Voronoi Tessellations and the Cosmic Web: Spatial Patterns and Clustering across the Universe

The spatial cosmic matter distribution on scales of a few up to more than a hundred Megaparsec displays a salient and pervasive foamlike pattern. Voronoi tessellations are a versatile and flexible mathematical model for such weblike spatial patterns. They would be the natural asymptotic result of an evolution in which low-density expanding void regions dictate the spatial organization of the Megaparsec Universe, while matter assembles in high-density filamentary and wall-like interstices between the voids. We describe the results of ongoing investigations of a variety of aspects of cosmologically relevant spatial distributions and statistics within the framework of Voronoi tessellations. Particularly enticing is the finding of a profound scaling of both clustering strength and clustering extent for the distribution of tessellation nodes, suggestive for the clustering properties of galaxy clusters. Cellular patterns may be the source of an intrinsic ``geometrically biased'' clustering.Comment: 10 pages, 9 figures, accepted for publication as long paper in proceedings Fourth International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), ed. C. Gold, IEEE Computer Society, July 2007. For high-res version see http://www.astro.rug.nl/~weygaert/tim1publication/vorwey.isvd07.pd

The Cosmic Foam and the Self-Similar Cluster Distribution

Voronoi Tessellations form an attractive and versatile geometrical asymptotic model for the foamlike cosmic distribution of matter and galaxies. In the Voronoi model the vertices are identified with clusters of galaxies. For a substantial range out to a scale in the order of the cellsize, their spatial two-point correlation function is a power-law with a slope $\gamma \approx 1.95$. This study presents recent results showing that subsets of vertices selected on the basis of their ``richness'', i.e. inflow volume, retain this power-law correlation behaviour. Interestingly, they do so with a ``clustering length'' $r_{\rm o}$ that is exactly linearly proportional to the average inter-vertex distance in the sample, thus forming a realization of the Szalay-Schramm prescription. For the geometry and structural patterns even more significant is the finding tessellation vertices display a similar linear increase for their correlation length $r_{\rm a}$, the coherence length at which $\xi(r_{\rm a})=0$. Such patterns therefore exhibit positive correlations out to distances considerably in excess of the cellsize. Most intriguing is the implication of self-similar scaling, while these results may be regarded as the presentation of a ``geometrical bias'' effect. A seemingly rigid structure appears to represent a flexible and useful geometric model for exploring the statistical and dynamical repercussions of the nontrivial cellular patterns in Megaparsec scale cosmic structure.Comment: Contribution to ``Large Scale Structure in the X-ray Universe'', Workshop Santorini, Greece, September 1999, eds. M. Plionis and I. Georgantopoulos (Editions Frontieres). 5 pages of LaTex including 2+3 postscript figures. Uses psfi

Large Scale Structure: Setting the Stage for the Galaxy Formation Saga

Over the past three decades the established view of a nearly homogeneuous, featureless Universe on scales larger than a few Megaparsec has been completely overhauled. In particular through the advent of ever larger galaxy redshift surveys we were revealed a galaxy distribution displaying an intriguing cellular pattern in which filamentary and wall-like structures, as well as huge regions devoid of galaxies, are amongst the most conspicuous morphological elements. In this contribution we will provide an overview of the present observational state of affairs concerning the distribution of galaxies and the structure traced out by the matter distribution in our Universe. In conjunction with the insight on the dynamics of the structure formation process obtained through the mapping of the peculiar velocities of galaxies in our local Universe and the information on the embryonic circumstances that prevailed at the epoch of Recombination yielded by the various Cosmic Microwave Background experiments, we seek to arrive at a more or less compelling theoretical framework of structure formation.The main aspects of this framework of the rise of structure through gravitational instability can probably be most readily appreciated through illustrative examples of various scenarios, as for instance provided by some current state-of-the-art N-body simulations. We will subsequently wrap up the observational and theoretical evidence for the emergence and evolution of structure in the Universe by sketching the stage for the ultimate Holy Grail of late 20th century astrophysics, understanding the saga of the formation of what arguably are the most prominent and at the same time intoxicatingly beautiful and intriguing denizens of our Cosmos, the {\it galaxies}.Comment: 25 pages, 7 figures. Invited Review at `The most distant radio galaxies' KNAW Colloquium, Amsterdam, October 1997, eds Best et al., Kluwer. 25 pages of LaTex including 7 postscript (bitmapped) figures. Uses knawproc.cl

The Cosmic Web: Geometric Analysis

The lecture notes describe the Delaunay Tessellation Field Estimator for Cosmic Web analysis. The high sensitivity of Voronoi/Delaunay tessellations to the local point distribution is used to obtain estimates of density and related quantities. The adaptive and minimum triangulation properties of Delaunay tessellations are utilized for interpolation to continuous volume-covering density and velocity fields. The adaptivity to the local density and geometry preserves three key characteristics of the cosmic matter distribution: 1) its hierarchical nature,2) its anisotropic weblike morphology, 3) the presence and structure of voids. The lecture notes discuss the mathematical context and background of the DTFE method, in essence a first order version of Natural Neighbour methods, smooth and higher order spatial interpolation techniques. This is followed by an outline of the DTFE procedure. A series of tests on spatially complex point distributions provides quantitative information on its performance and noise characteristics. We conclude with a few cosmological applications. The reconstruction of the spatial weblike patterns in N-body simulations and in the 2dFGRS provide a visual impression of DTFE's performance. The analysis of the velocity divergence pdf and maps obtained of the velocity flow in the Local Universe (implied by PSCz) illustrate its promise for probing the dynamics of cosmic structure. The Watershed Void Finder (Platen et al. 2007), for the study of voids, and Multiscale Morphology Filter (Aragon-Calvo et al. 2007), for identifying and analyzing weblike networks of filaments, sheets and clusters, are based upon DTFE density fields. We also mention SimpleX, a related random lattice radiative transfer formalism (Ritzerveld & Icke 2006).Comment: 128 pp., 52+1 figures, lecture notes summerschool "Data Analysis in Cosmology" (Valencia), eds. V. Martinez, E. Saar, E. Martinez-Gonzalez, M. Pons-Borderia, Springer-Verlag. For high-res version see http://www.astro.rug.nl/~weygaert/tim1publication/weyval2004.pd

Clusters and the Cosmic Web

We discuss the intimate relationship between the filamentary features and the rare dense compact cluster nodes in this network, via the large scale tidal field going along with them, following the cosmic web theory developed Bond et al. The Megaparsec scale tidal shear pattern is responsible for the contraction of matter into filaments, and its link with the cluster locations can be understood through the implied quadrupolar mass distribution in which the clusters are to be found at the sites of the overdense patches. We present a new technique for tracing the cosmic web, identifying planar walls, elongated filaments and cluster nodes in the galaxy distribution. This will allow the practical exploitation of the concept of the cosmic web towards identifying and tracing the locations of the gaseous WHIM. These methods, the Delaunay Tessellation Field Estimator (DTFE) and the Morphology Multiscale Filter (MMF) find their basis in computational geometry and visualization.Comment: 13 pages, 6 figures, appeared in proceedings workshop "Measuring the Diffuse Intergalactic Medium", eds. J-W. den Herder and N. Yamasaki, Hayama, Japan, October 2005. For version with high-res figures see http://www.astro.rug.nl/~weygaert/weywhim05.pd

Persistent topology of the reionisation bubble network. I: Formalism & Phenomenology

We present a new formalism for studying the topology of HII regions during the Epoch of Reionisation, based on persistent homology theory. With persistent homology, it is possible to follow the evolution of topological features over time. We introduce the notion of a persistence field as a statistical summary of persistence data and we show how these fields can be used to identify different stages of reionisation. We identify two new stages common to all bubble ionisation scenarios. Following an initial pre-overlap and subsequent overlap stage, the topology is first dominated by neutral filaments (filament stage) and then by enclosed patches of neutral hydrogen undergoing outside-in ionisation (patch stage). We study how these stages are affected by the degree of galaxy clustering. We also show how persistence fields can be used to study other properties of the ionisation topology, such as the bubble size distribution and the fractal-like topology of the largest ionised region.Comment: 18 pages, 12 figures, 1 table. Submitted to MNRA

Cold Flows and Large Scale Tides

Several studies have indicated that the local cosmic velocity field is rather cold, in particular in the regions outside the massive, virialized clusters of galaxies. If our local cosmic environment is taken to be a representative volume of the Universe, the repercussion of this finding is that either we live in a low-$\Omega$ Universe and/or that the galaxy distribution is a biased reflection of the underlying mass distribution. Otherwise, the pronounced nature of the observed galaxy distribution would be irreconcilable with the relatively quiet flow of the galaxies. Here we propose a different view on this cosmic dilemma, stressing the fact that our cosmic neighbourhood embodies a region of rather particular dynamical properties, and henceforth we are apt to infer flawed conclusions with respect to the global Universe. Suspended between two huge mass concentrations, the Great Attractor region and the Perseus-Pisces chain, we find ourselves in a region of relatively low density yet with a very strong tidal shear. This tidal field induces a local velocity field with a significant large-scale bulk flow but a low small-scale velocity dispersion. By means of constrained realizations of our local Universe, consisting of Wiener-filtered reconstructions inferred from the Mark III catalogue of galaxy peculiar velocities in combination with appropriate spectrally determined fluctuations, we study the implications for our local velocity field. We find that we live near a local peak in the distribution of the cosmic Mach number, $|v_{bulk}|/\sigma_v$, and that our local cosmic niche is located in the tail of the Mach number distribution function.Comment: Contribution to `Evolution of Large Scale Structure', MPA/ESO Conference, August 1997, eds. A. Banday & R. Sheth, Twin Press. 5 pages of LaTeX including 3 postscript figures. Uses tp.sty and psfi

A Hierarchy of Voids

We present a model for the distribution of void sizes and its evolution within the context of hierarchical scenarios of gravitational structure formation. For a proper description of the hierarchical buildup of the system of voids in the matter distribution, not only the "void-in-void" problem should be taken into account, but also that of the "void-in-cloud" issue. Within the context of the excursion set formulation of an evolving void hierarchy is one involving a "two-barrier" excursion problem, unlike the "one-barrier" problem for the dark halo evolution. This leads to voids having a peaked size distribution at any cosmic epoch, centered on a characteristic void size that evolves self-similarly in time, this in distinct contrast to the distribution of virialized halo masses which do not have a small-scale cut-off.Comment: 6 pages, 4 figures, to appear in the Proceedings of IAU Colloquium 195: "Outskirts of Galaxy Clusters: intense life in the suburbs", Torino Italy, March 200