Parameters from the XCS

The Xmm Cluster Survey: Forecasting cosmological and cluster scaling-relation parameter constraints

Abstract

We forecast the constraints on the values of , , and cluster scaling relation parameters which we expect to obtain from the XMM Cluster Survey (XCS). We assume a flat CDM Universe and perform a Monte Carlo Markov Chain analysis of the evolution of the number density of galaxy clusters that takes into account a detailed simulated selection function. Comparing our current observed number of clusters shows good agreement with predictions. We determine the expected degradation of the constraints as a result of self-calibrating the luminosity–temperature relation (with scatter), including temperature measurement errors, and relying on photometric methods for the estimation of galaxy cluster redshifts. We examine the effects of systematic errors in scaling relation and measurement error assumptions. Using only self-calibration, we expect to measure to (and to the same accuracy assuming flatness), and to , also constraining the normalization and slope of the luminosity–temperature relation to and per cent (at ) respectively in the process. Self-calibration fails to jointly constrain the scatter and redshift evolution of the luminosity–temperature relation significantly. Additional archival and/or follow-up data will improve on this. We do not expect measurement errors or imperfect knowledge of their distribution to degrade constraints significantly. Scaling-relation systematics can easily lead to cosmological constraints or more away from the fiducial model. Our treatment is the first exact treatment to this level of detail, and introduces a new ‘smoothed ML’ estimate of expected constraints.

keywords:
cosmological parameters – cosmology: observations – cosmology: theory – galaxies: clusters: general – methods: statistical – X-rays: galaxies: clusters

1 Introduction

The abundance of galaxy clusters as a function of mass and redshift can give a powerful constraint on cosmological models. Specifically, data on the evolution of the number density of galaxy clusters with redshift has been used to obtain direct estimates for both , the dispersion of the mass field smoothed on a scale of , and on , the present mean mass density of the Universe (Frenk et al. 1990; Oukbir & Blanchard 1992; Viana & Liddle 1996; Oukbir & Blanchard 1997; Henry 1997; Bahcall, Fan & Cen 1997; Eke et al. 1998; Reichart et al. 1999; Donahue & Voit 1999; Viana & Liddle 1999; Blanchard et al. 2000; Henry 2000; Borgani et al. 2001; Refregier, Valtchanov & Pierre 2002; Henry 2004; Gladders et al. 2007; Rozo et al. 007a). Furthermore, such data could be used to constrain the present energy density of a dark energy component, , and its equation of state (Wang & Steinhardt 1998; Haiman, Mohr & Holder 2001; Huterer & Turner 2001; Levine, Schulz & White 2002; Weller, Battye & Kneissl 2002; Battye & Weller 2003; Hu 2003; Majumdar & Mohr 2003, 2004; Wang et al. 2004; Lima & Hu 2005; Mantz et al. 2008), or more simply the present vacuum energy density associated with a cosmological constant, (Holder, Haiman & Mohr, 2001). Others have suggested using galaxy clusters to constrain particle physics beyond the Standard Model (e.g. Wang et al. 2005; Erlich, Glover & Weiner 2008), or modified-gravity models where it has been shown that e.g. the Dvali–Gabadadze–Porrati (DGP) modified-gravity model should be testable in coming surveys (Tang et al., 2006; Schäfer & Koyama, 2008). An alternative method to abundance evolution using X-ray galaxy clusters to constrain cosmology, is based on the gas mass fraction (e.g. Allen, Schmidt & Fabian 2002; Vikhlinin et al. 2003; Ettori, Tozzi & Rosati 2003; Rapetti, Allen & Weller 2005; Vikhlinin et al. 2006; Allen et al. 2008; Rapetti et al. 2008).

Galaxy cluster measurements are complementary to other cosmological constraints derived from the Cosmic Microwave Background (CMB) and distant Type Ia Supernovae observations, and thus help break degeneracies amongst the various cosmological parameters (Bahcall et al., 1999; Haiman et al., 2001; Huterer & Turner, 2001; Levine et al., 2002; Battye & Weller, 2003; Melchiorri et al., 2003; Wang et al., 2004).

Several surveys have been proposed with the explicit aim of significantly increasing the number of known distant clusters of galaxies. These proposals rely on a variety of detection methods across a wide range of wavelengths: the Sunyaev–Zel’dovich (SZ) effect in the millimeter (see Carlstrom, Holder & Reese 2002 for a review, and Juin et al. 2005 for a list of proposed surveys); galaxy overdensities in the visible/infrared (e.g. Gladders & Yee 2005; Hsieh et al. 2005; Rozo et al. 007b); bremsstrahlung emission by the intracluster medium (ICM) in the X-rays (e.g. Jahoda & the DUET collaboration, 2003; Haiman et al., 2005; Pierre et al., 2008). Galaxy cluster identification using weak lensing techniques is another possibility (e.g. Wittman et al., 2006), but is still in its infancy. Many of these proposals, in particular those regarding the detection of distant clusters through their X-ray emission, imply the building of new observing facilities such as eROSITA (Predehl et al., 2006), that will likely take many years to yield results. The cluster X-ray temperature is one of the best proxy observables in lieu of mass; it is a better estimator of the cluster mass than the cluster X-ray luminosity (e.g. Balogh et al., 2006; Zhang et al., 2006) (but more difficult to determine), and galaxy clusters are also most unambiguously identified in X-ray images. This makes X-ray-based galaxy cluster surveys those with the most accurately determined selection function. For all these reasons, we have undertaken to construct a galaxy cluster catalogue, called XCS: XMM Cluster Survey, based on the serendipitous identification of galaxy clusters in public XMM–Newton (XMM) data (Romer et al., 2001).

The aim of this paper is to forecast the expected galaxy cluster samples from the XCS and, based on those, its ability to constrain cosmology and cluster scaling relations using only self-calibration. Specifically, we consider the expected constraints on , and the luminosity–temperature relation for a flat Universe. Our results represent the statistical power expected to be present in the full XMM archive. This work builds upon previous efforts in several ways, and to a large extent constitutes the first coherent treatment of effects and methods previously only considered separately. Specifically, we combine all the following characteristics:

  1. we use a Monte Carlo Markov Chain (MCMC) approach and can thus characterize all degeneracies exactly (in contrast to Fisher matrix analyses),

  2. we include scatter in scaling relations in the parameter estimation (enabled by MCMC),

  3. we include a detailed, simulated selection function (essentially that of the XMM archive), not a simple hard flux/photon-count/mass limit,

  4. we include realistic photometric redshift errors, including degradation and catastrophic errors,

  5. we include temperature measurement errors, partly based on detailed simulations of XMM observations, and propagate the redshift errors to the temperature, and,

  6. we investigate quantitatively the effect on cosmological constraints from systematic errors in cluster scaling relation and measurement error characterization.

Our work builds on the galaxy cluster survey exploitation methods developed and studied primarily in Haiman et al. (2001); Holder et al. (2001); Levine et al. (2002); Hu & Kravtsov (2003); Hu (2003); Battye & Weller (2003); Majumdar & Mohr (2003, 2004); Lima & Hu (2004); Wang et al. (2004); Lima & Hu (2005). Forecasted cosmological constraints from XMM data have also been considered for the XMM–LSS survey in Refregier et al. (2002), but they did not take into account scaling-relation scatter or measurement errors, and used the Press–Schechter mass function. The most relevant precursors to this paper are Haiman et al. (2001) and Majumdar & Mohr (2004), who consider cosmological constraints expected from the Dark Universe Exploration Telescope (DUET) (Jahoda & the DUET collaboration, 2003) – a deg X-ray survey with flux limit erg s cm in the –- keV band. We extend the methodology of both papers through each of the six points above, either by more detailed modeling or by obtaining more robust results through the use of MCMC. Other relevant works are Huterer et al. (2004, 2006) and Lima & Hu (2007), who discuss photometric redshifts. We particularly complement these analyses through our detailed treatment/inclusion of measurement errors and selection effects. The recent work by Rapetti et al. (2008) takes an approach similar to ours in that they employ MCMC, include scaling-relation scatter and consider measurement errors, but focuses on combining future X-ray gas mass fraction measurements with SZ cluster and CMB power spectrum data.

The structure of this paper is as follows. We begin by reviewing the progress to date of the XCS and present the survey selection function (Sect. 2). Next, we present the models and methodology we use to derive constraints on cosmological parameters from the simulated XCS sample (Sects. 3 & 4). We then go on to the expected cluster distributions and, our estimates for the constraints on , , and cluster scaling relation parameters that we expect to obtain from the XCS using self-calibration, including the effect of temperature measurement errors and relying on photometric methods to obtain XCS galaxy cluster redshifts (Sect. 5). We discuss and summarize our findings in Sect. 6. Additional material setting out modeling details is provided in the Appendix.

2 The XMM Cluster Survey

2.1 Background and current status

XMM–Newton is the most sensitive X-ray spectral imaging telescope deployed to date. It is typically used in pointing mode, whereby it observes a single central target for a long period of time (the typical exposure time being kilo-seconds). The field of view of the XMM cameras is roughly half a degree across, so that a considerable area around the central target is observed ‘for free’ during these long pointings. Already many thousands of these pointings are available in the public XMM archive. The XCS is exploiting this archive by carrying out a systematic search for serendipitous detections of clusters of galaxies in the outskirts of XMM pointings (Romer et al., 2001). Once a cluster candidate has been selected from the archival imaging data, it is then followed up using optical imaging and/or optical spectroscopy, to confirm the indentification of the X-ray source and to measure redshifts (see Sect. 3.4). For those XCS clusters that were detected with sufficient counts, an X-ray spectroscopy analysis is carried out, again using the archival data, in order to measure the temperature of the hot intracluster medium (ICM). These temperatures can then be used to study cluster scaling relations and/or to estimate the mass of the cluster (see Sects. 3.2 & 3.3).

The XCS project is ongoing, but already more than XMM pointings have been analysed, yielding a cluster candidate catalogue numbering almost entries. So far, the XCS covers a combined area of deg suitable for cluster searching and for which optical follow-up has been completed; i.e. this area excludes overlapping and repeat exposures, regions of low Galactic latitude, the Magellanic clouds, and pointings with very extended central targets. Around clusters with photons and keV are present in this initial area. With many thousand more XMM pointings waiting to be analysed by the XCS, and a mission lifetime extending to , a conservative estimate for the final XCS area for cluster searching is deg. We use deg herein for XCS cosmology forecasting (see Table 1), assume a redshift range of , and temperatures of . We further limit our representative survey to clusters with photon counts (XCS hereafter), so that we can be sure to estimate X-ray temperatures with reasonable accuracy (see Sect. 3.5). The lower redshift limit is associated with cluster extents becoming too large, and the cosmic volume also becoming small. The maximum redshift is chosen so that the luminosity–temperature relation can still be reliably modelled/estimated (see Sect. 3.3.3). The temperature range is chosen such that we can expect i) a small contamination from galaxy groups (which typically have temperatures ), yet include as many of the numerous low-temperature clusters as possible, and ii) that clusters above the high-temperature limit are sufficiently rare that none can be expected. The final cluster catalogue (without the cut-offs defined above for XCS) will contain several thousand clusters out to a redshift of . The highest-redshift cluster discovered by the XCS so far is XMMXCS J2215.9-1738 at (Stanford et al., 2006; Hilton et al., 2007).

In addition to producing one of the largest samples of X-ray clusters ever compiled, the XCS will also be a valuable resource for cosmology studies (see Sect. 4). This is because the catalogue will be accompanied by a complete description of the selection function. In this work we make use of an initial XCS selection function that assumes simple models for the distribution of the ICM, and flat cosmologies (see below). Future cosmology analyses will take advantage of more sophisticated selection functions that are based on hydrodynamical simulations of clusters (Kay et al., 2007).

Survey XCS
Sky coverage 500 deg (serendipitous)
Redshift coverage 0.1 – 1.0
X-ray temperature coverage 2 – 8 keV
Min. photon count 500
X-ray flux limit By selection functiona
  • The flux limit is in the keV band, if defined as a probability of detection greater than or equal to per cent. See also Sect. 5.2.1 and Fig. 9.

Table 1: Survey specifications.
(a) Constant relation
(b) Self-similar relation
Figure 1: Selection function for our fiducial cosmology and different evolution. Values in the shaded region are extrapolated from those in the coloured region (), for which the selection function has been calculated explicitly.

2.2 The XCS selection function

Model

In order to properly model the selection function of a survey like the XCS, it is important to account for all of the observational variations present in real data. We can achieve this by placing a sample of fake surface-brightness profiles into real XMM Observation Data Files (ODFs). This ensures that our simulated images re-create real-life issues such as clusters lying on chip gaps and point-source contamination. The fake surface-brightness profiles are created as follows. We use an isothermal model with , core radius (close to the mean values of , obtained from a uniform ROSAT analysis of clusters from ; Ota & Mitsuda, 2004), and plasma metallicity . For a given cosmology we simulate sets of cluster parameters:

  • redshifts (linearly spaced )

  • luminosities (log. spaced erg s)

  • temperatures (linearly spaced keV)

For selection function determination, we drew on a list of ODFs that have already been processed by the XCS and have been deemed to be suitable for cluster searching (see above). Before each selection function run, a smaller list of ODFs is selected at random from the full set of . These ODFs are then copied from the main XCS archive to local processing nodes for temporary storage, to speed up the analysis. Tests have shown that with ODFs it is still possible to reproduce the variance in exposure time, target type, point source density, etc., inherent to the XCS. In the following we define a ‘selection function run’ as the analysis over the sets of cluster parameters and ODFs – a total of combinations.

For each of the different combinations of cluster parameters, the process proceeds as follows. First, to account for the fact that the XCS searches the entire field of view for serendipitous cluster detections, the centre of the fake surface-brightness profile is randomly positioned into a blank XMM–style ODF, with a uniform probability across the field of view. The profile is then convolved with the appropriate PSF model. For this purpose we use the two-dimensional medium-accuracy model1. At this stage, an ODF is chosen at random from the list of stored locally, into which the fake source will later be added. The profile is then assigned an absorbed count rate using a series of arrays calculated using Xspec (Arnaud, 1996). The arrays tabulate conversions from unabsorbed bolometric luminosity to absorbed count-rate as a function of temperature, redshift, hydrogen column density, and XMM camera/filter combination. The fake count-rate image is then multiplied by the exposure map of the chosen ODF to account for vignetting, masking and chip gaps. Finally, the fake cluster image is added to the original ODF at the chosen position, and the ODF is run through our source detection/classification pipeline to determine if the fake cluster passes our automated cluster-candidate selection process. For more details on the detection/classification pipeline, refer to Davidson et al. (in preparation). The process is repeated a total of one hundred times, so that we can build up an average XCS detectability for that parameter combination. Once the full set of combinations has been tested times each, the run is complete. We then change the cosmology inputs and start the entire sequence again. The process is very CPU intensive; each selection function run (of combinations) takes several weeks to run on a single node. For the forecasting work presented herein, we carried out seven selection function runs over the flat CDM cosmologies with . We limit ourselves to flat cosmologies as we use a flatness prior in the forecasting of cosmological constraints.

The resulting selection function is shown in Fig. 1 for the two luminosity–temperature relations (see Sect. 3.3.1) we consider. Note that the selection function in regions where we have not calculated it explicitly is extrapolated from the region where we have done so. Hence, its features in those extrapolated regions should only be considered a rough indication of its behaviour, particularly in the high-redshift, high-temperature region. This region is only relevant for including measurement errors, and since such high-temperature clusters are exceedingly rare, the uncertainty in this part of the selection function has no significant impact on our results2.

Uncertainty

The shape of the selection function is dependent on the cluster model employed, as described above. It is well known that clusters of galaxies have a range of morphologies, with core radii varying from many tens of kpc to a few hundred kpc and values varying generally between and (e.g. Reiprich & Böhringer, 2002; Ota & Mitsuda, 2004; Maughan et al., 2008).

To include the variation of cluster-model parameters in our analysis in a realistic manner, one would require i) a model for the distribution of such parameters among the cluster population (including correlations among parameters), and ii) a characterization of the selection function dependence on such parameters. Lacking either or both of these will produce some level of uncertainty in cluster number predictions and cosmological parameter constraints. However, assessing the level of such uncertainty of course requires a fiducial model (realizing i) and ii)) to compare with. As we do not currently have a realistic model for the model-parameter distribution among the cluster population, it is somewhat premature to carry out such an analysis. In actual data analysis, we intend to model this in detail. What we can currently do is to compare our standard selection function to one assuming that all clusters have the most extreme values of cluster-model parameters, leading to a gross overestimation of the overall uncertainty in cluster number predictions. We have carried this calculation out for clusters with temperatures typical for the underlying distribution at different redshifts, and describe it below. We again stress that its usefulness for estimating the actual uncertainty in the selection function is limited, as it does not take into account the actual distribution of clusters and their model parameters. Ultimately, we expect that the cosmological constraints we obtain would change little, even if a more realistic model/selection function was used. This is because the changes in the selection function would have a similar impact for the fiducial cosmological model, and for models in its neighbourhood.

We have tested the sensitivity of the selection function for clusters with photons to variations in the cluster core radius , between the values of  kpc and  kpc (recall the fiducial value used in this work is  kpc, see Sect. 2.2.1). For this we use mock clusters with typical temperatures of (and hence luminosities) for a given redshift (as predicted by our models, see Sects. 3 & 4). In the following, we refer to the relative difference in the selection function detectability, as this is most relevant to the relative difference in numbers of clusters. Our results show that, for most of the redshift range tested (), clusters with a core radius of kpc are easier to detect (as extended XMM sources), than those with smaller or larger core radii. However, the dependence is shallow; the relative uncertainty in the detectability is less than per cent up to a redshift of , across the entire range. At higher redshifts, the relative uncertainty approaches per cent. However, this subset of clusters constitutes only per cent of the total population. For higher-temperature clusters, the relative uncertainty drops back to around per cent at for .

In summary, our model for the cluster population is a simplification based on mean observational values of cluster-model parameters, and as a result will have somewhat differing detection properties compared to a real sample. To characterize such uncertainty requires modelling of the cluster-model parameter distribution, and the selection function dependence on those parameters. However, once such information becomes available, it will be included in the analysis and hence remove/reduce such uncertainty. As we do not currently have a realistic model of the cluster parameter distribution, we have determined the impact of assuming extreme structural cluster parameters, and found that at most the typical selection function uncertainty is of the order of per cent. These results agree with those of Burenin et al. (2007) for the 400d survey, which show that reasonable variations in cluster size, morphology and scaling relations induce an uncertainty in the detectability for a given flux of typically less than per cent.

3 From X-ray observables to mass

3.1 Modeling summary

Making predictions for X-ray cluster observations requires the modeling of scaling relations to relate temperature to mass, and temperature to luminosity. In addition, the observables will have uncertainties associated with them, which need to be taken into account. The following subsections detail our modeling assumptions, but we summarize them here for reference and orientation.

We first assume that we know a priori exactly how the cluster X-ray temperature relates to luminosity at the present time, and how this relation evolves with redshift. We then study how the constraints on cosmological parameters degrade if such an assumption is dropped. We consider four extra free parameters: two parameters to characterize the present-day, power-law, relation between cluster X-ray temperature and luminosity, another to describe its redshift evolution as a power of (), and lastly one for the logarithmic dispersion in the (assumed) Gaussian distribution of the intrinsic (redshift-independent) scatter in the relation between cluster X-ray temperature and luminosity.

In addition, we evaluate the full impact on the XCS’s ability to impose constraints on cosmological parameters that arises from assuming a dispersion in the Gaussian photometric redshift distribution of either or per cent about the true redshift, both with and without the presence of unaccounted-for catastrophic errors in the photometric redshift estimation procedure. Further, we will also determine the impact of a systematic mis-estimation of the assumed true dispersion in the photometric redshifts about the true redshift. Our aim is to test the impact of realistic assumptions regarding the distribution of photometric redshifts around the true redshift, and then determine by how much such impact increases by considering a worst-case scenario.

Similarly, we consider the impact of realistic X-ray temperature errors obtained from simulations based on the relevant XMM fields, as well as significantly larger errors corresponding to a worst-case scenario. Lastly, we consider the impact of incorrect assumptions about the cluster scaling relations on cosmological constraints.

Summary tables with our main cluster scaling relation and measurement error assumptions are given in Sect. 5. Detailed information on the mathematical treatment is given in the Appendix.

3.2 The X-ray temperature to mass relation

We need to assume a relation between cluster X-ray temperature and mass to be able to predict cluster distributions. The reason is that presently the effect of cosmological parameters on the galaxy cluster population can only be accurately predicted as a function of cluster mass (e.g. Reiprich & Böhringer, 2002). The X-ray temperature is one of the best proxy observables, as explained in the Introduction.

Evolution

We assume the self-similar prediction (e.g. Kaiser, 1986; Bryan & Norman, 1998; Voit, 2005a),

(1)

for the redshift dependence of the relation between cluster X-ray temperature and virial mass to hold for any combination of cosmological parameters, given that it is consistent with the most recent analyses of observational data (Ettori et al. 2004a, b; Arnaud, Pointecouteau & Pratt 2005; Kotov & Vikhlinin 2005, 2006; Vikhlinin et al. 2006; Zhang et al. 2006). Here is the cluster virial mass, while is the mean overdensity within the cluster virial radius with respect to the critical density. If the only relevant energy densities in the Universe are those associated with non-relativistic matter and a cosmological constant, then

(2)

with . (Note that we will restrict ourselves to a flat Universe, , in our analysis – see Sect. 4.2). Deviations from a self-similar mass–temperature relation will be considered in Sect. 5.5, as explained in the following Section.

Normalization

The constant of proportionality is set by demanding that for our fiducial cosmological model (with , see Sect. 4.2)

(3)

at for an X-ray temperature of 5 keV, where is the mass within a sphere centered on the cluster within which its mean density falls to 500 times the critical density at the cluster redshift. In this way, our fiducial cosmological model reproduces the local abundance of galaxy clusters as given by the HIFLUGCS catalogue (Reiprich & Böhringer 2002; Pierpaoli, Scott & White 2001; Viana et al. 2003). Note that such a normalization of the cluster X-ray temperature to mass relation happens to be very close to that directly derived from X-ray data by Arnaud et al. (2005) and Vikhlinin et al. (2006).

The conversion between and the halo mass, , will be carried out by using the formulae derived by Hu & Kravtsov (2003) under the assumption that the halo density profile is of the NFW type (Navarro, Frenk & White, 1995, 1996, 1997), and we will take the concentration parameter to be 5. This has been shown to provide a good description of the typical density profile in galaxy clusters (see Arnaud 2005 or Voit 2005b and references therein; Vikhlinin et al. 2006).

The normalization of the relation is subject to a number of uncertainties, the most important of which are the possible violation of hydrostatic equilibrium (Rasia, Tormen & Moscardini 2004; Nagai, Kravtsov & Vikhlinin 2007) and the possible difference between the spectroscopic X-ray temperature and the temperature of the electron gas (Mazzotta et al., 2004; Rasia et al., 2005; Vikhlinin, 2006). The precise level of these effects remains to be firmly established, but could be of the order in the normalization mass (e.g. Vikhlinin, 2006; Nagai et al., 2007). The scatter, as well as slope, could also be under-estimated due to these effects (Vikhlinin, 2006; Nagai et al., 2007). We make some estimates of all these systematic effects on cosmological constraints in Sect. 5.5.

Scatter

We assume that the intrinsic scatter in the relation between cluster X-ray temperature and mass has a Gaussian distribution (truncated at and re-normalized) with a redshift-independent dispersion of 0.10 about the logarithm of the temperature. This is motivated by both cluster X-ray data analysis (e.g. Arnaud et al., 2005; Vikhlinin et al., 2006; Zhang et al., 2006) and results from -body hydrodynamic simulations (e.g. Viana et al. 2003; Borgani et al. 2004; Balogh et al. 2006; Kravtsov, Vikhlinin & Nagai 2006). As explained in the preceding Section, we consider systematic deviations in the scatter in Sect. 5.5.

3.3 The X-ray luminosity to temperature relation

In order to understand how the XCS selection function depends on cluster mass, we need to know how cluster X-ray luminosity and temperature relate to cluster mass (see Sect. 2.2). In practice, we will use the relation between luminosity and temperature instead of that between luminosity and mass, in effect relating these two quantities via the temperature. This makes sense because the estimation of cluster mass from X-ray data is always based on the X-ray temperature, via the assumption of hydrostatic equilibrium, and not on the luminosity. Thus, while we always need, at least implicitly, to know how the cluster luminosity relates to temperature to derive the relation between the luminosity and mass from X-ray data, the reverse is not true.

As for the mass–temperature relation, assuming self-similarity leads to a specific prediction (Kaiser, 1986),

(4)

under which clusters with the same X-ray temperature are predicted to be more X-ray luminous if they have a higher redshift. We have chosen here to normalize the relation with respect to the local () relation. Based on this expression, we write the relation in the general form

(5)

and discuss below the assumptions made for the different parameters.

Evolution

We consider two possible fiducial scenarios, which bracket most observational results and theoretical expectations: either

  • no evolution ()     or

  • self-similar evolution ()

for the fiducial combination of cosmological parameters. The parameters and are defined above in equation (5). Presently, there is some uncertainty surrounding the redshift evolution of the relation between cluster X-ray luminosity and temperature. Essentially, what we know is how that relation behaves for redshifts below 0.3 (e.g. Ikebe et al. 2002; Novicki, Sornig & Henry 2002; Ota et al. 2006; Zhang et al. 2006). For higher redshifts, the data is still sparse, and the evidence contradictory, from claims that the relation between cluster X-ray luminosity and temperature barely evolves at all with redshift (Holden et al., 2002; Ettori et al., 2004a, b; Ota et al., 2006; Branchesi et al., 2007), to claims that its evolution is close to the self-similar prediction (Novicki et al., 2002; Vikhlinin et al., 2002; Lumb et al., 2004; Kotov & Vikhlinin, 2005; Maughan et al., 2006; Zhang et al., 2006; Hicks et al., 2008). Some authors argue that self-similarity remains viable at all redshifts, and that at least some of the observed discrepancies could be due to selection effects, as the Malmquist bias from scaling-relation scatter (also discussed below) could distort the deduced evolution if the sample selection is not sufficiently understood (e.g. Branchesi et al., 2007; Maughan, 2007; Pacaud et al., 2007; Nord et al., 2008). On the other hand, Hilton et al. (2007) argue for deviation from the self-similar prediction based on a set of high-redshift clusters combined with the recently discovered XCS cluster XMMXCS J2215.9-1738 at .

When the XCS catalogue becomes available, the relation between cluster X-ray luminosity and temperature, as a function of redshift, will be estimated jointly with the cosmological parameters, but for now we will have to rely on the limited information available.

Normalization & slope

We assume the local () relation between the cluster X-ray luminosity in the ROSAT keV band and temperature to be

(6)

as was derived in Viana et al. (2003) for a combination of cosmological parameters similar to those assumed for our fiducial cosmological model. The X-ray data used in Viana et al. (2003) was that of galaxy clusters present in the HIFLUGCS catalogue (Reiprich & Böhringer, 2002), and therefore the conversion between and X-ray bolometric luminosity is performed through a fit (derived by us) based on the values both quantities take for the galaxy clusters in HIFLUGCS,

(7)

As in Ikebe et al. (2002), the relation between the cluster X-ray luminosity and temperature derived in Viana et al. (2003) takes into account the fact that any flux-limited sample of galaxy clusters will be composed of objects which are on average more X-ray luminous than the mean luminosity of all existing galaxy clusters with the same redshift and X-ray temperature. This Malmquist type of bias increases with decreasing temperature, and thus ignoring it leads not only to an overestimation of the normalization of the relation between luminosity and temperature, but also to an underestimation of its slope.

Scatter

We assume that the intrinsic scatter in the relation between cluster X-ray luminosity (in the 0.1 to 2.4 keV band) and temperature has a redshift-independent Gaussian distribution (truncated at and re-normalized) about the logarithm of the X-ray luminosity, with dispersion (Ikebe et al., 2002; Viana et al., 2003). This is also close to what was found by Kay et al. (2007) in the CLEF simulation. Although Kay et al. also observe an evolution of the scatter with redshift, there is no strong observational evidence for or against such an evolution at present, and therefore we do not include it in our analysis.

The existence of intrinsic scatter in the relation between cluster luminosity and mass will effectively increase the observed number of galaxy clusters above any X-ray luminosity (or flux) threshold, relative to the case without scatter. This results from the steepness of the cluster mass function, due to which significantly more clusters have their X-ray luminosity scattered up than down across any given luminosity threshold. Therefore, intrinsic scatter between X-ray luminosity and mass can have a considerable impact on the predicted number of XCS clusters and on the estimation of the constraints the XCS will impose on cosmological parameters. This scatter can be considered as the combination of the scatter in the luminosity to temperature and temperature to mass relations, with clear observational evidence that the former dominates over the latter (Stanek et al., 2006; Zhang et al., 2006).

As higher redshifts are considered, it is expected that an increasing number of galaxy clusters will have undergone recent major mergers, not only leading to increased scatter in the cluster scaling relations but also making its distribution highly non-Gaussian, with long tails developing towards both high X-ray luminosity and, to a lesser degree, temperature, at fixed mass (Randall, Sarazin & Ricker, 2002). This has the potential to substantially affect the estimation of the constraints the XCS will be able to impose on cosmological parameters. There is a lack of high-redshift observational data in this regard and we are also not confident that we will detect, for the purposes of understanding this behaviour, many useful clusters at . We therefore chose to consider in the estimation procedure only those clusters in the mock XCS catalogues which have a redshift .

3.4 Photometric redshifts

The role of photometric redshifts

Redshifts are required for XCS clusters, both to place them correctly in the evolutionary sequence and to allow the measurement of X-ray temperatures from XMM spectra. With regard to the latter point, pure thermal bremsstrahlung spectra are essentially featureless (barring a high-energy cut-off), making them degenerate in temperature and redshift. Therefore, in the absence of independent redshift information, all one can measure from a typical XCS cluster spectrum would be a so-called apparent X-ray temperature, i.e. one scaled by , see Appendix A.1.2. As shown by Liddle et al. (2001), these apparent temperatures are not sufficient to allow one to measure cosmological parameters from cluster catalogues. As a result, optically-determined redshifts will be required for almost all clusters in the XCS catalogue (the exception being a tiny number that are detected with sufficient signal to noise to allow X-ray emission features, such as the Iron K complex at keV, to be resolved over the thermal continuum).

As is now typical for cluster surveys (e.g. Gladders & Yee, 2005), the XCS is relying heavily on the photometric redshift technique for its optical follow-up. This is because photometric redshifts are much more efficient, in terms of telescope time requirements, than spectroscopic redshifts. However, they have the disadvantage, over spectroscopic redshifts, that the redshift errors are larger and sometimes poorly understood. The XCS is using both public-domain photometry (e.g. from SDSS and 2MASS) and proprietary data from the NOAO–XCS Survey (NXS, Miller et al., 2006) to both optically confirm (as clusters) XCS candidates and to measure photometric redshifts. To date, more than XCS candidates have been optically confirmed in this way.

Errors on photometric redshifts must be accounted for when determining cosmological parameters from cluster surveys, and so we have included prescriptions for such errors in the forecasting work presented herein. Our prescriptions include both purely statistical errors and so-called catastrophic systematic errors. As shown by previous work (Huterer et al., 2004, 2006; Lima & Hu, 2007), purely statistical errors have a negligible impact on cosmological parameter constraints. By contrast, if catastrophic errors are not accounted for properly in the fitting, they could have a significant impact on cosmological parameter constraints. We note that previous work has concentrated only on the impact of redshift errors on the evolutionary sequence, whereas we have also included the impact of photometric errors on X-ray temperature determinations.

Figure 2: Realistic redshift error distributions at various redshifts. The upper right panel shows a magnification of the bottom-right distribution, highlighting the catastrophic-error part of the distribution.

Distribution

Following Huterer et al. (2004) we assume that the statistical error in the photometric redshifts of individual galaxy clusters has a Gaussian distribution about the true redshift, . In an attempt to reproduce the expected degradation with redshift of the absolute accuracy of cluster photometric redshift estimation methods, and in contrast to Huterer et al. (2004) (but in the same way as Lima & Hu 2007), we will assume the dispersion to be proportional to , normalized at to either or (Gladders & Yee, 2000; Gladders, 2004; Gladders & Yee, 2005; Gladders et al., 2007). Unaccounted-for systematic errors in the photometric redshift estimation procedure are much harder to model, because they can take a variety of guises. We will consider here one such type of error: catastrophic errors in the photometric redshift estimation procedure. The existence of unaccounted-for catastrophic errors will be modelled by assigning a random photometric redshift error to either a fraction or of the galaxy clusters, drawn from a Gaussian distribution that has four times the dispersion of the standard distribution, with the requirement that the photometric redshift error has to be at least away from the true redshift. The functional form of the redshift error distribution is given in Appendix A.1.

We label the case ‘realistic’ and the case ‘worst-case’ redshift errors. Examples of realistic redshift error distributions are shown in Fig. 2.

3.5 X-ray temperature

Estimating the measurement errors

Initial estimates (Liddle et al., 2001) showed that X-ray temperatures measured for XCS clusters are expected to have an associated measurement uncertainty of less than per cent at 1. However, these estimates were based on a photon count of and assume a single hydrogen column density over the XMM fields, and are therefore not directly applicable to our XCS sample. Hence, in order to estimate the temperature errors that will be present in the XCS statistical sample more accurately, we have conducted Monte Carlo simulations using the Xspec spectral fitting package (Arnaud, 1996). We created sets of fake spectra for the XMM–Newton EPIC PN and MOS instruments, from a mekal plasma model (Mewe, Lemen & van den Oord, 1986) multiplied by a wabs photo-electric absorption model (Morrison & McCammon, 1983). Responses for a mean off-axis angle were used and a mean background was added. The model was then fitted to each of the spectra to derive a temperature. A plasma metallicity of was used throughout, in accordance with the assumptions in our selection function calculations (see Sect. 2.2), and we assume a photon count of . This procedure was repeated for a range of input temperatures, redshifts and absorption columns. We then marginalize over the hydrogen absorption columns using the expected hydrogen column distribution for our XMM fields.

Figure 3: Mean fractional temperature errors from the simulations performed, for photons, and as marginalized over expected absorption columns for the XCS.
Figure 4: Realistic temperature error distributions at various redshifts and temperatures, based on our XMM–Newton simulations, for a photon count of .

The mean fractional temperature errors from our simulations are shown in Fig. 3. The largest influence on the temperature errors comes from the input temperature itself. Since metal lines in the spectrum provide much better constraints on the temperature than the shape of the bremsstrahlung continuum, and the fraction of line emission in the spectrum declines with increasing temperature, the errors are larger for hotter clusters. The effect of redshift on the errors is much smaller and itself temperature-dependent. For low-temperature systems at high redshifts, part of the X-ray spectrum is shifted out of the bottom of the XMM passband, increasing the errors. For high-temperature systems, the effect of increasing redshift is to shift the source spectrum to lower energies for which the XMM effective area is larger, thus decreasing the errors.

Distribution

The distribution of temperatures obtained in our simulations was fitted by an asymmetric Gaussian function to parametrize the temperature error distribution, with the fractional error given by a two-dimensional quadratic expression in temperature and redshift. We marginalize over the distribution of absorption columns found in XCS fields to obtain mean parameters for our asymmetric Gaussian error distribution. The exact functional form of the fitted error distribution is given in equation (23) in Appendix A.1.

We will label the case with according to our simulation results as ‘realistic’ and the case with three times this dispersion as ‘worst-case’ temperature errors. Examples of realistic temperature error distributions are shown in Fig. 4. Note that, as we are assuming that all detected clusters have a photon count of exactly , our error distributions represent a worst-case scenario in this regard.

4 From mass to cosmology and constraints

4.1 The mass function

Having connected our direct X-ray observables to cluster mass using the methodology in the preceding Section, we can then combine these relations with the mass function (below) to find the cluster distribution as a function of temperature and redshift. The differential number density of haloes in a mass interval about at redshift can be written as

(8)

where is the dispersion of the density field at some comoving scale and redshift , and the matter density.

Parametrization

It has been shown by Jenkins et al. (2001) that a good fit (accurate to better than 20 per cent) to the mass functions recovered from various large -body simulations, in the regime , is given by

(9)

where the halo mass is defined at a mean overdensity of 180 with respect to the background matter density, independently of the cosmological parameters assumed, or equivalently to a mean overdensity of with respect to the critical density. This result has been confirmed by Evrard et al. (2002); Hu & Kravtsov (2003); Klypin et al. (2003); Linder & Jenkins (2003); Reed et al. (2003); Łokas, Bode & Hoffman (2004); Warren et al. (2006); Reed et al. (2007), and we will therefore use this fit to estimate the expected comoving number density of haloes for any given combination of cosmological parameters. This also makes the like-for-like comparison with other cluster constraints straightforward, as most rely on the Jenkins mass function. The dispersion is calculated using a fit in analogue to Viana & Liddle (1999), which is accurate to within two percent for the range of halo masses relevant for this work, compared to the exact expression employing the BBKS transfer function (Bardeen et al., 1986). (In a real data analysis, this prescription would not be sufficiently accurate, however for forecasting purposes it is acceptable.)

4.2 Cosmology

We have already seen that cosmology enters into the prediction of cluster numbers as a function of temperature and redshift through the selection function, the cluster scaling relations, and the mass dispersion. Additionally, the cosmic volume will also enter as we need to multiply the differential distribution by this quantity (discussed in the following Section).

Figure 5: Flowchart for cluster predictions and forecast parameter estimation. The dash-enclosed area indicates the processes that enter in our calculations. Bi-directional dashed arrows are used to indicate the main circular relations, where information from one part is used to inform another, which then informs the first, and so on.

Parameters

We work within the Cold Dark Matter (CDM) paradigm, with a spectrum of primordial adiabatic Gaussian density perturbations. We assume that , , and . As we do not expect the XCS to have particularly competitive constraining power on , we restrict our analysis to the case of a flat universe, , in accordance with observations of e.g. the cosmic microwave background (Spergel et al., 2007). We take the present-day shape of the matter power spectrum to be well approximated by a CDM model with scale-invariant primordial density perturbations whose transfer function shape parameter is . This is the mean value obtained from different analyses of SDSS data (Szalay et al., 2003; Pope et al., 2004; Tegmark et al., 2004b; Eisenstein et al., 2005; Blake et al., 2007; Padmanabhan et al., 2007) and also perfectly compatible with the 3-year WMAP data (Spergel et al., 2007). We have checked that assuming is either 0.16 or 0.20 does not change our results. (In a real data analysis, using the shape parameter is too simplistic, but for forecasting purposes it is sufficient.) A summary of our cosmological parameter assumptions is given in Table 2.

Parameter Value Prior
Table 2: Cosmology assumptions used. Fiducial values are given first, followed by priors assumed in parameter estimation.

4.3 Combining observables and cosmology

As we have seen, our cluster distribution calculations involve many different steps and components. Importantly, they rely on both simulation and observational data, as well as direct theoretical input. A schematic overview of the relevant inputs, processes and outputs is shown in a flowchart form in Fig. 5. Collecting all components, the number of clusters in around is given by

(10)

where combines survey area and selection function. This expression ignores scatter in the scaling relations and measurement errors. A complete treatment is given in Appendix A.1. The remaining component for arriving at parameter constraints is the likelihood, which is described next.

Parameter Description
Colouring Pink Green Orange Blue
:
Normalization
flat, unrestricted
flat, unrestricted
Slope
flat, unrestricted
flat, unrestricted
Self-similarity exp.
Redshift exp.
Scatter
Max. scatter in units of
:
evolution self-similar, normalized to HIFLUGCS
Scatter
Max. scatter in units of
Table 3: Cluster scaling relation assumptions and their labelling. Fiducial values are given first, followed by priors assumed in parameter estimation below (usually in brackets). Note that the colour coding at the top of the table is used to indicate these fiducial models throughout. See Sects. 3.2 & 3.3 and the Appendix for details.

4.4 Likelihood

Turning our attention to using the cluster distribution prediction for cosmological constraints and forecasting, we need an expression for the likelihood of an observed catalogue of galaxy clusters. The likelihood for a given observed catalogue is simply the product of the Poisson probabilities of observing XCS clusters in the bin with widths centered at each of the positions,

(11)

where

(12)

is the expected number of XCS clusters in bin , taking into account sky coverage, survey selection function, and any uncertainties in scaling relations or measurements (see equations in Appendix A.1). We do not take into account the fact that the positions of galaxy clusters are spatially correlated, because the mean distance between XCS clusters is typically much larger than the observationally determined correlation length in the range (see e.g. Nichol et al. 1992; Romer et al. 1994; Collins et al. 2000; Gonzalez, Zaritsky & Wechsler 2002; Brodwin et al. 2007), as a result of the XMM pointings being scattered all over the sky. Even if the XCS area was contiguous, given the very large depth of the XCS, the impact of cluster spatial correlations on the estimation of cosmological parameters with the XCS galaxy cluster abundance data would be small (White, 2002; Hu & Kravtsov, 2003; Holder, 2006; Hu & Cohn, 2006).

Figure 6: Parameter constraints ( per cent confidence level) for a set of 10 random realizations of the catalogue Poisson distribution (dashed coloured lines) compared to the average-catalogue parameter constraint (solid black line). In the right-hand panel, each contour has been re-centered around its distribution mean. A constant relation and no or scatter was assumed.
Quantity Labels/assumptions
Redshift
X-ray temperature
Table 4: Summary of measurement error assumptions and their labelling. See Sects. 3.4 & 3.5 and the Appendix for details.

As we are seeking to obtain expected/typical constraints, in a sense a Maximum Likelihood (ML) point estimate, we use , where the asterisk denotes fiducial-model values. Using this ‘average-catalogue’ construction, we obtain an excellent estimate of the size and shape of the expected likelihood contours, but avoid the offset in the best fit away from the fiducial parameter values that is associated with, e.g., the most likely catalogue. Any random realization of a Poisson sample will exhibit such an offset. Examples can be seen in the right-hand panel of Fig. 6, where the results for the average-catalogue method is compared to those for random catalogue realizations. We wish to avoid offsets of this type as we are mainly interested in the shape and size of contours, or wish to separate possible biases from such an offset. This methodology is explained and motivated in detail in Appendix A.2. As stated above, Fig. 6 compares constraints derived using this method to constraints derived from a Poisson sample of mock catalogues. The results confirm that constraints derived using our methodology provide an excellent estimate of the expected constraints. Note that in future real data analyses this methodology cannot be used, and there will in general be some offset.

The exploration of the likelihood function in parameter space was carried out using a custom code based on standard Monte Carlo Markov Chain techniques (Gelman & Rubin 1992; Gilks, Richardson & Spiegelhalter 1996; Lewis & Bridle 2002; Verde et al. 2003; Tegmark et al. 2004a; Dunkley et al. 2005). The calculation of the integrals involved in the likelihood was done with the state-of-the-art numerical integration packages CUBPACK (Cools & Haegemans, 2003) and Cuba (Hahn, 2005).

(a) Underlying cluster distribution. Note that only the relation is relevant for the underlying distribution, and we therefore colour according to both assumptions with the same relation.
(b) Expected detections using selection function.
(c) Detected fraction per bin.
Figure 7: Expected cluster number count distributions for XCS, for no nor scatter and no evolution (pink), no nor scatter and self-similar evolution (green), and scatter and no evolution (orange), and and scatter and self-similar evolution (blue). Bin sizes are and .
(a) Underlying cluster distribution. Note that only the relation is relevant for the underlying distribution, and we therefore colour according to both assumptions with the same relation.
(b) Expected detections using selection function.
(c) Detected fraction of clusters per bin.
Figure 8: Expected cluster distributions for the XCS, for our four different cluster scaling relation cases.
Figure 9: Expected observational limits for the XCS (defined as ), for the simplest case of no scatter in the cluster scaling relations and a constant (pink dashed line) or self-similar (green dashed line) relation. Solid lines correspond to the hard temperature cut .
Figure 10: Changes in total number of clusters due to our different error assumptions, compared to no-errors distributions in Fig. b.

5 Results

5.1 Labelling

Generally, the colouring scheme in Table 3 will be used to indicate the fiducial cluster scaling relation model; see Sects. 3.2 and 3.3 as well as the Appendix. X-ray temperature and redshift errors will be indicated as ‘realistic’ or ‘worst-case’ according to Table 4; see Sects. 3.4 and 3.5 as well as the Appendix.

5.2 Expected cluster distributions

Without measurement errors

The expected D distributions of clusters for our four standard models are shown in Fig. a (underlying distributions), Fig. b (expected detections) and Fig. c (detection efficiency), where the selection function has been used to go from Fig. a to Fig. b. The expected redshift distributions and total cluster number counts are shown similarly in Fig. 8. Note that as the relation changes, so does the expected number of detected clusters, since we are more likely to detect a cluster the more luminous it is (and for a given temperature, the cluster luminosity increases with redshift for self-similar evolution). The underlying distribution however is of course not dependent on the relation. We find that XCS can be expected to find somewhere in the range of clusters for its projected area of deg and , . This corresponds to around per cent of the total number of clusters we would expect to detect with no photon count cut-off (effectively a -photon cut-off). This full set of XCS clusters will constitute a significant sample (relative to previous studies), representing around a quarter to a third of the actual clusters present in the observed fields. Going to higher redshifts, we roughly estimate that a minimum of clusters will be found at , of which at least should have photons.

The XCS DR1 currently covers an area of deg, for which clusters/groups with measured redshifts and photons have been identified from candidate extended sources (with photon counts). No temperature, redshift or other cuts have been applied to this set. In the current redshift sample of clusters with more than photons, approximately per cent have temperatures below keV and are therefore classified as groups. We therefore expect the final number of genuine  keV clusters we detect in this area to be in the range , depending on the fraction of ‘extended’ sources detected by the survey that turn out not to be clusters (blended point sources, etc.). We have here assumed that selection effects in the redshift follow-up do not significantly bias this number. For our fiducial cosmology, we find that the corresponding expected number of clusters is , the range corresponding to the lower and upper limits from our set of scaling-relation assumptions. These two ranges are clearly consistent. The lower predicted limit corresponds to no evolution, and hence – all else equal – the observational result might indicate a near-constant relation.

An overview of the expected observational limits of the XCS for mass, X-ray temperature, X-ray luminosity and X-ray flux (in the keV band), is given in Fig. 9. We have there defined the detection limit, through the selection function, as . These limits are thus the values above which we expect to detect, with a photon count of or above, at least half of the clusters. It is worth noting that the change in detection probability is slow as a function of X-ray temperature, and hence the concept of e.g. a single flux limit (which would correspond to a sharp transition between one and zero in the probability) is not suitable for defining the XCS sample. The underlying reason for this is that the XMM archive images occupy a range of different exposure times, hence individual flux limits. Caution is therefore advised when comparing Fig. 9 to similar plots based on a single flux or mass limit. For comparison, using to define the detection limit leads to a flux limit of  erg s cm, considerably lower than that shown in Fig. 9.

With measurement errors

Introducing measurement errors for redshift and X-ray temperature will introduce scattering of clusters across the redshift and temperature cut-offs. As the cluster distribution is not symmetric with respect to these cut-offs, there may be a net increase/decrease in the expected number of clusters as a result (a type of Malmquist bias). Furthermore, the measurement error distributions may also be asymmetric, as is our temperature error distribution. Note that the relevant ‘underlying’ cluster distributions for these purposes are the expected detections, shown in Fig. b.

The change in the expected total number of clusters as a result of different measurement error assumptions are shown in Fig. 10. We find that the effect of measurement errors on the number count is significantly less than the effect of intrinsic scaling-relation scatter (cf. Fig. b). This is not surprising since the scaling-relation scatter is based on the true underlying cluster distribution in Fig. a, a much steeper function than the expected detections in Fig. b.

We also see that only in the case of worst-case temperature errors is the Malmquist bias significant, and as we shall see later only in this case do the measurement errors give a significant bias in cosmological constraints, if unaccounted for. For realistic temperature errors, a net increase in clusters is seen, as the skewness of the temperature error distribution toward low temperatures (Fig. 4) is compensated by the somewhat larger number of low-temperature clusters scattering up in temperature at the low-temperature end. For worst-case temperature errors the temperature is very poorly constrained, and this compensatory effect is not sufficient to counteract the net decrease in number of clusters. Redshift errors tend to cause a loss of clusters at the low-redshift end, as the smaller cosmic volume at lower redshifts means more clusters scatter down in redshift than scatter up. However, the redshift errors also affect the temperature determination, so that low-temperature clusters scattering up could give a net increase. For realistic redshift errors the size of this induced error in temperature is per cent, which is too small to have a significant impact. For worst-case redshift errors, we see that for the case with no scaling-relation scatter, the induced temperature error of per cent reduces the loss of clusters compared to that for realistic redshift errors. For the case with scaling-relation scatter, this effect is not significant, presumably due to the much sharper drop in cluster numbers at low redshifts seen for these models (Fig. b), leading to the direct redshift error dominating.

(a) Known scaling relations, no scatter
(b) Self-calibration of relation, with scatter
Figure 11: Expected and per cent parameter constraints for XCS, without measurement errors. Stars denote the fiducial model assumed.

The fractional change in the number of clusters is very similar for the case with scaling-relation scatter as without such scatter. Hence, for the case with scatter, the statistical effect will tend to be larger since the difference to the clusters with no measurement errors relative to the Poisson error bars,

(13)

grows with the number of clusters (and scatter increases the number). Here, is the fractional change in the number of clusters. Based on this, we estimate that for all the models we consider, an upper limit on the fractional change in cluster count for a less than () bias in constraints is around () per cent, which compares favourably with the results for realistic errors in Fig. 10. (This comparison could be made more rigorous using the Kolmogorov–Smirnov test as in Haiman et al. (2001), but this treatment is sufficient for our purposes.) Due to computational limitations we have not calculated the change in number count for the case with scaling-relation scatter and both types of measurement errors, but based on the results obtained would expect them to be very similar (in fractional terms) to the results for the no-scatter case.

5.3 Constraints: without measurement errors

Known scaling relations, no scatter

For both choices of relation (constant and self-similar), the expected constraints are shown in Fig. a. We expect XCS to measure , in each case. The degeneracy differs somewhat between the two cases, for a constant approximately given by

(14)

and for a self-similar by

(15)

These degeneracies are somewhat different from previous studies, e.g. in Viana & Liddle (1999). That study however used only the total number of clusters above a certain temperature threshold to arrive at constraints. The orientation also depends on redshift depth (Levine et al., 2002). These constraints are better than what has been forecast for XMM–LSS (Refregier et al., 2002), but the comparison is not entirely like-for-like as they employ the Press–Schechter mass function and assume a rather different fiducial and . The constraints are also fairly competitive with what can be expected from other surveys using e.g. the South Pole Telescope (SPT), Planck or DUET (Majumdar & Mohr, 2004; Geisbüsch & Hobson, 2007), but in making this comparison one should note that we employ much more restrictive priors; the set of free parameters is not exactly the same.

Figure 12: Comparison of XCS to the case where all detections are used. A constant (left) or self-similar (right) relation, and no or scatter was assumed. Contours correspond to and per cent confidence levels.
evolution Known scaling relations, no scatter Self-calibration of , with scatter
Constant
Self-similar
Table 5: Expected parameter constraints for XCS when marginalized over all other parameters, without measurement errors.

The constraints in Fig. a are for a photon-count threshold of 500. Lowering the photon-count threshold so that more clusters are included in the sample should clearly affect the size of constraints. We find that using all detections (corresponding to an effective photon-count threshold of typically photons) improves 1D constraints by about per cent (Fig. 12). This corresponds to an increase in the number of clusters used of around ( per cent). For clusters with few photon counts the temperature errors will become very large (Liddle et al., 2001). Contamination from e.g. galaxy groups will also rise sharply with decreasing photon-count threshold, partly because clusters with low photon count will tend to have a low temperature. Hence, these estimates provide only upper limits on the possible constraint improvement. Taking error and contamination effects into account, it is likely that there would be only a weak improvement by including those XCS clusters expected to have a photon count below . However, follow-up observations with e.g. XMM or IXO (formerly XEUS) could improve the photon statistics of those clusters enough to make their inclusion in the analysis worthwhile. We discuss this in more detail in Sect. 6.

Self-calibration of relation, with scatter

Self-calibration is the process by which e.g. the relation can be constrained jointly with cosmological parameters using only the () cluster number counts (Hu, 2003; Lima & Hu, 2004; Majumdar & Mohr, 2004; Lima & Hu, 2005).

(a) Constant relation
(b) Self-similar relation. (As can be surmised from some of the D distributions, the marginalized and mean likelihoods approach each other very slowly in the MCMC due to the prior cutting the distribution, however the statistical properties of the distribution have converged appropriately.)
Figure 13: Expected and per cent parameter constraints for XCS, with scaling-relation scatter and no measurement errors, and fitting jointly with relation (self-calibration) for which reasonable priors on scatter and redshift evolution have been adopted. Solid lines correspond to marginalized likelihood, dotted lines and shading to mean likelihood. Stars denote the fiducial model assumed.
(a) Accounted-for single measurement errors
(b) Unaccounted-for single measurement errors
(c) Unaccounted-for combined measurement errors
Figure 14: Expected and per cent parameter constraints for XCS, for known scaling relations, no scatter, and with measurement errors. Stars denote the fiducial model assumed.
evolution Realistic errors Worst-case errors Realistic errors Worst-case errors
Constant
Self-similar
Table 6: Expected parameter constraints for XCS when marginalized over the other parameter, for known scaling relations, no scatter, and with accounted-for measurement errors.
evolution
Realistic
errors
Worst-case
errors
Realistic
errors
Worst-case
errors
Realistic
& errors
Worst-case
& errors
Constant
Self-similar
Table 7: Expected parameter constraints for XCS when marginalized over the other parameter, for known scaling relations, no scatter, and with unaccounted-for measurement errors.

We find that jointly fitting for the cosmological parameters and the relation, XCS will measure under our assumptions. The marginalized likelihood distributions are shown in Fig. b, and the full set of likelihood distributions in Fig. 13 (note that Fig. b is just the top triangles of these plots). The D parameter constraints are listed in Table 5. The constraints for the case of self-similar evolution appear narrower than for a constant . This is due to the redshift-evolution prior, explained below, significantly cutting the distribution. We thus believe the constant case to be most representative of the constraints we can expect. As expected, the constraints on and degrade when marginalizing over the four parameters (compared to Fig. a), but still remain relatively small. In comparison to the South Pole Telescope, Planck, and DUET (Majumdar & Mohr, 2004; Geisbüsch & Hobson, 2007), our constraints are still competitive (we lack comparable results for XMM–LSS, but expect to do better given our larger survey area and depth). However, if we were to consider self-calibration of the relation as well (rather than using an external description, as described in Sect. 3.2), those surveys would have more power than the XCS (using only archival XMM data) through the use of the cluster power spectrum (Majumdar & Mohr, 2004; Lima & Hu, 2004). In fact, we do not expect XCS to have any significant constraining power if the relation is self-calibrated as well. We show examples of the effects of systematics in Sect. 5.5. It has been shown (e.g. Majumdar & Mohr, 2004) that small follow-up samples can dramatically improve the situation. Therefore, weak-lensing/SZ follow-up and/or a contiguous e.g. XMM survey would be highly advantageous (see also Bergé et al., 2008; Pierre et al., 2008). Comparing to Fig. a, although we lose constraining power due to an increase in the number of parameters, since we are including scaling-relation scatter the number of clusters increases significantly which mitigates the degradation. Note that, as shown in Table 3, we fit the data to a power-law relation . Although the functional form for a self-similar used to generate data is different in principle, we have checked that a power law can approximate its redshift evolution very well.

Using number-count self-calibration, based only on archival XMM data (Fig. 13), we can constrain the normalization to (or per cent) and the slope to (or per cent). The self-calibration procedure is not able to jointly constrain the scatter and redshift evolution significantly. We have therefore imposed flat priors on these parameters, and to limit the distribution within reasonable bounds of a size reflecting the minimum accuracy to which we would hope to measure these parameters from our direct data, i.e. also taking into account the measured X-ray flux (see also Table 3).

Thus, the self-calibration power to constrain the relation is present in the data, but as can be seen in Fig. 13 there are strong degeneracies between parameters. The main degeneracy is that between and ; increasing can easily be offset by reducing , which also is easy to understand physically as they both effectively scale the cluster luminosities up or down, and corresponds to the observation by several authors (e.g. Branchesi et al., 2007; Maughan, 2007; Nord et al., 2008; Pacaud et al., 2007) that scatter can mimic evolution (also discussed in Sect. 3.3.1). The redshift evolution is also degenerate with the slope , which is thus itself degenerate with . Interestingly though, the cosmological parameters show little degeneracy with . It is the result of these degeneracies that all four parameters cannot be jointly constrained. Bayesian Complexity (Kunz, Trotta & Parkinson, 2006) suggests that at most five parameters (including and ) can be fully constrained, which is also what we find in practice. As one might expect, we will therefore have to rely on our direct measurement to constrain the scatter and evolution (as proposed by Verde, Haiman & Spergel 2002; Hu 2003; Battye & Weller 2003; Wang et al. 2004; Lima & Hu 2005).

The fact that our relatively generous priors on the scatter and evolution still restricts the distribution, affecting the size of cosmological constraints, also serves to illustrate a slightly different point of view: turning the problem around, and using complementary cosmological data to constrain e.g. and , thereby possibly also improving constraints on astrophysical parameters (as noted by e.g. Levine et al., 2002; Hu & Kravtsov, 2003; Hu, 2003).

5.4 Constraints: with measurement errors

Known scaling relations, no scatter

The effect on derived cosmological constraints from measurement errors in X-ray temperature and redshift is small. Taking into account knowledge of the error distributions in the data analysis, we find that the size of uncertainties increases somewhat compared to the no-errors case (see Fig. a and Table 6, cf. Fig. a and column 1 in Table 5). Interestingly, even with temperature or redshift errors of an unrealistically large magnitude, the effect on the constraints is small. As such, we expect the broadening of constraints due to measurement errors to be a minor effect compared to the effects of possible systematic errors. These findings are in agreement with what has already been found by e.g. Huterer et al. (2004, 2006); Lima & Hu (2007).

Figure 15: Expected and per cent parameter constraints from the XCS, for various cluster scaling relation assumptions inconsistent with the fiducial model used for generating the data. The different data and fitting assumptions are colour coordinated with the contours, and listed in the panel above the plot. The model parameters are the same as previously, and listed in Table 3. The corresponding cluster distributions in redshift and temperature can be found in Fig. b.
1
2
3
4
5
6
Table 8: Expected parameter constraints for XCS when marginalized over the other parameter, for systematic errors in the scaling-relation assumptions (Fig. 15). The different scenarios are numbered according to the order in which they are listed in Fig. 15.

The effect of ignoring temperature and redshift errors in the fitting procedure can to some extent model one such systematic; poor knowledge of the measurement error distributions. As can be seen in Fig. b, we find that when ignoring measurement errors in the fitting, for all combinations of single measurement errors (i.e. only or at a time), the difference in cosmological constraints compared to the fiducial model is within (and most are within ). For combined and measurement errors, the same is still true for realistic errors, but for a self-similar and worst-case errors the bias is larger than (see Fig. c). These results agree well with the expectations presented in Sect. 5.2.2, and thus suggest that a good estimate of the bias in cosmological constraints due to Malmquist-bias effects can be obtained by comparing the net Malmquist bias to the Poisson error of the total cluster number count (at least to roughly discriminate bias from bias). This is not that surprising as the shape of the cluster distribution does not differ much between such models, and thus the total number count carries most of the information (also noted in Haiman et al., 2001). The D constraints corresponding to Figs. b & c are listed in Table 7.

Self-calibration of relation, with scatter

Because of computational limitations we have not explicitly calculated cosmological constraints for self-calibration with measurement errors. We have however checked that when scatter is included in the data, the effect of temperature and redshift errors on the expected cluster distribution is very similar to the case where no scatter is included, see Fig. 10, and the discussion in the preceding Section and Sect. 5.2.2. We thus expect that the effect from measurement errors on constraints where scatter is included, with or without self-calibration, can be expected to be small or negligible – both in terms of bias if the errors are ignored, or broadening of error contours when errors are taken into account. We therefore believe that the self-calibration results for the case without measurement errors (Figs. b & 13, Table 5) should provide a good rough approximation of the expected self-calibration constraints with measurement errors. Note that this situation is bound to change once direct data is added to the procedure, as the temperature errors will then have a significant impact on the accuracy to which the evolution of the relation can be determined, hence setting the size of the constraints on and . One can therefore not conclude that temperature errors are largely unimportant for the cosmological constraints we will ultimately produce from the data, but an upper limit on the size is set by this work (see e.g. Verde et al., 2002; Hu, 2003; Battye & Weller, 2003).

5.5 Constraints: systematic biases

It is clear from the above sections that measurement errors in the guises we consider are not expected to be a major source of bias or degradation of constraints vis-à-vis the underlying cluster distribution. However, if incorrect assumptions as to the characteristics of the and relations are used when fitting the data, significant bias may occur, as seen in Figs. 15 & 16 and Tables 8 & 9.

Looking first at Fig. 15 (and Table 8 for the 1D marginalized constraints), the figure shows how both the size and best-fitting values of cosmological constraints are affected when ignoring scatter in the scaling relations, using an inappropriate relation, or both. The first case (from left in the panel above the plot) shows how using a self-similar to fit data coming from a constant leads to an overestimation of . Comparing the second, third and fourth cases, we can see that evolution and scaling-relation scatter all have a similar effect when unaccounted for, all leading to an overestimation of (and consequently underestimation of ). As they both have a similar effect, the self-similar evolution in the fifth case can mimic some of the unaccounted-for scatter, leading to a lesser overestimation than for the previous cases. On the other hand, the sixth and last case combines the two effects thus leading to a dramatic overestimation of . As such, this last case represents a worst-case scenario for this type of bias.

Figure 16: Expected and per cent parameter constraints from the XCS, for various mass–temperature-relation assumptions inconsistent with the fiducial model used for generating the data. The different data and fitting assumptions are colour coordinated with the contours, and listed in the panel above the plot. Throughout, a self-similar relation with scatter (as specified in Table 3) has been assumed.
1
2
3
4
5
6
Table 9: Expected parameter constraints for XCS when marginalized over the other parameter, for systematic errors in the relation assumptions (Fig. 16). The different scenarios are numbered according to the order in which they are listed in Fig. 16.

The other figure, Fig. 16 (and Table 9 for the 1D marginalized constraints), shows how constraint size and best-fitting values vary with systematic errors in the relation only. The first two cases (from left in the panel above the plot) illustrates significantly underestimating a scatter of or per cent (deviations similar to what might be expected according to Vikhlinin 2006). This leads to an overestimation of , and relatively narrow constraints, since scatter significantly increases the number of detected clusters. The largest impact seen in this figure comes from poor knowledge of the redshift evolution of the relation, seen in the second pair of contours. We consider a self-similar analyzed as constant in redshift, and a constant analyzed as self-similar. In both cases the deviation from the fiducial model is very significant, with the size of constraints also affected, due to the fiducial-model assumptions having a significant impact on the number of detected clusters. The third, and last, pair of contours show the effect of over- or underestimating the normalization mass by (this value agrees with what might be expected according to e.g. Vikhlinin 2006). Overestimation of the mass leads to an overestimation of , since the higher the assumed mass for a given temperature, the fewer the number of clusters at that temperature. Underestimation of the mass consequently also leads to an underestimation of .

In most cases, the constraints are more than away from the fiducial model. Referring back to the discussion on Poisson errors in Sect. 5.2.2 and applying that to the relevant cluster distributions (see Fig. b), this result is not surprising. We find that in terms of total number count Poisson error bars, the discrepancy between data and fitting assumptions are at least . These limitations will apply to any galaxy cluster survey employing cluster scaling relations to arrive at results, certainly all X-ray surveys, with the exact susceptibility to bias given by the combination of true cluster distribution and survey selection function. This stresses the importance of knowledge of the behaviour of the scaling relations in the form of self-calibration and/or separate follow-up information. For this, accurate knowledge of the selection function is necessary, so that scaling-relation scatter and evolution can be correctly distinguished (as pointed out in e.g. Pacaud et al., 2007).

6 Discussion and conclusions

6.1 The XCS forecast

The XMM Cluster Survey (XCS) will cover deg and is expected to produce one of the largest catalogues of galaxy clusters so far, with clusters having . Around per cent of these will belong to the XCS sample that have sufficient photons () for their X-ray temperature to be reliably estimated. In a rough approximation, we expect to find an additional or more clusters at , of which at least should have photons. We have proven the potential of the XCS with the recent discovery of the most distant galaxy cluster known, XMMXCS J2215.9-1738 at (Stanford et al., 2006; Hilton et al., 2007). Cluster redshifts are obtained from both public-domain photometry and the NOAO–XCS Survey (NXS, Miller et al., 2006). To date, more than XCS candidates have been optically confirmed. An initial observational area of deg (XCS DR1) contains in the range detected clusters with  keV and photons. This number is consistent with the theoretical expectations presented here for our fiducial models.

We have shown the power in determining both cosmological and astrophysical parameters expected from the XMM archive, using only self-calibration from the () distribution and taking detailed selection function, cluster distribution and measurement error modeling into account in a Monte Carlo Markov Chain (MCMC) setting. Inclusion of the selection function requires the specification of the luminosity–temperature relation, and thus enables us to also self-calibrate this relation. We also introduce and motivate a new ‘smoothed Maximum Likelihood estimate’ of the expected constraints, which can be regarded as intermediate between a Fisher matrix analysis and a full mock catalogue ensemble averaging in MCMC.

We expect the XCS to measure

for a flat CDM Universe, the uncertainty on also being that on . The cosmological constraints are similar to those already obtained using gas mass fraction measurements (e.g. Allen et al., 2002; Allen et al., 2008). They are better than those that can be expected from XMM–LSS (Refregier et al., 2002), because XCS covers more area than XMM–LSS (predicted maximum area of deg, but so far results for only deg have been published) and has a higher average exposure time. Our constraints are also somewhat competitive compared to expected constraints from e.g. the SPT, Planck, and DUET (Majumdar & Mohr, 2004; Geisbüsch & Hobson, 2007), except if self-calibration of the mass–temperature relation is also considered. The scatter and redshift evolution of the luminosity–temperature relation cannot be jointly constrained to a significant degree by the self-calibration data alone; additional data – archival XMM and/or follow-up – is needed to distinguish e.g. no evolution from self-similar evolution if the scatter is left as a free parameter. Like e.g. Levine et al. (2002); Hu & Kravtsov (2003); Hu (2003), we note that there is also potential to use this conversely, to let complementary cosmological data help constrain astrophysical parameters. We may return to this in future work.

6.2 Measurement errors

We include for the first time realistic temperature measurement errors, based on detailed Xspec simulations of the XMM fields, and propagate redshift errors to the temperature determination. The presence of realistic or worst-case measurement errors in X-ray temperature and redshift will have only a small impact on the accuracy to which cosmological parameters can be expected to be measured, of order in D confidence limits. Furthermore, we find that imperfect knowledge of the variances of measurement errors, or the presence of catastrophic photometric redshifts, should not produce significant bias in the cosmological constraints. We conclude that, under these assumptions, even ignoring the expected realistic measurement errors in the data analysis will provide a reasonable estimate of the true constraints. For the case where direct data is included in the analysis, the impact of measurement errors (including susceptibility to systematics) will be larger (Verde et al., 2002; Hu, 2003; Battye & Weller, 2003). The size of constraints forecast here provide an upper limit for that scenario.

It is already known (Huterer et al., 2004, 2006; Lima & Hu, 2007) that irreducible systematic errors in redshift estimation is a potential problem for cluster surveys, but we leave for future work the specific requirements for the XCS.

We do not yet take into account the variation of photon count with temperature/luminosity, and how that affects the size of temperature errors. Including this effect, instead of employing a lower threshold only, may well improve the size of our constraints. However the maximum improvement for self-calibration is small. For inclusion of direct data the importance will be larger.

6.3 Cluster scaling relations

The choice of relation itself has no significant impact on the size of cosmological constraints. In our considerations, we do not yet take into account the separate measurement to be performed by the XCS. In the final data analysis, the measurements will be jointly fitted with the cluster distribution. Hence, our expected constraints represent a worst-case scenario of no direct data on the relation. We plan to revisit the issue of the XCS measurement in the future. As an example, estimates for the DUET survey (Majumdar & Mohr, 2004) show that follow-up information on the relation can improve constraints by more than a factor of three.

We quantitatively show that making incorrect assumptions about the cluster scaling relations can typically result in at least a bias in cosmological constraints, a result which can be considered generic for all X-ray and SZ cluster surveys, and those optical surveys relying on cluster scaling relations. Thus, parametrizing the scaling relations appropriately and using self-calibration and/or follow-up information is crucial to arrive at robust results. This places high demands on precise characterization of the survey selection function to accurately distinguish scaling-relation evolution and scatter. That is not a problem for X-ray cluster surveys (as they generally have the best-understood selection functions), and shows the importance of the XCS measurement of the relation for cosmological applications. The XMM–LSS collaboration have already pointed this out, and obtained some first results (Pacaud et al., 2007). A potential pitfall however is the possible redshift evolution of the scatter, as observed in the CLEF simulation (Kay et al., 2007). This has not so far been considered in the literature, but is a possible source of bias that should be better understood. The future IXO mission (Bleeker & Mendez, 2002) will be of great importance for precision measurements of all details of the relation. The XCS will provide thousands of clusters for IXO to target.

An important source of uncertainty is the mass–temperature relation. We have shown quantitatively that, as for the luminosity–temperature relation, imperfect knowledge can easily lead to significant bias. Joint estimation of the mass–temperature relation will lead to broader constraints, and we do not expect the XCS to be able to constrain both the and relations simultaneously. Generally, it has been found that an accuracy of less than per cent in the relation will be needed, and that self-calibration (particularly if making use of the power spectrum, which XCS can not do) and/or small follow-up samples can achieve that (Holder et al., 2001; Haiman et al., 2001; Levine et al., 2002; Majumdar & Mohr, 2003, 2004; Wang et al., 2004; Lima & Hu, 2004, 2005). A recent development is the claim that the X-ray luminosity is a better mass proxy than previously thought (Maughan, 2007). This remains somewhat controversial, but could be worthwhile to consider. Its potentially low scatter and the prospect of including low-temperature clusters, for which the temperature cannot be accurately measured, makes this interesting. Likewise, employing the quantity (e.g. Kravtsov et al., 2006) would also be an interesting option to consider. We leave the XCS-specific details for future work.

It has also been noted by, amongst others, Younger et al. (2006) and Ascasibar & Diego (2008), that the choice of parametrization for the cluster scaling relations can have a significant impact on the size of cosmological constraints, and they argue that a physically-motivated form is beneficial. As also noted by Lima & Hu (2007), efforts in correlating physical properties of clusters, such as that of Shaw et al. (2006), could therefore be of great importance for the size of cosmological constraints, not just biases or astrophysics. However, as the observed dependence on parametrization appears to largely come from an degeneracy, and in this work we assume that , we do not expect this to be of importance for our results here.

6.4 Selection function

The cluster model assumptions made in our calculation of the selection function could have an impact on cluster detectability and thus cosmological constraints. For this reason, we have studied the selection function dependence on cluster structure parameters (for the beta model assumed). For core radii within reasonable bounds, the relative difference compared to our standard beta model is of the order per cent. This number is however an overestimation to the resulting overall uncertainty in cluster number predictions, as it does not include any detailed knowledge of the cluster population, in particular the detailed distribution of cluster-model parameters. Therefore, the significance is limited. We expect the real differences to be smaller. These results agree with those of Burenin et al. (2007) for the 400d survey, which show that reasonable variations in cluster size, morphology and scaling relations induce an uncertainty in the detectability for a given flux of typically less than per cent. In a future, real analysis, the cluster-model parameter distribution would be included along with a selection function dependence on these parameters, thereby also significantly reducing any uncertainty. We are currently studying the selection function further in this respect. Among other things we are applying the detection pipeline to the simulated clusters from CLEF (Kay et al., 2007), to understand the impact of various cluster properties (Hosmer et al., in preparation).

6.5 Sensitivity to fiducial model and priors

Dropping our assumption of spatial flatness, we do not expect particularly strong constraining power on , based on results such as Allen et al. (2002); Majumdar & Mohr (2004). One should note though that the utilization of X-ray galaxy clusters over a large range of redshifts has additional constraining power compared to Allen et al. (2002), as those results are based only on the gas mass fraction in nearby clusters; most of the constraining power on comes from (Holder et al., 2001; Levine et al., 2002). The constraints on and should broaden as a consequence of dropping the flatness assumption; however Majumdar & Mohr (2004) find that such increase tends to be fairly marginal. This increase could arguably also be alleviated by employing appropriate parametrizations for the cluster scaling relations, as discussed above in Sect. 6.3. This, as well as the constraining power on modified-gravity models, is a topic for further investigation.

The assumption of fixed values for the priors of some cosmological parameters, e.g. the scalar spectral index and the Hubble constant, is not realistic given the uncertainty that still exists regarding their true values. Relaxing those priors would increase the size of constraints, though probably not in a significant manner.

6.6 Requirements for a real analysis

Our methodology contains a number of simplifications that are sufficient for a forecast, but for a future real analysis will need to be modified. In particular, this applies to the calculation of the matter-field dispersion , which ought to be calculated employing e.g. Eisenstein & Hu (1998) or CMBFAST (Seljak & Zaldarriaga, 1996) for the transfer function. The uncertainty in the selection function will have to be better understood and included explicitly. In this context, understanding the distribution and impact of cluster structure parameters and their correlation is important. New mass function fits which are more accurate than Jenkins et al. (2001) should also be used, and the impact of uncertainty included. Uncertainty in the normalization chosen will have to be more carefully considered, as will possible systematic uncertainties in temperature definitions and in photometric redshifts. We will in the final analysis fit the relation jointly with cosmology (i.e. use flux data as well), thereby achieving some improvement on the constraints presented here.

6.7 The role of XCS, future surveys and outlook

The XCS is a complete investigation into the galaxy cluster content of the XMM archive. It will be a pathfinder survey for the many ongoing and planned galaxy cluster surveys, and in particular help guide design of new XMM observations and the upcoming X-ray missions eROSITA and IXO. The measurement from the XCS will be the most accurate so far, and will provide an important calibration for potentially all cluster surveys. Although the expected cosmological constraints may not be particularly tight compared to those from the combination of other cosmological data, they will provide important complementary, independent tests of the standard cosmological model from the largest X-ray cluster sample. Particularly, we expect to provide good independent constraints on .

For example, re-imaging of the XCS sample with XMM and/or IXO could improve temperature errors sufficiently, so that at least some of the remaining per cent of the XCS clusters not in the XCS could be used for constraints (this corresponds to no photon-count cut-off in our calculations – effectively a -photon cut-off). We find that an upper limit on the improvement in constraints is by (D) or per cent (D), which thus also represents the best one could possibly do with the current XMM archive using self-calibration only. Once we add direct data to the procedure, the lever arm from re-imaging will be more significant. The XMM–LSS collaboration argue (Pacaud et al., 2007) that the most efficient way to constrain the evolution is to increase the sample size, rather than improve temperature errors, and propose a future deg XMM survey with this rationale (Pierre et al., 2008). A complementary approach to additional observations would be to also use the luminosity–mass relation as mass proxy for those clusters for which the temperature determination is difficult (Maughan, 2007), or use the relatively new quantity advocated by e.g. Kravtsov et al. (2006).

The future for galaxy clusters as a precision and complementary cosmological tool looks increasingly promising, with a range of surveys planned or underway, and numerous simulations undertaken to understand the mass function and cluster physics. The XCS will produce one of the largest ever catalogues of galaxy clusters, providing valuable information on cosmology and cluster physics through the luminosity–temperature relation, beating a path for the many planned surveys. The interface between well-understood cluster physics and cosmology, cross-calibration, and complementary cosmological data will surely be important for constraining dark energy, the primordial power spectrum, and cluster physics over the next decade and beyond.

Acknowledgments

M.S. was partially supported by the Swedish Gålö, Gunvor & Josef Anér and C. E. Levin foundations and the Sir Richard Stapley Educational Trust. P.T.P.V. acknowledges the support of POCI2010 through the project POCTI/CTE-AST/58888/2004. A.K.R., E.L.-D., M.H. & N.M. were supported by PPARC/STFC, and M.D. in part by the RAS Hosie Bequest. M.S., M.D., K.S. & M.H. acknowledge additional financial support from their respective universities. A.K.R & K.S. acknowledge financial support from the XMM and Chandra guest observer programmes and from the NASA–LTSA award NAG-11634. The work by S.A.S. was performed under the auspices of the U.S. Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. This work is based on data obtained by XMM–Newton, an ESA science mission funded by contributions from ESA member states and from NASA. The research made use of the NASA/GSFC-supported Xspec software, and was conducted in co-operation with SGI/Intel utilizing the Altix 4700 supercomputer (COSMOS), also funded by HEFCE and STFC. We are grateful to Ronald Cools and the NINES group at KU Leuven for making the numerical integration package CUBPACK available to us. We thank Ben Maughan and Jochen Weller for useful discussions.

Appendix A Equations

a.1 Cluster counts

Ideal measurements

The expected number of clusters with temperatures between and at redshifts between and when measurements are assumed to be exact is given by

(16)

where is the actual number density of clusters in temperature and redshift, given by the convolution of the mass function with cluster scaling relations, their scatter (through ), cosmic volume and the survey selection function (including sky coverage):

(17)

The scaling-relation scatter probability distributions are assumed to be statistically independent,

(18)

each having a log-normal form:

(19)