  We forecast constraints on the amplitude of matter clustering sigma_8(z)
achievable with the combination of cluster weak lensing and number counts, in
current and next-generation weak lensing surveys. We advocate an approach,
analogous to galaxy-galaxy lensing, in which the observables in each redshift
bin are the mean number counts and the mean weak lensing profile of clusters
above a mass proxy threshold. The primary astrophysical nuisance parameter is
the logarithmic scatter between the mass proxy and true mass near the
threshold. For surveys similar to the Dark Energy Survey (DES), the Roman Space
Telescope High Latitude Survey (HLS), and the Rubin Observatory Legacy Survey
of Space and Time (LSST), we forecast aggregate precision on sigma_8 of 0.26%,
0.24%, and 0.10%, respectively, if the mass-observable scatter has an external
prior better than 0.01. These constraints would be degraded by about 20% for a
0.05 prior on scatter in the case of DES or HLS and for a 0.016 prior for LSST.
A one-month observing program with Roman Space Telescope targeting
approximately 2500 massive clusters could achieve a 0.5% constraint on
sigma_8(z=0.7) on its own, or a ~0.33% constraint in combination with the HLS.
Realizing the constraining power of clusters requires accurate knowledge of the
mass-observable relation and stringent control of systematics. We provide
analytic approximations to our numerical results that allow easy scaling to
other survey assumptions or other methods of cluster mass estimation.
