1. Quantitaive Morphology of the Cosmic Web
Sergei Shandarin
University of Kansas
Lawrence
05/21/07
IPAM

2. Outline Introduction: What is Cosmic Web?
Physics of structure formation
Field statistics vs. Object statistics
Shapes of superclusters and voids, percolation, genus
Minkowski functionals
Role of phases
Summary
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3. N-BODY SIMULATIONS (Efstathiou, Eastwood 1981)
Method: P^3M
N grid: ^3
N particles: or less
Initial conditions
(i) Poisson (Om=1, 0.15)
(ii) cells distribution (Om=1)
Boundary cond: Periodic
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4. Cosmic Web: first hints
Observations
Theory
N-body simulation
Gregory & Thompson 1978
Shandarin 1975
2D; Zel’dovich
Approximation
Klypin & Shandarin 1981, 1983
3D, N-body Simulation, HDM
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5. Observations
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6. Sloan Digital Sky Survey (SDSS)
2d projection of 3D galaxy distribution
in the wedge region.
The Earth is at the center;
the radius of the circle
is about 2,500 million light years.
Each dot represents a galaxy.
The galaxy number density field
is highly non-Gaussian.
The visual impression suggests
that many galaxies are concentrated
into filaments.
Present time
Gott III et al
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7. N-body: Millennium simulation
Springel et al. 2004
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8. Wilkinson Microwave Anisotropy Probe
Full sky map
(resolution ~10 arcmin)
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9. Dynamical model
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10. The Zel’dovich Approximation (1970)
The Zel’dovich approximation describes anisotropic collapse

11. ZA vs. Linear model Truncation means that in the initial conditions
P(k) is set to zero
at k > k_nl
ZA vs. Linear model
Truncated Linear
Linear
N-body ZA
Truncated ZA
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12. Dynamical model and archetypical structures
Zel’dovich approximation describes well the structures in the
quazilinear regime and predicts the archetypical structures:
* pancakes,
* filaments,
* clumps.
(Arnol’d, Shandarin, Zel’dovich 1982)
The morphological statistics has a goal to identify and quantify
these structures.
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13. in four cosmological models
N-body simulations
in four cosmological models
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14. Sensitivity to morphology (i.e. to shapes, geometry, topology, …)
Type of statistic Sensitivity to morphology
“blind”
1-point and 2-point functions
“cataract”
3-point, 4-point functions
Examples of statistics sensitive to morphology :
*Percolation (Shandarin 1983)
Minimal spanning tree (Barrow, Bhavsar & Sonda 1985)
*Global Genus (Gott, Melott, Dickinson 1986)
Voronoi tessellation (Van de Weygaert 1991)
*Minkowski Functionals (Mecke, Buchert & Wagner 1994)
Skeleton length (Novikov, Colombi & Dore 2003)
Various void statistics (Aikio, Colberg, El-Ad, Hoyle, Kaufman,
Mahonen, Piran, Ryden, Vogeley, …)
Inversion technique (Plionis, Ragone, Basilakos 2006)
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15. Superclusters and voids
are defined as the regions enclosed by isodensity surfaces
= 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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16. Examples of superclusters in LCDM simulation
(VIRGO consortium) by SURFGEN
Percolating i.e. largest supercluster
Sheth, Sahni, Sh, Sathyaprakash
2003, MN 343, 22
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17. Fitting voids by ellipsoids
Shandarin et al. 2004, MNRAS,
Examples of VOIDS
Fitting voids by ellipsoids
Shandarin,Feldman,Heitmann,Habib
2006, MNRAS
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18. Percolation Gaussian fields Non-linear evolution SC Voids Gauss
P = 1/k^2
P = k^4 SC
Voids
Gauss
Conspicuous connectedness of the web
in LCDM cosmology has two causes:
1) initial spectrum has considerable power at k<k_nl
2) nonlinear dynamics
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19. Superclusters vs. Voids
Red: super clusters = overdense Blue: voids = underdense
Solid: % of mass/volume
Dashed: 10% of mass/volume
Superclusters by mass
Voids by volume
dashed: the largest object
solid: all but the largest
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20. 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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21. SUPERCLUSTERS and VOIDS should be studied before percolation in the
corresponding phase occurs.
Individual SUPERCLUSTERS should be studied at the density contrasts
Individual VOIDS should be studied at the density contrasts
There are 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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22. Minkowski Functionals
Mecke, Buchert & Wagner 1994
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23. Set of Morphological Parameters
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24. Sizes and Shapes For each supercluster or void
Sahni, Sathyaprakash & Shandarin 1998
Basilakos,Plionis,Yepes,Gottlober,Turchaninov 2005
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25. N-body: Millennium simulation
Springel et al. 2004
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26. LCDM Superclusters vs Voids Median (+/-) 25% Top 25% log(Length)
(radius)
Breadth
Thickness
Shandarin, Sheth, Sahni 2004
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27. LCDM
Superclusters vs. Voids
Median (+/-)25%
Top 25%
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28. Correlation with mass (SC) or volume (V)
Genus
Green: at percolation
Red: just before percolation
Blue: just after percolation
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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29. Nonlinear evolution:
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30. Are there other “scales of nonlinearity”?
Ryden, Gramann (1991) showed that phases become significantly different
from the initial values on the scale of average displacement.
Fry, Melott, Shandarin 1993: scaling of 3-point functions with d_rms x
This scale can be computed from the initial power spectrum (Shandarin 1993)
t_0
n = -1 is critical
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31. Swap of amplitudes and phases
Courtesy of P. Coles
Globe
WMAP
Phases
Phases
Amplitudes
WMAP phases
Globe amplitudes
Globe phases
WMAP amplitudes
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32. Summary In the LCDM model there are at least 2 scales of nonlinearity:
thickness ~ breadth ~ 1/k_nl < d_rms ~ length of filaments
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
global genus curve shows no features or peculiarities.
Percolation and genus are different (independent?) characteristics of the web.
Morphological parameters (L,B,T, P,F) can discriminate models.
Voids have more complex topology than superclusters. Voids: G ~ 50; superclusters: G ~ a few
Phases are very important for patterns, but it’s not known how they are related to patterns
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