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We explore the kinematics of 36 rich RGH89 groups identified from the first two complete slices of the CfA redshift survey. These groups have more than five members identified by a friends-of-friends algorithm at a number density contrast delta rho/rho greater than or equal to 80. To examine the stability of the determination of the velocity dispersion for these systems, we compare results for the original 232 members with results for a larger redshift sample, including 334 fainter members in the redshift neighborhoods. On average, we double the number of group members in each system. The observed distribution of velocity dispersions is stable. In fact, the velocity dispersion based on the original members identified in the CfA redshift survey is a reliable predictor of the value for the enlarged sample in an individual group. The velocity dispersion is thus a stable physical parameter for discrimination among systems galaxies. A larger sample of groups, particularly one selected from a distance limited catalog, should provide an interesting constraint on models for the formation of large-scale structure. We take H(sub 0) = km/s/Mpc.

We have selected 36 loose groups of galaxies (RGH89) with at least five members, and with mean redshift average value of CZ is greater than 3200 km/s. These groups all lie within the first two slices of the CfA redshift survey 8(sup h) less than or equal to alpha less than or equal to 17(sup h) and 26.5 deg less than or equal to delta less than or equal to 38.5 deg). For each of these groups, we define the redshift-space neighborhood as a region centered on the group coordinates and delimited by a circle of projected radius R(sub cir) = 1.5/h Mpc on the sky, and by a velocity interval delta (sub cz) = 3000 km/s. Here we give the redshifts of 334 galaxies in these redshift-space neighborhoods. For completeness, we also give the redshifts of the 232 original members. These data include 199 new redshifts. We demonstrate that these samples of fainter galaxies significantly increase the number of members.

We compute the power spectrum of galaxy density fluctuations in a recently completed Southern Sky Redshift Survey of optically selected galaxies (SSRS2). The amplitude and shape of the SSRS2 power spectrum are consistent with results of the Center for Astrophysics redshift survey of the northern hemisphere (CfA2), including the abrupt change of slope on a scale of 30-50/h Mpc; these results are reproducible for independent volumes of space, and variations are consistent with the errors estimated from mock surveys. Taken together, the SSRS2 and the CfA2 form a complete sample of 14,383 galaxies which covers one-third of the sky. The power spectrum of this larger sample continues to rise on scales up to approximately 200/h Mpc, with weak evidence for flattening on the largest scales. The SSRS2 + CfA2 power spectrum and the power spectrum constraints implied by COBE are well matched by an Omega(h) is approximately 0.2, Omega + lambda(sub 0) = 1 cold dark matter model with minimal biasing of optically selected galaxies.

Void statistics of the galaxy distribution in the Center for Astrophysics Redshift Survey provide strong constraints on galaxy clustering in the nonlinear regime, i.e., on scales R equal to or less than 10/h Mpc. Computation of high-order moments of the galaxy distribution requires a sample that (1) densely traces the large-scale structure and (2) covers sufficient volume to obtain good statistics. The CfA redshift survey densely samples structure on scales equal to or less than 10/h Mpc and has sufficient depth and angular coverage to approach a fair sample on these scales. In the nonlinear regime, the void probability function (VPF) for CfA samples exhibits apparent agreement with hierarchical scaling (such scaling implies that the N-point correlation functions for N greater than 2 depend only on pairwise products of the two-point function xi(r)) However, simulations of cosmological models show that this scaling in redshift space does not necessarily imply such scaling in real space, even in the nonlinear regime; peculiar velocities cause distortions which can yield erroneous agreement with hierarchical scaling. The underdensity probability measures the frequency of 'voids' with density rho less than 0.2 -/rho. This statistic reveals a paucity of very bright galaxies (L greater than L asterisk) in the 'voids.' Underdensities are equal to or greater than 2 sigma more frequent in bright galaxy samples than in samples that include fainter galaxies. Comparison of void statistics of CfA samples with simulations of a range of cosmological models favors models with Gaussian primordial fluctuations and Cold Dark Matter (CDM)-like initial power spectra. Biased models tend to produce voids that are too empty. We also compare these data with three specific models of the Cold Dark Matter cosmogony: an unbiased, open universe CDM model (omega = 0.4, h = 0.5) provides a good match to the VPF of the CfA samples. Biasing of the galaxy distribution in the 'standard' CDM model (omega = 1, b = 1.5; see below for definitions) and nonzero cosmological constant CDM model (omega = 0.4, h = 0.6 lambda(sub 0) = 0.6, b = 1.3) produce voids that are too empty. All three simulations match the observed VPF and underdensity probability for samples of very bright (M less than M asterisk = -19.2) galaxies, but produce voids that are too empty when compared with samples that include fainter galaxies.

We describe and apply a method for directly computing the power spectrum for the galaxy distribution in the extension of the Center for Astrophysics Redshift Survey. Tests show that our technique accurately reproduces the true power spectrum for k greater than 0.03 h Mpc(exp -1). The dense sampling and large spatial coverage of this survey allow accurate measurement of the redshift-space power spectrum on scales from 5 to approximately 200 h(exp -1) Mpc. The power spectrum has slope n approximately equal -2.1 on small scales (lambda less than or equal 25 h(exp -1) Mpc) and n approximately -1.1 on scales 30 less than lambda less than 120 h(exp -1) Mpc. On larger scales the power spectrum flattens somewhat, but we do not detect a turnover. Comparison with N-body simulations of cosmological models shows that an unbiased, open universe CDM model (OMEGA h = 0.2) and a nonzero cosmological constant (CDM) model (OMEGA h = 0.24, lambda(sub zero) = 0.6, b = 1.3) match the CfA power spectrum over the wavelength range we explore. The standard biased CDM model (OMEGA h = 0.5, b = 1.5) fails (99% significance level) because it has insufficient power on scales lambda greater than 30 h(exp -1) Mpc. Biased CDM with a normalization that matches the Cosmic Microwave Background (CMB) anisotropy (OMEGA h = 0.5, b = 1.4, sigma(sub 8) (mass) = 1) has too much power on small scales to match the observed galaxy power spectrum. This model with b = 1 matches both Cosmic Background Explorer Satellite (COBE) and the small-scale power spect rum but has insufficient power on scales lambda approximately 100 h(exp -1) Mpc. We derive a formula for the effect of small-scale peculiar velocities on the power spectrum and combine this formula with the linear-regime amplification described by Kaiser to compute an estimate of the real-space power spectrum. Two tests reveal luminosity bias in the galaxy distribution: First, the amplitude of the pwer spectrum is approximately 40% larger for the brightest 50% of galaxies in volume-limited samples that have M(sub lim) greater than M*. This bias in the power spectrum is independent of scale, consistent with the peaks-bias paradigm for galaxy formation. Second, the distribution of local density around galaxies shows that regions of moderate and high density contain both very bright (M less than M* = -19.2 + 5 log h) and fainter galaxies, but that voids preferentially harbor fainter galaxies (approximately 2 sigma significance level).

We derive the luminosity function for different morphological types in the original CfA Redshift Survey (CfA1) and in the first two slices of the CfA Redshift Survey Extension (CfA2). CfA1 is a complete sample containing 2397 galaxies distributed over 2.7 steradians with m(sub z) less than or equal 14.5. The first two complete slices of CfA2 contain 1862 galaxies distributed over 0.42 steradians with m(sub z)=15.5. The shapes of the E-S0 and spiral luminosity functions (LF) are indistinguishable. We do not confirm the steeply decreasing faint end in the E-S0 luminosity function found by Loveday et al. for an independent sample in the southern hemisphere. We demonstrate that incomplete classification in deep redshift surveys can lead to underestimates of the faint end of the elliptical luminosity function and could be partially responsible for the difference between the CfA survey and other local field surveys. The faint end of the LF for the Magellanic spirals and irregulars is very steep. The Sm-Im luminosity function is well fit by a Schechter function with M*=-18.79, alpha=-1.87, and phi*=0.6x10(exp -3) for M(sub z) less than or equal to -13. These galaxies are largely responsible for the excess at the faint end of the general CfA luminosity function. The abundance of intrinsically faint, blue galaxies nearby affects the interpretation of deep number counts. The dwarf population increases the expected counts at B=25 in a no-evolution, q(sub 0)=0.05 model by a factor of two over standard no-evolution estimates. These dwarfs change the expected median redshift in deep redshift surveys by less than 10 percent . Thus the steep Sm-Im LF may contribute to the reconciliation of deep number counts with deep redshift surveys.

We compare the combined distribution of 31 group and 25 cluster velocity dispersions with the ensemble of 32 models for the formation and evolution of large-scale structure examined by Weinberg & Cole (1992). The models include Gaussian and non-Gaussian initial fluctuations, different power law spectra (n = -1, n = 0, n = -2, 'pancake'), flat (Omega = 1) and open (Omega = 0.2) cosmologies, and unbiased (b(sub 8) = 1) and biased (b(sub 8) = 2) galaxy formation. The set of initial conditions we test, although limited, samples enough parameter space to indicate which general classes of models are consistent with the data. The two Gaussian, n = -1 models which best approximate the standard and open Cold Dark Matter (CDM) models do not match the observed distribution of velocity dispersions; models with b(sub 8) = 2 and Omega = 1 ('standard') or b(sub 8) = 1 and Omega = 0.2 ('open') predict too large a ratio of low to high velocity dispersion systems. A 'COBE-normalized' CDM model with b(sub 8) = 1 and Omega = 1 produces clusters with velocity dispersions higher than those measured. All three models overestimate the total abundance of systems.

We use complete redshift surveys to study the redshift neighborhoods of 38 Hickson compact groups (HCGs). Twenty-nine of these HCGs (76%) are embedded in rich looser systems which we call HCG associations. Analysis of the redshift neighborhood of HCGs outside the CfA survey suggests that most HCGs are embedded in more extended physical systems. Rich loose groups extracted from the CfA survey (Ramella et al. (1994)) have physical properties similar to those of the HCG associations. These rich loose groups often contain compact configurations. N-body experiments (Diaferio (1994)) suggest that compact configurations analogous to HCGs form continually during the collapse of rich loose groups. These observational and numerical results suggest that rich loose groups are the birthplace of HCGs.

The small crossing time of compact groups of galaxies (t(sub cr)H(sub 0) approximately less than 0.02) makes it hard to understand why they are observable at all. Our dissipationless N-body simulations show that within a single rich collapsing group compact groups of galaxies continually form. The mean lifetime of a particular compact configuration if approximately 1 Gyr. On this time scale, members may merge and/or other galaxies in the loose group may join the compact configuration. In other words, compact configurations are continually replaced by new systems. The frequency of this process explains the observability of compact groups. Our model produces compact configurations (compact groups (CG's) with optical properties remarkably similar to Hickson's (1982) compact groups (HCG's): (1) CG's have a frequency distribution of members similar to that of HCG's; (2) CG's are approximately equals 10 times as dense as loose groups; (3) CG's have dynamical properties remarkably similar to those of HCG's; (4) most of the galaxy members of CG's are not merger remnants. The crucial aspect of the model is the relationship between CG's and the surrounding rich loose group. Our model predicts the frequency of occurrence of CG's. A preliminary analysis of 18 rich loose groups is consistent with the model prediction. We suggest further observational tests of the model.

We have measured 125 redshifts in 31 groups of galaxies observed with Einstein, and have compiled an additional 543 redshifts from the literature. There is a correlation between galaxy surface density and group velocity dispersion, with mu is proportional to sigma(exp 1.6 +/- 0.6), but the scatter about this relation is large. We examine the relationship between the group x-ray luminosity in the 0.3-3.5 keV band and the measured velocity dispersion. Richer groups follow the same relation as rich clusters (1982) with L(sub X) proportional to sigma(exp 4.0 +/- 0.6), but the relation flattens for lower luminosity systems which have velocity dispersions below 300 km/s. We suggest that the L(sub x)-sigma relation arises from a combination of extended cluster emission and emission associated with individual galaxies. The x-ray emission for the richer groups is dominated by emission from the intragroup medium, as for the richer clusters; emission from the poorer clusters is dominated by less extended emission associated with the individual group galaxies.

We study the topology of large-scale structure in the Center for Astrophysics Redshift Survey, which now includes approximately 12,000 galaxies with limiting magnitude m(sub B) is less than or equal to 15.5. The dense sampling and large volume of this survey allow us to compute the topology on smoothing scales from 6 to 20/h Mpc; we thus examine the topology of structure in both 'nonlinear' and 'linear' regimes. On smoothing scales less than or equal to 10/h Mpc this sample has 3 times the number of resolution elements of samples examined in previous studies. Isodensity surface of the smoothed galaxy density field demonstrate that coherent high-density structures and large voids dominate the galaxy distribution. We compute the genus-threshold density relation for isodensity surfaces of the CfA survey. To quantify phase correlation in these data, we compare the CfA genus with the genus of realizations of Gaussian random fields with the power spectrum measured for the CfA survey. On scales less than or equal to 10/h Mpc the observed genus amplitude is smaller than random phase (96% confidence level). This decrement reflects the degree of phase coherence in the observed galaxy distribution. In other words the genus amplitude on these scales is not good measure of the power spectrum slope. On scales greater than 10/h Mpc, where the galaxy distribution is rougly in the 'linear' regime, the genus ampitude is consistent with the random phase amplitude. The shape of the genus curve reflects the strong coherence in the observed structure; the observed genus curve appears broader than random phase (94% confidence level for smoothing scales less than or equal to 10/h Mpc) because the topolgoy is spongelike over a very large range of density threshold. This departre from random phase consistent with a distribution like a filamentary net of 'walls with holes.' On smoothing scales approaching approximately 20/h Mpc the shape of the CfA genus curve is consistent with random phase. There is very weak evidence for a shift of the genus toward a 'bubble-like' topology. To test cosmological models, we compute the genus for mock CfA surveys drawn from large (L greater than or approximately 400/h Mpc) N-body simulations of three variants of the cold dark matter (CDM) cosmogony. The genus amplitude of the 'standard' CDM model (omega h = 0.5, b = 1.5) differs from the observations (96% confidence level) on smoothing scales is less than or approximately 10/h Mpc. An open CDM model (omega h = 0.2) and a CDM model with nonzero cosmological constant (omega h = 0.24, lambda (sub 0) = 0.6) are consistent with the observed genus amplitude over the full range of smoothing scales. All of these models fail (97% confidence level) to match the broadness of the observed genus curve on smoothing scales is less than or equal to 10/h Mpc.

We have measured redshifts in the fields of the 31 R greater than 1 Abell clusters with z of 0.02-0.05 and absolute b exp II above 30 deg. At least ten of the fields are severely contaminated by superimposed velocity peaks. We derive the mean velocities and velocity dispersions of the 25 dense peaks in the sample. The abundance of peaks, 6.6 x 10 exp -6 h-cubed Mpc exp -3, is consistent with the mean number density of R above 1 Abell clusters. The range of velocity dispersions is 304-1346 km/s. The median dispersion is 718 km/s. The subset of eight systems with cD galaxies has a median velocity dispersion of 792 km/s, close to that of non-cD systems (626 km/s). When these data are combined with 16 cD cluster velocity dispersions from our previous study (Zabludoff et al., 1990, or ZHG) and Dunn (1991), eight of 25 cD galaxies have peculiar motions larger than half the cluster velocity dispersions. These findings further support the conclusions of Beers and Geller (1983), ZHG, and Dunn (1991), who argue that cD galaxies do not lie in the global kinematic center, but in local potential minima. If so, systems with speeding cD's are probably a guide to substructure in dynamically evolving systems.

We use the moments of the X-ray surface brightness distribution to constrain the dynamical state of a galaxy cluster. Using X-ray observations from the Einstein Observatory IPC, we measure the first moment FM, the ellipsoidal orientation angle, and the axial ratio at a sequence of radii in the cluster. We argue that a significant variation in the image centroid FM as a function of radius is evidence for a nonequilibrium feature in the intracluster medium (ICM) density distribution. In simple terms, centroid shifts indicate that the center of mass of the ICM varies with radius. This variation is a tracer of continuing dynamical evolution. For each cluster, we evaluate the significance of variations in the centroid of the IPC image by computing the same statistics on an ensemble of simulated cluster images. In producing these simulated images we include X-ray point source emission, telescope vignetting, Poisson noise, and characteristics of the IPC. Application of this new method to five Abell clusters reveals that the core of each one has significant substructure. In addition, we find significant variations in the orientation angle and the axial ratio for several of the clusters.

Groups are systems of galaxies with crossing times t(cr) much smaller than the Hubble time. Most of them have t(cr) less than 0.1/H0. The usual interpretation is that they are in virial equilibrium. We compare the data of the group catalog selected from the CfA redshift survey extension with different N-body models. We show that the distributions of kinematic and dynamical quantities of the groups in the CfA catalog can be reproduced by a single collapsing group observed along different line of sights. This result shows that (1) projection effects dominate the statistics of these systems, and (2) observed groups of galaxies are probably still in the collapse phase.

The characteristic diameter of the most prominent void in the redshift survey of de Lapparent et al. (1986) is measured. Distances and peculiar velocities to individual galaxies are derived, and it is shown that the void is approximately a 'Hubble Bubble' in which the near and far edges are separating with the general expansion of the universe. At the 3 sigma level, infall toward the Coma cluster is detected for a portion of the bubble wall. Limits on the net outflow from the void and infall into Coma are used to estimate Omega.

The power spectrum of the galaxy distribution in the Center for Astrophysics Redshift Survey (de Lapparent et al., 1986; Geller and Huchra, 1989; and Huchra et al., 1992) is measured up to wavelengths of 200/h Mpc. Results are compared with several cosmological simulations with Gaussian initial conditions. It is shown that the power spectrum of the standard CDM model is inconsistent with the observed power spectrum at the 99 percent confidence level.

Redshifts are acquired for a complete sample of 351 Abell clusters with tenth-ranked galaxy magnitudes (m10) less than or equal to 16.5, including 115 entirely new cluster redshifts. Analysis of the spatial distribution of these clusters reveals no clustering on scales larger than 75/h Mpc. The correlation length is 20.0 (+/-4.3)/h Mpc, consistent with the results from other surveys. The frequency of voids with radii of order 60/h Mpc or less is consistent with the form and amplitude of the observed two-point correlation function. There is no significant difference between the clustering properties of clusters with RC = 0 and RC not less than 1. A percolation analysis yields 23 superclusters, 17 of which are new. The superclusters are not significantly elongated in the radial direction; large-scale peculiar motions are of order 1000 km/s or less.

The galaxy distribution within the 'Great Wall', the most striking feature in the first three 'slices' of the CfA redshift survey extension is examined. The Great Wall is extracted from the sample and is analyzed by counting galaxies in cells. The 'local' two-point correlation function within the Great Wall is computed and the local correlation length, is estimated 15/h Mpc, about 3 times larger than the correlation length for the entire sample. The redshift distribution of galaxies in the pencil-beam survey by Broadhurst et al. (1990) shows peaks separated about by large 'voids', at least to a redshift of about 0.3. The peaks might represent the intersections of their about 5/h Mpc pencil beams with structures similar to the Great Wall. Under this hypothesis, sampling of the Great Walls shows that l approximately 12/h Mpc is the minimum projected beam size required to detect all the 'walls' at redshifts between the peak of the selection function and the effective depth of the survey.

Clustering properties of two samples from the CfA redshift survey, each containing about 2500 galaxies, are studied. A comparison of the velocity distributions via a K-S test reveals structure on scales comparable with the extent of the survey. The void probability function (VPF) is employed for these samples to examine the structure and to test for scaling relations in the galaxy distribution. The galaxy correlation function is calculated via moments of galaxy counts. The shape and amplitude of the correlation function roughly agree with previous determinations. The VPFs for distance-limited samples of the CfA survey do not match the scaling relation predicted by the hierarchical clustering models. On scales not greater than 10/h Mpc, the VPFs for these samples roughly follow the hierarchical pattern. A variant of the VPF which uses nearly all the data in magnitude-limited samples is introduced; it accounts for the variation of the sampling density with velocity in a magnitude-limited survey.

1D, 2D, and 3D simulations are used here to investigate the salient features in the evolution of void-filled cosmological structures in universes with arbitrary values of Omega. It is found that the growth of a void as a function of time decreases significantly at the time corresponding to Omega = 0.5. In models constructed in 2D and 3D, suitable initial conditions lead to cellular structure with faceted voids similar to those observed in redshift surveys. Matter compressed to planes flows more rapidly toward condensations at the intersections than would be expected for spherical infall. The peculiar streaming velocities for void diameters of 5000 km/s should be observable. The simulations provide a more physical basis and dynamics for the bubbly and Voronois tesselation models used to derive statistical properties of cellular large-scale structure.

Variations of the counts-in-cells with cell size are used here to define two statistical measures of large-scale clustering in three 6 deg slices of the CfA redshift survey. A percolation criterion is used to estimate the filling factor which measures the fraction of the total volume in the survey occupied by the large-scale structures. For the full 18 deg slice of the CfA redshift survey, f is about 0.25 + or - 0.05. After removing groups with more than five members from two of the slices, variations of the counts in occupied cells with cell size have a power-law behavior with a slope beta about 2.2 on scales from 1-10/h Mpc. Application of both this statistic and the percolation analysis to simulations suggests that a network of two-dimensional structures is a better description of the geometry of the clustering in the CfA slices than a network of one-dimensional structures. Counts-in-cells are also used to estimate at 0.3 galaxy h-squared/Mpc the average galaxy surface density in sheets like the Great Wall.

Present-day understanding of the large-scale galaxy distribution is reviewed. The statistics of the CfA redshift survey are briefly discussed. The need for deeper surveys to clarify the issues raised by recent studies of large-scale galactic distribution is addressed.

Redshifts have been measured for a complete sample of 145 Abell clusters of galaxies of median redshift 0.16 covering a 561 sq deg region at high Galactic latitude. Clustering on scales greater than about 70/h Mpc cannot be detected. A large void of diameter about 20,000 km/s seen in maps of the cluster distribution is consistent with smaller scale behavior of the correlation function of the survey. Large-scale peculiar motions are about 1000 km/s or less, a limit inconsistent with the claimed detection by Bahcall et al. (1986) of peculiar motions of about 2000 km/s.

Redshifts were measured for 96 galaxies in right ascension alpha between 8h and 17h declination delta between 30 and 31 deg, and with m(Zwicky) in the range 15.6-15.7. These correspond to 94 of the 96 entries in the Zwicky-Nilson merged catalog. The declination range delta between 29 deg and 31 deg is now complete to m(Zwicky) = 15.7. The structures in the first 6-deg-wide slice of the Center for Astrophysics redshift survey slice (delta between 26.5 and 32.5 deg are clearly defined in the 2-deg-wide slightly deeper sample; the fainter galaxies trace the structures defined by the brighter ones.

Velocity histograms, galaxy positions, and velocity dispersions are presented for 69 nearby Abell clusters. The shape of the cumulative distribution for R = 1 or more clusters does not match the predictions of standard CDM models for any biasing parameter b. The only models consistent with the median and maximum dispersions of the sample are those of b about 1.6-2.0 and in which clusters are identified in three dimensions. Velocity dispersions of high-redshift systems appear fundamentally different from those of nearby clusters. The median dispersion of the subset of 25 cD clusters is similar to that of non-cD systems. A substantial fraction of the cD galaxies have velocities significantly different from the mean of their parent clusters. Three of these have 4 s or more measured redshifts. These last two results suggest that the formation of cD galaxies is dominated by the local, rather than global, cluster environment.