Christian Marinoni
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Transcript Christian Marinoni
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
R. Descartes
Le monde ou Traité de la Lumière 1644
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
E T V 0
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)
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
• 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, ….)
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
d kz
z
o e
e
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 about
geometry 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:
dN
dV
dn
ddzd
ddz d
comoving mass
volume function
0 1 d M
dn
0.315
dM
M M dM
overall
normalization
( M h 2 )
exp [0.61 log( g z M )]3.8
power
spectrum (8, n)
growth
function
(m,,w)
Jenkins
et al. 2001
Hubble volume
N-body simulations
in three cosmologies
cf: Press-Schechter
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
The Standard Algorithm
Friends of Friends (percolation) method
(Huchra & Geller 1982)
-Define a set of linking/threshold parameters
{|| xi x j || D || yi y j || L; i j}
n
n
c
-Decide how to scale it with redshift [D(z),L(z)]
n( z )
L( z )
( L)dL
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
z=1.5
• Coverage:
0.7x0.7 sq. deg
(40x40 Mpc at z=1.5)
• Volume sampled:
2x106 Mpc3 (~CfA2)
(1/16th of final goal)
•Mean inter-galaxy
separation at z=0.8
<l>~4.3 Mpc (~2dF at z=0.1)
•Sampling rate: 1 over 3
galaxies down to I=24
z=0
Problems with the
percolation approach
ri , j n( z )
Local Universe
1/ 3
Deep 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
Algorithm
I Identify central galaxies of a clusters
c {max i [ V ]; i 0,1,..., N 1}
3D Voronoi Representation of a group with 10 galaxies
Real space
Redshift space
3D Voronoi Representation of a group with 10 galaxies
Is the densest Voronoi cell at the center of a cluster in Z-space?
Algorithm
II Determine the k-order Delaunay neighbours of
the peak within a fixed L.o.S. cylinder (R,L>>R)
This way we
recover a physical
quantity: the
cluster projected
central density
0
Algorithm
K-order Delaunay neighbours tells you how big
the underlying cluster is
Virial relationship
0 1 ( M ) v 2
R R ( 0 )
L L ( 0 )
Process all the N>K
Delaunay orders with
an inclusion-exclusion
logic (very fast)
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
Distance independent velocity dispersion
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
The
Density Field
(smoothing R=2Mpc)
2DFGRS/SDSS stop here
Marinoni et al. 2006
Filaments
Filaments
Walls
The
Density Field
(smoothing R=2Mpc)
2DFGRS/SDSS stop here
The Probability Distribution Function (PDF)
of galaxy overdensities
Probability of having a density fluctuation in the range
(,+d) within a sphere of radius R randomly located
in the survey volume
fR()
High density
Low density
Marinoni et al. 2006
Time Evolution of the galaxy PDF
The 1P-PDF of galaxy overdensities g ()
R
Z=1.1-1.5
Independent data
statistics
Z=0.7-1.1
Masked area
exclusion
Volume limited sample M<-20+5log h
Time Evolution of the galaxy PDF
The 1P-PDF of galaxy overdensities g ()
R
Z=1.1-1.5
• The PDF is different
at different cosmic
epochs
Z=0.7-1.1
• 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 M<-20+5log h
Theoretical Interpretation
Gravitational
instability
in an
Expanding
Universe
(r)
v 2 dV
r
Measuring the galaxy bias up to z=1.5 with the VVDS
Marinoni et al. 2005 A&A in press astro-ph/0506561
Bias: difference in distribution of DM and galaxy fluctuations
Linear Bias Scheme:
Our goal:
Strategy
g b
g b(z,,R)
(Kaiser 1984)
• Redshift evolution
• Non linearity
• Scale dependence
g(g )dg ( )d
Derive the biasing function
Marinoni & Hudson 2002
Ostriker et al. 2003
g g ( )
The PDF of galaxy overdensities g (): Shape
R
Coles & Jones 1991
Z=1.1-1.5
Z=0.7-1.1
bk k
g
k!
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
Conclusion
Reconstruction algorithm based on a virial definition
of custer of mass points:
2 T U 0
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