The XXL Survey. XIII. Baryon content of the bright cluster sample
D. Eckert, S. Ettori, J. Coupon, F. Gastaldello, M. Pierre, J.-B. Melin, A. M. C. Le Brun, I. G. McCarthy, C. Adami, L. Chiappetti, L. Faccioli, P. Giles, S. Lavoie, J. P. Lefevre, M. Lieu, A. Mantz, B. Maughan, S. McGee, F. Pacaud, S. Paltani, T. Sadibekova, G. P. Smith, F. Ziparo
AAstronomy & Astrophysics manuscript no. xxl_fgas c (cid:13)
ESO 2018November 5, 2018
The XXL Survey (cid:63)
XIII. Baryon content of the bright cluster sample
D. Eckert , , S. Ettori , , J. Coupon , F. Gastaldello , M. Pierre , J.-B. Melin , A. M. C. Le Brun , , I. G. McCarthy ,C. Adami , L. Chiappetti , L. Faccioli , P. Giles , S. Lavoie , J. P. Lefèvre , M. Lieu , A. Mantz , B. Maughan , S.McGee , F. Pacaud , S. Paltani , T. Sadibekova , G. P. Smith , and F. Ziparo Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerlande-mail:
[email protected] INAF - IASF-Milano, Via E. Bassini 15, 20133 Milano, Italy INAF - Osservatorio Astronomico di Bologna, Via Ranzani 1, 40127 Bologna, Italy INFN, Sezione di Bologna, viale Berti Pichat 6 /
2, 40127 Bologna, Italy Laboratoire AIM, IRFU / Service d’Astrophysique – CEA / DSM – CNRS – Université Paris Diderot, Bât. 709, CEA-Saclay, 91191Gif-sur-Yvette Cedex, France DSM / Irfu / SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK Université Aix Marseille, CNRS, LAM (Laboratoire d’Astrophysique de Marseille), UMR 7326, 13388, Marseille, France Department of Physics, H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK Department of Physics and Astronomy, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8P 1A1, Canada SEDI CEA Saclay, France School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA Argelander-Institut für Astronomie, University of Bonn, Auf dem Hügel 71, 53121 Bonn, GermanyNovember 5, 2018
ABSTRACT
Traditionally, galaxy clusters have been expected to retain all the material accreted since their formation epoch. For this reason, theirmatter content should be representative of the Universe as a whole, and thus their baryon fraction should be close to the Universalbaryon fraction Ω b / Ω m . We make use of the sample of the 100 brightest galaxy clusters discovered in the XXL Survey to investigatethe fraction of baryons in the form of hot gas and stars in the cluster population. Since it spans a wide range of mass (10 − M (cid:12) )and redshift (0 . − .
1) and benefits from a large set of multiwavelength data, the XXL-100-GC sample is ideal for measuringthe global baryon budget of massive halos. We measure the gas masses of the detected halos and use a mass–temperature relationdirectly calibrated using weak-lensing measurements for a subset of XXL clusters to estimate the halo mass. We find that the weak-lensing calibrated gas fraction of XXL-100-GC clusters is substantially lower than was found in previous studies using hydrostaticmasses. Our best-fit relation between gas fraction and mass reads f gas , = . + . − . (cid:16) M / M (cid:12) (cid:17) . + . − . . The baryon budget ofgalaxy clusters therefore falls short of the Universal baryon fraction by about a factor of two at r , MT . Our measurements require ahydrostatic bias 1 − b = M X / M WL = . + . − . to match the gas fraction obtained using lensing and hydrostatic equilibrium, whichholds independently of the instrument considered. Comparing our gas fraction measurements with the expectations from numericalsimulations, we find that our results favour an extreme feedback scheme in which a significant fraction of the baryons are expelledfrom the cores of halos. This model is, however, in contrast with the thermodynamical properties of observed halos, which mightsuggest that weak-lensing masses are overestimated. In light of these results, we note that a mass bias 1 − b = .
58 as required toreconcile
Planck
CMB and cluster counts should translate into an even lower baryon fraction, which poses a major challenge to ourcurrent understanding of galaxy clusters.
Key words.
X-rays: galaxies: clusters - Galaxies: clusters: general - Galaxies: groups: general - Galaxies: clusters: intraclustermedium - cosmology: large-scale structure
1. Introduction
Recent observations of the cosmic microwave background with
Planck were able to measure the relative amount of baryons anddark matter in the Universe with very high precision, indicatingthat baryons account for (15 . ± . (cid:63) Based on observations obtained with
XMM-Newton , an ESA sci-ence mission with instruments and contributions directly funded byESA Member States and NASA. of their deep gravitational wells, galaxy clusters are traditionallyexpected to have retained most of the material accreted since theformation epoch (White et al. 1993; Eke et al. 1998). For thisreason, the relative amount of baryons and dark matter in galaxyclusters should be close to the Universal value, provided thatthe measurement has been performed over a su ffi ciently largevolume inside which the e ff ects of baryonic physics can be ne-glected (Evrard 1997; Ettori et al. 2003). Recently, Semboliniet al. (2015) presented a comparison between 12 di ff erent non- Article number, page 1 of 19 a r X i v : . [ a s t r o - ph . C O ] D ec & A proofs: manuscript no. xxl_fgas radiative hydrodynamical codes, showing that in all cases withinan overdensity of 500 compared to the critical density the overallbaryon budget of galaxy clusters is close to the Universal value.In recent times, numerical simulations including baryonicphysics (cooling, star formation, and feedback from supernovaeand active galactic nuclei (AGNs), e.g. Planelles et al. 2013;Battaglia et al. 2013; Le Brun et al. 2014) have shown that en-ergy injection around the cluster formation epoch may be able toexpel some of the gas from the cores of halos (McCarthy et al.2011), leading to a depletion of baryons in the central regions.In particular, Planelles et al. (2013) has found that the deple-tion factor Y b = f bar / ( Ω b / Ω m ), where f bar denotes the clusterbaryon fraction, reaches the value Y b = .
85 at r . Le Brunet al. (2014) has shown that di ff erent AGN feedback implemen-tations have an impact on the depletion factor, i.e. models withstrong feedback tend to produce a lower baryon fraction, witha depletion factor ranging from 0.7 to 1.0 for M = M (cid:12) depending on the adopted setup. Therefore, robust observationalconstraints on the baryon fraction and its mass dependence arecrucial to calibrating the implementation of baryonic physics incosmological simulations.In the most massive halos, the majority of the baryons re-sides in the hot, ionised intracluster medium (ICM), which ac-counts for 80 −
90% of the baryons at this mass scale (e.g. Ar-naud & Evrard 1999; Ettori 2003; Vikhlinin et al. 2006; Prattet al. 2009). The observed gas fraction is generally found toincrease with radius because of non-gravitational energy input(Allen et al. 2004; Vikhlinin et al. 2006; Eckert et al. 2013) andit converges toward the value of ∼
13% at r . The stellar con-tent of cluster galaxies and intracluster light generally represents10-20% of the total baryonic mass (Lin & Mohr 2004; Gonza-lez et al. 2007; Andreon 2010, 2015; Mulroy et al. 2014). Theoverall baryon content of galaxy clusters is therefore found tobe close to the Universal value (Lin et al. 2003; Giodini et al.2009; Laganá et al. 2013). Interestingly, these studies observeda general trend of increasing gas fraction (Sun et al. 2009; Prattet al. 2009; Lovisari et al. 2015) and decreasing stellar fraction(e.g. Behroozi et al. 2010; Leauthaud et al. 2012; Coupon et al.2015) with increasing halo mass, which indicates a mass depen-dence of the star formation e ffi ciency, although the strength ofthis e ff ect is subject to some debate (e.g. Gonzalez et al. 2007;Budzynski et al. 2014).It must be noted, however, that most of the studies measuringthe hot gas fraction assumed that the ICM is in hydrostatic equi-librium to derive the total cluster mass, which may be biased lowin the presence of a significant non-thermal pressure contribution(e.g. Rasia et al. 2004; Nagai et al. 2007). Recent studies com-paring X-ray and weak-lensing mass estimates have found largediscrepancies between the results obtained with the two meth-ods, although significant di ff erences are found between the vari-ous studies (see Sereno & Ettori 2015, and references therein). Inlight of these results, a careful assessment of the cluster baryonbudget in a sample spanning a wide mass range is required.The XXL Survey (Pierre et al. 2015) is the largest observingprogramme undertaken by XMM-Newton . It covers two distinctsky areas (XXL-North and XXL-South) for a total of 50 squaredegrees down to a sensitivity of 5 × − ergs cm − s − for point-like sources ([0.5-2] keV band). Thanks to this combination ofarea and depth, the survey is ideally suited to measuring the over-all baryon content of massive halos. Indeed, the detected clusterscover a wide range of nearly two decades in cluster mass (Gileset al. 2015, hereafter Paper III). In addition, a wealth of high-quality optical and near-infrared data are available in the survey area (e.g. CFHTLS, WIRCam), which provides a robust mea-surement of the total and stellar mass of the detected clusters.In this paper, we exploit the combination of multiwavelengthdata for the brightest XXL clusters to perform a comprehensivecensus of the baryon fraction of dark-matter halos in the range10 − M (cid:12) . The paper is organised as follows. In Sect. 2 wepresent the available XMM-Newton data and the method devel-oped to estimate the gas mass. In Sect. 3 we describe the methodused to estimate the stellar fraction. Our main results are pre-sented in Sect. 4 and discussed in Sect. 5.Throughout the paper we assume a WMAP9 cosmology(Hinshaw et al. 2013) with Ω m = . Ω Λ = .
72, and H o =
70 km / s / Mpc. In this cosmology the cosmic baryon fraction is Ω b / Ω m = . ± . σ level. The quantities indicated with the subscript 500 refer toquantities integrated within an overdensity of 500 with respectto the critical density.
2. Data analysis
The XXL-100-GC cluster sample (Pacaud et al. 2015, here-after Paper II) is the sample of the 100 brightest clusters de-tected in the XXL Survey . It is particularly well suited for thestudy conducted here, as it benefits from a well-defined selec-tion function (see Paper II) and spans a broad range of mass(2 × − × M (cid:12) ) and redshift (0 . − . z < . M − T relation (see PaperIV), which allowed us to calculate an estimate for r (hereafter r , MT ). To validate this approach, in Appendix D we comparethe values of r , MT with those obtained from the weak-lensingmeasurements . XMM-Newton data and processing
We processed the XXL data using the XMMSAS package andcalibration files v10.0.2 and the data reduction pipeline X amin (Pacaud et al. 2006) to obtain cleaned event files for each ob-servation (see Paper II for details on the data reduction scheme).We extracted photon images in the [0.5-2.0] keV band for eachEPIC instrument and created co-added EPIC images by sum-ming the images obtained for each detector. We then created ex-posure maps using the XMMSAS task eexpmap for each EPICdetector and summed them, each weighted by its respective ef-fective area.The non-X-ray background (NXB) was estimated for eachobservation by measuring the count rate in several energy bandsin the unexposed corners of the three EPIC instruments. A tem-plate image of the NXB was created using a collection of closed-filter observations and scaled to match the count rates recorded XXL-100-GC data are available in computer readable form via theXXL Master Catalogue browser http://cosmosdb.iasf-milano.inaf.it/XXL and via the XMM XXL DataBase http://xmm-lss.in2p3.fr
Article number, page 2 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample in the corners. For the details of this procedure, see Leccardi &Molendi (2008b). We neglected the contribution of residual softprotons and fitted this contribution together with the sky back-ground components since the soft protons are funneled throughthe
XMM-Newton telescopes.
Surface brightness profiles were extracted for each cluster us-ing the P roffit package (Eckert et al. 2011). Specifically, wedefined annular regions of width 8 arcsec and accumulated thesky and NXB count rates in each annulus, taking vignetting andCCD gaps into account. The NXB was then subtracted from theobserved profile. The profiles were centred on the best-fit coor-dinates provided by X amin .To estimate the local cosmic X-ray background and Galac-tic foreground emission, we fitted the surface-brightness profilesbeyond 1 . r , MT2 from the X-ray peak with a constant andsubtracted the resulting value from the surface-brightness pro-files, propagating the uncertainties in the sky background and theNXB to the source profile. As a result, a source-only profile wasobtained for each cluster. In the cases for which the estimatedvalue of r , MT exceeds half of the XMM-Newton field of view( ∼
15 arcmin radius), the estimation of the local background maybe a ff ected by systematic uncertainties. For this reason, we ex-cluded the corresponding clusters (mainly low-redshift systems)from the analysis.To convert the observed surface brightness profiles into emis-sion measure profiles, we used the temperature measured within300 kpc from Paper III and simulated a single-temperature ab-sorbed thin-plasma model using the APEC code in X spec (Smithet al. 2001). The Galactic absorption was fixed to the 21cm valueas measured by Kalberla et al. (2005). The mean column densityis 2 . × cm − in the XXL-North field and 1 . × cm − in the XXL-South field. The metal abundance was fixed to thevalue of 0 . Z (cid:12) using the Anders & Grevesse (1989) solar abun-dances (Leccardi & Molendi 2008a), and the cluster redshift wasset to the spectroscopic value (see Paper II). This allowed us tocompute the conversion between count rate in the [0.5-2.0] keVrange and APEC norm, which is proportional to the emissionmeasure, Norm = − π d A (1 + z ) (cid:90) n e n H dV , (1)assuming constant temperature and metallicity. We note that thisconversion is largely insensitive to the temperature and metallic-ity as long as the temperature exceeds ∼ . ∼
10% of the sample), the X-ray emissiv-ity depends strongly on line emission and on the position of thebremsstrahlung cut-o ff . For these systems we thus expect a sys-tematic uncertainty of ∼
30% on the recovered emission measure(see Sect. 5.1). In principle the cluster emission might extend beyond 1 . r , MT .However, in the survey data no significant emission is detected be-yond r , MT , and we verified that the background measurements do notchange significantly when a larger radius is used. r/r -2 -1
10 1 ] - [ c m - h ( z ) H n -4 -3 -2 Fig. 1.
Self-similarly scaled gas density profiles for the XXL-100-GCclusters.
To estimate the gas density and gas mass profiles from the emis-sion measure, we proceeded in two steps: i) deprojecting thesource emission-measure profiles assuming spherical symmetryand ii) converting the resulting profile into gas density assumingconstant density inside each shell. We assumed an electron-to-proton ratio of 1.21 in a fully ionised astrophysical plasma. Thegas mass within a radius R could then be obtained by integratingthe resulting gas density profiles over the volume, M gas ( < r ) = µ m p (cid:90) r n gas ( r (cid:48) ) 4 π r (cid:48) dr (cid:48) , (2)with n gas = n e + n H = . n H the particle number density, µ = .
61 the mean molecular weight, and m p the proton mass.Since the emission region is optically thin, the observed X-ray emissivity is the result of the projection of the 3D emissiv-ity along the line of sight. Therefore, the observed emission-measure profiles must be deprojected to estimate the 3D gas den-sity profile and integrate Eq. 2. To deproject the profiles, we useda multiscale forward-fitting approach in which the signal is de-composed into a sum of King functions, which are then individ-ually deprojected assuming spherical symmetry. The method isdescribed in greater detail in Appendix B. To take the e ff ects ofthe XMM-Newton point spread function (PSF) into account, themodel was convolved with a PSF convolution matrix drawn fromthe latest calibration files (see Appendix C). A maximum like-lihood algorithm was then applied to fit the emission-measureprofiles, and the best-fit parameters were used to compute the 3Ddensity profile. In Fig. 1 we show the gas density profiles recov-ered using this technique for the entire sample, scaled by theirexpected self-similar evolution n ( z ) ∼ E ( z ) ≡ Ω m (1 + z ) + Ω Λ (Bryan & Norman 1998).To sample the parameter space, we used the Markov chainMonte Carlo (MCMC) code emcee (Foreman-Mackey et al.2013). We then drew the posterior distributions of gas densityand gas mass from the resulting Markov chains. The uncertain-ties on the values of r , MT were propagated to the posterior M gas distributions by randomly drawing a value of r , MT foreach MCMC step according to the uncertainties and computingthe gas mass within the corresponding radius.In Table A.1 we provide the gas masses recovered using thistechnique for the XXL-100-GC sample together with the basicproperties of the sample. We also provide the measurements of Article number, page 3 of 19 & A proofs: manuscript no. xxl_fgas ] [M gas,true M ] [ M g as , r ec M Fig. 2.
Gas mass within r , MT reconstructed from mock XMM-Newton observations as a function of the true 3D gas mass in cosmo-OWLSsimulations. The red line shows the 1-1 relation. Y X , = T × M gas , and f gas , = M gas , / M , MT , where M , MT was computed from the M , WL − T relation presentedin Paper IV. For the remainder of the paper, we restrict ourselvesto the systems for which the estimated value of r , MT does notexceed 8 arcmin, which a ff ords a robust estimation of the localbackground and hence of the gas mass. Our final sample com-prises 95 clusters in the redshift range 0.05-1.1. To validate the method, we used mock X-ray images cre-ated from the OverWhelmingly Large cosmological Simulation(cosmo-OWLS, Schaye et al. 2010; McCarthy et al. 2010; LeBrun et al. 2014) using the AGN-8.0 model (see Sect. 5.4) in afield comparable to the XXL Survey (Faccioli et al., in prep.).The mock images were then folded through the
XMM-Newton response and a realistic background was added to obtain simu-lated data that were as close as possible to real XXL observations(Valtchanov et al. 2001).We selected a sample of 61 bright clusters from these mockobservations and applied the method described above to recon-struct the gas mass. The true value of r was used to inte-grate the gas density profile. In Fig. 2 we show the reconstructedgas mass within r compared to the true 3D gas mass withinthe same radius. The reconstructed quantity traces remarkablywell the true value with no measurable bias ( M gas , rec / M gas , true = . ± .
01) and a low intrinsic scatter of 7%. The good agree-ment between the true and reconstructed gas masses demon-strates the reliability of the method used here and was also foundby other similar studies based on hydrodynamical simulations(e.g. Nagai et al. 2007; Rasia et al. 2011).
3. Stellar fraction
To estimate the total stellar content of the XXL-100-GC clus-ters, we adopted a non-parametric approach to measure the meannumber of galaxies as a function of halo mass. Unlike paramet-ric halo occupation distribution (HOD) techniques, here no as-sumption on the statistical distribution of host halos is required as the hot gas is a direct tracer of the dark matter halo posi-tions. We probed the distribution of satellites by means of thestacked galaxy overdensity as a function of distance from thecentral galaxy, and compared this value to the field galaxy den-sity.The stellar fractions are measured over 15 deg of the XXL-North field (corresponding to 34 systems in the XXL-100-GCsample) where galaxy photometric redshifts and stellar massesare taken from the sample of Coupon et al. (2015, and referencestherein). In brief, this sample is composed of deep optical pho-tometry from the Canada-France-Hawaii Telescope Legacy Sur-vey (CFHTLS, u , g , r , i , z AB ∼
25) complemented by medium-deep K -band photometry ( K AB ∼
22) from the WIRCam cameraat CFHT. Calibrated with 60 000 spectroscopic redshifts fromthe VIPERS (Guzzo et al. 2014) and VVDS surveys (Le Fèvreet al. 2005), these photometric redshifts reach a precision of0 . × (1 + z ) with less than 2% catastrophic failures up to z ∼ M (cid:12) , and the stellarpopulation synthesis templates from Bruzual & Charlot (2003).For the details of the procedure we refer the reader to Arnoutset al. (2013).To measure the projected overdensity of satellite galaxies inhalos, we use the two-point cross-correlation function betweenthe bright central galaxy (BCG) positions of the XXL-100-GCclusters from the sample of Lavoie et al. (2015), and galaxiesfrom the full photometric sample. To account for masking andedge e ff ects, we used the Landy & Szalay (1993) estimator tomeasure the projected two-point correlation function w ( R ); themean number of galaxies within a radius R away from the centrewrites N gal ( r ) = π η (cid:90) r (1 + w ( r (cid:48) )) r (cid:48) dr (cid:48) , (3)where R is the physical transverse distance from the centralgalaxy (at the cluster redshift) and η is the mean density of galax-ies in the field. Galaxies in the photometric sample are selectedusing their photometric redshift estimates within the redshiftrange spanned by the cluster sample, extended at low and highredshift ( ± .
1) to account for photometric redshift uncertainties.Here we cross-correlate the full cluster sample with the full pho-tometric sample. To increase the signal, galaxies can also be se-lected in a thin redshift slice around each cluster redshift; how-ever, as the background estimation becomes significantly noisierin consequence, no gain in signal-to-noise is observed.As the central galaxy does not correlate with the field galax-ies outside of the halo, the number of satellites is simply ex-pressed by the overdensity within r , MT compared to the field, N sat = π η (cid:90) r , MT w ( r (cid:48) ) r (cid:48) dr (cid:48) , (4)where r , MT is derived from the M − T relation from PaperIV. This method provides a measurement of the projected galaxyoverdensity, which might lead to a slight overestimation of theenclosed stellar mass because of projection e ff ects, since thecluster actually extends significantly beyond r , MT . Assumingthat the distribution of stellar mass follows that of the dark mat-ter, we used an NFW profile (Navarro et al. 1997) to estimate thefraction of the projected stellar mass located beyond r , MT . Wefound that this fraction is of the order of 15% for typical NFWparameters.The process is repeated for each galaxy sample selected bystellar mass in eight stellar mass bins in the range 10 − M (cid:12) , Article number, page 4 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample and the total stellar mass is obtained by integrating the num-ber of satellites over their stellar masses, and adding the stellarmass of the central galaxy. The errors are estimated from a jack-knife resampling of 64 sub-volumes. This accounts for Poissonerror, cosmic variance and intrinsic halo-to-satellite number dis-persion.We note that this method di ff ers from those relying on indi-vidual satellite identification using redshift probability distribu-tions (e.g. George et al. 2011). Given our photometric redshiftstatistical uncertainties, our cross-correlation estimator providesa more robust (although noisier) satellite number estimate.
4. Results
In a companion paper (Paper IV), we used weak-lensing shearmeasurements for a subsample of 38 XXL-100-GC clusters toderive the relation between X-ray temperature and halo massin XXL clusters. The relation was complemented with weak-lensing mass measurements from the COSMOS (Kettula et al.2013) and CCCP (Hoekstra et al. 2015) samples to increase thesample size to 95 systems and the dynamical range to 1-15 keV.The best-fit relation readslog E ( z ) M WL h − M (cid:12) = a + b log (cid:32) T (cid:33) , (5)with a = . + . − . and b = . + . − . , i.e. slightly steeper thanthe self-similar expectation. For the details on the mass measure-ment method and the derivation of the M − T relation, we referto Paper IV.The spectroscopic temperatures were measured for the entireXXL-100-GC within a fixed radius of 300 kpc and the halo massfor each cluster was estimated using Eq. 5, taking the scatter inthe relation into account (see Paper III for details). Using the 38objects with measured weak-lensing signal, we verified that themasses and radii measured using the M − T relation match thevalues expected from the lensing measurements (see AppendixD). M gas − T relation and gas mass fraction To estimate the average gas fraction and the relation between f gas , and M WL , we fitted the M gas − T relation with the relationlog E ( z ) M gas , h − / M (cid:12) = log N + α log (cid:32) T (cid:33) . (6)As we did for the M WL − T relation (see Paper IV), we fitted thedata using the Bayesian regression code of Kelly (2007) and theGibbs MCMC sampler. In Fig. 3 we show the relation between M gas , and T for the XXL-100-GC clusters together withtheir best-fitting relation. We measured log N = . ± . α = . + . − . . The relation is very tight, with a measuredintrinsic scatter σ int = . ± .
03 dex; however, the use of thetemperature to estimate r , MT introduces a small level of co-variance between the two quantities, which might lead to an un-derestimation of the intrinsic scatter. As shown in Fig. 3, ourbest-fit relation agrees very well with the relation derived by Ar-naud et al. (2007) using ten nearby relaxed clusters, which yields kT [keV] ] [ M g as , E ( z ) M XXL-100-GCArnaud+07
Fig. 3.
Gas mass within r , MT for the XXL-100-GC sample as a func-tion of their temperature within a fixed aperture of 300 kpc. The red lineand the red shaded area show the best-fit relation and its uncertainty. Theblue curve represents the relation of Arnaud et al. (2007). a slope of 2 . ± .
05 fully consistent with ours . The slope ofthe relation is steeper than the self-similar expectation (1.5; e.g.Bryan & Norman 1998) at more than 5 σ and steeper than theslope of the M − T relation, which indicates a dependence of thegas fraction on cluster mass.We searched for breaks in the relation by splitting our sampleinto several temperature ranges and performing the fitting proce-dure again. When considering only the systems with T > . α T > . = . ± .
16, compared to α T < . = . ± .
22 below 2.5 keV. Thus, we find no evidencefor a strong break in the relation.To infer the relation between f gas , and M WL , we combinedthe best-fit M − T and M gas − T relations. This method is prefer-able to directly fitting the f gas − M relation, since the M − T relation is a ff ected by a significant scatter which is propagatedto individual f gas , measurements. To recover the relation be-tween f gas and mass, we selected 10,000 MCMC realisations ofthe two relations and computed the resulting f gas − M relation ineach case. We then calculated the median and dispersion of thevalues of f gas , for a grid of M WL values. The relation obtainedin this way reads h − / f gas , = . + . − . × M , MT h − M (cid:12) . + . − . . (7)In the left panel of Fig. 4 we show the recovered relation and itsuncertainties, together with the individual f gas , values. In theright panel of Fig. 4 we compare our results with three state-of-the-art f gas − M relations obtained for group and cluster-scalesystems assuming that the ICM is in hydrostatic equilibrium:Sun et al. (2009, Chandra ), Lovisari et al. (2015,
XMM-Newton ),and Ettori (2015, a large sample of published results). We note In Arnaud et al. (2007) the temperatures were measured in the range[0 . − . r , i.e. excluding the core. See Paper III for a discussionof the e ff ects of excising (or not) the core region. Arnaud et al. (2007)integrated the gas mass within a hydrostatic-based r ; since M gas ∝ M / (see Sect. 5.2) the relation should depend mildly on the adoptedvalue of r . Article number, page 5 of 19 & A proofs: manuscript no. xxl_fgas ] [M M g as f m W / b W b Y m W / b W ] [M M g as f XXL-100-GCEttori15Sun+09Lovisari+15 m W / b W b Y m W / b W Fig. 4.
Relation between gas fraction and halo mass within r , MT for the XXL-100-GC sample. Left:
The red line and the red shaded area showthe best-fit relation and its uncertainty. The data points show the individual f gas estimates obtained using the M − T relation. The WMAP9 cosmicbaryon fraction is displayed in the grey shaded area, whereas the dashed magenta line indicates the cosmic baryon fraction corrected by thedepletion factor Y b = .
85 at r (Planelles et al. 2013). Right:
Same as in the left panel. For comparison, the solid curves show similar relationsobtained using hydrostatic masses (yellow, Sun et al. (2009); blue, Ettori (2015); green, Lovisari et al. (2015)). a clear o ff set between our weak-lensing calibrated f gas − M rela-tion and the hydrostatic relations, for all mass scales. The rela-tion also falls short of the cosmic baryon fraction Ω b / Ω m (PlanckCollaboration XIII 2015) by almost a factor of three at 10 M (cid:12) .In addition, we investigated the redshift dependence of ourmeasurements. Fitting the M gas − T relation only for the clos-est systems ( z < . α z < . = . ± . ff ect can be explained by the mass dependence ofthe gas fraction. To confirm this statement, we calculated the gasfraction expected from Eq. 7 at the median mass of each redshiftbin and compared with our measurements. This test closely re-produces the observed trend of increasing gas fraction with red-shift. Therefore, we find no evidence for an evolution of the gasfraction with redshift.The discrepancy reported here between X-ray and weak-lensing measurements of the gas fraction is independent ofthe instrument used for hydrostatic-based measurements. Re-cent calibration works reported discrepancies at the level of ∼
15% between the temperatures measured with
XMM-Newton and
Chandra (Nevalainen et al. 2010; Schellenberger et al. 2015).However, this systematic issue was observed mainly for high-temperature systems, for which the energy of the bremsstrahlungcut-o ff is hard to calibrate. The bulk of the clusters in the XXL-100-GC sample have observed temperatures in the range 2-5keV, where calibration uncertainties are much less important. To measure the stellar fraction, we applied the method presentedin Sect. 3 to the subset of clusters in the XXL-100-GC northernsample where CFHTLS and WIRCam photometry is available,and within the redshift range 0 . < z < . Fig. 5.
Mean gas fraction and inter-quartile ranges (red points) for theXXL-100-GC clusters in eight redshift bins normalised to the Univer-sal baryon fraction Ω b / Ω m . The blue diamonds show the gas fractionexpected from Eq. 7 at the median mass of each redshift bin in our sam-ple. The magenta shaded area shows the value expected at r from thesimulations of Planelles et al. (2013). most of the stellar mass is accounted for given our near-infraredcompleteness (10 M (cid:12) ; see Coupon et al. 2015): this thresholdcorresponds to the limit in stellar mass above which the vast ma-jority of the total stellar mass is encapsulated in massive clusters(see also van der Burg et al. 2014).This selection yields a subsample of 34 clusters for this anal-ysis (marked with an asterisk in Table A.1), further subdividedinto three temperature bins with approximately similar signal- Article number, page 6 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample ] [M * log M sa t N -1 kT<3.5 keV3.5
Non-parametric halo occupation distribution of satellite galaxiesin XXL-100-GC clusters in three temperature bins as a function of thegalaxy’s stellar mass for three temperature bins ( T < . . < T < . T > . to-noise ( T < . . < T < . T > . < R < r , MT for each stellar mass and tempera-ture bins. In Fig. 6 we show the number of satellite galaxies as afunction of stellar mass for the three temperature bins. The meanstellar mass for each temperature bin was then obtained by inte-grating the satellite numbers multiplied by their respective stellarmass in each bin, over the full stellar mass range. Although oursample is not complete below 10 M (cid:12) , low-mass galaxies con-tribute very little to the total stellar mass (van der Burg et al.2014). We used the mean temperature in each bin and the M − T relation to estimate the corresponding halo mass. The results ofthis analysis are provided in Table 1.In Fig. 7 we show the measured stellar fraction and the gasfraction as a function of halo mass. Our results are compared totwo literature studies combining galaxy clustering and galaxy-galaxy lensing in COSMOS ( z ∼ .
3, Leauthaud et al. 2012) andCFHTLenS / VIPERS ( z ∼ .
8, Coupon et al. 2015). Our mea-surements are in good agreement with the latter study, whereasa slight discrepancy is observed with the former, most probablyexplained by stellar mass measurement systematics compared toLeauthaud et al. (2012), as described in Sect. 5.3.1 of Couponet al. (2015).Our measurements yield a stellar-to-halo mass fraction ofabout 1% with little dependence on halo mass. This is clearlyinsu ffi cient to bridge the gap between the measured hot gas frac-tion and the cosmic value. Combining the fraction of baryons inthe form of hot gas and stars, we found f bar , = . ± . M (cid:12) halos, which is discrepant with the cosmic baryonfraction corrected by the depletion factor at a confidence level of6 . σ .We note that this estimate of the stellar fraction neglectsthe contribution of intracluster light (ICL) to the stellar contentof dark-matter halos. Intracluster light is usually found to con-tribute to a fraction of the total stellar mass of galaxy clusters(20 − ] [M M b a r f -2 -1 , XXL-100-GC gas f , XXL-100-GC star f , Leauthaud+12 star f , Coupon+15 star f Fig. 7.
Baryon fraction in the form of hot gas (red, this work) and stars.The cyan data points show the measurements obtained for the XXL-100-GC sample in three temperature bins (see Table 1), compared toliterature measurements at di ff erent redshifts ( z ∼ .
3, Leauthaud et al.(2012, green); z ∼ .
8, Coupon et al. (2015, blue)). The WMAP9 cos-mic baryon fraction is displayed in the grey shaded area, whereas thedashed magenta line indicates the cosmic baryon fraction corrected bythe depletion factor Y b = .
85 at r (Planelles et al. 2013). of XXL clusters only represents about half of the expected value,even when the total stellar content is accounted for. We investigated the distribution of intracluster gas using thePSF-corrected gas density profiles extracted following themethod described in Sect. 2.4 (see Fig. 1). We divided our sam-ple into four temperature (and hence, mass) bins ( T < < T < < T < T > bootstrap samplings of the population andcomputed the mean and standard deviation of the bootstrap sam-ples. In addition to the statistical uncertainties, this method takesthe intrinsic scatter in the population into account.The mean gas density profiles are shown in Fig. 8, scaledaccording to the self-similar expectation. We observe significantdi ff erences between the various temperature bins, with a gen-eral trend of higher gas density in the central regions for massiveclusters than for groups. This trend provides a clear confirma-tion of the dependence of the gas fraction on mass, since the gasfraction can be computed directly by integrating the self-similarscaled gas density profile (see Appendix D in Eckert et al. 2012).We note clear di ff erences in the shape of the gas density pro-files between the various samples. While at high mass the meanprofile shows a cored behaviour and a relatively steep outer gra-dient, as usually inferred for massive clusters (see e.g. Mohr et al.1999; Ettori & Fabian 1999; Eckert et al. 2012), galaxy groupsexhibit a cuspy profile and a flatter outer slope, in agreement withprevious studies (e.g. Helsdon & Ponman 2000; Sun et al. 2009).Interestingly, the gas density at r , MT appears to be roughly in-dependent of cluster mass (see also Sun 2012). This e ff ect couldindicate a mass dependence of entropy injection by AGN feed-back, which would lead to a more pronounced expansion of thegas atmosphere in low-mass systems (e.g. Ponman et al. 1999).Such di ff erences would gradually disappear in cluster outskirts. Article number, page 7 of 19 & A proofs: manuscript no. xxl_fgas
Table 1.
Results of the non-parametric halo occupation distribution analysis. (cid:104) T (cid:105) [keV] N obj M ∗ [ M (cid:12) ] M , MT [ M (cid:12) ] M gas , [ M (cid:12) ] f ∗ . ± .
65 21 (1 . ± . × . × (9 . ± . × . ± . . ± .
35 7 (3 . ± . × . × (1 . ± . × . ± . . ± .
21 6 (3 . ± . × . × (2 . ± . × . ± . Column description:
1. Mean temperature and standard deviation in the considered temperature bin. 2. Number of clusters per bin.3. Mean stellar mass per cluster. 4. Mean halo mass estimated using the M WL − T relation. 5. Mean gas mass per cluster (calculatedfor the same systems). 6. Mean stellar fraction. R/R -1
10 1 ] - [ c m - h ( z ) H n -4 -3 kT<2 keV2
10 1 R a t i o Fig. 8.
Mean self-similar scaled ion density profiles for the XXL-100-GC sample for four di ff erent temperature bins ( T < < T < < T < T > bootstrapsamplings of the populations. The bottom panel shows the ratio of themean gas density profiles to the high-temperature bin.
5. Discussion
As shown in Sect. 4.2 and highlighted in Fig. 7, our results arein substantial tension with the Universal baryon fraction within r , MT . Here we review the possible explanations for this result,both physical and instrumental. We investigated whether the subtraction of the background couldhave a significant impact on our gas mass measurements. Indeed,a systematic overestimation of the
XMM-Newton backgroundlevel would lower the observed surface brightness and bias therecovered gas mass low. In the soft band, residual soft protonscan typically a ff ect the measured background by ∼
20% (Lecca-rdi & Molendi 2008b; Kuntz & Snowden 2008). To investigatethe impact of such an uncertainty on our measurements, we de-creased the measured background level by 20% when computingthe emission measure and recalculated the gas masses. The re-sulting gas masses are only mildly a ff ected by such a change,with typical di ff erences at the 5% level.A further source of systematic uncertainty is the conversionbetween count rate and emission measure, which depends onthe temperature and metallicity of the plasma. For gas temper- atures exceeding 1.5 keV (83 systems out of 95), the soft-bandemissivity depends very weakly on temperature ( ∼
3% uncer-tainty) and the overall emissivity is dominated by the contin-uum, thus the metallicity does not play an important role. Forthe coolest systems, however, line emission and the position ofthe bremsstrahlung cut-o ff have a significant impact on the soft-band emissivity (see the discussion by Lovisari et al. 2015). Weestimate from the dependence of the cooling function on temper-ature and metal abundance that this e ff ect can have an impact aslarge as 30% on the soft-band emissivity, which translates into anuncertainty of 15% on the gas density and gas mass. To estimatethe impact of this systematic uncertainty on the gas fraction, wevaried the gas masses of the systems with T < . ±
15% and fitted again the M gas − T relation. We found thatthe slope of the relation varies in the range 1.92-2.11. The e ff ecton the gas fraction is less than the statistical uncertainties in theentire mass range.Our test of the gas mass measurement method using thecosmo-OWLS simulations (Sect. 2.5) shows that our method isable to reconstruct the 3D gas mass with no bias and little in-trinsic scatter (see Fig. 2), provided that the value of r used tointegrate the gas density profile is accurate. Since the uncertain-ties in r , MT were propagated to the gas mass (see Sect. 2.4),we conclude that potential systematics in the gas mass cannotreconcile our f gas measurements with the cosmological value. Inaddition, the excellent agreement observed between the M gas − T relation measured here and results from the literature (see Fig.3) indicates that our measurements of the gas temperature are inline with other samples. To cross-check our
XMM-Newton Y X , measurements, we ex-tracted measurements of the Sunyaev-Zeldovich (SZ) flux Y from Planck . This constitutes a powerful further check, since X-ray and SZ provide a measurement of the same physical quan-tity using two completely independent techniques. We extracted Y for each object of our sample following the method usedin Planck Collaboration X (2011); Planck Collaboration XII(2011); Planck Collaboration XI (2013). Namely, we adoptedthe SZ profile from Arnaud et al. (2010) and applied multifre-quency matched filters (Melin et al. 2006) scaled to the radii r , MT given in Table A.1 at the positions of the clusters.Unfortunately, only four clusters were detected individuallyby Planck at S / N > Y X , bins. For each Y X , , the individual variance is taken asthe mean of the square of the upper and lower errors given in Ta-ble A.1. The Y X , values are then converted into Mpc adopting Article number, page 8 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample
Table 2.
Binned values for
XMM-Newton Y X , = M gas , × T and Planck SZ Y . Y X , σ Y X , Y σ Y Column description:
1. Mean Y X , value; 2. Error on Y X , ;3. Stacked SZ flux Y ; 4. Error on Y . All measurements arein units of 10 − Mpc . Fig. 9.
Planck
SZ flux Y d A ( z ) versus Y X , for the XXL-100-GC sam-ple (red diamonds). The expected value 0.924 (Eq. 19 in Arnaud et al.2010) is the solid black line and the unity is shown as the dashed line.The inset presents the same data points in the lin-log plane to show alsothe low Y X , bins. The numerical values and error bars for the 8 pointsare given in Table 2. the factor C XSZ = . × −
19 Mpc M (cid:12) keV (Arnaud et al. 2010). Weaveraged the SZ flux similarly in the same Y X , bins. In Table 2we give the resulting binned Y X , and Y , and associated er-rors.The results presented in Table 2 are shown in Fig. 9. In thisFigure, we show the expected relation ( Y = . Y X , ) fromthe REXCESS sample (Eq. 19 of Arnaud et al. 2010) and theone-to-one relation. We can see that the values of Y measuredby Planck agree with the values of Y X , obtained with XMM-Newton . This provides an important additional test showing thatX-ray gas mass measurements are not subject to significant sys-tematic uncertainties. In light of these results, we conclude thatthe XXL galaxy cluster population does not appear to di ff er fromother X-ray selected cluster samples. An alternative possibility is that the weak-lensing calibrated M − T relation used here and presented in Paper IV is biasedhigh. For a discussion of the various systematic uncertainties af-fecting these measurements, we refer to Paper IV. We note, how-ever, that our M − T relation for XXL clusters agrees with the results of Kettula et al. (2015) combining CCCP, COSMOS andCFHTLS mass measurements. Therefore, any bias in our weak-lensing measurements would a ff ect as well these other data sets,and the low gas fraction observed here should be a generic con-sequence of current weak-lensing mass calibrations. The results presented in Fig. 4 clearly show a substantial ten-sion between the gas fraction (and, in turn, baryon fraction) ofthe XXL-100-GC clusters with our weak-lensing mass calibra-tion and the existing results based on hydrostatic masses. Thestraightforward interpretation is that the studies based on hy-drostatic masses are biased low by the presence of a significantnon-thermal pressure, which would result in an overestimatedgas fraction. The e ff ect of the mass bias on the f gas − M planeis twofold. Obviously, when increasing the total mass the aver-age gas fraction goes down by the same amount. On top of that,the increase in the total mass shifts the curve to the right, whichgiven the gradient in the f gas − M plane decreases the gas frac-tion at fixed mass further. On the other hand, the decrease of thegas fraction is mitigated by a slight increase in M gas , follow-ing from the increase in integration radius. This e ff ect is howevermild, since the slope of the gas density profile at r , MT is steep .By comparing the XXL-100-GC measurement withhydrostatic-based relations, we can work out the mean hydro-static bias and its mass dependence required to match the twocurves. To estimate the bias and its mass dependence, we tookthe measurement of Ettori (2015) as a reference point, since itincludes the largest sample of hydrostatic mass measurements,and expressed the bias as a power law,1 − b = M X M WL = − b + α log (cid:32) M WL M (cid:12) (cid:33) . (8)In Fig. 10 we show the two measurements and the bias requiredto reconcile the two measurements. By matching the two rela-tions, we find b = . + . − . and α = . + . − . .Therefore, in case the discrepancy between weak-lensingand hydrostatic-based measurements comes from a generalisedhydrostatic bias, we are observing a roughly mass-independenthydrostatic bias of 30%, which is consistent with the o ff set be-tween hydrostatic and lensing-based M − T relations in the lit-erature (see Fig. 8 of Paper IV). This result directly translatesinto significant di ff erences in the inferred gas fraction. As a fur-ther check, we recomputed the gas mass and gas fraction us-ing the M − T relations of Sun et al. (2009) and Lovisari et al.(2015) and compared the resulting f gas − M relation for XXLclusters with the results of these studies. The recovered gas frac-tion agrees well with the original measurements (see AppendixE). Therefore, we conclude that the low gas fractions reportedhere are a direct consequence of the increased halo masses ob-tained through weak lensing. Numerical simulations predict that the bias in hydrostatic massesinduced by the presence of non-thermal energy is of the or-der of ∼
15% on average (e.g. Rasia et al. 2004; Nagai et al.2007; Nelson et al. 2012). The bias could be as high as 30% incase the temperature structure in the ICM is strongly inhomoge-neous (Rasia et al. 2006), although the mean temperature of local At r , MT , the mean slope of the density profiles is d log ρ/ d log R ∼−
2; therefore M gas , ∝ r , MT ∝ M / , MT Article number, page 9 of 19 & A proofs: manuscript no. xxl_fgas ] [M M g as f Fig. 10. f gas − M relation within r , MT obtained for XXL clusters (redline) compared to Ettori (2015, blue). The green line in the inset showsthe recovered hydrostatic bias 1 − b = M X M WL required to bring the twomeasurements in accord. The dashed lines show the expected true gasfraction curves for constant values of 1 − b in the range [0.5-0.9]. clusters was found to agree with the mass-weighted temperature(Frank et al. 2013). The bias 1 − b = . + . − . recovered aboveis definitely on the high side of the expectations from numericalsimulations. On the other hand, this value is in line with a num-ber of recent weak-lensing measurements, such as Weighing theGiants (WtG, von der Linden et al. 2014b) and CCCP (Hoek-stra et al. 2015). Okabe & Smith (2015) presented weak-lensingmass measurements for the LoCuSS sample and found results ∼
10% lower than WtG, but in good agreement with CCCP andCLASH (Donahue et al. 2014). While WtG and CCCP indicatea bias of the order of 20-30% in the mass calibration adoptedby the
Planck team (Planck Collaboration XI 2011), Smith et al.(2015) showed that the LoCuSS data are consistent with no bias,depending on the method adopted to compute the sample aver-age. No consensus has thus been reached on the true value of thehydrostatic bias (see also Sereno & Ettori 2015; Applegate et al.2015). We remark that the
Planck calibration, which serves asa benchmark for the measurement of the hydrostatic bias in thestudies discussed here, is consistent with our calibration of thehydrostatic f gas − M relation (see Fig. 1 of Ettori 2015).As a word of caution, we note that systematic di ff erenceshave been reported in the literature between hydrostatic massesextracted by di ff erent groups and di ff erent instruments (Rozoet al. 2014; Donahue et al. 2014). Since our calculation relies ona benchmark hydrostatic f gas − M relation, the recovered hydro-static bias certainly depends on the adopted hydrostatic relation.The relation derived by Ettori (2015) is based on a large com-pilation of hydrostatic measurements (94 systems) obtained byseveral groups and several instruments. Therefore, it likely sum-marises our current knowledge of hydrostatic gas fraction mea-surements. Moreover, the XXL-100-GC sample is comprisedmainly of galaxy groups and poor clusters in the temperaturerange 2-5 keV, for which the systematic uncertainties associatedwith the temperature measurements are much less important thanfor massive clusters (Nevalainen et al. 2010; Schellenberger et al.2015). Nevertheless, systematic uncertainties in the adopted hy-drostatic relation cannot be excluded.Using the WtG weak-lensing mass calibration (von der Lin-den et al. 2014a), Mantz et al. (2015) derived a mean gas frac- tion of ∼ .
11 at r . This value is slightly lower than thehydrostatic-based calibrations discussed above, but higher thanwhat is measured here. Given that the WtG clusters are moremassive ( M > × M (cid:12) ) than for XXL-100-GC, the twostudies are broadly consistent, although less so than one wouldnaively expect from the similarity of the 1 − b values derived inthis work and from the direct comparison of WtG and Planck masses (von der Linden et al. 2014b).
The gas fraction of 5 .
5% for 10 M (cid:12) systems estimated here in-dicates that, once the stellar fraction is taken into account, evenat the high-mass end the baryon fraction within r , MT falls shortof the Universal baryon fraction by a factor of two. As shown inthe recent paper by Sembolini et al. (2015), which presents thecomparison between gas properties in a dozen di ff erent flavorsof non-radiative numerical simulations, such a result cannot beexplained by gravitational or hydrodynamical e ff ects. Baryonicphysics, and in particular feedback e ff ects, must therefore be in-voked to explain such a low gas fraction.We compared our results with the predictions of the cosmo-OWLS simulations (Le Brun et al. 2014; McCarthy et al. 2014),which is a large suite of hydrodynamical simulations (an exten-sion of the OverWhelmingly Large Simulations, OWLS, projectof Schaye et al. 2010) utilising the smoothed particle hydrody-namics (SPH) code GADGET-3 (Springel 2005). This set of simu-lations includes several runs with di ff erent gas physics (see Table1 of Le Brun et al. 2014, for details): i) a purely hydrodynamicrun neglecting the e ff ects of baryonic physics (hereafter NO-COOL); ii) a run including gas cooling, star formation and feed-back from supernovae (hereafter REF); iii) three runs includingAGN feedback. In the latter case, AGN feedback was modelledusing the Booth & Schaye (2009) model, where a fraction of theaccreted rest mass energy is used to increase the temperature ofneighbouring gas particles by an amount ∆ T heat . The black holesstore up accretion energy in a reservoir until it is su ffi cient toheat neighbouring gas by ∆ T heat . The three runs considered hereinclude a deposited temperature ∆ T heat of 10 K (hereafter AGN-8.0), 10 . K (AGN-8.5), and 10 . K (AGN-8.7). An increase in ∆ T heat will in general lead to more bursty (and energetic) feed-back, as more time is required between feedback events to storeup su ffi cient energy to heat the gas to a higher value of ∆ T heat .In the left-hand panel of Fig. 11 we show the comparisonin the f gas − M plane between the XXL-100-GC curve and theresults of the various cosmo-OWLS runs. As expected, the non-radiative run (NOCOOL) largely overestimates the observed gasfraction, since in this case the baryons condensate into dark-matter halos but do not form stars. The REF run is consistentwith the observations at high mass but overestimates the gas frac-tion in the group regime. However, because of the absence ofAGN feedback this run is a ff ected by the usual “cooling catas-trophe”, which results in a stellar fraction much above the ob-served one. The curves providing the best match to the data arethe AGN-8.5 and AGN-8.7 runs. In these cases, AGN drive ener-getic outflows from the progenitor groups and clusters (typicallyat z ∼ −
3, corresponding to the peak of the cosmic blackhole accretion rate density) which are not e ffi ciently recapturedlater on, resulting in groups and clusters with lower than Univer-sal baryon fractions within r (see McCarthy et al. 2011). Wenote, however, that these models predict a steeper trend of in-creasing f gas with halo mass compared to the data, which couldbe reproduced by invoking a mass dependence of the depositedheat ∆ T heat , although the di ff erent selection procedure for ob- Article number, page 10 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample ] [M M g as f XXL-100-GCNOCOOLREFAGN-8.0AGN-8.5AGN-8.7 [keV]
300 kpc T ] [ M g as , k p c M XXL-100-GCNOCOOLREFAGN-8.0AGN-8.5AGN-8.7
Fig. 11.
Left panel:
Gas fraction of XXL-100-GC galaxy clusters (dashed red curve and red shaded area) compared to cosmo-OWLS simulationswith di ff erent gas physics (non-radiative, NOCOOL; cooling and star formation, REF; AGN feedback with various energy injection, AGN-8.0,AGN-8.5, AGN-8.7). The grey shaded area shows the WMAP9 cosmic baryon fraction. Right panel:
Gas mass within 500 kpc as a function ofthe temperature inside 300 kpc for the XXL-100-GC sample (black points) compared to cosmo-OWLS simulations (same colour code). The reddashed curve and shaded area show the best fit to the data with a power law and its error envelope. served and simulated halos may play a role here. Therefore, ifthe results obtained in XXL-100-GC using weak-lensing calibra-tion are to be trusted, our measurements favour the most extremeAGN feedback schemes tested in this work.Interestingly, the AGN-8.0 model systematically overesti-mates the measured gas fraction. This run was found to providethe best match to hydrostatic-based gas fraction measurementsand to X-ray-only proxies, such as the gas density and entropyprofiles (see Le Brun et al. 2014). Conversely, in Le Brun et al.(2014) the AGN-8.5 and (particularly) AGN-8.7 models werestrongly disfavoured by X-ray-only proxies, in particular the gasentropy. Indeed, a strong AGN feedback leads to a substantialentropy injection in the surrounding ICM (Gaspari 2015), in ex-cess of what is observed in local galaxy groups (e.g. Ponmanet al. 1999; Sun et al. 2009). As a further test, we made the com-parison between the M gas − T relation measured here and theresults of the runs including various gas physics. In the right-hand panel of Fig. 11 we show the gas mass measured within afixed physical radius of 500 kpc as a function of the tempera-ture inside a 300 kpc radius. The XXL-100-GC data points arecompared to the simulation results obtained in the same physicalregions. In agreement with Le Brun et al. (2014), we find thatthe AGN-8.0 run closely reproduces the observed M gas − T data,confirming that X-ray measurements prefer a mild entropy injec-tion by AGN feedback. Conversely, the AGN-8.5 and AGN-8.7schemes provide an excessive heating to the gas, which leads toan overestimate of the temperature at fixed gas mass comparedto real systems. We emphasize that because of the use of fixedphysical apertures this result is independent of any scaling rela-tion.To summarise, we conclude that AGN feedback alone as isimplemented in current numerical simulations cannot reproduceconsistently the lensing-based f gas − M relation and the thermo-dynamical properties of the ICM, which challenges our under-standing of the ICM. This tension would be mitigated in case theweak-lensing mass calibration used here is biased high. Recently, the final results from the
Planck mission have been re-leased, providing benchmark cosmological constraints throughthe CMB power spectrum (Planck Collaboration XIII 2015) andthe number counts of SZ sources (Planck Collaboration XXIV2015). The
Planck team noted a tension between the results ob-tained with the two techniques, cluster counts preferring system-atically lower values of σ and Ω m than expected from the CMBpower spectrum. To alleviate this issue, two possible interpre-tations were put forward. The two measurements could be rec-onciled for a neutrino mass in the range (cid:80) m ν ∼ . − b = . ± .
04) inthe mass calibration used by the
Planck team for the analysis ofthe SZ cluster counts, which is based on hydrostatic masses.The results presented here clearly highlight the tension thata strong hydrostatic bias implies on the baryon fraction (seeFig. 10). Since our reference hydrostatic masses agree with the
Planck mass calibration (see Sect. 5.3), a mass bias in the range1 − b = .
58 would translate into an even lower baryon fractionthan is measured here. Such a bias would imply f gas , ∼ . M (cid:12) halos, which would pose a considerable challengefor our understanding of cluster physics and evolution. Even themost extreme feedback schemes adopted in numerical simula-tions would fail to reproduce such characteristics (see Sect. 5.4),even though such models have trouble reproducing the globalproperties of galaxy clusters. Therefore, the results presentedhere argue against the strong hydrostatic bias implied by Planck primary CMB. A satisfactory solution to the
Planck
CMB / SZdiscrepancy must therefore be able to explain at the same timethe baryon fraction of high-mass halos.
6. Conclusion
In this paper, we have presented a study of the baryon budgetof dark-matter halos in the mass range 10 − M (cid:12) using the Article number, page 11 of 19 & A proofs: manuscript no. xxl_fgas sample of the 100 brightest clusters discovered in the XXL Sur-vey. Our main results can be summarised as follows: – We developed a method to measure the gas mass fromXXL Survey data (Sect. 2.4). The method was calibratedusing mock X-ray images drawn from cosmological simu-lations and was found to reproduce the true gas mass withexcellent accuracy (see Fig. 2). The gas mass measure-ments for all the clusters in the XXL-100-GC sample areprovided in Table A.1. We also provide measurements of Y X , = T × M gas , , which were found to agree wellwith the stacked Planck
SZ flux. – The scaling relation between the gas mass within r , MT and the spectroscopic X-ray temperature is very tight andsignificantly steeper than the self-similar prediction (seeFig. 3), in excellent agreement with previous measurements(Arnaud et al. 2007). We recovered the relation betweenhot gas fraction and halo mass within r , MT by combiningour M gas − T relation with the M WL − T relation basedon an internal mass calibration using weak gravitationallensing (see Paper IV). The recovered relation reads h − / f gas , = . + . − . (cid:16) M , MT / h − M (cid:12) (cid:17) . + . − . . – We measured a mass-independent stellar fraction of ∼ f bar , = . ± .
008 in 10 M (cid:12) halos). – Comparing our best-fit f gas − M relation with results fromthe literature based on hydrostatic masses, we found that ourweak-lensing based gas fraction is significantly lower thanprevious hydrostatic measurements (Fig. 4). The tensionis particularly important around 10 M (cid:12) , where the bulkof XXL-100-GC systems lies. The two methods can bereconciled by considering a roughly mass-independenthydrostatic bias 1 − b = M X / M WL = . + . − . . This valueis on the high side of the expectations of numerical simu-lations (Rasia et al. 2004; Nagai et al. 2007); however, it isconsistent with recent studies such as WtG (von der Lindenet al. 2014b) and CCCP (Hoekstra et al. 2015). Therefore,the low gas fraction observed here directly follows from thehigher masses obtained through weak lensing. – We compared our f gas − M relation with the predictionsof cosmological simulations using di ff erent gas physics(cosmo-OWLS, Le Brun et al. 2014). We found that ourresults favour extreme AGN feedback schemes in which alarge fraction of the baryons is expelled from the potentialwell of dark matter halos. Such models are, however, intension with X-ray-only proxies such as the gas density andentropy profiles (Le Brun et al. 2014) and are not able toreproduce the relation between gas mass and temperatureof XXL clusters. Therefore, the results presented hereare challenging for current numerical simulations, andreconciling the observed gas fraction with the predictionswould require that weak-lensing masses be systematicallyoverestimated. – A mass bias 1 − b = . ± .
04, which is required to rec-oncile
Planck cluster counts with primary CMB, would fur-ther exacerbate the tension between f bar and the cosmologi- cal value, which would challenge our understanding of clus-ter physics. Therefore, a satisfactory solution to the tensionbetween CMB and cluster counts must also simultaneouslyexplain the low baryon fraction measured here for massivehalos. Acknowledgements.
XXL is an international project based on an XMM VeryLarge Program surveying two 25 deg extragalactic fields at a depth of ∼ × − ergs s − cm − in the [0.5-2] keV band. The XXL website is http://irfu.cea.fr/xxl . Multi-band information and spectroscopic follow-up of the X-ray sources are obtained through a number of survey programmessummarised at http://xxlmultiwave.pbworks.com/ . DE acknowledgessupport from the Swiss National Research Foundation. AMCLB acknowledgessupport from an internally funded PhD studentship at the Astrophysics ResearchInstitute of Liverpool John Moores University and from the French Agence Na-tionale de la Recherche under grant ANR-11-BD56-015. FP acknowledges sup-port from the BMBF / DLR grant 50 OR 1117, the DFG grant RE 1462-6 and theDFG Transregio Programme TR33.
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Article number, page 13 of 19 & A proofs: manuscript no. xxl_fgas
Table A.1.
Basic data
Cluster z T r , MT M gas , Y X , f gas , [keV] [kpc] [ M (cid:12) ] [keV M (cid:12) ]XLSSC 001 (cid:63) . + . − . ±
118 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 003 0.836 3 . + . − . ±
106 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 006 (cid:63) . + . − . ±
145 (4 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 010 0.330 2 . + . − . ±
113 (7 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 011 (cid:63) . + . − . ±
136 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 022 (cid:63) . + . − . ±
94 (6 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 023 0.328 2 . + . − . ±
96 (5 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 025 (cid:63) . + . − . ±
110 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 027 (cid:63) . + . − . ±
116 (8 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 029 1.050 4 . + . − . ±
98 (2 . + . − . ) × (9 . + . − . ) × . + . − . XLSSC 036 0.492 3 . + . − . ±
120 (2 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 041 (cid:63) . + . − . ±
98 (4 . + . − . ) × (8 . + . − . ) × . + . − . XLSSC 050 0.140 3 . + . − . ±
127 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 052 0.056 0 . + . − . ±
56 (6 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 054 (cid:63) . + . − . ±
105 (4 . + . − . ) × (8 . + . − . ) × . + . − . XLSSC 055 (cid:63) . + . − . ±
128 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 056 (cid:63) . + . − . ±
124 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 057 (cid:63) . + . − . ±
105 (7 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 060 (cid:63) . + . − . ±
158 (3 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 061 (cid:63) . + . − . ±
106 (7 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 062 0.059 0 . + . − . ±
61 (3 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 072 1.002 3 . + . − . ±
99 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 083 (cid:63) . + . − . ±
154 (3 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 084 0.430 4 . + . − . ±
202 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 085 0.428 4 . + . − . ±
176 (1 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 087 (cid:63) . + . − . ±
87 (2 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 088 (cid:63) . + . − . ±
120 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 089 (cid:63) . + . − . ±
171 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 090 (cid:63) . + . − . ±
72 (1 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 091 (cid:63) . + . − . ±
161 (5 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 092 (cid:63) . + . − . ±
138 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 093 (cid:63) . + . − . ±
123 (2 . + . − . ) × (8 . + . − . ) × . + . − . XLSSC 094 0.886 4 . + . − . ±
129 (1 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 095 0.138 0 . + . − . ±
64 (3 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 096 0.520 5 . + . − . ±
180 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 097 (cid:63) . + . − . ±
143 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 098 (cid:63) . + . − . ±
139 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 099 (cid:63) . + . − . ±
221 (4 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 100 0.915 4 . + . − . ±
143 (1 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 101 0.756 4 . + . − . ±
134 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 102 (cid:63) . + . − . ±
96 (2 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 103 (cid:63) . + . − . ±
172 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 104 (cid:63) . + . − . ±
188 (8 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 105 0.429 5 . + . − . ±
165 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 106 (cid:63) . + . − . ±
125 (2 . + . − . ) × (7 . + . − . ) × . + . − . Article number, page 14 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample
Table A.1. continued.
Cluster z T r , MT M gas , Y X , f gas , [keV] [kpc] [ M (cid:12) ] [keV M (cid:12) ]XLSSC 107 (cid:63) . + . − . ±
111 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 108 (cid:63) . + . − . ±
101 (5 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 109 (cid:63) . + . − . ±
151 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 110 0.445 1 . + . − . ±
76 (8 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 111 0.299 4 . + . − . ±
152 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 112 0.139 1 . + . − . ±
95 (4 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 113 (cid:63) . + . − . ±
78 (1 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 114 (cid:63) . + . − . ±
288 (1 . + . − . ) × (9 . + . − . ) × . + . − . XLSSC 115 (cid:63) . + . − . ±
113 (5 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 501 0.333 2 . + . − . ±
121 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 502 0.141 1 . + . − . ±
75 (3 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 503 0.336 2 . + . − . ±
97 (1 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 504 0.243 13 . + . − . ±
503 (4 . + . − . ) × (8 . + . − . ) × . + . − . XLSSC 505 0.055 1 . + . − . ±
92 (1 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 506 0.717 4 . + . − . ±
188 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 507 0.566 2 . + . − . ±
106 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 508 0.539 3 . + . − . ±
123 (2 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 509 0.633 4 . + . − . ±
143 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 510 0.395 2 . + . − . ±
106 (6 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 511 0.130 1 . + . − . ±
77 (3 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 512 0.402 3 . + . − . ±
128 (1 . + . − . ) × (6 . + . − . ) × . + . − . XLSSC 513 0.378 4 . + . − . ±
144 (3 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 514 0.169 1 . + . − . ±
86 (4 . + . − . ) × (6 . + . − . ) × . + . − . XLSSC 515 0.101 1 . + . − . ±
77 (2 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 516 0.866 4 . + . − . ±
114 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 517 0.699 3 . + . − . ±
118 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 518 0.177 1 . + . − . ±
75 (3 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 519 0.270 1 . + . − . ±
86 (3 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 520 0.175 2 . + . − . ±
113 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 521 0.807 4 . + . − . ±
131 (3 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 522 0.395 2 . + . − . ±
108 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 523 0.343 2 . + . − . ±
121 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 524 0.270 2 . + . − . ±
121 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 525 0.379 3 . + . − . ±
120 (2 . + . − . ) × (7 . + . − . ) × . + . − . XLSSC 526 0.273 2 . + . − . ±
115 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 527 0.076 3 . + . − . ±
211 (7 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 528 0.302 3 . + . − . ±
132 (1 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 529 0.547 3 . + . − . ±
119 (1 . + . − . ) × (6 . + . − . ) × . + . − . XLSSC 530 0.182 2 . + . − . ±
100 (5 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 531 0.391 4 . + . − . ±
214 (8 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 532 0.392 3 . + . − . ±
126 (9 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 533 0.107 2 . + . − . ±
111 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 534 0.853 4 . + . − . ±
138 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 535 0.172 2 . + . − . ±
112 (1 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 536 0.170 1 . + . − . ±
97 (4 . + . − . ) × (9 . + . − . ) × . + . − . Article number, page 15 of 19 & A proofs: manuscript no. xxl_fgas
Table A.1. continued.
Cluster z T r , MT M gas , Y X , f gas , [keV] [kpc] [ M (cid:12) ] [keV M (cid:12) ]XLSSC 537 0.515 4 . + . − . ±
159 (2 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 538 0.332 3 . + . − . ±
140 (6 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 539 0.184 1 . + . − . ±
89 (1 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 540 0.414 3 . + . − . ±
118 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 541 0.187 2 . + . − . ±
121 (8 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 542 0.402 6 . + . − . ±
170 (8 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 543 0.381 2 . + . − . ±
109 (1 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 544 0.095 2 . + . − . ±
114 (6 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 545 0.353 2 . + . − . ±
138 (5 . + . − . ) × (1 . + . − . ) × . + . − . XLSSC 546 0.792 3 . + . − . ±
110 (1 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 547 0.371 4 . + . − . ±
165 (9 . + . − . ) × (4 . + . − . ) × . + . − . XLSSC 548 0.321 1 . + . − . ±
62 (3 . + . − . ) × (3 . + . − . ) × . + . − . XLSSC 549 0.808 4 . + . − . ±
135 (1 . + . − . ) × (5 . + . − . ) × . + . − . XLSSC 550 0.109 1 . + . − . ±
71 (2 . + . − . ) × (2 . + . − . ) × . + . − . XLSSC 551 0.475 2 . + . − . ±
117 (8 . + . − . ) × (2 . + . − . ) × . + . − . Column description:
1. Cluster name. The clusters marked with an asterisk were used for the stellar fraction analysis (see Sect. 3);2. Cluster redshift (Paper II); 3. X-ray temperature within 300 kpc aperture (Paper III); 4. r , MT estimated using the XXL M WL − T relation (Paper IV); 5. Gas mass within r , MT (this work); 6. Integrated Compton parameter Y X , within r , MT (this work); 7.Hot gas fraction within r , MT (this work). Article number, page 16 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample
Appendix B: Multiscale forward-fitting deprojection
If the geometry of the emitting region is assumed to be known,the intrinsic emissivity profiles can in principle be recoveredfrom the observed projected emission-measure profiles by in-verting the projection kernel (e.g. Kriss et al. 1983). In practice,the problem is rendered complicated by the presence of noisein the original data. As for all inverse problems, the projectionkernel smoothes small-scale fluctuations, thus the inverse trans-formation has the opposite e ff ect and the noise can be greatlyamplified (see Lucy 1974, 1994). This e ff ect is particularly im-portant in the low signal-to-noise regime.Two main approaches usually exist to solve this problem: di-rect geometrical deprojection (e.g. Fabian et al. 1981; Kriss et al.1983; Morandi et al. 2007) and forward fitting using a paramet-ric form (e.g. Cavaliere & Fusco-Femiano 1976; Sarazin & Bah-call 1977; Pointecouteau et al. 2004; Mahdavi et al. 2007). Theadvantage of the former is that it makes no assumption on theshape of the intrinsic profile, thus it is in principle the most gen-eral. However, this method is very sensitive to measurement un-certainties, since small variations in the projected profile can begreatly magnified; therefore, the resulting profile is generally notsmooth. Moreover, error bars obtained through this techniqueare not statistically meaningful, given that the result dependson the adopted binning. Conversely, the latter can provide rig-orously determined error bars, but requires strong assumptionson the shape of the density profile since the chosen functionalform (usually an extended version of a beta model) must repro-duce the observed shape accurately. For a review of the variousexisting deprojection methods, we refer to Buote & Humphrey(2012).In this work, we propose a di ff erent approach to combine therigorous results provided by forward fitting with minimal pri-ors on the shape of the density profile. We suggest decomposingthe density profile into a sum of analytical multiscale functionswhich can be independently deprojected since the projection ker-nel is linear. Namely, the projected emission-measure profileswere fit with a sum of N King functions, EM ( s ) = N (cid:88) i = A i + (cid:32) sr c , i (cid:33) − β i + . , (B.1)where the core radii r c , i are fixed adaptively to reproduce theshape of the profile on a given scale and s is the projected cluster-centric distance. The normalisations and slopes are left free tovary while fitting. As is known from the beta model (Cavaliere& Fusco-Femiano 1976), these functional forms have the goodproperty that the corresponding deprojected function is analyt-ical. Given that the projection kernel is linear, this property ispreserved for any linear combination of King functions, i.e. n ( r ) = N (cid:88) i = n , i + (cid:32) rr c , i (cid:33) − β i . (B.2)Here, n , i is proportional to A i . Thus, the deprojected gas densitycan be easily obtained from the fitted parameters. In this partic-ular case, the normalisations of the deprojected profile n , i canbe evaluated directly from the projected normalisations (see e.g.Appendix A of Hudson et al. 2010), n , i = Γ (3 β ) Γ (3 β − . √ π r c , i A i , (B.3) Radius -2 -1
10 1 10 S u r f ace B r i gh t n ess -3 -2 -1 Fig. B.1.
Example of application of the multiscale forward-fitting tech-nique to simulated data (Eq. B.1). The red curve shows the total model,while the dashed lines represent the contribution of each individualcomponent. The units on both axes are arbitrary. where r c , i is expressed in physical units.Naturally, this method can be generalised to any type of basefunctions, given the problem of interest. In the particular case ofgalaxy cluster emissivity profiles, the choice of King functionsis easily justified.In Fig. B.1 we show a test of this method on simulated data.Specifically, we simulated an XMM-Newton pointing with anextended source with a random radius-dependent slope and ex-tracted the surface-brightness profile from the image. The sim-ulated surface-brightness profile was then fit with Eq. B.1 (redcurve). The dashed curves indicate the contribution of each com-ponent to the model.We applied the technique described here by including oneKing component for each set of four data points. The total modelthus has N / N is the number of data pointsin the emission-measure profile. The core radius of each compo-nent was fixed to the median radius of the corresponding block ofdata points. This choice allows a very large freedom to the fittedfunction and is su ffi cient to fit adequately any surface-brightnessprofile, provided that the following priors are verified: i) the gasdensity is smooth and decreases monotonically with radius and ii) the gas density is always ≥
0. The former condition should bevalid if the point sources and neighbouring clusters are properlyexcised, while the latter is always valid as long as the subtractionof the background is correct. Therefore, this approach combinesthe rigorous approach of forward fitting with weak priors on thecluster density profile.
Appendix C: Modelling the
XMM-Newton
PSF
Since high-redshift clusters are only slightly extended for
XMM-Newton , an accurate modelling of the instrument’s point-spreadfunction (PSF) is required to give a realistic estimate of thegas density profile, and hence gas mass. The observed image isgiven by the 2D convolution of the original image with the in-strumental PSF. Performing such a convolution directly is time-consuming since it requires computing a triple integral on the
Article number, page 17 of 19 & A proofs: manuscript no. xxl_fgas j i Fig. C.1.
PSF convolution matrix P computed from ray tracing (seetext). The value P i , j gives the probability that a photon originating frombin j is detected in bin i . fly. Thus, fitting a surface-brightness model accounting for PSFconvolution is lengthy and numerically di ffi cult.To alleviate this problem, we take advantage of the finitebinning of the surface-brightness profile to turn the continuousproblem into a discrete one. For a given radial binning { r i } Ni = weassume that in each bin the surface brightness is approximatelyconstant, which is a reasonable approximation for a bin width of8 arcsec. Then we define a convolution matrix P such that P i , j = Prob( j → i ) , (C.1)i.e. P i , j is the probability that a photon originating from bin j isdetected in bin i . The matrix is normalised such that (cid:80) Ni = P i , j = , ∀ j . Then from the model EM ( r i ) (see Eq. B.1), the convolvedmodel ˜ EM ( r i ) can be written as˜ EM i = N (cid:88) j = P i , j EM ( r j ) . (C.2)The convolved model ˜ EM is then fit to the observed surface-brightness profile.To compute the convolution matrix P , we adopt a ray trac-ing approach. Namely, for each radial bin, we simulate 10 ray-tracing photons, taking vignetting e ff ects and CCD gaps intoaccounts, and randomise the position of each simulated photonaccording to the properties of the XMM-Newton
PSF. For eacho ff -axis angle, we model the PSF as a King profile with the pa-rameters given in the latest XMM-Newton calibration files. Wethen compute the fraction of photons falling into each radial bin.An example of convolution matrix is shown in Fig. C.1.To validate this method, we applied it to the NW cold front ofA2142 (see Rossetti et al. 2013) and compared our results withthe values obtained from higher resolution
Chandra data (Ow-ers et al. 2009). Without PSF convolution, the broken power-lawmodel gives a poor fit to the data ( χ = . P null = . n in / n out = . ± .
03, which is inconsistent with the
Chan-dra value (2 . ± . χ = . /
55 d.o.f., P null = .
66) and a density jump n in / n out = . ± .
04, inagreement with the
Chandra measurement. This demonstratesthe ability of our method to take the
XMM-Newton
PSF into ac-count. [kpc] r
200 400 600 800 1000 1200 1400 1600 1800 [ k p c ] , M T r XXL-100-GCCOSMOSCCCP
Fig. D.1.
Comparison between the values of r obtained by using the M − T relation of Paper IV and the values measured through directly us-ing weak lensing for the XXL-100-GC (black), CCCP (red), and COS-MOS (blue) samples. The dashed black line represents unity. Appendix D: Comparing the M − T relation with theweak lensing data Since weak-lensing masses are available only for a fraction ofXXL-100-GC systems, our analysis rests on the assumption thaton average the masses and values of r , MT for our systems areconsistent with the values obtained through weak lensing. Totest this assumption, we used the weak-lensing data from XXL-100-GC, CCCP, and COSMOS used in Paper IV and appliedour M − T relation to the same systems. In the XXL-100-GCcase, only the clusters for which a significant mass measurementcould be obtained are considered here. In Fig. D.1 we show thecomparison between the values of r obtained from the M − T relation and the values measured directly through weak lensing.The two sets of values are consistent, leading to a mean ratio r , MT / r , WL = . ± . ff erence between the values of r , MT and the ones expectedfrom weak lensing. Appendix E: Gas fraction of the XXL sample usinghydrostatic M − T relations As a further check of the quality of the X-ray analysis for theXXL-100-GC sample, we calculated the cluster mass and r , MT using the hydrostatic-based mass-temperature relations of Sunet al. (2009, Tier1 + + clusters) and Lovisari et al. (2015). Inboth cases, we recomputed the gas mass within the correspond-ing aperture and calculated the gas fraction by fitting again the M gas − T relation using the gas masses measured within the mod-ified aperture and combining it with the adopted M − T rela-tion (see Sect. 4.2). In Fig. E.1 we show the resulting curves inthe f gas − M plane compared to the reference hydrostatic mea-surements. The results obtained for the XXL-100-GC sampleagree well with the literature measurements. Above 10 M (cid:12) , thegas fraction of XXL-100-GC clusters estimated using the Sunet al. (2009) relation slightly exceeds that measured by Sun et al.(2009). Di ff erences of this order are to be expected at the high-mass end, however, given that our sample only contains a smallnumber of systems beyond ∼ × M (cid:12) . Article number, page 18 of 19. Eckert et al.: The XXL Survey XIII. Baryon content of the bright cluster sample ] [M M g as f XXL-100, S09 M-T relationXXL-100, L15 M-T relationEttori15Sun+09Lovisari+15
Fig. E.1.