Extracting linear and nonlinear quasinormal modes from black hole merger simulations

In general relativity, when two black holes merge they produce a rotating (Kerr) black hole remnant. According to perturbation theory, the remnant emits "ringdown" radiation; a superposition of exponentials with characteristic complex frequencies that depend only on the remnant's mass and spin. While the goal of the black hole spectroscopy program is to measure the quasinormal mode frequencies, a knowledge of their amplitudes and phases is equally important to determine which modes are detectable, and possibly to perform additional consistency checks. Unlike the complex frequencies, the amplitudes and phases depend on the properties of the binary progenitors, such as the binary mass ratio and component spins. In this paper we develop a fitting algorithm designed to reliably identify the modes present in numerical simulations and to extract their amplitudes and phases. We apply the algorithm to over 500 binary black hole simulations from the public SXS numerical relativity simulation catalog, and we present fitting formulas for the resulting mode amplitudes and phases as functions of the properties of the progenitors. Crucially, our algorithm allows for the extraction of not only prograde fundamental modes and overtones, but also retrograde modes and second -order modes. We unveil interesting relations for the amplitude ratios of different modes. The fitting code and interactive versions of some of the plots are publicly available [1]. The results presented in this paper can be updated as more and better simulations become available.