A critical test of the successful Lambda-CDM picture for structure formation is the measurement of the power law exponent, gamma, of the centre of dark matter density profiles, predicted to lie in the range 1.0-1.5. Measurements of gamma derived from rotation curves of LSB galaxies appear to contradict CDM, but rely on assumptions that are difficult to verify (e.g. axisymmetry). We have recently demonstrated, using our new `semi-linear' inversion method, how strong gravitational lensing by galaxies can provide a clean and accurate measurement of gamma, free of such ambiguities. HST images of lensed non-AGN galaxies provide hundreds of resolution elements, each a constraint on the mass profile. Such lenses are exceedingly rare, but we have recently discovered new systems. We propose deep ACS-HRC observations of 3 systems to measure gamma in each, accurate to 0.15 (95% confidence) and to obtain an indication of its variation between galaxies. To establish the required number of orbits we have undertaken an end-to-end simulation of the problem, creating and analysing synthetic ACS images. Additionally the semi-linear method simultaneously reconstructs the pixelised source surface brightness distribution. Our simulations demonstrate that the fine sampling and small pixel scattering of the HRC, resolves the morphology of the sources with exquisite detail.

We will use ACS to observe the 3 Einstein ring targets in the F606W filter, for a total of 10 orbits on 0047-2808, 4 orbits on 1459-0055, and 4 orbits on 2309-0039. The feasibility of the experiment depends on; 1) the S/N of flux in the ring image, 2) PSF and CCD smearing of the image (ie. choice of camera), 3) the efficiency of our inversion algorithm to make optimal use of the information in the image. To properly address these effects, we have undertaken an extensive set of simulations to model the entire end-to-end problem for different integration times using both the HRC and WFC. The simulations are calibrated by the flux measured in our F555W WFPC2 images of 0047-2808 and verified by the Exposure Time Calculator. We allow for the noise characteristics, improved throughput, image pixel scale and image smearing (CCD+PSF) as given in the ACS handbook. The preferred filter for our deepest image is the F606W which includes the wavelength range covered by F555W, but is twice as broad. In the case of 0047-2808, because of the strong Lyman-alpha redshifted emission (lambda = 5588 Angstrom), we estimate that the ring flux in the broader filter is 1.7 times as great, while the flux from the lens galaxy and from the sky is twice as great. In order to discriminate between the currently popular competing theoretical dark matter profile models, our simulations show that we need the superior resoltuion of the HRC and the orbit allocations per target given above. With fewer orbits, this error grows relatively quickly whereas increasing the number of orbits does not reduce the uncertainty enough to justify the extra time. In contrast, for 0047-2808, 10 orbits with the WFC will give an error of 0.23 at 95% confidence. Although this is only ~50% larger than with the HRC, the higher resolution of the HRC will reveal far more detail in the reconstructed source. The Vega R-band brightnesses of the lensed sources of our targets are 0047-2808: 22.3mag, 1459-0055: 21.1mag, and 2309-0039: 21.0mag. On this basis we estimate that 4 orbits for the two brighter sources will provide similar constraints on the dark matter lens profile slope as the computed 10 orbits for 0047-2808. In reality the limits achieved will depend on the detailed structure revealed. Our inversion algorithm works optimally when pixels in the stacked image are independent of each other. For this reason, we require the same orientation per target. We have no restrictions on the absolute orientation of each target. We have split our orbits up into visits of 2 and 3 orbits to facilitate scheduling. 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