Orbital eccentricity in general relativity from catastrophe theory

Abstract: While the orbital eccentricity is a key feature of the gravitational two-body problem, providing an unambiguous definition in general relativity poses significant challenges. Despite such foundational issue, the eccentricity of binary black holes has important implications in gravitational-wave astronomy. We present a novel approach to consistently define the orbital eccentricity in general relativity, grounded in the mathematical field of catastrophe theory. Specifically, we identify the presence of catastrophes, i.e., breakdowns of the stationary-phase approximation, in numerical relativity waveforms and exploit them to develop a robust and fully gauge-invariant estimator of the eccentricity. Our procedure does not require orbital fitting and naturally satisfies the Newtonian limit. The proposed eccentricity estimator agrees with and generalizes a previous proposal, though with a fully independent derivation. We extract gauge-free eccentricity estimates from about 100 numerical relativity simulations and find that the resulting values are systematically lower compared to those reported alongside the simulations themselves.