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The idea that a massive foreground object would significantly distort the image of a background object is an old one. In 1937 Fritz Zwicky argued that galaxies would act as gravitational lenses, distorting and amplifying background objects.5 The light deflection is proportional to the mass in the lens. For a typical galaxy it's about 2 arcseconds. 6

In figure 2 the true angular distance between the distant galaxy and the centroid of the foreground cluster is phi. As a result of the bending of the traversing light by the cluster, the background source appears to us to be at the larger angular distance upsilon. The lensing displacement delta = upsilon - phi is related to the gravitational bending angle beta by

Equation 1

where and DLS and DS are the lens-source and observer-source distances.6 In the thin-lens approximation, we treat the cluster as a two-dimensional system, labeling the projected impact parameter of a ray by the two-dimensional vector r. Then the bending angle beta is given in terms of the projected two-dimensional mass density distribution sigma by

Equation 2

Because we don't know the true, unlensed position of the background source galaxy on the sky, we can't measure beta. But its shear - that is to say, the gradient of the gravitational bending - is observable. For most foreground mass distributions we can simplify the integral equation above. If the projected density distribution sigma(r) of the foreground cluster can be expressed as a sum of circularly symmetric components, the vector arguments reduce to scalar angles. In the simple case of a foreground point mass, beta = 4GM / rc2, where 2GM / c2 is the Schwarzschild radius of the mass M.

A source galaxy exactly behind this point mass would appear as an "Einstein ring" image of radius

Equation 3

where Msun is the mass of the Sun. If the lensing mass is elliptical or otherwise not circularly symmetric, this ring symmetry is broken. Therefore complete, circular Einstein rings are rare in the heavens. If the source angle theta is less than vE one sees two images of the source, separated by about 2vE.

How big are these effects? Single galaxies acting as lenses of about 1012 solar masses will produce multiple images with separations on the order of 3 arcseconds. Cluster lenses with masses of about 1014 Msun can produce image separations as large as an arcminute. For a gravitationally bound collection of luminous objects, the virial theorem lets one deduce the total mass, visible and invisible, from the observable velocities. The theorem states that the binding energy of the system equals its internal kinetic energy. It follows that <Deltav2>, the square of the velocity dispersion of observable components within a distance r of the center, equals GM(r) / r. For an equilibrated, isothermal distribution of gravitationally bound masses, the gravitational bending angle beta will just be 4pi <Deltav2> / c2. (Isothermality implies that mass density falls like r-2.) The velocity dispersion also gives the Einstein-ring size:

Equation 4

Many clusters of galaxies have measured velocity dispersions on the order of 103 km/sec. One can calculate the gravitational bending for fairly realistic mass distributions 7 for example a soft-core isothermal sphere, by substituting the appropriate projected density sigma(r) into the integral that gives beta(r). Generally the image distortion or, in the case of split images, the separation between images depends only on the mass distribution interior to the impact parameter r.

In 1973 William Press and James Gunn at Caltech pointed out that a cosmologically significant density of massive foreground objects would produce distorted and split images of background galaxies 8 The first gravitational lens was discovered six years later: a distant quasar split into at least two images by the gravity of a foreground galaxy. 9

In the 13 years since that first discovery, only eight more unambiguous split quasar images have been found, in searches that looked at a total of more than 4000 quasars. In several of these cosmic mirages the gravitational lensing clearly seems to be done by an isolated foreground galaxy dominated by its dark matter. The rarity of these lensed quasar images is consistent with the known abundance of quasars and foreground galaxies. The effective mass M and radius R of the foreground lensing galaxy can be deduced from the distortion of the quasar image. One finds M to be on the order of 1012 Msun and M/R around 3 x 1010 Msun / kpc. This is in accord with typical virial theorem results for heavy galaxies. 10 Dark matter appears to account for about 90% of M.

With so few gravitationally lensed quasar images in hand, we can't learn very much about the mass distributions in the lenses. These background quasars typically have redshifts on the order of 2. At these redshifts we can see a lot more galaxies than quasars. So if we want to find out as much as possible about dark matter in the foreground we must avail ourselves of the great abundance of background galaxies rather than the occasional quasar.

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