Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
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3.3 Caustics and Congruences

The angular diameter distance quoted in Equation 1 is only useful in a homogeneous universe. Inhomogeneous universes necessitate drastic approximations. Most treatments of this problem assume that the universe is uniform on the scale of the Hubble radius and that the relationship between redshift and cosmological time is the same as that in a FRW universe of similar mean density.

We are interested in the evolution of the cross section of a null geodesic congruence (a bundle of rays) as it propagates backward in time away from the observer, specifically in the mapping from the observer's angle space to proper distance perpendicular to some central or fiducial ray. Let us generalize Figure 5 and set the vector, xi(lambda)=(xi1, xi2), equal to this offset as measured all along the bundle, not just in the lens plane. The subscripts 1, 2 represent components with respect to an orthonormal basis parallel propagated along the fiducial ray, and lambda is the affine parameter. Next, define the complex number xi(lambda) = xi1 + i xi2. The angle between a ray in the congruence and the fiducial ray at the observer can also be represented as a complex number dotxi0, where a dot denotes differentiation with respect to lambda. We are now in a position to generalize the notion of angular diameter distance by defining a two component vector, D = (D1, D2), where both D1 and D2 are complex, using the general linear relation

Equation 9 9.

The real part of D1 measures the expansion of the ray while its imaginary part describes pure rotation. (In practice, rotation is usually small and D1 is approximately real.) D2 measures the shear. All the information about the local image distortion is contained in D. The conventional angular diameter distance, whose square is the ratio of the source area to the solid angle it subtends, is given by [| D1 | 2 - | D2 | 2] 1/2, and suffices for point sources where only the flux can be measured.

In an inhomogeneous universe containing Newtonian matter, D can be shown to evolve according to

Equation 10 10.

where the quantity R = -(1 + z)2(phi,11 + phi,22) = -4pi(1 + z)2Grho describes focusing by matter lying within the congruence with pr oper density rho, and F = -(1 + z)2(phi,11 - phi,22 + 2i phi,12) describes the influence of matter external to the congruence (e.g. Penrose 1966, Blandford et al. 1991). This formalism immediately gives expressions for the magnification tensors, [µ] (cf Equation 3), whose definition we can now generalize by identifying beta with the angle which would be subtended by the proper length xi in the source plane in a FRW universe of similar average density to the inhomogeneous universe under consideration. (See Ehlers & Schneider 1986 for an alternative choice of reference universe).

In the limiting case when all the matter in the universe apart from the lens is isolated from the congruence (D2 = 0), the lack of focusing by matter in the beam (save for the lens) compared to a FRW universe of the same Omega 0 increases the angular diameter distance of the source (Dashevskii & Zel'dovich 1965, Dyer & Roeder 1972, Nottale 1983, Nottale & Hammer 1984, Kasai et al. 1990). The increase is about 30 per cent for a source with zs ~ 2 in an Einstein-De Sitter universe. However, the cumulative shear caused by external matter usually produces a second order focusing which leads to a diminished net effect. In general, if multiple imaging is uncommon, the distribution of magnifications due to smoothly distributed matter is dominated by the convergence rather than shear (Lee & Paczynski 1990, Watanabe & Sasaki 1990). The total flux is always conserved when suitably averaged over all directions (Weinberg 1976, Peacock 1986).

If the focusing (say due to a lensing galaxy) is strong enough to make the rays cross along any congruence (Figure 8), then multiple images must form and we have an example of gravitational lensing. At the point where the rays cross, known as a conjugate point to the observer, the conventional angular diameter distance vanishes (| D1 | = | D2 |) and the formal magnification diverges. The locus of these conjugate points is a two-dimensional surface, a caustic sheet, to which the rays are tangent (see Blandford & Narayan 1986 for a schematic diagram showing the caustic sheets associated with an elliptical lens). Equivalently, we can think in terms of wavefronts normal to the rays merging at a caustic (Kayser & Refsdal 1983). For a source at a fixed redshift, the source plane intersects the caustic sheets at caustic lines. The images of these lines are known as critical curves (cf Figures 6, 7).

Figure 8. An infinitesimal conical bundle of rays is shown drawn backwards from an observer, past an elliptical lens, and touching two caustic sheets. The second caustic sheet, on the left, has a cusp line perpendicular to the plane of the diagram, while the first caustic sheet has a cusp in the plane. Representative cross sections of the bundle are indicated at the bottom. Where the bundle touches a caustic, its cross section degenerates to a straight line. Beyond this point, the bundle is ``inverted'' and a source located here will acquire two additional images. In general there could be many caustic sheets behind a complex lens, but with a single elliptical lens there are only two sheets (which may penetrate each other, cf Blandford & Narayan 1986).

In the generic situation, the caustic sheet corresponds to a fold caustic. When a source crosses a fold, an extra pair of images will either be created or destroyed. These image pairs will be stretched toward each other along a direction essentially perpendicular to the projection of the caustic on the sky (Blandford & Kovner 1988). Because of the stretching, the images will be bright; an example is the pair of bright images, A1A2, in Q1115+080. The magnifications of the two images will be inversely proportional to their separations and also inversely proportional to the square root of the distance of the source from the caustic (Benson & Cooke 1979, Ohanian 1983, Blandford & Narayan 1986, Kayser & Witt 1989). Therefore, for a fold caustic, the cross section, sigma ( > µ), for the magnification to be greater than µ has a universal scaling, sigma propto µ-2, for µ >> 1. Equivalently, the differential cross section scales as dsigma / dµ propto µ-3.

Every time a ray touches a caustic (grazes it tangentially), the associated image is inverted, i.e. its parity is reversed. (Polarization directions are parallel propagated and unaffected.) In Q0957+561, the A image is believed to be uninverted while the ray associated with the B image has touched one caustic, so the two images are approximate reflections of each other. A faint third image ought to be formed in the galaxy nucleus, inverted twice through roughly orthogonal planes, hence rotated through ~ 180°.

Fold surfaces meet at cusp lines, which correspond to a cusp caustic. Sources lying just inside cusps create three bright images (plus any additional images that are not associated with the cusp). Sources lying just outside cusps have one of their images highly brightened. In this region, the cross section for large µ scales as sigma (> µ) propto µ-5/2, or dsigma /dµ propto µ-7/2. Cusps are believed to play an important role in the luminous arcs. Cusp lines meet at points associated with higher order singularities, but these have not yet been identified in the observations. The closest point of the caustic to the observer is generically a cusp. When a source is located close to this point, the lens is said to be marginal and may produce one or three bright images (Narayan et al. 1984, Kovner 1987d, e).

In general, for a non singular lens, caustic surfaces separate regions with image multiplicities differing by two. Since far from the lens a source has but one image, therefore the total number of images has to be odd (Burke 1981, McKenzie 1985).

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