Gravitational waveforms from restriction theory and rapid-decay homology

TL;DR

The paper develops a systematic framework to compute frequency-domain gravitational waveforms from relativistic binary scattering across soft and post-Newtonian regimes by marrying differential-equation techniques with restriction theory and algebraic geometry on impact-parameter-space Fourier integrals. It derives recursion relations for tree-level waveforms and demonstrates how to push these methods toward higher-loop computations, aided by an $ abla$-form (epsilon-form) for the master integrals and rapid-decay homology to organize integration kernels. The authors show that Fourier master integrals obey an $oldsymbol\varepsilon$-factorized differential equation mixing Bessel- and exponential-type kernels, marking a first step toward uncovering the underlying analytic structure of the exact solution. The framework is validated by achieving PN expansions up to $igO(v^{30})$ and by matching soft and PN results from the literature, while outlining a path to fully analytic, higher-order waveform computations with potential applications to precision gravitational-wave templates and tests of compact-object structure.

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.

Figures (2)

• Figure 1: Kinematics of the five-point process.
• Figure 2: (Left) The twist $u(z_{3})=\mathrm{e}^{z_{3}}(\hat{w}_{2}^{2}-z_{3}^{2})^{-1/2}$ has branch points at $\pm\hat{w}_{2}$, connected by a red wavy branch cut. The exponential factor introduces an essential singularity at $z_{3}=\infty$, growing for $\mathrm{Re}\,(z_{3})\to +\infty$ and decaying for $\mathrm{Re}\,(z_{3})\to -\infty$. The rapid-decay homology group $H^{\mathrm{rd}}_{1}(X,u)$, with $X=\mathbb{C}\setminus\{\pm\hat{w}_{2}\}$, is two-dimensional and spanned by $\gamma_{1}$ (blue, a small loop encircling the cut) and $\gamma_{2}$ (green, a ray extending from $-\infty$ to $-\hat{w}_{2}$). (Right) The corresponding basis $\{\gamma_1',\gamma_2\}$ in the relative homology group $H_{1}^{\mathrm{rel}}(\tilde{X},Q)$. The dashed circles represent the boundary components $S_{p}^1$ at the blow-up points $p=\infty$ (black) and $p=\pm \hat{w}_2$ (red) in the oriented blow-up $\tilde{X}\subset \mathbb{CP}^1$. Their union defines the boundary set $Q$, where cycles can ends while preserving closedness under the boundary map $\partial_{\text{rel}}$. The decaying directions for $\gamma_{2}$ ($\tfrac{\pi}{2}<\phi<\tfrac{3\pi}{2}$) are indicated by the thick semicircle at complex infinity.