Selection function

In this final section on X-ray properties of XDCP cluster candidates, some important aspects concerning the selection efficiency of serendipitous extended sources with XMM- Newton are summarized.

Figure 9.3 displays the extended X-ray sources of the XDCP distant cluster candidate samplein the 0.5-2.0 keV flux versus core radius3_{plane (see Equ. 2.6), the two prime source}

3_{Core radiir}

parameters governing the detection probability. This empirical plot illustrates nicely the two main XMM limitations that constrain the accessible parameter space of distant cluster candidate sources (see Sect. 6.1). The XMMresolution limitsets a lower core radius cut-off at about 600 _{(vertical red line), whereas the background imposes a limit on the maximum} core radius for sources of given flux (lower red line). Note that the currently three most distant spectroscopically confirmed clusters in the Southern hemisphere (green diamonds) are well away from the sample limits, in particular concerning the accessible minimum flux threshold (vertical offset to the lower red line).

The selection function characterizes a galaxy cluster survey in terms of the extended source detection probability of clusters with given properties in the X-ray data. The most important cluster characteristics with prime influence on the detection rate are given in the following:

Source Flux: For an idealized, homogenous X-ray survey, the flux limit is the single characteristic that determines the detection probability of a source. The received X- ray fluxfX in a given energy band depends on the cluster luminosityLX, the spectral

shape governed by the gas temperatureTX, the redshiftz, and the galactic hydrogen

absorption column NH. Note that high cluster redshifts lead to severe cosmological

surface brightness dimming (Equ. 3.22) and can hence introduce selection biases at faint flux levels, where the surface brightness drops below the signal-to-noise threshold (for a real survey with background).

Core Radius: Once the background surface brightness reaches levels comparable to that of the source, the physical size of the cluster, characterized by the projected core radiusrc4 (see Equ. 2.6), becomes the second prime property governing the detection

rate. For further analytical considerations it is important to note that the source flux fraction encircled within the core radiusfX(r < rc) is constant for β-models,i.e.

it does not vary with the size of the source5_{. For extended sources whose SB-profiles}

follow the appliedβ= 2/3 model, this fraction isfX(r < rc)/fX(tot) = 0.293,i.e.about

30% of the flux originates from the core region.

Cooling Core Activity: Cooling core activity in clusters (see Sect. 2.3.5) is connected to very peaked inner SB profiles with smaller core radii than non-cooling core clusters (e.g.Chenet al., 2007). For possible high-redshift CCC, the peaked inner profile can have positive or negative effects on the detection probability. Moderate cooling core activity, as found for XMMU J2235.3−2557 at z= 1.393 (Santos et al., submitted), increases the SB contrast and hence improves the signal-to-noise ratio. However, if the core radius of a strong CCC drops below the XMM resolution limit, the extended source detection probability drops and the cluster might be missed.

4_{To avoid confusion with the off-axis angle Θ, the notation r}

c is used for the projected core radius

throughout the following discussion.

Cluster Morphology: For the detection process, cluster sources are usually idealized as spherically symmetric. This might not be a good approximation (e.g. Hashimoto et al., 2005), in particular at high redshifts, where clusters are either dynamically very young objects or are still in the formation process with a high merger frequency (see Fig. 3.4). Deviations from spherical symmetry and in particular cluster substructure will lower the detection probability, since the cluster emission might be mistaken for several point sources.

AGN Activity: If a cluster hosts an X-ray bright AGN, the chance for a misidentification as a point source increases. While only 5% of local clusters have bright AGN close to their center (B¨ohringeret al., 2004), the point source contamination for high-redshift systems can be significant (e.g. Stanford et al., 2001).

The relevantXMM instrumental and data characteristicsfor the detection of faint extended sources have been discussed in Sect. 6.1. As a function of detector position, i.e. off-axis angle Θ and azimuth φ, the key features are (i) the effective, vignetting corrected clean exposure time teff(Θ), (ii) the PSF FWHM(Θ) defining the minimum resolvable cluster

angular size rmin(Θ), (iii) the background surface brightness B(Θ, φ), and (iv) detector

defects (e.g. gaps, dead columns, hot pixels).

Close to the detection threshold for extended sources, XMM observations are, in very good approximation, always background limited (see Equ. 7.1). Assuming that the detec- tion threshold is defined by a minimum signal-to-noise ratio SNRmin in a given detection

aperture, e.g. the core radius of the source, then the limiting flux threshold scales as flim∝

q

B(Θ, φ)/qteff(Θ) with the exposure time teff and the background level B. Sim-

ilarly, the increased detection aperture of larger extended sources at a given background level has the scaling effect flim∝

q

πBr2

c∝rc. This expected linear dependance of the core

radius on the detection limit is reflected by the lower red line in Fig. 9.3. A priori unknown effects can be taken into account by introducing fudge factors for the cluster AGN and fore- ground flux contamination of nearby sources Acontam∼>1 and for possible morphological

cluster distortions Bmorph>∼1. By defining the local flux limit as the level where a cluster

with resolvable core radius rc> rmin(Θ) is detected with a specified probability, e.g. 50%,

then the combination of all factors yields flim(rc> rmin)∝rc

q

B(Θ, φ)

q

teff(Θ)

AcontamBmorph 6=const . (9.1)

The effectivelocal extended source flux limit is hence a function of the XMM field charac- teristics (teff, B), the location on the detector (Θ, φ), the sky position for contaminating

nearby sources Acontam(RA, DEC), and the cluster properties (rc,Bmorph). A single global

flux limit for background limited XMM serendipitous surveys is hencenot well defined. The variation of the cluster core radius over typical values in the local Universe of 50-500 kpc already introduces a difference in the limiting sensitivity of one order of magnitude. A stated flux limit for XMM surveys is only meaningful for a specified cluster size, e.g. for

core radii of rc= 1500, which can be resolved over the full FoV. All other factors combined

cause sensitivity variations across the XDCP survey of approximately another factor of ten from the deepest to the shallowest fields.

The currently best modelled selection function of all XMM-Newton surveys has been provided by the XMM-LSS project (see Sect. 4.3). For the published five square degrees of data with 29 clusters, Pacaudet al.(2006, 2007) have determined the detection probability of sources as a function of core radius from simulated data assuming symmetric β-model cluster profiles. As a result, the XMM-LSS survey has abandoned the concept of a single global flux limit, for the discussed reasons, and replaced it by local likelihood thresholds for the source extent anddetection significance.

In close collaboration with a team led by Joe Mohr at the University of Illinois, simu- lation efforts have started to determine the XDCP selection function for each of the 469 survey fields. The final goal is to go one step beyond the XMM-LSS modelling and use realistic artificial clusters from high-resolution simulations with different masses, redshifts, morphologies, and dynamical states. These simulated clusters will be placed in mock im- ages that take the XMM-Newton instrumental characteristics into account. After adding background and a point source population that follows the observed AGN number counts, the XDCP detection pipeline will be applied to multiple mock realizations of the same field. From the comparison of the detected extended source list with the cluster input catalog, the completeness and contamination functions can be evaluated for each field as a function of input cluster mass M, size rc, morphology, and redshift z.