United Kingdom  S. J. Rowlinson  University of Birmingham, Birmingham B15 2TT,
United Kingdom  Santosh Roy  Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  Soumen Roy  Institute for Gravitational and
Subatomic Physics (GRASP), Utrecht University, 3584 CC Utrecht, Netherlands
A. Royzman  The University of Utah, Salt Lake City, UT 84112, USA  D. Rozza
Università degli Studi di Sassari, I-07100 Sassari, Italy  INFN, Laboratori
Nazionali del Sud, I-95125 Catania, Italy  P. Ruggi  European Gravitational
Observatory (EGO), I-56021 Cascina, Pisa, Italy  K. Ruiz-Rocha  Vanderbilt
University, Nashville, TN 37235, USA  K. Ryan  LIGO Hanford Observatory,
Richland, WA 99352, USA  S. Sachdev University of Wisconsin-Milwaukee,
Milwaukee, WI 53201, USA  T. Sadecki  LIGO Hanford Observatory, Richland, WA
99352, USA  J. Sadiq IGFAE, Universidade de Santiago de Compostela, 15782
Spain  P. Saffarieh  Nikhef, 1098 XG Amsterdam, Netherlands  Department of
Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam,
Netherlands  S. Saha National Tsing Hua University, Hsinchu City, 30013
Taiwan, Republic of China  Y. Saito  Institute for Cosmic Ray Research (ICRR),
KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida City, Gifu
506-1205, Japan  K. Sakai  Department of Electronic Control Engineering,
National Institute of Technology, Nagaoka College, Nagaoka City, Niigata
940-8532, Japan  M. Sakellariadou King’s College London, University of London,
London WC2R 2LS, United Kingdom  S. Sakon  The Pennsylvania State University,
University Park, PA 16802, USA  F. Salces-Carcoba LIGO Laboratory, California
Institute of Technology, Pasadena, CA 91125, USA  L. Salconi  European
Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy  M. Saleem
University of Minnesota, Minneapolis, MN 55455, USA  F. Salemi Università di
Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy  INFN, Trento
Institute for Fundamental Physics and Applications, I-38123 Povo, Trento,
Italy  M. Sallé Nikhef, 1098 XG Amsterdam, Netherlands  A. Samajdar INFN,
Sezione di Milano-Bicocca, I-20126 Milano, Italy  E. J. Sanchez  LIGO
Laboratory, California Institute of Technology, Pasadena, CA 91125, USA  J. H.
Sanchez  California State University Fullerton, Fullerton, CA 92831, USA  L.
E. Sanchez  LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  N. Sanchis-Gual Departamento de Matemática da Universidade de
Aveiro and Centre for Research and Development in Mathematics and
Applications, 3810-183 Aveiro, Portugal  Departamento de Astronomía y
Astrofísica, Universitat de València, E-46100 Burjassot, València, Spain  J.
R. Sanders  Marquette University, Milwaukee, WI 53233, USA  A. Sanuy Institut
de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, 08028,
Spain  T. R. Saravanan  Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  N. Sarin  OzGrav, School of Physics &
Astronomy, Monash University, Clayton 3800, Victoria, Australia  A. Sasli
Department of Physics, Aristotle University of Thessaloniki, 54124
Thessaloniki, Greece  B. Sassolas  Université Lyon, Université Claude Bernard
Lyon 1, CNRS, Laboratoire des Matériaux Avancés (LMA), IP2I Lyon / IN2P3, UMR
5822, F-69622 Villeurbanne, France  H. Satari  OzGrav, University of Western
Australia, Crawley, Western Australia 6009, Australia  B. S. Sathyaprakash The
Pennsylvania State University, University Park, PA 16802, USA  Cardiff
University, Cardiff CF24 3AA, United Kingdom  O. Sauter University of Florida,
Gainesville, FL 32611, USA  R. L. Savage LIGO Hanford Observatory, Richland,
WA 99352, USA  V. Savant Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  T. Sawada Department of Physics, Graduate
School of Science, Osaka City University, Sumiyoshi-ku, Osaka City, Osaka
558-8585, Japan  H. L. Sawant  Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  S. Sayah  Université Lyon, Université Claude
Bernard Lyon 1, CNRS, Laboratoire des Matériaux Avancés (LMA), IP2I Lyon /
IN2P3, UMR 5822, F-69622 Villeurbanne, France  D. Schaetzl  LIGO Laboratory,
California Institute of Technology, Pasadena, CA 91125, USA  M. Scheel  CaRT,
California Institute of Technology, Pasadena, CA 91125, USA  J. Scheuer
Northwestern University, Evanston, IL 60208, USA  M. G. Schiworski OzGrav,
University of Adelaide, Adelaide, South Australia 5005, Australia  P. Schmidt
University of Birmingham, Birmingham B15 2TT, United Kingdom  S. Schmidt
Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University,
3584 CC Utrecht, Netherlands  R. Schnabel Universität Hamburg, D-22761
Hamburg, Germany  M. Schneewind  Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), D-30167 Hannover, Germany  Leibniz
Universität Hannover, D-30167 Hannover, Germany  R. M. S. Schofield
University of Oregon, Eugene, OR 97403, USA  A. Schönbeck  Universität
Hamburg, D-22761 Hamburg, Germany  B. W. Schulte  Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
Leibniz Universität Hannover, D-30167 Hannover, Germany  B. F. Schutz  Cardiff
University, Cardiff CF24 3AA, United Kingdom  Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
Leibniz Universität Hannover, D-30167 Hannover, Germany  E. Schwartz Cardiff
University, Cardiff CF24 3AA, United Kingdom  J. Scott SUPA, University of
Glasgow, Glasgow G12 8QQ, United Kingdom  S. M. Scott OzGrav, Australian
National University, Canberra, Australian Capital Territory 0200, Australia
M. Seglar-Arroyo Univ. Savoie Mont Blanc, CNRS, Laboratoire d’Annecy de
Physique des Particules - IN2P3, F-74000 Annecy, France  Y. Sekiguchi Faculty
of Science, Toho University, Funabashi City, Chiba 274-8510, Japan  D. Sellers
LIGO Livingston Observatory, Livingston, LA 70754, USA  A. S. Sengupta  Indian
Institute of Technology, Palaj, Gandhinagar, Gujarat 382355, India  D.
Sentenac  European Gravitational Observatory (EGO), I-56021 Cascina, Pisa,
Italy  E. G. Seo  The Chinese University of Hong Kong, Shatin, NT, Hong Kong
V. Sequino  Università di Napoli “Federico II”, I-80126 Napoli, Italy  INFN,
Sezione di Napoli, I-80126 Napoli, Italy  A. Sergeev  Institute of Applied
Physics, Nizhny Novgorod, 603950, Russia  G. Servignat  Laboratoire Univers et
Théories, Observatoire de Paris, 92190 Meudon, France  Y. Setyawati Institute
for Gravitational and Subatomic Physics (GRASP), Utrecht University, 3584 CC
Utrecht, Netherlands  T. Shaffer  LIGO Hanford Observatory, Richland, WA
99352, USA  M. S. Shahriar Northwestern University, Evanston, IL 60208, USA
M. A. Shaikh International Centre for Theoretical Sciences, Tata Institute of
Fundamental Research, Bengaluru 560089, India  B. Shams  The University of
Utah, Salt Lake City, UT 84112, USA  L. Shao Kavli Institute for Astronomy and
Astrophysics, Peking University, Haidian District, Beijing 100871, China  A.
Sharma  Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy  INFN,
Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy  P. Sharma  RRCAT,
Indore, Madhya Pradesh 452013, India  P. Shawhan University of Maryland,
College Park, MD 20742, USA  N. S. Shcheblanov NAVIER, École des Ponts, Univ
Gustave Eiffel, CNRS, Marne-la-Vallée, France  A. Sheela  Indian Institute of
Technology Madras, Chennai 600036, India  E. Sheridan  Vanderbilt University,
Nashville, TN 37235, USA  Y. Shikano Graduate School of Science and
Technology, Gunma University, Maebashi, Gunma 371-8510, Japan  Institute for
Quantum Studies, Chapman University, Orange, CA 92866, USA  M. Shikauchi
Research Center for the Early Universe (RESCEU), The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan  H. Shimizu Accelerator Laboratory, High
Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki
305-0801, Japan  K. Shimode Institute for Cosmic Ray Research (ICRR), KAGRA
Observatory, The University of Tokyo, Kamioka-cho, Hida City, Gifu 506-1205,
Japan  H. Shinkai Faculty of Information Science and Technology, Osaka
Institute of Technology, Hirakata City, Osaka 573-0196, Japan  T. Shishido
The Graduate University for Advanced Studies (SOKENDAI), Mitaka City, Tokyo
181-8588, Japan  A. Shoda Gravitational Wave Science Project, National
Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan
D. H. Shoemaker LIGO Laboratory, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  D. M. Shoemaker University of Texas, Austin, TX
78712, USA  S. ShyamSundar  RRCAT, Indore, Madhya Pradesh 452013, India  M.
Sieniawska  Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
D. Sigg LIGO Hanford Observatory, Richland, WA 99352, USA  L. Silenzi INFN,
Sezione di Perugia, I-06123 Perugia, Italy  Università di Camerino,
Dipartimento di Fisica, I-62032 Camerino, Italy  L. P. Singer NASA Goddard
Space Flight Center, Greenbelt, MD 20771, USA  D. Singh The Pennsylvania State
University, University Park, PA 16802, USA  M. K. Singh International Centre
for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru
560089, India  N. Singh Astronomical Observatory Warsaw University, 00-478
Warsaw, Poland  A. Singha Maastricht University, 6200 MD Maastricht,
Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  A. M. Sintes IAC3–IEEC,
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain  V. Sipala
Università degli Studi di Sassari, I-07100 Sassari, Italy  INFN, Laboratori
Nazionali del Sud, I-95125 Catania, Italy  V. Skliris  Cardiff University,
Cardiff CF24 3AA, United Kingdom  B. J. J. Slagmolen OzGrav, Australian
National University, Canberra, Australian Capital Territory 0200, Australia
T. J. Slaven-Blair  OzGrav, University of Western Australia, Crawley, Western
Australia 6009, Australia  J. Smetana  University of Birmingham, Birmingham
B15 2TT, United Kingdom  J. R. Smith California State University Fullerton,
Fullerton, CA 92831, USA  L. Smith  SUPA, University of Glasgow, Glasgow G12
8QQ, United Kingdom  R. J. E. Smith OzGrav, School of Physics & Astronomy,
Monash University, Clayton 3800, Victoria, Australia  J. Soldateschi
Università di Firenze, Sesto Fiorentino I-50019, Italy  INAF, Osservatorio
Astrofisico di Arcetri, I-50125 Firenze, Italy  INFN, Sezione di Firenze,
I-50019 Sesto Fiorentino, Firenze, Italy  S. N. Somala Indian Institute of
Technology Hyderabad, Sangareddy, Khandi, Telangana 502285, India  K. Somiya
Graduate School of Science, Tokyo Institute of Technology, Meguro-ku, Tokyo
152-8551, Japan  I. Song National Tsing Hua University, Hsinchu City, 30013
Taiwan, Republic of China  K. Soni Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  S. Soni LIGO Laboratory, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  V. Sordini  Université
Lyon, Université Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, UMR 5822,
F-69622 Villeurbanne, France  F. Sorrentino  INFN, Sezione di Genova, I-16146
Genova, Italy  N. Sorrentino Università di Pisa, I-56127 Pisa, Italy  INFN,
Sezione di Pisa, I-56127 Pisa, Italy  R. Soulard  Artemis, Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, F-06304 Nice, France  T.
Souradeep  Indian Institute of Science Education and Research, Pune,
Maharashtra 411008, India  Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  V. Spagnuolo  Maastricht University, 6200 MD
Maastricht, Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  A. P. Spencer
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom  M. Spera
Università di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova,
Italy  INFN, Sezione di Padova, I-35131 Padova, Italy  P. Spinicelli  European
Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy  A. K.
Srivastava  Institute for Plasma Research, Bhat, Gandhinagar 382428, India  V.
Srivastava  Syracuse University, Syracuse, NY 13244, USA  C. Stachie  Artemis,
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, F-06304 Nice,
France  F. Stachurski  SUPA, University of Glasgow, Glasgow G12 8QQ, United
Kingdom  D. A. Steer Université de Paris, CNRS, Astroparticule et Cosmologie,
F-75006 Paris, France  J. Steinlechner  Maastricht University, 6200 MD
Maastricht, Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  S.
Steinlechner Maastricht University, 6200 MD Maastricht, Netherlands  Nikhef,
1098 XG Amsterdam, Netherlands  N. Stergioulas  Department of Physics,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece  D. J. Stops
University of Birmingham, Birmingham B15 2TT, United Kingdom  K. A. Strain
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom  L. C. Strang
OzGrav, University of Melbourne, Parkville, Victoria 3010, Australia  G.
Stratta Istituto di Astrofisica e Planetologia Spaziali di Roma, 00133 Roma,
Italy  INFN, Sezione di Roma, I-00185 Roma, Italy  M. D. Strong  Louisiana
State University, Baton Rouge, LA 70803, USA  A. Strunk  LIGO Hanford
Observatory, Richland, WA 99352, USA  R. Sturani  International Institute of
Physics, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970,
Brazil  A. L. Stuver Villanova University, Villanova, PA 19085, USA  M.
Suchenek  Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
00-716, Warsaw, Poland  S. Sudhagar Inter-University Centre for Astronomy and
Astrophysics, Pune 411007, India  R. Sugimoto Department of Space and
Astronautical Science, The Graduate University for Advanced Studies
(SOKENDAI), Sagamihara City, Kanagawa 252-5210, Japan  Institute of Space and
Astronautical Science (JAXA), Chuo-ku, Sagamihara City, Kanagawa 252-0222,
Japan  H. G. Suh University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
A. G. Sullivan Columbia University, New York, NY 10027, USA  T. Z.
Summerscales Andrews University, Berrien Springs, MI 49104, USA  L. Sun
OzGrav, Australian National University, Canberra, Australian Capital Territory
0200, Australia  S. Sunil  Institute for Plasma Research, Bhat, Gandhinagar
382428, India  A. Sur Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, 00-716, Warsaw, Poland  J. Suresh Research Center for the Early
Universe (RESCEU), The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium  P. J.
Sutton Cardiff University, Cardiff CF24 3AA, United Kingdom  Takamasa Suzuki
Faculty of Engineering, Niigata University, Nishi-ku, Niigata City, Niigata
950-2181, Japan  Takanori Suzuki  Graduate School of Science, Tokyo Institute
of Technology, Meguro-ku, Tokyo 152-8551, Japan  Toshikazu Suzuki  Institute
for Cosmic Ray Research (ICRR), KAGRA Observatory, The University of Tokyo,
Kashiwa City, Chiba 277-8582, Japan  B. L. Swinkels Nikhef, 1098 XG Amsterdam,
Netherlands  A. Syx  Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000
Strasbourg, France  M. J. Szczepańczyk University of Florida, Gainesville, FL
32611, USA  P. Szewczyk Astronomical Observatory Warsaw University, 00-478
Warsaw, Poland  M. Tacca Nikhef, 1098 XG Amsterdam, Netherlands  H. Tagoshi
Institute for Cosmic Ray Research (ICRR), KAGRA Observatory, The University of
Tokyo, Kashiwa City, Chiba 277-8582, Japan  S. C. Tait SUPA, University of
Glasgow, Glasgow G12 8QQ, United Kingdom  H. Takahashi Research Center for
Space Science, Advanced Research Laboratories, Tokyo City University,
Setagaya, Tokyo 158-0082, Japan  R. Takahashi Gravitational Wave Science
Project, National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo
181-8588, Japan  S. Takano  Department of Physics, The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan  H. Takeda Department of Physics, The
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  M. Takeda  Department
of Physics, Graduate School of Science, Osaka City University, Sumiyoshi-ku,
Osaka City, Osaka 558-8585, Japan  C. J. Talbot  SUPA, University of
Strathclyde, Glasgow G1 1XQ, United Kingdom  C. Talbot  LIGO Laboratory,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  N. Tamanini
L2IT, Laboratoire des 2 Infinis - Toulouse, Université de Toulouse,
CNRS/IN2P3, UPS, F-31062 Toulouse Cedex 9, France  K. Tanaka  Institute for
Cosmic Ray Research (ICRR), Research Center for Cosmic Neutrinos (RCCN), The
University of Tokyo, Kashiwa City, Chiba 277-8582, Japan  Taiki Tanaka
Institute for Cosmic Ray Research (ICRR), KAGRA Observatory, The University of
Tokyo, Kashiwa City, Chiba 277-8582, Japan  Takahiro Tanaka Department of
Physics, Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan  A. J.
Tanasijczuk  Université catholique de Louvain, B-1348 Louvain-la-Neuve,
Belgium  S. Tanioka Institute for Cosmic Ray Research (ICRR), KAGRA
Observatory, The University of Tokyo, Kamioka-cho, Hida City, Gifu 506-1205,
Japan  D. B. Tanner  University of Florida, Gainesville, FL 32611, USA  D. Tao
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
L. Tao University of Florida, Gainesville, FL 32611, USA  R. D. Tapia  The
Pennsylvania State University, University Park, PA 16802, USA  E. N. Tapia San
Martín Nikhef, 1098 XG Amsterdam, Netherlands  C. Taranto  Università di Roma
Tor Vergata, I-00133 Roma, Italy  A. Taruya Yukawa Institute for Theoretical
Physics (YITP), Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan
J. D. Tasson Carleton College, Northfield, MN 55057, USA  R. Tenorio
IAC3–IEEC, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
J. E. S. Terhune Villanova University, Villanova, PA 19085, USA  L. Terkowski
Universität Hamburg, D-22761 Hamburg, Germany  H. Themann  California State
University, Los Angeles, Los Angeles, CA 90032, USA  M. P. Thirugnanasambandam
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India  M.
Thomas  LIGO Livingston Observatory, Livingston, LA 70754, USA  P. Thomas
LIGO Hanford Observatory, Richland, WA 99352, USA  S. Thomas  California State
University Fullerton, Fullerton, CA 92831, USA  D. Thompson  Carleton College,
Northfield, MN 55057, USA  E. E. Thompson  Georgia Institute of Technology,
Atlanta, GA 30332, USA  J. E. Thompson Cardiff University, Cardiff CF24 3AA,
United Kingdom  S. R. Thondapu  RRCAT, Indore, Madhya Pradesh 452013, India
K. A. Thorne  LIGO Livingston Observatory, Livingston, LA 70754, USA  E.
Thrane  OzGrav, School of Physics & Astronomy, Monash University, Clayton
3800, Victoria, Australia  Shubhanshu Tiwari University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland  Srishti Tiwari Inter-
University Centre for Astronomy and Astrophysics, Pune 411007, India  V.
Tiwari Cardiff University, Cardiff CF24 3AA, United Kingdom  A. M. Toivonen
University of Minnesota, Minneapolis, MN 55455, USA  A. E. Tolley University
of Portsmouth, Portsmouth, PO1 3FX, United Kingdom  T. Tomaru Gravitational
Wave Science Project, National Astronomical Observatory of Japan (NAOJ),
Mitaka City, Tokyo 181-8588, Japan  T. Tomura Institute for Cosmic Ray
Research (ICRR), KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida
City, Gifu 506-1205, Japan  M. Tonelli  Università di Pisa, I-56127 Pisa,
Italy  INFN, Sezione di Pisa, I-56127 Pisa, Italy  A. Torres-Forné
Departamento de Astronomía y Astrofísica, Universitat de València, E-46100
Burjassot, València, Spain  C. I. Torrie  LIGO Laboratory, California
Institute of Technology, Pasadena, CA 91125, USA  I. Tosta e Melo INFN,
Laboratori Nazionali del Sud, I-95125 Catania, Italy  E. Tournefier Univ.
Savoie Mont Blanc, CNRS, Laboratoire d’Annecy de Physique des Particules -
IN2P3, F-74000 Annecy, France  D. Töyrä  OzGrav, Australian National
University, Canberra, Australian Capital Territory 0200, Australia  A.
Trapananti Università di Camerino, Dipartimento di Fisica, I-62032 Camerino,
Italy  INFN, Sezione di Perugia, I-06123 Perugia, Italy  F. Travasso INFN,
Sezione di Perugia, I-06123 Perugia, Italy  Università di Camerino,
Dipartimento di Fisica, I-62032 Camerino, Italy  G. Traylor  LIGO Livingston
Observatory, Livingston, LA 70754, USA  J. Trenado Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona, Barcelona, 08028, Spain  M. Trevor
University of Maryland, College Park, MD 20742, USA  M. C. Tringali European
Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy  A. Tripathee
University of Michigan, Ann Arbor, MI 48109, USA  L. Troiano  Dipartimento di
Scienze Aziendali - Management and Innovation Systems (DISA-MIS), Università
di Salerno, I-84084 Fisciano, Salerno, Italy  INFN, Sezione di Napoli, Gruppo
Collegato di Salerno, I-80126 Napoli, Italy  A. Trovato INFN, Sezione di
Trieste, I-34127 Trieste, Italy  Dipartimento di Fisica, Università di
Trieste, I-34127 Trieste, Italy  L. Trozzo INFN, Sezione di Napoli, I-80126
Napoli, Italy  Institute for Cosmic Ray Research (ICRR), KAGRA Observatory,
The University of Tokyo, Kamioka-cho, Hida City, Gifu 506-1205, Japan  R. J.
Trudeau  LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  D. Tsai  National Tsing Hua University, Hsinchu City, 30013
Taiwan, Republic of China  K. W. Tsang  Nikhef, 1098 XG Amsterdam, Netherlands
Van Swinderen Institute for Particle Physics and Gravity, University of
Groningen, 9747 AG Groningen, Netherlands  Institute for Gravitational and
Subatomic Physics (GRASP), Utrecht University, 3584 CC Utrecht, Netherlands
T. Tsang Faculty of Science, Department of Physics, The Chinese University of
Hong Kong, Shatin, N.T., Hong Kong  J-S. Tsao  Department of Physics, National
Taiwan Normal University, sec. 4, Taipei 116, Taiwan  M. Tse LIGO Laboratory,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  R. Tso  CaRT,
California Institute of Technology, Pasadena, CA 91125, USA  S. Tsuchida
Department of Physics, Graduate School of Science, Osaka City University,
Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan  L. Tsukada  The Pennsylvania
State University, University Park, PA 16802, USA  D. Tsuna  Research Center
for the Early Universe (RESCEU), The University of Tokyo, Bunkyo-ku, Tokyo
113-0033, Japan  T. Tsutsui Research Center for the Early Universe (RESCEU),
The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  K. Turbang Vrije
Universiteit Brussel, 1050 Brussel, Belgium  Universiteit Antwerpen, 2000
Antwerpen, Belgium  M. Turconi  Artemis, Université Côte d’Azur, Observatoire
de la Côte d’Azur, CNRS, F-06304 Nice, France  C. Turski  Universiteit Gent,
B-9000 Gent, Belgium  D. Tuyenbayev Department of Physics, Graduate School of
Science, Osaka City University, Sumiyoshi-ku, Osaka City, Osaka 558-8585,
Japan  H. Ubach Institut de Ciències del Cosmos (ICCUB), Universitat de
Barcelona, Barcelona, 08028, Spain  A. S. Ubhi University of Birmingham,
Birmingham B15 2TT, United Kingdom  T. Uchiyama Institute for Cosmic Ray
Research (ICRR), KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida
City, Gifu 506-1205, Japan  R. P. Udall LIGO Laboratory, California Institute
of Technology, Pasadena, CA 91125, USA  A. Ueda  Applied Research Laboratory,
High Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki
305-0801, Japan  T. Uehara Department of Communications Engineering, National
Defense Academy of Japan, Yokosuka City, Kanagawa 239-8686, Japan  Department
of Physics, University of Florida, Gainesville, FL 32611, USA  K. Ueno
Research Center for the Early Universe (RESCEU), The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan  G. Ueshima  Department of Information and
Management Systems Engineering, Nagaoka University of Technology, Nagaoka
City, Niigata 940-2188, Japan  C. S. Unnikrishnan  Tata Institute of
Fundamental Research, Mumbai 400005, India  A. L. Urban  Louisiana State
University, Baton Rouge, LA 70803, USA  T. Ushiba Institute for Cosmic Ray
Research (ICRR), KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida
City, Gifu 506-1205, Japan  A. Utina Maastricht University, 6200 MD
Maastricht, Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  H. Vahlbruch
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover,
Germany  N. Vaidya LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA  G. Vajente LIGO Laboratory, California Institute of
Technology, Pasadena, CA 91125, USA  A. Vajpeyi  OzGrav, School of Physics &
Astronomy, Monash University, Clayton 3800, Victoria, Australia  G. Valdes
Texas A&M University, College Station, TX 77843, USA  M. Valentini The
University of Mississippi, University, MS 38677, USA  Università di Trento,
Dipartimento di Fisica, I-38123 Povo, Trento, Italy  INFN, Trento Institute
for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy  S.
Vallero  INFN Sezione di Torino, I-10125 Torino, Italy  V. Valsan University
of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA  N. van Bakel  Nikhef, 1098
XG Amsterdam, Netherlands  M. van Beuzekom Nikhef, 1098 XG Amsterdam,
Netherlands  M. van Dael Nikhef, 1098 XG Amsterdam, Netherlands  Eindhoven
University of Technology, 5600 MB Eindhoven, Netherlands  J. F. J. van den
Brand Maastricht University, 6200 MD Maastricht, Netherlands  Department of
Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam,
Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  C. Van Den Broeck
Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University,
3584 CC Utrecht, Netherlands  Nikhef, 1098 XG Amsterdam, Netherlands  D. C.
Vander-Hyde  Syracuse University, Syracuse, NY 13244, USA  A. Van de Walle
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  J. van
Dongen  Nikhef, 1098 XG Amsterdam, Netherlands  Department of Physics and
Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, Netherlands  H.
van Haevermaet Universiteit Antwerpen, 2000 Antwerpen, Belgium  J. V. van
Heijningen Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
J. Vanosky  LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  M. H. P. M. van Putten  Department of Physics and Astronomy,
Sejong University, Gwangjin-gu, Seoul 143-747, Republic of Korea  Z. van Ranst
Maastricht University, 6200 MD Maastricht, Netherlands  N. van Remortel
Universiteit Antwerpen, 2000 Antwerpen, Belgium  M. Vardaro  Institute for
High-Energy Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands
Nikhef, 1098 XG Amsterdam, Netherlands  A. F. Vargas  OzGrav, University of
Melbourne, Parkville, Victoria 3010, Australia  V. Varma Max Planck Institute
for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam,
Germany  M. Vasúth Wigner RCP, RMKI, H-1121 Budapest, Hungary  A. Vecchio
University of Birmingham, Birmingham B15 2TT, United Kingdom  G. Vedovato
INFN, Sezione di Padova, I-35131 Padova, Italy  J. Veitch SUPA, University of
Glasgow, Glasgow G12 8QQ, United Kingdom  P. J. Veitch OzGrav, University of
Adelaide, Adelaide, South Australia 5005, Australia  J. Venneberg Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), D-30167
Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover, Germany  G.
Venugopalan LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  P. Verdier Université Lyon, Université Claude Bernard Lyon 1,
CNRS, IP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France  D. Verkindt
Univ. Savoie Mont Blanc, CNRS, Laboratoire d’Annecy de Physique des Particules
- IN2P3, F-74000 Annecy, France  P. Verma  National Center for Nuclear
Research, 05-400 Świerk-Otwock, Poland  Y. Verma RRCAT, Indore, Madhya Pradesh
452013, India  S. M. Vermeulen Cardiff University, Cardiff CF24 3AA, United
Kingdom  D. Veske Columbia University, New York, NY 10027, USA  F. Vetrano
Università degli Studi di Urbino “Carlo Bo”, I-61029 Urbino, Italy  A. Viceré
Università degli Studi di Urbino “Carlo Bo”, I-61029 Urbino, Italy  INFN,
Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy  S. Vidyant
Syracuse University, Syracuse, NY 13244, USA  A. D. Viets Concordia University
Wisconsin, Mequon, WI 53097, USA  A. Vijaykumar International Centre for
Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru
560089, India  V. Villa-Ortega IGFAE, Universidade de Santiago de Compostela,
15782 Spain  J.-Y. Vinet  Artemis, Université Côte d’Azur, Observatoire de la
Côte d’Azur, CNRS, F-06304 Nice, France  A. Virtuoso  Dipartimento di Fisica,
Università di Trieste, I-34127 Trieste, Italy  INFN, Sezione di Trieste,
I-34127 Trieste, Italy  S. Vitale LIGO Laboratory, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  H. Vocca  Università di Perugia, I-06123
Perugia, Italy  INFN, Sezione di Perugia, I-06123 Perugia, Italy  E. R. G. von
Reis  LIGO Hanford Observatory, Richland, WA 99352, USA  J. S. A. von Wrangel
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover,
Germany  C. Vorvick LIGO Hanford Observatory, Richland, WA 99352, USA  S. P.
Vyatchanin Lomonosov Moscow State University, Moscow 119991, Russia  L. E.
Wade  Kenyon College, Gambier, OH 43022, USA  M. Wade Kenyon College, Gambier,
OH 43022, USA  K. J. Wagner Rochester Institute of Technology, Rochester, NY
14623, USA  R. C. Walet  Nikhef, 1098 XG Amsterdam, Netherlands  M. Walker
Christopher Newport University, Newport News, VA 23606, USA  G. S. Wallace
SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom  L. Wallace
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
J. Wang State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Innovation Academy for Precision Measurement Science and Technology
(APM), Chinese Academy of Sciences, Xiao Hong Shan, Wuhan 430071, China  J. Z.
Wang  University of Michigan, Ann Arbor, MI 48109, USA  W. H. Wang  The
University of Texas Rio Grande Valley, Brownsville, TX 78520, USA  R. L. Ward
OzGrav, Australian National University, Canberra, Australian Capital Territory
0200, Australia  J. Warner  LIGO Hanford Observatory, Richland, WA 99352, USA
M. Was Univ. Savoie Mont Blanc, CNRS, Laboratoire d’Annecy de Physique des
Particules - IN2P3, F-74000 Annecy, France  T. Washimi Gravitational Wave
Science Project, National Astronomical Observatory of Japan (NAOJ), Mitaka
City, Tokyo 181-8588, Japan  N. Y. Washington  LIGO Laboratory, California
Institute of Technology, Pasadena, CA 91125, USA  K. Watada  Christopher
Newport University, Newport News, VA 23606, USA  D. Watarai  University of
Tokyo, Tokyo, 113-0033, Japan.  J. Watchi Université Libre de Bruxelles,
Brussels 1050, Belgium  K. E. Wayt  Kenyon College, Gambier, OH 43022, USA  B.
Weaver  LIGO Hanford Observatory, Richland, WA 99352, USA  C. R. Weaving
University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom  S. A. Webster
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom  M. Weinert  Max
Planck Institute for Gravitational Physics (Albert Einstein Institute),
D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover,
Germany  A. J. Weinstein LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA  R. Weiss  LIGO Laboratory, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  C. M. Weller  University of Washington,
Seattle, WA 98195, USA  R. A. Weller Vanderbilt University, Nashville, TN
37235, USA  F. Wellmann  Max Planck Institute for Gravitational Physics
(Albert Einstein Institute), D-30167 Hannover, Germany  Leibniz Universität
Hannover, D-30167 Hannover, Germany  L. Wen  OzGrav, University of Western
Australia, Crawley, Western Australia 6009, Australia  P. Weßels  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), D-30167
Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover, Germany  K.
Wette OzGrav, Australian National University, Canberra, Australian Capital
Territory 0200, Australia  J. T. Whelan Rochester Institute of Technology,
Rochester, NY 14623, USA  D. D. White  California State University Fullerton,
Fullerton, CA 92831, USA  B. F. Whiting University of Florida, Gainesville, FL
32611, USA  C. Whittle LIGO Laboratory, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  O. S. Wilk  Kenyon College, Gambier, OH 43022, USA
D. Wilken Max Planck Institute for Gravitational Physics (Albert Einstein
Institute), D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167
Hannover, Germany  Leibniz Universität Hannover, D-30167 Hannover, Germany  C.
E. Williams  Carleton College, Northfield, MN 55057, USA  D. Williams SUPA,
University of Glasgow, Glasgow G12 8QQ, United Kingdom  M. J. Williams SUPA,
University of Glasgow, Glasgow G12 8QQ, United Kingdom  A. R. Williamson
University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom  J. L. Willis
LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
B. Willke Max Planck Institute for Gravitational Physics (Albert Einstein
Institute), D-30167 Hannover, Germany  Leibniz Universität Hannover, D-30167
Hannover, Germany  C. C. Wipf  LIGO Laboratory, California Institute of
Technology, Pasadena, CA 91125, USA  G. Woan SUPA, University of Glasgow,
Glasgow G12 8QQ, United Kingdom  J. Woehler  Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
Leibniz Universität Hannover, D-30167 Hannover, Germany  J. K. Wofford
Rochester Institute of Technology, Rochester, NY 14623, USA  I. A. Wojtowicz
Carleton College, Northfield, MN 55057, USA  D. Wong  University of British
Columbia, Vancouver, BC V6T 1Z4, Canada  I. C. F. Wong The Chinese University
of Hong Kong, Shatin, NT, Hong Kong  M. Wright  SUPA, University of Glasgow,
Glasgow G12 8QQ, United Kingdom  C. Wu National Tsing Hua University, Hsinchu
City, 30013 Taiwan, Republic of China  D. S. Wu Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
Leibniz Universität Hannover, D-30167 Hannover, Germany  H. Wu  National Tsing
Hua University, Hsinchu City, 30013 Taiwan, Republic of China  D. M. Wysocki
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA  L. Xiao LIGO
Laboratory, California Institute of Technology, Pasadena, CA 91125, USA  N.
Yadav  Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
00-716, Warsaw, Poland  T. Yamada  Accelerator Laboratory, High Energy
Accelerator Research Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan
H. Yamamoto LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  K. Yamamoto Faculty of Science, University of Toyama, Toyama City,
Toyama 930-8555, Japan  T. Yamamoto Institute for Cosmic Ray Research (ICRR),
KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida City, Gifu
506-1205, Japan  K. Yamashita  Graduate School of Science and Engineering,
University of Toyama, Toyama City, Toyama 930-8555, Japan  R. Yamazaki
Department of Physical Sciences, Aoyama Gakuin University, Sagamihara City,
Kanagawa 252-5258, Japan  F. W. Yang The University of Utah, Salt Lake City,
UT 84112, USA  K. Z. Yang University of Minnesota, Minneapolis, MN 55455, USA
L. Yang Colorado State University, Fort Collins, CO 80523, USA  Y.-C. Yang
National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China
Y. Yang Department of Electrophysics, National Yang Ming Chiao Tung
University, Hsinchu, Taiwan  Yang Yang  University of Florida, Gainesville, FL
32611, USA  M. J. Yap  OzGrav, Australian National University, Canberra,
Australian Capital Territory 0200, Australia  D. W. Yeeles  Cardiff
University, Cardiff CF24 3AA, United Kingdom  S.-W. Yeh  National Tsing Hua
University, Hsinchu City, 30013 Taiwan, Republic of China  A. B. Yelikar
Rochester Institute of Technology, Rochester, NY 14623, USA  J. Yokoyama
Research Center for the Early Universe (RESCEU), The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan  Department of Physics, The University of
Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  T. Yokozawa  Institute for Cosmic Ray
Research (ICRR), KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida
City, Gifu 506-1205, Japan  J. Yoo Cornell University, Ithaca, NY 14850, USA
T. Yoshioka  Graduate School of Science and Engineering, University of Toyama,
Toyama City, Toyama 930-8555, Japan  Hang Yu CaRT, California Institute of
Technology, Pasadena, CA 91125, USA  Haocun Yu LIGO Laboratory, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  H. Yuzurihara  Institute
for Cosmic Ray Research (ICRR), KAGRA Observatory, The University of Tokyo,
Kashiwa City, Chiba 277-8582, Japan  A. Zadrożny  National Center for Nuclear
Research, 05-400 Świerk-Otwock, Poland  M. Zanolin  Embry-Riddle Aeronautical
University, Prescott, AZ 86301, USA  S. Zeidler Department of Physics, Rikkyo
University, Toshima-ku, Tokyo 171-8501, Japan  T. Zelenova  European
Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy  J.-P. Zendri
INFN, Sezione di Padova, I-35131 Padova, Italy  M. Zevin University of
Chicago, Chicago, IL 60637, USA  M. Zhan  State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Innovation Academy for Precision
Measurement Science and Technology (APM), Chinese Academy of Sciences, Xiao
Hong Shan, Wuhan 430071, China  H. Zhang  Department of Physics, National
Taiwan Normal University, sec. 4, Taipei 116, Taiwan  J. Zhang OzGrav,
Australian National University, Canberra, Australian Capital Territory 0200,
Australia  L. Zhang  LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA  R. Zhang University of Florida, Gainesville, FL
32611, USA  T. Zhang  University of Birmingham, Birmingham B15 2TT, United
Kingdom  Y. Zhang  Texas A&M University, College Station, TX 77843, USA  C.
Zhao OzGrav, University of Western Australia, Crawley, Western Australia 6009,
Australia  G. Zhao  Université Libre de Bruxelles, Brussels 1050, Belgium  Y.
Zhao Institute for Cosmic Ray Research (ICRR), KAGRA Observatory, The
University of Tokyo, Kashiwa City, Chiba 277-8582, Japan  Gravitational Wave
Science Project, National Astronomical Observatory of Japan (NAOJ), Mitaka
City, Tokyo 181-8588, Japan  Yue Zhao  The University of Utah, Salt Lake City,
UT 84112, USA  Y. Zheng Missouri University of Science and Technology, Rolla,
MO 65409, USA  R. Zhou  University of California, Berkeley, CA 94720, USA  X.
J. Zhu OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800,
Victoria, Australia  Z.-H. Zhu Department of Astronomy, Beijing Normal
University, Beijing 100875, China  School of Physics and Technology, Wuhan
University, Wuhan, Hubei, 430072, China  A. B. Zimmerman University of Texas,
Austin, TX 78712, USA  M. E. Zucker  LIGO Laboratory, California Institute of
Technology, Pasadena, CA 91125, USA  LIGO Laboratory, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA  J. Zweizig LIGO Laboratory,
California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

Gravitational lensing by massive objects along the line of sight to the source
causes distortions of gravitational wave-signals; such distortions may reveal
information about fundamental physics, cosmology and astrophysics. In this
work, we have extended the search for lensing signatures to all binary black
hole events from the third observing run of the LIGO–Virgo network. We search
for repeated signals from strong lensing by 1) performing targeted searches
for subthreshold signals, 2) calculating the degree of overlap amongst the
intrinsic parameters and sky location of pairs of signals, 3) comparing the
similarities of the spectrograms amongst pairs of signals, and 4) performing
dual-signal Bayesian analysis that takes into account selection effects and
astrophysical knowledge. We also search for distortions to the gravitational
waveform caused by 1) frequency-independent phase shifts in strongly lensed
images, and 2) frequency-dependent modulation of the amplitude and phase due
to point masses. None of these searches yields significant evidence for
lensing. Finally, we use the non-detection of gravitational-wave lensing to
constrain the lensing rate based on the latest merger-rate estimates and the
fraction of dark matter composed of compact objects.

††software:  Analyses in this paper made use of LALSuite (LIGO Scientific
Collaboration and Virgo Collaboration, 2018), the GstLAL (Cannon et al., 2012;
Messick et al., 2017; Hanna et al., 2020; Sachdev et al., 2019) pipeline;
Bayesian inference with bilby (Ashton et al., 2019; Smith et al., 2020;
Romero-Shaw et al., 2020); as well as the packages NumPy (Harris et al.,
2020), SciPy (Virtanen et al., 2020), Astropy (Robitaille et al., 2013; Price-
Whelan et al., 2018), IPython (Perez & Granger, 2007), and ligo.skymap
(Singer, 2019). Plots were produced with Matplotlib (Hunter, 2007), and
Seaborn (Waskom et al., 2020).

##  1 Introduction

Gravitational lensing occurs when a massive object bends spacetime in a way
that alters the path or properties of a propagating wave. Gravitational
lensing is expected to affect gravitational waves, resulting, for example, in
repeated signals, (de-)magnification of the amplitude, phase shifts, and
beating patterns (Ohanian, 1974; Thorne, 1982; Deguchi & Watson, 1986; Wang et
al., 1996; Nakamura, 1998; Takahashi & Nakamura, 2003). The exact alteration
of the gravitational waveform depends on the nature of the lens system.

For massive lenses, gravitational lensing changes the GW amplitude without
affecting the frequency evolution (Wang et al., 1996; Dai & Venumadhav, 2017;
Ezquiaga et al., 2021). Moreover, such systems may also produce multiple
signals observed as repeated events separated by a time delay of minutes to
months for galaxies (Ng et al., 2018; Li et al., 2018; Oguri, 2018), and up to
years for galaxy clusters (Smith et al., 2018, 2017, 2019; Robertson et al.,
2020; Ryczanowski et al., 2020). The current-generation GW detector network
has a realistic chance of detecting the first lensed signal within its
operation period (Ng et al., 2018; Li et al., 2018; Oguri, 2018).

For low-mass lenses, such as stars or compact objects, microlensing introduces
beating patterns in the waveform (Deguchi & Watson, 1986; Nakamura, 1998;
Takahashi & Nakamura, 2003; Cao et al., 2014; Jung & Shin, 2019; Lai et al.,
2018; Christian et al., 2018; Dai et al., 2018; Diego, 2020). More generally,
a field of light lenses may produce even more complex patterns on the
gravitational waveform (Diego et al., 2019; Pagano et al., 2020; Cheung et
al., 2021). Under the right conditions and with sufficient knowledge about the
lens, these beating patterns may be observable with current-generation GW
detectors.

The detection of lensed GWs paves the way for numerous scientific pursuits,
including source localization (Hannuksela et al., 2020) and characterization
(Lai et al., 2018; Diego, 2020; Oguri & Takahashi, 2020), precision cosmology
(Sereno et al., 2011; Liao et al., 2017; Cao et al., 2019; Li et al., 2019b;
Hannuksela et al., 2020), and tests of general relativity (Baker & Trodden,
2017; Collett & Bacon, 2017; Fan et al., 2017; Goyal et al., 2021b; Ezquiaga &
Zumalacárregui, 2020). Indeed, the prospects for fundamental physics and
astrophysics have sparked a wide interest in searching for lensed GWs.
Previous work from the LIGO–Virgo Collaboration has considered a range of
strong and microlensing signatures for events in the first half of the third
observing run (O3a) (Abbott et al., 2021a). Nevertheless, these studies have
yielded no confident evidence for GW lensing.

In this work, we search for a variety of lensing signatures in the third LIGO
Scientific, Virgo, and KAGRA (LVK) Collaboration Gravitational-Wave Transient
Catalog (GWTC-3) (Abbott et al., 2021b) and study its implications for GW
lensing. In particular, we expand on the lensing results presented for the
first half of the third observing run of the LIGO–Virgo network (O3a) (Abbott
et al., 2021a) by including the signals found in the second half of the third
observing run (O3b) and by including additional analyses to further test the
lensing hypothesis and interpret their outcomes. First, we search for the
effects of strong lensing by studying the similarity and lensing evidence for
pairs of binary black hole (BBH) mergers. We consider both pairs of detected
mergers (super-threshold) and pairs formed by detected mergers and candidates
that nominally fall below the detection threshold (sub-threshold) with
consistent waveform morphologies. Second, we search for evidence of
microlensing induced by point-mass lenses. Finally, we constrain the expected
rate of lensed signals, black hole (BH) merger-rate density, and the fraction
of dark matter composed of compact objects.

It is important to note that GWTC-3 is a cumulative catalog describing all the
GW transients found in all observing runs to date: O1, O2, O3a, and O3b. O1
made observations between 2015 September 12 00:00 UTC to 2016 January 19 16:00
UTC, O2 between 2016 November 30 16:00 UTC to 2017 August 25 22:00 UTC, O3a
between 2019 April 1 15:00 UTC to 2019 October 1 15:00 UTC, and O3b between
2019 November 1 15:00 UTC to 2020 March 27 17:00 UTC.

Results of all analyses in this paper and associated data products can be
found in Abbott et al. (2021c). GW strain data (GWOSC, 2021) and posterior
samples (Abbott et al., 2021d) for all events from GWTC-3 are available from
the Zenodo platform or the Gravitational Wave Open Science Center (Abbott et
al., 2021e).

##  2 Data and Events

The analyses presented here expand on the lensing results from the first half
of O3 (also referred to as O3a) by documenting new results from the second
half of O3 (also referred to as O3b) using GWTC-3 (Abbott et al., 2021f). The
O3a lensing results paper (Abbott et al., 2021a) used the GWTC-2 catalog
(Abbott et al., 2021g). Since then, GWTC-2.1 (Abbott et al., 2021h) has
reclassified 2 of the candidates used in the O3a lensing paper as having a
probability of astrophysical origin of less than 0.5 and are not included in
the results described here, specifically GW190424_180648 and GW190909_114149.
GWTC-3 also includes 5 events that were identified by the O3a lensing sub-
threshold counterpart image search, namely GW190925_233845, GW190426_190642,
GW190725_184728, GW190805_211137, and GW190916_200658.

Various instrumental upgrades have led to more sensitive data in O3b, with a
median binary neutron star (BNS) inspiral ranges (Finn & Chernoff, 1993; Allen
et al., 2012a) of 115 Mpc in O3b compared to 108 Mpc in O3a for LIGO Hanford,
133 Mpc in O3b compared to 135 Mpc O3a for LIGO Livingston, and 51 Mpc in O3b
compared to 45 Mpc in O3a for Virgo (Abbott et al., 2021f). The duty factor
for at least one detector being online was 96.6%; for any two detectors being
online at the same time was 85.3%; and for all three detectors together was
51%. Further details regarding instrument performance and data quality for O3b
are available in Abbott et al. (2021f); Davis et al. (2021a); Acernese et al.
(2022).

The LIGO and Virgo detectors used a photon recoil-based calibration (Karki et
al., 2016; Cahillane et al., 2017; Viets et al., 2018) resulting in a complex-
valued, frequency-dependent detector response. Previous studies have
documented the systematic error and uncertainty bounds for O3b strain
calibration in LIGO (Sun et al., 2020, 2021) and Virgo (Acernese et al.,
2021).

Transient noise sources, referred to as glitches, contaminate the data and can
affect the confidence of candidate detections. Times affected by glitches and
other data quality issues are identified so that searches for GW events can
exclude (veto) these periods of poor data quality (Abbott et al., 2016a,
2020a; Davis et al., 2021b; Nguyen et al., 2021; Fiori et al., 2020). In
addition, several known persistent noise sources are subtracted from the data
using information from witness auxiliary sensors (Driggers et al., 2019; Davis
et al., 2019).

Candidate events, including those reported in Abbott et al. (2021f) and the
new candidates found by the search for sub-threshold counterpart images in
Sec. 3.1 of this paper, have undergone a validation process to evaluate if
instrumental artifacts could affect the analysis; this process is described in
detail in Sec. 5.5 of Davis et al. (2021b). This process can also identify
data quality issues that need further mitigation for individual events, such
as the subtraction of glitches (Cornish et al., 2021; Davis et al., 2022) and
non-stationary noise couplings (Vajente et al., 2020), before executing
parameter estimation (PE) algorithms. See Table XIV of Abbott et al. (2021f)
for the list of events requiring such mitigation.

The GWTC-3 catalog (Abbott et al., 2021f) contains 35 events from O3b in
addition to the 55 previous events from previous observing runs (Abbott et
al., 2021h) with a false-alarm rate (FAR) below two per year, and an expected
rate of contamination from detector noise less than 10–15% (Abbott et al.,
2021f). We neglect the potential contamination in this analysis. These events
were identified by four search pipelines: one minimally modeled transient
search cWB (Klimenko et al., 2004, 2005, 2006, 2011, 2016) and three matched-
filter searches GstLAL (Sachdev et al., 2019; Hanna et al., 2020; Messick et
al., 2017), Multi-Band Template Analysis (MBTA) (Adams et al., 2016; Aubin et
al., 2021), and PyCBC (Allen et al., 2012b; Allen, 2005; Dal Canton et al.,
2014; Usman et al., 2016; Nitz et al., 2017). Their parameters were estimated
through Bayesian inference using the bilby (Ashton et al., 2019; Smith et al.,
2020; Romero-Shaw et al., 2020) and RIFT (Pankow et al., 2015; Lange et al.,
2017; Wysocki et al., 2019) packages. Both the matched-filter searches and PE
use a variety of BBH waveform models which generally combine knowledge from
post-Newtonian theory, the effective-one-body formalism, and numerical
relativity (for general introductions to these approaches, see Blanchet, 2014;
Damour & Nagar, 2016; Palenzuela, 2020; Schmidt, 2020 and references therein).
The analyses in this paper rely on the same methods, and the specific waveform
models and analysis packages used are described in each section.

Of the 35 events from O3b, 31 are likely BBHs, while four have component
masses consistent with being below 3 $\text{M}_{\odot}$ (Abbott et al., 2021i,
f), thus potentially containing a neutron star. We consider these 35 events in
the analyses documented in this paper. Specifically, we use the following
input data sets for each analysis. The searches for sub-threshold counterpart
images in Sec. 3.1 cover the whole O3 strain data set, using the same data
quality veto choices as in Abbott et al. (2021f) but a strain data set
consistent with the PE analyses: the final calibration version of LIGO data
(Sun et al., 2021) with additional noise subtraction (Vajente et al., 2020).
The posterior-overlap analysis in Sec. 3.2 starts from the posterior samples
released with GWTC-3 (GWOSC, 2021). The joint-PE analyses in Sec. 3.3 and
microlensing analysis in Sec. 4 reanalyze the strain data in short segments
around the event times, available from the same data release, with data
selection and noise mitigation choices matching those of the PE analyses in
Abbott et al. (2021f).

##  3 Strong lensing

If a GW travels close enough to a massive lens, it will produce multiple
images, with the number of images depending on the lens profile and source
lens geometry. This regime is known as the strong-lensing limit. Each of these
lensed images $h_{j}^{\mathrm{L}}$ will have a change in its amplitude,
arrival time and phase compared to the emitted signal $h$ (Schneider et al.,
1992):

$h_{j}^{\mathrm{L}}(f)=\sqrt{|\mu_{j}|}\exp\left[{i2\pi f\Delta
t_{j}-i\mathrm{sign}(f)n_{j}\pi}\right]h(f)\,,$ (1)

for $n_{j}=0,1/2,1$ for type I, II and III images, which correspond to
different minima of the lensing potential. While the magnification $\mu_{j}$
and time delay $\Delta t_{j}$ do not affect the waveform morphology (they are
completely degenerate with the luminosity distance and coalescence time) the
frequency-independent lensing phase shift $n_{j}\pi$ could induce distortions
when the signal has multiple frequency components (Dai & Venumadhav, 2017;
Ezquiaga et al., 2021). In particular, this occurs for type II images since
type I does not have a phase shift, and type III only flips the overall sign,
which is degenerate with shifting the polarization angle by $\pi/2$. The
$\mathrm{sign}(f)$ term is only there to ensure that the time domain waveform
is real.

Making a distinction between effects that do and do not change the waveform
morphology, we divide our search into two parts. First, we search for pairs of
events consistent with the strong-lensing hypothesis. Some of these pairs will
have sufficiently strong amplitudes that can be identified as confident
detections (super-threshold) by the search pipelines used in Abbott et al.
(2021g, h, f), while others may have not been identified as signals (sub-
threshold) because of the relative de-magnification. Our searches will include
both sub- and super-threshold pairs. A pair is the minimum association, but
higher multiplicities are also possible. Then, we search for strong lensing
focusing on the distortion of type II images.

###  3.1 Sub-threshold Search

In this section, we describe the search for possible sub-threshold
counterparts of super-threshold detections from O3. We perform searches over
all O3 strain data following the rules for data selection described in Abbott
et al. (2021h) and Abbott et al. (2021f). A general search for GWs uses a
large template bank covering a broad parameter space as we have no prior
information about the signal subspace, resulting in a high trials factor and
hence incurring a high noise background. Sub-threshold (lensed) GWs with
smaller amplitudes will therefore be easily buried in the noise without being
identified as detections as they cannot pass the usual detection threshold.

To uncover these sub-threshold (lensed) signals, we have to effectively reduce
the noise background while keeping the targeted foreground (i.e. the signals)
constant (Li et al., 2019a; McIsaac et al., 2020; Dai et al., 2020). The
strong lensing hypothesis asserts that lensed GWs, super-threshold or sub-
threshold, coming from the same origin have identical waveforms apart from an
overall scaling factor and a Morse phase factor as described in Eq. 1, and
hence should have consistent inferred intrinsic masses and spins. 111 The
Morse phase factor for different image types has not been considered in the
search described here. Should a GW include detectable higher-order multipole
moments, then the Morse phase factor will cause complicated changes to the
waveforms, inducing a loss in the search sensitivity.  Therefore, we can
construct a reduced template bank with only templates that have masses and
spins similar to those of a target super-threshold detection. Using the
reduced bank lowers the trials factors and noise background and effectively
searches for previously unidentified possible sub-threshold lensed
counterparts to the target detection. For each known candidate from O3 with a
probability of astrophysical origin $p_{\text{astro}}>0.5$, we create a
reduced template bank using their respective public posterior mass and spin
samples released with GWTC-3 (GWOSC, 2021), ensuring that the templates will
match well with their respective target events while improving the ranking
statistics of the search for similar events, and hence potentially returning
new candidates that previously did not reach the threshold
$p_{\text{astro}}>0.5$ in GWTC-3. Details of how the reduced banks are
constructed can be found in Li et al. (2019a).

Given these template banks, we proceed with configurations and procedures as
outlined in Abbott et al. (2021a) to produce a priority list of potential
lensed candidates matching each target event, using GstLAL (Messick et al.,
2016; Sachdev et al., 2019) as the search pipeline. The list of candidates
obtained is again further vetted using a sky location consistency check
detailed in Wong et al. (2021); Abbott et al. (2021a) to ensure the candidates
have consistent sky location with the target event. To avoid false dismissal
at this step, we only veto candidates with an overlap in $90\%$ credible
region of the sky location $O_{\rm 90\%CR}=0$. All candidates with non-
vanishing localization overlap are kept for further follow-up with data
quality checks as discussed in Sec. 2.

In Table 1, we list the top five candidates from the _individual_ targeted
searches for counterparts of the detections reported in O3. As in the O3a
lensing paper (Abbott et al., 2021a), we do not assess in detail the
probability of astrophysical origin for each of these. It is also important to
note that the reported false-alarm rates (FARs) do not indicate how likely
each trigger is a lensed counterpart of the target event, but only how likely
noise produces a trigger with a ranking statistic higher or equal to that of
the candidate under consideration using these reduced template banks. Similar
to Abbott et al. (2021a), we account for the fact that we have analyzed $\sim
332$ days of data multiple times for a total of $76$ events, and set the FAR
threshold to be $1$ in $69$ years (i.e. $4.59\times 10^{-10}$ Hz). We followed
up on the top two candidates listed that passed the FAR threshold through
golum’s joint PE analysis (Janquart et al., 2021 , discussed in 3.3). The
results are included in Table 1. Since both pairs of candidates have mildly
_negative_ $\log_{10}$ coherence ratios, showing that there is no evidence
supporting the lensing hypothesis for either of these pairs, we did not
further follow them up with the more computationally intensive hanabi analysis
(Lo & Magaña Hernandez, 2021 , discussed in 3.3).

Table 1:  Top $5$ candidates from individual sub-threshold searches for
strongly-lensed counterpart images of O3 events from GWTC-3.

Target event | Lensed candidate (UTC) |  $\Delta t$ [days] | $(1+z)\mathcal{M}(M_{\odot})$ | $(1+z)\mathcal{M}^{\text{target}}(M_{\odot})$ | FAR $\left[\text{yr}^{-1}\right]$ |  $O_{\rm 90\%CR}$ [%] | $\log_{10}(\mathcal{C}^{\mathrm{L}}_{\mathrm{U}})$
---|---|---|---|---|---|---|---
GW$190930\\_133541$ | 19-08-05 13:43:48 | $-56.0$ | 10.2 | $9.86$ | 0.002 | $61.00\%$ | $-6.4$
GW$191204\\_171526$ | 19-08-05 13:43:48 | $-121.1$ | 10.2 | $9.66$ | 0.006 | $25.40\%$ | $-12.2$
GW$190828\\_065509$ | 19-11-12 12:13:18 | $76.2$ | 45.5 | $17.3$ | 0.023 | $18.00\%$ | -
GW$190725\\_174728$ | 19-08-05 13:43:48 | $10.8$ | 10.2 | $8.88$ | 0.038 | $39.50\%$ | -
GW$190828\\_065509$ | 20-02-06 07:24:59 | $162.0$ | 15.1 | $17.3$ | 0.154 | $36.00\%$ | -

###  3.2 Preliminary Identification of Lensed Pair Candidates

Multiple, non-overlapping images produced by strongly lensed GW signals have
identical phase evolution, and therefore their intrinsic parameters (as well
as their orbit’s inclination with respect to the line of sight) are expected
to have overlapping posteriors. In addition, the angular separations of images
(produced by galaxies or galaxy clusters) are several orders of magnitude
smaller than the uncertainties associated with their GW sky location. As a
result, their sky localisations will also overlap. (As in the previous
section, the Morse phase for different image types is not considered here.)

Under these assumptions, a Bayes-factor statistic ($\textsf{B}^{\rm L}_{\rm
U}$) that assesses the consistency between a lensed candidate pair’s posterior
distributions of intrinsic parameters, sky location, and inclination angle
(and thus acts as a discriminator between the lensed and unlensed hypotheses)
can be constructed (Haris et al., 2018). To convert this statistic to a false-
positive probability (FPP),222 FAR and FPP, while conceptually similar,
pertain to different contexts in this work. In particular, we use FPP
exclusively for significances associated with candidate lensed pairs to
discriminate them from unlensed pairs. On the other hand, a FAR is associated
with the significance assigned to individual candidate GW signal events.  a
background distribution of unlensed $\textsf{B}^{\rm L}_{\rm U}$ needs to be
estimated.

To that end, we conduct an injection campaign involving BBHs only, in which we
sample component masses $m_{1,2}$ from a power-law distribution (Abbott et
al., 2016b) in the range $(10$–$50M_{\odot})$. We assume that the redshift
distribution of BBHs is similar to population synthesis simulations of
isolated binary evolution (Belczynski et al., 2008, 2010; Dominik et al.,
2013; Eldridge et al., 2019; Bouffanais et al., 2021; Zevin et al., 2021). All
other parameters are sampled from uninformative prior distributions (Haris et
al., 2018). We inject the simulated signals into Gaussian noise with O3a
representative power spectral density (PSD) for a LIGO–Virgo detector network.
We compute $\textsf{B}^{\rm L}_{\rm U}~{}$for all possible pairs in this
injection set, and following Abbott et al. (2021a), we assign an FPP to a
candidate pair using its $\textsf{B}^{\rm L}_{\rm U}$. Candidate lensed pairs
involving BNS or neutron star-black hole (NSBH) events are not analyzed and
ranked.

We additionally employ a machine learning (ML)-based binary classification
scheme to rapidly provide a probability of class membership (lensed or
unlensed) for a given candidate BBH pair (Goyal et al., 2021a). Such an
analysis not only serves as an independent method to rank candidate pairs but
also provides a quantitative significance to pairs for which source-parameter
inference samples are unavailable.

Q-transform-based (Chatterji et al., 2004) time–frequency maps of strongly
lensed BBHs are expected to have similar shapes, although the signal energy in
each time-frequency tile will differ between images. Furthermore, as mentioned
earlier, their sky localisations will overlap. Exploiting these facts, ML
models that take Q-transforms and Bayestar (Singer & Price, 2016) sky
localisations as inputs, are built. These models use a DenseNet (Huang et al.,
2016) architecture (with several layers pre-trained on the ImageNet dataset;
Deng et al., 2009), and XGBoost (Chen & Guestrin, 2016) algorithms, trained on
lensed and unlensed BBH signals injected in Gaussian noise (for details on the
ML training set, see Goyal et al. (2021a)) The outputs of the individual
models are then combined to provide a probability that a candidate pair is
lensed or unlensed.

To convert this probability to an FPP, we construct a background distribution
of ML probabilities using a population of unlensed BBH events injected in
Gaussian noise characterized by the O3a representative PSD – the same as was
used for the posterior overlap statistic. This PSD is found to be sufficiently
similar to the averaged O3 PSD for the estimation of the background
distribution so as not to change the preliminary selection of candidate pairs.
The BBH population is identical to the one used by the posterior overlap
statistic analysis to construct its corresponding background distribution.
Furthermore, the sky localisations used to rank candidate pairs come from the
same PE analysis used to estimate the posterior overlap statistic 333Note that
Bayestar, which is used to assign ML probabilities to real-event candidate
pairs, is expected to provide sky localisations that are similar to those
provided by this PE analysis..

A plot comparing the FPPs assigned by the posterior overlap and ML analyses is
shown in Fig. 1. Candidates that have either a posterior-overlap-assigned FPP
or ML-assigned-FPP, (or both), that are smaller than $1\%$, are selected for
more comprehensive Bayesian analyses.

Figure 1:  The FPPs of each lensed candidate pair constructed from the set of
GW events that exceed an astrophysical probability (Farr et al., 2015; Kapadia
et al., 2020) threshold of $0.5$, as evaluated using the $\textsf{B}^{\rm
L}_{\rm U}~{}$ and ML classification statistics. Orange dashed lines that
correspond to an FPP threshold of $10^{-2}$, are also placed. Pairs whose
$\textsf{B}^{\rm L}_{\rm U}$-based or ML-based FPPs fall below this threshold
are selected for additional joint parameter estimation analyses.
$\textsf{B}^{\rm L}_{\rm U}<10^{-6}$ has been mapped to an FPP of 1, which is
reflected in the gap along the vertical axis between $0.4$ and $1$.

###  3.3 Joint Parameter Estimation

Similar to the analysis of O3a data (Abbott et al., 2021a), we perform a joint
PE analysis for the most relevant candidate lensing pairs. We follow up on the
pairs that display low FPP in their posterior overlap or ML classification
scheme. These are pairs within the whole of O3, but we only consider here
those with at least one event in O3b since pairs in O3a were studied in Abbott
et al. (2021a). We use two complementary pipelines: golum (Janquart et al.,
2021) and hanabi (Lo & Magaña Hernandez, 2021). Both pipelines use the nested
sampling algorithm dynesty (Speagle, 2020), and implement the joint PE with
the help of bilby (Ashton et al., 2019; Romero-Shaw et al., 2020).

golum (Janquart et al., 2021) is a joint PE tool where the workload is reduced
by analyzing the two images, under the lensed hypothesis, in two successive
stages. The first image is characterized by the same parameters of the
unlensed case (where the time of coalescence and the luminosity distance are
the observed ones) with an additional Morse factor. The second image is then
analyzed using (samples of) the posterior from the first image as the prior
and linking the parameters modified by lensing through three lensing
parameters: a time difference, a relative magnification, and a Morse factor
difference. The final coherence ratio $\mathcal{C}^{\mathrm{L}}_{\mathrm{U}}$
is the ratio of the product of the evidences for the two runs under the lensed
hypothesis and the product of evidences for the two images analyzed under the
unlensed hypothesis.

hanabi (Lo & Magaña Hernandez, 2021) first performs a joint inference on a
signal pair by constructing a joint likelihood function that is a product of
the likelihood function for each individual event, with a joint prior
distribution. The latter is defined for a set of joint parameters that can
simultaneously describe both signals if they are truly lensed, for example,
the masses and the spins, as well as a set of parameters that are different
for each of the signals such as the time of arrival, the apparent luminosity
distance, and the Morse phase factor associated to each of the lensed signals.
The joint parameter space is explored with the package hanabi.inference (Lo &
Magaña Hernandez, 2021). The inference result is then reweighted with an
astrophysically motivated prior distribution; for example, the astrophysical
prior distribution for the redshifted component masses would be dependent on
both the population model for the intrinsic BBH masses and the redshift
distribution of the sources. However, the true source redshift cannot be
determined from GW observations alone since the true source redshift is
degenerate with the magnification from strong lensing. To compute the Bayes
factor $\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}$, the source redshift, which
serves as a hyper-parameter for the signal pair, must be marginalized over.
Selection effects enter as a normalization constant to the marginal data
likelihood. This procedure is implemented in hanabi.hierarchical with the help
of gwpopulation (Talbot et al., 2019). The ratio of unnormalized evidences
calculated under the lensed hypothesis and the unlensed hypothesis using this
astrophysical prior is referred to as the population-weighted coherence ratio
$\left.\mathcal{C}^{\mathrm{L}}_{\mathrm{U}}\right|_{\mathrm{pop}}$, while the
ratio of normalized evidences that accounts for both population prior and
selection effects is referred to as the Bayes factor
$\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}$ in this analysis. We follow our
fiducial singular isothermal sphere (SIS) lensing model when computing the
magnification prior (Abbott et al., 2021a). This analysis however does not
impose any informative prior on the time delay or the image types from the
lensing model.

Both pipelines use IMRPhenomXPHM (Pratten et al., 2021) as the waveform model,
with an additional Morse phase applied to each of the waveform polarizations
in the frequency domain. Other inputs, such as the power spectral density
estimates and the calibration envelopes, are chosen to match the analyses done
in the GWTC-3 catalog paper (Abbott et al., 2021b). Following the same
prescriptions of the other analyses, we fix the BBH population model to the
Power-Law + Peak model for the primary masses and the merger rate history to
Madau–Dickinson star-formation rate (Madau & Dickinson, 2014) normalized by
the median GWTC-3 rate (Abbott et al., 2021j).

Taking advantage of golum’s rapid joint PE, we analyze the 75 pairs of
candidates highlighted by posterior overlap and ML. For each of them, we
compute the coherence ratio, which accounts for the probability ratio of the
lensed and unlensed hypotheses without including selection effects and
population priors. We find that there is a wide range of
$\log_{10}(\mathcal{C}^{\mathrm{L}}_{\mathrm{U}})$ values, with a peak
slightly above zero. This comes from the fact that this analysis considers
only triggers already flagged by the posterior overlap and ML analyses. As a
consequence, the analysis is biased towards the higher values. Nevertheless, a
significant proportion of events flagged with the ML pipeline and the
posterior overlap pipeline are disfavored, having
$\log_{10}(\mathcal{C}^{\mathrm{L}}_{\mathrm{U}})<0$. When comparing the
highest coherence ratio found in the data,
$\log_{10}(\mathcal{C}^{\mathrm{L}}_{\mathrm{U}})=2.5$, with a background of
unlensed events, we find that it is well within the expected values, with
$1\%$ of the background events having larger
$\mathcal{C}^{\mathrm{L}}_{\mathrm{U}}$. This background is computed for a
population of compact binaries that follows the mass, spin and redshift
distribution of GWTC-3 (Abbott et al., 2021j). This large number of positive
$\log_{10}(\mathcal{C}^{\mathrm{L}}_{\mathrm{U}})$ is consistent with the high
number of expected false alarms (Wierda et al., 2021; Çalışkan et al., 2022a).
For those pairs with the highest coherence ratio, we follow up with the hanabi
pipeline for a total of 17 pairs. Our main results are presented in Fig. 2,
where the left column indicates the event pairs and the horizontal axis their
$\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}$. There we can observe that none of the
event pairs shows support for the lensing hypothesis, i.e. all
$\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}<1$. The pair with highest
$\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}$ is GW190620_030421 – GW200216_220804,
for an evidence against lensing of $\sim 1/100$ with the fiducial merger rate
density model following the Madau-Dickinson star-formation rate. As a
robustness check of how using different merger rate density models would
change the results, we repeat the calculations using two more models, namely
$R_{\rm min}(z)$ and $R_{\rm max}(z)$ from our previous O3a analysis (Abbott
et al., 2021a) that minimally and maximally bracket many existing population-
synthesis results (Belczynski et al., 2008, 2010; Dominik et al., 2013;
Eldridge et al., 2019). We see that while the exact values for the Bayes
factor change with the use of different merger rate density models, the
conclusion remains that there is no support for the lensing hypothesis in any
of the event pairs analyzed. To further assess the significance of these pairs
we also include a color code to indicate the probability of having an
astrophysical origin $p_{\mathrm{astro}}^{\mathrm{pair}}$, defined as the
product of the highest $p_{\mathrm{astro}}$ of each event reported in the
GWTC-3 catalog paper (Abbott et al., 2021b) by different pipelines. In
conclusion, we find no evidence of multiply imaged events.

Figure 2:  Bayes factors $\mathcal{B}^{\mathrm{L}}_{\mathrm{U}}$ from hanabi
for the highest-ranked multiple-image candidate pairs. As a check on the
robustness of our results, we show the Bayes factors calculated using three
different merger rate density models, namely the fiducial model tracking the
Madau–Dickinson star-formation rate (Madau & Dickinson, 2014), and also the
$R_{\rm min}(z)$ and $R_{\rm max}(z)$ model introduced in Abbott et al.
(2021a). The color for each marker represents the value of $p_{\rm astro}^{\rm
pair}$ for each pair, which is the probability that both of the signals from a
pair are of astrophysical origins and not from terrestrial sources.

###  3.4 Type II image search

In addition to the search for strong-lensing identifying multiple images, we
also look for the distortions that lensing introduces in type II images
(Ezquiaga et al., 2021). This is because the frequency-independent phase shift
that each image acquires becomes a frequency-dependent time delay for
different frequency components. Therefore, for signals containing different
measurable spherical harmonic modes, as recently detected in GW190412 (Abbott
et al., 2020c), GW190814 (Abbott et al., 2020d), and other events (Abbott et
al., 2021b), the overall lensed waveform can be distorted. The extent of the
distortion is subject to the power in modes beyond the quadrupole radiation.
As a consequence, we do not expect to see these distortions in the majority of
the lensed events with current sensitivities. However, if not searched for,
they might be mistaken with deviations from general relativity (Ezquiaga et
al., 2022).

To look for these distortions, we use golum (Janquart et al., 2021). Within
GWTC-3 we identify 10 events whose posterior has some information about the
Morse phase, either by favoring or disfavoring the distortions of the type II
image by more than $4\%$ with respect to normality, i.e. the probability of
each image type $p(n_{j})$ is $p(n_{j})>0.37$ or $p(n_{j})<0.29$. We summarize
the evidence of one image type versus another in Fig. 3. Since only type II
images display waveform distortions, we only compute the Bayes factors of the
type-II-vs-I and the type-II-vs-III hypotheses. As can be seen in Fig. 3, only
a few events display a preference for one image type versus the other one.
This is expected given the signal-to-noise ratio (SNR) of these events and
their power in higher multipole moments. However, GW190412 and GW200129_065458
present higher evidence for type II images. For GW190412 we find a $\log_{10}$
Bayes factor for type II vs. I of $0.60\pm 0.16$ and for type II vs. III of
$0.22\pm 0.16$. For GW200129_065458 we find $0.38\pm 0.14$ and $0.24\pm 0.14$
for type II vs. I and type II vs. III respectively. These events have possible
super-threshold counterparts but those were discarded by the golum analysis.
In addition, we have also searched for sub-threshold triggers associated with
these events, but found none.

Figure 3:  Distribution of Bayes factors comparing different image type
hypotheses for the 10 most relevant events. We compare the probability of
being type II vs. type I (blue-solid histogram) and of being type II vs. type
III (orange-dashed histogram). Only type II images display waveform
distortions and for that reason, we do not compare type III vs. type I.

To assess the significance of the type II images, we follow up on GW190412 and
GW200129_065458 performing a simulation campaign of type I and type II images.
GW190412 simulations show that indeed this event has enough power in higher
multipole moments to favor the type II hypothesis so that it could
meaningfully test that hypothesis and would favor it if it were true. For
GW200129_065458, however, that is not the case. Moreover, GW200129_065458
might have a significant glitch under subtraction (Payne et al., 2022). The
preference of GW190412 for a type II image could be just a systematic effect
due to the waveform modeling, especially since this event falls in challenging
parts of the parameter space (Abbott et al., 2020c; Colleoni et al., 2021;
Hannam et al., 2021). For this reason, we repeat the analysis with different
waveform families from our fiducial IMRPhenomXPHM model (Pratten et al.,
2021). We find that the preference for a type II image remains when using
SEOBNRv4PHM (Ossokine et al., 2020) or IMRPhenomPv3HM (Khan et al., 2020). The
same conclusion holds when using different noise realizations for the
simulations. Details on these simulation campaigns can be found in Appendix A.

Although we find a mild preference for the type II image hypothesis in
GW190412, we find that this analysis cannot provide conclusive evidence of
strong lensing. However, our techniques and pipeline will be relevant for
future observing runs when high-SNR events display stronger evidence of
higher-order modes.

##  4 Microlensing Effects

When the characteristic wavelengths of GWs are comparable to the Schwarzschild
radius of a lens ($\lambda_{\mathrm{GW}}\sim
R^{\mathrm{lens}}_{\mathrm{Sch}}$), we may observe frequency-dependent
magnification of the waveform that can inform us about the lens model
(Takahashi & Nakamura, 2003; Cao et al., 2014; Jung & Shin, 2019; Lai et al.,
2018; Christian et al., 2018; Dai et al., 2018; Diego et al., 2019; Diego,
2020; Pagano et al., 2020; Cheung et al., 2021; Cremonese et al., 2021;
Çalışkan et al., 2022b). Since the GWs of sources such as BBHs sweeps through
a wide range of frequencies, these beating patterns can reveal the presence of
intervening microlenses. In the sensitive range of ground-based detectors,
these effects are expected for objects up to $\sim 10^{5}M_{\sun}$, which
includes stellar-mass objects and intermediate-mass BHs.

Objects that can cause these microlensing effects are predominantly found in
larger structures. Therefore we expect that realistic microlensing due to a
field of microlenses embedded in an external macromodel potential such as
galaxies and galaxy clusters causes complex effects on the unlensed waveforms
(Diego et al., 2019). While the effects of these systems on GW signals have
been studied (Diego, 2020; Cheung et al., 2021; Mishra et al., 2021; Yeung et
al., 2021), the resulting waveforms are computationally costly to evaluate.
Nevertheless, in the absence of specific knowledge of the matter distribution
along the travel path and to keep the problem computationally tractable, we
assume that the beating patterns are caused by isolated point masses as a
first approximation. In this case, the microlensed waveform
$h^{\mathrm{Micro}}$ can be related to the unlensed waveform $h^{\mathrm{U}}$
according to

$h^{\mathrm{\rm
Micro}}(f;\theta,M^{z}_{\mathrm{L}},y)=h^{\mathrm{U}}(f;\theta)\,F(f;M^{z}_{\mathrm{L}},y)\,,$
(2)

where $\theta$ represents the set of parameters defining an unlensed GW
signal, $M^{z}_{\mathrm{L}}=M_{\mathrm{L}}(1+z_{\mathrm{l}})$ is the
redshifted lens mass, $y$ is the dimensionless impact parameter, and
$F(f;M^{z}_{\mathrm{L}},y)$ is the frequency-dependent lensing magnification
factor (e.g., Takahashi & Nakamura, 2003).

Similar to Abbott et al. (2021a), we perform Bayesian inference on all events
from O3b using the unlensed signal model $h^{\mathrm{U}}$ and the microlensing
signal model $h^{\mathrm{Micro}}$. In particular, we use bilby (Ashton et al.,
2019; Romero-Shaw et al., 2020) and the nested sampling algorithm dynesty
(Speagle, 2020). Data products such as strain data and PSDs are the same as
for GWTC-3 and between the two signal models (Abbott et al., 2021b) For the GW
parameters, we use the same priors as GWTC-3, while the prior on the lens mass
$M^{z}_{\mathrm{L}}$ is log uniform in the range
$[1$–$10^{5}~{}\text{M}_{\odot}]$ and the prior on the impact parameter is
$p(y)\propto y$ between $[0.1,3]$. All events were analyzed using
IMRPhenomXPHM (Pratten et al., 2021).

The process yields posterior probability distributions of $\theta$ or
$\left\\{\theta,M^{z}_{\mathrm{L}},y\right\\}$ for the unlensed and lensed
signal models, respectively. Moreover, we compute the evidence ratio between
the microlensed and unlensed signal models, better known as the Bayes factor
$\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$.

Figure 4: Distribution of microlensing $\log_{10}$ Bayes factors
$\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ for all events in O3 (blue, solid
line) and simulated unlensed signals (orange, dashed line) from Abbott et al.
(2021a).

Fig. 4 shows the distribution of
$\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ for all the events in O3
and simulated unlensed signals from Abbott et al. (2021a). The distribution of
$\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ is primarily clustered
around 0 and the distribution for O3 events does not extend to significantly
higher values than the distribution for simulated signals. The marginalized
posteriors of the microlensing parameters are shown in Appendix B. We conclude
that there is no compelling evidence for the presence of microlensing
signatures.

##  5 Implications

In this section, we consider some of the implications that derive from the
search for lensing signatures. We first forecast the number of detectable
strongly lensed events based on the latest knowledge on the merger-rate
density (Sec. 5.1). Next, we infer upper limits on the strong lensing rate
using the non-detection of resolvable strongly lensed BBH events (Sec. 5.2).
Finally, we use the non-detection of microlensing to infer the compact dark
matter fraction in the Universe (Sec. 5.3).

###  5.1 Strong lensing rate

We predict the rate of lensing using the standard methods outlined in the
literature (Ng et al., 2018; Li et al., 2018; Oguri, 2018; Xu et al., 2021;
Mukherjee et al., 2021a; Wierda et al., 2021), at galaxy and galaxy-cluster
lens mass scales. To model the lens population, we need to choose a density
profile and a mass function. We adopt the SIS density profile for both
galaxies and galaxy clusters. Moreover, we use the velocity dispersion
function from the Sloan Digital Sky Survey (Choi et al., 2007) for galaxies
and the halo mass function from Tinker et al. (2008) for clusters which have
also been used in other lensing studies (e.g., Oguri & Marshall, 2010;
Robertson et al., 2020). The SIS profile can accurately describe lensing by
galaxies, but the mass distribution of clusters tends to be more complicated.
Nevertheless, Robertson et al. (2020) have demonstrated that the SIS model can
reproduce the lensing rate predictions from a study of numerically simulated
cluster lenses. Thus, we adopt the same model for both galaxies and galaxy
clusters.

Under the SIS model, we obtain two images with different magnifications and
arrival times. The rate of strong lensing is given by

$\begin{split}\mathcal{R}_{\rm
lens}=\int&\frac{\mathrm{d}N(M_{\mathrm{h}},z_{\mathrm{l}})}{\mathrm{d}M_{\mathrm{h}}}\frac{\mathrm{d}D_{\mathrm{c}}}{\mathrm{d}z_{\mathrm{l}}}\frac{\mathcal{R}_{\mathrm{m}}(z_{\mathrm{m}})}{1+z_{\mathrm{m}}}\frac{\mathrm{d}V_{\mathrm{c}}}{\mathrm{d}z_{\mathrm{m}}}\,\sigma(M_{\mathrm{h}},z_{\mathrm{l}},z_{\mathrm{m}},\rho,\rho_{\mathrm{c}})\\\
&\times
p(\rho|z_{\mathrm{m}})~{}\mathrm{d}\rho~{}\mathrm{d}z_{\mathrm{m}}~{}\mathrm{d}z_{\mathrm{l}}~{}\mathrm{d}M_{\mathrm{h}}\,,\end{split}$
(3)

where $\mathrm{d}N(M_{\mathrm{h}},z_{\mathrm{l}})/\mathrm{d}M_{\mathrm{h}}$ is
the differential comoving number density of lensing halos in a halo mass shell
at lens redshift $z_{\mathrm{l}}$, $D_{\mathrm{c}}$ and $V_{\mathrm{c}}$ are
the comoving distance and volume, respectively, at a given redshift,
$\mathcal{R}_{\mathrm{m}}(z_{\mathrm{m}})$ is the total comoving merger rate
density at redshift $z_{\mathrm{m}}$, (1+$z_{\mathrm{m}}$) accounts for the
cosmological time dilation, $p(\rho\mid z_{\rm m})$ is the distribution of SNR
at a given redshift, $\rho_{\mathrm{c}}$ is the network SNR threshold, and
$\sigma$ is the lensing cross-section which indicates, as a function of its
various arguments, how efficiently strong lensing will occur. We model the
mass distribution of BBHs following the results for the Power Law + Peak model
of Abbott et al. (2021j). We consider a merger rate density model that assumes
the Madau–Dickinson ansatz (Madau & Dickinson, 2014) that is consistent with
recent results from GWTC-3. Moreover, we make use of the absence of a detected
stochastic gravitational-wave background (SGWB) to further constrain the
merger rate density (Abbott et al., 2021j). For consistency with previous
analyses (e.g., Abbott et al., 2021k), we take the Hubble constant from Planck
2015 observations to be $H_{0}=67.9\ {\rm km\ s^{-1}\ Mpc^{-1}}$ (Ade et al.,
2016). Furthermore, we choose $\rho_{\mathrm{c}}=8$ as a point estimator of
the detectability of GW signals. We find this choice to be consistent with the
search results in Abbott et al. (2021g) and Sec. 3.1, and we estimate its
impact to be subdominant compared to other sources of uncertainty.

Table 2:  Expected fractional rates of observable lensed double events at current LIGO–Virgo sensitivity.  Merger Rate Density | Galaxies | Galaxy Clusters
---|---|---
Model | $R_{\rm D}$ | $R_{\rm S}$ | $R_{\rm D}$ | $R_{\rm S}$
GWTC-3+Stochastic | 1.9$-$11.0 $\times 10^{-4}$ | 5.0$-$19.5 $\times 10^{-4}$ | 0.8$-$4.4 $\times 10^{-4}$ | 2.0$-$7.6 $\times 10^{-4}$

In Table 2, we show our estimates for the relative rate of lensing expected to
be observed by the LIGO–Virgo network of detectors. The results are shown
separately for galaxy-scale and cluster-scale lenses. Furthermore, these rates
are calculated for events that are doubly lensed and for two cases: when only
a single event (i.e., the brighter one) is detected (S), and when both of the
doubly lensed events are detected (D). The expected fractional rate of lensing
(i.e. the lensed to unlensed rate) spans the range
$\mathcal{O}(10^{-4}$–$10^{-3})$, depending on the merger rate density
assumed. We estimate the fractional rate of observed double (single) events
for galaxy-scale lenses to lie in the range 1.9$-$11.0 $\times 10^{-4}$
(5.0$-$19.5 $\times 10^{-4}$ ). Similarly, for cluster-scale lenses, the
fractional rate is estimated to be in the range of 0.8$-$4.4 $\times 10^{-4}$
(2.0$-$7.6 $\times 10^{-4}$), typically lower than the rates on galaxy scales.
These estimates suggest that observing a lensed double image is unlikely at
the current sensitivity of the LIGO–Virgo network of detectors. Nevertheless,
at design sensitivity and with future upgrades, standard forecasts suggest
that the possibility of observing such events might become significant (Ng et
al., 2018; Li et al., 2018; Oguri, 2018; Xu et al., 2021; Mukherjee et al.,
2021a; Wierda et al., 2021). Compared with other lens models, our lensing
rates are consistent with those predicted for singular isothermal ellipsoid
(SIE) models (e.g., Oguri, 2018; Xu et al., 2021; Wierda et al., 2021).

###  5.2 Implications from the non-observation of strongly lensed events

The absence of any detections of strongly lensed GW events before and during
O3 provides a complementary way to constrain the merger rates of compact
objects at high redshift. The detection of individual GW events has enabled
measurement of the low redshift ($z<1$) merger rate (Abbott et al., 2021j).
However, the high redshift merger rate of GW sources is not yet measured
directly, and we have only been able to place an upper limit on it from the
absence of a detection of the SGWB (Abbott et al., 2021l). The absence of such
a detection naturally leads to a bound on the lensing rate expected from
GWTC-3 (Mukherjee et al., 2021b; Buscicchio et al., 2020).

By using the same power-law form for the merger rate as that used in Sec. 5.1,
but extended up to $z=2$, we obtain limits on the merger rate at redshift
$z>1$ from the absence of detections of strongly lensed events. The
corresponding constraints ($90\%$ credible intervals) are shown in Fig. 5 as
the cross-hatched region bounded by the dash-dotted curves. The changes in the
upper bound of the merger rates are driven by the absence of detected lensing
events, whereas the lower bound is driven by the low-redshift constraints on
the merger rate. For comparison, the current limits on the merger rate from
GWTC-3 up to redshift $z=1$ (Abbott et al., 2021j), with the bounding curves
extrapolated to higher redshifts $z>1$, are shown as the grey shaded region
bounded by the dotted curves. For further comparison, we have also plotted the
solid black curves which show the current constraints from the absence of
detection of the SGWB (Abbott et al., 2021l). The upper bounds on the merger
rate from lensing are more stringent than the bounds from GWTC-3 at high
redshift (Abbott et al., 2021j), and are also comparable with the bounds from
the SGWB for redshifts $z<1.2$. The slight difference between the constraints
on the merger rates at low redshift derived from the SGWB (Abbott et al.,
2021l) and from GWTC-3 (Abbott et al., 2021j) arise because the bounds from
the SGWB are obtained here using the previous constraints on the merger rate
at low redshift derived using GWTC-2 (Abbott et al., 2021m).

Figure 5:  Merger rate density as a function of redshift based on the GWTC-3
results without lensing constraints (grey) and with lensing constraints
(cross-hatching) included. For clarity, we show only the results for galaxy-
scale lenses. Because lensed detections may occur at higher redshifts than
unlensed events, their non-observation can be used to constrain the rate of
mergers at higher redshifts. The ‘No lensing’ results shown here do not
include constraints derived from the absence of an SGWB. The latter
constraints are shown separately by the solid black curves.

###  5.3 Constraints on compact dark matter from gravitational-wave
microlensing

Objects whose size is comparable to their gravitational radius, and that cause
microlensing effects on GW signals, could be candidates for dark matter.
Although their abundance is heavily constrained by several astronomical
observations (Carr & Kuhnel, 2020; Carr et al., 2020), the possibility of
their contributing to dark matter cannot be ruled out in several mass windows.

Here we use the non-observation of microlensing effects on the GW signals
detected by LIGO and Virgo to constrain the fraction of dark matter
contributed by compact objects in the mass range $\sim
10^{2}$–$10^{5}~{}{M_{\odot}}$ (Jung & Shin, 2019; Urrutia & Vaskonen, 2021;
Basak et al., 2021). The essential idea is that if a significant fraction of
dark matter is in the form of compact objects, they would introduce detectable
microlensing signatures on the GW signals that we observe.

Assuming that lensed and unlensed events occur as Poisson processes, we
compute the posterior distribution on the lensing fraction
($u\equiv{\Lambda_{\ell}}/\Lambda$), defined as the ratio of Poisson means of
lensed events to the total number of detected events. This is then used to
compute the posterior of the fraction of compact dark matter
(${f_{\mathrm{DM}}}\equiv\Omega_{\mathrm{CO}}/\Omega_{\mathrm{DM}}$) (Basak et
al., 2021). We take that a total of $N=67$ BBH mergers are detected during the
O3 run 444These are the events cataloged in GWTC-3 that do not contain a
neutron star component., and none of them is lensed (i.e., ${N_{\ell}}=0$). We
then estimate the posterior distribution of the lensing fraction $u$. Finally,
the posterior of ${f_{\mathrm{DM}}}$ can be computed as

$p({f_{\mathrm{DM}}}\mid\\{{N_{\ell}}=0,N\\})=p(u\mid\\{{N_{\ell}}=0,N\\})\left|\frac{\mathrm{d}u}{\mathrm{d}{f_{\mathrm{DM}}}}\right|,$
(4)

where $\mathrm{d}u/\mathrm{d}{f_{\mathrm{DM}}}$ is the Jacobian that relates
the observed fraction $u$ of lensed events to the compact dark matter fraction
${f_{\mathrm{DM}}}$ in the Universe.

We determine this Jacobian by simulating astrophysical populations of BBH
mergers lensed by point mass lenses (Basak et al., 2021).555 The simulations
are done assuming the O3b representative PSD and Gaussian noise. The Jacobian
is not expected to change significantly if real noise is used instead.  The
constraints we obtain depend upon the assumed distributions of the component
masses, spins and the redshifts of the mergers, which have considerable
uncertainties. We assume that the masses are distributed according to the
Power-law + Peak model of Abbott et al. (2021j) while spins are assumed to be
aligned/antialigned with the orbital angular momentum with magnitudes
distributed uniformly in (0, 0.99). We use the approximant IMRPhenomD (Khan et
al., 2016) to produce the waveforms. We consider different redshift
distributions of the mergers: uniform distribution in comoving volume, the
power-law model of Abbott et al. (2021j), the Madau-Dickinson model (Madau &
Dickinson, 2014), as well as some representative population-synthesis models
given by Dominik et al. (2013) and Belczynski et al. (2016). In our
simulations, compact objects are approximated by point mass lenses and
distributed uniformly in comoving volume. Binaries producing a network SNR of
8 or above in the LIGO–Virgo detectors are deemed detectable. In order to
reduce the computational cost of performing the simulations, we estimate
$\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ using an approximation to the
Bayes factor that is expected to be accurate in the high-SNR regime (Cornish
et al., 2011; Vallisneri, 2012). We then compute the fraction of detected
events that produce a $\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ larger than
the highest $\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ obtained from real
LIGO–Virgo events. This lensing fraction is computed as a function of the
${f_{\mathrm{DM}}}$, which is used to compute the Jacobian
$\mathrm{d}u/\mathrm{d}{f_{\mathrm{DM}}}$.

Figure 6:  The spread in the 90% upper limits on ${f_{\mathrm{DM}}}$ obtained
from the O3 events using 5 different redshift distribution models for BBH
mergers: Belczynski et al. (2016),Dominik et al. (2013), Madau & Dickinson
(2014), Abbott et al. (2021j) and uniform in comoving 4-volume, assuming a
monochromatic mass spectrum for the compact objects forming dark matter. The
lens mass is shown on the horizontal axis. The grey (black) shaded regions
correspond to the spread in ${f_{\mathrm{DM}}}$ upper bounds computed assuming
flat (Jeffreys) prior on $\Lambda$ and ${\Lambda_{\ell}}$. The upper and lower
curves bounding the spreads correspond to the most pessimistic (weakest) and
optimistic (strongest) upper limits, as determined from the set of assumed
redshift distributions, in each mass bin.

The largest value of the microlensing likelihood ratio obtained from GWTC-3
events is $\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}=0.799$. We
compute the fraction of simulated events with
$\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}\geq 0.799$, for different
lens masses. This allows us to compute the Jacobian
$\mathrm{d}u/\mathrm{d}{f_{\mathrm{DM}}}$ and thus the posterior on
${f_{\mathrm{DM}}}$. The 90% upper limits are shown as a function of the lens
mass (assuming a monochromatic spectrum) in Fig. 6. The bounds we obtain are
weaker than some of the existing constraints (Carr & Kuhnel, 2020; Carr et
al., 2020). The GW lensing bounds will improve significantly in the next few
years as the sensitivity of GW detectors improve (Abbott et al., 2018).
Assuming $\sim 300$ BBH detections in O4 and $\mathcal{O}(1000)$ detections in
O5, the constraints on ${f_{\mathrm{DM}}}$ will improve to $\sim 10^{-1}$ and
$\sim 10^{-2}$, respectively.

##  6 Concluding Remarks

We have extended the search for lensing signatures to all BBH candidates with
a probability of astrophysical origin higher than 0.5 from O3b (Abbott et al.,
2021b). While we have not observed any significant candidates for strongly
lensed events, we updated the constraints on the rate of such events from
several different analyses. First, we searched for sub-threshold repeated
signals associated with super-threshold events using reduced template banks
produced from the posterior probability distributions of the super-threshold
events. Interesting sub-threshold/super-threshold pairs and pairs formed from
two super-threshold events were further analyzed for their probability of
being from a single, strongly lensed source. For super-threshold/super-
threshold pairs, we calculated the degree of overlap between the posteriors of
the intrinsic parameters and sky location, which were obtained from Bayesian
inference. Moreover, we analyzed these pairs using a new analysis based on the
comparison of spectrograms through machine learning. Finally, pairs with
false-positive probability from either analysis smaller than $10^{-2}$ were
further studied by conducting full joint Bayesian inference analyses that take
population priors and selection effects into account. We found no pairs that
show significant evidence for strong lensing.

The events from O3b were also analyzed for distortions caused by the lens on
the gravitational waveform. First, we searched for the distortions that
lensing introduces on type II signals, which are in the form of a frequency-
independent phase shift (Morse phase). The Bayes factors for all events show
no evidence for type II signal distortions. Similarly, we searched for the
frequency-dependent distortions caused by point masses. None of the computed
Bayes factors show any significant signs of microlensing. For both analyses,
some events show interesting features in the posteriors for the Morse phase or
lens mass. However, follow-up analyses using simulated signals show no further
signs of the lensing nature of these features. Altogether, we found no
significant evidence for distortions of the gravitational waveforms that can
be attributed to lensing.

The lack of evidence for lensing is then used to infer properties of the
lensing rates and to set constraints on the dark matter fraction of (dark)
compact objects.

Finally, we note that our conclusions are based on estimates and assumptions
that are in line with other analyses from the LIGO–Virgo–KAGRA Collaboration
(Abbott et al., 2021f, j). It is possible to arrive at different conclusions
and interpretations if assumptions are chosen differently. Examples include
claims that almost all detections are strongly lensed if one assumes that
heavy BHs do not exist (Broadhurst et al., 2018, 2020a, 2020b). Data from the
upcoming observing runs are expected to further expand the catalog of GW
detections that can further shed light on the lensing of GWs (Abbott et al.,
2020b). Moreover, multi-messenger astronomy may provide significant input in
confirming and interpreting possible lensed GW signals (Wempe et al., 2022).

This material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. The authors
also gratefully acknowledge the support of the Science and Technology
Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS),
and the State of Niedersachsen/Germany for support of the construction of
Advanced LIGO and construction and operation of the GEO 600 detector.
Additional support for Advanced LIGO was provided by the Australian Research
Council. The authors gratefully acknowledge the Italian Istituto Nazionale di
Fisica Nucleare (INFN), the French Centre National de la Recherche
Scientifique (CNRS) and the Netherlands Organization for Scientific Research
(NWO), for the construction and operation of the Virgo detector and the
creation and support of the EGO consortium. The authors also gratefully
acknowledge research support from these agencies as well as by the Council of
Scientific and Industrial Research of India, the Department of Science and
Technology, India, the Science & Engineering Research Board (SERB), India, the
Ministry of Human Resource Development, India, the Spanish Agencia Estatal de
Investigación (AEI), the Spanish Ministerio de Ciencia e Innovación and
Ministerio de Universidades, the Conselleria de Fons Europeus, Universitat i
Cultura and the Direcció General de Política Universitaria i Recerca del
Govern de les Illes Balears, the Conselleria d’Innovació, Universitats,
Ciència i Societat Digital de la Generalitat Valenciana and the CERCA
Programme Generalitat de Catalunya, Spain, the National Science Centre of
Poland and the European Union – European Regional Development Fund; Foundation
for Polish Science (FNP), the Swiss National Science Foundation (SNSF), the
Russian Foundation for Basic Research, the Russian Science Foundation, the
European Commission, the European Social Funds (ESF), the European Regional
Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the
Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund
(OTKA), the French Lyon Institute of Origins (LIO), the Belgian Fonds de la
Recherche Scientifique (FRS-FNRS), Actions de Recherche Concertées (ARC) and
Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO), Belgium, the Paris Île-
de-France Region, the National Research, Development and Innovation Office
Hungary (NKFIH), the National Research Foundation of Korea, the Natural
Science and Engineering Research Council Canada, Canadian Foundation for
Innovation (CFI), the Brazilian Ministry of Science, Technology, and
Innovations, the International Center for Theoretical Physics South American
Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council
of Hong Kong, the National Natural Science Foundation of China (NSFC), the
Leverhulme Trust, the Research Corporation, the National Science and
Technology Council (NSTC), Taiwan, the United States Department of Energy, and
the Kavli Foundation. The authors gratefully acknowledge the support of the
NSF, STFC, INFN and CNRS for provision of computational resources. This work
was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS
Grant-in-Aid for Specially Promoted Research 26000005, JSPS Grant-inAid for
Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and
JP17H06364, JSPS Core-to-Core Program A. Advanced Research Networks, JSPS
Grantin-Aid for Scientific Research (S) 17H06133 and 20H05639 , JSPS Grant-in-
Aid for Transformative Research Areas (A) 20A203: JP20H05854, the joint
research program of the Institute for Cosmic Ray Research, University of
Tokyo, National Research Foundation (NRF), Computing Infrastructure Project of
Global Science experimental Data hub Center (GSDC) at KISTI, Korea Astronomy
and Space Science Institute (KASI), and Ministry of Science and ICT (MSIT) in
Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the National Science
and Technology Council (NSTC) in Taiwan under grants including the Rising Star
Program and Science Vanguard Research Program, Advanced Technology Center
(ATC) of NAOJ, and Mechanical Engineering Center of KEK.

##  Appendix A Type II simulation campaigns

Given the mild evidence of GW190412 and GW200129_065458 towards being a type
II strongly lensed image presented in Sec. 3.4, we follow up on these events
by doing an injection campaign where we simulate type I and type II images
similar to the events, and verify whether the posteriors recovered are
compatible with the distribution observed for the real events. These
injections are performed in different noise realizations and with different
waveform models. The observed feature could be caused by two main other
effects than a type II image: noise artifacts or systematic effects in the
waveform modeling. In the former, non-Gaussianities in the noise could be such
that they lead to the observation of spurious features, while in the latter
case, the specific combination of observed parameters could lead to some
systematic issues in fitting with the waveform model. Waveform systematics
might be especially important for these events since they lie in challenging
parts of the parameter space (Abbott et al., 2020c; Colleoni et al., 2021;
Hannam et al., 2021). Moreover, for GW200129_065458 Payne et al. (2022)
reports that there could be a significant glitch under subtraction.

To test for the noise-related features, we generate colored Gaussian noise
from the PSD around the time of the candidate and then inject the maximum
likelihood parameters coming from the parameter estimation and take the Morse
factor to be either the value for a type I or a type II image. In the first
step, we do this for only one noise realization for each event to see whether
we can reproduce similar features or not. For GW200129_065458, the injection
shows that the effect is too weak to be distinguishable from one image type to
the other, as can be seen in the uninformative posteriors of Fig. 7.

Figure 7:  Posterior distribution of the Morse phase for GW200129_065458. We
compare the real event posterior (solid-blue) with an injection campaign of
type I (dashed-orange) and type II (dotted-green) images. Type II images
correspond to $n_{1}=1/2$. For this event, the differences between the
distribution are small and make it difficult to learn anything additional
about the event. The Kolmogorov–Smirnov statistic is 0.07 for type I vs real,
and 0.08 for type II vs real.

As a consequence, no further investigation is done into this event. On the
other hand, for GW190412, the feature seen in the real data is compatible with
the one seen in the injection (see Fig. 8).

Figure 8:  Posterior distribution of the Morse phase for GW190412. We compare
the real event posterior (solid-blue) with an injection campaign of type I
(dashed-orange) and type II (dotted-green) images. Type II images correspond
to $n_{1}=1/2$. For this event, the peak seen in the real data and the one
seen for the type II image are compatible, hinting at a possible type II
image. In this case, the Kolmogorov–Smirnov statistic is 0.20 for type I vs
real, and 0.13 for type II vs real.

Given that the real-data results are compatible with the type II injection for
GW190412, we investigate further the noise hypothesis. For this purpose, we
take the maximum likelihood parameters and a Morse factor of 0 or $1/2$ and
inject the signal generated with the IMRPhenomXPHM (Pratten et al., 2021)
model in ten different noise realizations. We then repeat the analysis in the
same way as for the real signal and verify if we retrieve the same preference
for a type II image. For all the noise realizations used here, we see the same
behavior as in Fig. 8.

We perform an extra test by injecting the maximum likelihood parameters with a
given image type in the generated noise for different waveform models. We use
the IMRPhenomPv3HM (Khan et al., 2020) and the SEOBNRv4PHM (Ossokine et al.,
2020) model to generate the signal and use the IMRPhenomXPHM (Pratten et al.,
2021) model to recover it. This enables us to combine the two possible sources
of systematics. This way, we can verify whether a different noise combined
with a different model also leads to a preference for type II images. For all
the noise realizations and the two models used for the injections, we find
that the injections always recover the correct hypothesis, and the fact that
the real event supports type II is unlikely to be a result of noise or
waveform artifacts, as shown in Fig. 9.

Figure 9:  Comparison of the Morse factor distribution for the real event
(solid-purple) with the recovered posterior distribution for an injection made
with IMRPhenomPv3HM for a type I image (dashed-blue) and a type II image
(dotted-orange), and with an injection made with SEOBNRv4PHM for a type I
image (dashed-green) and a type II image (dotted-red). In all the cases, the
posterior distributions agree with the injected data, with the real event
resembling a type II image.

Although these tests do not discard the type II image hypothesis, they cannot
conclusively confirm it. To confirm the presence of lensing for this event
with a mild preference for a type II image, we would need additional evidence.
Therefore, we search for possible sub-threshold counterparts with the
methodology explained in Sec. 3.1. However, we find only marginal triggers.

In the end, these additional searches did not enable us to find any extra
evidence for lensing, while still not ruling out the possibility for GW190412
to be a type II image.

##  Appendix B Marginalized Posteriors of Microlensing Parameters

As a supplement to the distribution of $\log_{10}$ Bayes factors
$\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$ shown in Fig. 4, we show the
individual marginalized posterior distributions of redshifted lens mass
$M^{z}_{\mathrm{L}}$ and $\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$
(right vertical axis) in Fig. 10.

Figure 10: Marginalized posterior distributions of redshifted lens mass
$M^{z}_{\mathrm{L}}$ and $\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$
between microlensed and unlensed hypotheses.

The Bayes factors individually do not show clear evidence for microlensing by
point-mass lenses. However, several events show a narrow posterior
distribution of the redshifted lens mass. An example is GW200208_130117 (with
$\log_{10}\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}=0.8$), for which the
waveform corresponding to the maximum posterior for this event, with and
without lensing, is shown in Fig. 11.

Figure 11: The time-domain waveform corresponding to the maximum posterior of
GW200208_130117, with and without the microlensing hypothesis.

The beating pattern introduced by the point-mass lens is most visible as a
reduction of the amplitude for two cycles in the middle of the signal and an
increase in the amplitude before and after this reduction. We hypothesize that
short-duration noise fluctuations may have caused an apparent dip in the
signal, which in turn may have led to a distortion similar to a point-mass
lens beating pattern. This is corroborated by a low Bayes factor
$\mathcal{B}_{\mathrm{U}}^{\mathrm{Micro}}$, which concludes the data is
inconclusive about the microlensing hypothesis.

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