Dynamics & Dynamics & Multiscale Multiscale

Morphology Morphology

Cosmic Web Cosmic Web

Rien van de Weygaert

3rd KIAS Workshop, Oct 27-28, Seoul, 2008

Collaboration & References:Collaboration & References:

Pablo Araya-MeloPablo Araya-Melo

Miguel Aragon-Calvo Miguel Aragon-Calvo

Erwin Platen Erwin Platen

Emilio Romano-Diaz Emilio Romano-Diaz

Willem Schaap Willem Schaap

Bernard Jones Bernard Jones

Gert Vegter (comp. geometry & Gert Vegter (comp. geometry &

topology) topology)

Foams in Foams in NatureNature

Foams in Foams in NatureNature

The largest complex pattern found in nature is Megaparsec Scale:

the Cosmic Web

manifestation of the gravitational origin, emergence and growth of ALL structure in the cosmos …

The Cosmic The Cosmic WebWebStochastic

Spatial

Pattern of

Clusters,

Filaments &

Walls

around

Voids

in which matter,

(DM, gas, gal’s)

has agglomerated

The Cosmic The Cosmic WebWeb

Significance:Significance: Manifestation mildly nonlinear Manifestation mildly nonlinear clustering: clustering:

Transition stage between linear phase Transition stage between linear phase

and fully collapsed/virialized objectsand fully collapsed/virialized objects

Weblike configurations contain Weblike configurations contain

cosmological information: cosmological information: e.g. Void shapes & alignments (recent e.g. Void shapes & alignments (recent study J. Lee 2007)study J. Lee 2007)

Cosmic environment within which to Cosmic environment within which to understand the understand the

formation of galaxies.formation of galaxies.

Web Web AnalysisAnalysis Translate discrete galaxy/particle distribution into continuous density field/image:

- DTFE: Delaunay Tessellation Field Estimator

(Post)processing:

- filtering/smoothing image

- identifying clusters (peaks in image)

- identifying filaments/walls: * MMF & spine (Morse theory)

- tracing void regions: * watershed transform (Morse theory)

- topological aspects * alpha shapes

The Cosmic The Cosmic WebWebWeb Discretely Sampled:

By far, most information on the

Cosmic Web concerns

discrete samples:

• observational:

Galaxy Distribution

• theoretical:

N-body simulation particles

Power ofPower of

Tessellations Tessellations

Multiscale Multiscale MorphologyMorphologyFilter Filter

MMF dissection of cosmic web

into sheets, filaments, clusters …

filament

cluster

sheet

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Watershed Void Identification Watershed Void Identification

Alpha Alpha ShapesShapes

Alpha Shape ofcosmological simulation

left to right: alpha value increases.

LSSLSS

TopologyTopology

Cosmic WebCosmic Web

Evolution &Evolution &DynamicsDynamics

Dynamical Evolution Cosmic Dynamical Evolution Cosmic Web Web

● hierarchical structure formation

● anisotropic collapse

● void formation: asymmetry overdense vs.

underdense

Dynamical Evolution Cosmic Dynamical Evolution Cosmic Web Web

● Gravitational Instability:Gravitational Instability:

- any small initial deviation from sphericity of a collapsing cloud - any small initial deviation from sphericity of a collapsing cloud gets magnified gets magnified

- gravitational collapse proceeds along sequence: - gravitational collapse proceeds along sequence:

● collapse along smallest axis planar geometry wall

● collapse medium axis elongated filament

● full 3-D collapse clump halo

clump/halo virialization cosmic object

Anisotropic Anisotropic CollapseCollapse

(courtesy: A.

Kravtsov).

Formative agent of the Cosmic Web:Formative agent of the Cosmic Web:

Tidal strain induced my the Megaparsec Matter Distribution: - anisotropic collapse of structures - connection clusters-filaments: clusters main agent for stretching filaments

Ellipsoidal Ellipsoidal CollapseCollapse

Self-gravity

Internal tidal shear (due to shape)

External tidal shear

spherical

short axis

medium axis

long axis

Cosmic Web & Cosmic Web & ClustersClusters

Perseus Cluster (A426)

Tidal Constraints:Tidal Constraints:Example: elongated filamentary feature

Constrained Field:

Cosmic Web Theory:

Specification Cluster Node Locations & Orientation:

Outline Cosmic Web

Bond & Myers 1996

Bond et al. 1996

Spatial structure of Cosmic Web structure present in primordial density field.

Geometry of primordial field mainly filamentary (not Zel’dovich sheets), by sheer of statistics Gaussian field.

Filaments defined by tidal field imposed by cluster patches.

Incremental improvement of cosmic web spatial outline by inserting more and more clusters

Cosmic Web Theory :

Specification Cluster Node Locations & Orientation:

Outline Cosmic Web

Bond & Myers 1996

Bond et al. 1996

Cosmic Web Theory :

Specification Cluster Node Locations & Orientation:

Outline Cosmic Web

Bond et al. 1996

Spatial structure of Cosmic Web structure present in primordial density field.

Geometry of primordial field mainly filamentary (not Zel’dovich sheets), by sheer of statistics Gaussian field.

Filaments defined by tidal field imposed by cluster patches.

Incremental improvement of cosmic web spatial outline by inserting more and more clusters

Multiscale Nature Anisotropic Collapse

Structure arises over

a vast range of scales

- Small scales:

fully collapsed objects

- Very large scales:

still linear (& Gaussian) - Medium Mpc scales:

weblike features

Two analytical formalisms:

statistical Press-Sch.

dynamical PeakPatch

Hierarchical Pattern Hierarchical Pattern FormationFormation

Hierarchical Void Hierarchical Void Evolution:Evolution:

“Local” Excursion Set Approach

Two-barrier description:

- void merger

- void collapse

Peaked Void Size DistributionVoid Void MergingMerging

Void Void CollapseCollapse

Galaxy/HaloGalaxy/Halo&&

Web Web EnvironmentEnvironment

Halo Shape Halo Shape AlignmentsAlignmentsHalo & Halo & Galaxy Galaxy

EnvironmenEnvironmentaltal

Influences Influences

Halo ShapeHalo Shape

AlignmentAlignment

evolutionevolution Filaments: Haloes elongated along filamentFilaments: Haloes elongated along filament

Walls: Haloes elongated within plane wall Walls: Haloes elongated within plane wall (major axis in wall)(major axis in wall)

Orientation weakens in timeOrientation weakens in time

Effect stronger for low-mass haloesEffect stronger for low-mass haloes

Spinning the GalaxiesSpinning the Galaxies

Spinning the GalaxiesSpinning the Galaxies

Spinning Up

a collapsing protogalaxy

by Tidal Torque

Spinning the GalaxiesSpinning the Galaxies

Spinning the Galaxies:

Tidal Forces that also Shape Cosmic Web

Connection

Galaxy Spin

Cosmic Web

Cosmic Cosmic Web:Web:Environmental Environmental ImpactImpactHalo & Galaxy Formation:Halo & Galaxy Formation:

Environmental influences Environmental influences

Environmental Environmental dependence dependence

strongest wrt. strongest wrt. Alignments:Alignments:

- halo shape- halo shape

- halo spin direction- halo spin direction

To be understood from To be understood from the the

crucial role of tidal crucial role of tidal forces in forces in

shaping the Cosmic shaping the Cosmic WebWeb

Cosmic Cosmic Web:Web:Environmental Environmental ImpactImpactHalo & Galaxy Formation:Halo & Galaxy Formation:

Environmental influences Environmental influences

Halo Spin distribution in Halo Spin distribution in clusters, clusters,

filaments filaments & sheets & sheets

Environmental Environmental dependence on the halo dependence on the halo spin distributionspin distribution

To a large extent To a large extent influenced by presence influenced by presence of unbound particles of unbound particles

Halo Spin Halo Spin AlignmentsAlignments

Halo Spin Halo Spin AlignmentsAlignmentsHalo & Halo & Galaxy Galaxy

EnvironmenEnvironmentaltal

Influences Influences

Halo SpinHalo Spin

AlignmentAlignment

evolutionevolution Filaments:Filaments:High-mass haloes: always perpendicularHigh-mass haloes: always perpendicular

Low-mass haloes: starting perpendicular, they Low-mass haloes: starting perpendicular, they evolve to parallelevolve to parallel

Walls:Walls:High & Low-mass: always in plane of wallHigh & Low-mass: always in plane of wall

Filament-Galaxy alignment SDSS DR5

Filament-Galaxy alignment SDSS DR5

Jones et al. 2008 (in prep.) edge-on galaxies

Cosmic WebCosmic Web

AnalysisAnalysis

Discrete Point/Galaxy Distribution Continuous FieldDiscrete Point/Galaxy Distribution Continuous Field

(Sibson 1980, 1981; Watson 1992)(Sibson 1980, 1981; Watson 1992)

Natural NeighbourNatural Neighbour

InterpolationInterpolation

Exploiting the Adaptive Nature of Voronoi/Delaunay TessellationsExploiting the Adaptive Nature of Voronoi/Delaunay Tessellations

Point Sample Point Sample Continuous FieldContinuous Field

Aspects: ● Anisotropy of Aspects: ● Anisotropy of

Structural Features Structural Features

● ● Hierarchical Hierarchical

InfrastructureInfrastructure

● ● Voids, cq. empty Voids, cq. empty

regionsregions

● ● Inhomogeneous Inhomogeneous

SamplingSampling

Dual Tessellations

Voronoi Vertices: Centers Circumscribing Spheres 4 nuclei

Delaunay Tetrahedron

Voronoi Voronoi Delaunay Delaunay

Delaunay Tetrahedron:

Set of 4 nuclei,

circumscribing sphere

not containing

any other nuclei

Space-Covering Complete

Set Delaunay Tetrahedra:

Delaunay Tessellation

Voronoi Tessellation &

Delaunay Tessellation:

Duals

Delaunay Tessellation

Delaunay Tetrahedra:

Natural Multidimensional Interpolation Volume

• NN-neighbour Interpolation KernelNN-neighbour Interpolation Kernel

Linear KernelLinear Kernel

DTFEDTFE

Higher Order NN interpolation Linear DTFEHigher Order NN interpolation Linear DTFE

(Bernardeau & vdW 1996; Schaap & vdW 2000)(Bernardeau & vdW 1996; Schaap & vdW 2000)

Delaunay TessellationDelaunay Tessellation

Field Estimator Field Estimator (DTFE)(DTFE)

Large number of ObjectsLarge number of Objects

Dimensions higher > 2Dimensions higher > 2

DTFE DTFE Operational Virtues:Operational Virtues:

Volume-weighted (<> Volume-weighted (<> mass-weighted)mass-weighted)

Field UniversalityField Universality

Dynamic Range & Spatial Dynamic Range & Spatial ResolutionResolution

Shape & Morphology Shape & Morphology SensitivitySensitivity

Void (structure) detection & Void (structure) detection & renderingrendering

DTFE DTFE

Dynamic Range & Dynamic Range & ResolutionResolution

* DTFE resolves internal * DTFE resolves internal substructuresubstructure

over full range of over full range of scales presentscales present

in datain data

* Test: Soneira-Peebles * Test: Soneira-Peebles (fractal) model(fractal) model

DTFE DTFE

Shape & Morphology Shape & Morphology SensitivitySensitivity

* DTFE retains the elongated or * DTFE retains the elongated or flattened flattened

geometry of geometry of anisotropic featuresanisotropic features

Spine:Spine:

Singularities, Singularities, Morse complex & Morse complex &

watershedwatershed

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Spine of the Cosmic Web:Spine of the Cosmic Web: Morse complex & search for Morse complex & search for singularities singularities

Morse theory:topological structure well-behaved C2 field:

• singularities - minima, maxima, saddles• “flow” field:• critical net slope lines: connection singularities• saddles-maxima: spine of field • basin minima: watershed voids

Density Field Flow LinesDensity Field Flow Lines

Density Field Flow LinesDensity Field Flow Lines

Density Field Flow Lines:Density Field Flow Lines: SingularitiesSingularities

Density Field Flow Lines:Density Field Flow Lines: SingularitiesSingularities

Critical NetCritical Net saddles-maxima: ridgessaddles-maxima: ridges

Critical NetCritical Net saddles-maxima: ridgessaddles-maxima: ridges

Spine of the Cosmic Web Spine of the Cosmic Web

Spine of the Cosmic Web Spine of the Cosmic Web

Void Void Detection:Detection:

Watershed Void Watershed Void FinderFinder

Particle distr. DTFE densiy

Median Filtered Watershed transf.

WATERSHEDS : A cell is the union of points that are

topological closer to a certain minimum

Topological Distance: The path that connects two points via the

steepest slope: the path a water-droplet would take, when

running down a landscape

WatershedsWatershedsSegmentation:

A division of space in individual cellsA division of space in individual cells

Following the water-flow into the distinctcatchment basins.

Each basin belongingto one individualminima defines oneregion

Surface of Density Field

Local

MinimaLocal

Minima

Pierce the local Pierce the local minima, and let the minima, and let the

landscape sink landscape sink slowly in a tub of slowly in a tub of

water water

Every time two different flooding basins meet we draw a dividing wall

Flooding the Density Field

Final

segmentation

lines

Void

Patches

Particle distr.

Watershed transf.

Particle distr.DTFE density + WVF zoom-in:DTFE density +WVF +void orientation

void shape void orientationtidal shear orientation

MMF:MMF:

MultiscaleMultiscaleMorphology AnalysisMorphology Analysis

Multiscale Multiscale MorphologyMorphologyFilter Filter

MMF dissection of cosmic web

into sheets, filaments, clusters …

filament

cluster

sheet

Scale Space Pyramids

Gaussian smooth keeping same number of pixels.

The ensemble of images is referred to as a scale space stack: it is analysed as a single object.

Scale Space AnalysisScale Space AnalysisInspiration from Medical Imaging:

trace blood vessels, tumors, etc.

Florack, Kuijper et al.; Lindeberg et al.

Sato et al. 1997; Lorentz et al. 1997

Frangi et al. 1998 Multiscale vessel enhancement filter

scale

Scale Space AnalysisScale Space Analysis

scale

• Smooth the field over the range of relevant scales

• Select the characteristic scale of a particular (local) morphological element

Scale Space AnalysisScale Space Analysis

scale

• Smooth the field over the range of relevant scales

• Select the characteristic scale of a particular (local) morphological element

MultiscaleMultiscale

MorpholoMorphologygy

Filter Filter

(MMF)(MMF)

Aragon-Calvo, Jones,

van de Weygaert &

van der Hulst 2007

MMF MMF Filamentary Filamentary Cosmic WebCosmic Web

Ph.D. thesis

Miguel Aragon-Calvo

2007

Gray: filament particles

Black: feature compression

Gray: filament particles

Black: feature compression

Cosmic Web:Cosmic Web:Cluster-Filament Cluster-Filament ConnectionConnection

#filaments emerging from #filaments emerging from clusters clusters

Mass & Volume content Mass & Volume content

Web morphologiesWeb morphologies

Cosmic Web:Cosmic Web:Cluster-Filament Cluster-Filament ConnectionConnection

Density distribution Density distribution

Morph. elements Morph. elements

Mass & Volume content Mass & Volume content

Web morphologiesWeb morphologies

Cosmic WebCosmic Web Dynamics & Evolution

- Megaparsec Tidal Field (Cosmic Web theory Bond et al.) - determines weblike geometry & topology - intimate link clusters and web features - large-small scale interaction: alignment galaxy spins along filaments possible SDSS confirmation - alignment voids over > 30 Mpc

Analysis: Unbiased analysis anisotropy & hierarchy cosmic web: - Voronoi & Delaunay tessellations - (natural neighbour) DTFE density field reconstruction & interpolation - basis for geometric & topological analysis: - Morse complex, singularities & critical net: spine - Watershed : voids - Multiscale Morphology Filter : multiscale weblike patterns