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Quasinormal Modes from EFT of Black Hole Perturbations in Vector-Tensor Gravity

Shogo Tomizuka, Hajime Kobayashi, Naritaka Oshita, Kazufumi Takahashi, Shinji Mukohyama

TL;DR

This work formulates an EFT framework for vector-tensor gravity with a timelike vector and analyzes odd-parity black hole perturbations. It derives a second-order action describing two master variables, χ (tensor) and a (vector), which decouple on stealth Schwarzschild(-de Sitter) backgrounds and yield quasinormal mode frequencies that relate to GR by simple scalings. A key result is that, although the QNMs scale predictably, the observed ringdown can exhibit characteristic modulation because the metric perturbation mixes χ and a, offering a potential observational signature of vector-tensor gravity. The study provides a model-independent route to test such theories with gravitational-wave data and sets the stage for extending the analysis to the even-parity sector and to more general backgrounds, including scordatura effects.

Abstract

We study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike vector field based on the effective field theory (EFT) approach. We derive the quadratic Lagrangian written in terms of two master variables, corresponding to the tensor and vector gravitons, which are coupled in general, while they can be decoupled on a stealth Schwarzschild(-de Sitter) background. For the stealth Schwarzschild background, we find that the quasinormal mode frequencies for both degrees of freedom are obtained from those in general relativity by simple scaling. Nonetheless, due to the fact that the metric perturbation is a non-trivial linear combination of the two degrees of freedom with different QNM spectra, the ringdown gravitational waves may exhibit characteristic modulation that can in principle be a signature of vector-tensor gravity.

Quasinormal Modes from EFT of Black Hole Perturbations in Vector-Tensor Gravity

TL;DR

This work formulates an EFT framework for vector-tensor gravity with a timelike vector and analyzes odd-parity black hole perturbations. It derives a second-order action describing two master variables, χ (tensor) and a (vector), which decouple on stealth Schwarzschild(-de Sitter) backgrounds and yield quasinormal mode frequencies that relate to GR by simple scalings. A key result is that, although the QNMs scale predictably, the observed ringdown can exhibit characteristic modulation because the metric perturbation mixes χ and a, offering a potential observational signature of vector-tensor gravity. The study provides a model-independent route to test such theories with gravitational-wave data and sets the stage for extending the analysis to the even-parity sector and to more general backgrounds, including scordatura effects.

Abstract

We study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike vector field based on the effective field theory (EFT) approach. We derive the quadratic Lagrangian written in terms of two master variables, corresponding to the tensor and vector gravitons, which are coupled in general, while they can be decoupled on a stealth Schwarzschild(-de Sitter) background. For the stealth Schwarzschild background, we find that the quasinormal mode frequencies for both degrees of freedom are obtained from those in general relativity by simple scaling. Nonetheless, due to the fact that the metric perturbation is a non-trivial linear combination of the two degrees of freedom with different QNM spectra, the ringdown gravitational waves may exhibit characteristic modulation that can in principle be a signature of vector-tensor gravity.

Paper Structure

This paper contains 14 sections, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Review of the EFT
  4. Static and spherically symmetric background
  5. Odd-parity perturbations
  6. Odd-parity perturbations with l=2 or larger
  7. Master equation for tensor graviton
  8. Master equation for vector graviton
  9. Dipole perturbations
  10. Quasinormal mode frequency
  11. Tensor graviton
  12. Vector graviton
  13. Modulation of ringdown gravitational waves
  14. Conclusions

Figures (3)

  • Figure 1: Effective potential $V_{{\rm eff},T}$ of the stealth Schwarzschild background for the $l=2$ tensor graviton as a function of $r/r_{\rm H}$.
  • Figure 2: Effective potential $V_{{\rm eff},V}$ of the stealth Schwarzschild background for the $l=2$ vector graviton as a function of $r/r_{\rm H}$.
  • Figure 3: Schematic behavior of $|h_0|$ measured at a distant region $r=r_o$. The amplitude is normalized at $t=0$. We have chosen $\alpha_T=0.01$ and $\beta_V=0$, and taken into account only the $l=2$ fundamental QNMs of the tensor and vector gravitons. We see a non-trivial modulation in vector-tensor gravity (solid orange) due to a contribution of the vector graviton, which is in sharp contrast to the GR case (dashed blue).