Gravitational waves from the Papaloizou-Pringle instability in black hole-torus systems

Black hole (BH)–torus systems are promising candidates for the central engine of gamma-ray bursts (GRBs), and also possible outcomes of the collapse of supermassive stars to supermassive black holes (SMBHs). By three-dimensional general relativistic numerical simulations, we show that an $m=1$ nonaxisymmetric instability grows for a wide range of self-gravitating tori orbiting BHs. The resulting nonaxisymmetric structure persists for a timescale much longer than the dynamical one, becoming a strong emitter of large amplitude, quasiperiodic gravitational waves. Our results indicate that both, the central engine of GRBs and newly formed SMBHs, can be strong gravitational wave sources observable by forthcoming ground-based and spacecraft detectors.

💡 Research Summary

This paper investigates the nonlinear development of the Papaloizou‑Pringle instability (PPI) in black‑hole–torus (BH‑torus) systems and the resulting gravitational‑wave (GW) emission, using three‑dimensional general‑relativistic (GR) simulations. The authors construct equilibrium BH‑torus configurations in the puncture framework, with a non‑spinning BH (M_BH set to unity) surrounded by self‑gravitating tori whose mass ratios R = M_tor/M_BH range from 0.06 to 0.10. Two families of specific angular‑momentum profiles are explored: (i) constant j (the “C” models) and (ii) a non‑constant profile j ∝ Ω⁻¹⁄⁶ (the “NC” models). A Γ = 4/3 polytropic equation of state is adopted to mimic a radiation‑dominated or relativistic‑electron gas. The simulations employ the SACRA code with fixed‑mesh‑refinement (six refinement levels, finest grid spacing Δx₆ = 0.05 M_BH) and cover ≈20–40 orbital periods, sufficient to follow the linear growth, saturation, and post‑saturation phases of the instability.

The main findings are:

1. PPI Growth: In all models except NC06, the m = 1 non‑axisymmetric mode grows exponentially, with growth rates Im(ω₁) ≈ 5–25 % of the local angular velocity Ω_c. The growth is faster for larger torus masses and for constant‑j profiles; the NC06 model, which has a steeply decreasing j(r), remains stable.

2. Nonlinear Saturation and Persistence: After ≈10 orbital periods the instability saturates, the maximum rest‑mass density rises briefly, then settles to a new quasi‑equilibrium. Crucially, the m = 1 deformation survives for many tens of orbital periods, because the centre of mass of the BH‑torus system does not coincide with the BH horizon, leaving a persistent lopsided mass distribution.

3. Gravitational‑Wave Emission: The dominant GW mode is (l,m) = (2,2), analogous to a test particle orbiting a black hole. The wave signal consists of an initial burst at saturation followed by long‑lived quasiperiodic oscillations at the Keplerian frequency of the torus (∼100–200 Hz for a 10 M⊙ BH). The effective strain is defined as
   h_eff =