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

TL;DR

This paper clarifies the discrepancies in growth of perturbations in the Deser-Woodard nonlocal gravity model by comparing original nonlocal and localized formulations, showing they are consistent when initial conditions are aligned, and correcting previous numerical errors.

Contribution

It demonstrates that the nonlocal and localized formulations produce consistent perturbation solutions and resolves conflicting results by identifying and fixing a numerical error in earlier work.

Findings

Nonlocal and localized models yield the same perturbation solutions with identical initial conditions.

Different sub-horizon limit implementations cause transient differences but not in matter growth.

Correcting the numerical code shows suppressed growth in the nonlocal model as previously reported.

Abstract

We examine the origin of two opposite results for the growth of perturbations in the Deser-Woodard (DW) nonlocal gravity model. One group previously analyzed the model in its original nonlocal form and showed that the growth of structure in the DW model is enhanced compared to general relativity (GR) and thus concluded that the model was ruled out. Recently, however, another group has reanalyzed it by localizing the model and found that the growth in their localized version is suppressed even compared to the one in GR. The question was whether the discrepancy originates from an intrinsic difference between the nonlocal and localized formulations or is due to their different implementations of the sub-horizon limit. We show that the nonlocal and local formulations give the same solutions for the linear perturbations as long as the initial conditions are set the same. The different implementations of the sub-horizon limit lead to different transient behaviors of some perturbation variables; however, they do not affect the growth of matter perturbations at the sub-horizon scale much. In the meantime, we also report an error in the numerical calculation code of the former group and verify that after fixing the error the nonlocal version also gives the suppressed growth. Finally, we discuss two alternative definitions of the effective gravitational constant taken by the two groups and some open problems.