Dark matter imprints on a caustic encounter in an effective spinning black hole binary lens

Mohsen Fathi

REVIEW 3 major objections 5 minor 40 references

Dark matter imprints on a caustic encounter in an effective spinning black hole binary lens

T0 review · 3 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read A cored dark-matter correction to a spinning black-hole binary lens leaves a caustic timing shift near 0.0015 in orbital phase, and a vacuum binary fitted over the full orbit cannot fully mimic it.

desk verdict A transparent, well-scoped numerical demonstration that a cNFW correction can leave a residual caustic timing shift after a best-fit spinning vacuum comparison, but the headline shift is not yet shown to be numerically converged. read the letter →

arxiv 2607.23300 v1 pith:TPZRPCQ3 submitted 2026-07-25 gr-qc astro-ph.HE

Mohsen Fathi This is my paper

classification gr-qcastro-ph.HE PACS 04.70.Bw95.35.+d98.62.Sb

keywords gravitationallensingcausticsblackholebinarycoredNFWdarkmatterfinite-sourceresponsegravitomagneticspindeflectioneffectivelensmapcaustictiming

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 dark matter can leave a visible fingerprint in the lensing of a spinning black-hole binary. The author builds an effective lens map for the binary, adds a small cored dark-matter correction to the deflection law, and then lets a plain spinning vacuum binary try to imitate the result by fitting its scale, spin, separation, mass ratio, spin direction, and orbital phase over a full orbit. The imitation is good but incomplete: at the chosen caustic encounter the vacuum model still misses the dark-matter timing by about 0.0015 in orbital phase, with a local response residual near 1.3 percent. The general-interest point is that caustics — the sharp features of the lens map where images can appear — act as amplifiers: a tiny radial change in the lens law survives as a measurable shift in when a finite-sized source crosses one of them, so binary lensing could in principle trace dark-matter structure. The paper presents this as a numerical demonstration within a tested effective family, not as an observational claim or an exact binary-spacetime calculation.

What carries the argument

The load-bearing machinery is the effective quasi-stationary lens map β = θ − Σ_i(α_M,i + α_J,i), with mass and gravitomagnetic spin contributions per component. The dark-matter imprint rides inside the mass term through F_i(u_i) = 1 + g_h[μ(R_* u_i) − 1], where μ(R) is a cored radial profile — a cored variant of the standard NFW dark-matter halo, controlled by strength η and core scale λ_c. The paper stresses that μ is a phenomenological radial correction, not an exact relativistic halo solution. The comparison clock is a finite-source response C(Φ) built from the minimum distance between the source centre and the extracted caustic; its rising half-response crossings are the "encounters". T

What would settle it

Run the identical pipeline with a different radial profile in place of the cored μ(R) — a pure cusped NFW branch, or λ_c moved by a factor of two — and refit the vacuum parameters over the full orbit. If the aligned shift ΔΦ_tr,al drops below about 3 × 10^-4 (the change between the two numerical settings quoted in the paper), the claimed imprint is an artifact of the chosen function; if the shift persists or grows, caustic timing is a genuine discriminator of the lens's radial mass structure.

Watch

Extended reading notes

Core claim

Central claim: after a full-orbit fit of a spinning vacuum binary to the cNFW-corrected lens (parameters A, χ, d, q, γ, δΦ), the best model — equal mass, common spin, A = 0.0326, χ = 0.84, d = 0.340, γ = 25° — still leaves at the selected crossing near phase 0.08 an aligned timing shift of 1.5 × 10^-3 in orbital phase and a local window rms of 1.34 × 10^-2 (maximum residual 4.02 × 10^-2). A nearby resolution gives 1.8 × 10^-3 and 1.50 × 10^-2, so the quoted features do not disappear under a grid change — a disappearance check, not a formal convergence test. All three rising half-response crossings in one orbit were audited; the chosen event is not the strongest, with shifts ranging from −9.5

Load-bearing premise

The result stands or falls with the chosen cored profile μ(R) (Eq. 5) and its mapping into the deflection law through g_h = 6 and R_* = 40: a real halo with a steeper, cuspier, or differently cored radial dependence could leave a vanishing, sign-flipped, or fully absorbable caustic shift — the paper itself says "the numerical results therefore depend on this chosen function."

Editorial extensions

If this is right

Within the tested six-parameter vacuum family, no spinning vacuum binary fully mimics the cored dark-matter lens: after the full-orbit fit the aligned crossing shift (≈1.5 × 10^-3 in orbital phase) and the local response rms (≈1.3 × 10^-2) remain.
The feature survives a nearby numerical resolution change: the maximum local difference moves by less than 4 × 10^-5 and the aligned shift by about 3 × 10^-4, so the quoted local features do not disappear under a grid change (a disappearance check, not a formal convergence test).
The effect is not a single tuned event: all three rising half-response crossings in one orbit carry aligned shifts, from −9.5 × 10^-4 to +1.54 × 10^-2.
Phase-sensitive caustic differences could complement searches for quasi-periodic lensed starlight from supermassive black-hole binaries, offering a dark-matter-versus-vacuum diagnostic that does not require full image reconstruction.
The sign and size of the crossing shift change with halo strength, source radius, and binary separation (exploratory cases flip the sign), so applying the diagnostic to data requires event-by-event modelling rather than a universal threshold.

Reading between the lines

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

Reversed use of the same pipeline: once a physical mass and distance scale is assigned, the leftover shift measures how strongly the deflection law can deviate from vacuum — turning the residual into an upper bound on the strength η of any radial correction near a binary.
The dimensionless shift of ~10^-3 sets the required timing precision: resolving it observationally needs phase-tagged light curves with roughly one-part-per-thousand resolution of the orbital period, which the paper's own caveats (emission model, cadence, noise) leave for future work.
Direct testable extension: rerun the comparison with a cusped (NFW) or steeper profile instead of the cored one; if the residual changes sign or magnitude, caustic timing becomes a core-versus-cusp discriminator of halo structure, not just a dark-matter-versus-vacuum detector.
Because split-spin vacuum models slightly improve the windowed residual while worsening the full-orbit fit, the residual landscape may contain near-degenerate vacuum models; a continuous (non-grid) fit could either absorb the signal or sharpen it — the paper's own gridded search is finite.

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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

The paper asks a clean question: after a spinning vacuum binary is fitted to imitate a cNFW-corrected binary across the full orbit, does any local caustic signal survive? The answer, on the numbers presented, is yes: the selected crossing still shows an aligned shift of 1.5e-3 (1.8e-3 at a nearby resolution), with a stable max local residual of 4.02e-2. The author is honest about the scope — this is an effective quasi-static lens map, a phenomenological halo profile, and a finite grid search, not a first-principles binary spacetime or an observational claim. The all-crossing audit and the refusal to cherry-pick the largest shift are good practice. The comparison design is also fair: the vacuum has free scale, spin, separation, mass ratio, angle, and phase, fitted over the whole orbit, and the local window is not refit.

The main weakness is the resolution check, which the stress-test note correctly identifies. The production grid is 170x110 with 240 phases; the comparison is only to a coarser 160x104/200-phase setting. The aligned crossing shift moves by 20% (1.507e-3 to 1.816e-3), and the window rms by ~12%. Meanwhile the supplementary centroid convergence check needed a 480x308 grid to suppress a grid-dependent quantity, and the production grid is far coarser. That does not prove the quoted shift is wrong, but it does mean the phrase 'numerically resolved' is stronger than the evidence supports. A finer run — or at least a three-level check showing the shift settling — would nail it. The shift is also smaller than the orbital phase step and relies on interpolation between response samples, so the extraction method itself could be adding error.

The cNFW halo function is admittedly phenomenological, and the abstract and discussion both flag this. That is a limitation, not a hidden flaw. The reference list is fine: the spin-lens, caustic, and cNFW ingredients are all standard and cited appropriately.

Who gets value from this: people working on binary-lens caustics or DM imprints on lensing signatures, who want a proof-of-principle that a cored DM correction can survive a flexible vacuum fit. It is a modest step, not a breakthrough, but it is a solid and honest one. The author ships no code or data yet, though a public repository is promised; for a numerical letter that is a real cost, and I would want the scripts before fully trusting the residuals.

Bottom line: a fair and carefully hedged numerical study with one load-bearing gap — the convergence of the headline shift. It deserves a serious referee, and my own verdict would be revise-and-resubmit with a fine-resolution check and some better estimate of caustic-extraction error.

Referee Report

3 major / 5 minor

Summary. The paper considers a quasi-stationary effective binary lens composed of point-mass terms, a leading-order spin term, and a phenomenological cored-NFW radial correction μ(R). It asks whether a spinning vacuum binary, after fitting its lens scale A, common spin χ, separation d, mass ratio q, projected spin angle γ, and a global phase offset δΦ over the full orbit, can fully mimic the cNFW-modified lens. The central quantitative result is that in one selected caustic encounter the aligned crossing shift is ΔΦ_tr,al ≈ 1.5×10^-3 with local rms R_W ≈ 1.34×10^-2 and maximum residual 4.02×10^-2; a coarser nearby numerical setting gives ΔΦ_tr,al ≈ 1.8×10^-3 and R_W ≈ 1.50×10^-2. The paper reports all three rising half-response crossings, noting that the selected event is not the strongest, and it explicitly cautions that the cNFW form is a phenomenological input rather than a self-consistent halo solution. The conclusion is that a cored DM-inspired radial correction leaves a numerically resolved caustic difference after the tested vacuum comparison.

Significance. If the result holds, it is a useful proof-of-concept: caustic timing in binary-lens systems could in principle retain information about a cored DM component even after a broad vacuum-parameter fit. The paper has several genuine strengths: it clearly states the tested parameter family; it reports all rising crossings instead of cherry-picking the strongest one; it provides spin-reversal and centroid-convergence checks; and it repeatedly disclaims overinterpretation, including the lack of a self-consistent halo model and the non-observational nature of the response C(Φ). These features are helpful and should be preserved. However, the headline claim that the residual is 'numerically resolved' is currently supported only by a two-point resolution comparison that the supplement itself labels as not a formal convergence-order test, and the robustness of the shift under changes to source radius and to the global phase-fit convention is not yet demonstrated. The result is promising but not yet established at the standard required for a central claim.

major comments (3)

[Supplementary §5, Table 4; Supplementary §4, Table 3] The production run uses 240 orbital phases and a 170×110 extraction grid, while the only resolution comparison is to a coarser 200-phase, 160×104 grid. The supplement explicitly states that this is not a formal convergence-order test. Yet the aligned crossing shift changes by about 20% (from 1.507×10^-3 to 1.816×10^-3) and the window rms by about 12%. In contrast, the paper's own centroid convergence check (§4, Table 3) required a 480×308 grid to drive D_c down to 2.6×10^-3, a value comparable to the quoted shift. With production-grid caustic extraction at 170×110, the headline shift may be comparable to or smaller than the grid-induced caustic-position error. Thus the phrase 'numerically resolved caustic difference' in the Discussion is not yet supported. I ask for a production run at or above the resolution of the centroid check, ideally with a three-level phase/grid sequence so that a
[§3, Eqs. (8)-(9); Table 4] The 'aligned' crossing shift ΔΦ_tr,al is measured after a global phase offset δΦ is fitted over the full orbit and subtracted. At the selected event, the raw cNFW-vacuum phase separation is about 2.0×10^-3; after subtracting the fitted δΦ=5.0×10^-4 it becomes 1.5×10^-3. Because δΦ is a free parameter in the comparison, ΔΦ_tr,al is not a raw timing observable, and the global fit can absorb part of the DM-induced shift. The paper correctly reports both the raw and aligned crossings, but to support the central claim it should add a sensitivity test in which δΦ is fitted on the complement of the selected window (or on the other two crossings alone) and the local residual in the window is then recomputed. This would verify that the event's residual is not an artifact of the global phase subtraction.
[Supplementary §7, Table 6] The benchmark fixes R_s=0.030, but the exploratory fixed-reference variations show that changing R_s to 0.024 or 0.036 changes the crossing shift from +0.0090 to -0.0073, and changes the local rms and maximum residual by large factors. Although these exploratory rows use a different crossing convention and no refit, they demonstrate that the sign and magnitude of the claimed imprint depend sensitively on an input that is not varied in the main comparison. Since R_s enters the definition of C(Φ) through Eq. (7) and is not a fitted vacuum parameter, the central claim that a cNFW correction leaves a resolved caustic difference is currently established only for this specific source radius and response-convention choice. I request a robustness scan over R_s (and, where feasible, the logistic width 0.18R_s) with the same global-fitting procedure, or an explicit statement that the result is ben

minor comments (5)

[§3, Eq. (8)] The notation for the orbital average/norm in Eq. (8) is ambiguous: the expression mixes an average-like bracket with an L2-style exponent. Please define ⟨...⟩_orb explicitly (e.g., as a mean square over uniform phase samples).
[Fig. 1 caption] In the definition D = min[|I_cNFW − I_vac|/q0.997, 1], the symbol q0.997 denotes a percentile but q is already used for the mass ratio. Use a distinct symbol such as Q_0.997 to avoid confusion.
[Supplementary Table 6] The table is labeled as using a 'fixed-reference crossing convention' that should not be compared directly with globally aligned shifts. Please define this convention explicitly, since the reader cannot otherwise interpret the sign and magnitude changes.
[Data availability] The statement that scripts 'will be placed in a public repository before publication' is useful, but for review reproducibility a temporary repository link or DOI should be provided now.
[Discussion and conclusion] The final sentence 'A cored DM-inspired radial correction leaves a numerically resolved caustic difference' should be qualified to read 'for the adopted cNFW form, response function, and benchmark parameters', matching the caveats already present in the main text and supplement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported caustic residual is a post-fit diagnostic, not a fitted parameter renamed as a prediction.

full rationale

The paper makes no claim to derive a standalone prediction; it performs an explicit model comparison. The vacuum parameters (A, chi, d, q, gamma, delta_Phi) are fitted by minimizing the full-orbit residual R_orb (Eq. 8 / Supp. Eq. 4), and the reported quantities R_W, max|Delta C|, and Delta_Phi_tr,al are evaluated after that fit without a second local fit. The paper states this clearly: 'The phase offset is fitted over the whole orbit, rather than around the selected crossing' and 'We then evaluate the local residual without another fit.' The aligned crossing shift is therefore a residual diagnostic, not a fitted parameter relabeled as a prediction; it is defined as Phi_cNFW - Phi_vac,al after global alignment, so no equation-level identity makes the output equal to the input. The cNFW radial correction is an input (Eq. 5, adapted from Ref. [17]) and the paper explicitly cautions 'We do not treat it as an exact relativistic binary halo' and 'The numerical results therefore depend on this chosen function.' This is acknowledged model dependence, not circularity. There are no self-citations by the author; Refs. [15,16,17,18] are external, and no uniqueness theorem is imported from the author's own prior work. The resolution caveat ('this two-level comparison is not a formal convergence-order test') is a numerical-convergence limitation, not a circularity. The paper also reports all three rising crossings and notes that the selected event is not the largest, reducing selection-circularity concerns. Overall, the central claim is a self-contained numerical demonstration within a stated effective family, with no load-bearing step that reduces to its own inputs.

Assumptions & free parameters 16 free parameters · 6 assumptions · 0 invented entities

The central result is an output of a specific effective model. The vacuum comparison uses fitted parameters, and the DM correction is a phenomenological input. The ledger lists the many hand-chosen or fitted numbers and modeling assumptions the claim rests on.

free parameters (16)

A (lens scale) = 0.032 (cNFW benchmark); 0.0326 (vacuum best fit)
Overall deflection scale of the effective map; vacuum value obtained from grid minimization.
epsilon (softening) = 0.022
Smooths the center of the effective lens map; chosen by hand.
d (binary separation) = 0.340 (cNFW and best vacuum); scanned 0.335-0.345
Separation of the binary components; part of the fitted vacuum parameters.
q (mass ratio) = 1.000; scanned 0.8-1.2
Mass ratio m2/m1; best-fit from coarse scan.
chi (common spin) = 0.85 (cNFW); 0.84 (vacuum best fit); scanned 0.80-0.88
Dimensionless spin magnitude; fitted for the vacuum model.
gamma (projected spin angle) = 25 deg; scanned 15-35 deg
Projected spin direction; fitted in the coarse scan.
kappa_J (spin coefficient) = 0.18
Effective normalization of the spin term; chosen by hand, not fitted.
eta (halo strength) = 0.0025; exploratory 0.001
Strength of the cNFW correction; chosen by hand.
lambda_c (core scale) = 10
Core scale of the cNFW correction; chosen by hand.
g_h (halo gain) = 6
Maps the cNFW radial variation onto the effective lens; chosen by hand with no derivation.
R_star (radial mapping scale) = 40
Scale for mapping R_star*u_i into the cNFW function; chosen by hand.
R_s (source radius) = 0.030; exploratory 0.024 and 0.036
Finite-source radius appearing in the response C(Phi); chosen by hand.
Source trajectory parameters = 0.23, 0.085, phase 0.35
Amplitudes and phase offset of the source center path beta_s(Phi); chosen by hand.
delta_Phi (orbital phase offset) = 5.00e-4
Global phase offset fitted over the full orbit to align cNFW and vacuum responses; the reported aligned crossing shift is measured after subtracting this fitted offset.
Response width factor = 0.18*R_s = 0.0054
Logistic steepness in the definition of C(Phi); chosen by hand.
Local window half-width = 0.115
Window |Phi - Phi_mid| <= 0.115 used for the local residual; chosen by hand.

assumptions (6)

standard math Geometrical-optics point-mass lens equation: beta = theta - sum(alpha_M + alpha_J), with alpha_M proportional to A m_i F_i(u_i) r_i/u_i^2.
Standard gravitational lensing in the thin-lens/weak-deflection regime; Eqs. (2)-(3).
domain assumption Leading-order weak-field spin deflection formula for alpha_J (Eq. 6).
Assumes the Kerr spin effect is adequately represented by the weak-field dipole term of Refs. [7,8]; not an exact Kerr binary calculation.
domain assumption The cNFW correction mu(R) (Eq. 5) from Ref. [17] is a valid phenomenological representation of a cored DM halo's radial deflection.
The paper explicitly states it is used only as a phenomenological radial correction, not a self-consistent relativistic matter solution.
ad hoc to paper Mapping of the cNFW radial variation through g_h and R_star onto the effective lens (Eq. 4) is representative.
No derivation is given for the choices g_h=6 and R_star=40; the numerical result depends on this chosen mapping.
domain assumption Quasi-static binary approximation: at each orbital phase Phi the binary is an instantaneous lens map.
Neglects light-travel time, orbital motion during deflection, and radiation reaction; stated in the Effective lens map section.
ad hoc to paper Finite-source response C(Phi), defined via d_min and a logistic function with width 0.18R_s, is a meaningful proxy for caustic timing.
C is described as a smooth clock, not a calibrated observed flux; the 'timing shift' is a property of this constructed response.

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Cite this review

Pith. "Pith review of Dark matter imprints on a caustic encounter in an effective spinning black hole binary lens." pith.science (2026). https://pith.science/paper/TPZRPCQ3

@misc{pith2026260723300,
  author       = {Pith},
  title        = {Pith review of: Dark matter imprints on a caustic encounter in an effective spinning black hole binary lens},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPZRPCQ3}},
  note         = {Machine review of arXiv:2607.23300}
}

abstract

We ask whether a cored dark matter (DM) correction can leave a resolved caustic trace after a spinning vacuum binary is allowed to imitate it. The vacuum comparison varies lens scale, common spin, separation, mass ratio, projected spin angle, and one phase offset fitted over the full orbit. Nearby unequal component spins and projected directions are also tested. The best high-resolution fit remains the equal-mass, common-spin model. In the selected encounter, the local root mean square residual is $1.34\times10^{-2}$, the maximum residual is $4.02\times10^{-2}$, and the aligned crossing shift is $1.51\times10^{-3}$. A nearby resolution gives $1.50\times10^{-2}$, $4.02\times10^{-2}$, and $1.82\times10^{-3}$. All three rising half-response crossings are checked, and the selected event is not the largest. This is a numerical demonstration within a tested effective family, not an observational claim or an exact binary-spacetime calculation.

Figures

Figures reproduced from arXiv: 2607.23300 by the authors.

**Figure 1.** Image comparison at Φ = 0.080. Panels (a,b) show the best-fit vacuum and cNFW images, while panel (c) shows their normalized absolute difference. Panels (d,e) enlarge the same yellow-boxed region, and panel (f) shows the local difference. We plot D = min[|IcNFW − Ivac|/q0.997, 1], where q0.997 is the 99.7 percentile of the full-image difference. The colour scale is only a visual guide; it is not an absolute flux sca… view at source ↗

**Figure 2.** Finite-source response after the broader vacuum comparison. Left: the selected encounter and its half-response crossing. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 1.** The first local vacuum grid. Each panel fixes the separation and shows the phase-profiled rms residual [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

Figures from the paper (3 more)

**Figure 2.** Figure 2: Expanded coarse vacuum scan. Each cell shows the lowest full-orbit rms after fitting [PITH_FULL_IMAGE:figures/full_fig_p006_2.png]

**Figure 3.** Figure 3: Resolution comparison. Left: the vacuum and cNFW responses at the two numerical settings. Right: the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]

**Figure 4.** Figure 4: Full-orbit crossing check. Top: the cNFW response and the globally aligned vacuum response. Bottom: [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Works this paper leans on

40 extracted references · 11 linked inside Pith

[1]

Gravitational Lenses , publisher =

Schneider, Peter and Ehlers, J. Gravitational Lenses , publisher =. 1992 , doi =

1992
[2]

Classical and Quantum Gravity , volume =

Bartelmann, Matthias , title =. Classical and Quantum Gravity , volume =. 2010 , doi =. 1010.3829 , archivePrefix =

arXiv 2010
[3]

Astronomy and Astrophysics , volume =

Schneider, Peter and Weiss, Achim , title =. Astronomy and Astrophysics , volume =
[4]

Astronomy and Astrophysics , volume =

Erdl, Helmut and Schneider, Peter , title =. Astronomy and Astrophysics , volume =
[5]

Astronomy and Astrophysics , volume =

Dominik, Martin , title =. Astronomy and Astrophysics , volume =. 1999 , eprint =

1999
[6]

Scott and Petters, Arlie O

Gaudi, B. Scott and Petters, Arlie O. , title =. The Astrophysical Journal , volume =. 2002 , doi =. astro-ph/0112531 , archivePrefix =

arXiv 2002
[7]

Scott and Petters, Arlie O

Gaudi, B. Scott and Petters, Arlie O. , title =. The Astrophysical Journal , volume =. 2002 , doi =. astro-ph/0206162 , archivePrefix =

arXiv 2002
[8]

and Blandford, Roger D

Rauch, Kevin P. and Blandford, Roger D. , title =. The Astrophysical Journal , volume =. 1994 , doi =

1994

Show all 40 references

[9]

Monthly Notices of the Royal Astronomical Society , volume =

Sereno, Mauro , title =. Monthly Notices of the Royal Astronomical Society , volume =. 2003 , doi =. astro-ph/0307243 , archivePrefix =

2003 arXiv
[10]

Physical Review D , volume =

Sereno, Mauro and De Luca, Fabiana , title =. Physical Review D , volume =. 2006 , doi =. astro-ph/0609435 , archivePrefix =

2006 arXiv
[11]

Physical Review D , volume =

Sereno, Mauro and De Luca, Fabiana , title =. Physical Review D , volume =. 2008 , doi =. 0710.5923 , archivePrefix =

2008 arXiv
[12]

Physical Review D , volume =

Bozza, Valerio , title =. Physical Review D , volume =. 2008 , doi =. 0806.4102 , archivePrefix =

2008 arXiv
[13]

, title =

Patil, Mandar and Mishra, Priti and Narasimha, D. , title =. Physical Review D , volume =. 2017 , doi =. 1610.04863 , archivePrefix =

2017 arXiv
[14]

Black holes as telescopes: discovering supermassive binaries through quasiperiodic lensed starlight , journal =

Wang, Hanxi and Zumalac. Black holes as telescopes: discovering supermassive binaries through quasiperiodic lensed starlight , journal =. 2026 , doi =. 2506.16544 , archivePrefix =

2026
[15]

and Frenk, Carlos S

Navarro, Julio F. and Frenk, Carlos S. and White, Simon D. M. , title =. The Astrophysical Journal , volume =. 1997 , doi =. astro-ph/9611107 , archivePrefix =

1997 arXiv
[16]

Gravitational lensing by a massive object in a dark matter halo

Karamazov, Michal and Timko, Luk. Gravitational lensing by a massive object in a dark matter halo. I. Critical curves and caustics , journal =. 2021 , doi =. 2103.16965 , archivePrefix =

2021 arXiv
[17]

Gravitational lensing by a massive object in a dark matter halo

Karamazov, Michal and Heyrovsk. Gravitational lensing by a massive object in a dark matter halo. II. Shear, phase, and image geometry , journal =. 2022 , doi =. 2109.02495 , archivePrefix =

2022 arXiv
[18]

The European Physical Journal C , volume =

Senjaya, David , title =. The European Physical Journal C , volume =. 2026 , doi =

2026
[19]

Bolokhov, S. V. , title =. The European Physical Journal C , volume =. 2026 , doi =. 2512.06930 , archivePrefix =

2026
[20]

Schneider P., Ehlers J. Falco E. E. Gravitational Lenses (Springer, Berlin) 1992

1992
[21]

Gaudi B. S. Petters A. O. The Astrophysical Journal 574 2002 970

2002
[22]

Gaudi B. S. Petters A. O. The Astrophysical Journal 580 2002 468

2002
[23]

Rauch K. P. Blandford R. D. The Astrophysical Journal 421 1994 46

1994
[24]

De Luca F

Sereno M. De Luca F. Physical Review D 78 2008 023008

2008
[25]

Physical Review D 78 2008 063014

Bozza V. Physical Review D 78 2008 063014

2008
[26]

Monthly Notices of the Royal Astronomical Society 344 2003 942

Sereno M. Monthly Notices of the Royal Astronomical Society 344 2003 942

2003
[27]

De Luca F

Sereno M. De Luca F. Physical Review D 74 2006 123009

2006
[28]

Schneider P. Weiss A. Astronomy and Astrophysics 164 1986 237

1986
[29]

Schneider P

Erdl H. Schneider P. Astronomy and Astrophysics 268 1993 453

1993
[30]

Astronomy and Astrophysics 349 1999 108

Dominik M. Astronomy and Astrophysics 349 1999 108

1999
[31]

Narasimha D

Patil M., Mishra P. Narasimha D. Physical Review D 95 2017 024026

2017
[32]

Kocsis B

Wang H., Zumalac \'a rregui M. Kocsis B. Physical Review Letters 136 2026 061403

2026
[33]

F., Frenk C

Navarro J. F., Frenk C. S. White S. D. M. The Astrophysical Journal 490 1997 493

1997
[34]

Heyrovsk \'y D

Karamazov M., Timko L. Heyrovsk \'y D. The Astrophysical Journal 922 2021 72

2021
[35]

Heyrovsk \'y D

Karamazov M. Heyrovsk \'y D. The Astrophysical Journal 927 2022 101

2022
[36]

The European Physical Journal C 86 2026 130

Senjaya D. The European Physical Journal C 86 2026 130

2026
[37]

Bolokhov S. V. The European Physical Journal C 86 2026 576

2026
[38]

Classical and Quantum Gravity 27 2010 233001

Bartelmann M. Classical and Quantum Gravity 27 2010 233001

2010
[39]

Senjaya, Eur

D. Senjaya, Eur. Phys. J. C 86, 130 (2026)

2026
[40]

S. V. Bolokhov, Eur. Phys. J. C 86, 576 (2026)

2026

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [1]

Gravitational Lenses , publisher =

Schneider, Peter and Ehlers, J. Gravitational Lenses , publisher =. 1992 , doi =

1992

[2] [2]

Classical and Quantum Gravity , volume =

Bartelmann, Matthias , title =. Classical and Quantum Gravity , volume =. 2010 , doi =. 1010.3829 , archivePrefix =

arXiv 2010

[3] [3]

Astronomy and Astrophysics , volume =

Schneider, Peter and Weiss, Achim , title =. Astronomy and Astrophysics , volume =

[4] [4]

Astronomy and Astrophysics , volume =

Erdl, Helmut and Schneider, Peter , title =. Astronomy and Astrophysics , volume =

[5] [5]

Astronomy and Astrophysics , volume =

Dominik, Martin , title =. Astronomy and Astrophysics , volume =. 1999 , eprint =

1999

[6] [6]

Scott and Petters, Arlie O

Gaudi, B. Scott and Petters, Arlie O. , title =. The Astrophysical Journal , volume =. 2002 , doi =. astro-ph/0112531 , archivePrefix =

arXiv 2002

[7] [7]

Scott and Petters, Arlie O

Gaudi, B. Scott and Petters, Arlie O. , title =. The Astrophysical Journal , volume =. 2002 , doi =. astro-ph/0206162 , archivePrefix =

arXiv 2002

[8] [8]

and Blandford, Roger D

Rauch, Kevin P. and Blandford, Roger D. , title =. The Astrophysical Journal , volume =. 1994 , doi =

1994