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Hierarchical Clustering
Leopoldo InfantePontificia Universidad Católica de Chile
Reunión Latinoamericana de AstronomíaCórdoba, septiembre 2001
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Introduction
The Two-point Correlation Function
Clustering of Galaxies at Low Redshifts -SDSS results-
Evolution of Clustering -CNOC2 results-
Clustering of Small Groups of Galaxies
The ro - d diagram
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Rich Clusters
Groups
Galaxies
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How do we characterizeclustering?
Correlation Functions
and/or
Power Spectrum
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Random Distribution
1-Point
2-Point
N-Point
Clustered Distribution
2-Point
r
dV1
dV2
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Continuous Distribution
Fourier Transform
Since P depends only on k
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2-Dimensions - Angles
Estimators
In Practice
AA BB
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The co-moving Correlation Length
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Proper Correlation length
Proper Correlation distance
Clustering evolutionindex
Assumed Power Law 3-D Correlation Function
Assumed Power Law Angular Correlation Function
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Proper Correlation Length
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Inter-system Separation, d
V
Nn systems
3/11
n
dMean separation
of objects
Space density of galaxy systems
As richer systems are rarer, d scales with richness or mass
of the system
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CLUSTERING Measurements from Galaxy Catalogs
and Predictions from Simulations
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2-dF Catalog, 16.419 galaxies, south strip.
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Sloan Digital Sky SurveySloan Digital Sky Survey
•2.5m Telescope•Two Surveys
•Photometric•Spectroscopic
•Expect•1 million galaxies with spectra•108 galaxies with 5 colors
Current resultsCurrent resultsTwo nights Equatorial strip, 225 deg.2
2.5 million galaxies
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Mock Catalogs
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•Correlations on a given angular scale probe physical scales of all sizes.•Fainter galaxies are on average further away, so probe larger physical scales
Angular Clustering
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Power law over 2 orders of magnitudeCorrelation in faintest bin correspond to larger physical scales
less clustered
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CNOC2 SurveyCNOC2 Survey
Measures clustering evolution up to z 0.6 for Lateand Early type galaxies.
1.55 deg.2
~ 3000 galaxies 0.1 < z < 0.6
Redshifts for objects with Rc< 21.5Rc band, MR < -20 rp<10h-1Mpc
SEDs are determined from UBVRcIc photometry
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Projected
Correlation Length
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Clustering of Galaxy Clusters
Richer clusters are more strongly clustered.
Bahcall & Cen, 92, Bahcall & West, 92 ro=0.4 dc=0.4 nc
-1/3
However this has been disputed Incompleteness in cluster samples (Abell, etc.)
APM cluster sample show weaker trend
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N body simulations
• Bahcall & Cen, ‘92, ro dc
• Croft & Efstathiou, ‘94, ro dc but weaker
• Colberg et al., ‘00, (The Virgo Consortium)– 109 particles– Cubes of 2h-1Gpc (CDM) 3h-1Gpc (CDM)
CDM =1.0 =0.0 h=0.5 =0.21 8=0.6
CDM =0.3 =0.7 h=0.5 =0.17 8=0.9
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CDMdc = 40, 70, 100, 130 h-1Mpc
Dark matter
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Clustering and Evolution of
Small Groups of Galaxies
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• Objective: Understand formation and evolution
of structures in the universe, from individual galaxies, to galaxies in groups to clusters of galaxies.
• Main data: SDSS, equatorial strip, RCS, etc.• Secondary data: Spectroscopy to get redshifts.• Expected results: dN/dz as a function of z,
occupation numbers (HOD) and mass. Derive ro and d=n-1/3 Clustering Properties
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Bias
• The galaxy distribution is a bias tracer of the matter distribution.– Galaxy formation only in the highest peaks of density
fluctuations.
– However, matter clusters continuously.
• In order to test structure formation models we must understand this bias.
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Halo Occupation Distribution, HOD
Bias, the relation between matter and galaxy distribution, for a specific type of galaxy, is defined by:
The probability, P(N/M), that a halo of virial mass M
contains N galaxies.
The relation between the halo and galaxy spatial
distribution.
The relation between the dark matter and galaxy
velocity distribution.This provides a knowledge of the relation between galaxies and the
overall distribution of matter, the Halo Occupation Distribution.
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In practice, how do we measure HOD?
Detect pairs, triplets, quadruplets etc. n2 in
SDSS catalog.
Measure redshifts of a selected sample.
With z and N we obtain dN/dz
We are carrying out a project to find galaxies in smallgroups using SDSS data.
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Collaborators:
M. StraussN. BahcallJ. KnappM. VogeleyR. KimR. Lupton& Sloan consortium
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The DataEquatorial strip, 2.5100 deg2Seeing 1.2” to 2”Area = 278.13 deg2
Mags. 18 < r* < 20
Ngalaxies = 330,041
Note strips
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dlogN/dm=0.46Turnover at r* 20.8
De-reddened Galaxy Counts
Thin lines are counts on each of the 12 scanlines
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Selection of Galaxy Systems
Find all galaxies within angular separation 2”<<15” (~37h-1kpc) and 18 < r* < 20
Merge all groups which have members in common.
Define a radius group: RG
Define distance from the group o the next galaxy; RN
Isolation criterion: RG/RN 3
Sample
1175 groups with more than 3 members15,492 pairs
Mean redshift = 0.22 0.1
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Galaxy pairs, examples
Image imspection showsthat less than 3% are spurious
detections
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Galaxy groups, examples
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Main Results
A = 13.54 0.07 = 1.76
A = 4.94 0.02 = 1.77
arcsec arcsec
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galaxies
pairs
triplets
Secondary Results
•Triplets are more clustered than pairs•Hint of an excess at small angular scales
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Space Clustering Properties-Limber’s Inversion-
– Calculate correlation amplitudes from ()
– Measure redshift distributions, dN/dz
– De-project () to obtain ro, correlation lengths
– Compare ro systems with different HODs
CNOC2 SDSS
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The ro - d relation
3/11
n
d
Correlation scaleAmplitude of the
correlation function
Mean separationAs richer systems are rarer,
d scales with richness or mass of the system
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Rich Abell Clusters:•Bahcall & Soneira 1983•Peacock & West 1992•Postman et al. 1992•Lee &Park 2000
APM Clusters:•Croft et al. 1997•Lee & Park 2000
EDCC Clusters:Nichol et al. 1992
X-ray Clusters:•Bohringer et al. 2001•Abadi et al. 1998•Lee & Park 2000
Groups of Galaxies:•Merchan et al. 2000•Girardi et al. 2000
LCDM (m=0.3, L=0.7, h=0.7)SCDM (m = 1, L=0, h=0.5)Governato et al. 2000Colberg et al. 2000Bahcall et al. 2001
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CONCLUSIONSWe use a sample of 330,041 galaxies within 278 deg2, with
magnitudes 18 < r* < 20, from SDSS commissioning imagingdata.
We select isolated small groups.We determine the angular correlation function.
We find the following:
•Pairs and triplets are ~ 3 times more strongly clustered than galaxies.•Logarithmic slopes are = 1.77 ± 0.04 (galaxies and pairs)() is measured up to 1 deg. scales, ~ 9 h-1Mpc at <z>=0.22. No breaks.•We find ro= 4.2 ± 0.4 h-1Mpc for galaxies and 7.8 ± 0.7 h-1Mpc for pairs•We find d = 3.7 and 10.2 h-1Mpc for galaxies and pairs respectively.•LCDM provides a considerable better match to the data
Follow-up studiesdN/dz and photometric redshifts.
Select groups over > 1000 deg2 area from SDSS