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The redshift measurements have provided strong dynamical evidence for dark matter on scales from galaxies to superclusters of galaxies. But it is worthwhile to explore independent techniques that do not rest on assumptions about stellar and galactic orbits. My colleagues and I have chosen to look for the gravitational deflection of light from very distant sources by dark matter in the foreground. By this means we have sought to map and weigh the dark matter in several large clusters of galaxies. Our collaboration includes astronomers from Bell Labs, Princeton University, the University of Arizona, the University of Cambridge, the Institute for Advanced Study, the National Optical Astronomy Observatories and the Toulouse Observatory.

Arthur Eddington's famous solar eclipse expedition of 1919 was the first to measure gravitational lensing by a celestial body, thereby verifying the prediction of general relativity and making Einstein a household name. When a photon passes close by the Sun or any other sufficiently massive foreground system, visible or invisible, the gravitational bending of its path makes the source appear to be at an altered position. The bending angle is just twice the Schwarzschild radius of the massive body divided by the impact parameter. In this way, a clump of dark matter in the foreground will act as a gravitational lens, systematically distorting the images of more distant background galaxies. (See figure 1.)

Figure 1

Figure 1. Faint blue arcs circling a massive cluster of reddish-yellow galaxies are actually much more distant blue galaxies elongated by gravitational lensing as their light passes through the cluster Abell 2218, whose redshift is only 0.17. These distorted background images can provide a map of the mass of the foreground cluster, most of which is otherwise invisible dark matter. This CCD image, 4 arcminutes wide, was taken by Gary Bernstein and the author at the 4-meter telescope on Kitt Peak.

As shown in figure 2, light from a distant galaxy passing through a foreground mass concentration at a closest distance r from its center will be gravitationally bent through an angle beta = 4GM(r) / rc2, where M(r) is the total mass interior to the projected radius r. In addition to being displaced, the image of a sufficiently wide background galaxy can also be severely distorted. A galaxy of angular size 1 arcsecond seen through a foreground cluster may be elongated into a circular arc many arcseconds long.

Figure 2

Figure 2. Gravitational displacement and distortion of the image of a distant background galaxy (blue) by a compact foreground cluster of galaxies (orange). A light ray passing the cluster plane at an impact parameter vector r is gravitationally bent through an angle beta. Thus we see it displaced (through an angle delta) from theta, its true angular distance from the cluster centroid, to the larger angle v. Because of its finite width, the image also is distorted into a circular arc (of length a and width b) concentric with the cluster.

Suppose the lensing foreground mass is a cluster of galaxies whose dark mass component we seek to map out by measuring the distorted images of the background galaxies. Because a photon traverses the foreground cluster in a time much shorter than the typical orbital periods of the cluster galaxies, one is in effect taking a snapshot. It is not necessary to assume, as one must in the dynamical dark matter searches, that the cluster is an equilibrated bound state. Other mass estimates, based on x-ray flux maps, rest on assumptions about the state of the hot gas. Our technique does not even require the observation of any radiation from the system being weighed. Gravitational lensing could be used to discover mass concentrations that are entirely dark.

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