  We consider a theory of scalars minimally coupled to an auxiliary background
metric. The theory is generally covariant and subject to the constraint of
vanishing energy-momentum tensor. Eliminating the auxiliary metric leads to a
reparametrization invariant, non-polynomial, metric-independent action for the
scalar fields. Working in the limit of a large number of physical scalars, a
composite massless spin-2 state, the graviton, was identified in previous work,
in a two-into-two scalar scattering process. Here, we further explore the
possibility that dynamical emergent gravity is a natural feature of generally
covariant quantum field theories, by studying the self-interactions of the
emergent composite graviton. We show that the fine-tuning previously imposed to
ensure the vanishing of the cosmological constant, as well as the existence of
the massless spin-2 state, also assures that the emergent graviton's cubic
self-interactions are consistent with those of Einstein's general relativity,
up to higher-derivative corrections. We also demonstrate in a theory with more
than one type of scalar that the composite graviton coupling is universal.
