Structure formation in the Deser-Woodard nonlocal gravity model: a reappraisal

In this work, we extend previous analyses of the structure formation in the $f(\Box^{-1}R)$ model of nonlocal gravity proposed by Deser and Woodard (DW), which reproduces the background expansion of $\Lambda$CDM with no need of a cosmological constant nor of any dimensional constant beside Newton's one. A previous analysis based on redshift-space distortions (RSD) data concluded that the model was ruled out. In this work we revisit the issue and find that, when recast in a localized model, the DW model is not ruled out and actually gives a better fit to RSD data than $\Lambda$CDM. At the same time, the model predicts a slightly lower value of $\sigma_{8}$ than $\Lambda$CDM, in agreement with recent estimates based on lensing. We also produce analytical approximations of the two modified gravity functions and of $f\sigma_{8}(z)$ as a function of redshift. Finally, we also show how much the fit depends on initial conditions when these are generalized with respect to a standard matter-dominated era.