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Can Oscillatory and Persistent Nonlinearities Be Bridged in Black Hole Ringdown?

Jun-Xi Shi, Zhen-Tao He, Jiageng Jiao, Jing-Qi Lai, Caiying Shao, Yu Tian, Hongbao Zhang

Abstract

Quadratic quasinormal modes (QQNMs) and Christodoulou memory effect are key nonlinear phenomena in gravitational wave physics. QQNMs characterize the near zone nonlinear response of a perturbed black hole, whereas the memory effect is a nonlinear remnant imprinted at null infinity by outgoing radiation. This naturally raises the question of whether and in what sense the two can be bridged. We show that they are related through bridge coefficients which depend primarily on remnant black hole parameters during ringdown. Future space-based gravitational-wave detectors can probe this relation. These results provide a new avenue for testing gravity and a fresh perspective on the nonlinear regime of general relativity.

Can Oscillatory and Persistent Nonlinearities Be Bridged in Black Hole Ringdown?

Abstract

Quadratic quasinormal modes (QQNMs) and Christodoulou memory effect are key nonlinear phenomena in gravitational wave physics. QQNMs characterize the near zone nonlinear response of a perturbed black hole, whereas the memory effect is a nonlinear remnant imprinted at null infinity by outgoing radiation. This naturally raises the question of whether and in what sense the two can be bridged. We show that they are related through bridge coefficients which depend primarily on remnant black hole parameters during ringdown. Future space-based gravitational-wave detectors can probe this relation. These results provide a new avenue for testing gravity and a fresh perspective on the nonlinear regime of general relativity.
Paper Structure (9 sections, 25 equations, 4 figures, 1 table)

This paper contains 9 sections, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Quadratic quasinormal modes
  4. Memory effects
  5. Bridge coefficients of the nonlinearities
  6. tests with SXS waveforms
  7. observational prospects
  8. discussion and outlook
  9. Derivation of Eqs. (\ref{['eq:main_initial']})--(\ref{['eq:B2']})

Figures (4)

  • Figure 1: Dimensionless spin dependence of $B^{\mathrm{I}}$ from QNM self-coupling.
  • Figure 2: Dimensionless spin dependence of $B^{\mathrm{II}}$ from QNM cross-coupling.
  • Figure 3: Verification of the bridge coefficient $\Lambda$ for the $(2,2,0)\times(2,2,0)\rightarrow (4,4)$ coupling channel from the SXS catalog. The horizontal axis shows the dimensionless remnant spin, and the vertical axis shows the value of $\Lambda$ from dimensionless waveforms. Red error bars are $\Lambda$ values measured from a set of SXS BBH simulations spanning the main remnant dimensionless spin range, while the blue Yuste:2024, orange Khera:2025, and green Cheung:2024 dash-dotted curves are theoretical $\Lambda$ predictions obtained from the amplitude ratio $\mathcal{R}$ using three different methods. This figure confirms the bridge relation between the QQNM amplitude and the full accumulated memory strain in BBH ringdown waveforms.
  • Figure 4: Ratios of the memory strain sourced by an offspring QQNM to that sourced by a parent QNM, shown as $|\delta h^{(2)}/\delta h^{(1)}|$ versus the remnant spin $\chi_f$. Solid curves use $q=3.0000$ and dashed curves use $q=1.0001$. The legend labels each QQNM by its parent pair and specifies the parent mode used in the denominator. Shaded bands span the two mass ratios for the same component. Most ratios are below $5\times10^{-2}$ with weak spin dependence. The largest variation occurs for $(3,3,0)\times(2,-2,0)^{*}$, while $(2,2,0)\times(3,3,0)/(3,3,0)$ can approach unity at high $\chi_f$.