Diapositive 1

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• Cluster Identification and Reconstruction through
  Voronoi and Delaunay Tessellations

Christian Marinoni
Centre de Physique Théorique Marseille
The world a Jigsaw

Leiden, 6-10 March
2006
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R. Descartes
Le monde ou Traité de la Lumière 1644
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Outline

• What is a galaxy cluster

• A cluster finding tool based on 3D
  Voronoi-Delaunay geometry

• Tests of performances

• Reconstructing the overdensity PDF in the deep
  Universe

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The cluster includes the galaxies and any
material which is in the space between the
galaxies
Xray image (hot gas which shines in the X)
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Sunyaev-Zeldovich effect CMB photons through hot
electron cloud
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2D optical identification - look for red
galaxies - look for light deflection
(gravitational lenses) 2D X ray identification
- look for diffuse gas

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

• Need to identify peaks but also reconstruct
  individual
• galaxy membership
• Need to do this for very distant systems
• (faint objects, rare event statistics, .)

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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
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Maps appear to present Fingers of God pointing to
the earth as if we were the centre of the
universe
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Real space
z space
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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
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Galaxy Cluster Abundance tells us aboutgeometry
and energy content of the Universe
Clusters relatively simple objects. Evolution of
massive cluster abundance determined by gravity.
of clusters per unit area and z
mass function
comoving volume
mass function
Jenkins et al. 2001
Hubble volume N-body simulations in three
cosmologies cf Press-Schechter
growth function
power spectrum (?8, n)
overall normalization
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Outline

• What is a galaxy cluster

• A cluster finding tool based on 3D
  Voronoi-Delaunay geometry

• Tests of performances

• Reconstructing the overdensity PDF in the deep
  Universe

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• 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)

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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

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Sample Deep cone (2h Field
first-epoch data)
z1.5

• 7000 galaxies with secure redshifts, IAB?24
• Coverage
• 0.7x0.7 sq. deg
• (40x40 Mpc at z1.5)
• Volume sampled
• 2x106 Mpc3 (CfA2) (1/16th of final goal)

4300 Mpc

• Mean inter-galaxy separation at z0.8
• ltlgt4.3 Mpc (2dF at z0.1)
• Sampling rate 1 over 3 galaxies down to I24

z0
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Problems with the percolation approach
Local Universe
Deep Universe
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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
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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
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Algorithm
I Identify central galaxies of a clusters
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3D Voronoi Representation of a group with 10
galaxies
Real space
Redshift space
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3D Voronoi Representation of a group with 10
galaxies
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Is the densest Voronoi cell at the center of a
cluster in Z-space?
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Algorithm
II Determine the k-order Delaunay neighbours of
the peak within a fixed L.o.S. cylinder (R,LgtgtR)
This way we recover a physical quantity the
cluster projected central density
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Algorithm
K-order Delaunay neighbours tells you how big
the underlying cluster is
Virial relationship
Process all the NgtK Delaunay orders with an
inclusion-exclusion logic (very fast)
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Outline

• What is a galaxy cluster

• A cluster finding tool based on 3D
  Voronoi-Delaunay geometry

• Tests of performances

• Reconstructing the overdensity PDF in the deep
  Universe

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Distance independent velocity dispersion
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Outline

• What is a galaxy cluster

• A cluster finding tool based on 3D
  Voronoi-Delaunay geometry

• Tests of performances

• Reconstructing the overdensity PDF in the deep
  Universe

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The Density Field (smoothing
R2Mpc)
2DFGRS/SDSS stop here
Marinoni et al. 2006
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Filaments
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Filaments
Walls
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The Density Field (smoothing
R2Mpc)
2DFGRS/SDSS stop here
Marinoni et al. 2006
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The 1P-PDF of galaxy overdensities g (?)
Time Evolution of the galaxy PDF
R
Z1.1-1.5
Independent data statistics Masked area
exclusion
Z0.7-1.1

Volume limited sample Mlt-205log h
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The 1P-PDF of galaxy overdensities g (?)
Time Evolution of the galaxy PDF
R
Z1.1-1.5

• The PDF is different
• at different cosmic
• epochs

Z0.7-1.1

• 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 Mlt-205log h
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Theoretical Interpretation
Gravitational instability in an Expanding
Universe
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Measuring the galaxy bias up to z1.5 with the
VVDS Marinoni et al. 2005 AA in press
astro-ph/0506561
Bias difference in distribution of DM and galaxy
fluctuations ?
Linear Bias Scheme
(Kaiser 1984)

• Redshift evolution
• Non linearity
• Scale dependence

Our goal
Marinoni Hudson 2002 Ostriker et al. 2003
Strategy

• Derive the biasing function

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The PDF of galaxy overdensities g (?) Shape
R
Coles Jones 1991
Z1.1-1.5
Z0.7-1.1
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The biasing function 2) Shape b(?)
z

• Galaxy bias depends on redshift it encreases as
  z increases

• At present epochs galaxies form also in low
  density
• regions, while at high z the formation process
  is inhibited in
• underdensities

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