Abundance of moderate-redshift clusters in the cold plus hot dark matter model

Using a set of $pppm$ simulation which accurately treats the density evolution of two components of dark matter, we study the evolution of clusters in the Cold + Hot dark matter (CHDM) model. The mass function, the velocity dispersion function and the temperature function of clusters are calculated for four different epochs of $zle 0.5$. We also use the simulation data to test the Press-Schechter expression of the halo abundance as a function of the velocity dispersion $sigma_v$. The model predictions are in good agreement with the observational data of local cluster abundances ($z=0$). We also tentatively compare the model with the Gunn and his collaborators' observation of rich clusters at $zapprox 0.8$ and with the x-ray luminous clusters at $zapprox 0.5$ of the {it Einstein} Extended Medium Sensitivity Survey. The important feature of the model is the rapid formation of clusters in the near past: the abundances of clusters of $sigma_vge 700kms$ and of $sigma_vge 1200 kms$ at $z=0.5$ are only 1/4 and 1/10 respectively of the present values ($z=0$). Ongoing ROSAT and AXAF surveys of distant clusters will provide sensitive tests to the model. The abundance of clusters at $zapprox 0.5$ would also be a good discriminator between the CHDM model and a low-density flat CDM model both of which show very similar clustering properties at $z=0$.Comment: 21 pages + 6 figures (uuencoded version of the PS files), Steward Preprints No. 118