Nearly 7 years in the making, we have finally finished our first paper describing our second-order gravitational self-force results:

Second-order self-force calculation of gravitational binding energy

Adam Pound, Barry Wardell, Niels Warburton, Jeremy MillerSelf-force theory is the leading method of modeling extreme-mass-ratio inspirals (EMRIs), key sources for the gravitational-wave detector LISA. It is well known that for an accurate EMRI model, second-order self-force effects are critical, but calculations of these effects have been beset by obstacles. In this letter we present the first implementation of a complete scheme for second-order self-force computations, specialized to the case of quasicircular orbits about a Schwarzschild black hole. As a demonstration, we calculate the gravitational binding energy of these binaries.

Check out our paper on arXiv.