Advancing the Effective-One-Body Framework in the Test-Mass Limit

Abstract

We present SEOB-TML, an enhanced effective-one-body (EOB) framework for the test-mass limit, optimized for quasi-circular, spin-aligned binary black holes. On the dynamical side, we introduce a quadrupole-factorized (Q-factorized) prescription that maps the total energy flux-including horizon absorption-onto a single (2,2) mode baseline. This approach effectively captures higher-order multipole contributions without explicit mode summation, while simultaneously leading to a dramatic reduction in fractional flux errors. To ensure a smooth transition to the post-merger stage, we replace traditional next-to-quasicircular corrections with a phenomenological ansatz, enabling a flexible, mode-dependent attachment prescription. For the merger-ringdown stage, we utilize quasi-normal mode coefficients extracted from numerical waveforms via qnmfinder to explicitly model mode-mixing effects. These enhancements lead to a substantial reduction in residuals, capturing the complex physical modulations prominent in retrograde configurations. Additionally, we implement the (2,0) mode across the full waveform, further extending the model's physical coverage and accuracy. Overall, our framework generates highly accurate late inspiral-merger-ringdown waveforms for extreme-mass-ratio systems, significantly reducing dephasing and improving the near-merger reconstruction. We demonstrate the performance of SEOB-TML against the current state-of-the-art SEOBNRv5HM model, highlighting how our specialized developments extend the reliability of the EOB framework into the test-mass limit.

Figures (27)

• Figure 1: Absolute value of the fractional difference between the analytical EOB fluxes and FD Teukolsky data for spins $a \in \{-0.9, 0, 0.9, 0.95\}$. The 9PN $M$-factorized flux is shown in shades of blue, corresponding to truncation at $\ell_{\text{max}} = 8, 9, 10$, while the 9PN $Q$-factorized flux is shown in red. The $x$-axis corresponds to a dimensionless velocity parameter $x = \Omega^{2/3}$ and all fluxes are extended up to the ISCO. The dashed lines indicate the expected $x^{9.5}$ scaling. The $Q$-factorized flux effectively captures higher-multipole contributions, as evidenced by its low-frequency alignment with the $\ell_{\text{max}}=10$$M$-factorized flux. Furthermore, the $Q$-factorized prescription demonstrates superior accuracy in the strong-field regime across all considered spins.
• Figure 2: Absolute value of the fractional difference between $Q$-factorized fluxes at various PN orders and the FD Teukolsky flux for $a = 0.9$, all extended up to the ISCO. While increasing the PN order beyond 9PN yields only marginal improvement at low frequencies, it degrades the accuracy in the strong-field regime; for this reason, the 9PN flux (red) is adopted in this work.
• Figure 3: Ratio of the energy flux absorbed by the horizon, $F_H$, to the energy flux radiated to infinity, $F_{\infty}$, for different spin values, computed from the FD Teukolsky fluxes. The fluxes are extended up to $r = r_{\mathrm{LR}} + 0.01$, and the locations of the respective ISCOs are indicated by dots.
• Figure 4: Fractional error in the energy flux across the spin-frequency parameter space for four analytical prescriptions. The color map represents the absolute fractional difference $|F_{\text{tot}} / F_{\text{tot}}^{\text{Teuk}} - 1|$ relative to the numerical FD Teukolsky flux. The $x$-axis spans the dimensionless spin range $a \in [-0.993, 0.998]$, and the $y$-axis denotes the orbital frequency parameter $x$ rescaled by its value at the ISCO. Top left: The SEOBNRv5HM flux evaluated in the TML. Top right: 9PN $M$-factorized flux truncated at $\ell_{\max} = 10$. Bottom left: The proposed $Q$-factorized flux at infinity (Eq. \ref{['eq:Steff']}). Bottom right: The full $Q$-factorized flux including horizon absorption (Eq. \ref{['eq:SteffH']}). The inclusion of horizon absorption in the $Q$-factorized prescription dramatically reduces errors in the low-frequency regime, an improvement that extends to the ISCO for retrograde spins. While performance degrades for high prograde spins near the ISCO, the $Q$-factorized approach generally outperforms the $M$-factorized flux across the majority of the parameter space using significantly fewer multipolar modes.
• Figure 5: Evolution of the $\ell \neq m$ mode amplitudes (top panel) and the orbital frequency $\Omega$ (bottom panel) for various negative-spin configurations ($a = -0.5, -0.7, -0.9$). Solid, dash-dotted, and dashed lines correspond to decreasing spin magnitudes, as indicated in the legend. The time axis is centered at $t=0$, corresponding to the zero-crossing of $\Omega$.