Analyzing black-hole ringdowns with orthonormal modes

Abstract

The ringdown signal following a black hole (BH) merger can be modeled as a superposition of BH quasinormal modes (QNMs), offering a clean setup for testing gravitational theories. In particular, detecting multiple QNMs enables consistency checks of their frequencies and damping times, serving as a test of general relativity -- a technique known as black hole spectroscopy. However, incorporating additional QNMs introduces challenges such as increased parameter correlations and higher computational costs in data analysis. To address this, we propose an efficient Bayesian analysis method that applies the Gram-Schmidt algorithm to the QNMs. This reduces the correlation between the modes and enables analytic marginalization over the mode amplitudes. We validate our approach using damped sinusoids and numerical waveforms from the Simulating eXtreme Spacetimes catalog.

Figures (12)

• Figure 1: Top panel shows the template basis composed of the original (nonorthogonal) modes $v_{j,\alpha_k}^I(t)$, as presented in Eqs. \ref{['eq:template basis0']}--\ref{['eq:template basis3']}. We fix $M_f=68.2\,M_{\odot}$, $\chi_f=0.692$, and $t^I_{\mathrm{S}}=0$. The bottom panel shows the orthonormal template basis $\tilde{v}^I_{j,\alpha_k}(t)$ obtained by orthogonalizing the original basis. Here, we adopt the noise PSD estimated with the data around GW150914, available from LIGO-P1900011-v1. We focus solely on the $l=m=2$ modes with $D=2$ and include up to the second overtone, fixing the detector $I$ to Hanford.
• Figure 2: Posterior distributions for the remnant BH's mass, spin, and mode amplitudes obtained by analyzing the four-mode signal. The left (right) panel shows the results from the semianalytic (MCMC) method. The contours are color coded to represent the 1-$\sigma$ (39.3%), 2-$\sigma$ (86.5%), and 3-$\sigma$ (98.9%) credible regions. The black lines represent the injected values.
• Figure 3: Posterior distributions for mass, spin, and mode amplitudes obtained by analyzing the four-mode signal under the assumption of $D=2$ damped sinusoids per QNM. The left (right) panel shows the results from the semianalytic (MCMC) method.
• Figure 4: Posterior distributions for mass, spin, and mode amplitudes obtained from the semianalytic method by analyzing the eight-mode signal. The analysis starts from the peak time.
• Figure 5: Posterior distributions for the mode amplitudes obtained from the semianalytic method by analyzing the four-mode signal, evaluated using data segments starting at different times. The left (right) panel shows the result of the analysis assuming $D=4$ ($D=2$) damped sinusoids per QNM. The horizontal axes indicate the start time relative to the signal peak. Circles, thick vertical lines, and thin vertical lines denote the median, 1-$\sigma$ (68%), and 2-$\sigma$ (95%) credible intervals, respectively. The black solid lines represent the injected values. The oscillatory behavior of the true values, rather than a simple exponential decay, arises because the basis vectors are linear combinations of different QNM.