### 8. THE INVERSE PROBLEM

Given a lensing mass distribution and a background of faint galaxies, there is a straightforward recipe for calculating the distortion of the background. But our task - solving the inverse problem to approximate the lensing mass distribution - is more difficult. It can be done only for certain simple foreground mass distributions. Even then, the solution isn't so good at very small and very large distances from the lensing center.

Just how accurate are the foreground dark matter maps we deduce from the distorted background? Figure 5a shows a Monte Carlo simulation of a background field distorted by a fairly complex foreground lensing configuration: an equilateral triangle made of three spherically symmetrical, isothermal balls of mass. Figure 5b shows the foreground mass distribution our inverse algorithm deduced from figure 5a. It shows that we can reliably recover such complex mass distributions. But the technique fails for foreground mass components with velocity dispersions less than about 500 km/sec. For such meager lensing masses the coherent distortion signal is swamped by the random ellipticities of the background galaxies.

 Figure 5. The inverse problem, solved for a complex lensing mass configuration. Given the simulated background field distortion in a, a computer program tries to determine the simulated foreground mass distribution that did the gravitational lensing. Its solution b is a good approximation to the triangle of three isothermal mass spheres for which the lensing had actually been calculated.

Once we have found the lensing center, we can avail ourselves of a more direct method of solving for the mass M and core radius Rc of the dark matter. Assuming a circularly symmetric lens with a neatly parametrizable radial mass distribution, one can determine both M and Rc by a maximum-likelihood calculation. The tangential distortion function T(x,y) locates the lensing mass and determines its morphological shape on the sky.

Assuming that the faint blue arclets we see are gravitationally lensed background galaxies in the redshift range 0.5-1.5, we have obtained maximum-likelihood solutions for a class of plausible lens models. These solutions yield total cluster masses in good agreement with what one deduces from the observed velocity dispersion, and they provide at least an upper bound on the core radius of the dark matter distribution.

We want to study the distribution of dark matter at various stages in the evolution of clusters of galaxies. We therefore seek to obtain deep imaging exposures through a wide variety of rich foreground clusters with various optical and x-ray properties at various redshifts. Our survey of rich clusters will eventually employ CCD mosaics as large as 13 cm on a side, so that we will be able to construct dark matter maps extending more than a megaparsec from the center of the cluster under scrutiny.

Because most cluster galaxies are much redder than the faint background population, we can just about cancel out the foreground by subtracting an appropriately scaled red image from a blue image of the same field. Figure 6 is such a scaled blue-minus-red difference image of the rich cluster Abell 1689 and its surroundings. The faint blue galaxies of the background survive, as do a number of blue foreground stars. But the cluster galaxies have disappeared. We then reduce these difference images to a catalog of background galaxy images by means of automated procedures for detection, image splitting and photometry. 2 - 4 The positions, intensities and second moments of the faint blue galaxies are then passed on to software that calculates the net tangential alignment T(x,y) around any point (x,y) on the image.

 Figure 6. Blue-minus-red difference image for the cluster Abell 1689, whose redshift is 0.18. The intensity of the red-filtered image is adjusted so that subtracting it from the blue-filtered image cancels out the red foreground cluster galaxies while preserving the faint blue background galaxies whose distortion serves to measure dark matter in the foreground. A few bright blue stars in our own immediate neighborhood also survive the subtraction.