  We describe an infinite-parametric class of effective metric Lagrangians that
arise from an underlying theory with two propagating degrees of freedom. The
Lagrangians start with the Einstein-Hilbert term, continue with the standard
R^2, (Ricci)^2 terms, and in the next order contain (Riemann)^3 as well as
on-shell vanishing terms. This is exactly the structure of the effective metric
Lagrangian that renormalizes quantum gravity divergences at two-loops. This
shows that the theory underlying the effective field theory of gravity may have
no more degrees of freedom than is already contained in general relativity. We
show that the reason why an effective metric theory may describe just two
propagating degrees of freedom is that there exists a (non-local) field
redefinition that maps an infinitely complicated effective metric Lagrangian to
the usual Einstein-Hilbert one. We describe this map for our class of theories
and, in particular, exhibit it explicitly for the (Riemann)^3 term.
