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**Session 115 -- Cosmology and Dark Matter***Oral presentation, Saturday, January 15, 10:15-11:45, Salon III Room (Crystal Gateway)*

A new, three dimensional, shock capturing, hydrodynamic code (TVD) is utilized to determine the distribution of hot gas in a variety of cosmological models. Periodic boundary conditions are assumed: boxes with size $85h^{-1}$Mpc having cell size $0.31h^{-1}$Mpc, are followed in a simulation with $270^3=10^{7.3}$ cells. We identify the X-ray emitting clusters in the simulation boxes, compute the luminosity functions at several wavelength bands, the temperature functions and estimated sizes as well as the evolution of these quantities with redshift. Different models yield different results with the differences being far larger than the numerical inaccuracies involved. We find that COBE-normalized $\Omega=1$ CDM model pridicts an order of magnitude more bright X-ray clusters than observed, while a COBE-normalized CDM$+\Lambda$ model ($\Omega=0.45$, $\lambda=0.55$, $h=0.6$, $\Omega_b=0.042$, $\sigma_8=0.77$) and a PBI model ($\Omega=0.3$, $\lambda=0.0$, $h=0.65$, $\Omega_b=0.036$, $\sigma_8=1$) both yield cluster luminosity and temperature functions consistent with observations. The CDM$+\Lambda$ and PBI models predict an order of magnitude decline in the number density of bright ($h\nu = 2-10$keV) clusters from $z=0$ to $z=2$ in contrast to a slight increase in the number density for standard $\Omega=1$ CDM model. We find that the temperatures of clusters in the low $\Omega$ models tend to freeze out at later times while in the $\Omega=1$ CDM model there is a steep increase during the same interval of redshift. Equivalently, we find that $L^*$ of the Schechter fits of cluster luminosity functions peaks at some lower redshift ($z\sim0.3$ in the CDM$+\Lambda$ model, and $z\sim 1.0$ in the PBI model) while in the $\Omega=1$ CDM model it is a monotoically decreasing function of redshift. Both trends should be detectable even with a relatively ``soft" X-ray instrument such as ROSAT, providing a powerful discriminant between $\Omega=1$ and $\Omega<1$ models. Detailed computations of the luminosity functions in the range $L_x=10^{40}-10^{44}$erg/s in various energy bands are presented for both cluster cores ($r\le 0.5h^{-1}$Mpc) and total luminosities ($r<1h^{-1}$Mpc) to be used in comparison with ROSAT and other observational data sets. These show the above noted negative evolution for the lower $\Omega$ models. We find little dependence of core radius on cluster luminosity and a correlation between temperature on luminosity similar to observations in the lower $\Omega$ models. In contrast, the standard $\Omega=1$ model predicted temperatures which were significantly too high. Examining the ratio of gas to total mass in the clusters (which we find to be anti-biased by a factor close to unity in all the models), normalized to $\Omega_b h^2=0.015$, and comparing to observations, we conclude, in agreement with S. White, that the cluster observations argue for an open universe.