4.7 Ringing

Most studies of quasi-normal modes in black holes (black-hole ringing) have focused on the collapse of SMSs. A number of reviews already exist on this topic [162, 18, 17, 84] and we refer the reader to these texts for more details. For stellar-massed black holes, ringing is intimately connected to fragmentation. As we discussed in Section 4.6, although Duez et al. [73] found that fragmentation could lead to black-hole ringing with very strong GW amplitudes ≳ 10–21 at distances of 10 Mpc, FHH found that even with very optimistic accretion scenarios, such radiation will be of very low amplitude and beyond the upper frequency reach of LIGO-II (see [106] for details). In numerical studies, Nagar et al. [215, 216] argue that this accretion is not a superposition of quasi-normal modes (although see [9], indicating that the signal would be even lower than simple analytic estimates might predict). The strongest black-hole ringing signal will occur in SMBHs and the rest of this section will focus on the GW signal from these objects.

The outcome of SMS collapse can be determined only with numerical, relativistic 3D hydrodynamics simulations. Until recently, such simulations had been published only for nearly spherical collapse. The spherical simulations of Shapiro and Teukolsky [275] produced collapse evolutions that were nearly homologous. In this case, the collapse time τcoll is roughly the free-fall time at the horizon

( )1 ∕2 ( )− 1 -R3-- --M---- τcoll = 4πM = 14 s 106M ⊙ . (30 )
The peak GW frequency − 1 fGW = τcoll is then 10–2 Hz, if the mass of the star is 6 10 M ⊙. This is in the middle of LISA’s frequency band of 10–4 – 1 Hz [308, 95].

The amplitude h of this burst signal can be roughly estimated in terms of the star’s quadrupole moment

2M 2 h ≤ 𝜖----- Rd ( ) ( ) −18 --M---- ---d---- −1 ≤ 𝜖 ⋅ 1 × 10 106M 50 Gpc . (31 ) ⊙
Here d is the distance to the star and 𝜖 ∼ T ∕|W | is a measure of the star’s deviation from spherical symmetry. In this case, 𝜖 will be much less than one near the horizon, since the collapse is nearly spherical.

There are two possible aspherical collapse outcomes that have been studied. The first outcome is direct collapse to a SMBH. In this case, 𝜖 will be on the order of one near the horizon. Thus, according to Equation (31View Equation), the peak amplitude of the GW burst signal will be

( ) ( ) − 18 M d −1 hpk ≈ 1 × 10 106M--- 50-Gpc-- . (32 ) ⊙

Alternatively, the star may encounter the dynamical bar mode instability prior to complete collapse. Baumgarte and Shapiro [12Jump To The Next Citation Point] have estimated that a uniformly-rotating SMS will reach β ≈ 0.27 when R ∕M = 15. The frequency of the quasiperiodic gravitational radiation emitted by the bar can be estimated in terms of its rotation frequency to be

( )1 ∕2 fGW = 2fbar ∼ 2 GM--- R3 ( M ) −1 = 2 × 10− 3 Hz ------- , (33 ) 106M ⊙
when R ∕M = 15. The corresponding h pk, again estimated in terms of the star’s quadrupole moment, is
2M 2 hpk ≤ ----- Rd ( ) ( ) −1 −19 --M---- ---d---- ≤ 1 × 10 106M ⊙ 50 Gpc . (34 )
The LISA sensitivity curve is shown in Figure 30View Image (see [149] for details on the computation of this curve; a mission time of three years has been assumed). The GW signal from this dynamical bar mode could be detected with LISA.
View Image

Figure 30: A comparison between the GW amplitude h (f) for various sources and the LISA noise curve. See the text for details regarding the computations of h. The SMS sources are assumed to have masses of ∼ 106M ⊙ and be located at a luminosity distance of 50 Gpc. The bar-mode source is a dynamical bar mode.

Shibata and Shapiro [284] have published a fully general-relativistic, axisymmetric simulation of the collapse of a rapidly, rigidly-rotating SMS. They found that the collapse remained homologous during the early part of the evolution. An apparent horizon does appear in their simulation, indicating the formation of a black hole. Because of the symmetry condition used in their run, non-axisymmetric instabilities did not develop.

The collapse of a uniformly-rotating SMS has been investigated with post-Newtonian hydrodynamics, in 3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [259Jump To The Next Citation Point]. Their numerical scheme used a post-Newtonian approximation to the Einstein equations, but solved the fully relativistic hydrodynamics equations. Their initial model was an n = 3 polytrope.

The results of Saijo et al. (confirmed in conformally-flat simulations [256]) indicate that the collapse of a uniformly-rotating SMS is coherent (i.e., no fragmentation instability develops). The collapse evolution of density contours from their model is shown in Figure 31View Image. Although the work of Baumgarte and Shapiro [12] suggests that a bar instability should develop prior to black-hole formation, no bar development was observed by Saijo et al. They use the quadrupole approximation to estimate a mean GW amplitude from the collapse itself: −21 h = 4 × 10, for a 6 10 M ⊙ star located at a distance of 50 Gpc. Their estimate for fGW at the time of black-hole formation is −3 3 × 10 Hz. This signal would be detectable with LISA (see Figure 30View Image).

View Image

Figure 31: Meridional plane density contours from the SMS collapse simulation of Saijo, Baumgarte, Shapiro, and Shibata [259Jump To The Next Citation Point]. The contour lines denote densities ρ = ρc × d(1−i∕16), where ρc is the central density. The frames are plotted at (t∕tD, ρc, d) = (a) (5.0628 × 10–4, 8.254 × 10–9, 10–7), (b) (2.50259, 1.225 × 10–4, 10–5), (c) (2.05360, 8.328 × 10–3, 5.585 × 10–7), (d) (2.50405, 3.425 × 10–2, 1.357 × 10–7). Here t, tD, and M0 are the time, dynamical time (∘ --3---- = R e∕M, where Re is the initial equatorial radius and M is the mass), and rest mass. (Figure 15 of [259]; used with permission.)

Saijo et al. also consider the GW emission from the ringdown of the black-hole remnant. For the l = m = 2 quasi-normal mode of a Kerr black hole with a∕M = 0.9, they estimate the characteristic amplitude of emission to be h ≈ 1.2 × 10−20[(△EGW ∕M )∕10−4]1∕2 at fGW ∼ 2 × 10−2 Hz for an M = 106M ⊙ source located at a luminosity distance of 50 Gpc (see [174, 307, 285] for details). Here, △EGW ∕M is the radiated energy efficiency and may be −4 ≲ 7 × 10 [291]. This GW signal is within LISA’s range of sensitivity (see Figure 30View Image).


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