8.1 Strong lensing by galaxies

When multiple images of a background source are produced by a gravitational lens, one talks about strong lensing. In that case, most of the light bending occurs within a small range around the lens compared to the lens-source distance Dls and the observer-source distance Ds (where the distances are the usual luminosity distances in cosmology). In this thin-lens approximation57, the resulting deflection angle can be written as:
2 ∫ ∞ α = -2 ∇ ⊥Φdz, (108 ) c −∞
where Φ = − Ψ is the non-relativistic gravitational potential of Eq. 73View Equation (obeying a MONDian Poisson equation), and ∇ ⊥ denotes the two-dimensional gradient operator perpendicular to light propagation. The lens equation then relates the observed two-dimensional angular position of the source in the lens plane 𝜃 to its original angular position in the source plane β through:
Dls 𝜃 = β + ----α, (109 ) Ds
where it appears clearly that the expansion history will play an important role in converting redshifts to distances. It is also convenient to make the deflection angle α derive from a deflection potential ϒ in the lens-plane:
∫ ∞ ϒ(𝜃 ) = -2Dls--- Φ (Dl𝜃,z )dz. (110 ) c2DsDl −∞
If a source is much smaller than the angular scale on which the lens properties change, the lens equation 109View Equation can locally be linearized as:
β(𝜃) = β0 + 𝒜 (𝜃 )(𝜃 − 𝜃0 ), (111 )
where the inverse magnification matrix is
∂-β 𝒜 (𝜃) = ∂ 𝜃, where 𝒜11 = 1 − κ − γ1,𝒜12 = 𝒜21 = − γ2,𝒜22 = 1 − κ + γ1 (112 )
The convergence κ is directly given by the Laplacian of the deflection potential ϒ:
1 2 κ = -∇ ϒ. (113 ) 2
The Einstein radius is the radius within the lens-plane within which the mean convergence is ⟨κ⟩ = 1. The existence of a region where κ is of that order is sufficient to produce multiple images and is the definition of strong lensing. On the other hand, the shear components γ1,γ2 are given by
1 ( ∂2ϒ ∂2ϒ ) ∂2ϒ γ1 = -- ---2 − ---2 ,γ2 = ------. (114 ) 2 ∂ 𝜃1 ∂ 𝜃2 ∂𝜃1∂𝜃2
Due to Liouville’s theorem, gravitational lensing preserves the surface brightness, but it changes the apparent solid angle of a source. The resulting flux ratio between image and source can be expressed in terms of the magnification M,
− 1 2 2 2 M = (1 − κ ) − γ1 − γ2. (115 )
The flux ratio between two images A and B is fAB = AA ∕AB. Let us finally note that (i) the time-delay between the different images can be deduced directly from the lensing potential and depends on the Hubble constant and convergence at the Einstein radius, and that (ii) points in the lens plane where M −1 = 0 (infinite magnification) form closed curves called the critical curves. Their corresponding curves located in the source plane are called caustics. The location of the source with respect to caustics determines the number of images, a source outside of the outermost caustic producing only one image, while each caustic crossing changes the number of images by a factor of two. Spherically-symmetric models of galaxy lenses can never produce observed quadruple-imaged systems because the innermost caustic of spherical models degenerates into a point.

As outlined above, what differs from GR in all the relativistic MOND theories is the relation between the non-relativistic potential Φ and the underlying mass distribution of the lens ρ. However, different theories yield slightly different relations between Φ and ρ 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. 17View Equation in the static weak-field limit, TeVeS (Section 7.4) leads to the situation of Eq. 40View Equation, and BIMOND (Section 7.8) to Eq. 30View Equation. However, like in the case of rotation curves (see Figure 20View Image), 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. 17View Equation).

View Image

Figure 41: (a) The four images of the quasar Q2237+030 (known as the Einstein cross), gravitationally lensed by an isolated bulge-disk galaxy known as Huchra’s lens [197Jump To The Next Citation Point]. © ESA’s faint object camera on HST. (b) The empty squares denote the four observed positions of the images, and the filled square denotes the MOND-fit unique position of the source [419Jump To The Next Citation Point]. The critical curves for which M −1 = 0 in the lens plane are displayed in black, and their corresponding caustics in the source plane in red. Image reproduced by permission from [419Jump To The Next Citation Point].

The first studies of strong lensing by galaxies in relativistic MOND theories [93, 501, 507Jump To The Next Citation Point] 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. [507] also compared the predicted and observed flux ratios fAB. They used the α = 0 μ-function of Eq. 46View Equation, 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. [419Jump To The Next Citation Point] then improved the modelling method by considering analytic non-spherical models with locally–spherically-symmetric isopotentials on both sides of the symmetry plane z = 0, implying no curl field correction (S = 0) in Eq. 19View Equation. The MOND non-relativistic potential Φ can then analytically be written, and using Eq. 108View Equation, one can analytically compute the two components α1 and α2 of the deflection angle vector α 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 [419Jump To The Next Citation Point]). Using the lens equation (Eq. 109View Equation), 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 [419Jump To The Next Citation Point] to fit in MOND the famous quadruple-imaged system Q2237+030 known as the Einstein cross (see Figure 41View Image), a quasar gravitationally lensed by an isolated bulge-disk galaxy [197]. 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.[419Jump To The Next Citation Point] 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 μ-function (α = 0 in Eq. 46View Equation), and that this function has a tendency of slightly underpredicting stellar mass-to-light ratios in galaxy rotation curve fits [145], 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 [364] 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 μ-function (α = 1 in Eq. 46View Equation) has, on the contrary, been shown to yield perfectly acceptable fits [94] in accordance with the lensing fundamental plane [400].

Finally, the probability distribution of the angular separation of the two images in a sample of lensed quasars has been investigated by Chen [90Jump To The Next Citation Point, 91]. This important question has proven somewhat troublesome for the ΛCDM paradigm, but is well explained by relativistic MOND theories [90].

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