Exploring Cosmic Web Topology and Homology with Betti Numbers

1. LCDM vs. SUGRA

2. Betti Numbers : Dark Energy models

3. On the Alpha and Betti of the CosmosTopology and Homology of the Cosmic Web PratyushPranav Warsaw 12th-17th July

4. Rien van de Weygaert, GertVegter, Herbert Edelsbrunner,Changbom Park, Bernard Jones, PravabatiChingangbam, Michael Kerber, WojciechHellwing , Marius Cautun, Patrick Bos, Johan Hidding, MathijsWintraecken ,Job Feldbrugge, Bob Eldering, NicoKruithof, Matti van Engelen, ElineTenhave , Manuel Caroli, Monique Teillaud

5. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions

6. The Cosmic Web Stochastic Spatial Pattern of  Clusters,  Filaments &  Walls around  Voids in which matter & galaxies have agglomerated through gravity

7. Why Cosmic Web? Physical Significance:  Manifests mildly nonlinear clustering: Transition stage between linear phase and fully collapsed/virialized objects  Weblike configurations contain cosmological information: e.g. Void shapes & alignments (recent study J. Lee 2007)  Cosmic environment within which to understand the formation of galaxies.

8. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions

9. Genus, Euler & Betti • For a surface with c components, the genus G specifies  handles on • surface, and is related to the Euler characteristic () via: • where •  Euler characteristic 3-D manifold  & 2-D boundary manifold :

10. Genus, Euler & Betti •  Euler – Poincare formula • Relationship between Betti Numbers & Euler Characteristic :

11. Cosmic Structure Homology • Complete quantitative characterization of homology in terms of • Betti Numbers • Betti number k: - rank of homology groups Hp of manifold • - number of k-dimensional holes of an • object or shape • 3-D object, e.g. density superlevel set: • 0: -  independent components • 1: -  independent tunnels • 2: -  independent enclosed voids

12. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions

13. The Cosmic Web • Web Discretely Sampled: • By far, most information • on the Cosmic Web concerns • discrete samples: • observational: • Galaxy Distribution • theoretical: • N-body simulation particles

14. LSS Distance Function Density Function Filtration Lower-star Filtration Alphashapes Betti Numbers/Persistence

15. Alphashapes • Exploiting the topological information contained in the Delaunay Tessellation of the galaxy distribution • Introduced by Edelsbrunner & collab. (1983, 1994) • Description of intuitive notion of the shape of a discrete point set • subset of the underlying triangulation

16. Delaunay simplices within spheres radius 

17. DTFE • Delaunay Tessellation Field Estimator • Piecewise Linear representation • density & other discretely sampled fields • Exploits sample density & shape sensitivity of • Voronoi & Delaunay Tessellations • Density Estimates from contiguous Voronoi cells • Spatial piecewise linear interpolation by means of • Delaunay Tessellation

19. Persistence : search for topological reality Concept introduced by Edelsbrunner: Reality of features (eg. voids) determined on the basis of -interval between “birth” and “death” of features Pic courtsey H. Edelsbrunner

21. Persistence in the Cosmic Context • Natural description for hierarchical structure formation • Can probe structures at all cosmic-scale • Filtering mechanism – can be used to concentrate on structures persistent in a in a specific range of scales

22. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions

23. VoronoiKinematic Model: evolving mass distribution in Voronoi skeleton

24. Voids: Voronoi Evolutionary models Density function Distance function

25. Betti Space & Alpha Track

26. Void evolution Voronoi Points shift away from diagonal as voids grow General reduction in compactness of points on persistence diagram Fig : Persistence Diagram of Void Growth

27. Soneira-Peebles Model • Mimics the self-similarity of observed angular distribution of galaxies on sky • Adjustable parameters • 2-point correlation can be evaluated analytically Correlation function : Fractal Dimension :

28. Betti Numbers :Soneira-Peebles models Density function Distance function

29. Homology Analysisofevolving LCDM cosmology

30. Betti2:evolving void populations

31. LCDM void persistence

32. LCDM vs. SUGRA

33. Betti Numbers : Dark Energy models

34. Persistent LCDM Cosmic Web Death  Birth 

35. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions

36. Betti Numbers • Signals from all scales in a multi-scale distribution – suitable for hierarchical LSS. • Signals from different morphological components of the LSS – discriminator for filamentary/wall-like topology. Persistence • Persistence as a probe for analyzing the systematics of matter distribution as a function of single parameter “life interval” (hierarchy) • Persistence robust against small scale noise • Data doesn’t need to be smoothed.

37. Gaussian Random Fields:Betti Numbers Distinct sensitivity of Betti curves on power spectrum P(k): unlike genus (only amplitude P(k) sensitive)