Any modification to GR which is built entirely from diffeomorphism-preserving combinations of curvature invariants of the single metric can be written as GR plus a new matter sector. This is achieved by first identifying the gravitational dof present in the propagator, and then determining how these couple together: after which it is possible to `unmix' them from the gravity sector with a suitably chosen field redefinition.

For a large class of modified gravity theories, an invertible field redefinition can bring them into an Einstein frame, with the standard Einstein-Hilbert term of GR and a (potentially highly non-trivial) matter sector. This seems to indicate that one cannot meaningfully distinguish between these modified gravity effects and dark matter. A natural question is whether this extends to massive (bi)gravity theories, in which the graviton propagator is modified even at tree level, or there are multiple available curvature invariants.