We demonstrate how to construct purely metric modifications of gravity which agree with general relativity in the weak field regime appropriate to the Solar System, but which possess an ultra-weak field regime when the gravitational acceleration becomes comparable to $a_0\sim {10}^{-10}\mathrm{m}/{\mathrm{s}}^2$. In this ultra-weak field regime, the models reproduce the MOND force without dark matter and also give enough gravitational lensing to be consistent with existing data. Our models are nonlocal and might conceivably derive from quantum corrections to the effective field equations.

• Received 24 June 2011

DOI:https://doi.org/10.1103/PhysRevD.84.124054