---
_id: '1022'
abstract:
- lang: eng
  text: We introduce a multiscale topological description of the Megaparsec web-like
    cosmic matter distribution. Betti numbers and topological persistence offer a
    powerful means of describing the rich connectivity structure of the cosmic web
    and of its multiscale arrangement of matter and galaxies. Emanating from algebraic
    topology and Morse theory, Betti numbers and persistence diagrams represent an
    extension and deepening of the cosmologically familiar topological genus measure
    and the related geometric Minkowski functionals. In addition to a description
    of the mathematical background, this study presents the computational procedure
    for computing Betti numbers and persistence diagrams for density field filtrations.
    The field may be computed starting from a discrete spatial distribution of galaxies
    or simulation particles. The main emphasis of this study concerns an extensive
    and systematic exploration of the imprint of different web-like morphologies and
    different levels of multiscale clustering in the corresponding computed Betti
    numbers and persistence diagrams. To this end, we use Voronoi clustering models
    as templates for a rich variety of web-like configurations and the fractal-like
    Soneira-Peebles models exemplify a range of multiscale configurations. We have
    identified the clear imprint of cluster nodes, filaments, walls, and voids in
    persistence diagrams, along with that of the nested hierarchy of structures in
    multiscale point distributions. We conclude by outlining the potential of persistent
    topology for understanding the connectivity structure of the cosmic web, in large
    simulations of cosmic structure formation and in the challenging context of the
    observed galaxy distribution in large galaxy surveys.
acknowledgement: Part of this work has been supported by the 7th Framework Programme
  for Research of the European Commission, under FETOpen grant number 255827 (CGL
  Computational Geometry Learning) and ERC advanced grant, URSAT (Understanding Random
  Systems via Algebraic Topology) number 320422.
article_processing_charge: No
author:
- first_name: Pratyush
  full_name: Pranav, Pratyush
  last_name: Pranav
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Rien
  full_name: Van De Weygaert, Rien
  last_name: Van De Weygaert
- first_name: Gert
  full_name: Vegter, Gert
  last_name: Vegter
- first_name: Michael
  full_name: Kerber, Michael
  last_name: Kerber
- first_name: Bernard
  full_name: Jones, Bernard
  last_name: Jones
- first_name: Mathijs
  full_name: Wintraecken, Mathijs
  id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
  last_name: Wintraecken
  orcid: 0000-0002-7472-2220
citation:
  ama: Pranav P, Edelsbrunner H, Van De Weygaert R, et al. The topology of the cosmic
    web in terms of persistent Betti numbers. <i>Monthly Notices of the Royal Astronomical
    Society</i>. 2017;465(4):4281-4310. doi:<a href="https://doi.org/10.1093/mnras/stw2862">10.1093/mnras/stw2862</a>
  apa: Pranav, P., Edelsbrunner, H., Van De Weygaert, R., Vegter, G., Kerber, M.,
    Jones, B., &#38; Wintraecken, M. (2017). The topology of the cosmic web in terms
    of persistent Betti numbers. <i>Monthly Notices of the Royal Astronomical Society</i>.
    Oxford University Press. <a href="https://doi.org/10.1093/mnras/stw2862">https://doi.org/10.1093/mnras/stw2862</a>
  chicago: Pranav, Pratyush, Herbert Edelsbrunner, Rien Van De Weygaert, Gert Vegter,
    Michael Kerber, Bernard Jones, and Mathijs Wintraecken. “The Topology of the Cosmic
    Web in Terms of Persistent Betti Numbers.” <i>Monthly Notices of the Royal Astronomical
    Society</i>. Oxford University Press, 2017. <a href="https://doi.org/10.1093/mnras/stw2862">https://doi.org/10.1093/mnras/stw2862</a>.
  ieee: P. Pranav <i>et al.</i>, “The topology of the cosmic web in terms of persistent
    Betti numbers,” <i>Monthly Notices of the Royal Astronomical Society</i>, vol.
    465, no. 4. Oxford University Press, pp. 4281–4310, 2017.
  ista: Pranav P, Edelsbrunner H, Van De Weygaert R, Vegter G, Kerber M, Jones B,
    Wintraecken M. 2017. The topology of the cosmic web in terms of persistent Betti
    numbers. Monthly Notices of the Royal Astronomical Society. 465(4), 4281–4310.
  mla: Pranav, Pratyush, et al. “The Topology of the Cosmic Web in Terms of Persistent
    Betti Numbers.” <i>Monthly Notices of the Royal Astronomical Society</i>, vol.
    465, no. 4, Oxford University Press, 2017, pp. 4281–310, doi:<a href="https://doi.org/10.1093/mnras/stw2862">10.1093/mnras/stw2862</a>.
  short: P. Pranav, H. Edelsbrunner, R. Van De Weygaert, G. Vegter, M. Kerber, B.
    Jones, M. Wintraecken, Monthly Notices of the Royal Astronomical Society 465 (2017)
    4281–4310.
date_created: 2018-12-11T11:49:44Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2023-09-22T09:40:55Z
day: '01'
department:
- _id: HeEd
doi: 10.1093/mnras/stw2862
external_id:
  isi:
  - '000395170200039'
intvolume: '       465'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1608.04519
month: '01'
oa: 1
oa_version: Submitted Version
page: 4281 - 4310
publication: Monthly Notices of the Royal Astronomical Society
publication_identifier:
  issn:
  - '00358711'
publication_status: published
publisher: Oxford University Press
publist_id: '6373'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The topology of the cosmic web in terms of persistent Betti numbers
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 465
year: '2017'
...