Gravitational waves from oscillating accretion tori: Comparison between different approaches

Quasiperiodic oscillations of high density thick accretion disks orbiting a Schwarzschild black hole have been recently addressed as interesting sources of gravitational waves. The aim of this paper is to compare the gravitational waveforms emitted from these sources when computed using (variations of) the standard quadrupole formula and gauge-invariant metric perturbation theory. To this goal we evolve representative disk models using an existing general relativistic hydrodynamics code which has been previously employed in investigations of such astrophysical systems. Two are the main results of this work: First, for stable and marginally stable disks, no excitation of the black hole quasinormal modes is found. Second, we provide a simple, relativistic modification of the Newtonian quadrupole formula which, in certain regimes, yields excellent agreement with the perturbative approach. This holds true as long as back-scattering of GWs is negligible. Otherwise, any functional form of the quadrupole formula yields systematic errors $\sim 10\%$.

Figure 1

Left panel: Time evolution of the $\ell =2,m=0$ GWs for models A and B, computed using the Zerilli-Moncrief equation. The first 60 periods of a $t=100t_{\mathrm{orb}}$ simulation are shown. The oscillatory pattern in the waveforms is the response of the quasiperiodic oscillations of the disks. Right panel: Energy spectra for model A (solid line) and B (dashed line). Only fluid $p$-modes are visible in the spectra (enlarged in the inset, where a Hamming filter was used) which show no evidence of the excitation of the fundamental QNM of the black hole (its frequency is indicated by a vertical dotted line at $2M\omega =0.7473$).

Figure 2

Comparison between the $\ell =2$ gravitational waveforms for model B extracted using the quadrupole formula ${\mathrm{SQF}}_1$ (dotted line) and the gauge-invariant approach (solid line). The bottom panel shows the relative difference between the two normalized to the mean amplitude of the gauge-invariant method.

Figure 3

Comparison of the energy spectra for model B and a portion of the waveform when different definitions of the generalized quadrupole moment of the torus are used.

Figure 4

Model B: comparison between the Zerilli-Moncrief function, the ${\mathrm{SQF}}_1$ and the SF for different choices of the potential $\Phi$. The inset shows more periods of oscillations.