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Direct Waves in Black-Hole Binary Mergers: Insights from the Backwards One Body Model

Anuj Kankani, Sean T. McWilliams

Abstract

The merger-ringdown radiation from a black hole binary merger is accurately modeled by a sum of linear quasinormal modes (QNMs). Recently, a non-QNM ``direct wave" component of the radiation, associated with prompt emission from a plunging perturber, has been identified. Motivated by the behavior of null geodesics perturbed from the remnant light ring, the Backwards One Body (BOB) approach has been shown to model the full merger-ringdown radiation to high accuracy, while using only a minimal number of parameters. In this work, using the Pöschl--Teller potential, we first show how the BOB amplitude evolution can be recovered from the QNM pole contributions. We then apply rational filters to isolate the non-QNM content in BOB and numerical relativity waveforms. We show that BOB naturally captures the direct wave component of the merger radiation, explaining its accuracy near the waveform peak. Finally, we use BOB to show that the direct wave frequency is largely uncorrelated with the horizon frequency, even for high spin remnants, and instead tracks the News frequency at the time of the peak News amplitude.

Direct Waves in Black-Hole Binary Mergers: Insights from the Backwards One Body Model

Abstract

The merger-ringdown radiation from a black hole binary merger is accurately modeled by a sum of linear quasinormal modes (QNMs). Recently, a non-QNM ``direct wave" component of the radiation, associated with prompt emission from a plunging perturber, has been identified. Motivated by the behavior of null geodesics perturbed from the remnant light ring, the Backwards One Body (BOB) approach has been shown to model the full merger-ringdown radiation to high accuracy, while using only a minimal number of parameters. In this work, using the Pöschl--Teller potential, we first show how the BOB amplitude evolution can be recovered from the QNM pole contributions. We then apply rational filters to isolate the non-QNM content in BOB and numerical relativity waveforms. We show that BOB naturally captures the direct wave component of the merger radiation, explaining its accuracy near the waveform peak. Finally, we use BOB to show that the direct wave frequency is largely uncorrelated with the horizon frequency, even for high spin remnants, and instead tracks the News frequency at the time of the peak News amplitude.
Paper Structure (7 sections, 21 equations, 5 figures)

This paper contains 7 sections, 21 equations, 5 figures.

Table of Contents

  1. Introduction.---
  2. A QNM approach to BOB.---
  3. Studying BOB with Rational Filters.---
  4. Discussion.---
  5. Acknowledgements.--
  6. SN formalism
  7. Hypergeometric Function

Figures (5)

  • Figure 1: Top: Filtered NR News (SXS:BBH:0305 SXS:BBH:0305SXSCatalogData_3.0.0Boyle2025SXSsxs_cat3) waveform (solid black) compared to a filtered BOB waveform with the exact same QNMs removed (dotted red) and an "eikonal filtered" BOB waveform (dashed cyan). In light grey we plot the original NR waveform. Bottom: Instantaneous waveform frequency for the filtered waveforms. We also plot the qnm frequency (dashed green) and $w_H = 2\Omega_H$, where $\Omega_H$ is the horizon frequency (dashed red).
  • Figure 2: Top: Filtered NR News (SXS:BBH:4123 Boyle2025SXSSXSCatalogData_3.0.0sxs_cat3SXS:BBH:4123) waveform (solid black) compared to a filtered BOB waveform with the exact same QNMs removed (dotted red) and an "eikonal filtered" BOB waveform (dashed cyan). In light grey we plot the original NR waveform. Bottom: Instantaneous waveform frequency for the filtered waveforms. We also plot the qnm frequency (dashed green) and $w_H = 2\Omega_H$, where $\Omega_H$ is the horizon frequency (dashed red).
  • Figure 3: Top: Filtered NR News (SXS:BBH:2477 Boyle2025SXSSXSCatalogData_3.0.0sxs_cat3Yoo_2022SXS:BBH:2477) waveform (solid black) compared to a filtered BOB waveform with the exact same QNMs removed (dotted red) and an "eikonal filtered" BOB waveform (dashed cyan). In light grey we plot the original NR waveform. Bottom: Instantaneous waveform frequency for the filtered waveforms. We also plot the qnm frequency (dashed green) and $w_H = 2\Omega_H$, where $\Omega_H$ is the horizon frequency (dashed red).
  • Figure 4: Comparison of the mean "eikonal filtered" direct wave frequency (black circle) for the News with the BOB prediction for the News frequency at the time of the peak News amplitude (grey cross), $\omega_H = 2\Omega_H$, where $\Omega_H$ is the horizon frequency (red plus), real part of the $(s=-2,l=2,m=2,n=0)$ qnm frequency and $\omega_{\text{ISCO}} = 2\Omega_\text{ISCO}$ (blue star).
  • Figure 5: Comparison of sech$(t/\tau)$ (solid black) and $|F(t)|\text{sech}(t/\tau)$ where $F(t)$ is the hypergeometric function appearing in Eq. (\ref{['eq:dX_dt_propto_sech']}).