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