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LCDM vs. SUGRA
Betti Numbers : Dark Energy models
On the On the Alpha and Betti of the CosmosAlpha and Betti of the Cosmos
Topology and Homology of the Topology and Homology of the Cosmic Web Cosmic Web
Pratyush PranavPratyush Pranav
Warsaw 12Warsaw 12thth-17-17thth July July
Rien van de Weygaert, Gert Vegter, Herbert Edelsbrunner,Rien van de Weygaert, Gert Vegter, Herbert Edelsbrunner,Changbom Park, Bernard Jones, Pravabati Chingangbam, Michael Kerber, Changbom Park, Bernard Jones, Pravabati Chingangbam, Michael Kerber,
Wojciech Hellwing , Marius Cautun, Patrick Bos, Johan Hidding, Wojciech Hellwing , Marius Cautun, Patrick Bos, Johan Hidding, Mathijs Wintraecken ,Job Feldbrugge, Bob Eldering, Nico Kruithof, Mathijs Wintraecken ,Job Feldbrugge, Bob Eldering, Nico Kruithof,
Matti van Engelen, Eline Tenhave , Manuel Caroli, Monique Teillaud 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 WebThe Cosmic WebStochastic
Spatial
Pattern of
Clusters,
Filaments &
Walls
around
Voids
in which
matter & galaxies
have agglomerated
through gravity
Why Cosmic Web?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 objectsand 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 understandCosmic environment within which to understand
the formation of galaxies.the formation of galaxies.
LSS/Cosmic web
Topology/Homology(Euler chr., genus, Betti Numbers)
Methods
Models and Result
Conclusions
For a surface with c components, the genus G specifies handles on surface, and is related to the Euler characteristic () via:
where
Genus, Euler & BettiGenus, Euler & Betti
1( )
2G c M
1 2
1 1( )
2M dS
R R
0 1 2( ) 2M
1
2M M
Euler characteristic 3-D manifold & 2-D boundary manifold :
Genus, Euler & Genus, Euler & Betti Betti
Euler – Poincare formula
Relationship between Betti Numbers & Euler Characteristic :
0
1d
k
kk
Cosmic Structure HomologyCosmic 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
LSS/Cosmic web
Topology/Homology(Euler chr., genus, Betti Numbers)
Methods
Models and Result
Conclusions
The Cosmic WebThe Cosmic WebWeb 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
Betti Numbers/Persistence
Alphashapes Lower-star Filtration
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 simpliceswithin spheres radius
DTFEDTFE
• 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: Voronoi Kinematic Model:
evolving mass distribution in Voronoi skeleton
Voids: Voronoi Evolutionary models
Distance function Density function
Betti Space & Alpha Track
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
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 :
rr)(
)/1log(
)(loglim
0 r
rND
r
Betti Numbers :Soneira-Peebles models
Distance function Density function
Homology AnalysisHomology Analysis
ofof
evolving LCDM cosmologyevolving LCDM cosmology
Betti2:evolving void populations
LCDM void persistence
LCDM vs. SUGRA
Betti Numbers : Dark Energy models
Persistent LCDM Cosmic Web
Death
Birth
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)