By introducing an auxilliary field, it is possible to rewrite the Einstein-Hilbert action as purely cubic vertices. Not only does this greatly simplify the form of the Feynman rules, it allows new tree-level off-shell recursion relations to be derived. There exists a gauge in which all graviton interactions are proportional to the kinetic term. This construction can also be carried out on general curved backgrounds.

Interestingly, this cubic description of gravity also exhibits the two-fold Lorentz symmetry which was previously discussed in 1612.03927, motivated by the KLT and BCJ double copy relations. The double copy uses a particular cubic form for the Yang-Mills side, and it is tantalizing that gravity can also be written as a purely cubic theory. However, the twofold Lorentz symmetry is not manifest in these gravity variables, and so it remains unclear how best to formulate the double copy at the level of the action.