Morphological Statistics of the Cosmic Web

We report the {\em first} systematic study of the supercluster-void network in the $\Lambda$CDM concordance cosmology treating voids and superclusters on an equal footing. We study the dark matter density field in real space smoothed with the $L_s = 5$ \hm1 Mpc Gaussian window. Superclusters and voids are defined as individual members of over-dense and under-dense excursion sets respectively. We determine the morphological properties of the cosmic web at a large number of dark matter density levels by computing Minkowski functionals for every supercluster and void. At the adopted smoothing scale individual superclusters totally occupy no more than about 5% of the total volume and contain no more than 20% of mass if the largest supercluster is excluded. Likewise, individual voids totally occupy no more than 14% of volume and contain no more than 4% of mass if the largest void is excluded. The genus of individual superclusters can be $\sim 5$ while the genus of individual voids reaches $\sim 55$, implying significant amount of substructure in superclusters and especially in voids. Large voids are typically distinctly non-spherical.Comment: 6 pages, 4 figures, uses iaus.cls, Invited talk at IAU Colloquium 195 "Outskirts of galaxy clusters: intense life in the suburbs", Torino, Italy, March 12-16, 200

Tessellating the Universe: the Zel'dovich and Adhesion tiling of space

The adhesion approximation is a simple analytical model suggested for explanation of the major geometrical features of the observed structure in the galaxy distribution on scales from 1 to (a few)x100/h Mpc. It is based on Burgers' equation and therefore allows analysis in considerable detail. A particular version of the model that assumes the infinitesimal viscosity naturally results in irregular tessellation of the universe. Generic elements of the tessellation: vertices, edges, faces and three-dimensional tiles can be associated with astronomical objects of different kinds: clusters, superclusters and voids of galaxies. Point-like vertices contain the most of the mass and one-dimensional edges (filaments) are the second massive elements. The least massive are the two-dimensional faces and tiles (voids). The evolution of the large-scale structure can be viewed as a continuous process that transports mass predominantly from the high- to low-dimensional elements of the tessellation. For instance, the mass from the cells flows into faces, edges and vertices, in turn the mass from faces flows into edges and vertices, etc. At the same time, the elements of the tessellation themselves are in continuous motion resulting in mergers of some vertices, growth of some tiles and shrinking and disappearance of the others as well as other metamorphoses.Comment: 18 pages, 11 figure

Universality of the Network and Bubble Topology in Cosmological Gravitational Simulations

Using percolation statistics we, for the first time, demonstrate the universal character of a network pattern in the real space, mass distributions resulting from nonlinear gravitational instability of initial Gaussian fluctuations. Percolation analysis of five stages of the nonlinear evolution of five power law models reveals that all models show a shift toward a network topology if seen with high enough resolution. However, quantitatively, the shift is significantly different in different models: the smaller the spectral index ,n, the stronger the shift. On the contrary, the shift toward the "bubble" topology is characteristic only for the n <= -1 models. We find that the mean density of the percolating structures in the nonlinear density distributions generally is very different from the density threshold used to identify them and corresponds much better to a visual impression. We also find that the maximum of the number of structures (connected regions above or below a specified density threshold) in the evolved, nonlinear distributions is always smaller than in Gaussian fields with the same spectrum, and is determined by the effective slope at the cutoff frequency.Comment: The paper is 26 pages long. The latex file uses aasms.sty as a style file. There are 5 figures and 2 tables included

Topology of the Galaxy Distribution

The history and the major results of the study of the topology of the large-scale structure are briefly reviewed. Two techniques based on percolation theory and the genus curve are discussed. The preliminary results of the percolation analysis of the Wiener reconstruction of the IRAS $1.2 Jy$ redshift catalog are reported.Comment: Latex file with figures in postscript format, 8 page

Quasi-linear regime of gravitational instability as a clue to understanding the large-scale structure in the Universe

In the late seventies, an image of the large-scale structure in the Universe began to emerge as a result of the accumulation of the galaxy redshifts. Most of the galaxies are found to concentrate in large filaments and perhaps sheets leaving most of the volume empty. Similar structures were predicted theoretically in the frame of the adiabatic theory of galaxy formation (Zeldovich) and later in the hot dark matter cosmology. However, both scenarios have been ruled out by the observations. With these scenarios the dynamical part of the scenario was also erroneously rejected by many as well. In this talk, I derive the Zeldovich approximation from the exact dynamic equations and show that it is always better than the standard linear approximation. The advantage of the Zeldovich approximation is the greatest in the quasi-linear regime when delta(sub rms) is less than 1 (delta identical to delta(rho)/rho), but the displacement of the matter is essential. The range of scales in the quasi-linear regime depends upon the slope of the initial spectrum and increases with decreasing n, where n is the exponent, if the initial spectrum is approximated by a simple power law P(k) varies as k(exp n)