The uniqueness of the QNMs is related to the “no-hair” theorem of general
relativity according to which a black hole is completely specified by its mass and
spin^{9}.
Thus, observing QNMs would not only confirm the source to be a black hole, but would be an unambiguous
proof of the uniqueness theorem of general relativity.

The end state of a black hole binary will lead to the formation of a single black hole, which is initially highly distorted. Therefore, one can expect coalescing black holes to end their lives with the emission of QNM radiation, often called ringdown radiation. It was realized quite early on [164] that the energy emitted during the ringdown phase of a black-hole–binary coalescence could be pretty large. Although, the initial quantitative estimates [164] have proven to be rather high, the qualitative nature of the prediction has proven to be correct. Indeed, numerical relativity simulations show that about 1–2% of a binary’s total mass would be emitted in QNMs [300]. The effective one-body (EOB) model [98, 99], the only analytical treatment of the merger dynamics, gives the energy in the ringdown radiation to be about 0.7% of the total mass, consistent with numerical results. Thus, it is safe to expect that the ringdown will be as luminous an event as the inspiral and the merger phases. The fact that QNMs can be used to test the no-hair theorem puts a great emphasis on understanding their properties, especially the frequencies, damping times and relative amplitudes of the different modes that will be excited during the merger of a black hole binary and how accurately they can be measured.

QNMs are characterized by a complex frequency that is determined by three “quantum” numbers, (see, e.g., [78]). Here are indices that are similar to those for standard spherical harmonics. For each pair of there are an infinitely large number of resonant modes characterized by another integer . The time dependence of the oscillations is given by , where is a complex frequency, its real part determining the mode frequency and the imaginary part (which is always positive) giving the damping time: , defining the angular frequency and the damping time. The ringdown wave will appear in a detector as the linear combination of the two polarizations and , that is , and being the antenna pattern functions as defined in Equation (57). The polarization amplitudes for a given mode are given by

where is the angle between the black hole’s spin axis and the observer’s line of sight and is an unknown constant phase. The quality factor of a mode is defined as and gives roughly the number of oscillations that are observable before the mode dies out. Figure 8 [78] plots frequencies and quality factors for the first few QNMs as a function of the dimensionless spin parameter . The mode of a Schwarzschild black hole corresponding to , is given by For stellar-mass–black-hole coalescences expected to be observed in ground-based detectors the ringdown signal is a transient that lasts for a very short time. However, for space-based LISA the signal would last several minutes for a black hole of . In the latter case, the ringdown waves could carry the energy equivalent of converted to gravitational waves – a phenomenal amount of energy compared even to the brightest quasars and gamma ray bursts. Thus, LISA should be able to see QNMs from black hole coalescences anywhere in the universe, provided the final (redshifted) mass of the black hole is larger than about , as otherwise the signal lasts for far too short a time for the detector to accumulate the SNR. Berti et al. [78] have carried out an exhaustive study, in which they find that the LISA observations of
SMBH binary mergers could be an excellent testbed for the no-hair theorem. Figure 9 (left panel) plots the
fractional energy that must be deposited in the ringdown mode so that the event is observable at a
distance of 3 Gpc. Black holes at 3 Gpc with mass in the range of would be observable
(i.e., will have an SNR of 10 or more) even if a fraction of energy is in the ringdown phase.
Numerical relativity predicts that as much as 1% of the energy could be emitted as QNMs,
when two black holes merge, implying that the ringdown phase could be observed with an
SNR of 100 or greater all the way up to , provided their mass lies in the appropriate
range^{10}.
Furthermore, they find that at this redshift it should be possible to resolve the fundamental
mode. Since black holes forming from primordial gas clouds at could well be the
seeds of galaxy formation and large-scale structure, LISA could indeed witness their formation through out
the cosmic history of the universe.

Figure 9 (right panel) shows SNR-normalized errors (i.e., one-sigma deviations multiplied by the SNR) in the measurement of the various QNM parameters (the mass of the hole , its spin , the QNM amplitude and phase ) for the fundamental mode. We see that, for expected ringdown efficiencies of into the fundamental mode of an a-million–solar-mass black hole with spin at 3 Gpc (), the mass and spin of the black hole can measured to an accuracy of a tenth of a percent.

By observing a mode’s frequency and damping time, one can deduce the (redshifted) mass and spin of the black hole. However, this is not enough to test the no-hair theorem. It would be necessary, although by no means sufficient, to observe at least one other mode (whose damping time and frequency can again be used to find the black hole’s mass and spin) to see if the two are consistent with each other. Berti et al [78] find that such a measurement should be possible if the event occurs within a redshift of .

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