From the confluent Heun equation to a new factorized and resummed gravitational waveform for circularized, nonspinning, compact binaries

TL;DR

This work develops a novel factorized and resummed gravitational waveform for circularized, nonspinning binaries by solving the Teukolsky equation after mapping it to a Confluent Heun Equation (CHE). The CHE solution exposes external and internal tail factors that absorb all test-mass logs and transcendental numbers into exponentials and Γ-function structures, leaving residual amplitude and phase corrections as PN-truncated rational polynomials. The approach extends to comparable masses via a universal anomalous dimension of multipole moments, enabling a unified treatment across mass ratios and offering a pathway to improve EOB/NR-based waveform models, including potential Kerr generalizations. The paper also critically analyzes and refines the ILPZ and DIN factorization schemes, showing improved accuracy and a clearer analytic structure in the test-mass limit up to $10$PN and beyond. Overall, the method promises more efficient and accurate waveform modeling for gravitational-wave data analysis and NR-informed EOB formulations.

Abstract

We introduce a new factorized and resummed waveform for circularized, nonspinning, compact binaries that leverages on the solution of the Teukolsky equation once mapped into a confluent Heun equation. The structure of the solution allows one to identify new resummed factors that completely absorb all test-mass logarithms and transcendental numbers via exponentials and $Γ$-functions at any post-Newtonian (PN) order. The corresponding residual relativistic and phase corrections are thus polynomial with rational coefficients, that are in fact PN-truncated hypergeometric functions. Our approach complements the recent proposal of Ivanov et al. [Phys. Rev. Lett. 135 (2025) 14, 141401], notably recovering the corresponding renormalization group scaling of multipole moments from first principles and fixing the scaling constant. In the test mass limit, our approach (pushed up to 10PN) yields waveforms and fluxes that are globally more accurate than those obtained using the standard factorized approach of Damour et al. [Phys. Rev. D 79 (2009), 064004]. The method generalizes straightforwardly to comparable mass binaries implementing the new concept of universal anomalous dimension of multipole moments and might be eventually useful to improve current state of the art effective-one-body waveform models for coalescing binaries.

Figures (5)

• Figure 1: Choosing the PN accuracy of $\hat{2}$ from Eq. \ref{['eq:hat2']} in the test-mass limit. Top panel: $|\hat{h}_{22}|$ for a test-mass on circular orbits on a Schwarzschild black hole. Bottom panel: fractional difference between the exact $|\hat{h}_{22}|$, obtained numerically, and the analytical $|\hat{h}_{22}|$ obtained retaining both $(\rho_{22},\tilde{f}_{22})$ at 10PN accuracy, $\lambda_{\rm inst}^{\rm NS}$ in closed form with $i_{{\rm max},{\cal F}}=10$ but truncating $\hat{2}$ from Eq. \ref{['eq:hat2']} at 9PN ($i_{{\rm max},a}= 6$), 12PN ($i_{{\rm max},a}= 8$) and 15PN ($i_{{\rm max},a}= 10$). The 12PN truncation is the closest one to the numerical data.
• Figure 2: Effect of various approximations to the function $\lambda_{\rm inst}^{\rm NS}$ while keeping $\hat{2}$ at 12PN and both $(\rho_{22},\tilde{f}_{22}e^{i\tilde{\delta}_{22}})$ at 10PN accuracy. The red line is the same as Fig. \ref{['fig:hat2']} and it is very well represented by approximating $\lambda_{\rm inst}^{\rm NS}$ via Eq. \ref{['eq:expanded']} with $i_{{\rm max},{\cal F}}=2$.
• Figure 3: Performance of the new resummation for the full flux with all modes summed up to $\ell=8$ contrasted with the DIN procedure Damour:2008gu. The plot shows the relative differences with the exact flux obtained numerically for various analytical representations. We compare the effect of 8PN, 9PN or 10PN truncation of the residual amplitude corrections. The relative difference at $x_{\rm LSO}$ is $4.78\times 10^{-6}$ at 10PN for the new procedure versus $7.09\times 10^{-6}$ of DIN. Compare also with Fig. 8 of Ref. Nagar:2022fep that implements the DIN procedure at 22PN accuracy, that yields just $4\times 10^{-6}$ at $x_{\rm LSO}$.
• Figure 4: Performance of the new resummation with different truncations of $\hat{\hat{2}}$, taken from Eq. \ref{['eq:gamma_2m_univ']} with $\ell=2$, with respect to the DIN (resummed) function implemented in the state-of-the-art model TEOBResumS-Dalí. The effect of $\lambda_{\rm inst}^{\rm NS}$ is always negligible.
• Figure 5: Performance of the revisited ILPZ factorization described in Sec. \ref{['sec:ivanov_et_al']}: it is much closer to the DIN choice of TEOBResumS-Dalí, although the effect of the truncation of $\hat{\hat{\ell}}$ with $\ell=2$ is not negligible.