Statistic of Cosmic Web
Sergei Shandarin
University of Kansas
Lawrence
06/29/06
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“… understanding of what is taking place
or has taken place at an early time, is relevant…”
Bernard Jones
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Plan
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Introduction: What is Cosmic Web?
Field statistics v.s. Object statistics
Dynamical model
Minkowski functionals
Scales of LSS structure in Lambda CDM cosmology
How many scales of nonlinearity?
Substructure in voids
Summary
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1971 Peebles A&A 11, 377
Rotation of Galaxies and
the Gravitational Instability
Picture
Method:
N particles:
Direct Summation
90
Initial conditions
coordinates: Poisson
velocities: v=Hr(1-0.05) 30 internal
v=Hr(1+0.025) 60 external
Boundary cond: No particles
at R>R_0
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Method:
Direct Summation
1978 Peebles A&A 68, 345
N particles:
Stability of a Hierarchical
Clustering in the Distribution
Of Galaxies
256
Initial conditions
coordinates: Soneira, Peebles’ model
velocities: virial for each subclump
Boundary cond: Empty space
(*) Two types of particles (m=1, m=0)
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1979 Efstathiou, Jones MNRAS, 186,133
The Rotation of Galaxies:
Numerical investigation
Of the Tidal Torque Theory
Method:
Direct Summation
(Aarseth’ code)
N particles:
1000
Initial conditions
coordinates: Poisson
10 inner particles m=10
990 particles m=1
velocities: v=Hr
Boundary cond: No particles
at R>R_0
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1979 Aarseth, Gott III, Ed Turner
ApJ, 228, 664
Z=14.2
N-body Simulations of Galaxy
Clustering. I. Initial Conditions
and Galaxy Collapse Time
Method:
Direct Summation
(Aarseth’s code)
N particles:
Z=0
4000
Initial conditions
coordinates: On average 8 particles
are randomly placed on random 125 rods
This mimics P = k^(-1) spectrum
velocities: v=Hr
Boundary
cond: reflection on the sphere
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1980 Doroshkevich, Kotok, Novikov, Polyudov, Shandarin, Sigov
MNRAS, 192, 321
Two-dimensional Simulations of the Gravitaional System Dynamics
and Formation of the Large-Scale Structure of the Universe
Initial conditions: Growing mode,
Zel’dovich approximation
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1981 Efstathiou, Eastwood
MNRAS, 194, 503
On the Clustering of Particles in an Expanding Universe
Method:
P^3M
N grid:
32^3
N particles: 20000 or less
Initial conditions
(i) Poisson (Om=1, 0.15)
(ii) cells distribution (Om=1)
Boundary cond: Periodic
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1983 Klypin, Shandarin, MNRAS, 204, 891
Three-dimensional Numerical Model
of the Formation of Large-Scale Structure in the Universe
Method:
PM=CIC
N grid:
32^3
N particles: 32^3
Initial conditions: Growing mode,
Zel’dovich approximation
Boundary cond: Periodic
First time reported
at the Erici workshop
organized by Bernard
in 1981
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Cosmic Web: first hints
Observations
Gregory & Thompson 1978
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Simulations
Shandarin 1975
2D
Zel’dovich
Approximation
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Klypin & Shandarin 1981
3D
N-body Simulation
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1985 Efstathiou, Davis, Frenk, White ApJS, 57, 241
Numerical Techniques for Large Cosmological N-body Simulations
Methods:
PM, P^3M
Initial conditions: Growing mode, Zel’dovich approximation
A separate section is devoted to the description of generating initial
conditions (IV. SETTING UP INITIAL CONDITIONS” pp 248-250).
Quote:
Boundary cond: Periodic
Test of accuracy: comparison with 1D (ref to Klypin and Shand.)
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Lick catalogue
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Lick catalog
vs simulated
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Both distributions have similar
1-point, 2-point, 3-point, and 4-point correlation functions
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Soneira & Peebles 1978
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Einasto,
Klypin,
Saar,
Shandarin 1984
Redshift catalog
H.Rood, J.Huchra
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Field statistics v.s. ‘object’ statistics
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Sensitivity to morphology (i.e. to shapes, geometry, topology, …)
Type of statistic
Sensitivity to morphology
1-point and 2-point functions
3-point, 4-point
functions
“blind”
“cataract”
Examples of statistics sensitive to morphology :
*Percolation
Minimal spanning tree
*Global Genus
Voronoi tessellation
*Minkowski Functionals
Skeleton length
Various void statistics
Inversion technique
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(Shandarin 1983)
(Barrow, Bhavsar & Sonda 1985)
(Gott, Melott, Dickinson 1986)
(Van de Weygaert 1991)
(Mecke, Buchert & Wagner 1994)
(Novikov, Colombi & Dore 2003)
(Aikio, Colberg, El-Ad, Hoyle, Kaufman,
Mahonen, Piran, Ryden, Vogeley, …)
(Plionis, Ragone, Basilakos 2006)
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SDSS
slice
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Millennium simulation
Springel et al. 2004
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Dynamical model
* Nonlinear scale R_nl ~1/k_nl
* Small scales r < R_nl : hierarchical clustering
* Large scale r > R_nl : linear model
OR
* Large scale r > R_nl : Zel’dovich approximation
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Zel’dovich Approximation (1970)
in comoving coordinates
Density
potential
perturbations
is a symmetric tensor
Density becomes
are eigen values of
ZA: Examples of typical errors/mistakes
* ZA is a kinematic model and thus does not take into account gravity
* ZA can be used only in Hot Dark Matter model
( initial spectrum must have sharp cutoff on small scales)
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Coles et al 1993
ZA v.s. Eulerian linear model
Linear
N-body
Truncated Linear
Truncated ZA
ZA
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Coles et al 1993
ZA v.s. Eulerian linear model
Truncated Linear
N-body
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ZA
Truncated ZA
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Linear
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Dynamical model
k_nl = 4
k_c = 4
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k_c = 32
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k_c = 256
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Dynamical model
P ~ k^(-2)
P ~ k^0
P ~ k^2
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Little, Weinberg, Park 1991
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Melott, Shandarin 1993
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Dynamical model and archetypical structures
Zel’dovich approximation describes well the structures in the
quazilinear regime and therefore the archetypical structures
are pancakes, filaments and clumps. The morphological technique is
aimed to dettect and measure such structures.
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Superclusters and voids
are defined as the regions enclosed by isodensity surface
= excursion set regions
* Interface surface is build by SURFGEN algorithm, using linear interpolation
* The density of a supercluster is higher than the density of the boundary surface.
The density of a void is lower than the density of the boundary surface.
* The boundary surface may consist of any number of disjointed pieces.
* Each piece of the boundary surface must be closed.
* Boundary surface of SUPERCLUSTERS and VOIDS cut by volume boundary
are closed by corresponding parts of the volume boundary
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 CDM CDM
# of particles
2563
2563
Box size [h -1 Mpc] 239.5 239.5
zstart
50
30
0
1
0.3

0
0.7
Hubble const. h
0.5
 (initial spectrum) 0.21
 8 (normalization) 0.6
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0.7
0.21
0.9
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Superclusters in LCDM simulation (VIRGO consortium)
by SURFGEN
Percolating i.e. largest supercluster
Sheth, Sahni, Shandarin, Sathyaprakash
2003, MN 343, 22
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Superclusters vs.. Voids
Red: super clusters = overdense
Blue:
Ls  5h 1 Mpc
voids = underdense
Solid: 90% of mass/volume
Dashed: 10% of mass/volume
dashed: the largest object
solid: all but the largest
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Superclusters by mass
Voids by volume
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SUPERCLUSTERS and VOIDS should be studied before percolation in the
corresponding phase occurs.
Individual SUPERCLUSTERS should be studied
at the density contrasts   1.8
corresponding to filling factors FFC  0.07
Individual VOIDS should be studied at density contrasts
corresponding to filling factors FF  0.22
  0.5
V
There are practically only two very complex structures in between:
infinite supercluster and void.
CAUTION:
The above parameters depend on smoothing scale and filter
Decreasing smoothing scale i.e. better resolution
results in
growth of the critical density contrast for SUPERCLUSTERS
but decrease critical Filling Factor
decrease critical density contrast for VOIDS
but increase the critical Filling Factor
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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
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Minkowski Functionals
Volume :
V
Surface Area:
A   da
S
Integrated Mean Curvature :
Integrated Gaussian Curvature (EC):
where R1 and R 2
1
 1
1 
C      da
2 S  R1 R2 
1

2

S
1
da
R1 R2
Genus: G  1   / 2
are the principal curvature radii
Mecke, Buchert & Wagner 1994
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Set of Morphological Parameters
Partial Minkowski Functionals
vi volume of supercluster or void
ai area of the surface
ci integrated mean curvature
gi genus
MFs of percolating supercluster or void
V p , Ap , C p , G p
Global MFs:
V   vi , A   ai , C   ci , G   gi
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Percolation thresholds
are easy to detect
Gauss
Blue:
mass estimator
Red:
volume estimator
Green: area estimator
Magenta: curvature estimator
Gauss
Superclusters
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Voids
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Sizes
and
3V
Thickness: T 
A
A
Breadth: B 
C
Length:
C
L
4
Sphere:
T=B=L=R
Shapes
For each supercluster or void
SHAPEFINDERS
P=
B-T
B+T
Filamentarity: F =
L-B
L+B
Planarity:
Sahni, Sathyaprakash & Shandarin 1998
Sphere: P=F=0
Basilakos,Plionis,Yepes,Gottlober,Turchaninov 2005
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Toy Example: Triaxial Torus
For all
Genus = - 1 !
red points
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LCDM Superclusters vs Voids
Top 25%
Median (+/-) 25%
log(Length)
Breadth
Thickness
Shandarin, Sheth, Sahni 2004
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Are there olther “scales of nonlinearity”?
Fry, Melott, Shandarin 1993
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LCDM Superclusters vs. Voids
Top 25%
Median (+/-)25%
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Correlation with mass (SC)
or volume (V)
Green: at percolation
Red: just before percolation
Blue: just after percolation
SC
Genus
Planarity
Filamentarity
log(Length)
Breadth
Thickness
V
log(Genus)
Solid lines mark the radius
of sphere having
same volume as the object.
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Approximation of voids by ellipsoids:
uniform void has the same inertia
tensor as the uniform ellipsoid
Shandarin, Feldman, Heitmann,
Habib 2006
More examples of voids in the density destribution in LCDM
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SDSS mock catalog
Cole et al. 1998
Volume limited catalog
J. Sheth 2004
Smoothing scale
for density fields
LS  6h 1Mpc
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J. Sheth 2004
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Summary
LCDM: density field in real space seen with resolution 5/h Mpc displays filaments
but no isolated pancakes have been detected.
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/peculiarities.
Percolation and genus are different (independent?) characteristics of the web.
Morphological parameters (L,B,T, P,F) can discriminate models.
Voids defined as closed regions in underdense excursion set are different from common-view voids.
Why? 1) different definition, 2) uniform 5 Mpc smoothing, 3) DM distribution 4) real space
Voids have complex substructure. Isolated clumps may present along with filaments.
Voids have more complex topology than superclusters. Voids: G ~ 50; superclusters: G ~ a few
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