Gravitational Memory from Hairy Binary Black Hole Mergers

Gravitational-wave memory is a low-frequency, non-oscillatory component of the radiation field that provides a potentially powerful but as yet undetected probe of strong-field gravity. We present the first calculation of gravitational memory from full inspiral--merger--ringdown waveforms in a theory beyond general relativity, focusing on scalar-Gauss-Bonnet gravity as a theoretically well-motivated and numerically accessible extension of GR. Starting from the general memory formulas in Horndeski gravity, we derive explicit spin-weighted spherical-harmonic expressions for the tensor null memory in scalar-Gauss-Bonnet theory and evaluate them on existing numerical-relativity waveforms for both shift-symmetric and dynamically scalarizing binary black hole mergers. We find that the dominant effect is an indirect modification of the tensor memory through changes in the nonlinear merger dynamics, while the direct scalar contribution to the tensor memory remains suppressed by orders of magnitude for the systems considered in this work. For the largest deviations in our dataset, the final memory amplitude differs from the corresponding GR prediction by a few percent and by up to $\sim 4\%$ when compared to the GR template that minimizes the waveform mismatch in a detector-oriented analysis. We further show that including memory increases the mismatch between GR and scalar-Gauss-Bonnet waveforms by more than an order of magnitude, indicating that memory can provide complementary information for testing gravity with third-generation detectors, especially for low-mass binaries.