High order tail-terms of the Hadamard Green's function

One of the key sources of gravitational waves detected by LISA is expected to be extreme mass ratio inspirals (EMRIs). The mass ratio of such binary systems provides an excellent small parameter for perturbation theory. This allows one to linearise the Einstein field equations by expanding the full metric as a background metric of the primary, plus corrections in powers of the mass ratio. The linearised Einstein field equations describe a wave equation in the Lorenz gauge. The Green's function of the wave equation plays a key role in determining the motion of the secondary. Due to the nature of backscattering in curved spacetimes, the Green's function is not determined solely on the light cone but is instead governed by an integral over the entire past history of the worldline. This is known as the tail term. The Green's function can be expanded analytically near the worldline in a quasi-local Hadamard expansion. The expansion is given in terms of bi-tensors, which are defined in a neighbourhood near a given position x. Further covariant Taylor series expansions of these objects allow one to determine analytically the tail term’s contributions for early times near the worldline. Canonicalised higher-order tail terms for both the electromagnetic and gravitational cases have been calculated using a Mathematica package, “TInvar,” developed for canonicalising Riemann polynomials in the Fulling–King–Wybourne–Cummins (FKWC) basis.