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The galaxy cluster Ysz-Lx and Ysz-M relations from the WMAP 5-yr data

We use multifrequency matched filters to estimate, in the WMAP 5-year data, the Sunyaev-Zel'dovich (SZ) fluxes of 893 ROSAT NORAS/REFLEX clusters spanning the luminosity range Lx,[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 XMM-Newton observations of the REXCESS sample and numerical simulations. Although the clusters are individually only marginally detected, we are able to firmly measure the SZ signal (>10 sigma) when averaging the data in luminosity/mass bins. The comparison between the bin-averaged SZ signal versus luminosity and X-ray model predictions shows excellent agreement, implying that there is no deficit in SZ signal strength relative to expectations from the X-ray properties of clusters. Using the individual cluster SZ flux measurements, we directly constrain the Y500-Lx and Y500-M500 relations, where Y500 is the Compton y-parameter integrated over a sphere of radius r500. The Y500-M500 relation, derived for the first time in such a wide mass range, has a normalisation Y*500=[1.60 pm 0.19] 10^{-3} arcmin^2 at M500=3 10^{14} h^{1} Msun, in excellent agreement with the X-ray prediction of 1.54 10^{-3} arcmin^2, and a mass exponent of alpha=1.79 pm 0.17, consistent with the self-similar 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 self-similar evolution.


Introduction
Capability to observe the Sunyaev-Zel'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 high-quality SZ measurements of previously-known clusters (e.g., Mroczkowski et al. 2009;Plagge et al. 2009). In addition, large-scale 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 ) 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 S Z − Mass 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 frequency-dependent change in the CMB surface brightness in the direction of galaxy clusters that can be written as ∆i ν (n) = y(n) j ν (x), where j ν is a universal function of the dimensionless frequency x = hν/kT cmb . The Compton y-parameter is given by the integral of the electron pressure along the line-of-sight in the directionn, y = n kT e m e c 2 n e σ T dl, 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 y-parameter Y SZ -defined by dΩ ∆ i ν (n) = Y SZ j ν (x) -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 low-scatter 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 X-ray observations using Y X , the product of the gas mass and mean temperature 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. 2007Hinshaw et al. , 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 cross-correlating with the general galaxy distribution (Fosalba et al. 2003;Myers et al. 2004;Hernández-Monteagudo et al. 2004 or 'stacking' existing cluster catalogues in the optical or X-ray (Lieu et al. 2006;Afshordi et al. 2007; Atrio- Barandela et al. 2008;Bielby & Shanks 2007;Diego & Partridge 2009). These analyses indicate that an isothermal β-model is not a good description of the SZ profile, and some suggest that the SZ signal strength is lower than expected from the X-ray properties of the clusters (Lieu et al. 2006;Bielby & Shanks 2007).
Recent in-depth X-ray 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 Arnaud et al. 2009) that is best fit by a modified NFW profile. The isothermal β-model, on the other hand, does not provide an adequate fit. From this newly determined X-ray pressure profile, we can infer the expected SZ profile, y(r), and the Y SZ − M relation at low redshift ).
It is in light of this recent progress from X-ray 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. (2009). This profile was derived from REXCESS (Böhringer et al. 2007), a sample expressly designed to measure the structural and scaling properties of the local X-ray 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 signal-to-noise by adapting the filter scale to the expected characteristic size of each cluster. The size is estimated through the luminosity-mass 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 X-ray 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 β-model. Besides providing a direct constraint on these relations, the good agreement with X-ray predictions implies that there is in fact no deficit in SZ signal strength relative to expectations from the X-ray properties of these clusters.
The discussion proceeds as follows. We first present the WMAP 5-year data and the ROSAT cluster sample used, a combination of the REFLEX and NORAS catalogues. We then present the SZ model based on the X-ray-measured pressure profile (Sec. 3). In Sec. 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 X-ray predictions in Sect. 6 and 7 and then conclude in Sec. 8. Finally, we collect useful SZ definitions and unit conversions in the Appendices.
Throughout this paper, we use the WMAP5-only 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/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: M 500 = 4 3 π ρ crit (z) 500 r 3 500 .
2. The WMAP-5yr data and the NORAS/REFLEX cluster sample

The WMAP-5yr 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 zero-padding 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 noise-weighted average of the transfer functions of individual differential assemblies (a similar approach was used in Delabrouille et al. 2009).

The NORAS/REFLEX cluster sample and derived X-ray 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 All-Sky 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 ), 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 : 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. The final catalogue of 893 objects contains the position of the clusters (longitude and latitude), the measured redshift z, the derived X-ray luminosity L 500 , the mass M 500 and the angular extent θ 500 . The clusters uniformly cover the celestial sphere at Galactic latitudes above |b| > 20 deg. Their luminosities L 500 range from 0.002 to 35.0 10 44 erg/s, and their redshifts from 0.003 to 0.460. Figure 1 shows the masses M 500 as a function of redshift z for the cluster sample (red crosses). The corresponding corrected luminosities L 500 can be read on the right axis. The typical luminosity correction from measured L X to L 500 is about 10%. The progressive mass cut-off with redshift (only the most massive clusters are present at high z) reflects the flux limited nature of the sample.

The cluster SZ model
In this Section we describe the cluster SZ model, based on X-ray observations of the REXCESS sample combined with numerical simulations, as presented in Arnaud et al. (2009). We use the standard self-similar 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 .

Cluster shape
The dimensionless universal pressure profile is taken from Eq. B1 and Eq. B2 of Arnaud et al. (2009): 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 S Z flux from WMAP data for each ROSAT system fixing c 500 , α, β, γ to the above values, but leaving the normalisation free.

Normalisation
The model allows us to compute the physical pressure profile as a function of mass and z, thus the Y S Z -M 500 relation by integration of P(r) to r 500 . For the shape parameters given above, the normalisation parameter P 0 = 8.130 h 3/2 70 = 7.810 and the selfsimilar definition of P 500 (Arnaud et al. 2009, Eq. 5 and Eq. B2),

Fig. 2.
Left: Estimated SZ flux from a cylinder of aperture radius 5 × r 500 (Y cyl 5r500 ) as a function of the X-ray 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 1 σ error. Right: Red diamonds are the weighted average signal in 4 logarithmically-spaced 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 1 σ 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.
one obtains: arcmin 2 . Equivalently, one can write: Mpc 2 . Details of unit conversions are given in Appendix B. The mass dependence (M 5/3 500 ) 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 WMAP-measured values in Figs. 3, 4, 5 and 6.

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 signal-to-noise of a SZ cluster source by optimally filtering the data. The universal profile shape described in Sec. 3 is assumed, and we evaluate the effects of uncertainty in this profile as outlined in Sec. 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 X-ray data as described in Sec. 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 y-value 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 i th component is the map at frequency ν i . The maps contain the cluster SZ signal plus noise: 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 θ s (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 lineof-sight 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 . X-ray observations are typically well-constrained 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 X-ray prediction.
The multifrequency matched filters Ψ θ s (x) return a minimum variance unbiased estimate,ŷ o , of y o when centered on the cluster: where superscript t indicates a transpose (with complex conjugation when necessary). This is just a linear combination of the J.-B. Melin et al.: The galaxy cluster Y S Z -L X and Y S Z -M relations from the WMAP 5-yr data 5 maps, each convolved with its frequency-specific filter (Ψ θ s ) i . The result expressed in Fourier space is: with P(k) being the noise power spectrum, a matrix in frequency space with components P i j defined by The quantity σ θ s 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 Sec. 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 cross-spectra in annuli with width ∆l = 180.

Measurements of the SZ flux
The derived total WMAP flux from a cylinder of aperture radius 5×r 500 (Y cyl 5r500 ) for the 893 individual NORAS/REFLEX clusters is shown as a function of the measured X-ray luminosity L 500 in the left-hand panel of Fig. 2. The clusters are barely detected individually. The average signal-to-noise 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 right-hand panel of Fig. 2, we average the 893 measurements in four logarithmically-spaced 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 1/σ 2 θ s ). 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 Sec. 5.1. The SZ signal is clearly detected in the two highest luminosity bins (at 6.0 and 5.4 σ, 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 X-ray 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 Y cyl 5r500 into Y 500 as described in Appendix A. This allows a more direct comparison with the model derived from X-ray observations (Sec. 3). Before presenting the results, we first discuss the overall error budget.

Error due to dispersion in X-ray properties
The error σ θ s 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 X-ray 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 S Z 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. (2009, σ log P ∼ 0.10 beyond the core). Using new 100 MC realisations, we estimate this error by drawing a cluster profile in the log-log 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. (2009).
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.

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 sub-sample 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 6 J.-B. Melin et al.: The galaxy cluster Y S Z -L X and Y S Z -M relations from the WMAP 5-yr data 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.
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 X-ray 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 Sec. 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 S Z flux (see Fig. 11 of Arnaud et al. 2009). 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 M 5/3 500 (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).

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 cross-correlated 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 Figure 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 low-mass systems and unbound gas may contribute up to 16.3% +7% −6.4% 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. X-ray model We first consider bin averaged data, focusing on the luminosity range L 500 ∼ > 10 43 ergs/s where the SZ signal is significantly de- 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 X-ray based model predictions and are shifted to higher masses by 20% for clarity. The model is in very good agreement with the data. tected (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 equally-spaced logarithmic bins in X-ray 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 Sec. 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 Sec. 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 right-hand panel of Fig. 3 shows the ratio between the measured data points and those expected from the model. As discussed in Sec. 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.

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: using the statistical error on Y 500 given by the multifrequency matched filter. The total error is estimated by Monte Carlo (see Sec. 5.1) but is dominated by the statistical error. The results are presented in Table 2. As already stated in Sec. 6.1, the fitted values depend only weakly on the choice of L 500 -M 500 relation. Fig. 3. Left: Bin averaged SZ flux from a sphere of radius r 500 (Y 500 ) as a function of X-ray luminosity in a aperture of r 500 (L 500 ). The WMAP data (red diamonds and crosses), the SZ cluster signal expected from the X-ray 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. Table 2. Fitted parameters for the observed Y S Z -L 500 relation. The X-ray based model gives Y * L 500 = 0.89|1.07 × 10 −3 (h/0.719) −5/2 arcmin 2 , α L Y = 1.11|1.04 and β L Y = 2/3 for the REXCESS and intrinsic L 500 -M 500 relation, respectively.

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 X-ray based model. Furthermore, we fit the Y 500 -M 500 relation and compare it with the X-ray predictions. Figure 4 shows the bin averaged SZ flux measurement as a function of mass compared to the X-ray 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 Sec. 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 self-similar mass dependence M 5/3 500 . The expected selfsimilar behaviour of the new quantity Y 500 D ang (z) 2 /M 5/3 500 as a function of redshift is E(z) 2/3 (see Eq. 5). The right panel of Fig. 5 shows Y 500 D ang (z) 2 /M 5/3 500 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 self-similar 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.

Mass dependence and redshift evolution
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).  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 M 5/3 500 D ang (z) −2 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 1 σ statistical errors from WMAP data. The thin bars give the total 1 sigma errors.  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. Table 3. Fitted parameters for the observed Y S Z -M 500 relation. The X-ray based model gives Y * 500 = 1.54 × 10 −3 (h/0.719) −5/2 arcmin 2 , α Y = 5/3 and β Y = 2/3. Using the individual Y 500 measurements and M 500 estimated from the X-ray luminosity, we fit a relation of the form: 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. (2009), is slightly larger than the average mass of the sample (2.8|2.5 ×10 14 M ⊙ for the REXCESS-intrinsic L 500 -M 500 relation, respectively). We use a non-linear least-squares fit built on a gradient-expansion algorithm (IDL curvefit function). In the fitting procedure, only the statistical errors given by the matched multifilter (σ Y 500 ) are taken into account. The total errors on the final fitted parameters, taking into account uncertainties in X-ray properties, are estimated by Monte Carlo as described in Sec. 5. We first discuss the results obtained using the REXCESS L 500 -M 500 relation, which is expected to be close to the optimal case (see discussion in Sec. 5.2). First, we keep the mass and redshift dependence fixed to the self-similar expectation (α Y = 5/3, β Y = 2/3) and we fit only the normalisation. We obtain Y * 500 = 1.60 × 10 −3 (h/0.719) −2 arcmin 2 , in agreement with the X-ray prediction Y * 500 = 1.54 × 10 −3 (h/0.719) −5/2 arcmin 2 (at 0.4σ). Then, we relax the constraint on α Y and fit the normalisation and mass dependence at the same time. We obtain a value for α Y = 1.79, slightly greater than the self-similar expectation (5/3) by 0.8 σ. To study the redshift dependence of the effect, we fix the mass dependence to α Y = 5/3 and fit Y * 500 and β Y at the same time. We obtain a somewhat stronger evolution β Y = 1.05 than the self-similar expectation (2/3). The difference, however, is not significant (0.2σ). As already mentioned above (see also Fig. 5 right), the redshift leverage is too small to get interesting constraints on β Y .
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.

Discussion and conclusions
In this paper we have investigated the SZ effect and its scaling with mass and X-ray luminosity using WMAP 5-year data of the largest published X-ray-selected 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. (2009), derived from X-ray 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 high-quality SZ observations from SPT (Plagge et al. 2009). 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 X-ray data input, the measured and X-ray predicted SZ fluxes are essentially independent.
We studied the Y S Z − 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 X-ray model predictions based on the universal pressure profile derived by Arnaud et al. (2009) from REXCESS. An excellent agreement is found.
Using WMAP 3-year data, both Lieu et al. (2006) and Bielby & Shanks (2007) found that the SZ signal strength is lower than predicted given expectations from the X-ray properties of their clusters, concluding that that there is some missing hot gas in the intra-cluster medium. The excellent agreement between the SZ and X-ray properties of the clusters in our sample shows that there is in fact no deficit in SZ signal strength relative to expectations from X-ray 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 3-year data are missing with respect to the universal baryon fraction, but not with respect to the expectations from X-ray measurements. Afshordi et al. (2007) actually found good agreement between the strength of the SZ signal and the X-ray properties of their cluster sample, a conclusion that agrees with our results. This good convergence between SZ direct measurements and X-ray 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 self-similar 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 ). 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 (∝ D 2 ang (z) ). 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 self-similar 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 all-sky 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., XMM-Newton follow-up 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 S Z − L X and Y S Z − M relations, but also their evolution, opening the way for the use of SZ surveys for precision cosmology.