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We consider the structure formation in the nonlocal gravity model proposedrecently by Deser and Woodard (DW-2019 model), which can not only reproduce the$\Lambda$CDM cosmology without fine-tuning puzzle but also may provide ascreening mechanism for free. By using the direct numerical method of thereconstructing technique, the nonlocal distortion function $f(Y)$ is fixed as$f(Y)\simeq\boldsymbol{e}^{2.153(Y-16.97)}$ which has a small deviation withthe fitted function proposed by Deser and Woodard. Based on the numericalresults, we plotted the curve of the growth rate $f\sigma_8(z)$ under theDW-2019 model, which shows this model is not ruled out by the $f\sigma_8$ datafrom the Redshift-space distortions measurements. The evolving curve of thegrowth rate has a distinct plummet at $z\simeq0.39$, which is a greatdifference with the DW model. The possible reason is that the DW-2019 modelprovides the strong nonlocal effect, which behaves as the anisotropic effect toinfluence the matter perturbation in the low-redshift range. Finally thequalitative analysis of the screening mechanism is discussed by considering thespacial dependence of the nonlocal modification in the small-scale range.

Structure formation in the new Deser-Woodard nonlocal gravity model