06/29/06bernard's cosmic stories1 sergei shandarin university of kansas lawrence statistic of...

06/29/06 Bernard's Cosmic Stories 1

Sergei ShandarinSergei Shandarin

University of KansasLawrence

Statistic of Cosmic Web


“… understanding of what is taking place or has taken place at an early time, is relevant…”

Bernard Jones


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


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


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)


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


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


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


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


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


Cosmic Web: first hintsObservations

Simulations

Gregory & Thompson 1978Klypin & Shandarin 1981 3D N-body Simulation

Shandarin 1975 2D Zel’dovich Approximation


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.)


Lick catalogue

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


Einasto,Klypin,Saar,Shandarin 1984

Redshift catalog

H.Rood, J.Huchra


Field statistics v.s. ‘object’ statistics


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


SDSSslice


Millennium simulation

Springel et al. 2004


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


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)


ZA v.s. Eulerian linear model N-body

Truncated Linear

ZA

Truncated ZA

Linear

Coles et al 1993


ZA v.s. Eulerian linear model

N-body

Truncated Linear

ZA Truncated ZA Linear

Coles et al 1993


Dynamical modelk_nl = 4

k_c = 4 k_c = 32 k_c = 256


Dynamical model

P ~ k^(-2)

P ~ k^0

P ~ k^2


Little, Weinberg, Park 1991 Melott, Shandarin 1993


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.


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


-

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


Superclusters in LCDM simulation (VIRGO consortium)by SURFGEN

Sheth, Sahni, Shandarin, Sathyaprakash 2003, MN 343, 22

Percolating i.e. largest supercluster


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−=


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.


Genus vs. Percolation

Genus as a function of Filling Factor

PERCOLATION RatioGenus of the LargestGenus of Exc. Set

Red: SuperclustersBlue: VoidsGreen: Gaussian


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


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


Percolation thresholdsare easy to detect

Blue: mass estimatorRed: volume estimatorGreen: area estimatorMagenta: curvature estimator

Superclusters

Voids

Gauss

Gauss


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


Toy Example: Triaxial Torus

For all Genus = - 1 !

red points


Superclusters vs Voids

log(Length)

Breadth

Thickness

LCDM

Median (+/-) 25% Top 25%

Shandarin, Sheth, Sahni 2004


Are there olther “scales of nonlinearity”?

Fry, Melott, Shandarin 1993


Superclusters vs. VoidsLCDM

Median (+/-)25% Top 25%


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


More examples of voids in the density destribution in LCDM


SDSS mock catalogCole et al. 1998

Volume limited catalogJ. Sheth 2004

1

Smoothing scale

for density fields

6SL h Mpc−=


J. Sheth 2004


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

06/29/06bernard's cosmic stories1 sergei shandarin university of kansas lawrence statistic of...

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