Mining information from binary black hole mergers: a comparison of estimation methods for complex exponentials in noise

TL;DR

This paper investigates how to reliably extract quasinormal-mode parameters from noisy black hole merger ringdowns. It compares modern linear estimation techniques for sums of damped exponentials—Kumaresan-Tufts and matrix pencil methods—against standard nonlinear least-squares, using both synthetic damped sinusoids and numerical relativity merger waveforms. The results show that Prony-based methods offer comparable variance with reduced bias and notable practical advantages, such as not requiring initial guesses and directly handling complex signals. These methods hold promise for both diagnosing numerical relativity simulations and performing parameter estimation on real gravitational-wave data, including potential detection of non-linear effects and overtones.

Abstract

The ringdown phase following a binary black hole merger is usually assumed to be well described by a linear superposition of complex exponentials (quasinormal modes). In the strong-field conditions typical of a binary black hole merger, non-linear effects may produce mode coupling. Artificial mode coupling can also be induced by the black hole's rotation, if the radiation field is expanded in terms of spin-weighted spherical (rather than spheroidal) harmonics. Observing deviations from linear black hole perturbation theory requires optimal fitting techniques to extract ringdown parameters from numerical waveforms, which are inevitably affected by errors. So far, non-linear least-squares fitting methods have been used as the standard workhorse to extract frequencies from ringdown waveforms. These methods are known not to be optimal for estimating parameters of complex exponentials. Furthermore, different fitting methods have different performance in the presence of noise. The main purpose of this paper is to introduce the gravitational wave community to modern variations of a linear parameter estimation technique first devised in 1795 by Prony: the Kumaresan-Tufts and matrix pencil methods. Using "test" damped sinusoidal signals in Gaussian white noise we illustrate the advantages of these methods, showing that they have variance and bias at least comparable to standard non-linear least-squares techniques. Then we compare the performance of different methods on unequal-mass binary black hole merger waveforms. The methods we discuss should be useful both theoretically (to monitor errors and search for non-linearities in numerical relativity simulations) and experimentally (for parameter estimation from ringdown signals after a gravitational wave detection).

Figures (7)

• Figure 1: Real and imaginary parts of $rM\psi_{lm}$ for the two dominant multipoles. The binary has mass ratio $q=1.5$. The burst of radiation at early times is induced by the initial data. After the initial burst, the real and imaginary parts are simply related by a phase shift (i.e., the waveform is circularly polarized). The irregular behavior at late times is due to numerical noise.
• Figure 2: $|rM\psi_{lm}|$ for different multipoles. The high-resolution $l=m=2$, $l=m=3$ and $l=m=4$ wave amplitudes have maxima at $t_{\rm peak}/M=236.4,~238.4$ and $~235.7$, respectively. Wiggles at late times are due to numerical noise, mainly caused by reflections from the boundaries.
• Figure 3: QNM waveforms with and without Gaussian white noise. We normalize the time axis to the total duration of the signal $t_{\rm fin}$. The "pure" noiseless waveform (thick black line) has unit amplitude. Dashed (red), dotted (blue) and thin (green) lines are the same waveform superimposed to Gaussian white noise with $\sigma=10^{-4},~10^{-3},~10^{-2}$, respectively.
• Figure 4: Standard deviation (left) and bias (right) in the estimate of frequency, damping time and quality factor (top to bottom). All quantities are given as functions of the starting time of the fit $t_0$ (normalized by the duration of the signal $t_{\rm fin}$); each point is the result of a Monte Carlo simulation obtained by adding $N_{\rm noise}=100$ realizations of Gaussian white noise with zero mean and $\sigma=10^{-3}$ to the $j=0$ waveform of Fig. \ref{['fig:data']}. Solid (black), dashed (red) and dotted (blue) lines refer to the MP, KT and LM algorithms, respectively.
• Figure 5: Performance of different Prony methods in estimating the oscillation frequency $\omega$ (top) and damping factor $\alpha$ (bottom) for low and high-resolution runs (left and right, respectively). For concreteness we choose a binary with $q=1.5$ and consider the fundamental mode with $l=m=2$. Prony-type results refer to the complex waveform, while LM results are obtained by fitting the real part of the waveform only.