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Computing nonlinearity ratios using second order black hole perturbation theory

Jasveer Singh, Vardarajan Suneeta

TL;DR

This work develops and tests an analytic framework for computing nonlinearity ratios of quadratic quasinormal modes (QQNMs) in Schwarzschild black holes by combining a second-order perturbation theory setup with a WKB/matched-asymptotics approach. The method constructs first- and second-order Zerilli-type solutions, uses a Green’s-function formalism to extract QQNM amplitudes, and evaluates nonlinearity ratios both at infinity and on the horizon. It yields excellent agreement with numerical relativity for the $(2,2) imes(2,2) o(4,4)$ channel (NL∞ for the gravitational wave strain around 0.16–0.98 depending on normalization), while revealing a breakdown of the analytic steepest-descent approach for the $(2,0) imes(2,0) o(2,0)$ channel due to non-stationary phase and spatially truncated sources. The results also demonstrate robustness to regularization choices and matching-point choices, and they extend to QQNMs sourced by overtones, outlining a path toward precision nonlinear-ringdown analyses and future Kerr/higher-curvature generalizations.

Abstract

We revisit an analytical approximation scheme for computing nonlinearity ratios involving quadratic quasinormal modes (QQNMs). We compute these ratios for the general case when the QQNM is not one of the linear QNMs, for the $(l,m)$ channel $(2,2) \times (2,2) \to (4,4)$. We find an excellent match with numerical simulations. We also discuss where and why the method can fail, for example, for the channel $(2,0) \times (2,0) \to (2,0)$ where we can only get crude estimates for the nonlinearity ratio. Motivated by recent studies on nonlinear ringdown at the horizon, we also compute the nonlinearity ratios at the horizon. We find that the ratio both at the horizon and infinity is insensitive to different choices of regularization of the source term in the second order perturbations. We also discuss amplitudes of QQNMs sourced by linear overtones. Finally, we discuss the issues that must be resolved within this method to do precision analysis of nonlinear ringdown.

Computing nonlinearity ratios using second order black hole perturbation theory

TL;DR

This work develops and tests an analytic framework for computing nonlinearity ratios of quadratic quasinormal modes (QQNMs) in Schwarzschild black holes by combining a second-order perturbation theory setup with a WKB/matched-asymptotics approach. The method constructs first- and second-order Zerilli-type solutions, uses a Green’s-function formalism to extract QQNM amplitudes, and evaluates nonlinearity ratios both at infinity and on the horizon. It yields excellent agreement with numerical relativity for the channel (NL∞ for the gravitational wave strain around 0.16–0.98 depending on normalization), while revealing a breakdown of the analytic steepest-descent approach for the channel due to non-stationary phase and spatially truncated sources. The results also demonstrate robustness to regularization choices and matching-point choices, and they extend to QQNMs sourced by overtones, outlining a path toward precision nonlinear-ringdown analyses and future Kerr/higher-curvature generalizations.

Abstract

We revisit an analytical approximation scheme for computing nonlinearity ratios involving quadratic quasinormal modes (QQNMs). We compute these ratios for the general case when the QQNM is not one of the linear QNMs, for the channel . We find an excellent match with numerical simulations. We also discuss where and why the method can fail, for example, for the channel where we can only get crude estimates for the nonlinearity ratio. Motivated by recent studies on nonlinear ringdown at the horizon, we also compute the nonlinearity ratios at the horizon. We find that the ratio both at the horizon and infinity is insensitive to different choices of regularization of the source term in the second order perturbations. We also discuss amplitudes of QQNMs sourced by linear overtones. Finally, we discuss the issues that must be resolved within this method to do precision analysis of nonlinear ringdown.

Paper Structure

This paper contains 15 sections, 172 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Setting up the equations: review
  3. The Zerilli equation
  4. First-Order Solutions: review
  5. Review of Second-order Solutions
  6. The WKB Approximation of Schutz and Will
  7. General solution for $\phi_+(x,s)$ for $s \neq s_n$
  8. General solution for $\phi_-(x,s)$ for $s \neq s_n$
  9. Calculating non-linearities for $l=2$
  10. Nonlinearity ratio for $l=4$
  11. Gauge dependence in source
  12. Sensitivity to choice of matching point
  13. Adding overtones to the analysis
  14. Summary and Discussion
  15. Acknowledgments

Figures (5)

  • Figure 1: Diagram of $-Q(x)$
  • Figure 2: x-axis: contour rotated variable running from negative to positive infinity, with $x=0$ being the point about which we apply the saddle point approximation y-axis: argument of l=2 integrand after contour rotation
  • Figure 3: Contour of choice with $z$ being the complexified $(x'-x_0)$
  • Figure 4: x-axis: contour rotated variable running from negative to positive infinity, with $x=0$ being the point about which we apply the saddle point approximation y-axis: argument of $l=4$$I_1$ integrand after contour rotation
  • Figure 5: x-axis: contour rotated variable running from negative to positive infinity, with $x=0$ being the point about which we apply the saddle point approximation y-axis: argument of $l=4$$I_2$ integrand after contour rotation