Conformal and Einstein gravity from twistor actions

We use the embedding of Einstein gravity with cosmological constant into conformal gravity as a basis for using the twistor action for conformal gravity to obtain MHV scattering amplitudes not just for conformal gravity, but also for Einstein gravity on backgrounds with non-zero cosmological constant. The new formulae for the gravitational MHV amplitude with cosmological constant arise by summing Feynman diagrams using the matrix-tree theorem. We show that this formula is well-defined (i.e., is independent of certain gauge choices) and that it non-trivially reproduces Hodges' formula for the MHV amplitude in the flat-space limit. We give a preliminary discussion of a MHV formalism for more general amplitudes obtained from the conformal gravity twistor action in an axial gauge. We also see that the embedding of Einstein data into the conformal gravity action can be performed off-shell in twistor space to give a proposal for an Einstein twistor action that automatically gives the same MHV amplitude. These ideas extend naturally to N=4 supersymmetry.