4.4 Towards a galaxy cluster selection function for eROSITA
4.4.3 The eROSITA cluster selection function
As described in Chapter 2, by assuming a set of scaling relations one can obtain the luminosity, the gas temperature and physical extent of a galaxy cluster of a given mass at a given redshift. From these quantities, different galaxy cluster characteristics can be determined, such as the angular core radius and the X-ray flux. By knowing these two quantities, the detection probability of a galaxy cluster of any mass at any redshift can be predicted by using the detection efficiency curves displayed in Fig. 4.22. This information is essential for the calculation of the eROSITA cluster selection function.
Dr. Nicolas Clerc (MPE) has used the cluster detection efficiency curves obtained in this work (Fig.4.22) to compute the expected redshift distribution of the galaxy clusters that eROSITA will detect. In the following, this work is briefly presented as an illustration of the application of the obtained cluster detection efficiency.
Cosmological model and scaling relations
AΛCDM cosmological model is assumed, together with a flat Universe and a non-evolving dark energy. The cosmological parameters are set to the 5-year Wilkinson Microwave Anisotropy Probe (WMAP5) cosmology: h= 0.72, Ωm = 0.25, Ωb = 0.043, ΩΛ = 0.75, ns = 0.96, and σ8 = 0.79 (Dunkley et al. 2009). The Tinker et al. (2008) fit is used to describe the comoving halo number density as a function of mass, dn(M, z)/dM, i.e. the halo mass function (see Section2.2.3). This fit calculates the halo masses within r200b, i.e. the mass within a radius that encloses an overdensity 200 times the mean density of the Universe at a given redshift. However, the mass function is needed in terms of a mass defined with respect to the critical density of the Universe, since this is the mass definition that is used by the scaling relations. The conversion is performed using the approximate inversion equation from Hu & Kravtsov (2003) assuming an NFW mass profile (see Section2.2.2) and the concentration parameter model from Bullock et al. (2001).
As stated previously, at a given redshift a galaxy cluster can be described by its gas temperature and its bolometric luminosity. However, from the halo mass function, only the mass and redshift of the galaxy clusters are known. By using cluster scaling relations, the temperature and luminosity of the galaxy clusters can be obtained from the mass and redshift information. The cluster gas temperature is calculated using the M − T relation from Arnaud et al. (2005), and its bolometric luminosity using the LX− T relation from Pratt et al. (2009). Both relations include a self-similar redshift evolution. The intrinsic scatter of such relations was also taken into account to model the final cluster population. The β-model (see Section2.1.2) is used to describe the emission profile of galaxy clusters. The chosen values for this model are: β= 2/3 and the core radius, rc, is parameterised in terms of r500: xc = rc/r500, where r500is the radius that encloses an overdensity 500 times the critical density of the Universe at a given redshift. xc is taken constant at all redshifts and masses, with value xc = 0.15 (Böhringer et al. 2014). The core radius is an important parameter in this process since the derived cluster detection efficiency depends on it (see Fig.4.22).
The next step is to convert the galaxy cluster quantities (rc, T and LX) into observable ones. The core radius is transformed into angular scale by means of the angular diameter distance (see Section2.1.2). The total cluster count-rate or flux in the [0.5−2] keV energy band is calculated with the XSPEC spectral fitting package (Arnaud1996) by using an APEC thermal plasma emission model with a metallicity of
Figure 4.23: Comparison of the cosmological expectations between a flux-limited cluster sample (black solid line) and sample folded with a source selection function (from the synthetic sample of the simulated Equatorial sky region, red solid line). Left: Expected cluster selection function in the mass-redshift plane for the synthetic sample compared with the flux-limited one. The contours enclose the 30%, 60% and 90% of the total expected number of clusters. Right: Expected dn/dz for the synthetic sample compared with the flux-limited survey. Image provided by N. Clerc.
0.3 Z and a Hydrogen column density corresponding to the used simulated field (see Section4.3.1) folded through the eROSITA response matrices.
Model for the mass and redshift distributions
By using the above quantities, the expected galaxy cluster distribution, dn/dM/dz/dT/dLX, can be computed. First, the scaling relations are used to turn the halo mass function, dn/dM/dz, into dn/dM/dz/dLX/dT at a given redshift. The count-rate is estimated from each luminosity - temper- ature pair, and the core radius from each mass, M. The dn/dM/dz/dLX/dT distribution is ultimately based on the total count-rate and extent (core radius) of the galaxy clusters7. In this way, the cluster
detection efficiency is applied in the M − LX− T plane. By marginalizing over LXand T, one is able to recover dn/dM/dz with the selection function included. Finally, by integrating over all masses the dn/dz distribution is obtained.
The above method is applied to the simulated galaxy clusters that pass the selection of Fig.4.22. This will be referred as the eROSITA synthetic selection function. Figure 4.23 shows an example of the shape of dn/dz/dM and the redshift distribution of dn/dz for the simulated Equatorial field selection (left bottom panel of Fig.4.22). The results are compared with a flux-limited sample, which is assumed to have flux limit of 3.4 × 10−14erg s−1cm−2. This flux limit is taken from the eROSITA predictions
7The dn/dM/dz/dL
X/dT distribution depends on the assumption that galaxy cluster temperatures can be measured from their
collected X-ray photons, whose error depends mainly on the number of collected photons and the precision of the cluster redshift
4.4 Towards a galaxy cluster selection function for eROSITA
(see Merloni et al.2012). Both samples show very similar redshift distributions, with a peak at z= 0.2. Most of the expected clusters have masses between 3 × 1013h−1M and 1.5 × 1014h−1M .
From the synthetic cluster selection it is expected to detect 5 clusters per deg2, whereas from the flux- limited sample only 4.6 clusters per deg2 are expected. The synthetic cluster density gives more than 2 × 105galaxy clusters in the whole sky while the flux-limited cluster density gives less than 1.9 × 105. By taking as a reference a total sky coverage of 27, 145 deg2(excising ±20◦around the Galactic plane) the final number of expected clusters for the synthetic selection is ∼ 1.36 × 105, and ∼ 1.25 × 105for the flux-limited selection. However, the cluster selection function of the Equatorial field has a contamination level of 10%, i.e. 0.5 spurious clusters per deg2, increasing the number of clusters extended detections to ∼ 1.49 × 105. This additional contamination needs to be accounted for, either using an accurate determination of their distribution in the flux-extent plane, which can be included it in the cosmological modelling, or by establishing other strategies to clean the sample (e.g. via optical follow-up, when available).
X-ray observations are usually contaminated by background flares, which only impact the particle back- ground and have to be removed from the data. In the calculation of the total number of clusters the area covered by the Galactic plane was removed. Then, the remaining sky area has a larger exposure time than 1.6 ks. However, assuming that the data has to be cleaned from flares, the total exposure time is reduced. Therefore, as a first approximation, the calculation remains acceptable.
The 8% difference in the number of clusters between the synthetic cluster sample and the flux-limited sample can make a huge difference when using large samples of galaxy clusters to constrain cosmolo- gical parameters. Pillepich et al. (2012) has found that the constraints on the cosmological parameters can be improved up to 30% when they increase their sample from 9.32 × 104to 1.37 × 105clusters, i.e. in ∼ 4.5 × 104clusters.