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A Statistical Strategy for the Sunyaev-Zel'dovich Effect's Cluster Data

We present a statistical strategy for the efficient determination of the cluster luminosity function from the Sunyaev-Zel'dovich (SZ) effects survey. To determine the cluster luminosity function from the noise contaminated SZ map, we first define the zeroth-order cluster luminosity function as a discrepancy between the measured peak number density of the SZ map and the mean number density of noise. Then we demonstrate that the noise contamination effects can be removed by the stabilized deconvolution of the zeroth-order cluster luminosity function with the one-dimensional Gaussian distribution. We test this analysis technique against Monte-Carlo simulations, and find that it works quite well especially in the medium amplitude range where the conventional cluster identification method based on the threshold cut-off usually fails.


INTRODUCTION
Galaxy clusters are the biggest bound objects of the universe. Being rare and formed relatively late, the abundance of galaxy clusters depends sensitively on the background cosmology (Barbosa et al. 1996;Henry 1997;Bahcall & Fan 1998;Viana & Liddle 1999;Borgani et al. 1999;Fan & Chiueh 2001;Grego et al. 2001;Molnar et al. 2002). To find as many galaxy clusters as possible and to investigate the evolution of their abundance have thus become two of the most challenging tasks of the current observational cosmology. The cluster survey is categorized by the observing waveband as the X-ray, the optical, and the Sunyaev-Zel'dovich effect. In the past the X-ray or the optical surveys were favored due to their high rate of detecting clusters and relatively low-costs. However, thanks to recent development in technology, the Sunyaev-Zel'dovich effect is currently spotlighted as a powerful cosmological probe (Carlstrom et al. 1996;White et al. 1999;Holder et al. 2000;Lo et al. 2000) that can provide a statistically unbiased sample of clusters unlike the X-ray and the optical surveys.
The Sunyaev-Zel'dovich (SZ) effect represents the change in brightness of the cosmic microwave background (CMB) radiation caused by the interaction of the CMB photons with the ionized intra-cluster gas (Sunyaev & Zel'dovich 1972). The underlying physics of the SZ effect is the inverse Compton scattering: free electrons of the hot intra-cluster gas scatter off the CMB photons as they pass through the galaxy clusters, which results in the shift of the CMB photon frequency and a corresponding change in its radiation energy. Depending on whether the scattering of the CMB photons was by the random or systematic motions of the electrons, the SZ effect is called thermal or kinetic respectively. Here we focus on the thermal SZ effect since its contribution is order of magnitude stronger than the kinetic counterpart (Springel et al. 2001;Zhang et al. 2002).
The SZ effects (∆I f ) observed at a given frequency (f ) at a given position θ on the sky depend on the electron number density (n e ) and temperature (T e ) within the galaxy clusters that the CMB photons have encountered all along the path: with the integral taken along the CMB path in the line-of-sight direction (θ ≡ θ/| θ|). Here y is the cluster Comptonization parameter, and g f is the frequency-dependent spectral shape factor. Note that the SZ effects are spatially localized with the associated clusters along the line of sight, unlike the primordial CMB fluctuations.
Equation (1) implies that the amplitude of the SZ effects observed in a narrow frequency band with almost fixed g f can be quantified by the y-parameter. Provided that noise were absent in the SZ map, clusters could be counted as local maxima of y. In practice, however, noise always dominate the SZ map. A critical issue is how to eliminate the dominant noisecontamination effects. A common observational practice to identify true signals from the noise-contaminated field is to select only such peaks with amplitudes above some threshold (usually several times the noise standard deviation). Using this technique, however, one can select only high-amplitude signals for a given observational integration time. In order to increase the signal to noise ratio, one has to decrease the noise level by increasing the integration time, which is proportional only to the square of the the noise level.
Considering the usual high-costs of the SZ experiments, this conventional technique of signal selection is too inefficient to apply to the SZ cluster data . One may wish to find a more efficient statistical strategy that can allow us to determine the cluster luminosity function as quickly and accurately as possible. Powerful new generations of SZ instruments such as AMiBA (Array for Microwave Background Anisotropy, see Lo et al. 2001) are already in the pipeline. Expecting plenty of cluster data coming out in a few years, it is quite urgent in fact to develop such statistical strategies now.
In this Letter we develop a statistical strategy for the efficient determination of the cluster luminosity function, especially from interferometric SZ surveys using AMiBA as our model experiment.

SZ CLUSTER LUMINOSITY FUNCTION
The SZ cluster luminosity function, n cl (y), is defined as the number density of galaxy clusters with the associated cluster Compton parameter in the range of [y, y + dy]. Its cumulative function, N cl (≥ y) ≡ y n cl (y ′ )dy ′ , is connected to the cluster mass function (see eq. [7] in Barbosa et al. 1996), and thus can be used in principle as a cosmological discriminator (Holder et al. 2000;Fan & Chiueh 2001;Diego et al. 2002;Grego et al. 2001;Molnar et al. 2002;Benson et al. 2002). In practice, the direct conversion of the cluster luminosity function into the cluster mass function is fraught with the difficulties related to the rather large scatter in the correlation between cluster mass and the SZ effect strength (e.g., see Metzler 2002).
Anyway, our goal here is to find an efficient way to find n cl (y) from the observed SZ map that is expected to be significantly contaminated by noise. There are two different sources of noise: instrumental noise and primordial CMB fluctuations. The primordial fluctuations, however, turned out to be negligible in interferometric SZ cluster surveys (Zhang et al. 2002). In the following analysis, we concentrate on instrumental noise only.
Our model experiment, AMiBA, as an interferometric SZ survey, will employ the driftscan method to optimize the observations. For the detailed description of AMiBA and the drift-scan method, see Lo et al. (2000) and Pen et al. (2002) respectively. Among many advantages of the drift-scan method, it makes noise-analysis tractable: in a SZ map measured from the drift-scanned CMB sky, noise is Gaussian white. Therefore, a total SZ map measured by AMiBA will be a combination of non-Gaussian cluster sources with Gaussian white. In the following two subsections, we simulate a total SZ map by means of the Monte-Carlo method, and reconstruct the cluster luminosity function by eliminating the noise contamination effects from the SZ map.

Monte-Carlo Simulations of Drift-Scan SZ Maps
We have constructed a random field on a 2048 2 mesh in a periodic box of linear size 1 deg using the Monte-Carlo method, in such a way that the random field possesses the main statistical properties of a cleaned SZ map expected from AMiBA drift-scan survey over a unit area per a unit hour, assuming the flat-sky approximation.
By a cleaned SZ map, we mean the SZ map smoothed by a optimal filter: In Fourier u-space (u: the Fourier counterpart of θ, u ≡ |u|), the optimal filter for AMiBA drift-scan observations is given are the cluster intrinsic shape and the natural beam respectively. We use W F (u) = 1 u and , where the two angular scales, θ A and θ B , represent the size of the natural and primary beam respectively, related to each other by θ A θ B = 1 6 (Pen 2002, private communications). We first constructed a Gaussian random field with the white noise power spectrum and convolved it by W C (u), and rescaled the field by its rms fluctuations, Second, we simulated the cluster sources by generating a sparse set of two dimensional Gaussian functions of which peak locations and amplitudes were chosen randomly. Zhang et al. (2002) showed that the optimal scan rate for the purpose of AMiBA cluster search is around 150 hours per square degree, which could find 1 cluster every 8 hours. Thus, the total number of the cluster sources was set to be 20, with the expectation that the number of clusters per square degree from the AMiBA cluster search would be around the same number. We locate the cluster sources in the two-dimensional map deliberately so that they are not overlapping one another. The size of each cluster source, i.e., the length scale of each Gaussian function was chosen to be twice the pixel size, which is consistent with the expected AMiBA drift-scan map. The randomly chosen amplitudes of the cluster sources were in the range of [0, 5σ 0 ] where we would like to reconstruct the cluster luminosity function from the noise-contaminated map. The cluster amplitudes were chosen to be distributed exponentially, mimicking the real cluster distribution in this range (Zhang et al. 2002).
Finally, we obtained a simulated SZ map by combining the Gaussian noise field with the cluster sources. Then we identified the local maxima of the total field by selecting those pixels whose amplitudes exceed the amplitudes of their 8 closest-neighboring points, and count the number density of the total SZ map, n sz (ν), as a function of the rescaled amplitude, ν ≡ y/σ 0 .

Deconvolution Method
If there were no clusters, the measured SZ map would be just a map of Gaussian noise, whose mean number density of local maxima, n g (ν), is analytically derived to be (Longuet-Higgins 1957;Bond & Efstathiou 1987): Here σ 2 i = 1 (2π) 2 |W C (u)| 2 u 2i d 2 u for our noise spectrum, and R * ≡ √ 2(σ 1 /σ 2 ), γ ≡ σ 2 1 /(σ 0 σ 2 ). The difference between the peak number density of the total SZ map and equation (2) provides a zeroth-order approximation to the cluster luminosity function: n (0) cl (ν) = n sz (ν) − n g (ν). At high ν-tail (ν ≫ 1) where the mean number density of noise peaks drops practically to zero, n sz (ν) ≈ n (0) cl (ν) ≈ n cl (ν). At low-ν range (0 ≤ ν ≤ 2) where Gaussian noise strongly dominates, n sz (ν) ≈ n g (ν) and n The intriguing section is at the medium-ν range (ν ∼ 3), where n (0) cl (ν) includes not only the noise-scatter but also the noise-contaminated cluster peaks to a non-negligible degree. With the conventional cluster selection procedure based on the amplitude cut-off, the number density of cluster peaks in this intermediate range cannot be determined since the peaks in this range are all disregarded as noise. Thus, it is this medium-amplitude section where one needs a better analysis technique to find the number density.
Noise contaminates the cluster number density by changing its amplitude. Let x measure the noise-contamination effects on the peak amplitude such that ν = ν 0 + x where ν 0 and ν are the real and contaminated amplitude of a cluster peak respectively. For a Gaussian noise, x can be assumed to be a Gaussian variable. Now, the probability distribution of ν can be written as a convolution of the probability distributions of ν 0 and x. In terms of the number density, one can say where p(x) = 1 √ 2π e − x 2 2 and N tot sz and N tot cl are the total number of peaks of the SZ map and that of real clusters respectively.
Equation (3) implies that the cluster luminosity function, n cl (ν), can be found by the deconvolution of the n (0) cl (ν) and p(x). In theory, the deconvolution of n (0) cl (ν) and p(x) could be easily conducted : just dividing the Fourier transform of n (0) cl (ν) by that of p(x), performing its inverse Fourier transformation. In practice, however, the deconvolution process itself is very unstable (Press et al. 1992). To make the deconvolution stable and optimize the estimation of n cl (ν), we apply a Wiener filter, say, W P (ν), to n (0) cl (ν). Let P − and P + be the real and corrupted power spectrum of n (0) cl (ν) respectively, then the Wiener filter in the Fourier ν k -space (ν k : the Fourier counter part of ν) can be written as W P (ν k ) = P − (ν k ) P + (ν k ) . One can easily see that W P (ν k ) ≈ 1 where the error is negligible, and W P (ν k ) ≈ 0 where the error is dominant. Unfortunately, we cannot determine the exact functional form of W P (ν k ) since the only available quantity to us is P + (ν k ). Nevertheless, since the Wiener filter works in the least-square sense (Press et al. 1992), even a fairly reasonable approximation to W P (ν k ) can make it work quite well. Figure 1 plots P + (ν k ), showing there is a sharp boundary between the noise-dominant and negligible sections where P + (ν k ) has two distinct behaviors. Finding a turning point, ν k0 , by eye, we approximated the Wiener filter by a step function such that W P (ν k ) = Θ(|ν k | < ν k0 ), given the asymptotic behaviors of W P (ν k ).
Here is our recipe for the determination of the cluster luminosity function from the total SZ map: Construct n (0) cl (ν) on a one-dimensional discrete grid by counting the peak number density of the total SZ map and subtracting equation (3) from it. Calculate its power spectrum, P + (ν k ), by measuring the mean square amplitude of its Fourier transform. Plot P + (ν k ) to find the turning-point, ν k0 . Approximate W P (ν k ) as a step-function such that W P (ν k ) = Θ(|ν k | < ν k0 ). Convolve n (0) cl (ν) by W P (ν k ) and deconvolve it by the onedimensional Gaussian distribution, p(x).
In the upper panel of Figure 2, we show the cumulative cluster luminosity function N cl (≥ ν) (solid line) reconstructed from the SZ map with the above recipe and compare it with the real distribution (square dots). We also plot the cumulative peak number density of the total SZ map (long-dashed line), the cumulative noise mean number density (dashed line), and the cumulative zeroth-order cluster luminosity function (dotted line) for comparison. Figure  2 reveals that the reconstructed cluster luminosity function is indeed in good agreement with the real distribution, especially in the medium range of 2 ≤ ν ≤ 4. Note also that in the low-ν section (ν ≤ 1) noise dominates the SZ peaks while in the high-ν tail (ν ≥ 4) the total SZ peaks are mainly the cluster peaks as expected.
When determining n cl (ν), the total number of cluster peaks, N tot cl , are assumed to be given as priors. In case that N tot cl is not available, we can still determine the probability density distribution of cluster peaks, P cl (≥ ν) = N cl (≥ ν)/N tot cl . The lower panel of Figure  2 plots the cumulative probability density distributions that can be determined using our analysis technique without any prior. Again, the real and the reconstructed distributions agree with each other quite well. We tested our technique against different realizations of the SZ maps by varying the total number of cluster peaks and distribution shapes, and found it quite robust.

SUMMARY AND DISCUSSIONS
We have developed a useful analysis technique to determine the cluster luminosity function efficiently, using AMiBA, a drift-scan interferometric SZ survey, as a model experiment. We have simulated a total SZ map using the Monte-Carlo method, and counted the peak number density from it. The total SZ map is constructed by combining a Gaussian noise field with cluster sources. We noted that the peak number density of the SZ map at medium peak amplitude range has non-negligible contributions from the cluster peaks with noisecontamination effects included.
To determine the cluster number density, i.e., the cluster luminosity function, first we have measured the zeroth-order cluster luminosity function by subtracting the available mean noise number density from the peak number density of the total SZ map. We have quantified the noise-contamination effects included in the zeroth-order approximation by a single Gaussian variable, and found that the cluster luminosity function can be expressed as the deconvolution of the zeroth-order approximation by the one-dimensional Gaussian distribution.
We have stabilized the deconvolution process by convolving the zeroth-order approximation with a Wiener filter. The approximate functional form of the Wiener filter has been determined from the information of the power spectrum of the zeroth-order approximation. Finally by deconvolving the Wiener-filtered zeroth-order approximation of the cluster luminosity function we have determined the cluster luminosity function from the simulated SZ map. We have compared the reconstructed (cumulative) cluster luminosity function with the real one, and found good agreements between them, especially in the medium amplitude range where the conventional technique fails.
The consequence of the statistical strategy presented here is that it can allow us to find the cumulative distribution of the sources even when the number of the sources occupy only small fraction of the total number of maximum peaks. We also expect this statistical strategy to be applied to the construction of the cluster mass function in weak gravitational lensing analysis.
However, it is worth noting that the measurement accuracy of the cluster luminosity function may be improved on by improving the approximation accuracy of the Winer filter, and it is also worth noting that although our technique determines the cluster number density efficiently but it cannot select the cluster peaks from the SZ map. Furthermore, to examine the usefulness of our analysis technique in real practice, testing it against real SZ hydrodynamic simulations will be necessary. Our future work is in this direction.
We thank U. L. Pen for many helpful discussions on drift-scan method, and K. Yoshikawa for useful comments. This work was supported by the research grants of the JSPS fellowship .  The cumulative number density of local maxima as a function of the rescaled amplitude, ν. The total number of cluster peaks (N tot cl ) are assumed to be given as a prior. N tot cl = 20 in this figure. Lower: The cumulative probability distribution of local maxima as a function of the rescaled amplitude. Unlike the number density, the cluster probability distribution can be reconstructed without any prior.