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06/29/06 Bernard's Cosmic Stories 1
Sergei ShandarinSergei Shandarin
University of KansasLawrence
Statistic of Cosmic Web
06/29/06 Bernard's Cosmic Stories 2
“… understanding of what is taking place or has taken place at an early time, is relevant…”
Bernard Jones
06/29/06 Bernard's Cosmic Stories 3
Plan• 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
06/29/06 Bernard's Cosmic Stories 4
1971 Peebles A&A 11, 377
Rotation of Galaxies and the Gravitational InstabilityPicture
Method: Direct Summation
N particles: 90
Initial conditionscoordinates: Poissonvelocities: v=Hr(1-0.05) 30 internal v=Hr(1+0.025) 60 external
Boundary cond: No particles at R>R_0
06/29/06 Bernard's Cosmic Stories 5
1978 Peebles A&A 68, 345
Stability of a HierarchicalClustering in the DistributionOf Galaxies
Method: Direct Summation
N particles: 256
Initial conditionscoordinates: Soneira, Peebles’ modelvelocities: virial for each subclump
Boundary cond: Empty space
(*) Two types of particles (m=1, m=0)
06/29/06 Bernard's Cosmic Stories 6
1979 Efstathiou, Jones MNRAS, 186,133
The Rotation of Galaxies:Numerical investigationOf the Tidal Torque Theory
Method: Direct Summation (Aarseth’ code)
N particles: 1000
Initial conditionscoordinates: Poisson 10 inner particles m=10 990 particles m=1
velocities: v=Hr
Boundary cond: No particles at R>R_0
06/29/06 Bernard's Cosmic Stories 7
1979 Aarseth, Gott III, Ed TurnerApJ, 228, 664
N-body Simulations of GalaxyClustering. I. Initial Conditions and Galaxy Collapse Time
Method: Direct Summation (Aarseth’s code)
N particles: 4000
Initial conditionscoordinates: On average 8 particlesare randomly placed on random 125 rodsThis mimics P = k^(-1) spectrum velocities: v=Hr
Boundary cond: reflection on the sphere
Z=14.2
Z=0
06/29/06 Bernard's Cosmic Stories 8
1980 Doroshkevich, Kotok, Novikov, Polyudov, Shandarin, Sigov MNRAS, 192, 321
Two-dimensional Simulations of the Gravitaional System Dynamicsand Formation of the Large-Scale Structure of the Universe
Initial conditions: Growing mode, Zel’dovich approximation
06/29/06 Bernard's Cosmic Stories 9
1981 Efstathiou, Eastwood MNRAS, 194, 503On the Clustering of Particles in an Expanding Universe
Method: P^3M
N grid: 32^3N particles: 20000 or less
Initial conditions (i) Poisson (Om=1, 0.15) (ii) cells distribution (Om=1) Boundary cond: Periodic
06/29/06 Bernard's Cosmic Stories 10
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: 32^3N 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
06/29/06 Bernard's Cosmic Stories 11
Cosmic Web: first hintsObservations
Simulations
Gregory & Thompson 1978Klypin & Shandarin 1981 3D N-body Simulation
Shandarin 1975 2D Zel’dovich Approximation
06/29/06 Bernard's Cosmic Stories 12
1985 Efstathiou, Davis, Frenk, White ApJS, 57, 241Numerical Techniques for Large Cosmological N-body Simulations
Methods: PM, P^3M Initial conditions: Growing mode, Zel’dovich approximationA separate section is devoted to the description of generating initial conditions (IV. SETTING UP INITIAL CONDITIONS” pp 248-250).Quote:
Boundary cond: PeriodicTest of accuracy: comparison with 1D (ref to Klypin and Shand.)
06/29/06 Bernard's Cosmic Stories 15Soneira & Peebles 1978
Both distributions have similar 1-point, 2-point, 3-point, and 4-point correlation functions
Lick catalogvs
simulated
06/29/06 Bernard's Cosmic Stories 16
Einasto,Klypin,Saar,Shandarin 1984
Redshift catalog
H.Rood, J.Huchra
06/29/06 Bernard's Cosmic Stories 18
Sensitivity to morphology (i.e. to shapes, geometry, topology, …)
Type of statistic Sensitivity to morphology
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)
“cataract”
“blind”
3-point, 4-point functions
1-point and 2-point functions
06/29/06 Bernard's Cosmic Stories 22
Dynamical model
* Nonlinear scale R_nl ~1/k_nl
* Small scales r < R_nl : hierarchical clustering
* Large scale r > R_nl : linear model
* Large scale r > R_nl : Zel’dovich approximation
OR
Zel’dovich Approximation (1970)
in comoving coordinates
potential perturbations
Density
are eigen values of
is a symmetric tensor
Density becomes
06/29/06 Bernard's Cosmic Stories 24
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)
06/29/06 Bernard's Cosmic Stories 25
ZA v.s. Eulerian linear model N-body
Truncated Linear
ZA
Truncated ZA
Linear
Coles et al 1993
06/29/06 Bernard's Cosmic Stories 26
ZA v.s. Eulerian linear model
N-body
Truncated Linear
ZA Truncated ZA Linear
Coles et al 1993
06/29/06 Bernard's Cosmic Stories 30
Dynamical model and archetypical structures
Zel’dovich approximation describes well the structures in thequazilinear regime and therefore the archetypical structuresare pancakes, filaments and clumps. The morphological technique is aimed to dettect and measure such structures.
06/29/06 Bernard's Cosmic Stories 31
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
06/29/06 Bernard's Cosmic Stories 32
-
3
1
start
3
0
CDM
256
239
CDM
256
239.5
# of particles
Box size [h ]
z 30
.5
5
1
0
Mpc
τ
Ω
Λ
8
Hubble const. h (initial spectrum) (normali
0.30
zati
00.50.21
on) 0.
.70.70
.2
6 1
0.9σ
ΛΩ
Γ
06/29/06 Bernard's Cosmic Stories 33
Superclusters in LCDM simulation (VIRGO consortium)by SURFGEN
Sheth, Sahni, Shandarin, Sathyaprakash 2003, MN 343, 22
Percolating i.e. largest supercluster
06/29/06 Bernard's Cosmic Stories 34
Superclusters vs.. VoidsRed: super clusters = overdense Blue: voids = underdense
dashed: the largest objectsolid: all but the largest
Solid: 90% of mass/volume Dashed: 10% of mass/volume
Superclusters by massVoids by volume
15sL h Mpc−=
06/29/06 Bernard's Cosmic Stories 35
SUPERCLUSTERS and VOIDS should be studied before percolation in the corresponding phase occurs.
Individual SUPERCLUSTERS should be studied at the density contrasts corresponding to filling factors
Individual VOIDS should be studied at density contrastscorresponding to filling factors
1.8δ ≥0.07CFF ≤
0.5δ ≤−0.22VFF ≤
CAUTION: The above parameters depend on smoothing scale and filter Decreasing smoothing scale i.e. better resolution results ingrowth of the critical density contrast for SUPERCLUSTERS but decrease critical Filling Factor
decrease critical density contrast for VOIDSbut increase the critical Filling Factor
There are practically only two very complex structures in between: infinite supercluster and void.
06/29/06 Bernard's Cosmic Stories 36
Genus vs. Percolation
Genus as a function of Filling Factor
PERCOLATION RatioGenus of the LargestGenus of Exc. Set
Red: SuperclustersBlue: VoidsGreen: Gaussian
06/29/06 Bernard's Cosmic Stories 37
Minkowski Functionals
1 2
Surface Area:
1 1 1Integrated Mean
Volume :
Curvature :
2
Integrated
G
S
S
daR R
daA
C
V
⎛ ⎞= +⎜ ⎟
⎝ ⎠
=∫
∫∫
∫
Ò
Ò
1
1 2
2
1 1aussian Curvature (EC):
2
where R and R
a
Gen
re the p
us: 1 /
rincipal curvature ra
2
dii
S
daRR
G
χπ
χ
=
= −
∫∫Ò
Mecke, Buchert & Wagner 1994
06/29/06 Bernard's Cosmic Stories 38
Partial Minkowski Functionals volume of supercluster or void area of the surface integrated mean curvature genus
i
i
i
i
vacg
MFs of percolating supercluster or void , , , pp p pV A C G
,
Global MFs:
, , i i i ivV A a C c G g= = = =∑ ∑ ∑ ∑
Set of Morphological Parameters
06/29/06 Bernard's Cosmic Stories 39
Percolation thresholdsare easy to detect
Blue: mass estimatorRed: volume estimatorGreen: area estimatorMagenta: curvature estimator
Superclusters
Voids
Gauss
Gauss
06/29/06 Bernard's Cosmic Stories 40
Sizes and Shapes
Sphere:
C
4
T=B=L=R
3Thickness:
Breadth: B
Length:
VT
AAC
Lπ
=
=
=
Sphere: P=F=0
B - TPlanarity: P = B+
L
- BFilamentarity: F = L
SHAPEFINDER
T
S
+B
Sahni, Sathyaprakash & Shandarin 1998
For each supercluster or void
Basilakos,Plionis,Yepes,Gottlober,Turchaninov 2005
06/29/06 Bernard's Cosmic Stories 42
Superclusters vs Voids
log(Length)
Breadth
Thickness
LCDM
Median (+/-) 25% Top 25%
Shandarin, Sheth, Sahni 2004
06/29/06 Bernard's Cosmic Stories 43
Are there olther “scales of nonlinearity”?
Fry, Melott, Shandarin 1993
06/29/06 Bernard's Cosmic Stories 45
Correlation with mass (SC)or volume (V)
log(Length)BreadthThickness
PlanarityFilamentarity
Genus
log(Genus)
Green: at percolationRed: just before percolationBlue: just after percolation
Solid lines mark the radiusof sphere having same volume as the object.
SC
V
Approximation of voids by ellipsoids: uniform void has the same inertia tensor as the uniform ellipsoid
Shandarin, Feldman, Heitmann,Habib 2006
06/29/06 Bernard's Cosmic Stories 48
SDSS mock catalogCole et al. 1998
Volume limited catalogJ. Sheth 2004
1
Smoothing scale
for density fields
6SL h Mpc−=
06/29/06 Bernard's Cosmic Stories 50
SummaryLCDM: 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