Structure formation in the new Deser-Woodard nonlocal gravity model

We consider the structure formation in the nonlocal gravity model proposed recently by Deser and Woodard (DW-2019 model), which can not only reproduce the $\Lambda$CDM cosmology without fine-tuning puzzle but also may provide a screening mechanism for free. By using the direct numerical method of the reconstructing 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 with the fitted function proposed by Deser and Woodard. Based on the numerical results, we plotted the curve of the growth rate $f\sigma_8(z)$ under the DW-2019 model, which shows this model is not ruled out by the $f\sigma_8$ data from the Redshift-space distortions measurements. The evolving curve of the growth rate has a distinct plummet at $z\simeq0.39$, which is a great difference with the DW model. The possible reason is that the DW-2019 model provides the strong nonlocal effect, which behaves as the anisotropic effect to influence the matter perturbation in the low-redshift range. Finally the qualitative analysis of the screening mechanism is discussed by considering the spacial dependence of the nonlocal modification in the small-scale range.