strong coupling regime. The unwanted scalar mode might then be screened, and a different perturbative expansion at short distances might more closely approximate general relativity (see [@Hinterbichler] for a review). This mechanism has been confirmed for particular models (e.g., [@Babichev; @Kaloper; @DeFelice]), and with a very special choice of action, the Boulware-Deser ghosts may also be banished [@Hassan; @deRham; @RRosen]. But if the Vainshtein mechanism applies, it poses a new challenge: since the usual weak field approximation can no longer be trusted, one must work hard to find even a Newtonian limit for emergent gravity.

To a certain extent, this argument can be turned on its head: if one is certain that a massless, Lorentz invariant, purely spin two field has emerged with no lower spin partners, this strongly suggests the presence of diffeomorphism invariance or a similar symmetry. A symmetric rank two tensor field $h_{ab}$ contains components of spin zero, one, and two, and the only known Lorentz covariant way to project out the lower spins is with a gauge invariance. At linear order, the minimal requirement is invariance under “transverse diffeomorphisms,” diffeomorphisms generated by those vector fields $\xi^a$ for which $\partial_a\xi^a=0$ [@Bij; @Alvarez; @Blas]. This group may be 