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We present a general formalism for identifying the caustic structure of an evolving mass distribution in an arbitrary dimensional space. For the class of Hamiltonian fluids the identification corresponds to the classification of singularities in Lagrangian catastrophe theory. Based on this we develop a theoretical framework for the formation of the cosmic web, and specifically those aspects that characterize its unique nature: its complex topological connectivity and multiscale spinal structure of sheetlike membranes, elongated filaments and compact cluster nodes. The present work represents an extension of the work by Arnold et al., who classified the caustics for the 1- and 2-dimensional Zeldovich approximation. His seminal work established the role of emerging singularities in the formation of nonlinear structures in the universe. At the transition from the linear to nonlinear structure evolution, the first complex features emerge at locations where different fluid elements cross to establish multistream regions. The classification and characterization of these mass element foldings can be encapsulated in caustic conditions on the eigenvalue and eigenvector fields of the deformation tensor field. We introduce an alternative and transparent proof for Lagrangian catastrophe theory, and derive the caustic conditions for general Lagrangian fluids, with arbitrary dynamics, including dissipative terms and vorticity. The new proof allows us to describe the full 3-dimensional complexity of the gravitationally evolving cosmic matter field. One of our key findings is the significance of the eigenvector field of the deformation field for outlining the spatial structure of the caustic skeleton. We consider the caustic conditions for the 3-dimensional Zeldovich approximation, extending earlier work on those for 1- and 2-dimensional fluids towards the full spatial richness of the cosmic web.
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The $beta$-skeleton is a mathematical method to construct graphs from a set of points that has been widely applied in the areas of image analysis, machine learning, visual perception, and pattern recognition. In this work, we apply the $beta$-skeleton to study the cosmic web. We use this tool on observed and simulated data to identify the filamentary structures and characterize the statistical properties of the skeleton. In particular, we compare the $beta$-skeletons built from SDSS-III galaxies to those obtained from MD-PATCHY mocks, and also to mocks directly built from the Big MultiDark $N$-body simulation. We find that the $beta$-skeleton is able to reveal the underlying structures in observed and simulated samples without any parameter fine-tuning. A different degree of sparseness can be obtained by adjusting the value of $beta$; in addition, the statistical properties of the length and direction of the skeleton connections show a clear dependence on redshift space distortions (RSDs), cosmological effects and galaxy bias. We also find that the $N$-body simulation accurately reproduces the RSD effect in the data, while the MD-PATCHY mocks appear to underestimate its magnitude. Our proof-of-concept study shows that the statistical properties of the $beta$-skeleton can be used to probe cosmological parameters and galaxy evolution.
We investigate the characteristics and the time evolution of the cosmic web from redshift, z=2, to present time, within the framework of the NEXUS+ algorithm. This necessitates the introduction of new analysis tools optimally suited to describe the very intricate and hierarchical pattern that is the cosmic web. In particular, we characterize filaments (walls) in terms of their linear (surface) mass density. This is very good in capturing the evolution of these structures. At early times the cosmos is dominated by tenuous filaments and sheets, which, during subsequent evolution, merge together, such that the present day web is dominated by fewer, but much more massive, structures. We also show that voids are more naturally described in terms of their boundaries and not their centres. We illustrate this for void density profiles, which, when expressed as a function of the distance from void boundary, show a universal profile in good qualitative agreement with the theoretical shell-crossing framework of expanding underdense regions.
The cosmic web is one of the most striking features of the distribution of galaxies and dark matter on the largest scales in the Universe. It is composed of dense regions packed full of galaxies, long filamentary bridges, flattened sheets and vast low density voids. The study of the cosmic web has focused primarily on the identification of such features, and on understanding the environmental effects on galaxy formation and halo assembly. As such, a variety of different methods have been devised to classify the cosmic web -- depending on the data at hand, be it numerical simulations, large sky surveys or other. In this paper we bring twelve of these methods together and apply them to the same data set in order to understand how they compare. In general these cosmic web classifiers have been designed with different cosmological goals in mind, and to study different questions. Therefore one would not {it a priori} expect agreement between different techniques however, many of these methods do converge on the identification of specific features. In this paper we study the agreements and disparities of the different methods. For example, each method finds that knots inhabit higher density regions than filaments, etc. and that voids have the lowest densities. For a given web environment, we find substantial overlap in the density range assigned by each web classification scheme. We also compare classifications on a halo-by-halo basis; for example, we find that 9 of 12 methods classify around a third of group-mass haloes (i.e. $M_{rm halo}sim10^{13.5}h^{-1}M_{odot}$) as being in filaments. Lastly, so that any future cosmic web classification scheme can be compared to the 12 methods used here, we have made all the data used in this paper public.
The cosmic web is the largest scale manifestation of the anisotropic gravitational collapse of matter. It represents the transitional stage between linear and non-linear structures and contains easily accessible information about the early phases of structure formation processes. Here we investigate the characteristics and the time evolution of morphological components since. Our analysis involves the application of the NEXUS Multiscale Morphology Filter (MMF) technique, predominantly its NEXUS+ version, to high resolution and large volume cosmological simulations. We quantify the cosmic web components in terms of their mass and volume content, their density distribution and halo populations. We employ new analysis techniques to determine the spatial extent of filaments and sheets, like their total length and local width. This analysis identifies cluster and filaments as the most prominent components of the web. In contrast, while voids and sheets take most of the volume, they correspond to underdense environments and are devoid of group-sized and more massive haloes. At early times the cosmos is dominated by tenuous filaments and sheets, which, during subsequent evolution, merge together, such that the present day web is dominated by fewer, but much more massive, structures. The analysis of the mass transport between environments clearly shows how matter flows from voids into walls, and then via filaments into cluster regions, which form the nodes of the cosmic web. We also study the properties of individual filamentary branches, to find long, almost straight, filaments extending to distances larger than 100Mpc/h. These constitute the bridges between massive clusters, which seem to form along approximatively straight lines.
Voids form a prominent aspect of the Megaparsec distribution of galaxies and matter. Not only do they represent a key constituent of the Cosmic Web, they also are one of the cleanest probes and measures of global cosmological parameters. The shape and evolution of voids are highly sensitive to the nature of dark energy, while their substructure and galaxy population provides a direct key to the nature of dark matter. Also, the pristine environment of void interiors is an important testing ground for our understanding of environmental influences on galaxy formation and evolution. In this paper, we review the key aspects of the structure and dynamics of voids, with a particular focus on the hierarchical evolution of the void population. We demonstrate how the rich structural pattern of the Cosmic Web is related to the complex evolution and buildup of voids.
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