As outlined above, what differs from GR in all the relativistic MOND theories is the relation between the non-relativistic potential and the underlying mass distribution of the lens . However, different theories yield slightly different relations between and in the weak-field limit (see especially Sections 6.1 and 6.2). For instance, while GEA theories (Section 7.7) boil down to Eq. 17 in the static weak-field limit, TeVeS (Section 7.4) leads to the situation of Eq. 40, and BIMOND (Section 7.8) to Eq. 30. However, like in the case of rotation curves (see Figure 20), the differences are only minor outside of spherical symmetry (and null in spherical symmetry), and the global picture can be obtained by assuming a relation given by the BM equation (Eq. 17).
The first studies of strong lensing by galaxies in relativistic MOND theories [93, 501, 507] made use of the CfA-Arizona Space Telescope Lens Survey (CASTLES) and made a one-parameter–fit of the lens mass to the observed size of the Einstein radius, both for point-mass models and for Hernquist spheres (with observed core radius). Zhao et al.  also compared the predicted and observed flux ratios . They used the -function of Eq. 46, and concluded that reasonably good fits could be obtained with a lens mass corresponding to the expected baryonic mass of the lens. Shan et al.  then improved the modelling method by considering analytic non-spherical models with locally–spherically-symmetric isopotentials on both sides of the symmetry plane , implying no curl field correction () in Eq. 19. The MOND non-relativistic potential can then analytically be written, and using Eq. 108, one can analytically compute the two components and of the deflection angle vector as a function of the three parameters of the model, namely the lens-mass and two scale-lengths controlling the extent and flattening of the lens (see Eq. 18 of ). Using the lens equation (Eq. 109), one can then trace back light-rays for each observed image to the source plane and fit the lens parameters as well as its inclination, in order for the source position to be the same for each image. The quality of the fit is thus quantified by the squared sum of the source position differences. This notably allowed  to fit in MOND the famous quadruple-imaged system Q2237+030 known as the Einstein cross (see Figure 41), a quasar gravitationally lensed by an isolated bulge-disk galaxy . However, for three other quadruple-imaged systems of the CASTLES survey, the fits were less successful mostly because of the intrinsic limitations of the analytic model of Shan et al. at reproducing at the same time both a large Einstein radius and a large shear. What is more it does not take into account the effects of the environment in the form of an external shear, which is also often needed in GR to fit quadruple-imaged systems. For 10 isolated double-imaged systems in the CASTLES survey, the fits were much more successful58. However, for non-isolated systems, especially for those lenses residing in groups or clusters, the need for an external shear might be coupled to a need for dark mass on galaxy group scales (see Section 6.6.4 and Section 8.3).
Due to the fact that all the above models were using the Bekenstein -function ( in Eq. 46), and that this function has a tendency of slightly underpredicting stellar mass-to-light ratios in galaxy rotation curve fits , it was claimed that this was a sign for a MOND missing mass problem in galaxy lenses [152, 153, 262]. While such a missing mass is indeed possible, and even corroborated by some dynamical studies  of galaxies residing inside clusters (i.e., the small-scale equivalent of the problem of MOND in clusters), for isolated systems with well-constrained stellar mass-to-light ratio, the use of the simple -function ( in Eq. 46) has, on the contrary, been shown to yield perfectly acceptable fits  in accordance with the lensing fundamental plane .
Finally, the probability distribution of the angular separation of the two images in a sample of lensed quasars has been investigated by Chen [90, 91]. This important question has proven somewhat troublesome for the CDM paradigm, but is well explained by relativistic MOND theories .
Living Rev. Relativity 15, (2012), 10
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