MNRAS 000, 000–000 (2023) Preprint 16 August 2023 Compiled using MNRAS L A T E X style file v3.0 Follow-up Analyses to the O3 LIGO-Virgo-KAGRA Lensing Searches Editorial Team: J. Janquart, 1,2⋆ M. Wright, 3 † S. Goyal, 4 Analysts: J. C. L. Chan, 5 A. Ganguly, 4,6 ´ A. Garr ´ on, 7 D. Keitel, 7,8 A. K. Y. Li, 9 A. Liu, 10 R. K. L. Lo, 9 A. Mishra, 6 A. More, 6,11 H. Phurailatpam, 10 P. Prasia, 6 Contributors: P. Ajith, 4,12 S. Biscoveanu, 13,14 P. Cremonese, 7 J.R. Cudell, 15 J. M. Ezquiaga, 5 J. Garcia-Bellido, 16 O. A. Hannuksela, 10 K. Haris, 1,2 I. Harry, 8 M. Hendry, 3 S. Husa, 7 S. Kapadia, 6 T. G. F. Li, 17,18,10 I. Maga˜ na Hernandez, 19,20 S. Mukherjee, 21 E. Seo, 3 C. Van Den Broeck, 1,2 J. Veitch 3 1 Institute for Gravitational and Subatomic Physics (GRASP), Department of Physics, Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands 2 Nikhef – National Institute for Subatomic Physics, Science Park, 1098 XG Amsterdam, The Netherlands 3 SUPA, School of Physics and Astronomy, University of Glasgow, Scotland 4 International Centre for Theoretical Science, Tata Institute of Fundamental Research, Bangalore—560089, India 5 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 6 The Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India 7 Departament de F´ ısica, Universitat de les Illes Balears, IAC3–IEEC, E-07122 Palma, Spain 8 University of Portsmouth, Institute of Cosmology and Gravitation, Portsmouth PO1 3FX, United Kingdom 9 LIGO, California Institute of Technology, Pasadena, CA 91125, USA 10 Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 11 Kavli Institute for the Physics and Mathematics of the Universe, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8583, Japan 12 Canadian Institute for Advanced Research, CIFAR Azrieli Global Scholar, MaRS Centre, West Tower, 661 University Ave, Toronto, ON M5G 1M1, Canada 13 LIGO Laboratory, Massachusetts Institute of Technology, 185 Albany St, Cambridge, MA 02139, USA 14 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA 15 STAR Institute, Bˆ atiment B5, Universit´ e de Li` ege, Sart Tilman B4000 Li` ege, Belgium 16 Instituto de F´ ısica Te´ orica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 17 Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium 18 Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium 19 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 20 McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 21 Department of Astronomy & Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India 16 August 2023 ABSTRACT Along their path from source to observer, gravitational waves may be gravitationally lensed by massive objects leading to distortion in the signals. Searches for these distortions amongst the observed signals from the current detector network have already been carried out, though there have as yet been no confident detections. However, predictions of the observation rate of lensing suggest detection in the future is a realistic possibility. Therefore, preparations need to be made to thoroughly investigate the candidate lensed signals. In this work, we present some follow-up analyses that could be applied to assess the significance of such events and ascertain what information may be extracted about the lens-source system by applying these analyses to a number of O3 candidate events, even if these signals did not yield a high significance for any of the lensing hypotheses. These analyses cover the strong lensing, millilensing, and microlensing regimes. Applying these additional analyses does not lead to any additional evidence for lensing in the candidates that have been examined. However, it does provide important insight into potential avenues to deal with high-significance candidates in future observations. 1 INTRODUCTION Gravitational lensing of gravitational waves (GWs) happens when they pass nearby a massive object and the deformation of space- time caused by that object modifies their propagation. The observed modifications depend on the exact properties of the lens and include repeated events, phase shifts, changes in amplitude, beating patterns and distortions (Ohanian 1974; Thorne 1982; Deguchi & Watson 1986; Wang et al. 1996; Nakamura 1998; Takahashi & Nakamura 2003). When the lens is an extended high-mass object (e.g. a galaxy or galaxy cluster), the GW frequency evolution is unaffected as the geometric optics limit applies and results in multiple signals— called images—with different magnification, phase shift, and time delay (Wang et al. 1996; Dai & Venumadhav 2017; Ezquiaga et al. 2021). The relative time delays between the images range from min- © 2023 The Authors arXiv:2306.03827v4 [gr-qc] 15 Aug 2023