Gravitational lensing of distant field galaxies by rich clusters - I. Faint galaxy redshift distributions

{}From deep optical images of three clusters selected by virtue of theirX-ray luminosity and/or optical richness (1455+22; $z=0.26$, 0016+16; $z=0.55$and 1603+43; $z=0.89$), we construct statistically-complete samples of faintfield galaxies ($I \leq 25$) suitable for probing the effects of gravitationallensing. By selecting clusters across a wide redshift range we separate theeffects of the mean redshift distribution of the faint field population wellbeyond spectroscopic limits and the distribution of dark matter in the lensingclusters. A significant lensing signature is seen in the two lower redshiftclusters whose X-ray properties are well-constrained. Based on these anddynamical data, it is straightforward to rule out field redshift distributionsfor $I \leq 25$ which have a significant low redshift excess compared to the noevolution prediction, such as would be expected if the number counts at faintlimits were dominated by low-$z$ dwarf systems. The degree to which we canconstrain any high redshift tail to the no evolution redshift distributiondepends on the distribution of dark matter in the most distant lensing cluster.In the second paper in this series, we use the lensing signal to reconstructthe full two-dimensional mass distribution in the clusters and, together withhigh resolution X-ray images, demonstrate that their structural properties arewell-understood. The principal result is therefore the absence of a dominantlow-$z$ dwarf population to $I \leq25$.Comment: 13 pages (no figures), LaTex (MN style), postscript figures available via anonymous ftp in users/irs/figs/paper1.tar.gz on astro.caltech.edu, PAL-IRS-