On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

TL;DR

The paper develops a comprehensive multi-mode framework for gravitational-wave spectroscopy of massive black holes with LISA, combining a rigorous QNM formalism with Fisher-matrix parameter estimation to assess single- and multi-mode detectability and measurement accuracy. It analyzes how redshift, detector noise, and ringdown energy ε_rd shape the SNR and the ability to infer black-hole mass and spin from ringdown signals. The work also discusses the energy distribution among modes, the challenges of mode excitation modeling, and the resolvability requirements needed to test the no-hair theorem, highlighting when two or more QNMs are necessary for a robust no-hair test. Overall, the study provides actionable insights into when LISA can perform black-hole spectroscopy and what observational requirements are needed for rigorous tests of general relativity in the strong-field regime.

Abstract

Newly formed black holes are expected to emit characteristic radiation in the form of quasi-normal modes, called ringdown waves, with discrete frequencies. LISA should be able to detect the ringdown waves emitted by oscillating supermassive black holes throughout the observable Universe. We develop a multi-mode formalism, applicable to any interferometric detectors, for detecting ringdown signals, for estimating black hole parameters from those signals, and for testing the no-hair theorem of general relativity. Focusing on LISA, we use current models of its sensitivity to compute the expected signal-to-noise ratio for ringdown events, the relative parameter estimation accuracy, and the resolvability of different modes. We also discuss the extent to which uncertainties on physical parameters, such as the black hole spin and the energy emitted in each mode, will affect our ability to do black hole spectroscopy.

Figures (21)

• Figure 1: Value of $\epsilon_{\rm rd}$ required to detect the fundamental mode with $l=m=2$ (detection being defined by a SNR of $10$) at $D_L=3~$Gpc. For illustrative purposes here we pick the fundamental mode with $l=m=2$, but the dependence on $(n,l,m)$ is very weak. The three curves correspond to $j=0$ (solid), $j=0.8$ (dashed) and $j=0.98$ (dot-dashed), where $j=J^2/M=a/M$ is the dimensionless angular momentum parameter of the hole. The "pessimistic" prediction from numerical simulations of head-on collisions is $\epsilon_{\rm rd}=10^{-3}$ (as marked by the dashed horizontal line), so we should be able to see all equal-mass mergers with a final black hole mass larger than about $\sim 10^5~M_\odot$ (the vertical line is just a guide to the eye). The dip in the curves is a consequence of white-dwarf confusion noise in the LISA noise curve.
• Figure 2: Errors (multiplied by the signal-to-noise ratio $\rho$) in measurements of different parameters for the fundamental $l=m=2$ mode as functions of the angular momentum parameter $j$. Solid (black) lines give $\rho \sigma_j$, dashed (red) lines $\rho \sigma_M/M$, dot-dashed (green) lines $\rho \sigma_A/A$, dot-dot-dashed (blue) lines $\rho \sigma_{\phi}$, where $\sigma_k$ denotes the estimated rms error for variable $k$, $M$ denotes the mass of the black hole, and $A$ and $\phi$ denote the amplitude and phase of the wave.
• Figure 3: "Critical" SNR $\rho_{\rm crit}$ required to resolve the fundamental mode ($n=0$) from the first overtone ($n'=1$) with the same angular dependence ($l=l'$, $m=m'$). We assume the amplitude of the overtone is one tenth that of the fundamental mode. Solid lines refer to $m=l,..,1$ (bottom to top), the dotted line to $m=0$, and dashed lines to $m=-1,..,-l$ (bottom to top).
• Figure 4: "Critical" SNR $\rho_{\rm both}$ required to resolve both the frequency and the damping time of the fundamental mode ($n=0$) from the first overtone ($n'=1$) with the same angular dependence ($l=l'$, $m=m'$). We assume the amplitude of the overtone is one tenth that of the fundamental mode. Solid lines refer to $m=l,..,1$ (top to bottom), the dotted line to $m=0$, and dashed lines to $m=-1,..,-l$ (top to bottom, unless indicated). In the color versions, we used black for the modes with $l=|m|$, red for those with $0<|m|<l$ and blue for $m=0$. For $l=m$ the critical SNR grows monotonically as $j \to 1$.
• Figure 5: Frequency $f_{lmn}$ (left) and quality factor $Q_{lmn}$ (right) for the fundamental modes with $l=2,~3,~4$ and different values of $m$. Solid lines refer to $m=l,..,1$ (from top to bottom), the dotted line to $m=0$, and dashed lines refer to $m=-1,..,-l$ (from top to bottom). Quality factors for the higher overtones are lower than the ones we display here.