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REVIEW 4 major objections 8 minor 43 references

Dark Matter Surrounded Quartic Square-root Horndeski Black Hole: Thermodynamics, Optical Properties and Quasinormal Oscillations

T0 review · 4 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Comparing the predicted shadow with the measured Sgr A* image forces the Horndeski and perfect-fluid dark-matter parameters to be tiny, leaving general relativity as the effective description near the photon sphere.

desk verdict Standard toolkit applied to an asserted metric: the combined Horndeski–PFDM spacetime is never shown to solve the field equations, and the EHT bounds contain sign errors that contradict the paper's own tables. read the letter →

arxiv 2506.15763 v1 pith:5TQ3JGFS submitted 2025-06-18 gr-qc

Mrinnoy M. Gohain , Kalyan Bhuyan This is my paper · ORCID
classification gr-qc
keywords HorndeskigravityblackholeshadowperfectfluiddarkmatterquasinormalmodesthermodynamicsnullgeodesicsSgrA*WKBapproximation
topics Dark Matter
open problems Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a static black hole in quartic `square-root` Horndeski gravity—a scalar-tensor theory in which the scalar field couples non-minimally to curvature through a square-root kinetic term—immersed in a perfect-fluid dark-matter halo. The model's lapse function carries three extra parameters beyond the mass: $\beta$, $\eta$, and $b$. The paper shows how these parameters shift horizon structure, temperature, specific heat, photon orbits, shadow size, and scalar quasinormal ringing. Its main result is that matching the predicted shadow radius to the measured Sgr A* shadow forces $\beta < -0.5929$, $\eta < -0.3263$, and $b < 0.01059$ at $1\sigma$, so the spacetime near the photon sphere is effectively Schwarzschild. A companion thermodynamic result is that small-horizon states are locally stable but globally disfavoured, with no Hawking-Page phase transition.

What carries the argument

The load-bearing object is the combined lapse function $f(r)=1-\frac{2M}{r}-\frac{\beta^2}{2\eta r^2}+\frac{b}{r}\ln(r/b)$ for the quartic square-root Horndeski black hole immersed in perfect-fluid dark matter. Everything else flows from this function: the Hawking temperature $T=f'(r_h)/(4\pi)$, the specific heat $C$, the Helmholtz free energy $F$, the photon-sphere equation $f'(r_p)r_p-2f(r_p)=0$, the shadow radius $r_s=r_p/\sqrt{f(r_p)}$, and the scalar-field effective potential used in the sixth-order WKB quasinormal-mode calculation. The paper's parameter bounds come from combining the shadow-radius formula with the observed fractional deviation $\delta$.

What would settle it

Substitute the combined lapse function $f(r)=1-\frac{2M}{r}-\frac{\beta^2}{2\eta r^2}+\frac{b}{r}\ln(r/b)$ and a scalar-field profile into the Horndeski field equations together with the perfect-fluid dark-matter energy-momentum tensor. If the equations are not satisfied identically for some $(\beta,\eta,b)$, the combined metric is not a solution of the theory and the shadow-based bounds do not constrain Horndeski gravity.

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Extended reading notes

Core claim

The paper's central claim is that the combined metric $f(r)=1-\frac{2M}{r}-\frac{\beta^2}{2\eta r^2}+\frac{b}{r}\ln(r/b)$ produces observable signatures that current shadow measurements can already bound. Solving the photon-sphere condition $f'(r_p)r_p-2f(r_p)=0$ and using $r_s=r_p/\sqrt{f(r_p)}$, the paper compares $r_s/M$ with the $1\sigma$ interval $4.55 \lesssim r_s/M \lesssim 5.22$ derived from the fractional shadow deviation $\delta=-0.060 \pm 0.065$. This comparison forces $\beta<-0.5929$, $\eta<-0.3263$, and $b<0.01059$ at $1\sigma$ (with looser $2\sigma$ bounds), implying that any perfect-fluid dark-matter component is negligible within a few gravitational radii of Sgr A* and any Horndeski-induced departure from Schwarzschild is sub-percent. At the same time, the thermodynamic analysis shows the black hole is locally stable for small horizons but globally unstable, with no Hawking-Page transition.

Load-bearing premise

The paper assumes that adding the Horndeski term and the dark-matter term in the metric yields an exact solution of the theory's field equations; if the combined spacetime is not a genuine solution, the thermodynamic, shadow, and ringing results describe a metric the theory does not admit.

Editorial extensions

If this is right

  • If the bounds hold, any spherically symmetric dark-matter halo around Sgr A* has negligible gravitational backreaction inside the photon sphere, so current shadow measurements cannot distinguish it from vacuum.
  • The Horndeski couplings are pinned to a narrow window near the Schwarzschild limit, so future improvements in shadow resolution will either shrink the window further or reveal a deviation.
  • In this model, black holes with small horizons are locally thermodynamically stable but never globally preferred, and no Hawking-Page phase transition occurs.
  • Quasinormal-mode oscillation frequencies and damping rates move in opposite directions with $\beta$ and $\eta$, while both grow with the dark-matter parameter $b$, giving a potential ringdown signature for modified gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct field-equation check of the combined ansatz would promote the quoted bounds from constraints on a metric template to constraints on Horndeski gravity itself.
  • The same shadow-comparison pipeline applied to the M87* image or to future higher-resolution Galactic-center data would tighten the bounds, since each parameter enters the shadow radius monotonically.
  • A rotating version of the metric would turn the circular shadow into an oval, letting shape measurements break the degeneracy among $\beta$, $\eta$, and $b$.
  • In the eikonal limit the quasinormal frequencies should match the photon-sphere orbital frequency and Lyapunov exponent, so checking that correspondence would connect the WKB table to the geodesic analysis.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 8 minor

Summary. The manuscript studies the static spherically symmetric metric f(r)=1-2M/r-β²/(2ηr²)+(b/r)ln(r/b), presented as a quartic square-root Horndeski black hole immersed in perfect-fluid dark matter (PFDM), with parameters β, η, and b. It computes the Hawking temperature, ADM mass, specific heat, entropy, and free energy; analyzes null geodesics and the photon sphere; computes the shadow radius and compares it with Keck/VLTI-based EHT constraints on Sgr A* to obtain 1σ and 2σ bounds on β, η, and b; and calculates scalar quasinormal frequencies using sixth-order WKB approximation. The central claims are that small-horizon black holes are locally stable but never globally preferred, that increasing β and b enlarges the photon sphere and shadow while increasing η mildly shrinks them, and that EHT shadow observations require the Horndeski and PFDM parameters to be very small, leaving general relativity as the effective description near the photon sphere.

Significance. If the combined metric in Eq. (8) were a genuine exact solution of the Horndeski field equations with a PFDM source, the paper would provide a useful connection between modified gravity, dark matter, and EHT observations. The analysis is largely transparent: explicit expressions for thermodynamic quantities, a table of critical impact parameters and photon radii, a table of WKB frequencies, and an external comparison to EHT data with no parameter fitting, so the constraints are not circular. The quantitative shadow and QNM tables are in principle reproducible. However, the significance is fully conditional on the existence of the claimed solution and on resolving the internal inconsistencies in the thermodynamics, in the parameter-bound derivation, and in the reported parameter trends, each of which is problematic in the current text.

major comments (4)
  1. [II, Eq. (8)] The combined metric is asserted without derivation. Equation (6) is cited to [5] for the pure Horndeski solution, and the PFDM term is introduced by invoking [27], a general-relativistic PFDM paper. In a scalar-tensor theory, adding a matter source changes both the metric and the scalar-field equations; the scalar profile that solves the pure Horndeski system is not shown to survive, and no new profile is supplied. The authors need to demonstrate explicitly that f(r) in Eq. (8), together with a specified scalar profile and the PFDM stress tensor (7), satisfies the full Horndeski field equations, or provide a derivation from an existing exact solution. Until this is done, the thermodynamic quantities, the shadow and photon-sphere results in Tables I-V, and the QNM frequencies describe an arbitrary metric rather than the claimed Horndeski-PFDM theory.
  2. [III, Eqs. (13)-(14)] The entropy and the free energy are mutually inconsistent. Eq. (13) states S_BH = π r_h²/4, but the Helmholtz free energy in Eq. (14) follows from F = M - T S only if S_BH = π r_h², which is also the horizon area/4. With the stated value, Eq. (14) would have a different coefficient of β²/(η r_h), namely -9/32 instead of -3/8. This internal inconsistency affects the thermodynamic stability analysis and the claim that no Hawking-Page transition occurs (Fig. 5 and Section VII). The authors should correct Eq. (13) and verify all subsequent thermodynamic expressions.
  3. [V.A, Fig. 14 and Table I] The quoted 1σ bounds β < -0.5929, η < -0.3263, b < 0.01059 are not consistent with the text's conclusion that the parameters are very small, and their inequality directions appear to conflict with the data presented. Table I shows that, with the fixed values used in Fig. 14, the critical impact parameter is far below the EHT 1σ window (4.55M to 5.22M): for example, b_crit = 3.39M for (β,η,b) = (-0.5,-0.5,0.3). The bound β < -0.5929 permits β = -0.7, -0.8, for which the shadow is even smaller, so as written it does not confine the model to the EHT-allowed region. In addition, β²/(2|η|) evaluated at the quoted 1σ limits is about 0.54, not the 0.06 stated near Eq. (45). The authors should display the curves r_s/M versus each parameter over a range that actually enters the EHT band and quote the inequalities in the correct direction.
  4. [Table I vs. abstract and Section VII] The claimed monotonic dependence of the shadow on the model parameters is opposite to the values in Table I. For fixed β = -0.5, η = -0.5, increasing b from 0.1 to 0.7 decreases b_crit from 4.09145 to 3.06305; for fixed η = -0.5, b = 0.3, increasing |β| from 0.3 to 0.6 decreases b_crit from 3.56247 to 3.26334. The abstract states that increases in the dark matter parameter b enlarge the shadow, and Section VII states that 'increasing |β| or b consistently enlarges the critical impact parameter and photon sphere radius'. These statements are contradicted by the paper's own Table I. The qualitative summary of the geodesic and shadow results must be corrected, or the table and figures must be recomputed.
minor comments (8)
  1. [II, Eq. (3)] The sentence 'where η and β' is incomplete; it should state that they are constants and specify the sign condition required for a real scalar-field solution.
  2. [III, Eq. (13)] The text says the entropy follows the usual Bekenstein-Hawking form, but writes π r_h²/4; the horizon area/4 is π r_h², so the factor must be corrected.
  3. [Table V] The table and the surrounding text use 'η1' and 'β1' where η and β are meant; these should be renamed for clarity.
  4. [Fig. 3] In the first panel of Fig. 3, the legend lists β = -0.5, η = -0.5 twice and does not show a distinct η = -0.4 curve; the legend should be checked.
  5. [Fig. 14] The axis labels in Fig. 14 appear garbled in the compiled text, with missing symbols such as η, β, and b; the figure should be regenerated.
  6. [V.A] The phrase 'upper bounds' is misleading, because β < -0.5929 and η < -0.3263 are lower bounds on |β| and |η|, not upper bounds on the parameters; the wording should be reconciled with the actual inequalities.
  7. [VII] The conclusion's 'sub-percent departure' statement is inconsistent with the stated β²/(2|η|) ≲ 0.06, which corresponds to a six-percent correction; the numbers should be reconciled.
  8. [References] References [2] and [21] mix arXiv identifiers and DOIs with journal information; the reference list should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's predictions follow from an adopted metric and are benchmarked against external EHT data.

full rationale

The manuscript computes thermodynamic quantities, photon-sphere and shadow radii, and quasinormal-mode frequencies directly from the adopted lapse function in Eq. (8), which is imported from prior work by Babichev et al. and Xu et al. rather than re-derived here. Importing a solution ansatz is an assumption and a correctness risk, but it is not circular: the paper does not define any predicted quantity in terms of itself, and no parameter is fitted to the quantity it later claims to predict. The EHT Keck/VLTI shadow constraints are an external benchmark, not an output of the model, and the parameter intervals quoted in Section V are obtained by inverting the model's shadow formula against that external data. The references to Horndeski and PFDM solutions are not self-citations and are not used to forbid alternative models; they supply the starting metric. Similarly, the WKB quasinormal-mode calculation is a standard, independent method applied to the same metric, and the reported relative errors are internal convergence checks rather than circular validation. No self-definitional step, fitted-input-called-prediction step, or load-bearing self-citation chain can be exhibited from the text, so a non-circularity score of 0 is appropriate.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The model relies on three model parameters β, η, b that are carried over from prior literature. The main extra axiom is the validity of the combined metric, which is not derived. No new particles or entities are introduced.

free parameters (3)
  • β
    Horndeski coupling parameter, varied by hand in the analysis (e.g., -0.5); not fitted but constrained by EHT shadow range.
  • η
    Horndeski coupling parameter, varied by hand and required to share the sign of β; constrained by EHT.
  • b
    PFDM parameter, varied by hand (e.g., 0.3); constrained to be very small by EHT.
assumptions (3)
  • ad hoc to paper The combined metric f(r)=1-2M/r - β²/(2ηr²)+(b/r)ln(r/b) is an exact solution of the Horndeski field equations with PFDM energy-momentum tensor.
    Stated in Section II with citations [5,27] but no derivation or verification of field equations.
  • domain assumption The entropy is given by the Bekenstein-Hawking area law S=A/4.
    Used in Section III for free energy; not justified for a non-minimally coupled scalar-tensor theory where Wald entropy can differ.
  • domain assumption The 6th-order WKB approximation is accurate for the scalar QNM frequencies.
    Standard method; the paper reports convergence estimates but no comparison to exact or other numerical methods.

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Cite this review

Pith. "Pith review of Dark Matter Surrounded Quartic Square-root Horndeski Black Hole: Thermodynamics, Optical Properties and Quasinormal Oscillations." pith.science (2026). https://pith.science/paper/5TQ3JGFS

@misc{pith2026250615763,
  author       = {Pith},
  title        = {Pith review of: Dark Matter Surrounded Quartic Square-root Horndeski Black Hole: Thermodynamics, Optical Properties and Quasinormal Oscillations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TQ3JGFS}},
  note         = {Machine review of arXiv:2506.15763}
}
abstract

In this work, we study a special form of Horndeski solutions, viz. quartic "square-root" Horndeski black hole immersed in a perfect-fluid dark-matter halo, by examining its thermodynamics, null geodesic shape, optical shadow, and quasinormal ringdown spectrum. The model is characterized by three parameters, namely $\beta$, $\eta$ (non-minimal Horndeski coupling parameters), and $b$ (perfect fluid dark matter parameter), which collectively determine horizon properties and observational effects. To study thermodynamic stability, we used the specific heat and free energy arguments, with which we demonstrated that small horizon states are locally stable but are never globally preferred. Analytic solutions of null geodesics reveal the radius of the photon sphere and the critical impact parameter, proving that increases in the dark matter parameters and the Horndeski parameter $\beta$ enlarge both the photon sphere and the subsequent shadow, whereas increase in the Horndeski coupling $\eta$ causes a mild diminishing in the shadow radius. Numerical ray tracing verifies these qualitative trends in the apparent shadow. Using the 6$^{th}$ - order WKB approximation method, we also calculate the scalar quasinormal modes and determine that oscillation frequencies and damping rates behave oppositely according to the sign and magnitude of each parameter. By comparing the shadow radius with the recent Event Horizon Telescope constraints on the Sgr A*, we find a narrow window of parameter space that agrees with observed data. In other words, the coupling parameters should be very small. These measurements restrict modified gravity impacts within realistic astrophysical contexts.

Figures

Figures reproduced from arXiv: 2506.15763 by the authors.

Figure 1
Figure 1. FIG. 1. The lapse function is plotted for different combinations of the parameters [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The temperature is plotted for various values of the Horndeski parameters and the PFDM parameter. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The mass is plotted with respect to horizon radius for different combinations of the parameters [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The specific heat is plotted for different values combinations of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The free energy [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The effective potential for null-geodesics is shown for various combinations of the model parameters. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Geodesics of Null rays for different values of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Geodesics of Null rays for different values of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Geodesics of Null rays for different values of [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Phase portrait of [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Phase portrait of [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Phase portrait of [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The plot shows the shadow images for different values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The shadow radius is plotted against the model parameters [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The effective potential is plotted for different [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Plot of real and imaginary components for varying [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Plot of real and imaginary components for varying [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Plot of real and imaginary components for varying [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]

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