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

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##### Citation

Bhagwat, S., Forteza, F. J., Pani, P., & Ferrari, V. (2020). Ringdown overtones,
black hole spectroscopy and, no-hair theorem tests.* Physical Review D,* (4): 044033. doi:10.1103/PhysRevD.101.044033.

Cite as: https://hdl.handle.net/21.11116/0000-0004-E6D3-A

##### Abstract

Validating the black-hole no-hair theorem with gravitational-wave

observations of compact binary coalescences provide 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 with

discussions on some conceptual issues associated with black-hole spectroscopy

with overtones.

observations of compact binary coalescences provide 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 with

discussions on some conceptual issues associated with black-hole spectroscopy

with overtones.