We apply topological measures of clustering such as percolation and genus curves (PC & GC) and shape statistics to a set of scale free N-body simulations of large scale structure. Both genus and percolation curves evolve with time reflecting growth of non-Gaussianity in the N-body density field. The amplitude of the genus curve decreases with epoch due to non-linear mode coupling, the decrease being more noticeable for spectra with small scale power. Plotted against the filling factor GC shows very little evolution - a surprising result, since the percolation curve shows significant evolution for the same data. Our results indicate that both PC and GC could be used to discriminate between rival models of structure formation and the analysis of CMB maps. Using shape sensitive statistics we find that there is a strong tendency for objects in our simulations to be filament-like, the degree of filamentarity increasing with epoch.

New and improved data on the gravitational lens systems discovered so far are compared with the theoretical predictions of Gott, Park, and Lee (1989, GPL). Systems lensed by a single galaxy, compatible with assumptions of GPL, support flat or near-flat geometry for the universe. But the statistical uncertainty is too large to draw any definite conclusion. We need more lens systems. Also, the probability of multiple image lensing and mean separation of the images averaged over the source distribution are calculated for various cosmological models. Multiple-image lens systems and radio ring systems are compared with the predictions. Although the data reject exotic cosmological models, it cannot discriminate among conventional Friedmann models yet.

Dark matter in various size of scales is reviewed briefly. The evidence of dark matter in dwarf spheroidal galaxies is still uncertain. However there is no doubt about the existence of dark matter in larger scales. Many proposed candidates for dark matter are still speculative. Several possibilities of direct detection of dark matter are proposed.

Recent redshift surveys suggest that most galaxies may be distributed on the surfaces of bubbles surrounding large voids. To investigate the quantitative consistency of this qualitative picture of large-scale structure, we study analytically the clustering properties of galaxies in a universe filled with spherical shells. In this paper, we report the results of the calculations for the spatial and angular two-point correlation functions of galaxies. With ∼20 ∼20 of galaxies in clusters and a power law distribution of shell sizes, nsh(R)∼R−α nsh(R)∼R−α , α≃4 α≃4 , the observed slope and amplitude of the spatial two-point correlation function ξgg(r) ξgg(r) can be reproduced. (It has been shown that the same model parameters reproduce the enhanced cluster two-point correlation function, ξcc(r) ξcc(r) ). The corresponding angular two-point correlation function w(θ) w(θ) is calculated using the relativistic form of Limber's equation and the Schecter-type luminosity function. The calculated w(θ θ ) agrees with the observed one quite well on small separations (θ≲2deg θ≲2deg ).

We investigated the "tilting" of the Universe, i.e., a non-Doppler origin of the dipole moment of the cosmic background radiation (CBR). Superhorizon-sized isocurvature, rotational and true vacuum bubble perturbations are considered. We show that the more natural way of the "tilting" the Universe is via the true vacuum bubble perturbation. Nevertheless, due to the small filling fraction of the bubbles of viable extended inflationary models, we find that the probability of the real occurrence in the Universe is quite insignificant.

Energy density evolution of primordial black holes(PBHs) due to quantum gravitational tunneling effect in the very early Universe is calculated for the four cases of GUTs(grand unified theories) (SM, SUSY SM, SUSY SU(5), SU(5)). For the three of them (SM, SUSY SM, SUSY SU(5)), it is confirmed that there are a considerable amount of PBHs and so it may give a firm support to Lindley's paper(1981) in which he tried to solve the baryon asymmetry problem. It is shown that the formation of PBHs increases the cosmic scale factor R and decreases the total energy density ρt ρt faster than in the usual radiation dominated era.

Godel models of the universe filled with fluid are studied in the framework of the Einstein-Cartan theory of gravitation. It is assumed that the models admit a group of motions simply transitive on space-time. The combined effects of spin and rotation(vorticity) are studied with a particular attention to whether the held equations impose any restriction on alignement of spin direction (a polarized spin distribution). The solutions are found explicitly in a closed form, which show that spin components are vanishingly small except in the direction of z-axis (the compass of inertia) in which they can assume an arbitrary distribution.