Quantifying Systematic Biases in Black Hole Spectroscopy

TL;DR

The paper tackles systematic biases in black hole ringdown spectroscopy by introducing linear signal analysis (LSA) as an efficient surrogate to full Bayesian inference. It models ringdown as a sum of damped sinusoids and uses the Fisher information matrix and a bias formula to predict parameter shifts caused by unmodeled effects like overtones, quadratic QNMs, and tails, under flat-noise assumptions. The authors demonstrate strong agreement between LSA and Bayesian MCMC at moderate to high SNR (around 50) and perform extensive parameter-space studies on QNM phase differences, start times, and a zoo of unmodeled modes, revealing substantial biases that depend on phase and timing. They discuss validity limits, extensions to realistic detectors, and future applications, arguing that LSA can efficiently identify, quantify, and mitigate systematic uncertainties to improve the robustness of black hole spectroscopy and tests of GR for current and future gravitational-wave observatories.

Abstract

How long after the merger of two black holes can one rely on linear perturbation theory, and how many quasinormal modes are in the ringdown? Such questions suggest that black hole spectroscopy suffers from systematic uncertainties that potentially spoil ringdown analyses, both from high-accuracy simulations and in data from gravitational wave detectors. In this work, we demonstrate that linear-signal analysis is a powerful tool for quantifying biases, allowing for detailed explorations that are computationally too expensive for traditional Bayesian injection and recovery approaches. We quantify the validity of the Fisher information matrix and bias formula by comparing it to robust but slow Bayesian sampling. Working with flat noise in the time domain, statistical errors and systematic biases can mostly be detected analytically. Due to its efficiency, we provide detailed parameter space analyses for potentially unmodeled small contributions from overtones, quadratic modes, and tails. We find linear signal analysis well suited for predicting biases in simple ringdown models at intermediate signal-to-noise ratios (SNRs) of order 50 when unmodeled effects are small. It is also valuable in explaining ongoing issues in extracting quasinormal modes from high-precision simulations, as one can understand them as high-SNR signals. Therefore, this approach offers promising prospects for improving ringdown models by efficiently identifying and incorporating systematic uncertainties, ultimately enhancing the accuracy and robustness of black hole spectroscopy.

Figures (10)

• Figure 1: Ringdown signal and model: We show the fundamental mode $\omega_{220}$ along with the contribution of an unmodeled overtone $\omega_{221}$, unmodeled quadratic QNM $\omega_{220 \times 220}$, and unmodeled power-law tail. Numerical values of the parameters are reported in Table \ref{['table1']}.
• Figure 2: Comparison of LSA versus Bayesian analysis: Here we compare the theory-specific (top panel) and theory-agnostic (bottom panel) biases for three unmodeled effects (different colors) described in the main text. In each panel, we compare posteriors obtained using the Bayesian (thin dashed lines) and LSA (thick solid lines) methods, respectively. The values of the injected parameters are shown for reference (black crosshair). In this example, the signal has SNR=50.
• Figure 3: Bias ratios as a function of QNM phase differences: In all panels, we show the bias ratio originating from an unmodeled QNM with a relative amplitude of $0.1$ to the $\omega_{220}$ fundamental mode as a function of their phase difference $\Delta \phi$ for SNR=50. In the top row, we consider $\omega_{221}$, and in the bottom row, we consider the quadratic $\omega_{220\times220}$ as an unmodeled QNM. Left and right columns correspond to theory-specific and theory-agnostic models, respectively.
• Figure 4: Bias ratios as a function of starting time: In all panels, we show the bias ratio originating from an unmodeled QNM with a relative amplitude of $0.1$ to the $\omega_{220}$ fundamental mode as a function of the starting time $t_\text{start}-t_\text{peak}$ for SNR=50. The top, middle, and bottom rows show biases from unmodeled $\omega_{221}$, quadratic $\omega_{220\times220}$, and power-law tail contributions, respectively. Left and right columns correspond to theory-specific and theory-agnostic models, respectively.
• Figure 5: Exploring biases from unmodeled QNMs: Here we show biases of various unmodeled linear QNMs described in the main text using LSA. The contours indicate 70 % (light gray) and 90 % (dark gray) probability levels using the LSA prediction (Fisher) of the theory-specific fundamental mode, i.e., the marginalized errors on the mass and spin. The true value is shown in the center (black cross). In all cases, biases are computed one at a time with a relative mode amplitude of $A_{\ell m n}/A_{220}=0.1$ at SNR=50. The top panel shows modes with phase difference $\Delta \phi=0$, and the bottom panel with $\Delta \phi=\pi$.