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

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.

💡 Research Summary

The paper presents a novel factorized and resummed gravitational‑wave (GW) model for circular, non‑spinning compact binaries by exploiting a deep analytical connection between the Teukolsky master equation and the confluent Heun equation (CHE). Starting from the Teukolsky equation for perturbations of a Schwarzschild black hole, the authors perform a change of variables (Z = 2Mr) and introduce the small parameter X = 4iMω. In this new representation the homogeneous Teukolsky equation becomes a CHE with three singular points (regular at Z = 1, ∞ and irregular at Z = 0).

Two independent solutions are constructed: G₁(Z), valid for the physical domain Z∈(0,1), and G₀(Y) with Y = X/Z, which is exact in 1/r (a “post‑Minkowskian” expansion). Both solutions are expressed as a linear combination of a hypergeometric seed Hα (α = ±) and its derivative, multiplied by polynomials Pα and Ĥα in X and inverse powers of Z or Y. The coefficients of these polynomials are determined recursively and can be written explicitly in terms of the orbital frequency ω and a complex monodromy parameter a, which encodes the renormalized angular momentum (the so‑called universal anomalous dimension).

A crucial observation is that the ratio gα(X) = G₁α/G₀α depends only on X, not on the radial variable. This allows the authors to absorb every logarithmic term and every transcendental constant (π², ζ(3), etc.) that normally appear in post‑Newtonian (PN) expansions into simple exponentials and Γ‑functions. Consequently, the GW multipolar waveform hℓm can be written in the familiar factorized form

hℓm = hℓm^{(N)} · Tℓm · e^{iδℓm} · ρℓm,

where the “tail factor” Tℓm is now a closed‑form product of exponentials and Γ‑functions that resums all universal logarithms. The residual phase correction δℓm and amplitude correction ρℓm are pure rational‑coefficient polynomials; mathematically they are PN‑truncated hypergeometric functions (₂F₁).

The paper then connects this construction to the recent effective‑field‑theory (EFT) work of Ivanov, Liu, Pereira, and Zhang (ILPZ, Phys. Rev. Lett. 135 2025). ILPZ introduced the concept of a universal anomalous dimension for multipole moments, which governs the renormalization‑group scaling of the waveform. By identifying the monodromy parameter a with the anomalous dimension, the authors recover the ILPZ scaling from first principles and fix the otherwise arbitrary scaling constant.

Numerical tests are performed in the test‑mass limit (μ≪M). The authors push the new waveform to 10PN order, compute the associated energy flux, and compare against high‑precision numerical solutions of the Teukolsky equation. Across the entire frequency band the new model shows relative errors below 10⁻⁴, outperforming the standard Damour‑Nagar (DIN) factorization, especially at high frequencies where the DIN tail factor underestimates the flux.

Finally, the authors generalize the approach to comparable‑mass binaries. By promoting the anomalous dimension to a function of the symmetric mass ratio ν, they define a “universal anomalous dimension” that can be inserted into the same exponential‑Γ resummation. They demonstrate the method on the known 4.5PN multipolar waveform, obtaining a compact expression that can be directly incorporated into effective‑one‑body (EOB) models such as TEOBResumS.

In summary, the paper delivers three major advances: (1) a systematic mapping of the Teukolsky equation to a CHE that enables exact resummation of all logarithmic and transcendental contributions; (2) a reformulation of the residual PN series as rational‑coefficient polynomials (truncated hypergeometric functions); and (3) a first‑principles derivation of the universal anomalous dimension, providing a unified framework for both test‑mass and comparable‑mass binaries. The resulting waveform promises to improve the accuracy of state‑of‑the‑art EOB models and to be valuable for upcoming GW observatories such as LISA and the Einstein Telescope.