We investigate the ability of three reconstruction techniques to analyze and investigate weblike features and geometries in a discrete distribution of objects. The three methods are the linear Delaunay Tessellation Field Estimator (DTFE), its higher
order equivalent Natural Neighbour Field Estimator (NNFE) and a version of Kriging interpolation adapted to the specific circumstances encountered in galaxy redshift surveys, the Natural Lognormal Kriging technique. DTFE and NNFE are based on the local geometry defined by the Voronoi and Delaunay tessellations of the galaxy distribution. The three reconstruction methods are analysed and compared using mock magnitude-limited and volume-limited SDSS redshift surveys, obtained on the basis of the Millennium simulation. We investigate error trends, biases and the topological structure of the resulting fields, concentrating on the void population identified by the Watershed Void Finder. Environmental effects are addressed by evaluating the density fields on a range of Gaussian filter scales. Comparison with the void population in the original simulation yields the fraction of false void mergers and false void splits. In most tests DTFE, NNFE and Kriging have largely similar density and topology error behaviour. Cosmetically, higher order NNFE and Kriging methods produce more visually appealing reconstructions. Quantitatively, however, DTFE performs better, even while computationally far less demanding. A successful recovery of the void population on small scales appears to be difficult, while the void recovery rate improves significantly on scales > 3 h-1Mpc. A study of small scale voids and the void galaxy population should therefore be restricted to the local Universe, out to at most 100 h-1Mpc.
We present the SpineWeb framework for the topological analysis of the Cosmic Web and the identification of its walls, filaments and cluster nodes. Based on the watershed segmentation of the cosmic density field, the SpineWeb method invokes the local
adjacency properties of the boundaries between the watershed basins to trace the critical points in the density field and the separatrices defined by them. The separatrices are classified into walls and the spine, the network of filaments and nodes in the matter distribution. Testing the method with a heuristic Voronoi model yields outstanding results. Following the discussion of the test results, we apply the SpineWeb method to a set of cosmological N-body simulations. The latter illustrates the potential for studying the structure and dynamics of the Cosmic Web.
We study the topology of the Megaparsec Cosmic Web on the basis of the Alpha Shapes of the galaxy distribution. The simplicial complexes of the alpha shapes are used to determine the set of Betti numbers ($beta_{rm k},k=1,...,D$), which represent a c
omplete characterization of the topology of a manifold. This forms a useful extension of the geometry and topology of the galaxy distribution by Minkowski functionals, of which three specify the geometrical structure of surfaces and one, the Euler characteristic, represents a key aspect of its topology. In order to develop an intuitive understanding for the relation between Betti numbers and the running $alpha$ parameter of the alpha shapes, and thus in how far they may discriminate between different topologies, we study them within the context of simple heuristic Voronoi clustering models. These may be tuned to consist of a few or even only one specific morphological element of the Cosmic Web, ie. clusters, filaments or sheets.
We review and discuss aspects of Cosmic Voids that form the background for our Void Galaxy Survey (see accompanying paper by Stanonik et al.). Following a sketch of the general characteristics of void formation and evolution, we describe the influenc
e of the environment on their development and structure and the characteristic hierarchical buildup of the cosmic void population. In order to be able to study the resulting tenuous void substructure and the galaxies populating the interior of voids, we subsequently set out to describe our parameter free tessellation-based watershed void finding technique. It allows us to trace the outline, shape and size of voids in galaxy redshift surveys. The application of this technique enables us to find galaxies in the deepest troughs of the cosmic galaxy distribution, and has formed the basis of our void galaxy program.
