Astron Astrophys Rev (2020)28:7 https://doi.org/10.1007/s00159-020-00129-w(0123456789().,-volV)(0123456789().,-volV) REVIEW ARTICLE Cluster–galaxy weak lensing Keiichi Umetsu1 Received: 2 July 2020 / Accepted: 7 October 2020 Ó The Author(s) 2020 Abstract Weak gravitational lensing of background galaxies provides a direct probe of the projected matter distribution in and around galaxy clusters. Here, we present a self- contained pedagogical review of cluster–galaxy weak lensing, covering a range of topics relevant to its cosmological and astrophysical applications. We begin by reviewing the theoretical foundations of gravitational lensing from first principles, with a special attention to the basics and advanced techniques of weak gravitational lensing. We summarize and discuss key findings from recent cluster–galaxy weak- lensing studies on both observational and theoretical grounds, with a focus on cluster mass profiles, the concentration–mass relation, the splashback radius, and implications from extensive mass-calibration efforts for cluster cosmology. Keywords Cosmology: theory Á Dark matter Á Galaxies: clusters: general Á Gravitational lensing: weak Contents 1 Introduction............................................................................................................................... 2 Theory of gravitational lensing................................................................................................ 2.1 Bending of light in an asymptotically flat spacetime .................................................... 2.2 Lens equation................................................................................................................... 2.3 Cosmological lens equation............................................................................................. 2.4 Flat-sky approximation...................................................................................................... 2.5 Multiple lens equation....................................................................................................... 2.6 Thin-lens equation ............................................................................................................. 3 Basics of cluster weak lensing................................................................................................... 3.1 Weak-lensing mass reconstruction.................................................................................... 3.2 E/B decomposition............................................................................................................. & Keiichi Umetsu [email protected] 1 Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan 123 7 Page 2 of 106 K. Umetsu 3.3 Flexion................................................................................................................................ 3.4 Shear observables .............................................................................................................. 3.5 Tangential- and cross-shear components .......................................................................... 3.6 Reduced tangential shear................................................................................................... 3.7 Aperture mass densitometry.............................................................................................. 4 Standard shear analysis methods ............................................................................................... 4.1 Background source selection............................................................................................. 4.2 Tangential shear signal...................................................................................................... 4.3 Lens mass modeling .......................................................................................................... 4.4 Shear likelihood function .................................................................................................. 4.5 Stacked weak-lensing estimator ........................................................................................ 4.6 Quadrupole shear ............................................................................................................... 5 Magnification bias...................................................................................................................... 5.1 Magnified source counts ................................................................................................... 5.2 Magnification observables................................................................................................. 5.3 Nonlinear effects on the source-averaged magnification bias ......................................... 5.4 Observational systematics and null tests .......................................................................... 6 Recent advances in cluster weak-lensing observations ............................................................ 6.1 Cluster mass distribution ................................................................................................... 6.2 The concentration–mass relation....................................................................................... 6.3 Splashback radius .............................................................................................................. 6.4 Mass calibration for cluster cosmology............................................................................ 7 Conclusions................................................................................................................................. References......................................................................................................................................... 1 Introduction The propagation of light rays from a distant source to the observer is governed by the gravitational field of local inhomogeneities, as well as by the global geometry of the universe (Schneider et al. 1992). Hence, the images of background sources carry the imprint of gravitational lensing by intervening cosmic structures. Observations of gravitational lensing phenomena can thus be used to study the mass distribution of cosmic objects dominated by dark matter and to test models of cosmic structure formation (Blandford and Narayan 1992). Galaxy clusters represent the largest class of self-gravitating systems formed in the 14À15 À1 universe, with typical masses of M 10 h M . In the context of the standard structure formation scenario, cluster halos correspond to rare massive local peaks in the primordial density perturbations (e.g., Tinker et al. 2010). Galaxy clusters act as powerful cosmic lenses, producing a variety of detectable lensing effects from strong to weak lensing (Kneib and Natarajan 2011), including deflection, shearing, and magnifying of the images of background sources (e.g., Umetsu et al. 2016). The critical advantage of cluster gravitational lensing is its ability to study the mass distribution of individual and ensemble systems independent of assumptions about their physical and dynamical state (e.g., Clowe et al. 2006). Weak gravitational lensing is responsible for the weak shape distortion, or shear, and magnification of the images of background sources due to the gravitational field of intervening massive objects and large-scale structure (Bartelmann et al. 2001; Schneider 2005; Umetsu 2010; Hoekstra 2013; Mandelbaum 2018). Weak shear 123 Cluster–galaxy weak lensing Page 3 of 106 7 lensing by galaxy clusters gives rise to levels of up to a few 10% of elliptical distortions in images of background sources. Thus, the weak shear lensing signal, as measured from small but coherent image distortions in galaxy shapes, can provide a direct measure of the projected mass distribution of galaxy clusters (e.g., Kaiser and Squires 1993; Fahlman et al. 1994; Okabe and Umetsu 2008). On the other hand, lensing magnification can influence the observed surface number density of background galaxies seen behind clusters, by enhancing their apparent fluxes and expanding the area of sky (e.g., Broadhurst et al. 1995, 2005b; Taylor et al. 1998; Umetsu et al. 2011b; Chiu et al. 2020). The former effect increases the source counts above the limiting flux, whereas the latter reduces the effective observing area in the source plane, thus decreasing the observed number of sources per unit solid angle. The net effect, known as magnification bias, depends on the intrinsic faint-end slope of the source luminosity function. In this paper, we present a self-contained pedagogical review of weak gravitational lensing of background galaxies by galaxy clusters (cluster–galaxy weak lensing), highlighting recent advances in our theoretical and observational understanding of the mass distribution in galaxy clusters. We shall begin by reviewing the theoretical foundations of gravitational lensing (Sect. 2), with special attention to the basics and advanced techniques of cluster–galaxy weak lensing (Sects. 3, 4, and 5). Then, we highlight and discuss key findings from recent cluster– galaxy weak-lensing studies (Sects. 6), with a focus on cluster mass distributions (Sect. 6.1), the concentration–mass relation (Sect. 6.2), the splashback radius (Sect. 6.3), and implications from extensive mass-calibration efforts for cluster cosmology (Sect. 6.4). Finally, conclusions are given in Sect. 7. There have been a number of reviews of relevant subjects (e.g., Blandford and Narayan 1992;