Shapes and Alignments of Galaxy Cluster Halos

We present distribution functions and spatial correlations of the shapes of dark matter halos derived from Hubble Volume simulations of a LambdaCDM universe. We measure both position and velocity shapes within spheres encompassing mean density 200 times the critical value, and calibrate small-N systematic errors using Poisson realizations of isothermal spheres and higher resolution simulations. For halos more massive than 3x10^{14} Msun/h, the shape distribution function peaks at (minor/major, intermediate/major) axial ratios of (0.64,0.76) in position, and is rounder in velocity, peaking at (0.72,0.82). Halo shapes are rounder at lower mass and/or redshift; the mean minor axis ratio in position follows <c/a>(M,z) = c_{15,0} [1-\alpha\ln(M/10^{15}Msun/h)] (1+z)^{-\epsilon}, with c_{15,0}=0.631 \pm 0.001, \alpha=0.023 \pm 0.002 and \epsilon=0.086 \pm 0.004. Position and velocity principal axes are well aligned in direction, with median alignment angle $22^\circ$, and the axial ratios in these spaces are correlated in magnitude. We investigate mark correlations of halo pair orientations using two measures: a simple scalar product shows $\ge 1$ alignment extending to $30 \hinv \mpc$ while a filamentary statistic exhibits non-random alignment extending to scales $\sims 200 \hinv \mpc$, ten times the sample two-point correlation length and well into the regime of negative two-point correlation. Cluster shapes are unaffected by the large-scale environment; the shape distribution of supercluster members is consistent with that of the general population.