I am working with 48 re-simulated clusters provided by Joe Mohr. They are the same ones used in aph/9912364. I am trying to produce curves of limiting mass as a function of redhsift for several different telescope diameters which Joe can use to produce cosmological parameter constraint plots for each diameter to help us argue why an 8 meter dish is essential...
First I concatenate the 48 sims, 3 projections each into one big 144 cell image. Here it is both linear and log scale:
Next I take this and add a realization of the CMB (according to current models):
Next I convolve with the beam (in these plots 1.3 arcmin) and add pixel white noise appropriate to 10 uK/beam. i.e. If you take the pixel noise alone and convolve with the beam you will get a pixel rms of 10.
Next I apply an optimal filter to reextract the SZ input image according to the simple prescription given in Numerical Recipies book. This attempts to filter out the noise (CMB+pixel noise) *and* deconvolve the beam. To compose the filter one needs to know the power spectrum of the signal one is after. In this example I just use the input image, but realistically this is of course cheating. However, assuming a slightly incorrect power spectrum for the signal of interest will just cause the filter to be slightly non-optimal. I don't think this is a big issue.
Here are the power spectra.
Now comes the part I'm not sure about - I want to determine the significance at which each cluster is detected. The approach which I have taken is as follows. Take the distribution of pixel values in the filtered image and fit a Gaussian in the cental region. Then take the image, subtract the offset of the Gauss fitm and rescale by it's width - to make a "significance image".
Now I want to fit each cluster. I take the peak position in each SZ image input cell as the location for each cluster. I then fit a 2D Gaussian to the region around this peak in each cell of the significance image. I allow the width of the Gaussian to be free. I subtract each fit from the significance image so that I can gauge the adequacy of approximating the shape as Gaussian. The final result is shown below - we can see that the super massive clusters leave some residuals but basically a Gaussian is a pretty good approximation to the cluster shapes in the filtered images.
Alright so far so good - but what is the detection significance for these clusters? Right now I am using the peak height of the Gaussian fit to the significance image. Here it is plotted versus cluster mass (mass scaled to h=0.65) for three different beam sizes.
The clusters are well resolved in the sense that the fitter selects Gaussian sigma parameters several times the beam. The selected width is a function of beam size, but is rather constant with peak height for higher significance detections.
One might think that a better estimate of the cluster mass is the integral of the Gaussian fit rather than it's peak height. But since for a given beam the width is rather constant it's not going to improve the correlation.
In fact the filter makes things very hard to understand. It's different for each beam width, but also it will have a different effect on clusters of different widths, filtering out a different fraction of their flux. This is presumably why the exponent in the peak height vs. mass fits above is not 5/3 as expected.
Probably although the optimal filter scheme is a reasonable way to find candidate cluster positions it may well be better to go back and fit them in the unfiltered image. In that case the nature of the pixel noise contribution is well understood. The CMB is not - it does introduce pixel-pixel covariance, but it can probably largely be modelled as a constant offset in y for the region around the cluster, or maybe as a 2D slope in that region...
The bottom line is that my detection-significance versus mass plot above shows more scatter than aph/9912364. This may be right - or my method above may be flawed. Also the implied mass limit for 1.3 arcmin beam, 10 uK/beam noise is around 4e15 solar masses which is apparently close to what Bill and co got?
I have a catlog of supposed cluster positions and masses for one of the Springel, White, Hernquist, astro-ph/0008133 1 deg^2 maps so I will try and make a similar (very low statistics plot) from that to see if it's similar...