6.4 Black hole spectroscopy

An important question relating to the structure of a black hole is its stability. Studies that began in the 1970s [309381382398298360361299] showed that a black hole is stable under external perturbation. A formalism was developed to study how a black hole responds to generic external perturbations, which has come to be known as black hole perturbation theory [113]. What we now know is that a distorted Kerr black hole relaxes to its axisymmetric state by partially emitting the energy in the distortion as gravitational radiation. The radiation consists of a superposition of QNMs, whose frequency and damping time depend uniquely on the mass M and spin angular momentum J of the parent black hole and not on the nature of the external perturbation. The amplitudes and damping times of different modes, however, are determined by the details of the perturbation and are not easy to calculate, except in some simple cases.

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 [164Jump To The Next Citation Point] 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 [300Jump To The Next Citation Point]. The effective one-body (EOB) model [98Jump To The Next Citation Point99Jump To The Next Citation Point], 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.

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Figure 8: Normal mode frequencies (left) and corresponding quality factors (right) of fundamental modes with l = 2,3,4, as a function of the dimensionless black hole spin j, for different values of m = l,...,0,...− l (for each l, different line styles from top to bottom correspond to decreasing values of m). Figure reprinted with permission from [78Jump To The Next Citation Point]. Ⓒ External LinkThe American Physical Society.

QNMs are characterized by a complex frequency ω that is determined by three “quantum” numbers, (l,m, n) (see, e.g., [78Jump To The Next Citation Point]). Here (l,m ) are indices that are similar to those for standard spherical harmonics. For each pair of (l,m ) there are an infinitely large number of resonant modes characterized by another integer n. The time dependence of the oscillations is given by exp(iωt), where ω is a complex frequency, its real part determining the mode frequency and the imaginary part (which is always positive) giving the damping time: ω = ωlmn + i∕τlmn, ωlmn = 2πflmn defining the angular frequency and τlmn the damping time. The ringdown wave will appear in a detector as the linear combination h(t) of the two polarizations h+ and h ×, that is h(t) = F+h+ + F×h ×, F+ and F× being the antenna pattern functions as defined in Equation (57View Equation). The polarization amplitudes for a given mode are given by

( ) h = A-(flmn,-Qlmn,𝜖rd)(1 + cos2ι)exp −-πflmnt- cos(2πf t + φ ), + r Qlmn lmn lmn ( ) h× = A-(flmn,-Qlmn,𝜖rd)2cos ιexp −-πflmnt-- sin(2πflmnt + φlmn ), (111 ) r Qlmn
where ι is the angle between the black hole’s spin axis and the observer’s line of sight and φ lmn is an unknown constant phase. The quality factor Qlmn of a mode is defined as Qlmn = ωlmn τlmn ∕2 and gives roughly the number of oscillations that are observable before the mode dies out. Figure 8View Image [78Jump To The Next Citation Point] plots frequencies and quality factors for the first few QNMs as a function of the dimensionless spin parameter j = J ∕M 2. The mode of a Schwarzschild black hole corresponding to l = 2,m = n = 0, is given by
310M-⊙- −4 -M---- f200 = ±1.207 × 10 M Hz, τ200 = 5.537 × 10 10M s. (112 ) ⊙
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 M = 107M ⊙. In the latter case, the ringdown waves could carry the energy equivalent of 5 10 M ⊙ 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 106M ⊙, as otherwise the signal lasts for far too short a time for the detector to accumulate the SNR.

Berti et al. [78Jump To The Next Citation Point] 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 9View Image (left panel) plots the fractional energy 𝜖rd 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 M in the range of 106 –108M ⊙ would be observable (i.e., will have an SNR of 10 or more) even if a fraction 𝜖 ≃ 10 −7M rd 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 z ∼ 10, provided their mass lies in the appropriate range10. Furthermore, they find that at this redshift it should be possible to resolve the fundamental l = 2, m = 2 mode. Since black holes forming from primordial gas clouds at z = 10 –15 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.

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Figure 9: The smallest fraction of black hole mass in ringdown waveforms that is needed to observe the fundamental mode at a distance of 3 Gpc (left) for three values of the black hole spin, j = 0 (solid line) j = 0.80 (dashed line) and j = 0.98 (dot-dashed line) and the error in the measurement of the various parameters as a function of the black hole spin for the same mode (right). Figure reprinted with permission from [78Jump To The Next Citation Point]. Ⓒ External LinkThe American Physical Society.

Figure 9View Image (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 M, its spin j, the QNM amplitude A and phase φ) for the fundamental l = m = 2 mode. We see that, for expected ringdown efficiencies of 𝜖rd ≃ 10 −2M into the fundamental mode of an a-million–solar-mass black hole with spin j = 0.8 at 3 Gpc (ρ ∼ 2000), 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 z ∼ 0.5.

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