Testing cosmological models is worthwhile as long as the theoretical assumptions, on which these models are based, are true. In this report observational material testing the Cosmological Principle separately for geometry and substratum, is discussed without referring to any given model of the Universe.

Evidence that rich clusters of galaxies contain hot (T = 108 K) intracluster gas is reviewed. Such gas contributes little to Ω (0.003) but it has been argued that Ω must be less than 0.05 for true intergalactic gas, if accretion of more gas than is observed in rich clusters is to be avoided. This argument is reviewed.

If the de Vaucouleurs' groups are bound by intracluster gas, T is expected to be 105 to 107 K and the contribution to Ω is ≃ 1. Since the clumping factor C is estimated to be ≃ 7, the resulting value of Ω2C is ≃ 7. This does not violate the observed diffuse soft X-ray background intensity. Gas should be sought in such groups. Smoothly distributed gas with 107 < T < 3 × 108 K and Ω = 1 is not ruled out by direct observations.

The history of the cosmological constant and the Lemaître models is reviewed briefly. Using recent cosmological observations, it is found that the cosmological constant if non-zero must be in absolute value less than 2 × 10-56 cm-2. The predictions of the Lemaître models are compared with modern observations. It is shown that Lemaître models without evolution fail to reproduce the observed radio source counts. The existence of quasars with large redshift (z > 2.5) is shown to be strong evidence against the Lemaître models.

A preliminary report is given of recent work with A. Sandage on the Hubble constant. Through a chain of distance indicators in Sc and Ir galaxies (cepheids, brightest stars, H iiregions, and luminosity classes) the distance scale is carried beyond any possible local anisotropy of the velocity field. Special care is taken to allow for the dependence of the intrinsic properties of the distance indicators on the size of the parent galaxy, and for the effect of the Malmquist correction. H0 is found to be 55 ± 7 km s-1 Mpc-1; within the errors no systematic changes with distance were found.

A formal value of the deceleration constant q0 = 1 ± 1 was recently derived by Sandage (1972a) and Sandage and Hardy (1973). The most important correction to this value is probably the luminosity evolution of galaxies, which tends to push q0 below 0.5. The ensuing evidence for an open universe is also favored by independent arguments.

Evidence for non-cosmological redshifts is reviewed. It is shown that all current statistical tests favor association of QSRs with galaxies close by, in distance, to our own. It is possible that all QSRs originate from galaxies of much lower redshift. It is shown that in the four cases where QSRs fall projected closest to bright galaxies, that in all four cases the galaxies show evidence of physical interaction. Evidence for high redshift, compact and peculiar companion galaxies is reviewed. From the individual associations of high redshift QSRs and companions, an empirical continuity of observed characteristics is shown between compactness (youth) and excess redshift. Some theoretical explanations for intrinsic redshifts are mentioned.

Evidence for superclustering of galaxies includes the local supercluster, the large deviation of the observed surface distribution of clusters of galaxies from a random one, the cell size dependence of the index of clumpiness, the results of covariance function tests for correlations between different cluster centers and between clusters and individual galaxies, and a power-spectrum analysis. All tests give results that indicate or are compatible with the existence of inhomogeneities in the distribution of matter in space with linear diameters of 50 to 100 Mpc. If these are dynamically stable superclusters, they probably have internal velocity dispersions in the range 1000 to 3000 km s-1. The possible effects of superclustering on dynamical studies of individual clusters are discussed briefly.

Following the advice of Professor Zel'dovich, I will begin with two platitudes – first, when we consider the implications of the counts of radio sources, we must look at all the counts of radio sources at all frequencies; second, when we attempt to evaluate the implications of the source counts, we should use proper cosmological models and not rely on the local Euclidean model.

Counts of radio sources at 5 GHz (6 cm wavelength) have been derived from a number of surveys including a new strong source survey. The source counts do not appear to differ markedly from an integral number-flux relation having a slope of − 1.5 between 5 and 5 × 103 sources per steradian, and show a sharp drop at source densities smaller than 5 sr-1. On the basis of the form of the number counts and the observed anisotropies in the distribution of sources and of their spectra, the cosmological significance of the source counts is questioned. In particular, the evidence for strong cosmic evolution appears weaker than is generally thought unless the cosmological origin of the quasar redshifts is assumed. Measurements of the radio spectra of the sources suggest a dependence of the spectral curvature on flux density.

Using new observational data at optical (Jagiellonian Field) and radio (GB region) frequencies, the distributions of galaxies and radio sources are analysed.

The results of the analysis show that there is a significant non-random surface distribution of galaxies irrespective of the scale of the field investigated. The observed clumpiness does not allow us to introduce discrete hierarchical structures. The distribution of galaxies shows rather a continuous hierarchical structure and for the description of the distribution of galaxies, some density indices should be used instead of imprecisely defined clusters of galaxies.

In the radio domain, the distribution of radio sources in the GB region and the number-flux density relations which come from the different sub-regions are investigated. The results of our investigation show that the surface density of radio sources at several flux density levels and the number-flux density relations vary from place to place in the GB region and the significance level of the observed variations is at least 1%.

The most important conclusion which can be drawn from the GB data is that results obtained from the observations of a single selected region of the sky cannot be generalized to obtain information about the whole population of radio sources in a given flux density range, and in addition the region itself – even chosen at random – ought not to be considered as a typical one.

With reference to theory published earlier, formulas are given for the estimation of (i) abundances of morphological types among field galaxies, (ii) of selection probabilities, and (iii) of ‘space luminosity functions’. Strictly, the theory applies to ‘homogeneous classes’ of galaxies. This term designates a category of galaxies, say C, so finely defined that the probability, say Φ(m | C), that a galaxy of category C will be included in the catalogue depends on its photographic apparent magnitude m and on nothing else. The practical use of the theory is illustrated on data in the HMS Catalogue. It appears that certain combinations of the Hubble morphological types satisfy the definition of a homogeneous class. Such, for example, is the case for combinations of ellipticals E0–E3 and, separately, of spirals Sc, Scp, SBc. However, the combination of these two categories is not a homogeneous class.

In order to validate the theory empirically, calculations were performed to predict the abundances of eight combinations of morphological types among cluster galaxies listed in the HMS Catalogue, each combination being treated as a distinct homogeneous class. Additional hypotheses underlying these calculations are: (a) abundances of morphological types, (b) luminosity functions of these types, and (c) selection probabilities for cluster galaxies coincide with those for field galaxies. A comparison with the observations, reaching the value of z = 0.07, is satisfactory. This tends to validate the combination of formulas (i), (ii), (iii) with the additional hypotheses (a), (b) and (c). Incidentally, the result tends to support the steady state cosmology.

Experimental measurements of the intensity of the submillimeter background are reviewed. The latest results all indicate a low background; it is not yet possible to say whether or not the spectrum in this range is blackbody.

It is now generally accepted that the microwave background radiation, discovered in 1965 (Penzias and Wilson, 1965; Dicke et al., 1965), is cosmological in origin. Measurements of the spectrum of the radiation, discussed earlier in this volume by Blair, are consistent with the idea that the radiation is in fact a relic of a hot, dense, initial state of the Universe – the Big Bang. If the radiation is cosmological, measurements of both its spectrum and its angular distribution are capable of providing important – and remarkably precise – cosmological data.

In keeping with the title of Symposium No. 63, I would like to confront the assembly with a short review of observational upper limits on the amplitude of intensity fluctuations in the microwave background radiation on angular scales of less than one degree. These fluctuations are presently of interest for at least two important cosmological concerns:

Many of the cosmological models currently under discussion and theories of the origin of galaxies which involve antimatter, strong turbulence and so on result in significant energy release during the evolution of the Universe. It is evident that significant energy production should lead to distortions of the spectrum of the relic radiation. The absence of noticeable deviations from the Planckian spectrum enables us to set limits to the energy release in the early Universe (102 < z< 108). But in order to have a clear picture of possible distortions, let us first review the idealized situation.

Perhaps the most challenging problem confronting a cosmologist is to reconcile the observed large-scale structure of the Universe with the Friedmann-Lemaître cosmological models that have gained such widespread acceptance in recent years (cf. however the alternative viewpoint, as exemplified in this Symposium by Arp and others). In this review, I shall look anew at the spectrum of density inhomogeneities that survive decoupling of matter and radiation at z ~ 1000 and provide the primordial fluctuations that can eventually generate galaxies. A closely related matter, that of the associated fluctuations in the background radiation, is discussed elsewhere in this volume by Doroshkevich, Sunyaev and Zel'dovich.

The results of improved calculations of the abundances of the nuclei produced in big-bang models of the early universe are presented. In addition to the standard model, other possible universes are considered, including the recent statistical bootstrap theory of Carlitz, Frautschi, and Nahm. Some conclusions which can be drawn about the nature of the early universe, depending upon whether the observed deuterium and helium are of galactic or cosmological origin, are presented.

A short review of the theory of the formation of galaxies and clusters of galaxies is given within the framework of the non-linear theory of gravitational instability. Probable initial conditions are discussed as well as processes which bring about the origin of galaxies and clusters of galaxies. Possible observational tests are discussed.