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Near-future, space-based, radio- and gravitational-wave interferometrymissions will enable us to rigorously test whether the Kerr solution of generalrelativity accurately describes astrophysical black holes, or if it requiressome kind of modification. At the same time, recent work has greatly improvedour understanding of theories of gravity that modify the Einstein-Hilbertaction with terms quadratic in the curvature, allowing us to calculate blackhole solutions to (essentially) arbitrary order in a slow-rotation expansion.Observational constraints of such quadratic gravity theories require thecalculation of observables that are robust against the expansion order of theblack hole solution used.  We carry out such a study here and determine the accuracy with respect toexpansion order of ten observables associated with the spacetime outside arotating black hole in two quadratic theories of gravity,dynamical-Chern-Simons and scalar-Gauss-Bonnet gravity. We find that for allbut the most rapidly rotating black holes, only about the first eight terms inthe spin expansion are necessary to achieve an accuracy that is better than thestatistical uncertainties of current and future missions.

Accuracy of the slow-rotation approximation for black holes in modified gravity in light of astrophysical observables