1. LCDM vs. SUGRA

2. Betti Numbers : Dark Energy models

3. On the Alpha and Betti of the Cosmos Topology and Homology of the Cosmic Web Pratyush Pranav Warsaw 12 th -17 th July

4. Rien van de Weygaert, Gert Vegter, Herbert Edelsbrunner, Changbom Park, Bernard Jones, Pravabati Chingangbam, Michael Kerber, Wojciech Hellwing, Marius Cautun, Patrick Bos, Johan Hidding, Mathijs Wintraecken,Job Feldbrugge, Bob Eldering, Nico Kruithof, Matti van Engelen, Eline Tenhave, 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: Physical Significance:  Manifests mildly nonlinear clustering:  Manifests mildly nonlinear clustering: Transition stage between linear phase Transition stage between linear phase and fully collapsed/virialized objects and fully collapsed/virialized objects  Weblike configurations contain  Weblike configurations contain cosmological information: cosmological information: e.g. Void shapes & alignments (recent study J. Lee 2007) e.g. Void shapes & alignments (recent study J. Lee 2007)  Cosmic environment within which to understand  Cosmic environment within which to understand the formation of galaxies. the formation of galaxies.

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

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

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

11. Cosmic Structure Homology 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 FunctionDensity Function Filtration Betti Numbers/Persistence Alphashapes Lower-star Filtration

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. Voronoi Kinematic Model: evolving mass distribution in Voronoi skeleton

24. Voids: Voronoi Evolutionary models Distance function Density function

25. Betti Space & Alpha Track

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

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 Distance function Density function

29. Homology Analysis of evolving LCDM cosmology Homology Analysis of evolving LCDM cosmology

30. Betti 2 : 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)