18 Feb 2002 - South Pole SZ Scope Simulations IV

This page is an extension of my 21 Dec 2001 and 1 Jan 2002 entries but expanded to cluster images at several redshifts rather than just z=0.5 as before.

To do this I have interpolated images output from the cluster simulations at various redshifts onto a grid big enough to fit z=0.2. This means at greater z the images under-fill the available box. The following plots illustrate this. They show the SZE on a log scale for the set of 144 cluster image projections at z=0.2 and z=0.5. The pixel size is approx 0.2 arcmin.

I then proceed basically as before except now there are 5 sky "images", one for each redshift z=0.2,0.5.1.0,1.5,2.3.

I first convert SZ y to T uK at 150 GHz, realize (vanilla LCDM) CMB and add the two.

The redshift in all these five panel plots increases from left to right and top to bottom.

Next I convolve with a given beam size (here 1.2 arcmin FWHM) and add pixel noise of a given level (here 10 uK/beam):

Next I take the power spectra of total noise and beam smeared SZE and construct an optimal filter according to the simple prescription given in Numerical Recipes book; if S is the power spectrum of the beam smeared SZE and N is the "total noise", which in this case is the beam smeared CMB plus the pixel noise, then the filter F = S / (S+N). Note that the left axis is log and the right is linear.

The output of this filter is shown below (also including beam deconvolution but this makes little difference):

Note that I construct the filter using the cluster images at all 5 redshifts.

To access the significance of cluster detections I histogram the pixel values (again over all redshifts):

I then fit the histogram in the region about the peak, and divide the images by the resulting sigma value to get "significance images".

I next just go in and take the value of the significance image in the pixels where the original cluster images peaked. The size of the crosses is proportional to this value:

I have made various experiments fitting functions to the post filter cluster peaks. A 2D circular Gaussian is a pretty good fit - beta model fits are not appropriate for the post filter images. Presumably a parameter can be found that correlates better with the cluster mass than the crude peak height that I have used here...

The following plot shows the cluster "detection significance" plotted versus mass in solar masses for each of the five redshifts. The fit is a simple power law S = a * M^b. At the larger redshifts the correlation is quite good.

Next I repeat the above process for a variety of beam sizes and noise levels taking the value of the power law fit at a level of 5 sigma as some kind of "typical mass to give a 5 sigma detection". Note that the apparent "CMB floor" for an ideal detector is down around 1e14 solar masses. Also note that for the baseline 1.2 arcmin fwhm beam and 10 uK/beam noise the limiting mass is around 4e14 solar, compared to around 2e14 calculated for the SZA in aph/9912364 using the same simulations (but without CMB).

I also have a 1 square degree image as described in aph/0008133 kindly provided by Martin White along with a catalog of clusters in the image derived from the full simulation box. The red crosses show the catalog locations for all entries with mass greater than 3e13 solar, with the cross size proportional to the mass (most massive is 3e14):

I have then followed the same procedure as above except that here it doesn't make sense to break down by redshift as there are few distant clusters in such a small image. The following plot show the limiting mass when varying beam width and noise level:

It appears the flat line for zero noise is correct - if CMB is the only noise the effect of the beam can be perfectly de-convolved and the reconstructed image is independant of beam size.

The above is naive in a variety of ways including, but not limited to:

• I have assumed that the instrument delivers a perfect image of the sky with only simple uncorrelated pixel noise, and that the image comtains all spactial frequencies. In reality low frequencies are lost to common mode rejection. More importantly the map real map will be very far from perfectly "cross linked".
• I have formlulated my filter, from and applied it to, an equal number of clusters at 5 redshifts. In reality we will not know the power spectrum of the cluster signal we are searching for.
• In reality the clusters will not be arranged on a nice regular grid as here, and confusion effects may play some role. Probably pretty small though.
• I am told the cluster simulations I used are much too cuspy. Not clear if will have much effect though...

• Put in some simulation of the effects of gain variation and atmospheric noise on the map making process.
• Look at how to get a detection significance observable that correlates better with cluster mass (less scatter).
• Maybe wavelet filters are better than the Fourier filter I am using?...
• Get the cluster catalogs for the remaining 14 of the SWH images to improve the statistics, and maybe start to be able to break it down by redshift for that sim.
• Get other (more recent) sets of individual cluster simulations and see if they give the same limiting masses.