Matched-filtering and parameter estimation of ringdown waveforms

TL;DR

The paper addresses detecting ringdown gravitational waves and inferring black-hole parameters, including tests of the no-hair theorem, by integrating numerical-relativity results with matched-filtering analyses across LISA and ground-based detectors. It demonstrates that higher multipoles are significantly excited and that traditional single-mode templates can incur non-negligible event losses and biased parameter estimates, motivating a two-stage search strategy and the use of multi-mode or Prony-based methods for parameter estimation. It develops quantitative criteria for resolving multiple quasinormal modes and for amplitude resolvability, showing that second-generation detectors and LISA can perform no-hair tests with realistic signal-to-noise ratios. The findings have practical implications for detecting intermediate-mass black holes and testing Kerr black-hole geometry with upcoming gravitational-wave observatories.

Abstract

Using recent results from numerical relativity simulations of non-spinning binary black hole mergers we revisit the problem of detecting ringdown waveforms and of estimating the source parameters, considering both LISA and Earth-based interferometers. We find that Advanced LIGO and EGO could detect intermediate-mass black holes of mass up to about 1000 solar masses out to a luminosity distance of a few Gpc. For typical multipolar energy distributions, we show that the single-mode ringdown templates presently used for ringdown searches in the LIGO data stream can produce a significant event loss (> 10% for all detectors in a large interval of black hole masses) and very large parameter estimation errors on the black hole's mass and spin. We estimate that more than 10^6 templates would be needed for a single-stage multi-mode search. Therefore, we recommend a "two stage" search to save on computational costs: single-mode templates can be used for detection, but multi-mode templates or Prony methods should be used to estimate parameters once a detection has been made. We update estimates of the critical signal-to-noise ratio required to test the hypothesis that two or more modes are present in the signal and to resolve their frequencies, showing that second-generation Earth-based detectors and LISA have the potential to perform no-hair tests.

Figures (9)

• Figure 1: Owen's minimal match, as defined below Eq. (\ref{['ff']}), for different Earth-based detectors. For this illustrative calculation we assume that the Kerr parameter of the final black hole is $j=0.6$, and that the relative amplitude of the second mode is ${\cal A}=0.3$. We also set the phases in Eq. (\ref{['twomode']}) to be $\phi_1=\phi_2=0$ (thick lines) or $\phi_1=0$, $\phi_2=\pi$ (thin lines). The black circle and the red square mark two cases that we study in more detail below: a $M_0=100~M_\odot$ black hole as observed by LIGO and a $M_0=200~M_\odot$ black hole as observed by Advanced LIGO, respectively.
• Figure 2: Left: ringdown SNR for LIGO, Advanced LIGO, Virgo and EGO at $100$ Mpc. Thin lines refer to the EF convention, thick lines to the FH convention (see text). Right: luminosity distance (in Mpc) for detectability of the ringdown signal with a SNR of 10 (for clarity, here we only show results for the FH convention).
• Figure 3: $\theta$-dependent angular functions in Eq. (\ref{['response']}).
• Figure 4: Quality factor (top panel) and frequency (middle panel) maximizing the FF, and corresponding event loss (bottom panel), for Earth-based detectors. Solid and dashed lines are the template's frequency and quality factor maximizing the FF, and the corresponding event loss. Horizontal lines show the frequency and quality factor of the fundamental mode with $l=m=2$.
• Figure 5: Same as Fig. \ref{['fig:LIGO']} for LISA. In the left panel we choose $\phi_1=\phi_2=0$, as for Earth-based detectors. In the right panel we also consider a "dephased" QNM superposition with $\phi_1=0$, $\phi_2=\pi$ (green, dot-dashed lines).