Accretion-induced quasinormal mode excitation of a Schwarzschild black hole

By combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations, we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a Schwarzschild black hole. The matter models considered include quadrupolar dust shells and thick accretion disks, permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger. We focus on the interference pattern appearing in the energy spectra of the emitted gravitational waves and on the amount of excitation of the quasinormal modes of the accreting black hole. We show that, quite generically in the presence of accretion, the black-hole ringdown is not a simple superposition of quasinormal modes, although the fundamental mode is usually present and often dominates the gravitational-wave signal. We interpret this as due to backscattering of waves off the nonexponentially decaying part of the black-hole potential and to the finite spatial extension of the accreting matter. Our results suggest that the black-hole QNM contributions to the full gravitational-wave signal should be extremely small and possibly not detectable in generic astrophysical scenarios involving the accretion of extended distributions of matter.

Figure 1

Comparison between the outcome of a scattering problem over the Regge-Wheeler (solid lines) and the Pöschl-Teller (dashed lines) potential. The waveforms are extracted at $r_{\mathrm{obs}}=200M$ and refer to a choice of the parameters given by (from top to bottom and from left to right): $\sigma =M$ and $r_c=100M$, $\sigma =9.5M$ and $\sigma =11.5M$ with $r_c=50M$. The wider the pulse, the larger the differences between the two solutions. The top left panel shows a comparison between the profiles of the Regge-Wheeler and the Pöschl-Teller potentials.

Figure 2

The $\ell =2$ gravitational Zerilli-Moncrief function extracted at $r_{\mathrm{obs}}=500M$ for a dust shell falling from $r_0=15M$ with $\sigma /M=0.050$. Left panel: the complete waveform. Right panel: investigation of the presence of the black-hole QNMs: the dashed line was obtained through a nonlinear fit with the first five modes and the comparison with the computed waveform is shown on both a linear and a logarithmic vertical scale.

Figure 3

Left panel: Energy spectra and their typical interference fringes as computed from the waveforms produced by shells infalling from $r_0=15M$. Right panel: The same as in the left panel but only for the ringdown phase, showing the excitation of the $\ell =2$ black-hole QNMs. See text for details.

Figure 4

Infalling shells of dust: waveforms (left panels) and energy spectra (right panels) for two representative values of $r_0$ with $\sigma /M=0.050$. For the sake of comparison, the right panels show in dashed lines the $\ell =2$ energy spectra for a particle plunging radially onto the black hole (rescaled by a convenient factor).

Figure 5

Dependence of the characteristic frequency in the energy spectrum on the initial position of the shell.

Figure 6

Conformally-flat initial data and spurious bursts of radiation: energy spectra obtained by solving the Hamiltonian constraint with $\beta =0$ only (dashed line) and when the initial gravitational-wave pulse is additionally eliminated (solid line). The shell has $\sigma /M=0.158$, it falls from $r_0=7.5M$ and the signal is extracted at $r_{\mathrm{obs}}=500M$. The inset shows the corresponding initial part of the waveform.

Figure 7

Main panel: Total energy radiated in gravitational waves ($\ell =2$ multipole) as a function of the initial location $r_0$ of the center of a shell with $\sigma /M=0.158$. Inset panel: Total energy versus $\sigma$ for shells falling from $r_0=15M$ (open circles). The filled circles refer to the energy contribution coming from the black-hole ringdown only.

Figure 8

Model $D_0$. Left panel: time evolution of the equatorial rest-mass density profiles for a velocity perturbation $\eta =0.3$. Right panel: time evolution of the center of mass in the equatorial plane. The disk disappears in the black hole at $t\approx 0.75t_{\mathrm{orb}}$.

Figure 9

Model $D_0$: gravitational waveform (extracted at $r_{\mathrm{obs}}=200M$). The presence of two ringings is evident. The first one is related to the gravitational-wave content present in the initial profile of the Zerilli-Moncrief equation. The second ringing (magnified in the inset) is determined by the object crossing the peak of the potential. The dashed line, proportional to $u^{-7}$, shows that the late-time behavior of the waveform is properly captured.

Figure 10

Model $D_1$: time evolution of the projection of the center of mass in the equatorial plane. After an initial transient, the dynamics is characterized by constant-amplitude quasiperiodic oscillations. The inset shows the mass-accretion rate.

Figure 11

Model $D_1$. Left panel: gravitational waveforms (extracted at $r_{\mathrm{obs}}=500M$) versus retarded time. $h_{+}^{20}$ polarization (even-parity) superposed to the waveform extracted using the ${\mathrm{SQF}}_1$ (dashed line). Right panel: Energy spectrum for the $\ell =2$ and $\ell =3$ (odd-parity) modes. The highly damped oscillation in the early part of the $\ell =2$ waveform (from ${\Psi }_{20}^{(\mathrm{e})}$) results in a broad-band peak which is not related to the QNMs frequencies of the black hole.

Figure 12

Model $D_2$. Left panel: the motion of the center of mass on the equatorial plane and the mass-accretion rate (in the inset). Right panel: the qudrupole gravitational waveforms. The gauge-invariant (extracted at $r_{\mathrm{obs}}=500M$) and the ${\mathrm{SQF}}_1$ waveforms are shown on the same plot for comparison.