Charged Superradiant Instability of Spherically Symmetric Regular Black Holes in de Sitter Spacetime: Time- and Frequency-Domain Analysis

We investigate the superradiant instability of Ayón-Beato-García-de Sitter (ABG-dS) black holes under massless charged scalar perturbations using both time-domain evolutions and frequency-domain computations. We show that the instability occurs only for the spherically symmetric mode with $\ell=0$, whereas asymptotically flat ABG black holes remain stable in the massless limit, which underscores the essential role of the cosmological horizon in providing a confining boundary. We further study the dependence of the growth rate on the cosmological constant $Λ$, the scalar charge $q$, and the black hole charge $Q$, finding that it reaches a maximum at intermediate values of $Λ$ and $q$ and increases monotonically with $Q$. Compared with Reissner-Nordström-de Sitter black holes, ABG-dS black holes exhibit distinct instability characteristics due to the modified electrostatic potential induced by nonlinear electrodynamics.

💡 Research Summary

The paper investigates the superradiant instability of Ayón‑Beato‑García‑de Sitter (ABG‑dS) regular black holes when perturbed by a mass‑less, charged scalar field. Using both time‑domain numerical evolutions and frequency‑domain eigenvalue calculations, the authors demonstrate that instability appears exclusively in the spherically symmetric ℓ = 0 mode; all higher multipoles remain stable. The presence of a positive cosmological constant Λ introduces a cosmological horizon that naturally confines amplified modes, a feature absent in asymptotically flat ABG black holes, which stay stable in the mass‑less limit.

The metric function f(r) and the electric potential Φ(r) of the ABG‑dS solution are presented, highlighting the role of nonlinear electrodynamics in regularizing the spacetime at r = 0. The allowed parameter space (black‑hole charge Q versus Λ) is mapped out, showing three horizons (inner, event, cosmological) for appropriate choices. The Klein–Gordon equation with minimal coupling to the electromagnetic field yields an effective potential V_I(r) for time‑domain work and V_II(r) for frequency‑domain analysis. Both potentials contain a term –q²Φ², a centrifugal barrier ℓ(ℓ+1)/r², and contributions from the metric derivative.

In the time‑domain study, a second‑order finite‑difference scheme evolves an initial Gaussian pulse on a tortoise‑coordinate grid. The Courant factor Δt/Δr* = 0.5 ensures numerical stability, and boundaries are placed far enough to avoid spurious reflections. Results show that for fixed Q and scalar charge q, the perturbation grows exponentially only when Λ is below a critical value (corresponding to λ ≈ 2.06, where λ ≡ 10⁻³/Λ). The growth rate peaks near λ ≈ 2.58, then declines as Λ→0, confirming that the well‑depth of V_I(r) governs the instability. Increasing Q monotonically enhances the growth rate, while varying q yields a non‑monotonic behavior: the rate rises for small q, reaches a maximum around q ≈ 0.82, and then falls for larger q. For ℓ ≥ 1, the centrifugal term lifts the potential, preventing the formation of a trapping well; consequently, all higher‑ℓ evolutions remain bounded and oscillatory.

The frequency‑domain analysis solves the Schrödinger‑type radial equation with ingoing boundary conditions at the event horizon and outgoing conditions at the cosmological horizon. The direct integration method yields complex frequencies ω. Superradiance requires ω_cc < Re ω < ω_c, where ω_c = q Φ(r_h) and ω_cc = q Φ(r_c). Unstable modes satisfy Im ω > 0. The computed spectra reproduce the time‑domain findings: Im ω becomes positive for λ ≳ 2.06, peaks near λ ≈ 2.58, and vanishes as Λ→0. The dependence on Q mirrors the time‑domain result—larger Q produces larger Im ω—while the q‑dependence is again non‑monotonic with a peak near q ≈ 0.8. The real part of ω stays close to the lower superradiant bound ω_cc throughout.

A direct comparison with Reissner‑Nordström‑de Sitter (RN‑dS) black holes reveals distinct features. RN‑dS becomes unstable at a larger Λ (λ ≈ 1.55) and attains a higher peak growth rate, indicating a stronger superradiant amplification. However, for sufficiently small Λ both models converge to similar, small positive Im ω values, reflecting the disappearance of the trapping well in the Λ → 0 limit. The differences are attributed to the modified electrostatic potential in ABG‑dS arising from nonlinear electrodynamics, which softens the potential well and reduces the amplification efficiency.

In summary, the work establishes that regular black holes sourced by nonlinear electrodynamics can exhibit superradiant instabilities when embedded in de Sitter space, but only for the monopole scalar mode. The instability is controlled by a delicate interplay among the cosmological constant, black‑hole charge, and scalar charge. The findings underscore the essential confining role of the cosmological horizon and illustrate how nonlinear electromagnetic corrections modify the instability landscape compared to the standard RN‑dS case. Prospective extensions include adding scalar mass, rotation, alternative nonlinear electrodynamics models, and multi‑field interactions to explore richer instability phenomena.