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One of the central questions of cosmology and the evolution of galaxies is the role played by dark matter. As the universe expands, regions with a sufficient overdensity of dark matter will tend to collapse gravitationally, generating structures that survive today in the clustered and filamented distribution of galaxies and its underlying dark matter. If we knew the distribution of dark matter at any epoch in the evolution of the universe, we could calculate it at some other epoch and examine its physical implications. Was the dark matter that now resides in galactic clusters assembled along with the galaxies, or did pre-existing dense clumps of dark matter help to seed the formation of galaxies and clusters of galaxies?

Some dark matter candidates would have performed this seeding task more efficiently than others. Models in which the thermal motion of dark matter particles is too relativistic, for example, don't produce enough structure on the 10-kiloparsec scale typical of galaxies. (A parsec, abbreviated pc, is about 3 light-years.) On the other hand, models in which dark matter particles are too cold produce too much structure on the megaparsec scale of clusters of galaxies. For example, if relativistic neutrinos weighing no more than a few electron volts dominated the mass density, mass could not clump gravitationally on the galactic length scale. The free-streaming light neutrinos would prevent the growth of galaxies.

The line-of-sight component of a galaxy's velocity is measured by its Doppler shift z, the fractional wavelength displacement of its spectral features. In the universal Hubble expansion all distant galaxies are redshifted. The more distant the galaxy, the greater is its redshift. Light from a galaxy at a redshift of z ident Delta lambda / lambda = 1, for example, has been on its way to us for half the age of the universe. In cosmology it is often convenient to use z as the indicator of distance.

Of course cosmological redshifts are modified by internal motion within galaxies and clusters. Within a cluster of galaxies there is a dispersion about the mean redshift related to the local gravitational potential. Dynamically determining the mass of a cluster from its internal motion requires that one measure the redshifts of many of its galaxies. Gravitational lensing gives us a second, independent determination of the mass-of such a cluster, thus providing a valuable check on the virial and orbit assumptions underlying dynamical mass estimates. Furthermore, because clusters of galaxies are dominated by dark matter, the detailed distribution of mass in a cluster may provide a useful constraint on our conjectures about the nature of dark matter. One wants to know on what distance and time scales it clumps and how its spatial distribution relates to that of the cluster's luminous matter.

Our preliminary lens studies appear to confirm the large dark masses in rich galaxy clusters suggested by the virial theorem calculations using velocity dispersion data. We find that the dark matter distribution in a cluster looks like a smoothed distribution of the galaxies in the cluster. In the rich clusters we have studied thus far, there is evidence that the dark matter distribution has a soft core of up to 70 kpc radius. Because we don't yet know well enough how fast the density falls off outside this core, it is difficult to estimate the total dark mass associated with a rich cluster. If, for example, the density fell off like r-2 with no outer cutoff, the mass integral would diverge!

These gravitational lensing investigations promise to do more than just complement the dynamical studies of galactic clusters. Looking for lensing distortions of background galaxies in the "empty" regions between foreground clusters, we may well find concentrations of dark matter where there is no luminous matter to be seen.

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