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

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 05/21/07 IPAM

N-BODY SIMULATIONS (Efstathiou, Eastwood 1981) 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 05/21/07 IPAM

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 05/21/07 IPAM

Observations 05/21/07 IPAM

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. 2003 05/21/07 IPAM

N-body: Millennium simulation Springel et al. 2004 05/21/07 IPAM

Wilkinson Microwave Anisotropy Probe Full sky map (resolution ~10 arcmin) 05/21/07 IPAM

Dynamical model 05/21/07 IPAM

The Zel’dovich Approximation (1970) The Zel’dovich approximation describes anisotropic collapse

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 05/21/07 IPAM

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. 05/21/07 IPAM

in four cosmological models N-body simulations in four cosmological models 05/21/07 IPAM

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) 05/21/07 IPAM

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 05/21/07 IPAM

Examples of superclusters in LCDM simulation (VIRGO consortium) by SURFGEN Percolating i.e. largest supercluster Sheth, Sahni, Sh, Sathyaprakash 2003, MN 343, 22 05/21/07 IPAM

Fitting voids by ellipsoids Shandarin et al. 2004, MNRAS, Examples of VOIDS Fitting voids by ellipsoids Shandarin,Feldman,Heitmann,Habib 2006, MNRAS 05/21/07 IPAM

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 05/21/07 IPAM

Superclusters vs. Voids Red: super clusters = overdense Blue: voids = underdense Solid: 90% of mass/volume Dashed: 10% of mass/volume Superclusters by mass Voids by volume dashed: the largest object solid: all but the largest 05/21/07 IPAM

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 05/21/07 IPAM

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 05/21/07 IPAM

Minkowski Functionals Mecke, Buchert & Wagner 1994 05/21/07 IPAM

Set of Morphological Parameters 05/21/07 IPAM

Sizes and Shapes For each supercluster or void Sahni, Sathyaprakash & Shandarin 1998 Basilakos,Plionis,Yepes,Gottlober,Turchaninov 2005 05/21/07 IPAM

N-body: Millennium simulation Springel et al. 2004 05/21/07 IPAM

LCDM Superclusters vs Voids Median (+/-) 25% Top 25% log(Length) (radius) Breadth Thickness Shandarin, Sheth, Sahni 2004 05/21/07 IPAM

LCDM Superclusters vs. Voids Median (+/-)25% Top 25% 05/21/07 IPAM

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. 05/21/07 IPAM

Nonlinear evolution: 05/21/07 IPAM

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 05/21/07 IPAM

Swap of amplitudes and phases Courtesy of P. Coles Globe WMAP Phases Phases Amplitudes WMAP phases Globe amplitudes Globe phases WMAP amplitudes 05/21/07 IPAM

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 05/21/07 IPAM