  We present a comparison of X-ray and optical luminosities and luminosity
functions of cluster candidates from a joint optical/X-ray survey, the ROSAT
Optical X-Ray Survey. Completely independent X-ray and optical catalogs of 23
ROSAT fields (4.8 deg2) were created by a matched-filter optical algorithm and
by a wavelet technique in the X-ray. We directly compare the results of the
optical and X-ray selection techniques. The matched-filter technique detected
74% (26 out of 35) of the most reliable cluster candidates in the
X-ray-selected sample; the remainder could be either constellations of X-ray
point sources or z>1 clusters. The matched-filter technique identified
approximately 3 times the number of candidates (152 candidates) found in the
X-ray survey of nearly the same sky (57 candidates). While the estimated
optical and X-ray luminosities of clusters of galaxies are correlated, the
intrinsic scatter in this relationship is very large. We can reproduce the
number and distribution of optical clusters with a model defined by the X-ray
luminosity function and by an LX Lambda cl relation if H0=75 km s-1 Mpc-1 and
if the LX Lambda cl relation is steeper than the expected LX Lambda 2cl. On
statistical grounds, a bimodal distribution of X-ray luminous and X-ray faint
clusters is unnecessary to explain our observations. Follow-up work is required
to confirm whether the clusters without bright X-ray counterparts are simply
X-ray faint for their optical luminosity because of their low mass or youth or
are a distinct population of clusters that do not, for some reason, have dense
intracluster media. We suspect that these optical clusters are low-mass
systems, with correspondingly low X-ray temperatures and luminosities, or that
they are not yet completely virialized systems.
