Annu. Rev. Astron. Astrophys. 2002. 40: 539-577
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3.3. Strategies and Selection Functions for X-ray Surveys

Ideally, one would like to use selection criteria based on X-ray properties alone to construct a flux-limited sample with a simple selection function. The task of separating clusters from the rest of the X-ray source population is central to this work. At the ROSAT flux limit (~ 1 × 10-14 erg cm-2 s-1 for clusters) ~ 10% of extragalactic X-ray sources are galaxy clusters. A program of complete optical identification is very time consuming, as only spectroscopy can establish in many cases whether the X-ray source is associated with a real cluster. The EMSS and NEP samples, for example, were constructed in this way. In some cases, the hardness ratio (a crude estimate of the source's X-ray spectral energy distribution) is used to screen out sources which are incompatible with thermal spectra or to resolve source blends. With the angular resolution provided by ROSAT, however, it became possible to select clusters on the basis of their spatial extent. This is particularly feasible with pointed observations, as opposed to all-sky survey data which are characterized by a broader PSF and shallower exposures, so that faint and/or high redshift clusters are not always detected as extended (e.g. Ebeling et al. 1997, Böhringer et al. 2001).

In constructing RASS based samples (shaded circles in Figure 4) most of the authors had to undertake a complete optical identification program of ~ 104 sources using POSS plates or CCD follow-up imaging in order to build a sample of cluster candidates. Whereas a sizable fraction of these systems can be readily identified in previous cluster catalogs (primarily Abell's), spectroscopy is needed to measure redshifts of newly discovered systems or to resolve ambiguous identifications. We recall that optically selected, X-ray confirmed samples, such as the X-ray Brightest Abell-like Clusters (XBACS, Ebeling et al. 1996), while useful for studying optical-X-ray correlations, lead to incomplete flux-limited samples. Many of the low X-ray luminosity systems (poor clusters or groups) are missed in the optical selection even though they lie above the X-ray flux limit of the RASS.

Most of the ROSAT serendipitous surveys (dark circles in Figure 4) have adopted a very similar methodology but somewhat different identification strategies. Cluster candidates are selected from a serendipitous search for extended X-ray sources above a given flux limit in deep ROSAT-PSPC pointed observations. Moderately deep CCD imaging in red passbands (or in near-IR for the most distant candidates) is used to reveal galaxy overdensities near the centroid of X-ray emission. Extensive spectroscopic follow-up programs associated with these surveys, have lead to the identification of roughly 200 new clusters or groups, and have increased the number of clusters known at z > 0.5 by approximately a factor of ten.

An essential ingredient for the evaluation of the selection function of X-ray surveys is the computation of the sky coverage: the effective area covered by the survey as a function of flux. In general, the exposure time, as well as the background and the PSF are not uniform across the field of view of X-ray telescopes (owing to to their inherent optical design), which introduces vignetting and a degradation of the PSF at increasing off-axis angles. As a result, the sensitivity to source detection varies significantly across the survey area so that only bright sources can be detected over the entire solid angle of the survey, whereas at faint fluxes the effective area decreases. An example of survey sky coverage is given in Figure 5 (left). By integrating the volume element of the Friedmann-Robertson-Walker metric, dV / dOmega dz(z, Omegam, Omegalambda) (e.g. Carroll et al. 1992), over these curves one can compute the volume that each survey probes above a given redshift z, for a given X-ray luminosity (LX = 3 × 1044 erg s-1 appeq L*X, the characteristic luminosity, in the figure). The resulting survey volumes are shown in Figure 5 (right). By normalizing this volume to the local space density of clusters (phi*, see below) one obtains the number of L* volumes accessible in the survey above a given redshift. Assuming no evolution, this yields an estimate of the number of typical bright clusters one expects to discover.

Figure 5a Figure 5b

Figure 5. (Left) sky coverage as a function of X-ray flux of several serendipitous surveys; (Right) corresponding search volumes, V( > z), for a cluster of given X-ray luminosity (LX = 3 × 1044 erg s-1 [0.5 - 2 keV] appeq L*X). On the right axis the volume is normalized to the local space density of clusters, phi*.

By covering different solid angles at varying fluxes, these surveys probe different volumes at increasing redshift and therefore different ranges in X-ray luminosities at varying redshifts. The EMSS has the greatest sensitivity to the most luminous, yet rare, systems but only a few clusters at high redshift lie above its bright flux limit. Deep ROSAT surveys probe instead the intermediate-to-faint end of the XLF. As a result, they have lead to the discovery of many new clusters at z > 0.4. The RDCS has pushed this search to the faintest fluxes yet, providing sensitivity to the highest redshift systems with LX ltapprox L*X even beyond z = 1. The WARPS, and particularly the 160 deg2 survey have covered larger areas at high fluxes thus better studying the bright end of the XLF out to z appeq 1.

Particular emphasis is given in these searches to detection algorithms that are designed to examine a broad range of cluster parameters (X-ray flux, surface brightness, morphology) and to deal with source confusion at faint flux levels. The traditional detection algorithm used in X-ray astronomy for many years, the sliding cell method, is not adequate for this purpose. A box of fixed size is slid across the image, and sources are detected as positive fluctuations that deviate significantly from Poissonian expectations based on a global background map (the latter being constructed from a first scan of the image). Although this method works well for point-like sources, it is less suited to extended, low-surface brightness sources, which can consequently be missed leading to a significant incompleteness in flux-limited cluster samples.

The need for more general detection algorithms, not only geared to the detection of point sources, became important with ROSAT observations, which probe a much larger range in surface brightness than previous missions (e.g. Einstein). A popular alternative approach to source detection and characterization developed specifically for cluster surveys is based on wavelet techniques (e.g. Rosati et al. 1995, Vikhlinin et al. 1998b, Lazzati et al. 1999, Romer et al. 2000). Wavelet analysis is essentially a multi-scale analysis of the image based on an quasi-orthonormal decomposition of a signal via the wavelet transform which enables significant enhancement of the contrast of sources of different sizes against non-uniform backgrounds. This method, besides being equally efficient at detecting sources of different shapes and surface brightnesses, is well-suited to dealing with confusion effects, and allows source parameters to be measured without knowledge of the background. Another method that has proved to be well-suited for the detection of extended and low surface brightness emission is based on Voronoi Tessellation and Percolation (VTP, Scharf et al. 1997 and references therein).

Besides detection algorithms, which play a central role in avoiding selection effects, there are additional caveats to be considered when computing the selection function of X-ray cluster surveys. For example, the sky coverage function (Figure 5) depends not only on the source flux but in general on the extent or surface brightness of cluster sources (Rosati et al. 1995, Scharf et al. 1997, Vihklinin et al. 1998). This effect can be tested with extensive simulations, by placing artificial clusters (typically using beta-profiles) in the field and measuring the detection probability for different cluster parameters or instrumental parameters.

More generally, as in all flux-limited samples of extended sources (e.g. optical galaxy surveys), one has to make sure that the sample does not become surface brightness (SB) limited at very faint fluxes. As the source flux decreases, clusters with smaller mean SB have a higher chance of being missed, because their signal-to-noise is likely to drop below the detection threshold. SB dimming at high redshifts (SB propto (1 + z)-4) can thus create a serious source of incompleteness at the faintest flux levels. This depends critically on the steepness of the SB-profile of distant X-ray clusters, and its evolution. Besides simulations of the detection process, the most meaningful way to test these selection effects is to verify that derived cluster surface or space densities do not show any trend across the survey area (e.g. a decrease in regions with higher background, low exposures, degraded PSF). The task of the observer is to understand what is the fiducial flux limit above which the sample is truly flux-limited and free of SB effects. This fiducial flux limit is typically a factor of 2-3 higher than the minimum detectable flux in a given survey.

An additional source of sample contamination or misidentification may be caused by clusters hosting X-ray bright AGN, or by unrelated point sources projected along the line of sight of diffuse cluster emission. The former case does not seem to be a matter of great concern, because bright AGN have been found near the center of clusters in large compilations (Böhringer et al. 2001) in less than 5% of the cases. The latter effect can be significant in distant and faint ROSAT selected clusters, for which high resolution Chandra observations (Stanford et al. 2001, 2002) have revealed up to 50% flux contamination in some cases.

Concerning selection biases, a separate issue is whether, using X-ray selection, one might miss systems that, although virialized, have an unusually low X-ray luminosity. These systems would be outliers in the LX - M or LX - T relation (Section 5.2). Such hypothetical systems are at odds with our physical understanding of structure formation and would require unusual mechanisms that would (a) lead galaxies to virialize but the gaseous component not to thermalize in the dark matter potential well, (b) allow energy sources to dissipate or remove the gas after collapse, or (c) involve formation scenarios in which only a small fraction of the gas collapses. Similarly, systems claimed to have unusually high mass-to-optical luminosity ratio, M / L, such as MG2016+112 from ASCA observations (Hattori et al. 1998) have not held up. MG2016+112 was later confirmed to be an ordinary low mass cluster at z = 1 by means of near-infrared imaging (Benitez et al. 1999) and spectroscopic (Soucail et al. 2001) follow-up studies. Chartas et al. (2001) have completely revised the nature of the X-ray emission with Chandra observations. Comparing optical and X-ray techniques for clusters' detection, Donahue et al. (2001) carried out an optical/X-ray joint survey in the same sky area (ROXS). They found no need to invoke an X-ray faint population of massive clusters.

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