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

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).