Statistical measures of large-scale structure

To quantify clustering in the large-scale distribution of galaxies and to test theories for the formation of structure in the universe, we apply statistical measures to the Center for Astrophysics Redshift Survey. This survey is complete to the magnitude limit {dollar}msb{lcub}B(rm o){rcub}{dollar} = 15.5 over two contiguous regions which cover one-quarter of the sky and include {dollar}sim{dollar}11,000 galaxies. The salient features of these data are voids with diameter 30-50{dollar}hsp{lcub}-1{rcub}{dollar} Mpc and coherent dense structures with a scale {dollar}sim{dollar}100{dollar}hsp{lcub}-1{rcub}{dollar} Mpc.; We compute the power spectrum of galaxy density fluctuations on scales up to {dollar}sim{dollar}200{dollar}hsp{lcub}-1{rcub}{dollar} Mpc. The redshift-space power spectrum has slope {dollar}n approx -2.1{dollar} on scales {dollar}<{dollar}25{dollar}hsp{lcub}-1{rcub}{dollar} Mpc and slope {dollar}n approx -1.1{dollar} on scales {dollar}30 < lambda < 100hsp{lcub}-1{rcub}{dollar} Mpc. Comparison with N-body simulations of cosmological models rules out the "standard" Cold Dark Matter model {dollar}(Omega = 1, b = 1.5{dollar}, {dollar}sigmasb8 = 1){dollar} at the 99% confidence level because this model has insufficient power on scales {dollar}lambda > 30hsp{lcub}-1{rcub}{dollar} Mpc. An unbiased open universe CDM model {dollar}(Omega h = 0.2){dollar} and a biased CDM model with non-zero cosmological constant {dollar}(Omega h = 0.24, lambdasb0 = 0.6){dollar} match the observed power spectrum. The amplitude of the power spectrum depends on the luminosity of galaxies in the sample; bright {dollar}(L > Lsp*){dollar} galaxies are more strongly clustered than faint galaxies. The paucity of bright galaxies in low-density regions may explain this dependence.; To measure the topology of large-scale structure, we compute the genus of isodensity surfaces of the smoothed density field. The shape and amplitude of the genus-threshold density relation measure phase coherence in these data. On scales in the "non-linear" regime, {dollar}le{dollar}10{dollar}hsp{lcub}-1{rcub}{dollar} Mpc, the high- and low-density regions are multiply-connected over a broad range of density threshold, as in a filamentary net. On smoothing scales {dollar}>{dollar}10{dollar}hsp{lcub}-1{rcub}{dollar} Mpc, the topology is consistent with statistics of a Gaussian random field. Simulations of CDM models fail to produce the observed coherence of structure on non-linear scales ({dollar}>{dollar}95% confidence level).; We compute the Void Probability Function and the underdensity probability, which measures the frequency of regions with density contrast {dollar}deltarho/barrho = -0.8.{dollar} These statistics depend on the entire hierarchy of n-point correlation functions. The redshift-space VPF is consistent with hierarchical scaling models on non-linear scales, {dollar}le{dollar}10{dollar}hsp{lcub}-1{rcub}{dollar} Mpc. However, tests with N-body simulations show that evidence for hierarchical scaling in redshift space does not necessarily imply scaling behavior in real space. The underdensity probability depends strongly on the luminosity of galaxies; underdense regions are significantly more common {dollar}(>{dollar}2{dollar}sigma){dollar} in bright {dollar}(L > Lsp*){dollar} galaxy samples than in samples which include fainter galaxies. CDM simulations match the observed void statistics of bright galaxies, but produce voids which are too empty to match the distribution of faint galaxies.