arxiv: 2604.24813 · v1 · submitted 2026-04-27 · 🌀 gr-qc

Quasinormal Modes, Greybody Factors and Rigorous Bounds for Quantum Oppenheimer-Snyder Black Hole with Quintessential Dark Energy and a String Clouds

W. Sajjad , A. Zahid , M. A. Muawia , M. Azam This is my paper

Pith reviewed 2026-05-08 02:19 UTC · model grok-4.3

classification 🌀 gr-qc

keywords quasinormal modesgreybody factorsquantum Oppenheimer-Snyder black holequintessential dark energystring cloudsWKB approximationscalar perturbationsvector perturbations

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The pith

Quasinormal modes and greybody factors of the quantum Oppenheimer-Snyder black hole with quintessence and string clouds are computed via sixth-order WKB, with greybody factors shown to depend strongly on the deformation parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the effective potentials for scalar and vector perturbations around a black hole metric that includes quantum corrections to the Oppenheimer-Snyder collapse plus quintessence dark energy and a cloud of strings. It then applies the sixth-order WKB approximation to extract the quasinormal mode frequencies and examines the greybody factors that describe wave transmission through the potential barrier. A sympathetic reader would care because these quantities determine the late-time ringing of the spacetime and the fraction of incident waves that escape to infinity, both of which are in principle measurable. The analysis shows that the string-cloud, quantum-deformation, and quintessence parameters each shift the frequencies and alter the greybody factors, and it supplies explicit bounds on those factors in the scalar case.

Core claim

What carries the argument

the sixth-order WKB approximation applied to the effective potentials for scalar and vector perturbations derived from the quantum Oppenheimer-Snyder metric that incorporates quintessence and string-cloud terms

If this is right

The quasinormal frequencies shift when the string-cloud parameter, quantum-deformation parameter, or quintessence parameter is varied.
Greybody factors for both scalar and vector perturbations change measurably with each of the three parameters.
Strict upper and lower bounds on the greybody factors exist for scalar perturbations once the parameters are fixed.
The shape of the effective potential barrier is altered by the same three parameters, affecting both oscillation and damping rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the computed frequencies match future gravitational-wave ringdown data, the model parameters could be bounded observationally.
The reported bounds on greybody factors imply corresponding limits on the fraction of energy that can be radiated to infinity in scalar channels.
The same WKB procedure could be repeated for higher-spin perturbations to test whether the parameter influences remain qualitatively similar.

Load-bearing premise

The sixth-order WKB approximation accurately reproduces the quasinormal frequencies for the effective potentials that arise from this particular combination of quantum deformation, quintessence, and string clouds.

What would settle it

A numerical solution of the radial wave equation for the same metric that yields quasinormal frequencies lying outside the error band of the reported sixth-order WKB values.

Figures

Figures reproduced from arXiv: 2604.24813 by A. Zahid, M. A. Muawia, M. Azam, W. Sajjad.

**Figure 1.** Figure 1: The behavior of function S ˆ¯(r) vs. r for different values of ˜σe, ˜ae, and h˜ e with fixed values of Mˇ¯ = 1, and ˆ̟e = −2/3. 7 view at source ↗

**Figure 2.** Figure 2: The scalar potential V ¯ˇ sc(r) (left panel) and the electromagnetic potential V ¯ˇ em(r) (right panel) versus r for different multipole moment ly values with fixed values of Mˇ¯ = 1, ˜ae = 0.11, ˜σe = 0.21, h˜ e = 0.012, and ̟ˆ e = −2/3. by the BH, they tend to decrease the GFs of lower-energy waves when the potential barrier is larger and broader, and, consequently, more radiation is reflected and less i… view at source ↗

**Figure 3.** Figure 3: The scalar potential V ¯ˇ sc(r) (left panel) and the electromagnetic potential V ¯ˇ em(r) (right panel) versus r for different quantum correction parameter ˜σe values with fixed values of Mˇ¯ = 1, ˜ae = 0.11, h˜ e = 0.012, ly = 2 and, ˆ̟e = −2/3 view at source ↗

**Figure 4.** Figure 4: The scalar potential V ¯ˇ sc(r) (left panel) and the electromagnetic potential V ¯ˇ em(r) (right panel) vs. r for various string parameter ˜ae values with fixed values of Mˇ¯ = 1, ˜σe = 0.21, h˜ e = 0.012, ly = 2 and, ˆ̟e = −2/3. h ˜ e = 0.013 h ˜ e = 0.017 h ˜ e = 0.025 h ˜ e = 0.031 2 3 4 5 6 -0.2 -0.1 0.0 0.1 r - ˇ Vsc(r) . . h ˜ e = 0.013 h ˜ e = 0.017 h ˜ e = 0.025 h ˜ e = 0.031 2 3 4 5 6 -0.2 -0.1 0.… view at source ↗

**Figure 5.** Figure 5: The scalar potential V ¯ˇ sc(r) (left panel) and the electromagnetic potential V ¯ˇ em(r) (right panel) vs. r for various quintessence state parameter h˜ e values, where, Mˇ¯ = 1, ˜σe = 0.21, ˜ae = 0.11, ly = 2 and, ˆ̟e = −2/3. as the cloud of strings parameter ˜ae and the quintessence parameter h˜ e are increased, the potential barrier to the effective potential decreases. This decreases the barrier heig… view at source ↗

**Figure 6.** Figure 6: Scalor (upper panel) and electromagnetic (lower panel) qu view at source ↗

**Figure 7.** Figure 7: Scalor (upper panel) and electromagnetic (lower panel) qu view at source ↗

**Figure 8.** Figure 8: Angular velocity ψˆ (left panel) and Lyapunov exponent Eˆ (right panel) versus the string cloud parameter ˜ae with a fixed parameters are Mˇ¯ = 1, ly = 2, ˜σe = 0.5, and, h˜ e = 0.03. 0.015 0.020 0.025 0.030 0.155 0.160 0.165 0.170 h ˜ e ψ , 0.015 0.020 0.025 0.030 0.140 0.145 0.150 0.155 0.160 h ˜ e E ? ? view at source ↗

**Figure 9.** Figure 9: Angular velocity ψˆ (left panel) and Lyapunov exponent Eˆ (right panel) versus quintessence parameter h˜ e with a fixed parameters are Mˇ¯ = 1, ly = 2, ˜σe = 0.5, and, ˜ae = 0.03. 18 view at source ↗

**Figure 10.** Figure 10: Greybody factors of the massless scalar (left panel) an view at source ↗

**Figure 11.** Figure 11: Greybody factors of the massless scalar (left panel) an view at source ↗

**Figure 12.** Figure 12: Rigorous constraints on GFs are obtained at different va view at source ↗

read the original abstract

We obtained quasinormal modes and greybody factors of the quantum Oppenheimer-Snyder (QOS) black hole with quintessential dark energy and string clouds. We compute the effective potentials of the scalar and the vector perturbation fields and analyze their graphical behavior. For this, we compute quasinormal mode frequencies of the QOS black hole, that is surrounded by quintessence dark energy and a cloud of strings and this is done by 6th-order Wentzel-Kramers-Brillouin (WKB) approximation method. We observe the quasinormal frequencies by analyzing both the quintessence parameters and the string cloud parameters. We also examine the GFs by analyzing the role of string cloud ($\tilde{a}_{e}$), quantum deformation ($\tilde{\sigma}_{e}$), and quintessence ($\tilde{h}_{e}$) parameters. It is found that GFs are significantly influenced by the parameters of $\tilde{a}_{e}$, $\tilde{\sigma}_{e}$, and $\tilde{h}_{e}$. In the case of scalar perturbation, we also give strict limits of the GFs and check the influence of the parameters of $\tilde{a}_{e}$, $\tilde{\sigma}_{e}$, and $\tilde{h}_{e}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

WKB-based QNM and greybody calculations for this specific quantum-quintessence-string-cloud black hole model lack any validation of the approximation accuracy.

read the letter

This paper takes the quantum Oppenheimer-Snyder black hole, adds quintessential dark energy and a cloud of strings, and then computes the quasinormal modes and greybody factors for scalar and vector perturbations. The main tool is the sixth-order WKB approximation applied to the effective potentials derived from the metric. They vary the three parameters—string cloud ã_e, quantum deformation σ̃_e, and quintessence h̃_e—and show how these change the quasinormal frequencies and the greybody factors. They also give some strict limits for the greybody factors in the scalar case and discuss the graphical shape of the potentials. This is all done in a straightforward way following common practices in the field. The calculations seem consistent with what one would expect from similar papers on modified black holes. The parameter scans provide a picture of how each matter component influences the ringdown and the scattering. However, the approach has a clear limitation in that the WKB method is used without any tests against more precise techniques. For potentials that include the extra contributions from quantum effects, quintessence, and strings, the standard WKB can deviate more than usual. Without error estimates or comparisons to Leaver's method or numerical integration of the wave equation, the specific numbers and the strength of the parameter effects remain uncertain. This is not a minor issue because the central results depend directly on those frequencies and factors. Overall, the work is a standard extension to a particular model rather than a breakthrough in method or insight. It would be of interest to people studying quasinormal modes in spacetimes with multiple matter fields. A serious referee could evaluate it, especially if the derivations of the potentials are clear in the text, but they would probably ask for validation of the approximation or at least a discussion of its expected accuracy. I would not cite this in my own work soon, and it is not something I would bring to a reading group meeting. Still, it deserves to go through peer review rather than being desk rejected outright.

Referee Report

2 major / 2 minor

Summary. The manuscript computes quasinormal mode frequencies and greybody factors for scalar and vector perturbations of the quantum Oppenheimer-Snyder black hole with quintessential dark energy and string clouds. Effective potentials are derived from the metric, plotted, and used to obtain QNMs via the sixth-order WKB approximation while scanning the parameters ã_e (string cloud), σ̃_e (quantum deformation), and h̃_e (quintessence). The influence of these parameters on the frequencies and GFs is analyzed, with strict limits reported for the scalar greybody factors.

Significance. If the numerical results hold, the work adds to the catalog of perturbation spectra in modified black-hole spacetimes that combine quantum corrections, dark-energy terms, and string-cloud contributions. The parameter scans and graphical analysis of potentials offer qualitative trends that could inform future gravitational-wave phenomenology. No machine-checked derivations, reproducible code, or falsifiable predictions beyond the WKB outputs are provided.

major comments (2)

[Abstract and QNM computation section] The sixth-order WKB approximation is applied directly to the effective potentials without any validation against exact methods (Leaver continued fractions, time-domain integration) or higher-order error estimates. The potentials contain additional 1/r^3 and exponential contributions from the three parameters; the standard WKB accuracy assumptions therefore do not automatically carry over, so the reported frequencies and the claimed parameter influences rest on an untested approximation.
[Greybody factors section] The 'strict limits' on scalar greybody factors are obtained from the same unvalidated potentials and transmission integrals. No quantitative uncertainty or cross-check against numerical integration of the wave equation is supplied, so the stated significant influence of ã_e, σ̃_e, and h̃_e on the GFs inherits the same uncontrolled systematic error.

minor comments (2)

[Introduction and metric section] The tilded notation for the three free parameters should be introduced with explicit definitions and distinguished from similar symbols in the broader literature on quintessential or string-cloud spacetimes.
[Figures] Figure captions for the effective potentials and greybody-factor plots should state the fixed values of the remaining parameters and the range of the plotted variable to allow direct reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on quasinormal modes and greybody factors for the quantum Oppenheimer-Snyder black hole with quintessential dark energy and string clouds. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract and QNM computation section] The sixth-order WKB approximation is applied directly to the effective potentials without any validation against exact methods (Leaver continued fractions, time-domain integration) or higher-order error estimates. The potentials contain additional 1/r^3 and exponential contributions from the three parameters; the standard WKB accuracy assumptions therefore do not automatically carry over, so the reported frequencies and the claimed parameter influences rest on an untested approximation.

Authors: We agree that the manuscript does not include cross-validation of the sixth-order WKB results against exact methods such as Leaver's continued fractions or time-domain integration, nor does it provide higher-order error estimates. The additional 1/r^3 and exponential terms in the potentials from the string cloud, quantum deformation, and quintessence parameters mean that the usual accuracy benchmarks for WKB may not fully apply. The sixth-order WKB remains a widely used approximation in the literature for exploring parameter dependence in modified black-hole spacetimes, and the qualitative trends we report follow directly from the shapes of the effective potentials that are plotted and analyzed in the paper. To address the concern, we will revise the manuscript to include an explicit discussion of the limitations of the WKB approach in this setting and note that the results should be viewed as indicative of trends rather than high-precision values. revision: partial
Referee: [Greybody factors section] The 'strict limits' on scalar greybody factors are obtained from the same unvalidated potentials and transmission integrals. No quantitative uncertainty or cross-check against numerical integration of the wave equation is supplied, so the stated significant influence of ã_e, σ̃_e, and h̃_e on the GFs inherits the same uncontrolled systematic error.

Authors: We acknowledge that the strict limits on the scalar greybody factors are computed from the same effective potentials without supplying quantitative uncertainty estimates or independent numerical integration of the wave equation. Consequently, while the parameter dependence on ã_e, σ̃_e, and h̃_e is clearly visible through the modifications to the potentials and the resulting transmission integrals, the absolute values of the bounds carry the systematic uncertainty associated with the WKB approximation. The term 'strict limits' in the manuscript refers to the analytic bounds derived under the given potential and the integral expressions employed. In the revised manuscript we will add a clarifying paragraph that emphasizes the indicative nature of the reported influences and the absence of direct numerical cross-checks. revision: partial

Circularity Check

0 steps flagged

No circularity detected; QNMs and GFs computed directly from metric via standard WKB

full rationale

The paper first states the QOS metric with parameters for quantum deformation, quintessence, and string clouds, then derives the effective potentials for scalar and vector perturbations from the line element in the standard way. Quasinormal frequencies follow by direct application of the 6th-order WKB formula to those potentials, and greybody factors are obtained from the corresponding transmission integrals. No parameters are fitted to the output data, no self-citations supply load-bearing uniqueness theorems or ansatze, and no known empirical pattern is merely relabeled. All reported values are therefore genuine forward computations from the input metric and the chosen approximation method rather than reductions to the inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

The central results rest on the validity of the underlying spacetime metric as a solution to Einstein's equations with the added quantum, quintessence, and string-cloud terms, plus the applicability of the WKB series to its perturbation potentials.

free parameters (3)

string cloud parameter ã_e
Model parameter varied to examine its effect on frequencies and greybody factors
quantum deformation parameter σ̃_e
Model parameter varied to examine its effect on frequencies and greybody factors
quintessence parameter h̃_e
Model parameter varied to examine its effect on frequencies and greybody factors

axioms (2)

domain assumption The quantum Oppenheimer-Snyder metric supplemented by quintessential dark energy and string clouds is a valid exact solution of the field equations
Taken as the background geometry for deriving effective potentials
domain assumption The sixth-order WKB approximation yields sufficiently accurate quasinormal frequencies for the potentials of this spacetime
Directly employed to obtain the reported mode frequencies

invented entities (1)

Quantum Oppenheimer-Snyder black hole with quintessential dark energy and string clouds no independent evidence
purpose: Spacetime model combining quantum collapse, dark energy, and string matter
No independent observational or theoretical evidence supplied in the abstract

pith-pipeline@v0.9.0 · 5539 in / 1674 out tokens · 65235 ms · 2026-05-08T02:19:35.362682+00:00 · methodology

discussion (0)

Reference graph

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