Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov

We use the loop-by-loop Baikov representation to investigate the geometries in Feynman integrals contributing to the classical dynamics of a black-hole two-body system in the post-Minkowskian expansion of general relativity. These geometries determine the spaces of functions to which the corresponding Feynman diagrams evaluate. As a proof of principle, we provide a full classification of the geometries appearing up to three loops, i.e. fourth post-Minkowskian order, for all diagrams relevant to the conservative as well as the dissipative dynamics, finding full agreement with the literature. Moreover, we show that the non-planar top topology at four loops, which is the most complicated sector with respect to integration-by-parts identities, has an algebraic leading singularity and thus can only depend on non-trivial geometries through its subsectors.