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We examine the origin of two opposite results for the growth of perturbationsin the Deser-Woodard (DW) nonlocal gravity model. One group previously analyzedthe model in its original nonlocal form and showed that the growth of structurein the DW model is enhanced compared to general relativity (GR) and thusconcluded that the model was ruled out. Recently, however, another group hasreanalyzed it by localizing the model and found that the growth in theirlocalized version is suppressed even compared to the one in GR. The questionwas whether the discrepancy originates from an intrinsic difference between thenonlocal and localized formulations or is due to their differentimplementations of the sub-horizon limit. We show that the nonlocal and localformulations give the same solutions for the linear perturbations as long asthe initial conditions are set the same. The different implementations of thesub-horizon limit lead to different transient behaviors of some perturbationvariables; however, they do not affect the growth of matter perturbations atthe sub-horizon scale much. In the meantime, we also report an error in thenumerical calculation code of the former group and verify that after fixing theerror the nonlocal version also gives the suppressed growth. Finally, wediscuss two alternative definitions of the effective gravitational constanttaken by the two groups and some open problems.

Revival of the Deser-Woodard nonlocal gravity model: Comparison of the original nonlocal form and a localized formulation