Geometry and Topology

1
Geometry and Topology of the Cosmic Web
Sergei Shandarin
University of Kansas Lawrence CITA Toronto
1/1-7/15, 2006
2
Outline

• Introduction
• Cosmological model. What is Cosmic Web?
• Gravitational Instability
• Zeldovich approximation,Adhesion Approximation
• Morphological Statistics for LSS
• Percolation. Genus. Minkowski Functionals.
  Shapefinders
• Summary and remaining questions

3
History of the Universe


4
CMB Power and Anisotropy Spectra
COBE
5
Wilkinson MAP Full sky map (resolution 10
arcmin)

6
CMB power spectrum
Percival 2006
7
APM
8
2dFGRS
9
SDSS
Gott III etal. astro-ph/0310571
10
Power spectrum from galaxy surveys
wavelength
11
Cosmological Parameters





12
Marinoni et al 2005
13
Cosmic Web first hintsObservations
Simulations
Shandarin 1975 2D Zeldovich
Approximation
Gregory Thompson 1978
Klypin Shandarin 1983 3D
N-body Simulation
14
(No Transcript)
15
Changing Variables
16
(No Transcript)
17
Zeldovich Approximation (2D)
N-body simulations (2D)
versus
18
Adhesion Approximation
Gurbatov, Saichev, Shandarin 1985, 1987
Burgers equation
Graphical solution in 1D
Parabola
Fig. from Williams, Heavens, Peacock, Shandarin,
1991
19
Evolution in 1D
x
Fig. from Sahni, Sathyaprakash Shandarin 1994
20
Graphical solution in 2D
Initial (i.e. linear) potential
Paraboloid in 2D and 3D
Fig from Kofman, Pogosyan, Shandarin 1990
21
Evolution in 2D
Fig. from Sahni, Sathyaprakash, Shandarin 1994
22
Comparison of Adhesion Approximatin with N-body
Simulation in 2D (tessellation)
Kofman, Pogosyan, Shandarin, Melott 1994
23
Adhesion Approximation mock redsift survey
Weinberg Gunn 1990
Jones 1999 Adhesion model for two component
system
24
What kind of tessellation is this?
Peaks of the initial potential
25
Morphological Statistics for Cosmic Web (Cosmic
Foam?)
26
(No Transcript)
27
Filling Factor of overdense regions
DM density must be smoothed!
28
Lorenson Cline 1987 Sheth, Sahni, Shandarin,
Sathyaprakash 2004
29
Superclusters in LCDM simulation (VIRGO
consortium) by SURFGEN
Percolating i.e. largest supercluster
Sheth, Sahni, Shandarin, Sathyaprakash 2003, MN
343, 22 astro-ph/0210136
30
VOIDS
Shandarin et al. 2004, MNRAS,
31
Approximation of voids by ellipsoids
uniform void has the same
inertia
tensor as the uniform ellipsoid
Shandarin, Feldman, Heitmann, Habib 2006
32
Minkowski Functionals
Introduction to cosmology Mecke, Buchert
Wagner 1994
33
Set of Morphological Parameters
34
Sizes and Shapes
For each supercluster or void
Sahni, Sathyaprakash Shandarin 1998
Convex boundaries !
35
Toy Example Triaxial Torus
Sahni, Sathyaprakash Shandarin 1998
36
Superclusters vs.. Voids
Red super clusters overdense
Blue voids underdense
Solid 90 Dashed 10 Superclusters
by mass Voids by volume
dashed the largest object solid all but
the largest
Shandarin, Sheth, Sahni 2004
37
Superclusters vs. Voids
Red super clusters overdense
Blue voids underdense
Solid 90 Dashed 10 Superclusters
by mass Voids by volume
dashed the largest object solid all but
the largest
38
Blue mass estimator Red volume
estimator Green area estimator Magenta
curvature estimator
Percolation thresholds
Gauss
Gauss
Superclusters
Voids
39
LCDM
Superclusters vs. Voids
Length Breadth Thickness
Planarity Filamentarity
Mass Volume Density
40
Correlation with mass (SC) or volume (V)
Genus (SC)
Green at percolation Red just before
percolation Blue just after percolation
Planarity Filamentarity
log(Length) Breadth Thickness
log(Genus) (V)
Solid lines mark the radius of sphere having
same volume as the object.
41
SDSS mock catalog Cole et al. 1998
Volume limited catalog J. Sheth 2003
42
Cumulative probability functions
Top curves TCDM Bottom curves LCDM
J. Sheth 2003
43
Summary and questions
Adhesion model approximates the structure as a
tessellation (Not Voronoi) Cosmic foam!? How to
find the skeleton from galaxy distribution?
Voronoi diagrams?
LCDM density field in real space seen with
resolution 5/h Mpc displays filaments but
no isolated pancakes have been detected so far!
Smoothing? Web has both characteristics
filamentary network and bubble structure

(at different density thresholds !)
At percolation number of superclusters/voids,
volume, mass and other parameters of the largest
supercluster/void rapidly change (phase
transition) but
genus curve shows no features/
peculiarites. Percolation and genus are
different (independent?) characteristics of the
web.
Morphological parameters (L,B,T, P,F) can
discriminate models. EC Genus --gt Betti
numbers?
Voids have complex substructure. Isolated
structures are possible along with
tunnels. Voids have more complex topology than
superclusters. Voids G50 superclusters Ga few
44
(No Transcript)
45
Genus vs. Percolation
Red Superclusters Blue Voids Green Gaussian
Genus as a function of Filling Factor
PERCOLATION Ratio Genus of the
Largest Genus of Exc. Set
46
At percolation number of superclusters/voids and
volume, mass and other parameters of the
largest supercluster/void rapidly
change but (global) genus curve shows no
peculiarity