Effective metric Lagrangians from an underlying theory with two propagating degrees of freedom

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 $R^2$, $(\mathrm{\text{Ricci}}{)}^2$ terms, and in the next order contain $(\mathrm{\text{Riemann}}{)}^3$ 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 (nonlocal) 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 $(\mathrm{\text{Riemann}}{)}^3$ term.