The Ringdown and the Tide: Fingerprints of Dark Matter Halo Profiles

Arkadip Bhowmik; Avijit Chowdhury; Sayan Chakrabarti

REVIEW 2 major objections 4 minor 123 references

The Ringdown and the Tide: Fingerprints of Dark Matter Halo Profiles

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A single redshift integral controls the ringdown and tidal response of black holes in dark-matter halos, with ringdown and tidal data probing opposite sides of the halo.

desk verdict Useful new eikonal and tidal machinery for DM halos, but the perturbation closure is misderived – the QNM and TLN predictions are conditional until that's fixed. read the letter →

arxiv 2608.07678 v1 pith:7KWH6ZS5 submitted 2026-08-07 gr-qc

Arkadip Bhowmik , Avijit Chowdhury , Sayan Chakrabarti This is my paper · ORCID

classification gr-qc MSC 83C5783C35 PACS 04.70.-s04.30.-w95.35.+d

keywords quasinormalmodesdarkmatterhalostidalLovenumbersblackholeperturbationtheoryRegge-WheelerequationEinsteinclustergravitationalwaveringdown(alphabetagamma)densityprofile

topics 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 asks whether gravitational-wave observables from a black hole embedded in a dark-matter halo respond only to the total halo mass, or also to the halo's shape. Working with anisotropic Einstein-cluster halos and generalized $(\alpha,\beta,\gamma)$ density profiles, it shows that the axial ringdown spectrum shifts by a single common factor $e^{-I/2}$: both the oscillation frequency and the damping rate are redshifted by the same amount, equal to the light-ring shift, where $I$ is a positive integral over the halo mass distribution outside the inner cut-off. The same construction gives a nonzero axial tidal Love number proportional to an outer-weighted mass moment, so the ringdown and the tidal response are sensitive to opposite parts of the halo. Because configurations that are degenerate in ringdown generally differ in tidal response, combining the two could distinguish environmental effects from genuine deviations from vacuum black-hole geometry.

What carries the argument

The central object is the halo redshift integral $I = \int_{4M_{\mathrm{BH}}}^{\infty} \frac{2m_h(r)}{[r-2m(r)][r-2M_{\mathrm{BH}}]}\,dr$, which encodes the full profile dependence of the ringdown shift. The derivation rests on the fact that inside the DM-free cavity ($r \le 4M_{\mathrm{BH}}$) the axial potential is exactly $e^{-I}$ times the Schwarzschild Regge-Wheeler potential, so the master equation maps to the Schwarzschild problem with $\omega \to e^{I/2}\omega_{\mathrm{DM}}$. The eikonal identity $\lambda_c = \Omega_c$ follows because the local derivatives of the redshift function at the light ring retain their Schwarzschild values and only the global factor $f(r_c)=e^{-I}/3$ changes. For the tidal sector, the machinery is the Riccati equation for $y = r h_0'/h_0$; linearization about the pure growing branch yields $\kappa_l$ as an outer-weighted mass moment, with the truncation radius $R_{99}$ supplying the outer scale.

What would settle it

Take a fixed $(\alpha,\beta,\gamma)$ halo and solve the axial perturbation system without imposing the irrotational closure, keeping $\delta u$ and $\delta w$ as independent variables; if the fundamental $l=2$ quasinormal frequency differs from $e^{-I/2}\omega_{\mathrm{Sch}}$ by more than the estimated subleading corrections (fractional shift of order $10^{-3}$), the central result depends on that closure.

Watch

Extended reading notes

Core claim

The paper constructs static, spherically symmetric black holes dressed with anisotropic dark-matter halos modeled as Einstein clusters with generalized $(\alpha,\beta,\gamma)$ density profiles, and studies their axial gravitational perturbations. It shows that the entire ringdown spectrum is a uniformly redshifted Schwarzschild spectrum, $\omega_{\mathrm{DM}} = e^{-I/2} \omega_{\mathrm{Sch}}$, where $I$ is the halo redshift integral of Eq. (46) built from the enclosed halo mass outside the light ring. Because the same factor multiplies the real and imaginary parts, the fractional shifts of the oscillation frequency and the damping rate are equal and coincide with the shifts of the light-ring frequency and Lyapunov exponent. The paper also derives the static axial tidal Love number to leading order in halo compactness, finding $\kappa_l \propto \int m_h'(r) r^{2l}\,dr$; for $l=2$ this is $(4\pi/5)\int \bar\rho(r) r^6\,dr$, dominated by the outermost halo and the truncation radius. Since the ringdown shift is inner-weighted and the tidal response outer-weighted, the two observables break the compactness-shape degeneracy.

Load-bearing premise

The main results assume the halo fluid is irrotational, which sets the fluid velocity perturbation and the auxiliary spatial vector perturbation to zero; the paper acknowledges this but does not quantify how much the effective potential and tidal response would change under a different choice.

Editorial extensions

If this is right

Every axial quasinormal mode of the halo-dressed black hole is redshifted by the same factor $e^{-I/2}$, so frequency ratios and the quality factor keep their Schwarzschild values; any measured halo shift is a single number.
Ringdown alone cannot disentangle halo compactness from profile shape, because all parameter combinations yielding the same $I$ give the same spectrum.
The axial tidal Love number is controlled by the outer halo and grows roughly as the fourth power of the truncation radius, so it varies oppositely to the ringdown redshift with the profile slopes.
A joint measurement of ringdown and tidal response can separate halo configurations that are degenerate in either observable alone.
For fixed outer slope and halo mass, the redshift saturates as the inner slope approaches the outer slope, giving a maximum shift that any $(\alpha,\beta,\gamma)$ halo of that mass can produce.

Reading between the lines

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

Beyond the paper, if the common redshift factor survives a more general fluid closure, the same $I$ should also control photon-ring observables, since the light-ring frequency and Lyapunov exponent shift identically; this gives a testable link between ringdown and very long baseline interferometry of the shadow.
Beyond the paper, the opposite radial weightings suggest a concrete observational program: in an extreme-mass-ratio inspiral embedded in a dense halo, inspiral phasing constrains the outer-weighted tidal moment while the final ringdown constrains the inner-weighted $I$; the paper leaves finite-radius perturbers to future work, but this division of labor follows directly from its results.
Beyond the paper, a Bayesian joint fit of both observables to mock signals, with correlated halo parameters $(\alpha,\beta,\gamma,z,r_t)$, would quantify how much residual degeneracy remains beyond the pair-wise analysis shown in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things about this paper. The new analytic machinery is genuinely useful: the exact eikonal redshift integral I, the identity λ_c = Ω_c = Ω_Sch e^{-I/2}, and the closed-form axial TLN κ_l ∝ ∫ ρ r^{2l+2} dr are clean results that follow from the background geometry and are cross-checked numerically. But there is a real derivation error in Sec. III. The authors claim the irrotationality condition implies δu=0, and use that to drop the source terms and get the Regge-Wheeler equation (35). Substituting their own Eq. (34) into Eq. (20) gives δu^θ = -h0 S/(√f r^2), which is nonzero unless h0=0. The source-free equation actually comes from the separate choice δw=0. So the QNM potential and the TLN equation are conditional on an unmotivated closure, not on irrotationality. The summary mentions alternative prescriptions could change TLNs but doesn't quantify it, and doesn't flag the same issue for the QNM spectrum.

What's good: the paper avoids fitting circularity; the redshift integral and TLN are derived, not fitted. The WKB and time-domain frequencies agree well (Table I). The saturation mechanism and the degeneracy structure are thoughtful. The complementary weighting of ringdown (inner) and tidal response (outer) is a nice physical observation that will be worth pursuing.

The flaw is load-bearing, not cosmetic. The claim that 'the entire ringdown spectrum responds through a single redshift integral' is only as solid as the master equation, and that equation rests on the closure. The eikonal identity stands alone from geometry, but the transfer to the actual QNM spectrum goes through V_DM = e^{-I} V_Sch, which is derived from the flawed step. A referee should ask for a re-derivation with an explicit choice of U and W, or a physical justification for δu=0, and a quantitative estimate of how the spectrum and Love numbers change under alternative closures.

This paper deserves peer review, not a desk reject. The framework is solid enough and the results are important if the closure is resolved. It's a conditional accept, in my view, with the closure issue as the main revision. I'd bring it to the reading group.

Referee Report

2 major / 4 minor

Summary. The paper studies axial gravitational perturbations of a Schwarzschild black hole embedded in a generalized (α,β,γ) Einstein-cluster dark matter halo. It derives an effective Regge–Wheeler equation, computes quasinormal frequencies with sixth-order WKB and time-domain evolutions, and derives the static axial tidal Love number analytically to leading order in halo compactness. The central quantitative claims are that (i) the halo shifts both the real and imaginary parts of every QNM by the same factor, ω_DM = e^{-I/2} ω_Sch, with I the redshift integral of Eq. (46), and (ii) the axial TLN is proportional to an outer-weighted mass moment, so ringdown and tidal observables probe complementary halo regions and can break compactness–shape degeneracies. The paper also derives exact light-ring relations, studies the dependence on α, β, γ, and compares the predicted shifts with representative current ringdown precision inferred from GW250114.

Significance. If the central derivation were sound, this would be a valuable contribution. The exact light-ring identity λ_c = Ω_c and the single-redshift-integral formula provide a clean, parameter-free explanation of halo-induced QNM shifts; the analytic TLN formula, Eq. (79), is explicit and tested numerically; the WKB and time-domain results are cross-validated; and the GW250114 comparison is presented as illustrative rather than as a fit. No parameters are tuned to the target observables, which is a genuine strength. The proposed complementarity between an inner-weighted observable (ringdown) and an outer-weighted observable (tidal response) is an interesting and falsifiable idea. However, the derivation of the perturbation equation is not internally consistent as written, and both central predictions inherit that inconsistency, so the significance is conditional on a repaired closure.

major comments (2)

[Sec. III, Eqs. (20), (30), (34)–(36)] The derivation of the source-free master equation is not consistent as written. Eq. (34) fixes U in terms of h0, but substituting Eq. (34) into Eq. (20) gives δu^θ = -h_0 S_θ^{lm}/(√f r^2), which is nonzero for any nontrivial axial perturbation. The statement that 'from Eq. (34) it follows δu^μ = 0' is therefore incorrect; imposing δu = 0 would force U = 0, and then Eq. (34) would require h0 = 0 in the matter region. In addition, setting W = 0 in the sourced Regge–Wheeler equation (30) yields Eq. (35) with the potential (31), not the potential (36); the replacement of 4πrρ/(r−2m) by m'(r)/r^2 is not derived from any stated step. Since the WKB, Prony, and tidal computations all use Eq. (36), the central relation ω_DM = e^{-I/2}ω_Sch and the Love-number formula (79) are presently conditional on an unmotivated closure. Please derive Eq. (36) from a consistent fluid-perturbation closure and gauge, or quantify how the QNM frequencies and Love numbers change under alternative axial closures (e.g., U = 0 versus the irrotational U of Eq. (34)).
[Sec. VI, Eq. (65), and Summary] The tidal Love number calculation inherits the same closure issue. The static master equation (65) is obtained by substituting the same irrotational condition, Eq. (34), into Eq. (32), so the resulting κ_l is not robust to the choice of fluid-perturbation closure. The Summary correctly notes that alternative prescriptions could modify magnetic-type Love numbers, but no sensitivity estimate is provided, and the analogous caveat for the QNM potential is absent. Because the complementarity claim in Sec. VII compares the closure-dependent δω and κ_l, a one-parameter exploration of closures, or at least a comparison of the U = 0 case with the Eq. (34) case, is needed to establish that the degeneracy-breaking result is a physical property of the halo rather than an artifact of the chosen closure.

minor comments (4)

[Sec. III] The sentence 'we have three unknown functions, namely, h0, h1, U and W' lists four functions; this should read 'four unknown functions'.
[Sec. V, Fig. 12 and Table I] The time-domain evolutions and Prony fits use M_halo ~ 5M_BH and a0 ~ 50M_BH, corresponding to compactness z ~ 0.1, whereas the main quantitative results use z = 10^{-6}–10^{-4}. The agreement in Table I therefore validates the numerical scheme in a different compactness regime and does not directly test the linear-in-z scaling shown in Fig. 3; a low-compactness time-domain test would strengthen the comparison.
[Eq. (79) and Ref. [121]] The claimed independent confirmation of Eq. (79) by a variation-of-parameters computation is deferred to Ref. [121], which is listed as 'in preparation'; the derivation should be self-contained or the dependence on unpublished work should be stated explicitly in the text.
[Throughout] There are several typographical errors, including 'the ,omotonic increase' in Appendix B3 and 'Fig. 0' in the Summary (which should refer to Fig. 1). A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM and TLN results are derived from the background halo geometry rather than fitted, and the central redshift relation is an analytic consequence of the vacuum-cavity potential scaling.

full rationale

The central derivation chain is self-contained. The light-ring redshift f(rc)=(1/3)e^{-I} follows from integrating the background field equations (Appendix A1), with I defined directly from the halo mass function and not tuned to any QNM or Love number. Appendix B then proves omega_DM=e^{-I/2}omega_Sch by showing V_DM=e^{-I}V_Sch on the vacuum cavity r<=4M_BH (Eq. B3), rescaling the tortoise coordinate (Eq. B6), and mapping the boundary-value problem to the Schwarzschild one; the WKB and time-domain computations are numerical checks rather than inputs. The axial TLN formula (Eq. 79) is obtained by linearizing the Riccati equation and matching across the truncation radius, with the density integral following from m'_h=4pi r^2 rho; it is confirmed numerically and not assumed from the ringdown sector. The GW250114 comparison is explicitly illustrative, and no parameter is fitted to reproduce the QNM or TLN results. The only caveats are physical assumptions rather than circular reductions: the axial closure delta u=delta w=0 is adopted from Refs. [56,95] in Sec. III, and the Summary acknowledges that alternative prescriptions could modify magnetic-type TLNs; Ref. [121] is an in-preparation confirmation but is not load-bearing because the analytic derivation and numerical checks are present. Thus the paper's predictions are not equivalent to their inputs by construction, and there is no significant circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities. Its results depend on the Einstein-cluster modeling of the halo, the inner cutoff, the closure for axial fluid perturbations, and the truncation prescription for the TLN. The halo parameters are inputs, not fitted constants.

free parameters (6)

Inner slope gamma = varied 0-3 (input)
Controls inner density falloff of the (alpha,beta,gamma) profile; the paper finds it dominates the QNM redshift. Not fitted to data.
Outer slope beta = varied 0.25-5 (input)
Controls large-radius falloff; sets truncation behavior. Input parameter.
Transition sharpness alpha = varied 0.25-2.5 (input)
Controls profile transition; found subleading in QNM shift but important for TLN via R99.
Compactness z = Mhalo/a0 = 10^-6 to 10^-4 (input)
Determines halo strength; the paper explores the shift scaling with z but does not fit it.
Scale radius a0 = 10^5 M_BH or 10^7 M_BH (input)
Sets halo extent; varied to change z at fixed Mhalo.
Tidal truncation radius rt = 5 a0 for beta <= 3; R99 for beta > 3 (input)
Needed for TLN; chosen from mass-convergence criterion. A modeling choice, not fitted.

assumptions (6)

domain assumption Einstein cluster averaging of collisionless particles on circular geodesics produces an anisotropic fluid with vanishing radial pressure and tangential pressure pt = m/(2(r-2m)) rho (Eq. 7).
Standard construction (Refs [54,55]) that underlies the entire background geometry.
ad hoc to paper Axial fluid perturbations are closed by setting delta u = delta w = 0 (irrotational closure), yielding a source-free Regge-Wheeler equation.
This is a choice rather than a derivation; the text's claim that irrotationality forces delta u = 0 is incorrect. Alternative closures would modify the potential and TLN.
domain assumption DM density vanishes for r <= 4 M_BH (inner cutoff), the radius of the marginally bound circular orbit.
Based on Gondolo-Silk capture argument; the paper shows the choice matters for cuspy profiles (Fig. 8).
domain assumption Configurations are restricted to m(r) < r/3, excluding additional light-ring pairs, so the WKB single-barrier assumption holds.
Stated in Sec. II; the QNM calculation uses WKB which assumes one potential barrier.
domain assumption For beta <= 3 the halo is truncated at rt = 5 a0; for beta > 3 at R99 (99% mass radius).
Needed for finite TLN; the TLN is dominated by the truncation radius, so results depend on this choice.
standard math Standard QNM boundary conditions: purely ingoing at horizon, outgoing at infinity.
Defines the QNM spectrum; not specific to this paper.

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Pith. "Pith review of The Ringdown and the Tide: Fingerprints of Dark Matter Halo Profiles." pith.science (2026). https://pith.science/paper/7KWH6ZS5

@misc{pith2026260807678,
  author       = {Pith},
  title        = {Pith review of: The Ringdown and the Tide: Fingerprints of Dark Matter Halo Profiles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KWH6ZS5}},
  note         = {Machine review of arXiv:2608.07678}
}

abstract

Astrophysical black holes (BH) are not isolated, but embedded in matter supplied by their host galaxies. We study how the shape of a surrounding dark matter (DM) halo modifies the ringdown and tidal response of an asymptotically flat, static, spherically symmetric BH. The halo is modelled as an anisotropic Einstein cluster with vanishing radial pressure and a generalized $(\alpha,\beta,\gamma)$ density profile, supplemented by an inner cut-off near the BH and, where required, an outer tidal truncation. We derive the axial gravitational perturbation equation and compute the quasinormal mode (QNM) spectrum using sixth-order Wentzel-Kramers-Brillouin (WKB) methods and time-domain evolutions. The halo redshifts both the oscillation frequency and the damping rate, by an amount set not only by the halo compactness but also by the profile parameters: the inner slope $\gamma$ dominates for centrally concentrated halos, while $\alpha$ and $\beta$ give subleading but profile-dependent corrections. In the eikonal limit, the shift is governed by a single redshift integral encoding the halo mass distribution outside the light ring, explaining the close correspondence between the QNM frequencies, the light ring frequency, and the Lyapunov exponent. We also show that different combinations of compactness and profile shape can yield nearly degenerate ringdown spectra. Time-domain evolutions confirm the WKB frequencies and display the expected late-time Price law decay, with an intermediate tail controlled by the outer density falloff for slowly decaying profiles. Finally, we compute the static axial tidal Love number and show that it probes the halo with a radial weighting different from the ringdown sector. The combined ringdown and tidal response therefore provides a possible way to distinguish environmental effects from genuine deviations of the vacuum BH geometry.

Figures

Figures reproduced from arXiv: 2608.07678 by the authors.

**Figure 2.** Effective potential in the axial sector for the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Fractional deviation in the real QNM frequency and [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: Fractional deviations from Schwarzschild values, plotted on a logarithmic scale, as functions of the halo parameter [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Fractional deviations from Schwarzschild values, plotted on a logarithmic scale, as functions of the halo parameter [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Fractional deviations from Schwarzschild values, plotted on a logarithmic scale, as functions of the halo parameter [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: Degeneracy in the fractional deviations of the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 9.** Figure 9: Three-dimensional parameter space of the gener [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: Difference in the fundamental (l = 2, n = 0) QNM oscillation frequency and damping rate computed using halo profiles with inner cut-offs at 2MBH and 4MBH. The figure shows the magnitude of the differences on a logarithmic scale. producing comparable deviations. This co…

**Figure 12.** Figure 12: Evolution of the Gaussian wave packet for different [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 11.** Figure 11: Comparison of the predicted fractional QNM [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 13.** Figure 13: The 99% mass radius, R99 over the (γ, β) plane, at fixed a0 = 105MBH, for α = 0.5, 1, 2. The white wedge is the unphysical region γ ≥ β. R99 is largest for shallow profiles (small γ, β → 3 +) and smallest for steep ones, spanning several decades across the shape param…

**Figure 14.** Figure 14: The ratio R99/a0 as a function of a0 for representative convergent profiles. For β = 4 the ratio is nearly independent of a0, while for β = 3.5 it varies substantially, reflecting the growing influence of the fixed inner cut-off 4MBH as the outer slope approaches the…

**Figure 15.** Figure 15: The dimension-full response κ2 (top) and the dimensionless Love number [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: Variation of |k B 2 | with the outer slope β, at fixed α = γ = 1.The left panel shows the variation for the mass divergent profiles (β < 3) with fixed truncation radius at rt = 5a0. In contrast, the right panel plots the same with β > 3, and truncation radius rt = R99…

**Figure 17.** Figure 17: Variation of |k B 2 | with the inner slope γ (left) and the transition sharpness α (right). In the left panel, α = 1 and β = 4, while in the right panel β = 4 and γ = 1. In both panels, the truncation radius is chosen as rt = R99 for different combinations of a0 and z…

**Figure 18.** Figure 18: Representative contour map illustrating the com [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]

**Figure 4.** Figure 4: The boundary β > γ marks the edge of the physically allowed region. Breaking this degeneracy needs either subleading corrections to the uniform potential scaling or independent astrophysical priors on the halo profile. 5. Eikonal - Correspondence 6 It is interesting to…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

123 extracted references · 5 canonical work pages

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Equation(83) further solidifies the choice ofMADM as the normalizing length scale and provides an analytic anchor for the shape trends discussed below. C. Dependence on compactness and halo parameters At fixed shape and scale radius, the truncation radius R99 (or5 a0) is fixed, so the response is strictly linear in the halo mass,κl∝M halo∝z , as shown in ...
[2]

The mass function therefore satisfiesm(rc) =MBH, ¯ρ(rc) = 0, and m′(rc) = 0at the light ring rc = 3MBH

Redshift function at the light ring The DM density vanishes forr≤ 4MBH. The mass function therefore satisfiesm(rc) =MBH, ¯ρ(rc) = 0, and m′(rc) = 0at the light ring rc = 3MBH. The local geometry atrc is identical to the Schwarzschild case. The halo affects only the global value off(rc). Integrating Eq. (6) outward from rc and imposing asymptotic flatness ...
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Universal identityλ c = Ωc We now derive the identity quoted as Eq. (47). We define Vl(r) = f(r)/r2 and g(r) = f(r)(1− 2m/r), so dr∗/dr=g −1/2. SinceV ′ l (rc) = 0, d2Vl dr2∗ ⏐⏐⏐⏐ rc =g(r c)V′′ l (rc).(A4) At rc = 3MBH, using m(rc) = MBH, m′(rc) = 0, and f(rc) = 1 3e−I, g(rc) =f(r c) ( 1− 2MBH rc ) = f(rc) 3 .(A5) Differentiating f′/f = 2m/[r(r− 2m)]at rc...
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Profile dependence of the halo integral We now make precise the mechanisms described quali- tatively in Sec. IVA. 22 a. Inner slopeγ Near the inner cut-off.Setting s≡r− 4MBH and expanding the right hand side of Eq. (5) fors≪MBH: 4πr2−γ ( 1− 4MBH r ) = 4π(4MBH)1−γs+O(s 2/MBH). (A11) The constant term cancels, somh vanishes quadratically, mh(r) = 2π¯ρ0aγ 0 ...
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This is a bookkeeping device rather than a new physical parameter: the actual halo always hasλ = 1, and every result below is evaluated there

Compactness scaling and degeneracy At fixed(α,β,γ ), rescalingMhalo→λM halo at fixed a0 maps mh(r)→λm h(r)at every fixed r, turning the single numberI, defined for one fixed halo, into a function I(λ)that can be differentiated. This is a bookkeeping device rather than a new physical parameter: the actual halo always hasλ = 1, and every result below is eva...
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Eikonal - Correspondence 6 It is interesting to note that for most of the DM distributions studied in the present work, we observeδΩc to be a concave function ofI, implying(I ′)2 >2I′′. 24 DM Profile l From LR From WKB-6 0.5, 3.5, 1 2 0.3848872−0.0962218i 0.3736068−0.0888880i 3 0.5773308−0.0962218i 0.5994232−0.0926994i 10 1.9244360−0.0962218i 1.9967205−0....
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Potential scaling and exact frequency relation The Regge–Wheeler potential governing axial pertur- bations, Eq. (36), reads V(r) =f(r) [l(l+ 1) r2 − 6m(r) r3 + m′(r) r2 ] .(B1) This potential is defined on the full domain r∈ [2MBH,∞ ), with or without the halo. Its functional form changes atr = 4MBH because the halo density¯ρ(r), and hence m(r)and f(r), i...
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(A16) and (B9),δω = 1−e−I/2

Compactness dependence From Eqs. (A16) and (B9),δω = 1−e−I/2. For fixed halo shape at small compactnessz =Mhalo/a0, the halo integral satisfiesI∝z(Section A4), so δω≈ 1 2I∝z(z≪1),(B14) predicting a linear growth ofδω with compactness on a log-log plot, with slope close to unity, matching the initial rise in Fig. 3. AsMhalo grows,I grows faster than linear...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Equation(83) further solidifies the choice ofMADM as the normalizing length scale and provides an analytic anchor for the shape trends discussed below. C. Dependence on compactness and halo parameters At fixed shape and scale radius, the truncation radius R99 (or5 a0) is fixed, so the response is strictly linear in the halo mass,κl∝M halo∝z , as shown in ...

[2] [2]

The mass function therefore satisfiesm(rc) =MBH, ¯ρ(rc) = 0, and m′(rc) = 0at the light ring rc = 3MBH

Redshift function at the light ring The DM density vanishes forr≤ 4MBH. The mass function therefore satisfiesm(rc) =MBH, ¯ρ(rc) = 0, and m′(rc) = 0at the light ring rc = 3MBH. The local geometry atrc is identical to the Schwarzschild case. The halo affects only the global value off(rc). Integrating Eq. (6) outward from rc and imposing asymptotic flatness ...

[3] [3]

Universal identityλ c = Ωc We now derive the identity quoted as Eq. (47). We define Vl(r) = f(r)/r2 and g(r) = f(r)(1− 2m/r), so dr∗/dr=g −1/2. SinceV ′ l (rc) = 0, d2Vl dr2∗ ⏐⏐⏐⏐ rc =g(r c)V′′ l (rc).(A4) At rc = 3MBH, using m(rc) = MBH, m′(rc) = 0, and f(rc) = 1 3e−I, g(rc) =f(r c) ( 1− 2MBH rc ) = f(rc) 3 .(A5) Differentiating f′/f = 2m/[r(r− 2m)]at rc...

[4] [4]

Profile dependence of the halo integral We now make precise the mechanisms described quali- tatively in Sec. IVA. 22 a. Inner slopeγ Near the inner cut-off.Setting s≡r− 4MBH and expanding the right hand side of Eq. (5) fors≪MBH: 4πr2−γ ( 1− 4MBH r ) = 4π(4MBH)1−γs+O(s 2/MBH). (A11) The constant term cancels, somh vanishes quadratically, mh(r) = 2π¯ρ0aγ 0 ...

[5] [5]

This is a bookkeeping device rather than a new physical parameter: the actual halo always hasλ = 1, and every result below is evaluated there

Compactness scaling and degeneracy At fixed(α,β,γ ), rescalingMhalo→λM halo at fixed a0 maps mh(r)→λm h(r)at every fixed r, turning the single numberI, defined for one fixed halo, into a function I(λ)that can be differentiated. This is a bookkeeping device rather than a new physical parameter: the actual halo always hasλ = 1, and every result below is eva...

[6] [6]

Eikonal - Correspondence 6 It is interesting to note that for most of the DM distributions studied in the present work, we observeδΩc to be a concave function ofI, implying(I ′)2 >2I′′. 24 DM Profile l From LR From WKB-6 0.5, 3.5, 1 2 0.3848872−0.0962218i 0.3736068−0.0888880i 3 0.5773308−0.0962218i 0.5994232−0.0926994i 10 1.9244360−0.0962218i 1.9967205−0....

[7] [7]

(36), reads V(r) =f(r) [l(l+ 1) r2 − 6m(r) r3 + m′(r) r2 ] .(B1) This potential is defined on the full domain r∈ [2MBH,∞ ), with or without the halo

Potential scaling and exact frequency relation The Regge–Wheeler potential governing axial pertur- bations, Eq. (36), reads V(r) =f(r) [l(l+ 1) r2 − 6m(r) r3 + m′(r) r2 ] .(B1) This potential is defined on the full domain r∈ [2MBH,∞ ), with or without the halo. Its functional form changes atr = 4MBH because the halo density¯ρ(r), and hence m(r)and f(r), i...

[8] [8]

(A16) and (B9),δω = 1−e−I/2

Compactness dependence From Eqs. (A16) and (B9),δω = 1−e−I/2. For fixed halo shape at small compactnessz =Mhalo/a0, the halo integral satisfiesI∝z(Section A4), so δω≈ 1 2I∝z(z≪1),(B14) predicting a linear growth ofδω with compactness on a log-log plot, with slope close to unity, matching the initial rise in Fig. 3. AsMhalo grows,I grows faster than linear...