Cluster Cosmology

h1. SPT 150 GHz luminosity function and SZE contamination

h2. Cluster and galaxy sample

We build luminosity function from the SPT AGNs (at 150 and 90 GHz) and SUMSS galaxy catalog (at 843 MHz). We use MCXC catalog of galaxy clusters with a total number of 1734 clusters out of which 139 and 333 clusters are there in SPT and SUMSS region respectively. 

h3. SPT AGN sample

There are 4769 SPT point sources in the present catalog (all_fields_list_brady_astrom_corrected.txt) with information about the S/N as well as Flux (in mJy). Out of these point sources we chose those which have counterparts in the SUMSS catalog within the positional uncertainty of the SPT point sources at 3-sigma level. This allow us to select the AGN sample from the SPT point sources.

The positional uncertainty is {{latex(\sigma_{total}^2 = \sigma_{sys}^2 + ((FWHM_{beam})/S2N)^2)}}, where the FWHM is 1.6' at 90 GHz, 1' at 150 GHz and 0.8' at 220 GHz. Also, sigma_sys is about 10". We chose the smallest sigma_total out of 90, 150 or 220 GHz. The 3-sigma level is chosen by plotting the surface density of the SUMSS galaxies at different sigma levels as in the figure below (at 3-sigma level the SUMSS surface density is equivalent to background density).

!SUMSS_in_SPT_surface_density.png!

There are 3446 SPT point sources (AGNs) which have SUMSS counterparts at 3-sigma level.

h3. SUMSS sample

There are 211,050 galaxies in SUMSS catalog (2007). More information about the catalog can be found here: [[http://www.physics.usyd.edu.au/sifa/Main/SUMSS]]

h2. Luminosity function

h3. Estimation of the total number of PS or galaxies in a luminosity bin for each cluster

* We do not have the redshift information about the SPT PS and SUMSS galaxies, so we assume that they are at the redshift of the cluster in concern. In order to construct the luminosity function we take a logarithmic luminosity bin and loop over all the clusters which are there in the SPT or SUMSS region.

* For each cluster we use its redshift to calculate luminosity distance and the K-correction for the luminosities. Using mass and redshift we calculate the radius and theta_200 for the cluster. In MCXC catalog mass and radius of the cluster are given as the region where the overdensity is 500 times the critical density of the universe and we use NFW profile and Duffy et al. to change them to M_200cr and R_200cr.

* We use a flux cut of 10 mJy, 6 mJy and 30 mJy for the SPT 90 GHz, SPT 150 GHz and SUMSS samples, which are found from the flux histograms (logN-logS plots) for these samples. We find the luminosity cut for each cluster corresponding to these flux cut.

* For each cluster we find all matching radio galaxies within the theta_200, convert the given flux of SPT (both at 90 and 150 GHz) and SUMSS sample to luminosity and count those galaxies whose luminosity lies in the logarithmic luminosity bin we took.

h3. Background estimation for a luminosity bin for each cluster

* For the background estimation we use the field logN-logS plots and find the number of PS or galaxies in the logarithmic flux bin that corresponds to the logarithmic luminosity bin we took. In order to find background for each cluster we multiply this field number with the surface area of that cluster (pi*theta_200^2). 

We do this for all the clusters and stack the total number and background number of galaxies for each logarithmic luminosity bin. The subtraction of the background number from the total number for each logarithmic luminosity bin gives us the number of PS or galaxies within theta_200 and at the redshift of the clusters. This is then normalized by the total M_200cr of the clusters which contributed to each luminosity bin to get <N_PS>. This can be further divided by the luminosity bin size to get the luminosity function as dn/dlogP.

h3. SPT luminosity function

!luminosity_func_SPT150.png!

As described before the luminosity function is normalized by the total M_200cr of the clusters contributing to a luminosity bin. Another way is to normalize it with the total volume of the contributing clusters. There is a complication when placing the LF in units of Mpc^-3 (volume). Because we define the virial region R_200cr as the region with overdensity of 200 with respect to critical density, there is then a natural redshift sensitivity.  That is cluster virial region densities will scale as E^2(z). So normalizing by total mass is a better choice as  the number of galaxies per unit mass is about the same independent of the redshift. Luminosity function at 90 GHz fluxes for the sample AGN sample is also similar to this one.

h3. SUMSS luminosity function

!luminosity_func_SUMSS.png!

h3. Comparison between SUMSS and SPT luminosity functions

SPT point source sample seems to be a low redshift sample as we get the luminosity function from MCXC clusters at z<0.1 (70 clusters) equivalent to that from all MCXC clusters (139) in the SPT region of 2500 deg^2. Following is the plot with MCXC clusters at z<0.1 which is comparable to the LF from all MCXC clusters in SPT region (see figure in the SPT luminosity function for comparison). For MCXC clusters at z>0.1, we do not observe the SPT point sources in almost all of the luminosity bins.

!luminosity_func_SPT150_z_0_1.png!

However, SUMSS galaxy sample is extended to higher redshifts as we observe the luminosity function with high luminosity sources at higher redshifts. Following are the plots for SUMSS galaxies.

!luminosity_func_SUMSS_z_lt_0_1.png!

!luminosity_func_SUMSS_z_gt_0_1.png!

h2. SZ Contamination

For estimating the SZ contamination, we change the luminosity function to a function of flux. The luminosity functions described above are basically the probability distribution functions which tell us the probability of finding a PS or galaxy within the theta_200 of the cluster per unit luminosity and per unit mass at different frequencies (150 GHz, 90 GHz and 843 MHz). So the number of point sources that a cluster of a given mass M_cluster (and redshift z_cluster) has is given by first integrating the luminosity function in different luminosity bins (i.e in our case simply multiplying dn/dlogP with the size of the bin) to get the <N_PS>, which is then multiplied by the M_cluster. 

Another important aspect is the degree of contamination which is quantified as s=PS_flux/|SZE_flux|. So if s is 0.1, this means that the cluster is 10% contaminated by the PSs and if s is 1, this means that cluster is totally contaminated and will not appear in the SZ cluster catalog. PS_flux is calculated for different luminosity bins in the luminosity function at a redshift z_cluster. SZE flux is calculated using Arnaud et al. (2010).

Following is the plot of the probability of contamination for a cluster with