  High-resolution N-body simulations of four popular Cold Dark Matter
cosmologies (LCDM, OCDM, QCDM, and tilted SCDM), each containing 10^5 clusters
of galaxies in a cubic gigaparsec volume, are used to determine the evolution
of the cluster mass function from z=3 to z=0. The large volume and high
resolution of these simulations allow an accurate measure of the evolution of
cosmologically important (but rare) massive clusters at high redshift. The
simulated mass function is presented for cluster masses within several radii
typically used observationally (R=0.5, 1.0, and 1.5 Mpc/h, both comoving and
physical) in order to enable direct comparison with current and future
observations. The simulated evolution is compared with current observations of
massive clusters at redshifts 0.3<z<0.8. The Omega_m=1 tilted SCDM model, which
exhibits very rapid evolution of the cluster abundance, produces too few
clusters at z>0.3 and no massive clusters at z>0.5, in stark contradiction with
observations. The Omega_m=0.3 models- LCDM, OCDM, and QCDM- all exhibit
considerably weaker evolution and are consistent with current data. Among these
low density models, OCDM evolves the least. These trends are enhanced at high
redshift and can be used to discriminate between flat and open low density
models. The simulated mass functions are compared with the Press-Schechter
approximation. Standard Press-Schechter predicts too many low mass clusters at
z=0, and too few clusters at higher redshift. We modify the approximation by a
simple parameterization of the density contrast threshold for collapse, which
has a redshift dependence. This modified Press-Schechter approximation provides
a good fit to the simulated mass functions.
