Cluster Identification and Reconstruction through Voronoi and Delaunay Tessellations

Centre de Physique Thorique Marseille

Christian Marinoni Leiden, 6-10 March 2006The world a Jigsaw

R. Descartes Le monde ou Trait de la Lumire 1644

What is a galaxy cluster A cluster finding tool based on 3D Voronoi-Delaunay geometry Tests of performancesOutline Reconstructing the overdensity PDF in the deep Universe

The cluster includes the galaxies and any material which is in the space between the galaxiesXray image (hot gas which shines in the X)

Sunyaev-Zeldovich effect: CMB photons through hot electron cloud

2D optical identification - look for red galaxies - look for light deflection (gravitational lenses)

2D X ray identification - look for diffuse gas

Need to identify peaks but also reconstruct individual galaxy membership

Need to do this for very distant systems (faint objects, rare event statistics, .)

Groups are weak enhancements in the overall clustering pattern need to increase the detection S/N adding third dimension (depth)

Not so easy to work in 3D.

Galaxies are observed in redshift space (z) and not physical space (d)

Hubble laws: relation between cosmological redshift (due to cosmic expansion)and galaxy distances

Redshift z not entirely cosmological. Also doppler contributions due to peculiar velocities generated by local gravitational phenomena add to observed redshift

Hubble law breaks down On small scales velocities lead to elongated structures called Fingers of God clusters are smeared out along the line of sight

Maps appear to present Fingers of God pointing to the earth as if we were the centre of the universe

Real space z space

Groups are perverse examples of the topological effect of the algorithm used Kirshner 1977

We strongly believe that it will never be possible to assign Individual galaxies to groups or field in a definitive way. Any Such approach cannot possibly yield reliable results

Faber & Gallagher 1979

Galaxy Cluster Abundance tells us aboutgeometry and energy content of the Universegrowthfunctionpowerspectrum (8, n)Jenkinset al. 2001comoving volumemassfunction# of clusters per unit area and z:mass function: overallnormalization Hubble volumeN-body simulationsin three cosmologiescf: Press-SchechterClusters relatively simple objects. Evolution of massivecluster abundance determined by gravity.

What is a galaxy cluster A cluster finding tool based on 3D Voronoi-Delaunay geometry Tests of performancesOutline Reconstructing the overdensity PDF in the deep Universe

The Standard Algorithm Friends of Friends (percolation) method (Huchra & Geller 1982)

Define a set of linking/threshold parameters

Decide how to scale it with redshift [D(z),L(z)]

Problems with the percolation approach 3 arbitrary (non-physical, non-local) parameters

Over-merging of structures in dense regions (destroy sub-cluster elements)

Objects are linked by bridge galaxies and not by the cluster gravitational potential

Sample: Deep cone (2h Field: first-epoch data) ~7000 galaxies with secure redshifts, IAB24

Coverage:0.7x0.7 sq. deg (40x40 Mpc at z=1.5)

Volume sampled:2x106 Mpc3 (~CfA2) (1/16th of final goal)4300 MpcMean inter-galaxy separation at z=0.8 ~4.3 Mpc (~2dF at z=0.1)

Sampling rate: 1 over 3 galaxies down to I=24 z=0z=1.5

Problems with the percolation approachLocal UniverseDeep Universe

We want..

Adaptive algorithm (no global parameters)

which implements physical (not simply geometrical) prescriptions

which reconstructs not only the rare high density peaks but the whole hierarchy from small groups to rich clusters

Minimize contamination and fake detection

Assess completeness of the reconstruction scheme

Voronoi Diagram is a geometric structure that can be used to performe non parametric data smoothing:Natural way to measure packing

Identification of galaxy peaks in the galaxy distribution A Delaunay mesh describes the ensemble of neighboring galaxies: natural way to define cluster members

Reconstruction of neighborod relationships

AlgorithmI Identify central galaxies of a clusters

3D Voronoi Representation of a group with 10 galaxiesReal spaceRedshift space

3D Voronoi Representation of a group with 10 galaxies

Is the densest Voronoi cell at the center of a cluster in Z-space?

AlgorithmII Determine the k-order Delaunay neighbours of the peak within a fixed L.o.S. cylinder (R,L>>R)This way we recover a physicalquantity: the cluster projectedcentral density

AlgorithmK-order Delaunay neighbours tells you how big the underlying cluster is Virial relationshipProcess all the N>KDelaunay orders with an inclusion-exclusion logic (very fast)

What is a galaxy cluster A cluster finding tool based on 3D Voronoi-Delaunay geometry Tests of performancesOutline Reconstructing the overdensity PDF in the deep Universe

Distance independent velocity dispersion

What is a galaxy cluster A cluster finding tool based on 3D Voronoi-Delaunay geometry Tests of performancesOutline Reconstructing the overdensity PDF in the deep Universe

2DFGRS/SDSS stop hereThe Density Field (smoothing R=2Mpc)Marinoni et al. 2006

Filaments

FilamentsWalls

2DFGRS/SDSS stop hereThe Density Field (smoothing R=2Mpc)Marinoni et al. 2006

The 1P-PDF of galaxy overdensities g ()

RZ=0.7-1.1Z=1.1-1.5Volume limited sample M

The 1P-PDF of galaxy overdensities g ()

R The PDF is different at different cosmic epochsZ=0.7-1.1Z=1.1-1.5 Systematic shift of the peak towards low density regions as a function of cosmic time Cosmic space becomes dominated by low density regions at recent epochs Volume limited sample M

Theoretical InterpretationGravitational instability in an Expanding Universe

Bias: difference in distribution of DM and galaxy fluctuations Measuring the galaxy bias up to z=1.5 with the VVDS Marinoni et al. 2005 A&A in press astro-ph/0506561Linear Bias Scheme:(Kaiser 1984)Our goal: Redshift evolution Non linearity Scale dependenceStrategy Derive the biasing functionMarinoni & Hudson 2002Ostriker et al. 2003

The PDF of galaxy overdensities g (): ShapeRZ=0.7-1.1Z=1.1-1.5Coles & Jones 1991

Galaxy bias depends on redshift: it encreases as z increases The biasing function: 2) Shape b()z At present epochs galaxies form also in low density regions, while at high z the formation process is inhibited in underdensities

Reconstruction algorithm based on a virial definition of custer of mass points:ConclusionOnly two parameters (with immediate physical interpretation) Minimizes spourious distance-dependent effectsWide dynamical range: perform optimally over the whole systems mass range from small groups To rich clusters

Cosmological importance of clusters2 problems with optical identification of groupsNot only identify but also reconstruct and thus measure physical propertiesSince this affects only redshift and not positions on the sky, the stretching occours only radially (hence why fingers point back to the observer)Peculiar velocities are random strtching out a clusters in redshift spaceFoG long thin filaments in z space points directly back to the observerWe know we are not priviledge observer in the universeThus this effect must be due to local physics and not to Cosmology

We know we are not priviledge observer in the universeThus this effect must be due to local physics and not to Cosmology

We know we are not priviledge observer in the universeThus this effect must be due to local physics and not to Cosmology

Thys systems has the same physics but Different geometry

Inserire funzione di selezione Inserire sampling rate

Inserire funzione di selezione Inserire sampling rate

Matter fluctuations vs Galaxy fluctuations: the solid line represent the biasing function. Within this scheme thus bias is not a scalar but a function which allowUs to Assumes that clusters are sets of points in VIRIAL EQUILIBRIUM