Black-hole hair loss: learning about binary progenitors from ringdown signals

TL;DR

This paper investigates how the ringdown phase of black-hole mergers, modeled as a superposition of quasi-normal modes with amplitudes $A_{\ell m}$, can be used to test general relativity and infer progenitor properties. By fitting ringdown signals from numerical-relativity simulations of non-spinning binaries (up to mass ratio $q=11$) and deriving analytical fits for mode amplitudes, the authors show that relative mode strengths encode the progenitor mass ratio and final BH parameters. They assess detectability with LISA, ET, and aLIGO, demonstrating very high SNRs for LISA and substantial SNRs for ET, enabling measurements of final mass $M$, spin $j$, and inclination $\iota$, as well as the mass ratio $q$ of the progenitor binary. The work provides a practical framework and parametrizations for a multi-mode no-hair test and highlights the potential of future space- and ground-based detectors to constrain BH demographics and GR in the strong-field regime.

Abstract

Perturbed Kerr black holes emit gravitational radiation, which (for the practical purposes of gravitational-wave astronomy) consists of a superposition of damped sinusoids termed quasi-normal modes. The frequencies and time-constants of the modes depend only on the mass and spin of the black hole - a consequence of the no-hair theorem. It has been proposed that a measurement of two or more quasi-normal modes could be used to confirm that the source is a black hole and to test if general relativity continues to hold in ultra-strong gravitational fields. In this paper we propose a practical approach to testing general relativity with quasi-normal modes. We will also argue that the relative amplitudes of the various quasi-normal modes encode important information about the origin of the perturbation that caused them. This helps in inferring the nature of the perturbation from an observation of the emitted quasi-normal modes. In particular, we will show that the relative amplitudes of the different quasi-normal modes emitted in the process of the merger of a pair of nonspinning black holes can be used to measure the component masses of the progenitor binary.

Figures (9)

• Figure 1: This plot shows the relative luminosities, or radiated power, in modes $(2,2),$$(3,3),$$(4,4)$, and $(2,1).$$L_{lm}$ represent the luminosities (in units $c=G=1$ in which luminosity is dimensionless) and $r_{lm}$ denote the ratios $r_{lm}=L_{lm}/L_{22}.$ The different panels correspond to systems with different mass ratios as indicated in the panel. Note that as the mass ratio increases, the luminosity in each mode decreases but the amplitudes of all higher-order modes relative to the $(2,2)$-mode increase. We have omitted --- both in the figure and in this work --- the next most dominant modes, $(5,5)$, $(3,2)$, $(4,3)$, $(6,6)$ and $(5,4)$ as they are generally less than one percent as luminous as the $(2,2)$ mode. (see, however, Pan et al Pan:2011gk).
• Figure 2: Evolution of the first few dimensionless mode frequencies $f_{\ell m}=M\omega_{\ell m}$ as a function of the dimensionless time $t/M,$ for different values of the mass ratio $q$ of the progenitor binary. Also shown in arbitrary units is the luminosity in the 22 mode. All mode frequencies, especially $f_{22}$ and $f_{33},$ stop evolving and stabilise soon after the binary merges to form a single black hole. The waveform is assumed to contain a superposition of only quasi-normal modes a duration $10M$ after the luminosity in 22 mode reaches its peak.
• Figure 3: This plot shows the amplitudes as a function of the mass ratio for different modes at the peak of the luminosity of the 22 mode (circles), an epoch $10M$ and $15M$ after the peak (respectively, squares and triangles). We have plotted the absolute amplitude $\alpha_{22}$ of the 22 mode and ratio of the sub-dominant mode amplitudes $\alpha_{\ell m}/\alpha_{22}$ relative to 22 (cf. Table \ref{['table:table1']}). The solid lines are the best fits [cf. Eqs. (\ref{['eq:a22']})-(\ref{['eq:a44']})] to the amplitudes at $10M$ after the peak luminosity.
• Figure 4: The signal-to-noise ratio integrand for LISA for a quasi-normal mode signal that is composed of 22, 21 and 33 modes --- the three most dominant ones. The source is assumed to be at a red-shift of $z=1$ and the various angles are as in Table \ref{['tab:params']}. The left panel corresponds to a black hole of mass $M=5\times 10^6\,M_\odot$ and the right panel to a black hole of mass $M=10^7\,M_\odot.$ In both cases the mass ratio of the progenitor binary is taken to be $q=10.$
• Figure 5: Signal-to-noise ratio (SNR) in Advanced LIGO (top set of four panels) and Einstein Telescope (bottom set of four panels) as a function of the black hole's mass $M$ and progenitor binary's mass ratio $q$ for different modes. Most of the contribution to the SNR comes from the 22 mode but other modes too have significant contributions, 33 being more important than 21. The source is assumed to be at a distance of 1 Gpc and various angles are as in Table \ref{['tab:params']}.