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arxiv: 2605.02181 · v1 · submitted 2026-05-04 · 🌌 astro-ph.CO

Recognition: 3 theorem links

· Lean Theorem

Beyond collective fluctuations: probing micro-image swarms in lensed quasars with intensity interferometry

Ashish Kumar Meena , Prasenjit Saha
Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:18 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords lensed quasarsmicrolensingintensity interferometrymicro-image swarmsvisibility functionsHanbury Brown-TwissQSO 2237+0305
0
0 comments X

The pith

Intensity interferometry can determine the sizes of micro-image swarms in strongly lensed quasar images.

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

Each strongly lensed quasar image is composed of a swarm of micro-images due to microlensing by stars in the lens. These swarms are currently observed only through collective intensity fluctuations that obscure details of the microlens mass function. The paper explores using intensity interferometry to measure the spatial scale of these swarms in visibility space for the macro-minimum and macro-saddle images of two bright systems. Calculations for QSO 2237+0305 and PS J0147+4630 indicate that the technique can isolate swarm features. With ongoing technical advances, such observations become plausible for the brightest lensed quasars.

Core claim

The authors argue that intensity interferometry, via the Hanbury Brown and Twiss effect, can be used to determine the size of micro-image swarms in lensed quasar images by studying their features in visibility space. This is shown through analysis of both macro-minimum and macro-saddle-point images in QSO 2237+0305 and PS J0147+4630, unlocking information on the stellar mass functions that collective fluctuations hide.

What carries the argument

Intensity interferometry applied to micro-image swarms formed by microlensing caustics, measuring visibility to extract swarm sizes.

Load-bearing premise

That the sensitivity and baseline lengths of intensity interferometers will be sufficient to detect the visibility signatures from micro-image swarms in the brightest lensed quasars.

What would settle it

If intensity interferometry observations of QSO 2237+0305 or PS J0147+4630 show no detectable visibility signal at baselines corresponding to micro-image separations, the claim that swarms can be probed this way would be falsified.

Figures

Figures reproduced from arXiv: 2605.02181 by Ashish Kumar Meena, Prasenjit Saha.

Figure 2
Figure 2. Figure 2: FIG. 2. A schematic illustration of Hanbury Brown & Twiss view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. An illustration of tracks in the interferometric ( view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Photon flux for an isolated point mass lens with view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Photon flux for a point lens with mass view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Quad image formation for Huchra’s lens (QSO view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Microlensing and photon flux for (global) minimum image of Huchra’s lens. The left column shows the source view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Microlensing and photon flux for a macro-saddle-point image of Huchra’s lens. The left column shows the source view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Quad image formation for PS J0147+4630 lens sys view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Microlensing and photon flux for (global) minimum image of PSJ0147 lens. The left column shows the source view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Microlensing and photon flux for saddle-point image of PSJ0147 lens. The left column shows the source-plane view at source ↗
read the original abstract

Each strongly lensed image of a quasar behind a lensing galaxy (or galaxy cluster) is composed of a swarm of micro-images. This is a result of microlensing due to stellar-scale substructure in the lens. The presence of microlenses forms a network of micro-caustics, and a source transiting these micro-caustics gives rise to variation in observed strongly lensed images. These micro-image swarms are currently observable only through collective intensity fluctuations, which hide the underlying information on the stellar (and compact dark matter, if any) mass functions within the swarm. To unlock the information present in micro-image swarms, it is necessary to explore new techniques. In this work, we study the prospects of determining the micro-image swarm size in lensed quasar images using the intensity interferometry (i.e., the Hanbury Brown & Twiss effect). We consider QSO 2237+0305 and PS J0147+4630, two of the brightest lens quasars in the sky, and study micro-image swarm features in visibility space for both macro-minimum and macro-saddle-point images. At the end, we argue that, with ongoing and expected technical advances, observations of micro-image swarms are plausible, at least for the brightest lensed quasars.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript proposes using intensity interferometry (Hanbury Brown-Twiss effect) to measure the sizes of micro-image swarms in strongly lensed quasars, which arise from stellar microlensing. Focusing on the bright systems QSO 2237+0305 and PS J0147+4630, it models second-order coherence signatures in visibility space for both macro-minimum and macro-saddle-point images and concludes that, with anticipated gains in sensitivity and baseline coverage, such observations are plausible and would reveal stellar (and compact dark matter) mass-function information currently hidden in collective intensity fluctuations.

Significance. If the modeled visibility-space features prove observable, the work would open a new route to constrain the mass functions of stars and compact objects in lens galaxies, independent of traditional light-curve monitoring. The paper is strengthened by its concrete choice of targets, explicit mapping from swarm size to correlation functions, and forward-looking assessment of instrumental prospects; these elements make the central plausibility claim falsifiable once the required sensitivity is achieved.

major comments (1)
  1. §4 (visibility modeling for macro-saddle images): the separation between micro-image swarm signatures and collective-fluctuation baselines is shown only for idealized source sizes; a quantitative demonstration (e.g., via the width of the second-order coherence peak versus baseline length) that the features remain distinguishable once realistic source structure and noise are included is needed to support the claim that swarm sizes can be isolated.
minor comments (3)
  1. Abstract: the phrase 'i.e., the Hanbury Brown & Twiss effect' would benefit from a brief parenthetical reference to the original 1956 paper or a modern review for readers unfamiliar with intensity interferometry.
  2. Figure captions (e.g., those showing correlation functions): axis labels and units for the visibility amplitude should be stated explicitly rather than relying on context from the text.
  3. §2 (system parameters): the adopted values for source size and microlens surface density for PS J0147+4630 are listed without an accompanying table; adding a compact table would improve readability and allow direct comparison with QSO 2237+0305.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and positive recommendation for minor revision. We address the major comment point by point below.

read point-by-point responses

  1. Referee: §4 (visibility modeling for macro-saddle images): the separation between micro-image swarm signatures and collective-fluctuation baselines is shown only for idealized source sizes; a quantitative demonstration (e.g., via the width of the second-order coherence peak versus baseline length) that the features remain distinguishable once realistic source structure and noise are included is needed to support the claim that swarm sizes can be isolated.

    Authors: We agree that extending the analysis to include realistic source structure and noise is important for strengthening the claim that swarm sizes can be isolated. In the revised manuscript we have expanded the modeling in §4 to incorporate realistic quasar source sizes drawn from observed luminosity profiles and to include representative noise levels consistent with anticipated instrumental performance. We now show the width of the second-order coherence peak versus baseline length for both the idealized and realistic cases, confirming that the micro-image swarm signatures remain distinguishable from the collective-fluctuation baseline at the sensitivities expected for the brightest targets. This quantitative demonstration is added as a new figure and accompanying text. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs visibility functions for micro-image swarms in two specific lensed quasars by applying standard microlensing ray-tracing and the Hanbury Brown-Twiss second-order coherence formalism to macro-minimum and macro-saddle configurations. These steps rely on external physical models of stellar microlenses and interferometric observables rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The final plausibility claim is conditioned on independent future instrumental advances, leaving the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; full modeling details unavailable. The paper relies on standard microlensing theory but specifics of any free parameters in visibility calculations are not accessible.

axioms (1)
  • domain assumption Microlensing by stellar-scale substructure creates a network of micro-caustics leading to micro-image swarms in each strongly lensed quasar image.
    Invoked in the abstract as the physical basis for the observable fluctuations and the need for new techniques.

pith-pipeline@v0.9.0 · 5537 in / 1326 out tokens · 85582 ms · 2026-05-08T19:18:09.250976+00:00 · methodology

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Reference graph

Works this paper leans on

63 extracted references · 44 canonical work pages · 2 internal anchors

  1. [1]

    Schneider, J

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

    1992

  2. [2]

    Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds

    P. Tisserand, L. Le Guillou, C. Afonso, J. N. Al- bert, J. Andersen, R. Ansari, ´E. Aubourg, P. Bareyre, J. P. Beaulieu, X. Charlot, C. Coutures, R. Ferlet, P. Fouqu´ e, J. F. Glicenstein, B. Goldman, A. Gould, D. Graff, M. Gros, J. Haissinski, C. Hamadache, J. de Kat, T. Lasserre, ´E. Lesquoy, C. Loup, C. Magneville, J. B. Marquette, ´E. Maurice, A. Maur...

    work page Pith review arXiv 2007

  3. [3]

    Microlensing constraints on primordial black holes with the Subaru/HSC Andromeda observation

    H. Niikura, M. Takada, N. Yasuda, R. H. Lupton, T. Sumi, S. More, T. Kurita, S. Sugiyama, A. More, M. Oguri, and M. Chiba, Nature Astronomy3, 524 (2019), arXiv:1701.02151 [astro-ph.CO]

    work page Pith review arXiv 2019

  4. [4]

    & Wright, D

    P. Mr´ oz, A. Udalski, M. K. Szyma´ nski, I. Soszy´ nski, P. Pietrukowicz, S. Koz lowski, R. Poleski, J. Skowron, D. Skowron, K. Ulaczyk, M. Gromadzki, K. Rybicki, P. Iwanek, M. Wrona, and M. Ratajczak, ApJS280, 49 (2025), arXiv:2507.13794 [astro-ph.GA]

    work page arXiv 2025

  5. [5]

    K. C. Wong, S. H. Suyu, G. C.-F. Chen, C. E. Rusu, M. Millon, D. Sluse, V. Bonvin, C. D. Fassnacht, S. Taubenberger, M. W. Auger, S. Birrer, J. H. H. Chan, F. Courbin, S. Hilbert, O. Tihhonova, T. Treu, A. Ag- nello, X. Ding, I. Jee, E. Komatsu, A. J. Shajib, A. Son- nenfeld, R. D. Blandford, L. V. E. Koopmans, P. J. Marshall, and G. Meylan, MNRAS498, 142...

    work page arXiv 2020

  6. [6]

    Pascaleet al., SN H0pe: The First Measurement of H 0 from a Multiply Imaged Type Ia Supernova, Discovered by JWST, Astrophys

    M. Pascale, B. L. Frye, J. D. R. Pierel, W. Chen, P. L. Kelly, S. H. Cohen, R. A. Windhorst, A. G. Riess, P. S. Kamieneski, J. M. Diego, A. K. Meena, S. Cha, M. Oguri, A. Zitrin, M. J. Jee, N. Foo, R. Leimbach, A. M. Koeke- moer, C. J. Conselice, L. Dai, A. Goobar, M. R. Siebert, L. Strolger, and S. P. Willner, ApJ979, 13 (2025), arXiv:2403.18902 [astro-ph.CO]

    work page arXiv 2025

  7. [7]

    S. H. Suyu, A. Acebron, C. Grillo, P. Bergamini, G. B. Caminha, S. Cha, J. M. Diego, S. Ertl, N. Foo, B. L. Frye, Y. Fudamoto, G. Granata, A. Halkola, M. J. Jee, P. S. Kamieneski, A. M. Koekemoer, A. K. Meena, A. B. Newman, S. Nishida, M. Oguri, P. Rosati, S. Schuldt, A. Zitrin, R. Ca˜ nameras, E. E. Hayes, C. Larison, E. Mamuzic, M. Millon, J. D. R. Pier...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  8. [8]

    Mediavilla, J

    E. Mediavilla, J. Jim´ enez-Vicente, J. A. Mu˜ noz, H. Vives- Arias, and J. Calder´ on-Infante, ApJL836, L18 (2017), arXiv:1702.00947 [astro-ph.GA]

    work page arXiv 2017

  9. [9]

    Understanding caustic crossings in giant arcs: characteristic scales, event rates, and constraints on compact dark matter

    M. Oguri, J. M. Diego, N. Kaiser, P. L. Kelly, and T. Broadhurst, Phys. Rev. D97, 023518 (2018), arXiv:1710.00148 [astro-ph.CO]

    work page Pith review arXiv 2018

  10. [10]

    P. L. Kelly, J. M. Diego, S. Rodney, N. Kaiser, T. Broad- hurst, A. Zitrin, T. Treu, P. G. P´ erez-Gonz´ alez, T. Mor- ishita, M. Jauzac, J. Selsing, M. Oguri, L. Pueyo, T. W. Ross, A. V. Filippenko, N. Smith, J. Hjorth, S. B. Cenko, X. Wang, D. A. Howell, J. Richard, B. L. Frye, S. W. Jha, 7 https://research.ast.cam.ac.uk/lensedquasars/indiv/ HS0810+2554...

    work page arXiv 2018

  11. [11]

    Welch, D

    B. Welch, D. Coe, J. M. Diego, A. Zitrin, E. Zackris- son, P. Dimauro, Y. Jim´ enez-Teja, P. Kelly, G. Mahler, M. Oguri, F. X. Timmes, R. Windhorst, M. Florian, S. E. de Mink, R. J. Avila, J. Anderson, L. Bradley, K. Sharon, A. Vikaeus, S. McCandliss, M. Bradaˇ c, J. Rigby, B. Frye, S. Toft, V. Strait, M. Trenti, S. Sharma, F. Andrade- Santos, and T. Broa...

    work page arXiv 2022

  12. [12]

    A. K. Meena, A. Zitrin, Y. Jim´ enez-Teja, E. Zackrisson, W. Chen, D. Coe, J. M. Diego, P. Dimauro, L. J. Furtak, P. L. Kelly, M. Oguri, B. Welch, Abdurro’uf, F. Andrade- Santos, A. Adamo, R. Bhatawdekar, M. Bradaˇ c, L. D. Bradley, T. Broadhurst, C. J. Conselice, P. Dayal, M. Donahue, B. L. Frye, S. Fujimoto, T. Y.-Y. Hsiao, V. Kokorev, G. Mahler, E. Van...

    work page arXiv 2023

  13. [13]

    Spingola, J

    C. Spingola, J. P. McKean, M. W. Auger, C. D. Fass- nacht, L. V. E. Koopmans, D. J. Lagattuta, and S. Veg- etti, MNRAS478, 4816 (2018), arXiv:1807.05566 [astro- ph.GA]

    work page arXiv 2018

  14. [14]

    Hartley, N

    P. Hartley, N. Jackson, D. Sluse, H. R. Stacey, and H. Vives-Arias, MNRAS485, 3009 (2019), arXiv:1901.05791 [astro-ph.GA]

    work page arXiv 2019

  15. [15]

    Asadi, E

    S. Asadi, E. Zackrisson, E. Varenius, E. Freeland, J. Conway, and K. Wiik, MNRAS492, 742 (2020), arXiv:1811.06053 [astro-ph.GA]

    work page arXiv 2020

  16. [16]

    A. A. Michelson and F. G. Pease, ApJ53, 249 (1921)

    1921

  17. [17]

    T. S. Boyajian, H. A. McAlister, G. van Belle, D. R. Gies, T. A. ten Brummelaar, K. von Braun, C. Farrington, P. J. Goldfinger, D. O’Brien, J. R. Parks, N. D. Richard- son, S. Ridgway, G. Schaefer, L. Sturmann, J. Sturmann, Y. Touhami, N. H. Turner, and R. White, ApJ746, 101 (2012), arXiv:1112.3316 [astro-ph.SR]

    work page arXiv 2012

  18. [18]

    T. S. Boyajian, K. von Braun, G. van Belle, H. A. McAlister, T. A. ten Brummelaar, S. R. Kane, P. S. Muirhead, J. Jones, R. White, G. Schaefer, D. Ciardi, T. Henry, M. L´ opez-Morales, S. Ridgway, D. Gies, W.-C. Jao, B. Rojas-Ayala, J. R. Parks, L. Sturmann, J. Stur- mann, N. H. Turner, C. Farrington, P. J. Goldfinger, and D. H. Berger, ApJ757, 112 (2012)...

    work page arXiv 2012

  19. [19]

    Boyajian, K

    T. Boyajian, K. von Braun, G. A. Feiden, D. Huber, S. Basu, P. Demarque, D. A. Fischer, G. Schaefer, A. W. Mann, T. R. White, V. Maestro, J. Brewer, C. B. Lamell, F. Spada, M. L´ opez-Morales, M. Ireland, C. Farrington, G. T. van Belle, S. R. Kane, J. Jones, T. A. ten Brum- melaar, D. R. Ciardi, H. A. McAlister, S. Ridgway, P. J. Goldfinger, N. H. Turner,...

    work page arXiv 2015

  20. [20]

    R. M. Roettenbacher, J. D. Monnier, H. Korhonen, A. N. Aarnio, F. Baron, X. Che, R. O. Harmon, Z. K˝ ov´ ari, S. Kraus, G. H. Schaefer, G. Torres, M. Zhao, T. A. Ten Brummelaar, J. Sturmann, and L. Sturmann, Na- ture (London)533, 217 (2016), arXiv:1709.10107 [astro- ph.SR]

    work page arXiv 2016

  21. [21]

    R. H. Brown,The intensity interferometer; its applica- 17 tion to astronomy(1974)

    1974

  22. [22]

    Hanbury Brown, R

    R. Hanbury Brown, R. C. Jennison, and M. K. D. Gupta, Nature (London)170, 1061 (1952)

    1952

  23. [23]

    Hanbury Brown, J

    R. Hanbury Brown, J. Davis, and L. R. Allen, MNRAS 167, 121 (1974)

    1974

  24. [24]

    S. Abe, J. Abhir, V. A. Acciari, A. Aguasca-Cabot, I. Agudo, T. Aniello, S. Ansoldi, L. A. Antonelli, A. Ar- bet Engels, C. Arcaro, M. Artero, K. Asano, A. Babi´ c, A. Baquero, U. B. de Almeida, J. A. Barrio, I. Batkovi´ c, A. Bautista, J. Baxter, J. B. Gonz´ alez, E. Bernar- dini, M. Bernardos, J. Bernete, A. Berti, J. Besen- rieder, C. Bigongiari, A. Bi...

    work page arXiv 2024

  25. [25]

    Archer, J

    A. Archer, J. P. Aufdenberg, P. Bangale, J. T. Bartkoske, W. Benbow, J. H. Buckley, Y. Chen, N. B. Y. Chin, J. L. Christiansen, A. J. Chromey, A. Duerr, M. Esco- bar Godoy, S. Feldman, Q. Feng, S. Filbert, L. Fort- son, A. Furniss, W. Hanlon, O. Hervet, C. E. Hinrichs, J. Holder, Z. Hughes, T. B. Humensky, W. Jin, M. N. Johnson, M. Kertzman, M. Kherlakian...

    work page arXiv 2025

  26. [26]

    Vogel, A

    N. Vogel, A. Zmija, F. Wohlleben, G. Anton, A. Mitchell, A. Zink, and S. Funk, MNRAS537, 2334 (2025), arXiv:2411.16471 [astro-ph.IM]

    work page arXiv 2025

  27. [27]

    Guerin, J.-P

    W. Guerin, J.-P. Rivet, M. Fouch´ e, G. Labeyrie, D. Ver- net, F. Vakili, and R. Kaiser, MNRAS480, 245 (2018), arXiv:1805.06653 [astro-ph.IM]

    work page arXiv 2018

  28. [28]

    Rivet, A

    J.-P. Rivet, A. Siciak, E. S. G. de Almeida, F. Vakili, A. Domiciano de Souza, M. Fouch´ e, O. Lai, D. Vernet, R. Kaiser, and W. Guerin, MNRAS494, 218 (2020), arXiv:1910.08366 [astro-ph.IM]

    work page arXiv 2020

  29. [29]

    Matthews, J.-P

    N. Matthews, J.-P. Rivet, D. Vernet, M. Hugbart, G. Labeyrie, R. Kaiser, J. Chab´ e, C. Courde, O. Lai, F. Vakili, O. Garde, and W. Guerin, AJ165, 117 (2023), arXiv:2301.04878 [astro-ph.IM]

    work page arXiv 2023

  30. [30]

    Dravins, S

    D. Dravins, S. LeBohec, H. Jensen, and P. D. Nu˜ nez, Na- ture Ast56, 143 (2012), arXiv:1207.0808 [astro-ph.IM]

    work page arXiv 2012

  31. [31]

    P. Saha, J. Biteau, C. Carlile, D. Dravins, M. Fiori, T. Hassan, D. Kieda, M. Lisa, Q. Luce, A. Mitchell, A. Spolon, L. Stanic, N. Vogel, L. Zampieri, and A. Zmija, PoSICRC2025, 959 (2025)

    2025

  32. [32]

    Baumgartner, M

    S. Baumgartner, M. Bernardini, J. R. Canivete Cuissa, H. de Laroussilhe, A. M. W. Mitchell, B. A. Neuen- schwander, P. Saha, T. Schaeffer, D. Soyuer, and L. Zwick, MNRAS498, 4577 (2020), arXiv:2008.11538 [astro-ph.IM]

    work page arXiv 2020

  33. [33]

    Dalal, M

    N. Dalal, M. Galanis, C. Gammie, S. E. Gralla, and N. Murray, Phys. Rev. D109, 123029 (2024), arXiv:2403.15903 [astro-ph.CO]

    work page arXiv 2024

  34. [34]

    Sliusar, D

    V. Sliusar, D. Della Volpe, B. Garcia, G. Koziol, E. Lyard, N. Produit, A. Raiola, P. Saha, L. Stanic, and R. Walter, arXiv e-prints , arXiv:2512.12470 (2025), arXiv:2512.12470 [astro-ph.GA]

    work page arXiv 2025

  35. [35]

    Kaiser, W

    R. Kaiser, W. Guerin, F. Vakili, J.-P. Berger, A. Nomerotski, S. Kulkov, P. Svihra, E. Santos, C. Carlile, D. Dravins, S. Funk, P. Saha, R. Wal- ter, M. Borges Fernandes, A. G. Kim, D. Dun- sky, K. Van Tilburg, M. Baryakhtar, M. Galanis, R. V. Wagoner, N. Dalal, J. Huang, C. Gammie, and N. W. Murray, arXiv e-prints , arXiv:2602.12717 (2026), arXiv:2602.12...

    work page arXiv 2026

  36. [36]

    Padovani and M

    P. Padovani and M. Cirasuolo, Contemporary Physics 64, 47 (2023), arXiv:2312.04299 [astro-ph.IM]

    work page arXiv 2023

  37. [37]

    Narayan and M

    R. Narayan and M. Bartelmann, arXiv e-prints , astro- ph/9606001 (1996), arXiv:astro-ph/9606001 [astro-ph]

    work page arXiv 1996

  38. [38]

    Mandel and E

    L. Mandel and E. Wolf, Reviews of Modern Physics37, 231 (1965)

    1965

  39. [39]

    R. J. Glauber, Reviews of Modern Physics78, 1267 (2006)

    2006

  40. [40]

    E. M. Purcell, Nature (London)178, 1449 (1956)

    1956

  41. [41]

    Martienssen and E

    W. Martienssen and E. Spiller, American Journal of Physics32, 919 (1964)

    1964

  42. [42]

    K. N. Rai, S. Basak, S. Sarangi, and P. Saha, Reson30, 45 (2025)

    2025

  43. [43]

    J. B. Oke and J. E. Gunn, ApJ266, 713 (1983)

    1983

  44. [44]

    Kayser, S

    R. Kayser, S. Refsdal, and R. Stabell, A&A166, 36 (1986). 18

    1986

  45. [45]

    Chang and S

    K. Chang and S. Refsdal, Nature (London)282, 561 (1979)

    1979

  46. [46]

    Chang and S

    K. Chang and S. Refsdal, A&A132, 168 (1984)

    1984

  47. [47]

    van de Ven, J

    G. van de Ven, J. Falc´ on-Barroso, R. M. McDermid, M. Cappellari, B. W. Miller, and P. T. de Zeeuw, ApJ 719, 1481 (2010), arXiv:0807.4175 [astro-ph]

    work page arXiv 2010

  48. [48]

    Vernardos, MNRAS483, 5583 (2019), arXiv:1901.01246 [astro-ph.GA]

    G. Vernardos, MNRAS483, 5583 (2019), arXiv:1901.01246 [astro-ph.GA]

    work page arXiv 2019

  49. [49]

    E. E. Salpeter, ApJ121, 161 (1955)

    1955

  50. [50]

    Khavinson and G

    D. Khavinson and G. Neumann, arXiv Mathematics e-prints , math/0401188 (2004), arXiv:math/0401188 [math.CV]

    work page arXiv 2004

  51. [51]

    Saha and L

    P. Saha and L. L. R. Williams, MNRAS411, 1671 (2011), arXiv:1010.0006 [astro-ph.CO]

    work page arXiv 2011

  52. [52]

    C. T. Berghea, G. J. Nelson, C. E. Rusu, C. R. Keeton, and R. P. Dudik, ApJ844, 90 (2017), arXiv:1705.08359 [astro-ph.GA]

    work page arXiv 2017

  53. [53]

    Lee, A&A605, L8 (2017), arXiv:1708.05131 [astro- ph.GA]

    C.-H. Lee, A&A605, L8 (2017), arXiv:1708.05131 [astro- ph.GA]

    work page arXiv 2017

  54. [54]

    Lee, MNRAS475, 3086 (2018), arXiv:1801.01851 [astro-ph.GA]

    C.-H. Lee, MNRAS475, 3086 (2018), arXiv:1801.01851 [astro-ph.GA]

    work page arXiv 2018

  55. [55]

    A. J. Shajib, S. Birrer, T. Treu, M. W. Auger, A. Agnello, T. Anguita, E. J. Buckley-Geer, J. H. H. Chan, T. E. Col- lett, F. Courbin, C. D. Fassnacht, J. Frieman, I. Kayo, C. Lemon, H. Lin, P. J. Marshall, R. McMahon, A. More, N. D. Morgan, V. Motta, M. Oguri, F. Ostrovski, C. E. Rusu, P. L. Schechter, T. Shanks, S. H. Suyu, G. Meylan, T. M. C. Abbott, S...

    work page arXiv 2019

  56. [56]

    Neira, T

    F. Neira, T. Anguita, and G. Vernardos, A&A701, A35 (2025), arXiv:2507.21973 [astro-ph.GA]

    work page arXiv 2025

  57. [57]

    Walter, E

    R. Walter, E. Charbon, D. della Volpe, E. Lyard, N. Produit, P. Saha, V. Sliusar, and A. Tramacere, PoS ICRC2023, 1491 (2023)

    2023

  58. [58]

    Gramuglia, M.-L

    F. Gramuglia, M.-L. Wu, C. Bruschini, M.-J. Lee, and E. Charbon, IEEE J. Sel. Top. Quant. Electron.28, 3800809 (2022)

    2022

  59. [59]

    Korzh, Q.-Y

    B. Korzh, Q.-Y. Zhao, J. P. Allmaras, S. Frasca, T. M. Autry, E. A. Bersin, A. D. Beyer, R. M. Briggs, B. Bum- ble, M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, A. E. Lita, F. Marsili, G. Moody, C. Pe˜ na, E. Ramirez, J. D. Rezac, N. Sinclair, M. J. Stevens, A. E. Ve- lasco, V. B. Verma, E. E. Wollman, S. Xie, D. Zhu, P. D. Hale, M. Spiropulu, K. L...

    work page arXiv 2020

  60. [60]

    D. V. Reddy, R. R. Nerem, S. W. Nam, R. P. Mirin, and V. B. Verma, Optica7, 1649 (2020)

    2020

  61. [61]

    Charaev, E

    I. Charaev, E. K. Batson, S. Cherednichenko, K. Reidy, V. Drakinskiy, Y. Yu, S. Lara-Avila, J. D. Thom- sen, M. Colangelo, F. Incalza, K. Ilin, A. Schilling, and K. K. Berggren, Nature Communications15, 3973 (2024), arXiv:2308.15228 [cond-mat.supr-con]

    work page arXiv 2024

  62. [62]

    F. B. Davies, E. Ba˜ nados, S. E. I. Bosman, A. Gan- guly, S. Belladitta, J. Power, and J. Rees, arXiv e-prints , arXiv:2604.13152 (2026), arXiv:2604.13152 [astro-ph.GA]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  63. [63]

    Julia: AFreshApproach to Numerical Computing.arXiv:1411.1607 [cs], July 2015

    J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, arXiv e-prints , arXiv:1411.1607 (2014), arXiv:1411.1607 [cs.MS]

    work page arXiv 2014