Ringdown overtones, black hole spectroscopy, and no-hair theorem tests

TL;DR

This paper examines the feasibility of using overtones in black hole ringdown signals to test the no-hair theorem, analyzing modeling robustness, bias, and the signal-to-noise ratio needed for resolving overtones with current and future detectors.

Contribution

It provides a systematic analysis of overtone modeling, bias estimation, and the resolvability criteria for black hole spectroscopy using gravitational-wave data.

Findings

Overtones are crucial for black hole spectroscopy in nearly-equal mass binaries.

Resolving overtones requires a high signal-to-noise ratio, above ~30.

A computational method is proposed to optimize the start time of spectroscopic analysis.

Abstract

Validating the black-hole no-hair theorem with gravitational-wave observations of compact binary coalescences provides a compelling argument that the remnant object is indeed a black hole as described by the general theory of relativity. This requires performing a spectroscopic analysis of the post-merger signal and resolving the frequencies of either different angular modes or overtones (of the same angular mode). For a nearly-equal mass binary black-hole system, only the dominant angular mode ($l=m=2$) is sufficiently excited and the overtones are instrumental to perform this test. Here we investigate the robustness of modelling the post-merger signal of a binary black hole coalescence as a superposition of overtones. Further, we study the bias expected in the recovered frequencies as a function of the start time of a spectroscopic analysis and provide a computationally cheap procedure to choose it based on the interplay between the expected statistical error due to the detector noise and the systematic errors due to waveform modelling. Moreover, since the overtone frequencies are closely spaced, we find that resolving the overtones is particularly challenging and requires a loud ringdown signal. Rayleigh's resolvability criterion suggests that in an optimistic scenario a ringdown signal-to-noise ratio larger than $\sim 30$ (achievable possibly with LIGO at design sensitivity and routinely with future interferometers such as Einstein Telescope, Cosmic Explorer, and LISA) is necessary to resolve the overtone frequencies. We then conclude by discussing some conceptual issues associated with black-hole spectroscopy with overtones.