Towards a purely gravitational effective theory of dark matter

We stress that there can be no doubt about this model reproducing all the successes of CDM in cosmology. Those successes include the anisotropies of the cosmic microwave background, baryon acoustic oscillations, and linearized structure formation. All of this must come out right because this model was constructed by using the separate conservation of the CDM stress tensor to express it as a nonlocal functional of the metric. The only way this model can be falsified is by showing that CDM interacts with fields other than gravity, the evidence for which is weak. This model should put to rest the frequent claims that no modified gravity theory can supplant dark matter. It also demonstrates that the answer to the question of Pardo and Spergel about the price of abandoning dark matter: the price is nonlocality...

One thing our model does not do is to agree with the nonlocal extension of MOND [this refers to earlier work by the authors] which was constructed to reproduce the baryonic Tully-Fisher relation, with sufficient weak lensing, in gravitationally bound structure... It would be desirable to have a single formalism which connects both regimes.

It would also be desirable to derive these nonlocal models from fundamental theory. We believe they might arise from resumming the secular logarithms that are induced by loops of inflationary gravitons and which must eventually become nonperturbatively large during a prolonged period of inflation

mitchell porter said:

TL;DR Summary: Constructing a purely gravitational, nonlocal action that reproduces CDM cosmology

In a nutshell, as explained after equation 34, by adding a nonlocal self-interaction to the metric, they can reproduce the effects normally attributed to CDM.