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Gravitational waveforms from restriction theory and rapid-decay homology

Giacomo Brunello1,2,3,4,*, Vsevolod Chestnov5,9,†, Giulio Crisanti6,‡, Mathieu Giroux7,8,§, and Sid Smith1,2,6,∥

  • *Contact author: giacomo.brunello@sns.it
  • †Contact author: vsevolod.chestnov@maths.ox.ac.uk
  • ‡Contact author: g.crisanti@ed.ac.uk
  • §Contact author: giroux@ias.edu
  • ∥Contact author: sid.smith@ed.ac.uk
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Phys. Rev. D 113, 085011 – Published 15 April, 2026

DOI: https://doi.org/10.1103/ynk9-ykw7

Abstract

We present a systematic framework for computing frequency-domain gravitational waveforms from relativistic binary scattering in different asymptotic regimes. The method yields a controlled series expansion that can in principle be extended to arbitrary order in the relevant kinematic parameter. By combining differential-equation techniques with restriction theory and algebraic-geometry methods for impact-parameter-space Fourier integrals, we derive recursion relations that generate the leading-order (tree-level) waveform in both the soft-emission and post-Newtonian regimes, establishing a proof of principle for extending the approach to higher-loop computations. Finally, following constraints from rapid-decay homology, we show that the Fourier integrals underlying the waveform satisfy ϵ-form differential equations mixing Bessel- and exponential-type kernels, marking a first step toward uncovering the analytic structure of the exact solution.

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Phys. Rev. D 113, 085011– Published 15 April, 2026
  • Received 29 December 2025
  • Accepted 4 March 2026
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