A background galaxy will be multiply imaged if it lies within the caustic curves, and if the lens through which it is observed is dense enough to break the wave front coming from the galaxy into several pieces. The number of multiple images produced is the number of solutions to the lens equation (eqn. 5.6). This can be estimated via catastrophe theory (Zeeman, 1977; Erdl and Schneider, 1993), which stipulates that two additional images are produced per caustic line crossing, and predicts that there will always be an odd number of images for a non-singular mass distribution (Burke, 1981). However not all of these images are readily observable: some may be demagnified or obscured by the presence of a cluster galaxy.
5.3.1 Examples of multiple image configurations
The predicted geometry of multiple images is well prescribed and although critical lines are vir- tual and thus cannot be directly mapped, the multiple images which straddle them are usually readily identifiable in high resolution images. Simple patterns of images that are easily recog- nisable are tangential pairs or radial pairs (e.g. Miralda-Escude and Fort, 1993). The radially distorted images are found near radial critical lines, and similarly for images that are tangentially distorted.
In the case of tangentially distorted multiple images, there are two main configurations: fold andcusp, with the difference based on where the source lies in relation to the tangential caustic line. A fold configuration occurs when, in the source plane, the galaxy lies close to and inside the tangential caustic along one of the edges. This leads to two images placed symmetrically across the tangential critical line, with a third image, demagnified on the opposite side of the critical line. A cusp configuration is similar to that of the fold, however the source now lies close to an asteroid spike of the tangential caustic line. Three images are clustered towards the end of the semi-major axis of the tangential critical curve, sometimes appearing as an almost continuous arc. An example of fold and cusp configurations are shown in panels 10 and 6 of Figure 5.3, respectively.
For sources close to the radial caustic line, the image deformation is radial resulting inradial
Figure 5.3: Multiple image configurations produced by a simple elliptical mass profile. Panel (S) shows the caustic lines in the source plane and the source image positions 1 to 10 relative to the caustic. Panel (I) shows the image of the source without lensing. Panels (1) to (10) show the resulting lensed images for the various source positions given in (S). Some of these configurations are well-known and are named as follows: (3) radial arc, (6) cusp arc, (8) Einstein cross, (10) fold arc. Source: Kneib and Natarajan (2011).
arcs when two images straddle the radial critical line and almost merge, with the third counter image appearing on the opposite side of the critical line. The configuration of radial multiple images is heavily dependent on the shape of the mass profile at the centre of the lens – the more peaked the profile, the closer the radial image will be to the centre. Thus for singular mass profiles the counter radial image could be demagnified to such an extent it disappears entirely.
In this case, the radial configuration is identifiable only as an image pair. An example of a radial arc configuration is shown in panel 3 of Figure 5.3.
The above patterns of fold arcs, cusp arcs and radial arcs should be visible for clusters with one dominant mass clump (e.g. Fort et al., 1992; Mellier et al., 1993; Natarajan et al., 1998;
Smith et al., 2001, 2003). Besides the configurations mentioned above, Figure 5.3 shows a range of multiple-image configurations produced by a single elliptical mass distribution. Furthermore, bimodal mass distributions can produce straight arcs (e.g. Pello et al., 1991; Kneib et al., 1994) and triplets (e.g. Kneib et al., 1993; B´ezecourt et al., 1999; Limousin et al., 2012). Multiple image systems with higher multiplicities are created when the mass distribution exhibits a very complex structure with several massive core halos (Richard et al., 2014; Johnson et al., 2014;
Lam et al., 2014; Jauzac et al., 2014, 2015b; Coe et al., 2015; Diego et al., 2015d). Each addi- tional deforming mass clump typically adds two images to a simple configuration, provided the mass clump is well positioned relative to the central core. The set of seven multiple images of the E/S0 galaxy in Abell 2218 is an impressive example of such a system (Kneib et al., 1996).
Other exotic configurations are also possible (see e.g. Cabanac et al., 2007; Bolton et al., 2008;
Limousin et al., 2008; Shin and Evans, 2008; Orban de Xivry and Marshall, 2009; Collett and Bacon, 2015).
5.3.2 Multiple image identification
When galaxy clusters act as gravitational lenses, an analysis of the detailed configuration of the multiply-imaged sources can constrain the inner core of the cluster mass distribution. The cor- rect identification of the multiple images represents a mandatory step in order to obtain reliable information on the mass distribution of the cluster. In order to do this, one needs to be able to
identify these multiple images. This can be done via their several distinct properties.
The giant arcs as seen in the cases of Abell 370 (Soucail, 1987) and CL2244-02 (Lynds and Petrosian, 1986) are the traditionally identified sets of multiple images, however an observed arc is not necessarily a result of multiple images from a single source. Some giant arcs are only a single image which has been distorted, and multiple-image arcs may have more than one back- ground source contributing to the arc components (Ellis et al., 1991; Coe et al., 2010). One of the ways to identify the latter case is by the image colours and morphologies. Since lensing affects all wavelengths in the same geometric way, multiple images of the same source can be identified by the similarity of their colours, or by their shared brightness at a specific wavelength. Another useful property of multiple images is symmetry, in particular mirror symmetrywhere a counter image appears as the flipped version of a galaxy image with extremely similar morphology. This is best visible in high-resolution images in which the individual structure of the background galaxies can be resolved. The symmetries of multiple images are related to the parities of the eigenvalues of the amplification matrix. Every time a critical line is crossed, the parity of the image changes, resulting in the image appearing to be a “vertically” or “horizontally” flipped version of the counter image, depending on which side of the parity pair changes (Blandford and Narayan, 1986; Schneider et al., 1992).
The above methods for visually identifying multiply imaged systems are often used to iden- tify multiple image candidates, some of which may be spurious identifications. The potential multiple images can be confirmed or rejected by detailed modelling of the cluster lens. Well calibrated lens models, usually based on a few unambiguous visual system detections, can be used to predict the position of a counter image as well as estimate the redshift of the source being multiply lensed (Kneib et al., 1993, 1996; Jauzac et al., 2014). The larger the number of multiply-imaged systems identified, the more constrained the lens model becomes in the region of those systems.