Hierarchical Clustering Leopoldo Infante Pontificia Universidad Católica de Chile Reunión...
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Transcript of Hierarchical Clustering Leopoldo Infante Pontificia Universidad Católica de Chile Reunión...
Hierarchical Clustering
Leopoldo InfantePontificia Universidad Católica de Chile
Reunión Latinoamericana de AstronomíaCórdoba, septiembre 2001
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
Rich Clusters
Groups
Galaxies
How do we characterizeclustering?
Correlation Functions
and/or
Power Spectrum
Random Distribution
1-Point
2-Point
N-Point
Clustered Distribution
2-Point
r
dV1
dV2
Continuous Distribution
Fourier Transform
Since P depends only on k
2-Dimensions - Angles
Estimators
In Practice
AA BB
The co-moving Correlation Length
Proper Correlation length
Proper Correlation distance
Clustering evolutionindex
Assumed Power Law 3-D Correlation Function
Assumed Power Law Angular Correlation Function
Proper Correlation Length
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
CLUSTERING Measurements from Galaxy Catalogs
and Predictions from Simulations
2-dF Catalog, 16.419 galaxies, south strip.
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
Mock Catalogs
•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
Power law over 2 orders of magnitudeCorrelation in faintest bin correspond to larger physical scales
less clustered
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
Projected
Correlation Length
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
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
CDMdc = 40, 70, 100, 130 h-1Mpc
Dark matter
Clustering and Evolution of
Small Groups of Galaxies
• 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
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.
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.
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.
Collaborators:
M. StraussN. BahcallJ. KnappM. VogeleyR. KimR. Lupton& Sloan consortium
The DataEquatorial strip, 2.5100 deg2Seeing 1.2” to 2”Area = 278.13 deg2
Mags. 18 < r* < 20
Ngalaxies = 330,041
Note strips
dlogN/dm=0.46Turnover at r* 20.8
De-reddened Galaxy Counts
Thin lines are counts on each of the 12 scanlines
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
Galaxy pairs, examples
Image imspection showsthat less than 3% are spurious
detections
Galaxy groups, examples
Main Results
A = 13.54 0.07 = 1.76
A = 4.94 0.02 = 1.77
arcsec arcsec
galaxies
pairs
triplets
Secondary Results
•Triplets are more clustered than pairs•Hint of an excess at small angular scales
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
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
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
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