Evolution of clusters in cold plus hot dark matter models

We use N-body simulations to study the evolution of galaxy clusters over the redshift interval 0 less than or equal to z less than or equal to 0.5 in cosmological models with a mixture of cold and hot dark matter. Five different techniques are utilized: the cluster-cluster correlation function, axial ratios and quadrupoles of the dark matter distribution in individual clusters, virial properties, and density profiles. We find that the correlation function for clusters of the same mass limit was larger and steeper at high redshifts. The slope increases from 1.8 at z = 0 to 2.1 at z = 0.5. The comoving correlation length r(c) scales with the mass limit M within a comoving radius 1.5 h(-1) Mpc and the redshift z as r(c) approximate to 20(1 + z)(M/M*)(1/3), where M* = 3 x 10(14) h(-1) M(.). When the correlation length is normalized to the mean cluster separation d(c), it remains almost constant: r(c) approximate to (0.45-0.5)d(c). For clusters of small masses (M < 2 x 10(14) h(-1) M(.)), there is an indication that r(c) goes slightly above this relation, with the constant of proportionality being similar to 0.55-0.6. Anisotropy of density distribution in a cluster shows no change with redshift, with axial ratios remaining constant at similar to 1.2. In other words, clusters at present are as elongated as they were at z = 0.5. While the anisotropy of clusters does not change with time, the density profile shows visible evolution: the slope of the density profile changes from gamma approximate to -3.5 at z = 0.5 to gamma approximate to -2.5 at the present. We find that the core of a cluster remains essentially the same over time but the density of the outlying regions increases noticeably. The virial relation M similar to v(2) is a good approximation, but there is a large fraction of clusters with velocity dispersions greater than given by this relation, and clusters with the same rms velocities have smaller masses in the past, by a factor of 2 at z = 0.5.