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6.4 Numerical predictions of GW emission

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 [220] produced collapse evolutions that were nearly homologous. In this case, the collapse time tcoll is roughly the free-fall time at the horizon

( )1/2 ( ) -1 -R3-- --M----- tcoll = 4pM = 14 s 106Mo. . (7)
The peak GW frequency - 1 fGW = tcoll is then -2 10 Hz, if the mass of the star is 6 10 Mo .. This is in the middle of LISA’s frequency band of 10-4 -1 Hz [24376].

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

2M 2 h < e----- Rd ( ) ( )- 1 -18 --M----- ---d---- < e .1× 10 106 Mo . 50 Gpc . (8)
Here d is the distance to the star and e ~ T/|W | is a measure of the star’s deviation from spherical symmetry. In this case, e will be much less than one near the horizon, since the collapse is nearly spherical.

There are two possible aspherical collapse outcomes that can be discussed. The first outcome is direct collapse to a SMBH. In this case, e will be on the order of one near the horizon. Thus, according to Equation 8View Equation, the peak amplitude of the GW burst signal will be

( ) ( )- 1 hpk ~ 1× 10-18 --M----- ---d---- . (9) 106 Mo . 50 Gpc

Alternatively, the star may encounter the dynamical bar mode instability prior to complete collapse. Baumgarte and Shapiro [16Jump To The Next Citation Point] have estimated that a uniformly rotating SMS will reach b ~ 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 --6----- , (10) 10 Mo .
when R/M = 15. The corresponding hpk, again estimated in terms of the star’s quadrupole moment, is
2M 2 hpk < ----- Rd ( ) ( ) -1 - 19 --M----- ---d---- < 1 × 10 106Mo . 50 Gpc . (11)
The LISA sensitivity curve is shown in Figure 18View Image (see [119] for details on the computation of this curve; a mission time of 3 years has been assumed). The GW signal from this dynamical bar-mode could be detected with LISA.
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Figure 18: 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 be located at a luminosity distance of 50 Gpc. The bar-mode source is a dynamical bar-mode.
Shibata and Shapiro [228] 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 were unable to 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 [208Jump 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 [207]UpdateJump To The Next Update Information) 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 19View Image. Although the work of Baumgarte and Shapiro [16Jump To The Next Citation Point] suggests that a bar instability should develop prior to BH 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 Mo . star located at a distance of 50 Gpc. Their estimate for fGW at the time of BH formation is -3 3× 10 Hz. This signal would be detectable with LISA (see Figure 18View Image).

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Figure 19: Meridional plane density contours from the SMS collapse simulation of Saijo, Baumgarte, Shapiro, and Shibata [208Jump To The Next Citation Point]. The contour lines denote densities r = rc × d(1-i/16), where rc is the central density. The frames are plotted at (t/t D, r c, d)=(a)(5.0628 × 10- 4, 8.254 × 10 -9, - 7 10), (b)(2.50259, -4 1.225 × 10, -5 10), (c)(2.05360, - 3 8.328 × 10, -7 5.585 × 10), (d)(2.50405, 3.425 × 10- 2, 1.357 × 10-7), respectively. Here t, tD, and M0 are the time, dynamical time (V~ ------- = R3e/M, where Re is the initial equatorial radius and M is the mass), and rest mass. (Figure 15 of [208Jump To The Next Citation Point]; 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 = 106Mo . source located at a luminosity distance of 50 Gpc (see [149242229] for details). Here, /_\ EGW/M is the radiated energy efficiency and may be <~ 7× 10 -4 [233]. This GW signal is within LISA’s range of sensitivity (see Figure 18View Image).

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