Hierarchical Clustering

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Hierarchical Clustering

• Leopoldo Infante
• Pontificia Universidad Católica de Chile

Reunión Latinoamericana de Astronomía Córdoba,
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
Bias
Groups
Bias
Galaxies
Bias
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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
dV1
Clustered Distribution
r
2-Point
dV2
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Continuous Distribution
Fourier Transform
Since P depends only on k
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In Practice
2-Dimensions - Angles ?
Estimators
B
A
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The co-moving Correlation Length
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Assumed Power Law 3-D Correlation Function
Proper Correlation distance
Clustering evolution index
Proper Correlation length
Assumed Power Law Angular Correlation Function
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Proper Correlation Length
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Inter-system Separation, d
Space density of galaxy systems
Mean separation of objects
As richer systems are rarer, d scales with
richness or mass of the system
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CLUSTERING Measurements from Galaxy
Catalogsand Predictions from Simulations
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2-dF Catalog, 16.419 galaxies, south strip.
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• Sloan Digital Sky Survey
• 2.5m Telescope
• Two Surveys
• Photometric
• Spectroscopic
• Expect
• 1 million galaxies with spectra
• 108 galaxies with 5 colors

• Current results
• Two nights
• Equatorial strip, 225 deg.2
• 2.5 million galaxies

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Mock Catalogs
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Angular Clustering

• Correlations on a given angular scale probe
  physical scales of all sizes.
• Fainter galaxies are on average further away, so
  probe larger physical scales

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• Power law over 2 orders of magnitude
• Correlation in faintest bin correspond to larger
  physical scales
• ? less clustered

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CNOC2 Survey Measures clustering evolution up to
z ? 0.6 for Late and Early type galaxies. 1.55
deg.2 3000 galaxies 0.1 lt z lt 0.6 Redshifts
for objects with Rclt 21.5 Rc band, MR lt -20 ?
rplt10h-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 ? ro0.4
  dc0.4 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 h0.5 ?0.21 ?80.6
?CDM ?0.3 ?0.7 h0.5 ?0.17 ?80.9
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?CDM dc 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 dn-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. n?2 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
small groups using SDSS data.
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Collaborators M. Strauss N. Bahcall J. Knapp M.
Vogeley R. Kim R. Lupton Sloan consortium
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Note strips

• The Data
• Equatorial strip, 2.5?100 deg2
• Seeing ? 1.2 to 2
• Area 278.13 deg2
• Mags. 18 lt r lt 20
• Ngalaxies 330,041

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De-reddened Galaxy Counts
Thin lines are counts on each of the 12 scanlines
dlogN/dm0.46 Turnover at r? 20.8
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• Selection of Galaxy Systems
• Find all galaxies within angular separation
  2lt?lt15 (37h-1kpc)
• and 18 lt r lt 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
members 15,492 pairs Mean redshift 0.22 ? 0.1
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Galaxy pairs, examples
Image imspection shows that less than 3 are
spurious detections
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Galaxy groups, examples
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Main Results
arcsec
arcsec
A? 13.54 ? 0.07 ? 1.76
A? 4.94 ? 0.02 ? 1.77
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Secondary Results
galaxies
pairs
triplets

• Triplets are more clustered than pairs
• Hint of an excess at small angular scales

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Space Clustering Properties-Limbers Inversion-

• Calculate correlation amplitudes from ?(?)
• Measure redshift distributions, dN/dz
• De-project ?(?) to obtain ro, correlation lengths
• Compare ro systems with different HODs

SDSS
CNOC2
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The ro - d relation
Amplitude of the correlation function
Correlation scale
Mean separation As 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

LCDM (?m0.3, ?L0.7, h0.7) SCDM (?m 1, ?L0,
h0.5) Governato et al. 2000 Colberg et al.
2000 Bahcall et al. 2001

• Groups of Galaxies
• Merchan et al. 2000
• Girardi et al. 2000

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CONCLUSIONS
We use a sample of 330,041 galaxies within 278
deg2, with magnitudes 18 lt r lt 20, from SDSS
commissioning imaging data. 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 ltzgt0.22. 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 studies dN/dz and photometric
redshifts. Select groups over gt 1000 deg2 area
from SDSS