We present the modeled counts for the expected Sunyaev-Zel'dovich micr
owave sources associated with clusters of galaxies, predicted for expe
riments with arcminute-scale spatial resolution, assuming self-similar
cluster evolution, for different spectra of the primordial density fl
uctuations and values of the cosmological density parameter OMEGA. Our
simulations show that the source counts should be a powerful test of
the evolution of very high redshift clusters. Experiments with 1'-2' s
patial resolution, with moderate sensitivity but covering a large area
of the sky, would be most effective for studying the SZ source popula
tion. Recent arcminute-scale radio experiments, the OVRO RING experime
nt and VLA deep imaging, achieved sensitivy and sky coverage close to
that needed for the detection of negative sources associated with very
distant clusters. From the absence of cluster detections in these exp
eriments, we rule out, with 90% confidence, models with OMEGA < 0.3 an
d n = +1 as predicting too many bright sources; or there is no hot gas
in clusters more distant than z(max) = 5 in such models. If the singl
e negative source detected in the RING experiment is a distant cluster
, the OMEGA = 1, n = -2 model also may be ruled out as it predicts too
few sources. The new generation of telescopes, including the new SUZI
E and Ryle instruments, will soon be able to detect distant clusters.
The cluster population in the past has been modeled by scaling the obs
erved present-day sample of X-ray clusters back to high redshifts, an
approach which makes the best use of the observed cluster gas paramete
rs, and makes the simulations robust to the assumed evolution at very
early epochs. Although the pure self-similar model may be incompatible
with the variety of observed evolutionary effects, we show that reaso
nable modifications to the intracluster gas history in that model, pro
posed to reconcile the self-similar evolution of cluster mass and the
observed evolution of their X-ray luminosity, do not considerably chan
ge our microwave predictions made using the pure self-similar model. T
hat is, the results of our simulations are applicable to the wide clas
s of evolutionary models in which the cluster gas mass times gas tempe
rature evolves as the dark mass times cluster virial temperature.