Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
Copyright © 1992 by . All rights reserved

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Although basic geometrical optics is all that is required to describe gravitational lenses, the development of the subject has generated some fresh physical insights. Several complementary formalisms have been derived (cf Nityananda 1990a, b), but we shall confine our attention to those that are most appropriate for application to cosmology. Schneider et al. (1992) provide a more complete treatment of the subject.

3.1 Ray Deflections and the Lens Equation

Figure 5. Basic ray geometry of gravitational lensing. A light ray from a source S at redshift zs is incident on a deflector or lens L at redshift zd with impact parameter xi relative to some fiducial lens ``center''. Assuming the lens is thin compared to the total path length, its influence can be described by a deflection angle dot alpha (xi) (a two-vector) suffered by the ray on crossing the ``lens plane''. The deflection ray reaches the observer O, who sees the image of the source apparently at position theta on the sky. The true direction of the source, i.e. its position on the sky in the absence of the lens, is indicated by beta. Also shown are the angular diameter distances Dd, Ds, Dds, separating the source, deflector, and observer.

The basic ray geometry is shown in Figure 5. For the moment, consider a homogeneous Friedmann-Robertson-Walker (FRW) universe of cosmological density parameter Omega0 in which angular diameter distances D (zi, zj) relate proper lengths xij located at redshift zj to angles thetai subtended by these lengths when observed from redshift zi < zj:

Equation 1a

Equation 1b 1.

where H0 = 100h km s-1 Mpc-1 is the Hubble constant. For a source at redshift zs and a single deflector at redshift zd, we follow convention and set Dd = D(0, zd), Ds = D (0, zs), and Dds = D (zd, zs). Elementary geometry gives the lens equation (e.g. Refsdal 1964a), which connects the source position beta and the image position theta (Figure 5),

Equation 2 2.

where the reduced deflection angle alpha(theta) and the true deflection angle alphahat (xi) are related by

Equation 2a

Deflection angles of interest in astrophysical applications are always small. We are therefore justified in using the weak field approximation of General Relativity. For a thin lens, alphahat = delpsi / c2, where psi(xi) is twice the 2D Newtonian potential obtained by solving the 2D Poisson equation, del2 psi(xi) = 8pi GSigma(xi), corresponding to surface mass density Sigma(xi). [A weak gravitational lens is, in its essentials, equivalent to a flat space deflector with refractive index 1 - 2phi / c2, where phi is the 3D Newtonian potential with respect to infinity (Eddington 1919).]

From the structure of Equation 2 it is clear that, for a given lens, there is a unique source position beta for each image position theta. However, the converse is not true. For a nontrivial deflection law alpha(theta), it is possible to find more than one solution theta satisfying Equation 2 for a given beta, thus giving rise to multiple imaging. (In general, gravitational lenses are not like simple optical lenses which have alphahat (xi) varying linearly with xi.)

The magnification tensor of the ith image relative to the unlensed source is

Equation 3 3.

which is a symmetric 2 x 2 matrix. This is not directly measurable, but the relative magnification between two images, described by the transformation matrix [µ(ij)] = [µ(i)] [µ(j)]-1 can be estimated when the images are resolved. This is in general nonsymmetric and provides four independent observables. The flux magnification associated with the ith image is the absolute value of the determinant of the magnification tensor, i.e. µ(i) = | [µ(i)] |. For unresolved images, only the relative flux magnifications, µij = | [µ(ij)] |, are measurable.

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