matter to gravity, much less to compare couplings of more than one species of matter, the problem remains almost completely open.

Self-coupling \[self\]
----------------------

One aspect of universal coupling deserves special attention: we observe gravity’s coupling to its own energy to occur at the same universal strength as its coupling to matter [@Will; @LLR]. This self-coupling implies that the interaction is nonlinear, and, in fact, it can be used to determine the nonlinear terms, giving another route to the Einstein field equations [@Feynman; @Kraichnan; @Deser; @Deser2].

This property places requirements on emergent models beyond the linear approximation. Obtaining the correct linear behavior—a massless spin two excitation, even with the correct coupling to matter—is not sufficient to show that one has a model with gravity. Indeed, there are known examples (e.g., [@Zhang; @Wen]) in which the correct nonlinear behavior seems to require a good deal of fine tuning.

If one can obtain Lorentz invariance and diffeomorphism invariance, however, these provide some very helpful constraints. As Kraichnan first showed [@Kraichnan], if one starts with a massless spin two field $h_{ab}$ on a manifold with a flat metric $\eta_{ab}$ and assumes that its field equations can be derived from a Lorentz invariant, diffeomorphism 