Three ways to estimate Ω from the motions of nearby galaxies are reviewed: (i) comparisons can be made between observed local streaming motions and observed density fluctuations, (ii) a timing argument can give a value of f(Ω) = to Ho, and (iii) the motions in groups provide solid evidence regarding the amount of matter that is distributed like galaxies on a scale of 0.5 Mpc. In each case, the evidence suggests the Universe is open, but only by a factor of 3–5 and, given the uncertainties, a closed Universe definitely is not precluded.

This paper discusses the recent measurement of the number of galaxies vs. redshift and flux and presents new results pertaining to the two dimensionless geometrical quantities that describe the geometry of the conventional big-bang cosmology, the density parameter Ω and the dimensionless form λ = Λ/(3H02) of the cosmological constant. In contrast to the classical redshift-magnitude test as applied to the brightest galaxies in clusters, this new method is able to separate the effects of evolution from geometrical effects and is therefore able to measure the geometry of space. The 95% confidence limits are Ω - λ = 0.9−0 5+0 7 and −1.5 < Ω + λ < 7.1. The principal conclusions are these: (1) For both λ = 0 and inflationary models of the universe, this measurement and primordial nucleosynthesis imply a large density of nonbaryonic matter. (2) Hubble's constant H0 and the age of the universe τ are constrained by 0.60 < H0τ < 0.88 (95% confidence).

We have used a new distance estimator for elliptical galaxies to determine the peculiar velocities, with respect to a uniform Hubble flow, of approximately 400 galaxies. The relative distances of five clusters in common with those of Aaronson et al. (1981, 1986), based on the infrared Tully-Fisher relation for spirals, are in good agreement.

We do not see the reflex of the Local Group motion with respect to the microwave background out to recession velocities of 6000 km s−1. Rather, the frame of elliptical galaxies appears to be moving with respect to the microwave background with a velocity of 600 km s−1 towards 1 = 312°, b = +6°. This motion is consistent with a re-analysis of the Rubin et al. (1976) data on the magnitude-diameter relation for ScI galaxies and with the nearby and cluster samples of Aaronson et al. (1982, 1986).

Infrared wavelengths are free of several of the problems that plague optical galaxy surveys. At high galactic latitude ≥99% of 60μ sources in the IRAS Point Source Catalog, after deletion of obvious stars, are galaxies. At lower latitudes care has to be taken to avoid confusion with emission from interstellar dust (the ‘cirrus’). IRAS galaxies have been used to determined the direction of the gravitational acceleration acting on the Local Group due to galaxies and clusters within about 200 Mpc. This agrees well with the direction of the microwave background dipole. The density of matter in the universe, distributed like IRAS galaxies, needed to account for the observed velocity of the Local Group, corresponds to Ωo = 1.0 ± 0.2. In the standard hot Big Bang model, 90–95% of this matter would have to be non-baryonic.

IRAS galaxies are significantly less clustered than optically selected galaxy samples.

In the last decade, the inter-relation between the density structure and velocity perturbations in the local Virgo Supercluster (VSC) has provided a major means of determining the cosmological density parameter Ωo. The underlying dynamics, the so-called Virgocentric Flow Model (VFM), assumes that the relevant peculiar gravitational field in the VSC is dominated by the monopole term, due to material inside the VSC. Over the past year, however, this premise has been called into question, on both observational and theoretical grounds. It has become clear that the contributions to the local gravitational field from beyond the VSC cannot be neglected. The major new developments are summarized here. A more detailed review is given elsewhere (Yahil 1987).

The angular size - redshift (θ - z) relation can in principle be used to discriminate between world models because the angular size subtended by a rigid rod is quite a sensitive function of cosmology, specially at z ≳ 0.5. The test is simpler to apply to objects for which a metric diameter is measured than to objects with isophotal diameters (Sandage 1961). It was first suggested by Hoyle (1958) at the Paris symposium on Radio Astronomy, that the separation between the two lobes of extragalactic radio sources such as Cyg-A, could be used for performing such a test. In an Einstein-de Sitter Universe sources like Cyg-A cannot have angular sizes ≲ 15 arcsec (the minimum occuring at z = 1.25) whereas in the Steady State Universe their sizes should asymptotically approach a value near 4 arcsec at large redshifts. It was not until the early seventies that the test was actually applied to samples of radio quasars with redshifts of upto ∼ 2 (Legg 1970; Miley 1971; Wardle & Miley 1974). The angular sizes were found to show a large scatter due to a wide distribution of physical sizes and the projection effects associated with the essentially linear radio structures. The upper envelope to the θ -values (which would be expected to show much less scatter) nevertheless appeared to fall off monotonically with increasing z, more or less like the Euclidean relation θ ∝ z−1. The θ - z test thus appeared to be incompatible with the predictions of uniform world models in which the linear sizes of quasars are independent of epoch.

Evidence is presented that radio sources are more bent at larger redshifts. This effect is interpreted as due to interaction of the radio source with an ambient circumgalactic medium which must then have been clumpier at early epochs. It is suggested that the increased clumpiness could be associated with the late stages of galaxy formation and possibly also with CIV absorption line systems.

Following a brief review of the early developments in the field, we discuss a few new advances in the distribution of bright galaxies which occured after the extensive review paper by Oort (1983) We show that the large scale structure, shells, filaments and/or sheets, may bias the determination of the velocity dispersion in clusters of galaxies and that the boundaries of the voids may often be biassed by the clusters and groups velocity dispersion.

Of interest are the very large structures selected on catalogs of clusters of galaxies. The “Local Structure” claimed by Tully seems to be somewhat flattened and about parallel to the plane of the Local Supercluster. If the structure is real the alignement is relevant in relation to the physical mechanisms at work at the time of formation. Noticeable progress has been done in the measurement of the large scale velocity field. Large scale motions may somewhat bias the study of the topology of the Universe.

Relevant work has be done on the shape of the boundaries of the voids and observational work is progressing to detect faint galaxies in voids and determine their charactestics. This is important also in relation to the theory of biassed galaxy formation. To better focus the observational problem and eventually related biasses, we give some statistics on dwarf galaxies in the Virgo cluster.

Two slices of the Center for Astrophysics (CfA) redshift survey extension are now complete. The survey indicates that galaxies are distributed on the thin surfaces of “bubble-like” structures. The voids in the survey have diameters as large as 5,000 km s−1. These structures challenge theories for the formation of large-scale structure in the universe and suggest new approaches to several problems in the field.

A simple study of the galaxian surface density maps of the nearby Universe, obtained from CGCG, MCG and ESO catalogues, has allowed to evidence 3 new large-scale structures. Analysis of the redshifts of the galaxies in the corresponding zones shows that two of them are probably real three-dimensional entities.

The structure within the distribution of rich clusters is described. In particular, the Pisces-Cetus region is shown to contain a dense supercluster, a long filament of galaxies and clusters, and a ring of 12 Abell clusters surrounding a diameter void.

The amount of structure present among the Abell clusters out to redshift z = 0.085 has been compared with numerical supercomputer simulations (with 643 particles) of the isothermal, neutrino, and cold particle models for large-scale structure, assuming a flat universe and H = 50 km sec−1 Mpc−1. High-density clusters of particles were identified in each simulation. Correlation and percolation tests were then used to compare the spatial distribution of these high-density points with the apparent superclustering among Abell clusters. While all of the models had some small superclusters (the neutrino model has too many), none came very close to possessing the extremely extensive structures found in the Abell clusters (generally, disagreement by 2σ or more).

A second set of simulations used the cold particle model with Ω = 0.2 and 0.5. The structures found in these simulations were certainly larger than those of the Ω = 1.0 cold particle case, but still > 2σ too small in comparisons with the Abell clusters.

Attempts to measure structural parameters of superclusters are described. Catalogues of superclusters have been prepared using a percolation algorithm applied to a complete sample of UGC galaxies brighter than the 14.5 magnitude. Both a two dimensinal (projected) and a three dimensional approach have been tried. The (conservative) result is that superclusters are certainly not spherical. Probably the departure from sphericity is towards elongated cigar-like structers. No evidence of relative alignements of superclusters has been found.

We present the results of an analysis of fifty-five bright clusters clusters of galaxies as a homogeneous sample taken from a new effort to catalogue galaxy clusters in the Southern Hemisphere. The positions of some 21,000 galaxies in clusters have been catalogued along with visual magnitudes, morphological types, position angles of extended objects and pertinent remarks. For all of the clusters, various cluster parameters have been determined and form the basis of comparative studies for these fundamental aggregates of matter in the universe.

The evidence for the existence of very large scale structures, ∼ 100h−1Mpc in size, as derived from the spatial distribution of clusters of galaxies is summarized. Detection of a ∼ 2000 kms−1 elongation in the redshift direction in the distribution of the clusters is also described. Possible causes of the effect are peculiar velocities of clusters on scales of 10–100h−1Mpc and geometrical elongation of superclusters. If the effect is entirely due to the peculiar velocities of clusters, then superclusters have masses of order 1016.5M⊙ and may contain a larger amount of dark matter than previously anticipated.

To study distribution of galaxies and voids, the rectangular box under study is divided into cubic cells, and mean density of particles in cells is derived. For any density level cells can be divided into ‘filled’ or ‘empty’ ones if their density is higher or lower than a threshold density. The length and volume of the largest connected system, as well as the number of systems of connected cells are derived for observed, model and random samples. The comparison of results demonstrates that galaxy formation is biased, supercluster-void topology is sponge-like in a wide threshold density interval, and that there are no isolated galaxies in voids.

Recent results concerning the galaxy distribution at scales < 100 h−1 Mpc (Ho = 100 h kms−1 Mpc−1) show a number of characteristics which cannot be described by conventional statistical models. Correlation functions, for instance, can in no way give account of the presence of voids of the cellular (or spongy) appearance of the local galaxy distribution (M. Geller, this conference). There is clearly a need for new kinds of statistical models and statistical indicators.

Analyses for complete samples of galaxies and clusters of galaxies showed that the two-point correlation function of galaxy-galaxy and cluster-cluster have form of power law with power indices about −1.8 (Peebles, 1980; Bahcall and Soneira, 1983). Because the completeness of a sample means that we have observed completely the objects brighter than a given apparent magnitude in certain sky region. But, we know that a complete sample will be lack of faint objects at distant region. If we attempt to avoid the influences of any non-intrinsic properties on the analysis, we have to use the samples which are complete in certain interval of absolute magnitude.

Faint galaxy data are reviewed in the context of standard evolutionary models and our understanding of the statistical properties of galaxy populations. The differences in number magnitude counts from group to group can largely be understood via fluctuations induced by large scale clustering. However count slopes still present convincing arguments for an extra component of faint blue galaxies beyond B ∼21. Colours provide an useful tool in estimating redshift distributions but the uncertainties are large for the bluest galaxies of interest. However, new faint object redshift surveys are now underway and promise to determine definitively the nature of this extra component. We discuss preliminary results from one of these surveys. Neither distant luminous galaxies nor intrinsically faint nearby galaxies appear to be very numerous to B = 21.5. Many of the galaxies with 0.2 < z < 0.4 show spectral signatures of prominent star-formation. If such objects are somehow related to the excess counts, the traditional redshift-dependent evolutionary theory may require revision.

This paper reports preliminary results of two long-term redshift surveys that are near completion. One consists of nearly 400 redshifts of field galaxies; the sample is faint enough (B ≲ 22) to test models of galaxy luminosity and color evolution (and possibly cosmology) and to search for the presence of very large-scale structures among distant field galaxies. The other consists of over 60 spectra of quasar candidates of similar faintness; the identifications and redshifts of bona-fide quasars provide strong constraints on the evolution and shape of the luminosity function of distant quasars. Almost all of the observations have been made with the 4m telescope at Kitt Peak National Observatory with the Cryogenic Camera in multiaperture mode. The spectral range covered 4500Å to 7500Å with 15Å FWHM resolution and 4Å per pixel; simultaneous exposures of an hour or two were made for about 10 objects within the 5 arcmin field of view.