1. Observational Test of Halo Model: an empirical approach Mehri Torki Bob Nichol

2. Two types of pairs: both particles in same halo, or particles in different halos ξ dm (r) = ξ 1h (r) + ξ 2h (r) All physics can be decomposed similarly: influences from within halo, versus from outside (Sheth 1996) (lahmu.phyast.pitt.edu/~sheth/courses/allahabad/halomodel.ppt) The Halo-model of clustering

3. Halo Model as a tool to extract Cosmology Galaxies n(M), Mass & Size Cosmology ()

4. 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

5. 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. Group membership Z=0.07

6. 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 Mass estimation

7. 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. Determination

9. 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 Tests of

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

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

12. Total galaxy occupancy of `C4` Halo Occupation Distribution (HOD)

13. 0 Collister & Lahav 2004, Berlind & Weinberg 2002 Red galaxies All galaxies ) Halo Occupation Distribution

14. RedAll Faint Bright Investigating HOD as a function of galaxy properties

15. 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.

16. 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? Future work

17. 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!)

20. Mass dependence

21. 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

22. 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 Luminosity Function

23. 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.

24. Luminosity Function 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

25. 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

26. 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.

27. 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

28. 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.

29. for X-ray detections 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 0.94 1.15 1.20

30. 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.

31. Profile fitting 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

32. Z=0.07

34. 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 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

35. Scatter about HOD Is it Poisson as has been assumed? Expression by Gehrels (1986): Mean value of the observed sigma over the predicted one for

36. 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.

38. 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 Measure HOD with SDSS+C4 Provides the `Halo Occupation Distribution’ (HOD)

39. 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. Red galaxies All galaxies2.9 0.1 2.6 0.1 Tests of HOD

40. 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. Measuring mass distribution &

41. X-ray

43. 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 Tests of

44. 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.