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 spin9. 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
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
range10.
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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