  We derive new representations of the Einstein-Hilbert action in which
graviton perturbation theory is immensely simplified. To accomplish this, we
recast the Einstein-Hilbert action as a theory of purely cubic interactions
among gravitons and a single auxiliary field. The corresponding equations of
motion are the Einstein field equations rewritten as two coupled first-order
differential equations. Since all Feynman diagrams are cubic, we are able to
derive new off-shell recursion relations for tree-level graviton scattering
amplitudes. With a judicious choice of gauge fixing, we then construct an
especially compact form for the Einstein-Hilbert action in which all graviton
interactions are simply proportional to the graviton kinetic term. Our results
apply to graviton perturbations about an arbitrary curved background spacetime.
