Gravitational waveforms from restriction theory and rapid-decay homology

TL;DR

This paper introduces a new framework combining differential equations, restriction theory, and algebraic geometry to compute gravitational waveforms from binary scattering, enabling systematic higher-order calculations and revealing their analytic structure.

Contribution

It develops a systematic method for deriving frequency-domain gravitational waveforms using advanced mathematical techniques, extending to higher orders and analyzing their analytic properties.

Findings

Derived recursion relations for leading-order waveforms

Established differential equations with Bessel and exponential kernels

Proved the approach's potential for higher-loop computations

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 epsilon-form differential equations mixing Bessel- and exponential-type kernels, marking a first step toward uncovering the analytic structure of the exact solution.