Bayesian model selection for testing the no-hair theorem with black hole ringdowns

TL;DR

This paper explores how Bayesian model selection can be used with gravitational wave data to test the no-hair theorem by detecting deviations in black hole quasi-normal mode frequencies and decay times.

Contribution

It introduces a Bayesian framework to assess the detectability of deviations from general relativity in black hole ringdowns using future gravitational wave detectors.

Findings

ET can detect <1% frequency deviations for 500 Msun black holes at 4 Gpc

NGO can detect ~0.1% deviations for 10^8 Msun black holes at 30 Gpc

Decay time deviations are harder to identify than frequency deviations

Abstract

General relativity predicts that a black hole that results from the merger of two compact stars (either black holes or neutron stars) is initially highly deformed but soon settles down to a quiescent state by emitting a superposition of quasi-normal modes (QNMs). The QNMs are damped sinusoids with characteristic frequencies and decay times that depend only on the mass and spin of the black hole and no other parameter - a statement of the no-hair theorem. In this paper we have examined the extent to which QNMs could be used to test the no-hair theorem with future ground- and space-based gravitational-wave detectors. We model departures from general relativity (GR) by introducing extra parameters which change the mode frequencies or decay times from their general relativistic values. With the aid of numerical simulations and Bayesian model selection, we assess the extent to which the presence of such a parameter could be inferred, and its value estimated. We find that it is harder to decipher the departure of decay times from their GR value than it is with the mode frequencies. Einstein Telescope (ET, a third generation ground-based detector) could detect departures of <1% in the frequency of the dominant QNM mode of a 500 Msun black hole, out to a maximum range of 4 Gpc. In contrast, the New Gravitational Observatory (NGO, an ESA space mission to detect gravitational waves) can detect departures of ~ 0.1% in a 10^8 Msun black hole to a luminosity distance of 30 Gpc (z = 3.5).