Quasinormal ringing of Kerr black holes: The excitation factors

Distorted black holes radiate gravitational waves. In the so-called ringdown phase radiation is emitted as a discrete set of complex quasinormal frequencies, whose values depend only on the black hole’s mass and angular momentum. Ringdown radiation could be detectable with large signal-to-noise ratio by the Laser Interferometer Space Antenna (LISA). If more than one mode is detected, tests of the black hole nature of the source become possible. The detectability of different modes depends on their relative excitation, which in turn depends on the cause of the perturbation (i.e. on the initial data). A universal, initial data-independent measure of the relative mode excitation is encoded in the poles of the Green’s function that propagates small perturbations of the geometry (“excitation factors”). We compute the excitation factors for general-spin perturbations of Kerr black holes. We find that for corotating modes with $l=m$ the excitation factors tend to zero in the extremal limit, and that the contribution of the overtones should be more significant when the black hole is fast rotating. We also present the first analytical calculation of the large-damping asymptotics of the excitation factors for static black holes, including the Schwarzschild and Reissner-Nordström metrics. This is an important step to determine the convergence properties of the quasinormal mode expansion.

Figure 1

Integration contour for the vibrating string problem. Crosses mark zeros of the Wronskian $W$, corresponding to the normal frequencies of the system.

Figure 2

Integration contour to invert Eq. (3.6). The shaded area is the branch cut and crosses mark zeros of the Wronskian $W$ (the QNM frequencies).

Figure 3

Real part (left) and imaginary part (right) of the scalar QNEFs for the fundamental mode with $l=2$. Solid lines correspond to $|m|=2$, dashed lines to $|m|=1$, and the dotted line to $m=0$. The lines displaying oscillations for large $j$ have $m>0$.

Figure 4

Path of the scalar QNEFs for the fundamental mode (left) and first overtone (right) with $l=2$ as $j$ varies in the range $0\le j\le 0.996$. Line styles are the same as in Fig. 3.

Figure 5

Path of the electromagnetic QNEFs for the fundamental mode (left) and first overtone (right) with $l=1$ as $j$ varies in the range $0\le j\le 0.996$. The top panels refer to the SN formalism, the bottom panels to the Teukolsky formalism. Line styles are the same as in Fig. 3.

Figure 6

Real part (left) and imaginary part (right) of the gravitational QNEFs with $l=2$ and different values of the overtone index $n$, indicated in the inset. All plots refer to the SN formalism. Line styles are the same as in Fig. 3.

Figure 7

Path of the gravitational QNEFs for the fundamental mode (left) and first overtone (right) with $l=2$ as $j$ varies in the range $0\le j\le 0.996$. The top panels refer to the SN formalism, the bottom panels to the Teukolsky formalism. Line styles are the same as in Fig. 3.

Figure 8

Modulus of the SN QNEFs for the fundamental mode and first overtone with $l=m=2$. Solid lines are for gravitational perturbations, dashed lines for scalar perturbations.

Figure 9

Time-domain single-mode waveforms for scalar perturbations of a Kerr black hole with $j=0.98$. Each panel shows results for different values of $(l,m,n)$. Dotted lines correspond to initial data located extremely close to the horizon, at $r^S=0.6$. Solid lines are obtained for initial data localized relatively far away from the black hole, at $r^S=2$. Dashed lines show the response for initial data localized at the point of minimal excitation, that we determined to be: $r^S\simeq 0.86$ ($l=m=2$, $n=0$); $r^S\simeq 0.87$ $(l=m=2,n=1)$.