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LCDM vs. SUGRA Betti Numbers : Dark Energy models On the Alpha and Betti of the Cosmos Topology and Homology of the Cosmic Web Pratyush Pranav Warsaw 12th-17th July 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 LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions The Cosmic Web Stochastic Spatial Pattern of Clusters, Filaments & Walls around Voids in which matter & galaxies have agglomerated through gravity 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. LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions Genus, Euler & Betti æ For a surface with c components, the genus G specifies handles on surface, and is related to the Euler characteristic ( ) via: 1 G c (M ) 2 where 1 (M ) 2 Euler characteristic 3-D manifold 1 R1R2 dS & 2-D boundary manifold 1 M M 2 (M ) 2 0 1 2 : Genus, Euler & Betti Euler – Poincare formula Relationship between Betti Numbers & Euler Characteristic d : 1 k k 0 k 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: 1: 2: - independent components independent tunnels independent enclosed voids LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions 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 LSS Distance Function Density Function Filtration Alphashapes Lower-star Filtration Betti Numbers/Persistence 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 Delaunay simplices within spheres radius 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 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 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 LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions Voronoi Kinematic Model: evolving mass distribution in Voronoi skeleton Voids: Voronoi Evolutionary models Distance function Density function Betti Space & Alpha Track 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 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 : (r) r Fractal Dimension : log N (r ) D lim r0 log(1 / r ) Betti Numbers :Soneira-Peebles models Distance function Density function Homology Analysis of evolving LCDM cosmology Betti2: evolving void populations LCDM void persistence LCDM vs. SUGRA Betti Numbers : Dark Energy models Persistent Death Birth LCDM Cosmic Web LSS/Cosmic web Topology/Homology (Euler chr., genus, Betti Numbers) Methods Models and Result Conclusions 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. Gaussian Random Fields: Betti Numbers Distinct sensitivity of Betti curves on power spectrum P(k): unlike genus (only amplitude P(k) sensitive)