Issue 
A&A
Volume 525, January 2011



Article Number  A139  
Number of page(s)  11  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201013999  
Published online  09 December 2010 
The galaxy cluster Y_{SZ}−L_{X} and Y_{SZ}−M relations from the WMAP 5yr data
^{1}
DSM/Irfu/SPP, CEA/Saclay, 91191
GifsurYvette Cedex,
France
email: jeanbaptiste.melin@cea.fr
^{2}
APC – Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet,
75205
Paris Cedex 13,
France
email: bartlett@apc.univparis7.fr
^{3}
APC – CNRS, 10
rue Alice Domon et Léonie Duquet, 75205
Paris Cedex 13,
France
email: delabrouille@apc.univparis7.fr
^{4}
DSM/Irfu/SAp, CEA/Saclay, 91191
GifsurYvette Cedex,
France
email: monique.arnaud@cea.fr; rocco.piffaretti@cea.fr; gabriel.pratt@cea.fr
Received:
6
January
2010
Accepted:
21
October
2010
We use multifrequency matched filters to estimate, in the WMAP 5year data, the SunyaevZel’dovich (SZ) fluxes of 893 ROSAT NORAS/REFLEX clusters spanning the luminosity range L_{X,[0.1−2.4] keV} = 2 × 10^{41}−3.5 × 10^{45} erg s^{1}. The filters are spatially optimised by using the universal pressure profile recently obtained from combining XMMNewton observations of the REXCESSsample and numerical simulations. Although the clusters are individually only marginally detected, we are able to firmly measure the SZ signal (>10σ) when averaging the data in luminosity/mass bins. The comparison between the binaveraged SZ signal versus luminosity and Xray model predictions shows excellent agreement, implying that there is no deficit in SZ signal strength relative to expectations from the Xray properties of clusters. Using the individual cluster SZ flux measurements, we directly constrain the Y_{500}−L_{X} and Y_{500}−M_{500} relations, where Y_{500} is the Compton yparameter integrated over a sphere of radius r_{500}. The Y_{500}−M_{500} relation, derived for the first time in such a wide mass range, has a normalisation at M_{500} = 3 × 10^{14} h^{1} M_{⊙}, in excellent agreement with the Xray prediction of 1.54 × 10^{3} arcmin^{2}, and a mass exponent of α = 1.79 ± 0.17, consistent with the selfsimilar expectation of 5/3. Constraints on the redshift exponent are weak due to the limited redshift range of the sample, although they are compatible with selfsimilar evolution.
Key words: cosmology: observations / galaxies: clusters: general / galaxies: clusters: intracluster medium / cosmic background radiation / Xrays: galaxies: clusters
© ESO, 2010
1. Introduction
Capability to observe the SunyaevZel’dovich (SZ) effect has improved immensely in recent years. Dedicated instruments now produce high resolution images of single objects (e.g. Kitayama et al. 2004; Halverson et al. 2009; Nord et al. 2009) and moderately large samples of highquality SZ measurements of previouslyknown clusters (e.g., Mroczkowski et al. 2009; Plagge et al. 2010). In addition, largescale surveys for clusters using the SZ effect are underway, both from space with the Planck mission (Valenziano et al. 2007; Lamarre et al. 2003) and from the ground with several dedicated telescopes, such as the South Pole Telescope (Carlstrom et al. 2009) leading to the first discoveries of clusters solely through their SZ signal (Staniszewski et al. 2009). These results open the way for a better understanding of the SZMass relation and, ultimately, for cosmological studies with large SZ cluster catalogues.
The SZ effect probes the hot gas in the intracluster medium (ICM). Inverse Compton scattering of cosmic microwave background (CMB) photons by free electrons in the ICM creates a unique spectral distortion (Sunyaev & Zeldovich 1970, 1972) seen as a frequencydependent change in the CMB surface brightness in the direction of galaxy clusters that can be written as , where j_{ν} is a universal function of the dimensionless frequency x = hν/kT_{cmb}. The Compton yparameter is given by the integral of the electron pressure along the lineofsight in the direction , (1)where σ_{T} is the Thomson cross section.
Most notably, the integrated SZ flux from a cluster directly measures the total thermal energy of the gas. Expressing this flux in terms of the integrated Compton yparameter Y_{SZ} – defined by – we see that Y_{SZ} ∝ ^{∫} dΩ dln_{e}T_{e} ∝ ^{∫} n_{e}T_{e}dV. For this reason, we expect Y_{SZ} to closely correlate with total cluster mass, M, and to provide a lowscatter mass proxy.
This expectation, borne out by both numerical simulations (e.g., da Silva et al. 2004; Motl et al. 2005; Kravtsov et al. 2006) and indirectly from Xray observations using Y_{X}, the product of the gas mass and mean temperature (Nagai et al. 2007; Arnaud et al. 2007; Vikhlinin et al. 2009), strongly motivates the use of SZ cluster surveys as cosmological probes. Theory predicts the cluster abundance and its evolution – the mass function – in terms of M and the cosmological parameters. With a good mass proxy, we can measure the mass function and its evolution and hence constrain the cosmological model, including the properties of dark energy. In this context the relationship between the integrated SZ flux and total mass, Y_{SZ}−M, is fundamental as the required link between theory and observation. Unfortunately, despite its importance, we are only beginning to observationally constrain the relation (Bonamente et al. 2008; Marrone et al. 2009).
Several authors have extracted the cluster SZ signal from WMAP data (Bennett et al. 2003; Hinshaw et al. 2007, 2009). However, the latter are not ideal for SZ observations: the instrument having been designed to measure primary CMB anisotropies on scales larger than galaxy clusters, the spatial resolution and sensitivity of the sky maps render cluster detection difficult. Nevertheless, these authors extracted the cluster SZ signal by either crosscorrelating with the general galaxy distribution (Fosalba et al. 2003; Myers et al. 2004; HernándezMonteagudo et al. 2004, 2006) or “stacking” existing cluster catalogues in the optical or Xray (Lieu et al. 2006; Afshordi et al. 2007; AtrioBarandela et al. 2008; Bielby & Shanks 2007; Diego & Partridge 2009). These analyses indicate that an isothermal βmodel is not a good description of the SZ profile, and some suggest that the SZ signal strength is lower than expected from the Xray properties of the clusters (Lieu et al. 2006; Bielby & Shanks 2007).
Recent indepth Xray studies of the ICM pressure profile demonstrate regularity in shape and simple scaling with cluster mass. Combining these observations with numerical simulations leads to a universal pressure profile (Nagai et al. 2007; Arnaud et al. 2010) that is best fit by a modified NFW profile. The isothermal βmodel, on the other hand, does not provide an adequate fit. From this newly determined Xray pressure profile, we can infer the expected SZ profile, y(r), and the Y_{SZ}−M relation at low redshift (Arnaud et al. 2010).
It is in light of this recent progress from Xray observations that we present a new analysis of the SZ effect in WMAP with the aim of constraining the SZ scaling laws. We build a multifrequency matched filter (Herranz et al. 2002; Melin et al. 2006) based on the known spectral shape of the thermal SZ effect and the shape of the universal pressure profile of Arnaud et al. (2010). This profile was derived from REXCESS (Böhringer et al. 2007), a sample expressly designed to measure the structural and scaling properties of the local Xray cluster population by means of an unbiased, representative sampling in luminosity. Using the multifrequency matched filter, we search for the SZ effect in WMAP from a catalogue of 893 clusters detected by ROSAT, maximising the signaltonoise by adapting the filter scale to the expected characteristic size of each cluster. The size is estimated through the luminositymass relation derived from the REXCESS sample by Pratt et al. (2009).
We then use our SZ measurements to directly determine the Y_{SZ}−L_{X} and Y_{SZ}−M relations and compare to expectations based on the universal Xray pressure profile. As compared to the previous analyses of Bonamente et al. (2008) and Marrone et al. (2009), the large number of systems in our WMAP/ROSAT sample allows us to constrain both the normalisation and slope of the Y_{SZ}−L_{X} and Y_{SZ}−M relations over a wider mass range and in the larger aperture of r_{500}. In addition, the analysis is based on a more realistic pressure profile than in these analyses, which were based on an isothermal βmodel. Besides providing a direct constraint on these relations, the good agreement with Xray predictions implies that there is in fact no deficit in SZ signal strength relative to expectations from the Xray properties of these clusters.
The discussion proceeds as follows. We first present the WMAP 5year data and the ROSAT cluster sample used, a combination of the REFLEX and NORAS catalogues. We then present the SZ model based on the Xraymeasured pressure profile (Sect. 3). In Sect. 4, we discuss our SZ measurements, after first describing how we extract the signal using the matched filter. Section 5 details the error budget. We compare our measured scaling laws to the Xray predictions in Sects. 6 and 7 and then conclude in Sect. 8. Finally, we collect useful SZ definitions and unit conversions in the Appendices.
Throughout this paper, we use the WMAP5only cosmological parameters set as our “fiducial cosmology”, i.e. h = 0.719, Ω_{M} = 0.26, Ω_{Λ} = 0.74, where h is the Hubble parameter at redshift z = 0 in units of 100 km s^{1}/Mpc. We note h_{70} = h/0.7 and E(z) is the Hubble parameter at redshift z normalised to its present value. M_{500} is defined as the mass within the radius r_{500} at which the mean mass density is 500 times the critical density, ρ_{crit}(z), of the universe at the cluster redshift: .
2. The WMAP5yr data and the NORAS/REFLEX cluster sample
2.1. The WMAP5 yr data
We work with the WMAP full resolution coadded five year sky temperature maps at each frequency channel (downloaded from the LAMBDA archive^{1}). There are five full sky maps corresponding to frequencies 23, 33, 41, 61, 94 GHz (bands K, Ka,Q,V,W respectively). The corresponding beam full widths at half maximum are approximately 52.8, 39.6, 30.6 21.0 and 13.2 arcmins. The maps are originally at HEALPix^{2} resolution nside = 512 (pixel = 6.87 arcmin). Although this is reasonably adequate to sample WMAP data, it is not adapted to the multifrequency matched filter algorithm we use to extract the cluster fluxes. We oversample the original data, to obtain nside = 2048 maps, by zeropadding in harmonic space. In detail, this is performed by computing the harmonic transform of the original maps, and then performing the back transform towards a map with nside = 2048, with a maximum value of ℓ of ℓ_{max} = 750, 850, 1100, 1500, 2000 for the K,Ka,Q,V,W bands respectively. The upgraded maps are smooth and do not show pixel edges as we would have obtained using the HEALPix upgrading software, based on the tree structure of the HEALPix pixelisation scheme. This smooth upgrading scheme is important as the high spatial frequency content induced by pixel edges would have been amplified through the multifrequency matched filters implemented in harmonic space.
In practice, the multifrequency matched filters are implemented locally on small, flat patches (gnomonic projection on tangential maps), which permits adaptation of the filter to the local conditions of noise and foreground contamination. We divide the sphere into 504 square tangential overlapping patches (100 deg^{2} each, pixel = 1.72 arcmin). All of the following analysis is done on these sky patches.
The implementation of the matched filter requires knowledge of the WMAP beams. In this work, we assume symmetric beams, for which the transfer function b_{ℓ} is computed, in each frequency channel, from the noiseweighted average of the transfer functions of individual differential assemblies (a similar approach was used in Delabrouille et al. 2009).
2.2. The NORAS/REFLEX cluster sample and derived Xray properties
We construct our cluster sample from the largest published Xray catalogues: NORAS (Böhringer et al. 2000) and REFLEX (Böhringer et al. 2004), both constructed from the ROSAT AllSky Survey. We merge the cluster lists given in Tables 1, 6 and 8 from Böhringer et al. (2000) and Table 6 from Böhringer et al. (2004) and since the luminosities of the NORAS clusters are given in a standard cold dark matter (SCDM) cosmology (h = 0.5, Ω_{M} = 1), we converted them to the WMAP5 cosmology. We also convert the luminosities of REFLEX clusters from the basic ΛCDM cosmology (h = 0.7, Ω_{M} = 0.3, Ω_{Λ} = 0.7) to the more precise WMAP5 cosmology. Removing clusters appearing in both catalogues leaves 921 objects, of which 893 have measured redshifts. We use these 893 clusters in the analysis detailed in the next section.
The NORAS/REFLEX luminosities L_{X}, measured in the soft [0.1–2.4] keV energy band, are given within various apertures depending on the cluster. We convert the luminosities L_{X} to L_{500}, the luminosities within r_{500}, using an iterative scheme. This scheme is based on the mean electron density profile of the REXCESS cluster sample (Croston et al. 2008), which allows conversion of the luminosity between various apertures, and the REXCESS L_{500}−M_{500} relation (Pratt et al. 2009), which implicitly relates r_{500} and L_{500}. The procedure thus simultaneously yields an estimate of the cluster mass, M_{500}, and the corresponding angular extent θ_{500} = r_{500}/D_{ang}(z), where D_{ang}(z) is the angular distance at redshift z. In the following we consider values derived from relations both corrected and uncorrected for Malmquist bias. The relations are described by the following power law models^{3}: (2)where the normalisation C_{M}, the exponent α_{M} and the dispersion (nearly constant with mass) are given in Table 1. The L_{500}−M_{500} relation was derived in the mass range [10^{14}−10^{15}] M_{⊙}. These limits are shown in Fig. 1. Note that we assume the relation is valid for lower masses.
Values for the parameters of the L_{X} − M relation derived from REXCESS data (Pratt et al. 2009; Arnaud et al. 2010).
Fig. 1 Inferred masses for the 893 NORAS/REFLEX clusters as a function of redshift. The cluster sample is flux limited. The right vertical axis gives the corresponding Xray luminosities scaled by E(z)^{−7/3}. The dashed blue lines delineate the mass range over which the L_{500}−M_{500} relation from Pratt et al. (2009) was derived. 

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3. The cluster SZ model
In this section we describe the cluster SZ model, based on Xray observations of the REXCESS sample combined with numerical simulations, as presented in Arnaud et al. (2010). We use the standard selfsimilar model presented in their Appendix B. Given a cluster mass M_{500} and redshift z, the model predicts the electronic pressure profile. This gives both the SZ profile shape and Y_{500}, the SZ flux integrated in a sphere of radius r_{500}.
3.1. Cluster shape
The dimensionless universal pressure profile is taken from Eqs. (B1) and (B2) of Arnaud et al. (2010): (3)where x = r/r_{s} with r_{s} = r_{500}/c_{500} and c_{500} = 1.156, α = 1.0620, β = 5.4807, γ = 0.3292 and with P_{500} defined in Eq. (4) below.
This profile shape is used to optimise the SZ signal detection. As described below in Sect. 4, we extract the Y_{SZ} flux from WMAP data for each ROSAT system fixing c_{500}, α, β, γ to the above values, but leaving the normalisation free.
3.2. Normalisation
The model allows us to compute the physical pressure profile as a function of mass and z, thus the Y_{SZ}−M_{500} relation by integration of P(r) to r_{500}. For the shape parameters given above, the normalisation parameter and the selfsimilar definition of P_{500} (Arnaud et al. 2010, Eq. (5) and Eq. (B2)), (4)one obtains: (5)where . Equivalently, one can write: (6)where . Details of unit conversions are given in Appendix B. The mass dependence () and the redshift dependence (E(z)^{2/3}) of the relation are selfsimilar by construction. This model is used to predict the Y_{500} value for each cluster. These predictions are compared to the WMAPmeasured values in Figs. 3–6.
4. Extraction of the SZ flux
4.1. Multifrequency matched filters
We use multifrequency matched filters to estimate cluster fluxes from the WMAP frequency maps. By incorporating prior knowledge of the cluster signal, i.e., its spatial and spectral characteristics, the method maximally enhances the signaltonoise of a SZ cluster source by optimally filtering the data. The universal profile shape described in Sect. 3 is assumed, and we evaluate the effects of uncertainty in this profile as outlined in Sect. 5 where we discuss our overall error budget. We fix the position and the characteristic radius θ_{s} of each cluster and estimate only its flux. The position is taken from the NORAS/REFLEX catalogue and θ_{s} = θ_{500}/c_{500} with θ_{500} derived from Xray data as described in Sect. 2.2. Below, we recall the main features of the multifrequency matched filters. More details can be found in Herranz et al. (2002) or Melin et al. (2006).
Consider a cluster with known radius θ_{s} and unknown central yvalue y_{o} positioned at a known point x_{o} on the sky. The region is covered by the five WMAP maps M_{i}(x) at frequencies ν_{i} = 23, 33, 41, 61, 94 GHz (i = 1,...,5). We arrange the survey maps into a column vector M(x) whose ith component is the map at frequency ν_{i}. The maps contain the cluster SZ signal plus noise: (7)where N is the noise vector (whose components are noise maps at the different observation frequencies) and j_{ν} is a vector with components given by the SZ spectral function j_{ν} evaluated at each frequency. Noise in this context refers to both instrumental noise as well as all signals other than the cluster thermal SZ effect; it thus also comprises astrophysical foregrounds, for example, the primary CMB anisotropy, diffuse Galactic emission and extragalactic point sources. T_{θs}(x − x_{o}) is the SZ template, taking into account the WMAP beam, at projected distance (x − x_{o}) from the cluster centre, normalised to a central value of unity before convolution. It is computed by integrating along the lineofsight and normalising the universal pressure profile (Eq. (3)). The profile is truncated at 5 × r_{500} (i.e. beyond the virial radius) so that what is actually measured is the flux within a cylinder of aperture radius 5 × r_{500}.
Xray observations are typically wellconstrained out to r_{500}. Our decision to integrate out to 5 × r_{500} is motivated by the fact that for the majority of clusters the radius r_{500} is of order the Healpix pixel size (nside = 512, pixel = 6.87 arcmin). Integrating only out to r_{500} would have required taking into account that only a fraction of the flux of some pixels contributes to the true SZ flux in a cylinder of aperture radius r_{500}. We thus obtain the total SZ flux of each cluster by integrating out to 5 × r_{500}, and then convert this to the value in a sphere of radius r_{500} for direct comparison with the Xray prediction.
Fig. 2 Left: estimated SZ flux from a cylinder of aperture radius 5 × r_{500} () as a function of the Xray luminosity in an aperture of r_{500} (L_{500}), for the 893 NORAS/REFLEX clusters. The individual clusters are barely detected. The bars give the total 1σ error. Right: Red diamonds are the weighted average signal in 4 logarithmicallyspaced luminosity bins. The two high luminosity bins exhibit significant SZ cluster flux. Note that we have divided the vertical scale by 30 between Fig. left and right. The thick and thin bars give the 1σ statistical and total errors, respectively. Green triangles (shifted up by 20% with respect to diamonds for clarity) show the result of the same analysis when the fluxes of the clusters are estimated at random positions instead of true cluster positions. 

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The multifrequency matched filters Ψ_{θs}(x) return a minimum variance unbiased estimate, , of y_{o} when centered on the cluster: (8)where superscript t indicates a transpose (with complex conjugation when necessary). This is just a linear combination of the maps, each convolved with its frequencyspecific filter (Ψ_{θs})_{i}. The result expressed in Fourier space is: (9)where with P(k) being the noise power spectrum, a matrix in frequency space with components P_{ij} defined by . The quantity σ_{θs} gives the total noise variance through the filter, corresponding to the statistical errors quoted in this paper. The other uncertainties are estimated separately as described in Sect. 5.1. The noise power spectrum P(k) is directly estimated from the maps: since the SZ signal is subdominant at each frequency, we assume N(x) ≈ M(x) to do this calculation. We undertake the Fourier transform of the maps and average their crossspectra in annuli with width Δl = 180.
4.2. Measurements of the SZ flux
The derived total WMAP flux from a cylinder of aperture radius 5 × r_{500} () for the 893 individual NORAS/REFLEX clusters is shown as a function of the measured Xray luminosity L_{500} in the lefthand panel of Fig. 2. The clusters are barely detected individually. The average signaltonoise ratio (S/N) of the total population is 0.26 and only 29 clusters are detected at S/N > 2, the highest detection being at 4.2. However, one can distinguish the deviation towards positive flux at the very high luminosity end.
In the righthand panel of Fig. 2, we average the 893 measurements in four logarithmicallyspaced luminosity bins (red diamonds plotted at bin center). The number of clusters are 7, 150, 657, 79 from the lowest to the highest luminosity bin. Here and in the following, the bin average is defined as the weighted mean of the SZ flux in the bin (weight of ). The thick error bars correspond to the statistical uncertainties on the WMAP data only, while the thin bar gives the total errors as discussed in Sect. 5.1. The SZ signal is clearly detected in the two highest luminosity bins (at 6.0 and 5.4σ, respectively). As a demonstrative check, we have undertaken the analysis a second time using random cluster positions. The result is shown by the green triangles in Fig. 2 and is consistent with no SZ signal, as expected.
In the following sections, we study both the relation between the SZ signal and the Xray luminosity and that with the mass M_{500}. We consider Y_{500}, the SZ flux from a sphere of radius r_{500}, converting the measured into Y_{500} as described in Appendix A. This allows a more direct comparison with the model derived from Xray observations (Sect. 3). Before presenting the results, we first discuss the overall error budget.
Fig. 3 Left: bin averaged SZ flux from a sphere of radius r_{500} (Y_{500}) as a function of Xray luminosity in a aperture of r_{500} (L_{500}). The WMAP data (red diamonds and crosses), the SZ cluster signal expected from the Xray based model (blue stars) and the combination of the Y_{500}−M_{500} and L_{500}−M_{500} relations (dash and dotted dashed lines) are given for two analyses, using respectively the intrinsic L_{500}−M_{500} and the REXCESS L_{500}−M_{500} relations. As expected, the data points do not change significantly from one case to the other showing that the Y_{500}L_{500} relation is rather insensitive to the finer details of the underlying L_{500}−M_{500} relation. Right: ratio of data points to model for the two analysis. The points for the analysis undertaken with the intrinsic L_{500}−M_{500} are shifted to lower luminosities by 20% for clarity. 

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5. Overall error budget
5.1. Error due to dispersion in Xray properties
The error σ_{θs} on Y_{500} given by the multifrequency matched filter only includes the statistical SZ measurement error, due to the instrument (beam, noise) and to the astrophysical contaminants (primary CMB, Galaxy, point sources). However, we must also take into account: 1) uncertainties on the cluster mass estimation from the Xray luminosities via the L_{500}−M_{500} relation, 2) uncertainties on the cluster profile parameters. These are sources of error on individual Y_{500} estimates (actual parameters for each individual cluster may deviate somewhat from the average cluster model). These deviations from the mean, however, induce additional random uncertainties on statistical quantities derived from Y_{500}, i.e. bin averaged Y_{500} values and the Y_{500}−L_{500} scaling relation parameters. Their impact on the Y_{500}−M_{500} relation, which depends directly on the M_{500} estimates, is also an additional random uncertainty.
The uncertainty on M_{500} is dominated by the intrinsic dispersion in the L_{500}−M_{500} relation. Its effect is estimated by a Monte Carlo (MC) analysis of 100 realisations. We use the dispersion at z = 0 as estimated by Pratt et al. (2009), given in Table 1. For each realisation, we draw a random mass log M_{500} for each cluster from a Gaussian distribution with mean given by the L_{500}−M_{500} relation and standard deviation σ_{log L−log M}/α_{M}. We then redo the full analysis (up to the fitting of the Y_{SZ} scaling relations) with the new values of M_{500} (thus θ_{s}).
The second uncertainty is due to the observed dispersion in the cluster profile shape, which depends on radius as shown in Arnaud et al. (2010, σ_{log P} ~ 0.10. Using new 100 MC realisations, we estimate this error by drawing a cluster profile in the loglog plane from a Gaussian distribution with mean given by Eq. (3) and standard deviation depending on the cluster radius as shown in the lower panel of Fig. 2 in Arnaud et al. (2010).
The total error on Y_{500} and on the scaling law parameters is calculated from the quadratic sum of the standard deviation of both the above MC realisations and the error due to the SZ measurement uncertainty.
5.2. The Malmquist bias
The NORAS/REFLEX sample is flux limited and is thus subject to the Malmquist bias (MB). This is a source of systematic error. Ideally we should use a L_{500}−M_{500} relation which takes into account the specific bias of the sample, i.e. computed from the true L_{500}−M_{500} relation, with dispersion and bias according to each survey selection function. We have an estimate of the true, ie MB corrected, L_{500}−M_{500} relation, from the published analysis of REXCESS data (Table 1). However, while the REFLEX selection function is known and available, this is not the case for the NORAS sample. This means that we cannot perform a fully consistent analysis. In order to estimate the impact of the Malmquist Bias we thus present, in the following, results for two cases.
In the first case, we use the published L_{500}−M_{500} relation derived directly from the REXCESS data, i.e. not corrected for the REXCESS MB (hereafter the REXCESS L_{500}−M_{500} relation). Note that the REXCESS is a subsample of REFLEX. Using this relation should result in correct masses if the Malmquist bias for the NORAS/REFLEX sample is the same as that for the REXCESS. The Y_{500}−M_{500} relation derived in this case would also be correct and could be consistently compared with the Xray predicted relation. We recall that this relation was derived from pressure and mass measurements that are not sensitive to the Malmquist bias. However L_{500} would remain uncorrected so that the Y_{500}−L_{500} relation derived in this case should be viewed as a relation uncorrected for the Malmquist bias. In the second case, we use the MB corrected L_{500}−M_{500} relation (hereafter the intrinsic L_{500}−M_{500} relation). This reduces to assuming that the Malmquist bias is negligible for the NORAS/REFLEX sample. The comparison of the two analyses provides an estimate of the direction and amplitude of the effect of the Malmquist bias on our results. The REXCESS L_{500}−M_{500} relation is expected to be closer to the L_{500}−M_{500} relation for the NORAS/REFLEX sample than the intrinsic relation. The discussions and figures correspond to the results obtained when using the former, unless explicitly specified.
Fitted parameters for the observed Y_{SZ}–L_{500} relation.
The choice of the L_{500}−M_{500} relation has an effect both on the estimated L_{500}, M_{500} and Y_{500} values and on the expectation for the SZ signal from the NORAS/REFLEX clusters. However, for a cluster of given luminosity measured a given aperture, L_{500} depends weakly on the exact value of r_{500} due to the steep drop of Xray emission with radius. As a result, and although L_{500} and M_{500} (or equivalently r_{500}) are determined jointly in the iterative procedure described in Sect. 2.2, changing the underlying L_{500}−M_{500} relation mostly impacts the M_{500} estimate: L_{500} is essentially unchanged (median difference of ~0.8%) and the difference in M_{500} simply reflects the difference between the relations at fixed luminosity. This has an impact on the measured Y_{500} via the value of r_{500} (the profile shape being fixed) but the effect is also small (<1%). This is due to the rapidly converging nature of the Y_{SZ} flux (see Fig. 11 of Arnaud et al. 2010). On the other hand, all results that depend directly on M_{500}, namely the derived Y_{500}−M_{500} relation or the model value for each cluster, that varies as (Eq. (5)), depend sensitively on the L_{500}−M_{500} relation. M_{500} derived from the intrinsic relation is higher, an effect increasing with decreasing cluster luminosity (see Fig. B2 of Pratt et al. 2009).
5.3. Other possible sources of uncertainty
The analysis presented in this paper has been performed on the entire NORAS/REFLEX cluster sample without removal of clusters hosting radio point sources. To investigate the impact of the point sources on our result, we have crosscorrelated the NVSS (Condon et al. 1998) and SUMMS (Mauch et al. 2003) catalogues with our cluster catalogue. We conservatively removed from the analysis all the clusters hosting a total radio flux greater than 1 Jy within 5 × r_{500}. This leaves 328 clusters in the catalogue, removing the measurements with large uncertainties visible in Fig. 2 left. We then performed the full analysis on these 328 objects up to the fitting of the scaling laws, finding that the impact on the fitted values is marginal. For example, for the REXCESS case, the normalisation of the Y_{500}−M_{500} relation decreases from 1.60 to 1.37 (1.6 statistical σ) and the slope changes from 1.79 to 1.64 (1 statistical σ). The statistical errors on these parameters decrease respectively from 0.14 to 0.30 and from 0.15 to 0.40 due to the smaller number of remaining clusters in the sample.
The detection method does not take into account superposition effects along the line of sight, a drawback that is inherent to any SZ observation. Thus we cannot fully rule out that our flux estimates are not partially contaminated by low mass systems surrounding the clusters of our sample. Numerical simulations give a possible estimate of the contamination: Hallman et al. (2007) suggest that lowmass systems and unbound gas may contribute up to of the SZ signal. This would lower our estimated cluster fluxes by ~1.5σ.
6. The Y_{SZ}–L_{500} relation
6.1. WMAP SZ measurements vs. Xray model
We first consider bin averaged data, focusing on the luminosity range L_{500}_{~} > 10^{43} ergs/s where the SZ signal is significantly detected (Fig. 2 right). The left panel of Fig. 3 shows Y_{500} from a sphere of radius r_{500} as a function of L_{500}, averaging the data in six equallyspaced logarithmic bins in Xray luminosity. Both quantities are scaled according to their expected redshift dependence. The results are presented for the analyses based on the REXCESS (red diamonds) and intrinsic (red crosses) L_{500}−M_{500} relations. For the reasons discussed in Sect. 5.2, the derived data points do not differ significantly between the two analyses (Fig. 3 left), confirming that the measured Y_{500}−L_{500} relation is insensitive to the finer details of the underlying L_{500}−M_{500} relation.
We also apply the same averaging procedure to the model Y_{500} values derived for each cluster in Sect. 3. The expected values for the same luminosity bins are plotted as stars in the lefthand hand panel of Fig. 3. The Y_{500}−L_{500} relation expected from the combination of the Y_{500}−M_{500} (Eq. (5)) and L_{500}−M_{500} (Eq. (2)) relations is superimposed to guide the eye. The righthand panel of Fig. 3 shows the ratio between the measured data points and those expected from the model. As discussed in Sect. 5.2, the model values depend on the assumed L_{500}−M_{500} relation. The difference is maximum in the lowest luminosity bin where the intrinsic relation yields ~40% higher value than the REXCESS relation (Fig. 3 left panel). The SZ model prediction and the data are in good agreement, but the agreement is better when the REXCESS L_{500}−M_{500} is used in the analysis (Fig. 3 right panel). This is expected if indeed the agreement is real and the effective Malmquist bias for the NORAS/REFLEX sample is not negligible and is similar to that of the REXCESS.
6.2. Y_{500}–L_{500} relation fit
Working now with the individual flux measurements, Y_{500}, and L_{500} values, we fit an Y_{500}−L_{500} relation of the form: (12)using the statistical error on Y_{500} given by the multifrequency matched filter. The total error is estimated by Monte Carlo (see Sect. 5.1) but is dominated by the statistical error. The results are presented in Table 2. As already stated in Sect. 6.1, the fitted values depend only weakly on the choice of L_{500}−M_{500} relation.
Fig. 4 Estimated SZ flux Y_{500} (in a sphere of radius r_{500}) as a function of the mass M_{500} averaged in 4 mass bins. Red diamonds are the WMAP data. Blue stars correspond to the Xray based model predictions and are shifted to higher masses by 20% for clarity. The model is in very good agreement with the data. 

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Fig. 5 Evolution of the Y_{500}M_{500} relation. Left: the WMAP data from Fig. 4 are divided into three redshift bins: z < 0.08 (blue diamonds), 0.08 < z < 0.18 (green crosses), z > 0.18 (red triangles). We observe the expected trend: at fixed mass, Y_{500} decreases with redshift. This redshift dependence is mainly due to the angular distance (Y_{500} ∝ D_{ang}(z)^{2}). The stars give the prediction of the model. Right: we divide Y_{500} by and plot it as a function of z to search for evidence of evolution in the Y_{500}−M_{500} relation. The thick bars give the 1σ statistical errors from WMAP data. The thin bars give the total 1 sigma errors. 

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Fig. 6 Left: zoom on the >5 × 10^{13} M_{⊙} mass range of the Y_{500}−M_{500} relation shown in Fig. 4. The data points and model stars are now scaled with the expected redshift dependence and are placed at the mean mass of the clusters in each bin. Right: ratio between data and model. 

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7. The Y_{SZ}–M_{500} relation and its evolution
In this section, we study the mass and redshift dependence of the SZ signal and check it against the Xray based model. Furthermore, we fit the Y_{500}−M_{500} relation and compare it with the Xray predictions.
7.1. Mass dependence and redshift evolution
Figure 4 shows the bin averaged SZ flux measurement as a function of mass compared to the Xray based model prediction. As expected, the SZ cluster flux increases as a function of mass and is compatible with the model. In order to study the behaviour of the SZ flux with redshift, we subdivide each of the four mass bins into three redshift bins corresponding to the following ranges: z < 0.08, 0.08 < z < 0.18, z > 0.18. The result is shown in the left panel of Fig. 5. In a given mass bin the SZ flux decreases with redshift, tracing the D_{ang}(z)^{2} dependence of the flux. In particular, in the highest mass bin (10^{15} M_{⊙}), the SZ flux decreases from 0.007 to 0.001 arcmin^{2} while the redshift varies from z < 0.08 to z > 0.18. The mass and the redshift dependence are in good agreement with the model (stars) described in Sect. 3.
Since the D_{ang}(z)^{2} dependence is the dominant effect in the redshift evolution, we multiply Y_{500} by D_{ang}(z)^{2} and divide it by the selfsimilar mass dependence . The expected selfsimilar behaviour of the new quantity as a function of redshift is E(z)^{2/3} (see Eq. (5)). The right panel of Fig. 5 shows as a function of redshift for the three redshift bins z < 0.08, 0.08 < z < 0.18, z > 0.18. The points have been centered at the average value of the cluster redshifts in each bin. The model is displayed as blue stars. Since the model has a selfsimilar redshift dependence and E(z)^{2/3} increases only by a factor of 5% over the studied redshift range, the model stays nearly constant. The blue dotted line is plotted through the model and varies as E(z)^{2/3}. The data points are in good agreement with the model, but clearly, the redshift leverage of the sample is insufficient to put strong constraints on the evolution of the scaling laws.
We now focus on the mass dependence of the relation. We scale the SZ flux with the expected redshift dependence and plot it as a function of mass. The result is shown in Fig. 6 for the high mass end. The figure shows a very good agreement between the data points and the model, which is confirmed by fitting the relation to the individual SZ flux measurements (see next section).
7.2. Y_{500}−M_{500} relation fit
Using the individual Y_{500} measurements and M_{500} estimated from the Xray luminosity, we fit a relation of the form: (13)The results are presented in Table 3 for the analysis undertaken using the REXCESS and that using the intrinsic L_{500}−M_{500} relation. The pivot mass 3 × 10^{14}h^{1} M_{⊙}, close to that used by Arnaud et al. (2010), is slightly larger than the average mass of the sample (2.82.5 × 10^{14} M_{⊙} for the REXCESSintrinsic L_{500}−M_{500} relation, respectively). We use a nonlinear leastsquares fit built on a gradientexpansion algorithm (IDL curvefit function). In the fitting procedure, only the statistical errors given by the matched multifilter (σ_{Y500}) are taken into account. The total errors on the final fitted parameters, taking into account uncertainties in Xray properties, are estimated by Monte Carlo as described in Sect. 5.
Fitted parameters for the observed Y_{SZ}–M_{500} relation.
As cluster mass estimates depend on the assumption of the underlying L_{500}−M_{500} relation, so does the derived Y_{500}−M_{500} relation as well. However, the effect is small. The normalisation is shifted from (1.60 ± 0.14 stat[±0.19 tot]) 10^{3} arcmin^{2} to ^{(}1.37 ± 0.12 stat[±0.17 tot]^{)} 10^{3} arcmin^{2} when using the intrinsic L_{500}−M_{500} relation. The difference is less than two statistical sigmas, and for the mass exponent, it is less than one.
8. Discussion and conclusions
In this paper we have investigated the SZ effect and its scaling with mass and Xray luminosity using WMAP 5year data of the largest published Xrayselected cluster catalogue to date, derived from the combined NORAS and REFLEX samples. Cluster SZ flux estimates were made using an optimised multifrequency matched filter. Filter optimisation was achieved through priors on the pressure distribution (i.e., cluster shape) and the integration aperture (i.e., cluster size). The pressure distribution is assumed to follow the universal pressure profile of Arnaud et al. (2010), derived from Xray observations of the representative local REXCESS sample. This profile is the most realistic available for the general population at this time, and has been shown to be in good agreement with recent highquality SZ observations from SPT (Plagge et al. 2010). Furthermore, our analysis takes into account the dispersion in the pressure distribution. The integration aperture is estimated from the L_{500}−M_{500} relation of the same REXCESS sample. We emphasise that these two priors determine only the input spatial distribution of the SZ flux for use by the multifrequency matched filters; the priors give no information on the amplitude of the measurement. As the analysis uses minimal Xray data input, the measured and Xray predicted SZ fluxes are essentially independent.
We studied the Y_{SZ}−L_{X} relation using both bin averaged analyses and individual flux measurements. The fits using individual flux measurements give quantitative results for calibrating the scaling laws. The bin averaged analyses allow a direct quantitative check of SZ flux measurements versus Xray model predictions based on the universal pressure profile derived by Arnaud et al. (2010) from REXCESS. An excellent agreement is found.
Using WMAP 3year data, both Lieu et al. (2006) and Bielby & Shanks (2007) found that the SZ signal strength is lower than predicted given expectations from the Xray properties of their clusters, concluding that that there is some missing hot gas in the intracluster medium. The excellent agreement between the SZ and Xray properties of the clusters in our sample shows that there is in fact no deficit in SZ signal strength rel ative to expectations from Xray observations. Due to the large size and homogeneous nature of our sample, and the internal consistency of our baseline cluster model, we believe our results to be robust in this respect. We note that there is some confusion in the literature regarding the phrase “missing baryons”. The “missing baryons” mentioned by Afshordi et al. (2007) in the WMAP 3year data are missing with respect to the universal baryon fraction, but not with respect to the expectations from Xray measurements. Afshordi et al. (2007) actually found good agreement between the strength of the SZ signal and the Xray properties of their cluster sample, a conclusion that agrees with our results. This good convergence between SZ direct measurements and Xray data is an encouraging step forward for the prediction and interpretation of SZ surveys.
Using L_{500} as a mass proxy, we also calibrated the Y_{500}−M_{500} relation, finding a normalisation in excellent agreement with Xray predictions based on the universal pressure profile, and a slope consistent with selfsimilar expectations. However, there is some indication that the slope may be steeper, as also indicated from the REXCESS analysis when using the best fitting empirical M_{500}Y_{X} relation (Arnaud et al. 2010). M_{500} depends on the assumed L_{500}−M_{500} relation, making the derived Y_{500}−M_{500} relation sensitive to Malmquist bias which we cannot fully account for in our analysis. However, we have shown that the effect of Malmquist bias on the present results is less than 2σ (statistical).
Regarding evolution, we have shown observationally that the SZ flux is indeed sensitive to the angular size of the cluster through the diameter distance effect. For a given mass, a low redshift cluster has a bigger integrated SZ flux than a similar system at high redshift, and the redshift dependence of the integrated SZ flux is dominated by the angular diameter distance ( ). However, the redshift leverage of the present cluster sample is too small to put strong contraints on the evolution of the Y_{500}−L_{500} and Y_{500}−M_{500} relations. We have nevertheless checked that the observed evolution is indeed compatible with the selfsimilar prediction.
In this analysis, we have compensated for the poor sensitivity and resolution of the WMAP experiment (regarding SZ science) with the large number of known ROSAT clusters, leading to selfconsistent and robust results. We expect further progress using upcoming Planck allsky data. While Planck will offer the possibility of detecting the clusters used in this analysis to higher precision, thus significantly reducing the uncertainty on individual measurements, the question of evolution will not be answered with the present RASS sample due to its limited redshift range. A complementary approach will thus be to obtain new high sensitivity SZ observations of a smaller sample. The sample must be representative, cover a wide mass range, and extend to higher z (e.g., XMMNewton followup of samples drawn from Planck and ground based SZ surveys). This would deliver efficient constraints not only on the normalisation and slope of the Y_{SZ}−L_{X} and Y_{SZ}−M relations, but also their evolution, opening the way for the use of SZ surveys for precision cosmology.
Since we consider a standard selfsimilar model, we used the power law relations given in Appendix B of Arnaud et al. (2010). They are derived as in Pratt et al. (2009) with the same luminosity data but for masses derived from a standard slope M_{500}−Y_{X} relation.
Acknowledgments
The authors wish to thank the anonymous referee for useful comments. J.B. Melin thanks R. Battye for suggesting introduction of the h dependance into the presentation of the results. The authors also acknowledge the use of the HEALPix package (Górski et al. 2005) and of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. We also acknowledge use of the Planck Sky Model, developed by the Component Separation Working Group (WG2) of the Planck Collaboration, for the estimation of the radio source flux in the clusters and for the development of the matched multifilter, although the model was not directly used in the present work.
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Appendix A: SZ flux definitions
In this Appendix, we give the definitions of SZ fluxes we used. Table A.1 gives the equivalence between them. In this paper, we mainly use Y_{500} as the definition of the SZ flux. This flux is the integrated SZ flux from a sphere of radius r_{500}. It can be related to Y_{nr500}, the flux from a sphere of radius n × r_{500} by integrating over the cluster profile: (A.1)where P(r) is given by Eq. (3). The ratio Y_{nr500}/Y_{500} is given in Table A.1 for n = 1, 2 ,3 ,5 ,10.
In practice, an experiment does not directly measure Y_{500} but the SZ signal of a cluster integrated along the line of sight and within an angular aperture. This corresponds to the Compton parameter integrated over a cylindrical volume. In Sect. 4, we estimate , the flux from a cylinder of aperture radius 5 × r_{500} using the matched multifilter. Given the cluster profile, we can derive Y_{nr500} from : (A.2)The ratio is given in Table A.1 for n = 1, 2 ,3 ,5 ,10. In the paper, we calculate Y_{500} from .
Equivalence of SZ flux definitions
Appendix B: SZ units conversion
In this Appendix, we provide the numerical factor needed for the SZ flux units conversion and derive the relation between the recently introduced Y_{X} parameter and the SZ flux Y_{SZ}. The latter will allow readers to easily convert between SZ fluxes given in this paper and those reported in other publications.
Given the definition of SZ flux: (B.1)where Ω_{nr500} is the solid angle covered by n × r_{500}, and the fact that the Compton parameter y is unitless, the observational units for the SZ flux are those of a solid angle and usually given in arcmin^{2}.
The SZ flux can be also computed in units of Mpc^{2} and the conversion is given by (B.2)where D_{ang}(z) is the angular distance to the cluster.
The Xray analogue of the integrated SZ Comptonisation parameter is Y_{X} = M_{gas,500}T_{X} whose natural units are M_{⊙} keV, where M_{gas,500} is the gas mass in r_{500} and T_{X} is the spectroscopic temperature excluding the central 0.15 r_{500} region (Kravtsov et al. 2006). To convert between Y_{SZ} and Y_{X}, we first have (B.3)where σ_{T} is the Thomson cross section (in Mpc^{2}), m_{e}c^{2} the electron mass (in keV), T_{e}(r) the electronic temperature (in keV) and n_{e}(r) the electronic density. By assuming that the gas temperature T_{g}(r) is equal to the electronic temperature T_{e}(r) and writing the gas density as ρ_{g}(r) = μ_{e}m_{p}n_{e}(r), where m_{p} is the proton mass and μ_{e} = 1.14 the mean molecular weight per free electron, one obtains: (B.4)where, as in Arnaud et al. (2010), we defined (B.5)The mass weighted temperature is defined as: (B.6)and the factor A = T_{MW}/T_{X} takes into account for the difference between mass weighted and spectroscopic average temperatures. Arnaud et al. (2010) find A ~ 0.924.
All Tables
Values for the parameters of the L_{X} − M relation derived from REXCESS data (Pratt et al. 2009; Arnaud et al. 2010).
All Figures
Fig. 1 Inferred masses for the 893 NORAS/REFLEX clusters as a function of redshift. The cluster sample is flux limited. The right vertical axis gives the corresponding Xray luminosities scaled by E(z)^{−7/3}. The dashed blue lines delineate the mass range over which the L_{500}−M_{500} relation from Pratt et al. (2009) was derived. 

Open with DEXTER  
In the text 
Fig. 2 Left: estimated SZ flux from a cylinder of aperture radius 5 × r_{500} () as a function of the Xray luminosity in an aperture of r_{500} (L_{500}), for the 893 NORAS/REFLEX clusters. The individual clusters are barely detected. The bars give the total 1σ error. Right: Red diamonds are the weighted average signal in 4 logarithmicallyspaced luminosity bins. The two high luminosity bins exhibit significant SZ cluster flux. Note that we have divided the vertical scale by 30 between Fig. left and right. The thick and thin bars give the 1σ statistical and total errors, respectively. Green triangles (shifted up by 20% with respect to diamonds for clarity) show the result of the same analysis when the fluxes of the clusters are estimated at random positions instead of true cluster positions. 

Open with DEXTER  
In the text 
Fig. 3 Left: bin averaged SZ flux from a sphere of radius r_{500} (Y_{500}) as a function of Xray luminosity in a aperture of r_{500} (L_{500}). The WMAP data (red diamonds and crosses), the SZ cluster signal expected from the Xray based model (blue stars) and the combination of the Y_{500}−M_{500} and L_{500}−M_{500} relations (dash and dotted dashed lines) are given for two analyses, using respectively the intrinsic L_{500}−M_{500} and the REXCESS L_{500}−M_{500} relations. As expected, the data points do not change significantly from one case to the other showing that the Y_{500}L_{500} relation is rather insensitive to the finer details of the underlying L_{500}−M_{500} relation. Right: ratio of data points to model for the two analysis. The points for the analysis undertaken with the intrinsic L_{500}−M_{500} are shifted to lower luminosities by 20% for clarity. 

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In the text 
Fig. 4 Estimated SZ flux Y_{500} (in a sphere of radius r_{500}) as a function of the mass M_{500} averaged in 4 mass bins. Red diamonds are the WMAP data. Blue stars correspond to the Xray based model predictions and are shifted to higher masses by 20% for clarity. The model is in very good agreement with the data. 

Open with DEXTER  
In the text 
Fig. 5 Evolution of the Y_{500}M_{500} relation. Left: the WMAP data from Fig. 4 are divided into three redshift bins: z < 0.08 (blue diamonds), 0.08 < z < 0.18 (green crosses), z > 0.18 (red triangles). We observe the expected trend: at fixed mass, Y_{500} decreases with redshift. This redshift dependence is mainly due to the angular distance (Y_{500} ∝ D_{ang}(z)^{2}). The stars give the prediction of the model. Right: we divide Y_{500} by and plot it as a function of z to search for evidence of evolution in the Y_{500}−M_{500} relation. The thick bars give the 1σ statistical errors from WMAP data. The thin bars give the total 1 sigma errors. 

Open with DEXTER  
In the text 
Fig. 6 Left: zoom on the >5 × 10^{13} M_{⊙} mass range of the Y_{500}−M_{500} relation shown in Fig. 4. The data points and model stars are now scaled with the expected redshift dependence and are placed at the mean mass of the clusters in each bin. Right: ratio between data and model. 

Open with DEXTER  
In the text 
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