Johan Hidding

Alpha, betti and the megaparsec universe: on the topology of the cosmic web

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, α. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of α, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the Cosmic Web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field. Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the Cosmic Web.

Betti Numbers of Gaussian Fields

We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three-and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; β? is the number of connected regions, β 1 is the number of circular holes (i. e., complement of solid tori) and β₂ is the number of three-dimensional voids (i.e., complement of three-dimensional excursion regions). Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. β? dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). β₁ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and β₂ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum n in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as n decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even through the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.

The Zeldovich & Adhesion approximations, and applications to the local universe

The Zeldovich approximation (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher order derivatives. We show that the A 3 lines predict the formation of filaments in a two-dimensional model. We continue with applications of the adhesion model to visualise structures in the local (z <0.03) universe.

Adhesion and the Geometry of the Cosmic Web

We present a new way to formulate the geometry of the Cosmic Web in terms of Lagrangian space. The Adhesion model has an ingenious geometric interpretation out of which the spine of the Cosmic Web emerges naturally. Within this context we demonstrate a deep connection of the relation between Eulerian and Lagrangian space with that between Voronoi and Delaunay tessellations.

The cosmic spiderweb: equivalence of cosmic, architectural and origami tessellations

For over 20 years, the term ‘cosmic web’ has guided our understanding of the large-scale arrangement of matter in the cosmos, accurately evoking the concept of a network of galaxies linked by filaments. But the physical correspondence between the cosmic web and structural engineering or textile ‘spiderwebs’ is even deeper than previously known, and also extends to origami tessellations. Here, we explain that in a good structure-formation approximation known as the adhesion model, threads of the cosmic web form a spiderweb, i.e. can be strung up to be entirely in tension. The correspondence is exact if nodes sampling voids are included, and if structure is excluded within collapsed regions (walls, filaments and haloes), where dark-matter multistreaming and baryonic physics affect the structure. We also suggest how concepts arising from this link might be used to test cosmological models: for example, to test for large-scale anisotropy and rotational flows in the cosmos.