Cluster Cosmology

h1. SPT 150 GHz luminosity function and SZE contamination

h2. Cluster and galaxy sample

We build luminosity function from the SPT AGN sample (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 regions 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 the 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 whole SPT point source sample (which may have dusty galaxies also).

The positional uncertainty is 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 depending upon the FWHM at 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 its background density).

There are 3446 (72%) SPT point sources (AGNs) which have SUMSS counterparts at 3-sigma level.

!SUMSS_in_SPT_surface_density.png!

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 AGNs 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 regions where the overdensity is 500 times the critical density of the universe so 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 cuts for each cluster corresponding to these flux cut. This is an approximation and we will find flux cut from the survey depths by the next meeting.

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

* To correct for the background population, we use the field logN-logS plots and find the number of AGNs in the logarithmic flux bin which 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 background subtracted population 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 AGNs within theta_200 and at the redshift of the clusters. This is then normalized by the total mass M_200cr of the clusters which contributed to each luminosity bin to get <N_PS>, which 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. However, there is a complication when placing the LF in units of Mpc^-3 (volume), as 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 the 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.

The luminosity function is first fitted using an expression given in the Condon et al. (2002) and is then extrapolated to higher luminosity bins. The error bars on the luminosity function at these luminosity bins are just showing the 2-sigma upper limit from the Poisson's distribution, considering the fact that there were no AGN's detected in these bins. 

h3. SUMSS luminosity function

!luminosity_func_SUMSS.png!

h3. Looking at cluster subsets

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 the SPT region (see figure in the SPT luminosity function section 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 is the plot for SUMSS galaxies at z<0.1 and z>0.1.

!luminosity_func_SUMSS_z_lt_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 AGNs per unit luminosity and per unit mass at different frequencies (150 GHz, 90 GHz and 843 MHz). So the probability of finding AGNs in a cluster of given mass M_cluster (and redshift z_cluster) is obtained 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 point sources 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 finding AGNs in a given flux bin and within a cluster of mass M_500cr=4.e14 and 1.e15 vs the degree of contamination 's'. 

Note: We use M_500cr here as Arnaud et al. relations for Y_SZ are for M_500cr, we change it to M_200cr using NFW profile and Duffy et al. concentration values, M_cluster here is equivalent to the M_200cr.

!contamination1.png!

Another way of seeing contamination is to integrate over the degree of contamination for same mass cluster at different redshifts. In the following plot we show the probability of finding AGN's within a cluster of mass M_500cr=4.e14 and 1.e15 at different redshifts such that the degree of contamination is above 10%. 

!contamination2.png!