Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web
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Betti numbers of Gaussian random fields carry strictly more information concerning the random field than the Euler characteristic alone.
“More recently, Park et al. (2013) and van de Weygaert et al. (2011) have shown that the Betti numbers of Gaussian random fields carry strictly more information concerning the random field than the Euler characteristic alone.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as van de Weygaert et al. (2011)
The slope of the power spectrum affects the shape of the curves of the Betti numbers as the threshold r changes, while the shape of the curve of Euler characteristics is independent of it.
“Specifically, it is found that an invariant of the Gaussian field called the slope of the power spectrum affects the shape of the curves of the Betti numbers (as the threshold r changes), while the shape of the curve of Euler characteristics is independent of it.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as van de Weygaert et al. (2011)
van de Weygaert et al. (2011) used the α complex at various scales to study the Betti numbers produced by various stochastic process models, including some proposed for cosmological problems.
“This method is used by van de Weygaert et al. (2011) to study the Betti numbers produced by various stochastic process models, including some which are explicitly proposed for cosmological problems.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as van de Weygaert et al. (2011)
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References 78 · Cited by 84
- Authors
- van de Weygaert, R., Vegter, G., Edelsbrunner, H., Jones, B. J. T., Pranav, P., Park, C., Hellwing, W. A., Eldering, B., Kruithof, N., Bos, E. G. P., Hidding, J., Feldbrugge, J., ten Have, E., van Engelen, M., Caroli, M., & Teillaud, M.
- Affiliations
- University of Groningen (NL)
- Institute of Science and Technology Austria (AT)
- Korea Institute for Advanced Study (KR)
- Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw (PL)
- Institut national de recherche en sciences et technologies du numérique (FR)
- ORCID
- Venue
- Lecture Notes in Computer Science
- Publisher
- Springer Berlin Heidelberg
- Published
- 2011
- Issue
- 6970
- Pages
- 60-101
- ISSN
- 0302-9743 (ISSN-L)
- 1611-3349
- OpenAlex
- W2111169612
- Cited by
- 84
- References
- 78
- Citations by year
- 2026: 4 · 2025: 3 · 2024: 9 · 2023: 6 · 2022: 4 · 2021: 11 · 2020: 4 · 2019: 11 · 2018: 7 · 2017: 5 · 2016: 4 · 2015: 5 · 2014: 5 · 2013: 5 · 2012: 1
- FWCI
- 3.02
- Type
- conference paper
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Lecture notes in computer science (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion)
- arXiv (Cornell University)
- Funders
- European Commission (255827)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Galaxies: Formation, Evolution, Phenomena
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- Betti numbers, alpha shapes, persistent homology, Delaunay tessellation, galaxy distribution, dark energy, primordial non-Gaussianity, LCDM, hierarchical structure formation, superlevel sets, simplicial complexes
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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