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.
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