tensor, finding that such terms lead to instabilities in cosmological perturbations, and concludes that viable models should depend only on the Ricci scalar.

Contribution

It introduces and analyzes a class of non-local gravity models with Weyl-square terms, showing their limitations and viability compared to Ricci scalar-based models.

Findings

Weyl-square non-local term causes tensor perturbation instabilities.

Ricci scalar-based non-local models are cosmologically viable.

Non-local Weyl-square term does not affect background evolution.

Abstract

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.