- The paper demonstrates dynamical stability for massless scalar and electromagnetic perturbations, with all computed quasinormal modes showing negative imaginary parts across the tested parameter space.
- Using sixth-order WKB calculations, eikonal null-geodesic checks, and effective-potential analysis, the study finds that string clouds and quintessence lower barrier heights and reduce oscillation frequencies and damping rates, while quantum corrections produce the opposite trend.
- The paper finds that greybody transmission and rigorous scalar lower bounds respond consistently to the model parameters, indicating enhanced Hawking emission from string clouds or quintessence and suppressed transmission from stronger quantum deformation.
Background and motivation
This paper analyzes the perturbative response of the quantum Oppenheimer–Snyder (QOS) black hole — a loop-quantum-gravity-inspired regularization of gravitational collapse — when the hole is simultaneously surrounded by quintessential dark energy and a cloud of strings. The spacetime is static and spherically symmetric with metric function
S^(r)=1−a~e​−r2M​+r4σ~e​M2​−r3ϖ^e​+1h~e​​,
where σ~e​ encodes the quantum deformation, a~e​ the string-cloud contribution, and h~e​ (with equation-of-state parameter ϖ^e​=−2/3) the quintessence field. The horizon structure shows two horizons for all parameter choices examined; increasing σ~e​ expands the inner horizon substantially while leaving the outer one nearly unchanged, whereas increasing a~e​ or h~e​ lowers the metric function and increases both radii.
The study combines three complementary diagnostics: effective potentials of scalar and electromagnetic perturbations, quasinormal mode (QNM) frequencies computed via the sixth-order WKB approximation and cross-checked against the eikonal null-geodesic method, and greybody factors (GFs) obtained both from WKB transmission coefficients and from rigorous lower bounds in the tradition of Visser and Boonserm.
Effective potentials
Massless scalar and vector fields reduce, after separation of variables, to a Regge–Wheeler-type Schrödinger equation on the tortoise coordinate, with the spin-dependent effective potential
Veff​(r)=S^(r)[r2l(l+1)​+(ζ−ζ2)r21−S^(r)​+(1−ζ)r1​drdS^(r)​],
where ζ=0 for scalar and σ~e​0 for electromagnetic perturbations. The qualitative behavior is consistent across both spins: the barrier height grows with multipole number σ~e​1 and with the quantum correction parameter σ~e​2, while it decreases as either the string-cloud parameter σ~e​3 or the quintessence parameter σ~e​4 increases. A persistent asymmetry is that scalar barriers always exceed their electromagnetic counterparts at fixed parameters, implying stronger coupling of scalar modes to this geometry. These trends directly set expectations for the spectra: higher σ~e​5 or σ~e​6 should yield faster, more strongly damped ringing; larger σ~e​7 or σ~e​8 should produce slower, longer-lived oscillations with enhanced low-frequency transmission.
Quasinormal modes
Using the sixth-order WKB formula of Iyer–Will and Konoplya for the fundamental overtone (σ~e​9) and first overtone (a~e​0) at a~e​1, the authors tabulate complex frequencies for varying a~e​2 and a~e​3 at a~e​4, a~e​5. Representative values:
| Parameter |
Scalar (a~e​6) |
Electromagnetic (a~e​7) |
| a~e​8 |
a~e​9 |
h~e​0 |
| h~e​1 |
h~e​2 |
h~e​3 |
| h~e​4 |
h~e​5 |
h~e​6 |
| h~e​7 |
h~e​8 |
h~e​9 |
Two results carry the main physical weight. First, all computed frequencies have negative imaginary parts for both spins and all parameter values scanned, establishing dynamical stability of the QOS black hole against scalar and electromagnetic perturbations throughout the explored parameter space. Second, both Reϖ^e​=−2/30 and ϖ^e​=−2/31 decrease monotonically with increasing ϖ^e​=−2/32 and ϖ^e​=−2/33: exotic matter fields soften the effective potential, lengthening the ringdown while weakening damping. The quintessence effect is somewhat stronger than the string-cloud effect over comparable ranges (e.g., the scalar fundamental frequency drops by roughly 11% across the ϖ^e​=−2/34 scan versus roughly 4% across the ϖ^e​=−2/35 scan).
As an independent check, the eikonal-limit QNMs are computed via unstable circular null geodesics using the Cardoso et al. correspondence between angular velocity and Lyapunov exponent. The resulting curves for ϖ^e​=−2/36 and ϖ^e​=−2/37 versus ϖ^e​=−2/38 and ϖ^e​=−2/39 reproduce the WKB trends qualitatively and quantitatively, validating the numerical accuracy of the sixth-order WKB treatment for this spacetime. It should be noted that this agreement is expected to be strongest at large σ~e​0, so the close match at the plotted parameters supports but does not formally bound the WKB error at σ~e​1.
Greybody factors
Transmission coefficients are extracted from the WKB relation σ~e​2 under standard scattering boundary conditions. The GFs exhibit two opposing parametric trends that mirror the potential analysis:
- Suppressing factors: increasing σ~e​3 or σ~e​4 raises and widens the centrifugal/quantum-corrected barrier, reducing transmission probability and hence the Hawking flux reaching infinity.
- Enhancing factors: increasing σ~e​5 or σ~e​6 lowers the barrier, increasing tunneling probability, absorption cross-sections, and the observable Hawking intensity relative to the classical Schwarzschild case.
The paper acknowledges a limitation of the method here: the sixth-order WKB approximation loses accuracy at very low frequencies, where near-total reflection drives GFs toward zero; the authors argue this does not materially affect integrated emission rates, though no quantitative error estimate is provided.
Rigorous bounds
For scalar perturbations, the Visser–Boonserm bound is evaluated analytically between the event horizon σ~e​7 and cosmological horizon σ~e​8:
σ~e​9
with a~e​0 given in closed form involving logarithmic terms from the quintessence contribution and inverse-power terms from the quantum deformation. The resulting lower bounds respond to the model parameters exactly as the WKB-computed GFs do: they tighten with growing a~e​1 and a~e​2 and loosen with growing a~e​3 and a~e​4. This agreement provides an independent, assumption-light confirmation of the semi-analytical WKB trends, since the bound derivation does not rely on the WKB expansion itself. The bound analysis is restricted to the scalar sector on the grounds that electromagnetic GFs behave nearly identically — a claim asserted graphically rather than proven analytically.
Limitations and open questions
Several caveats qualify the results. The stability conclusion covers only massless scalar and electromagnetic fields at low overtone number; gravitational (a~e​5) and fermionic perturbations are not computed, even though the general spin-dependent potential is written down. All scans fix a~e​6 and a~e​7, leaving the dependence on the quintessential equation-of-state parameter unexplored. The claimed equivalence of scalar and electromagnetic greybody behavior is supported only visually. Finally, the work remains entirely classical-perturbative on a fixed quantum-corrected background; no backreaction of the quintessence field or string cloud on the geometry is modeled.
Conclusion
The paper establishes that the QOS black hole dressed with quintessence and a string cloud is dynamically stable to scalar and electromagnetic perturbations, with ringdown frequencies and damping rates monotonically reduced by both exotic-matter parameters, and with quantum corrections acting oppositely. Greybody factors and their rigorous analytic lower bounds agree in trend, jointly indicating enhanced Hawking transmission in the presence of quintessence and string clouds and suppressed transmission under stronger quantum deformation. The principal open question left by the analysis is whether the same stability and spectral trends persist for gravitational perturbations and for other quintessential equations of state.
