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)