Observational Test of Halo Model: an empirical approach

Mehri Torki

Bob Nichol

   

  

                                                                 

                                                                                        

                                                                

                                                                                        

                                                                

Two types of pairs: both particles in same halo, or particles in Two types of pairs: both particles in same halo, or particles in different halosdifferent halos

ξξdmdm(r) =(r) = ξξ1h1h(r) + (r) + ξξ2h2h(r)(r) All physics can be decomposed similarly: influences from within All physics can be decomposed similarly: influences from within

halo, versus from outside (Sheth 1996)halo, versus from outside (Sheth 1996)

(lahmu.phyast.pitt.edu/~sheth/courses/allahabad/halomodel.ppt)

The Halo-model of clustering

Halo Model as a tool to extract Cosmology

Galaxies

MN |

n(M)

m 8,

Mass & Size

Cosmology ( )

•Largest spectroscopic cluster catalogue ever made.•Contains galaxy clusters found in

the SDSS DR3 spectroscopic database.•1106 clusters.•Clusters are found in a seven

dimensional space.•Galaxies within clusters are co-evolving.•Thus, galaxies will not only cluster in position but also in

colour.

http://www.ctio.noao.edu/~chrism/current/research/C4/dr3

The SDSS-C4 Galaxy Cluster Catalogue

•We examine 94795 galaxies.

•Redshift range of 0.03 < z < 0.13

•Using all the galaxies projected

within

• And of the cluster centres.

•Absolute magnitude range of -24 < < -21.2

•Colour-cut:

we look at the radial profile of all the galaxies

within

of the red sequence for each cluster.

Mpc4 1h

rM

2

v4

Group membership

Z=0.07

• Mass comes from the scaling relationship determined from

the simulation presented in Miller et al. 2005

• (summed optical r-band luminosity ) is a powerful tool

-superior to the galaxy line-of-sight velocity dispersion

-or the richness

rL

Mass estimation

•It is not possible to measure directly the radius at which

cluster has a mass over density of

measure space over density of

is radius where mean number density of galaxies

= 200 critical density

•Calculate for 1106 clusters in ‘C4’ by building the radial density profile of a certain mass and a certain r-band.

•Stack all the galaxies in 4 bins of mass.

•Determine for each bin of mass.

M

N

N

200R

200R

200R

200R

Determination

Check value of the mean space density of field

Check the effect of misidentifying the cluster centre

Check values by using only `good centres’ (von der Linden et al. in prep)

Check for X-ray detection

Check the colour constraint

Check the fit to NFW profile

200R

200R

200R

Tests of

•Sheldon et al. (in prep.) have derived lensing profiles for clusters of galaxies in SDSS.

Compare results

Finn et al. (in prep.) have determined as: 200R

Mpc)1(

1

km/s100047.2 1

703200

h

zR

m

x

Total galaxy occupancy of `C4`

Halo Occupation Distribution

(HOD)

MN |

13.072.0

19.024.13

)/( 0MM cutMM

cutMM 0

Collister & Lahav 2004,

Berlind & Weinberg 2002

MhM 10/log(

Red galaxies

All galaxies

18.088.0

12.012.13

)

Halo Occupation Distribution

Red All

Faint

Bright

12.034.13 17.046.13

15.068.0

14.027.13 13.048.13

11.083.0

16.068.0 11.085.0

Investigating HOD as a function of galaxy properties

•Motivated by halo model, we use `C4’ to make a direct and

empirical determination of HOD from the known halos

(clusters).

•Compared to recent lensing work by Sheldon et al. (in

prep.) & found remarkable agreement in size of radii.

Summary & Conclusion

•Found a good fit to our galaxy radial distribution provided by

NFW.

•We have a stable HOD with respect to the colour & luminosity.

200R

•Try to find an analytic equation for our mass function.

•Combine our HOD parameters with galaxy clustering measurements to better constrain cosmological parameters as and .

•Study HOD as a function of local environment.

•Compare HOD with other measurements of a cluster and group mass like X-ray parameters.

•Compare our results with the mock SDSS catalogue to ensure that the catalogues are a fair representation of the SDSS.

•Improve our results with latest SDSS and C4 catalogues.

•Compare properties of galaxies as colour, luminosity, morphology for different HODs to see which properties of galaxies in a halo change?

8

Future work

The Holy Grail

Halo model provides natural framework within which to discuss, interpret most measures of clustering; it is the natural language of galaxy ‘bias’

The Halo Grail(phrase coined by Jasjeet Bagla!)

Mass dependence

•The model come up recently is the best used for the statistical analysis and understanding the large datasets as SDSS survey.•All the mass in the universe is assumed to be allocated in individual units called haloes.•Specifically provides the `Halo Occupation Distribution’ (HOD) which is a function telling us how dark matter halos populate with galaxies.•In contrast with the previous work which used the galaxy correlation functions to constrain HOD, we use known halos; clusters of galaxies to determine HOD. •Matter distribution can be studied in two steps: the distribution of the mass within every halo and the spatial distribution of the haloes

Halo Model as a tool to extract cosmology: an empirical approach

•LF relates the number density of galaxies to absolute magnitude.

•The form of LF is fitted by a universal Schechter function of the form (Schechter, 1976):

•To determine the mean space density of field we use the r-band LF of Blanton et al. (2003a) and integrating down to = -21.2

•Correcting for h by:

•Correcting for the filter by:

•The value of the mean space density of field is 0.0045

rM

]4084.0)([3943.002217.0 1.01.00.0 irrr

Luminosity Function

Testing Luminosity Function

•We used the r-band LF of Blanton et al. (2003a) in order to derive the mean space density of field.

•We test if we get the same distribution in that band pass for our sample.

•We make this distribution for the absolute magnitude range of

of the whole SDSS database.

•We find that for the galaxies in the absolute magnitude between -21 and -18 (as we go toward fainter galaxies), the number density of galaxies decrease.

•which is exactly where we are not complete!

•Our data is in redshift and

•Having considered -21.2 as our limit of completeness, there is no disagreement in the distribution we have achieved.

1224 rM

13.003.0 z 2.2124 rM

Luminosity Function

M)10exp(10)10ln(4.0M)M( )M-M(4.0)1)(MM(4.0*

** dd

7.010

1 )(log5 hh MhM

•LF relates the number density of galaxies to absolute magnitude.

•The form of LF is fitted by a universal Schechter function of the form (Schechter, 1976):

•To determine the mean space density of field we use the r-band LF of Blanton et al. (2003a) and integrating down to = -21.2

•Correcting for h by:

•Correcting for the filter by:

•The value of the mean space density of field is 0.0045

rM

]4084.0)([3943.002217.0 1.01.00.0 irrr-33Mpch

Testing value of the mean space density of field

We determine simply the value for N (number of galaxies in DR3 in spectroscopic area of 4188 sq. deg in the ) divided by the volume of our chosen sample.

N = 94795

S = Total sky area =

F = Fraction of sky covered = 0.1

V1 = Volume of the sphere in redshift 0.13

V2 = Volume of the sphere in redshift 0.03

V = (V1-V2). F

N / V = 0.0042

13.003.0 z

4)/180( 2

3Mpc-33Mpch

Test the effect of misidentifying the cluster centre

•Check if we are in the right centre, otherwise it cause different radial profile and hence different value for .

•There are three methods for finding the cluster centre; BCG, MEAN and GEOM cluster centroid measurements.

•BCG: position of the brightest galaxy in the cluster, we think is best to use because this method is relied on observations that clusters host a population of early type galaxies with small dispersion in colour.

•MEAN: coordinates of the galaxy with the highest density.

•GEOM: `luminosity weighted mean centroids’, theses are cluster centres using all galaxies within 1 calculating a luminosity weighted average (in r-band) for RA and DEC of them.

•We find that by using other measurements of the cluster centroid there is no significant change in values of .

200R

Mpc1h

200R

•We also recalculate our estimates using the clusters with only

`good’ centres.

•For ‘good’ centres we use the list of `C4’ clusters with corrected BCG centres (von der Linden et al. in prep), they claimed that SDSS photometry of BCGs underestimates the flux and they correct for it.

•We use this list to remove the `bad BCGs’ from `C4’.

•We find that there is no significant difference in our estimates of

200R

200R

Test the colour constraint

In the algorithm used to identify the galaxies around each cluster, we add this constraint in the sense that galaxies are clustered in colour & space.

We looked at the radial profile of all galaxies within of the red sequence for each cluster.

We may miss some galaxies.

By relaxing this colour-cut to and we evaluate the impact on the value of

We also vary -21.2 (limit of completeness) to brighter & fainter galaxies.

2

4 6200R

for X-ray detections200R

•Calculating the virial radius is crucial for our work.

•The X-ray detection is very accurate to measure the radii.

•We match `NORAS’ to `C4’ in order to find which cluster has X-ray detection, the X-ray selected clusters are taken from Bohringer et al. (2000).

•We find 40 overlapped clusters.

•With the same formalism explained before we derive 200R

2 4 6 0.94 1.15 1.20

Radial distribution of galaxies in groups

•We determine the projected galaxy density profile given mass from stacking groups scaled by their virial radius.

•Calculate the distance from cluster centre to each galaxy.

•Express them in units of (divide each distance to virial radius of each cluster).

•Stack them once in 4 bins of mass and then for the whole sample.

•Calculate the number of galaxies in radial bins divided by surface of each bin.

•Correct for the effect of fibre collision.

200R

Profile fitting

svir rr /

2)/1(/)(

ss

s

rrrrr

c•NFW profile is described as the ‘universal density profile’ expressed in terms of by the formula:

•Best-fitting NFW concentration parameters are:

•This means that the criteria used in ‘C4’ clusters provides a good definition for the member galaxies and the clusters have the same shape with and without the colour-cut.

2.9 0.1

2.6 0.1

All galaxies

Red

Z=0.07

0.38 0.44 0.46

0.49 0.55 0.56

0.61 0.71 0.73

0.79 0.94 0.96

0.42 0.46 0.47

0.54 0.59 0.60

0.69 0.75 0.76

0.84 0.97 0.99

rM 2 4 6

0.44 0.48 0.49

0.57 0.63 0.64

0.72 0.78 0.81

0.89 1.07 1.09

-20.7

-21.2

-21.7

Scatter about HOD

Is it Poisson as has been assumed?

75.01 n

Expression by Gehrels (1986):

Mean value of the observed sigma over the predicted one

96.0 1410Mfor

6.0 for 1410M

Summary & Conclusion

•We found a good fit to our galaxy radial distribution provided by NFW profile and obtain c almost the same for galaxies with & without the colour-cut.

•This makes us feel confident with criteria used in `C4’ ; clusters keep their shape.

•Analysing our HOD for all, red, bright & faint galaxies shows that does not depend on the type of the galaxies

•Thus we have a stable HOD with respect to the colour & luminosity.

7.17r

Take as empirical an approach as possible.Directly measure the radial and HOD of galaxies in the ‘C4 catalogue’.For investigating our HOD , we need to calculate

mass and size

Directly determine size-mass for clusters with a model independent method.Stack systems in bins measure the distribution of galaxies in clusters over a wide range of masses virial size

MN |

Measure HOD with SDSS+C4

Provides the `Halo Occupation Distribution’ (HOD)

•Determine the projected galaxy density profile.

•NFW profile is described as the ‘universal density profile’ expressed in terms of by :

•Best-fitting NFW concentration parameters are:

•This means the criteria used in ‘C4’ clusters provides a good definition for the member galaxies:

clusters have the same shape with and without the colour-cut.

2)/1(/)(

ss

s

rrrrr

c svir rr /

Red galaxies

All galaxies 2.9 0.1

2.6 0.1

Tests of HOD

•An important cosmological aim is to constrain

, its average density

, amplitude of its power spectrum p(k)

• More formally we want to know what the halo mass

function looks like in cosmology.

•Following halo model formalism, apply it to the ‘C4

Catalogue’ using SDSS data set.

m

8

Measuring mass distribution

&

1313 105101 M

1413 101105 M

1414 102101 M

1514 101102 M

Mpc42.0 1h

Mpc54.0 1h

Mpc69.0 1h

Mpc84.0 1h

X-ray

1313 105101 M

1413 101105 M

1414 102101 M

1514 101102 M

Mpc42.0 1h

Mpc54.0 1h

Mpc69.0 1h

Mpc84.0 1h X-ray

Check luminosity function

Check value of the mean space density of field

Check the effect of misidentifying the cluster centre

Check values by using only `good centres’ (von der Linden et al. in prep)

Check for X-ray detection

Check the colour constraint

Check the fit to NFW profile

200R

200R

200R

Tests of

•Motivated by halo model, we use `C4’ to make a direct and empirical

determination of HOD from the

known halos (clusters).

•We have tested this extensively.

•Compared to recent lensing work by Sheldon et al. (2006) &

found remarkable agreement in size of radii.

Summary & Conclusion

•Found a good fit to our galaxy radial distribution provided by NFW.

•This makes us feel confident with criteria used in `C4’.

•We have a stable HOD with respect to the colour & luminosity.

200R

Mpc42.0 1h

Mpc54.0 1h Mpc69.0 1h Mpc84.0 1h