Hyperboloidal method for frequency-domain self-force calculations

Gravitational self-force theory is the leading approach for modelling gravitational wave emission from small mass-ratio compact binaries. This method perturbatively expands the metric of the binary in powers of the mass ratio. The source for the perturbations depends on the orbital configuration, calculational approach, and/or the order the perturbative expansion is carried too. These sources fall into three broad classes: distributional, extended and compact, and non-compact. The latter, in particular, is important for emerging second-order in the mass ratio calculations. Traditional frequency domain approaches employ the variation of parameters method and compute the perturbation on constant time slices of the spacetime with numerical boundary conditions supplied at finite radius from series expansions of the asymptotic behaviour. This approach has been very successful but the boundary conditions calculations are tedious and the approach is not well suited to non-compact sources where homogeneous solutions must be computed at all radii.  In this talk, I outline an alternative approach where the spacetime is foliated by horizon-penetrating hyperboloidal slices. Further compactifying the coordinates along these slices allows for simple treatment of the boundary conditions. We implement this approach with a multi-domain spectral solver with analytic mesh refinement and present results for the scalar-field self-force on circular orbits as an example problem. We find the method works efficiently for all three classes of sources encountered in self-force calculations and has some distinct advantages over the traditional approach.