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Transactions on Computational Science XIV pp 60–101Cite as

Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web

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Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 6970))

Abstract

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.

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Authors and Affiliations

  1. Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    Rien van de Weygaert, Bernard J. T. Jones, Pratyush Pranav, E. G. P. (Patrick) Bos, Johan Hidding & Job Feldbrugge

  2. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Gert Vegter, Bob Eldering, Nico Kruithof & Matti van Engelen

  3. IST Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Herbert Edelsbrunner

  4. School of Physics, Korea Institute for Advanced Study, Seoul, 130-722, Korea

    Changbom Park

  5. Interdisciplinary Centre for Mathematical and Computational Modeling, University of Warsaw, ul. Pawinskiego 5a, 02-106, Warsaw, Poland

    Wojciech A. Hellwing

  6. Stratingh Institute for Chemistry, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands

    Eline ten Have

  7. Géométrica, INRIA Sophia Antipolis-Méditerranée, route des Lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    Manuel Caroli & Monique Teillaud

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  1. Rien van de Weygaert
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  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. Exascala Ltd., Unit 9, 97 Rickman Drive, B15 2AL, Birmingham, UK

    C. J. Kenneth Tan

  3. Département des sciences géomatiques, Université Laval Centre de Recherche en Géomatique, G1V 0A6, Québec, QC, Canada

    Mir Abolfazl Mostafavi

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van de Weygaert, R. et al. (2011). Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web. In: Gavrilova, M.L., Tan, C.J.K., Mostafavi, M.A. (eds) Transactions on Computational Science XIV. Lecture Notes in Computer Science, vol 6970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25249-5_3

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