# Presentation: 116027

## Hierarchical Galaxy Clustering in the 2dFGRS

- Peder Norberg (ETHZ)
- in collaboration with
- Carlton Baugh, Darren Crotton, Enrique Gaztanaga
- &
- the 2dFGRS Team
- Santa Fe, July 2004

## Contents: Counts-in-Cells Analysis of 2dFGRS

- Overview of the 2dFGRS
- Counts-in-Cells & hierarchical scaling model
- Methodology of our analysis
- volume limited samples
- incompleteness corrections
- error estimation & effects of superstructures
- Hierarchical clustering in 2dFGRS:
- 2nd order to 6th order
- luminosity segregation for 2nd to 4th order
- Conclusions

## 2dFGRS in a Nutshell

- The 2dF galaxy redshift survey (2dFGRS) input catalogue is selected from *UKST photographic plates*: SGP & NGP (main strips) + 100 random fields (spread over full APM survey).
- Magnitude limited survey: 14.0 ≤ bJ ≤ ~19.4, with σ(bJ) ~ 0.12 mag.; galaxy completeness ~91%; stellar contamination ~6%; (B-R) from SCOS.
- 225’000 unique galaxy redshifts over ~1500 sq. deg., with average spectroscopic completeness ~85%; median redshift ~0.11
- 2dFGRS is since June’03 in the public domain!

## 	What does CiC consist in? Just counting the number of elements in a set of cells of volume V (for spheres: radius R) and create the associated count probability distribution function (CPDF):

	where NN is the number of cells containing N galaxies out of a total number of cells thrown down NT.

- 	where NN is the number of cells containing N galaxies out of a total number of cells thrown down NT.

- Counts-in-Cells (CiC: I)

## Counts-in-Cells (CiC: II)

- The moments of the CPDF are given by:
- 	where  N is the mean number of galaxies obtained from the CPDF:
- The logarithm of the moment generating function is equal to the cumulant generating function. The cumulant, **p**, is simply the volume averaged reduced p-point correlation function:

## The hierarchical scaling model

- 	If gravitational instability is important in shaping large scale structure, then one expects the following scaling relation:
- 	where the volume averaged reduced p-point correlation function is:
- 	**NB: the volume average p-point correlation function is independent of position with the assumption of *****statistical homogeneity***** and *****isotropy****** *****of the cosmic density field: *****not true on small scales in redshift space*****! ****S****3**** is usually called ****skewness****, whereas ****S****4**** is referred as ****kurtosis****.**

## Real-Redshift Space & Small-Large Scales

**log cell size: 0.3 to 50 Mpc/h**

**Skewness**

**r-space**

**z-space**

- Hoyle et al. (2000)

## Counts-in-Cells (CiC: III)

- We opt for massively oversampling the survey volume, by randomly throwing down 25 106 spheres of radius going from R=0.5 Mpc/h to 30 Mpc/h
- Issues:
- selection function for the spheres: we use volume limited samples with constant radial mean density.
- survey geometry (drill holes, edge effects) and spectroscopic completeness are accounted for: we apply a volume correction (instead of a number correction) and sphere which are reduced by more than 50% are discarded; hence spheres of a given size can by construction only contribute to their own bin!
- Extremely correlated errors: principal component analysis shows that the first 2 or 3 eigenvectors are responsible for ~90% of the variance => PCA essential for the fits!

## Incompleteness corrections

**Log of Hierarchical Amplitudes: p=3,4,5**

**Log of sphere radius: 1.0 to 30 Mpc/h**

- Full mock (ie. no incompleteness)

- Sampled mock
- no incompleteness correction

## Counts-in-Cells (CiC: III)

- We opt for massively oversampling the survey volume, by randomly throwing down 25 106 spheres of radius going from R=0.5 Mpc/h to 30 Mpc/h
- Issues:
- selection function for the spheres: we use volume limited samples with constant radial mean density.
- survey geometry (drill holes, edge effects) and spectroscopic completeness are accounted for: we apply a volume correction (instead of a number correction) and sphere which are reduced by more than 50% are discarded; hence spheres of a given size can by construction only contribute to their own bin!
- Extremely correlated errors: principal component analysis shows that the first 2 or 3 eigenvectors are responsible for ~90% of the variance => PCA essential for the fits!

## Methodology: sample selection 

- High completeness sectors in SGP & NGP: effective area of 1140 sq. deg., containing >190’000 galaxy redshifts.
- 5 volume limited samples (two below L*, one L* sample, two above L*):
- each one magnitude wide, defined through a bright and faint absolute magnitude.
- not fully independent volumes: the brighter the characteristic luminosity, the more independent the sample is with respect to the fainter ones.

## Volume Limited Samples

**Absolute Magnitude (b****J ****)**

**redshift**

## Sample Properties

## Errors on the skewness estimated from 22 Hubble Volume mock catalogues

**Fractional rms error**

**Skewness**

**Log of sphere radius: 1.0 to 30 Mpc/h**

## Comparison between CDM and 2dFGRS: L* sample in z-space

**Log of sphere radius: 0.3 to 30 Mpc/h**

**Log of ********p****: p=2,3,4,5,6**

## Higher Order Clustering: p=2 to 6

**Log of sphere radius: 0.3 to 30 Mpc/h**

**Log of ********p****: p=2,3,4,5,6**

## Hierarchical Scaling: p=3,4,5,6

**Log of variance => 30 to 0.3 Mpc/h**

**Log of ********p****: p=3,4,5,6**

## Hierarchical Amplitudes: S3 to S5

- Hierarchical Amplitudes: S3 to S5

**Log of Hierarchical Amplitudes: p=3,4,5**

**Log of sphere radius: fit between 0.8 & 8 Mpc/h**

## Projected galaxy density in L* volume limited sample

- de

**15 Mpc/h**

**3 Mpc/h**

**  ****3**

**10**

**-1**

**-1**

- ****

- **3**

## Effects of the superstructures on the hierarchical amplitudes

**CPDF for R=10 Mpc/h**

**NB: the large difference in the CPDF**
**for the 3 VLS is due to the change in**
**the mean density only!**

## Sample Properties

## Some ‘evidence’ for luminosity dependent higher order clustering

**Skewness**

**Kurtosis**

## Conclusions

- 2dFGRS galaxies follow a hierarchical clustering model up to _6__th__ order__ __in redshift space_ (in each luminosity bin).
- Hierarchical amplitudes are *approximately *independent of cell radius used to smooth the galaxy distribution on intermediate scales  (~2 < R/Mpc/h < ~7).
- Presence of rare and extreme superstructures in the galaxy distribution can _**strongly influence**_ results on larger scales (R > ~7 Mpc/h).
- Skewness (S3) and Kurtosis (S4) show both a _*weak linear*_ dependence on log luminosity.
- Further details in astro-ph/0401405 (Baugh etal) and in astro-ph/0401434 (Croton etal).

## Recent Results from 2dFGRS:a quick & biased summary

-  Voids & Hierarchical scaling models       (Croton et al. 2004; astro-ph/0401406)
-  Luminosity functions by local environment (Croton et al 2004; in prep.)
-  Luminosity content of 2PIGG groups         (Eke et al. 2004; astro-ph/0402566)
-  Clustering of 2PIGG galaxy groups        (Padilla et al. 2004; astro-ph/0402577)
-  Stochastic relative biasing between galaxies (Wild et al. 2004; astro-ph/0404275)

## Conclusions (bis)

- Galaxy populations in voids and clusters differ significantly; voids are dominated by faint late-types, whereas clusters show an excess of bright early-types compared to the *mean*.
- Characteristic galaxy luminosity (L*) is larger for more massive 2PIGG groups; 2PIGG data show a robust trend of increasing M/L with increasing group luminosity; the typical bj-band M/L increases by a factor of 5 between L_bj=1010 Lsol to those 100 times more luminous.
- 2PIGG groups show a steady increase in clustering strength with luminosity; the most luminous groups are 10 times more strongly clustered than the full 2PIGG sample.
- Stochasticity in 2dFGRS is detected and it declines with increasing cell size; the nonlinearity of the biasing is less than 5% on all scales.
- Further details in Croton et al., Eke et al., Padilla et al. & Wild et al. or just ask some questions...