Mass profiles and concentration-dark matter relation in X-ray luminous galaxy clusters
S. Ettori, F. Gastaldello, A. Leccardi, S. Molendi, M. Rossetti, D. Buote, M. Meneghetti
aa r X i v : . [ a s t r o - ph . C O ] A ug Astronomy&Astrophysicsmanuscript no. cm c (cid:13)
ESO 2018October 25, 2018
Mass profiles and c − M DM relationin X-ray luminous galaxy clusters S. Ettori , , F. Gastaldello , , , A. Leccardi , , S. Molendi , M. Rossetti , D. Buote , and M. Meneghetti , INAF, Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy INAF, IASF, via Bassini 15, I-20133 Milano, Italy Universit`a degli Studi di Milano, Dip. di Fisica, via Celoria 16, I-20133 Milano, Italy Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 Occhialini FellowReceived 24 June 2010 / Accepted 14 September 2010
ABSTRACT
Context.
Galaxy clusters represent valuable cosmological probes using tests that mainly rely on measurements of cluster masses andbaryon fractions. X-ray observations represent one of the main tools for uncovering these quantities.
Aims.
We aim to constrain the cosmological parameters Ω m and σ using the observed distribution of the both values of the concen-trations and dark mass within R and of the gas mass fraction within R . Methods.
We applied two different techniques to recover the profiles the gas and dark mass, described according to the Navarro,Frenk & White (1997, ApJ, 490, 493) functional form, of a sample of 44 X-ray luminous galaxy clusters observed with
XMM-Newton in the redshift range . − . . We made use of the spatially resolved spectroscopic data and of the PSF–deconvolved surface bright-ness and assumed that hydrostatic equilibrium holds between the intracluster medium and the gravitational potential. We evaluatedseveral systematic uncertainties that affect our reconstruction of the X-ray masses. Results.
We measured the concentration c , the dark mass M and the gas mass fraction in all the objects of our sample, pro-viding the largest dataset of mass parameters for galaxy clusters in the redshift range . − . . We confirm that a tight correlationbetween c and M is present and in good agreement with the predictions from numerical simulations and previous observations.When we consider a subsample of relaxed clusters that host a low entropy core, we measure a flatter c − M relation with a totalscatter that is lower by 40 per cent. We conclude, however, that the slope of the c − M relation cannot be reliably determined from thefitting over a narrow mass range as the one considered in the present work. From the distribution of the estimates of c and M ,with associated statistical (15–25%) and systematic (5–15%) errors, we used the predicted values from semi-analytic prescriptionscalibrated through N-body numerical runs and obtain σ Ω . ± . = 0 . ± . (at σ level, statistical only) for the subsampleof the clusters where the mass reconstruction has been obtained more robustly and σ Ω . ± . = 0 . ± . for the subsampleof the 11 more relaxed LEC objects. With the further constraint from the gas mass fraction distribution in our sample, we break thedegeneracy in the σ − Ω m plane and obtain the best-fit values σ ≈ . ± . ( . ± . when the subsample of the more relaxedobjects is considered) and Ω m = 0 . ± . . Conclusions.
We demonstrate that the analysis of the distribution of the c − M − f gas values represents a mature and compet-itive technique in the present era of precision cosmology, even though it needs more detailed analysis of the output of larger sets ofcosmological numerical simulations to provide definitive and robust results. Key words. galaxies: cluster: general – intergalactic medium – X-ray: galaxies – cosmology: observations – dark matter.
1. Introduction
The distribution of the total and baryonic mass in galaxy clus-ters is a fundamental ingredient to validate the scenario of struc-ture formation in a Cold Dark Matter (CDM) Universe. Withinthis scenario, the massive virialized objects are powerful cosmo-logical tools able to constrain the fundamental parameters of agiven CDM model. The N − body simulations of structure for-mation in CDM models indicate that dark matter halos aggre-gate with a typical mass density profile characterized by only2 parameters, the concentration c and the scale radius r s (e.g.Navarro et al. 1997, hereafter NFW). The product of these twoquantities fixes the radius within which the mean cluster den-sity is 200 times the critical value at the cluster’s redshift [i.e. Send offprint requests to : S. Ettori
Correspondence to : [email protected] R = c × r s and the cluster’s volume V = 4 / πR isequal to M / (200 ρ c,z ) , where M is the cluster gravitatingmass within R ]. With this prescription, the structural proper-ties of DM halos from galaxies to galaxy clusters are dependenton the halo mass, with systems at higher masses less concen-trated. Moreover, the concentration depends upon the assemblyredshift (e.g. Bullock et al. 2001, Wechsler et al. 2002, Zhao etal. 2003, Li et al. 2007), which happens to be later in cosmolo-gies with lower matter density, Ω m , and lower normalization ofthe linear power spectrum on scale of h − Mpc, σ , implyingless concentrated DM halos of given mass. The concentration –mass relation, and its evolution in redshift, is therefore a strongprediction obtained from CDM simulations of structure forma-tion and is quite sensitive to the assumed cosmological parame-ters (NFW; Bullock et al. 2001; Eke, Navarro & Steinmetz 2001;Dolag et al. 2004; Neto et al. 2007; Macci`o et al. 2008). In thiscontext, NFW, Bullock et al. 2001 (with revision after Macci`o et
1. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters al. 2008) and Eke et al. 2001 have provided simple and powerfulmodels that match the predictions from numerical simulationsand allow comparison with the observational measurements.Recent X-ray studies (Pointecouteau, Arnaud & Pratt 2005;Vikhlinin et al. 2006; Voigt & Fabian 2006; Zhang et al. 2006;Buote et al. 2007) have shown good agreement between observa-tional constraints at low redshift and theoretical expectations. Byfitting 39 systems in the mass range between early-type galaxiesup to massive galaxy clusters, Buote et al. (2007) confirm withhigh significance that the concentration decreases with increas-ing mass, as predicted from CDM models, and require a σ , thedispersion of the mass fluctuation within spheres of comovingradius of 8 h − Mpc, in the range . − . (99% confidence)definitely in contrast to the lower constraints obtained, for in-stance, from the analysis of the WMAP
Chandra inthe redshift range . − . , Schmidt & Allen (2007) high-light a possible tension between the observational constraintsand the numerical predictions, in the sense that either the relationis steeper than previously expected or some redshift evolutionhas to be considered. Comerford & Natarajan (2007) compiled alarge dataset of observed cluster concentration and masses, find-ing a normalization higher by at least 20 per cent than the re-sults from simulations. In the sample, they use also strong lens-ing measurements of the concentration concluding that these aresystematically larger than the ones estimated in the X-ray band,and 55 per cent higher, on average, than the rest of the clusterpopulation. Recently, Wojtak & Łokas (2010) analyze kinematicdata of 41 nearby ( z < . ) relaxed objects and find a normal-ization of the concentration – mass relation fully consistent withthe amplitude of the power spectrum σ estimated from WMAP1data and within σ from the constraint obtained from WMAP5.In this work, we use the results of the spectral analysis pre-sented in Leccardi & Molendi (2008) for a sample of 44 X-rayluminous galaxy clusters located in the redshift range . − . with the aim to (1) recover their total and gas mass profiles, (2)constraining the cosmological parameters σ and Ω m throughthe analysis of the measured distribution of c , M and bary-onic mass fraction in the mass range above M ⊙ . We notethat this is the statistically largest sample for which this studyhas been carried on up to now between z = 0 . and z = 0 . .The outline of our work is the following. In Section 2, we de-scribe the dataset of XMM-Newton observations used in our anal-ysis to recover the gas and total mass profiles with the techniquespresented in Section 3. In Section 4, we present a detailed dis-cussion of the main systematic uncertainties that affect our mea-surements. We investigate the c − M relation in Section 5.By using our measurements of c and M , we constrain thecosmological parameters σ and Ω m , breaking the degeneracybetween these parameters by adding the further cosmologicalconstraints from our estimates of the cluster baryon fraction, asdiscussed in Section 6. We summarize our results and draw theconclusion of the present study in Section 7. Throughout thiswork, if not otherwise stated, we plot and tabulate values esti-mated by assuming a Hubble constant H = 70 h − km s − Mpc − and Ω m = 1 − Ω Λ = 0 . , and quote errors at the 68.3per cent ( σ ) level of confidence.We list here in alphabetic order, with the adopted acronyms,the work to which we will refer more often in the present study:Bullock et al. (2001 – B01); Dolag et al. (2004 – D04); Eke,Navarro & Steinmetz (2001 – E01); Leccardi & Molendi (2008 – LM08); Macci`o et al. (2008 – M08); Navarro, Frenk & White(1997 – NFW); Neto et al. (2007 – N07).
2. The dataset
Leccardi & Molendi (2008) have retrieved from the
XMM-Newton archive all observations of clusters available at the end ofMay 2007 (and performed before March 2005, when the CCD6of EPIC-MOS1 was switched off) and satisfying the selectioncriteria to be hot ( kT > . keV), at intermediate redshift( . < z < . ), and at high galactic latitude ( | b | > o ). Upperand lower limits to the redshift range are determined, respec-tively, by the cosmological dimming effect and the size of theEPIC field of view ( ′ radius). Out of 86 observations, 23 wereexcluded because they are highly affected by soft proton flares(see Table 1 in LM08) and have cleaned exposure time less than16 ks when summing MOS1 and MOS2. Furthermore, 15 ob-servations were excluded because they show evidence of recentand strong interactions (see Table 2 in LM08). The spectral anal-ysis of the remaining 48 exposures, for a total of 44 clusters, ispresented in LM08 and summarized in the next subsection. InTable 1, we present the list of the clusters analyzed in the presentwork. We use gas temperature profiles measured by LM08. A detaileddescription of how the profiles were obtained and tested againstsystematic uncertainties can be found in their paper. Here webriefly review some of the most important points. Unlike mosttemperature estimates the one reported in LM08 have been se-cured by performing background modelling rather than back-ground subtraction. Great care and considerable effort has goneinto building an accurate model of the EPIC background, both interms of its instrumental and cosmic components. Unfortunatelythe impossibility of performing an adequate monitoring of the pn instrumental background during source observation resultedin the exclusion of this detector from the analysis. Therefore,we adopt the measurements obtained from the two MOS instru-ments (M1 and M2, hereafter) independently in the followinganalysis.The impact of small errors in the background estimates ontemperature and normalization estimates was tested both by per-forming Monte-Carlo simulations (a-priori tests) and by check-ing how results varied for different choices of key parameters(a-posteriori tests). The detailed analysis allowed to track sys-tematic errors and provide an error budget including both statis-tical and systematic uncertainties.The two profiles have been analyzed both independently andafter they were combined as described below. M1 and M2 arecross-calibrated to about 5% (Mateos et al 2009). The largest dis-crepancy appears to be in the high energy range (above 4.5 keV),leading to a general tendency where M2 returns slightly softerspectra than M1. Since a similar comparison between M2 and pnshows that the latter returns even softer spectra, the M2 experi-ment may be viewed as returning spectra which are intermediatebetween M1 and pn in the . − keV band. As consequenceof that, a systematic shift between the M1 and M2 temperatureprofiles is present, meaning that an higher measurements is ob-tained with M1. This shift is not very sensitive to the value of thetemperature, but instead manifests itself as a difference betweenM1 and M2 in the shape of the radial temperature profile. Usingas reference the value of gas temperature measured with M2, weestimate the median deviation in the different radial bins to be
2. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 1.
Sample of the galaxy clusters.
Cluster Other name z Core Cl. Entropy Cl. X-ray refs.RXCJ0003.8+0203 Abell2700 0.092 ICC MEC Pr07, Cr08Abell3911 – 0.097 NCC HEC Sn08Abell3827 – 0.098 ICC MEC Sn08RXCJ0049.4-2931 AbellS0084 0.108 ICC MEC Si09Abell2034 – 0.113 NCC HEC Ke03, Ba07RXCJ1516.5-0056 Abell2051 0.115 NCC HEC Pr07, Cr08RXCJ2149.1-3041 Abell3814 0.118 CC LEC Cr08, Le08RXCJ1516.3+0005 Abell2050 0.118 NCC HEC Pr07, Cr08RXCJ1141.4-1216 Abell1348 0.119 CC LEC Pr07, Cr08RXCJ1044.5-0704 Abell1084 0.132 CC LEC Pr07, Cr08Abell1068 RXCJ1040.7+3956 0.138 CC LEC Wi04, Sn08RXCJ2218.6-3853 Abell3856 0.138 NCC MEC Pr07, Cr08RXCJ0605.8-3518 Abell3378 0.141 CC LEC Pr07, Cr08, Sn08RXCJ0020.7-2542 Abell22 0.142 NCC HEC Pr07, Cr08Abell1413 RXCJ1155.3+2324 0.143 ICC MEC Vi05, Ba07, Sn08, Ca09RXCJ2048.1-1750 Abell2328 0.147 NCC HEC Pr07, Cr08RXCJ0547.6-3152 Abell3364 0.148 NCC HEC Pr07, Cr08Abell2204 RXC J1632.7+0534 0.152 CC LEC Mo07, Sa09RXCJ0958.3-1103 Abell907 0.153 CC LEC Vi05, Cr08RXCJ2234.5-3744 Abell3888 0.153 NCC HEC Cr08RXCJ2014.8-2430 RXCJ2014.8-24 0.161 CC LEC Cr08RXCJ0645.4-5413 Abell3404 0.167 ICC MEC Cr08Abell2218 – 0.176 NCC HEC Go04, Ba07Abell1689 – 0.183 ICC MEC Pe98, An04, Ca09Abell383 – 0.187 CC LEC Vi05, Ca09, Zh10Abell209 – 0.206 NCC MEC Ca09Abell963 – 0.206 ICC MEC Sm05, Ba07, Ca09Abell773 – 0.217 NCC HEC Go04, Mo07, Ca09Abell1763 – 0.223 NCC HEC Du08, Ca09Abell2390 – 0.228 CC LEC Vi05, Mo07, Ca09, Zh10Abell2667 – 0.230 CC LEC Ca09RXCJ2129.6+0005 – 0.235 CC LEC Ca09, Zh10Abell1835 – 0.253 CC LEC Mo07, Zh10RXCJ0307.0-2840 Abell3088 0.253 CC LEC Fi05, Zh06Abell68 – 0.255 NCC HEC Zh10E1455+2232 RXCJ1457.2+2220 0.258 CC LEC Sn08RXCJ2337.6+0016 – 0.273 NCC HEC Fi05, Zh06, Zh10RXCJ0303.8-7752 – 0.274 NCC HEC Zh06RXCJ0532.9-3701 – 0.275 CC ? MEC Fi05, Zh06RXCJ0232.2-4420 – 0.284 Cool core remnant ? MEC Fi05, Zh06ZW3146 RBS0864 0.291 CC LEC Mo07RXCJ0043.4-2037 Abell2813 0.292 NCC HEC Zh06RXCJ0516.7-5430 AbellS0520 0.295 NCC HEC Zh06RXCJ1131.9-1955 Abell1300 0.307 NCC HEC Fi05, Zh06
Notes.
We quote the name of the object, the redshift adopted and the classification based on their X-ray properties.(
Core Cl. ): cool cores (CC), intermediate systems (ICC) and non-cool cores (NCC).(
Entropy Cl. ): as in Leccardi et al. (2010), low (LEC), medium (MEC) and high (HEC) entropy cores characterizing clusters with stronger coolingcores and more relaxed structure (LEC), more disturbed objects (HEC) and systems with intermediate properties (MEC).(
X-ray refs. ): Baldi et al. (2007, Ba07); Cavagnolo et al. (2009, Ca09); Croston et al. (2008, Cr08); Finoguenov et al. (2005, Fi05); Govoni et al.(2004, Go04); Kempner et al. (2003, Ke03); Morandi et al. (2007, Mo07); Pratt et al. (2007, Pr07); Sanderson et al. (2009, Sa09); Sivanandamet al. (2009, Si09); Snowden et al. (2008, Sn08); Vikhlinin et al. (2005, Vi05); Wise et al. (2004, Wi04); Zhang et al. (2006, Zh06); Zhang et al.(2010, Zh10). . < r/R < . , . < r/R < . and r/R > . , respectively. Forthis purpose, R as defined in Tab. 3 in Leccardi & Molendi(2008) is considered. An error of the same amount is propagatedin quadrature with the statistical error.
3. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 1. ( Left)
Surface brightness profile in the . − . keV band (black filled circles) of Abell1835 compared with the profiles ofthe background components. The open diamonds show the count rate predicted from the background spectral model in the annulus10–12 arcmin and rescaled for the mean vignetting correction of 0.472 at those radii: the instrumental component (NXB; green), thephoton component (CXB + galactic foregrounds; blue) and the total background (sky + instrumental; red). The dashed lines showthe background profiles that we have used in our analysis: the “photon” background (blue), which is constant and corresponds to thevalue in the outer annulus rescaled to the center, and the instrumental background profile (green), increasing with radius in order toconsider the over-correction of this component. The red dashed line shows the total background that we have subtracted from oursource plus background profile, with its associated one σ statistical error (red dotted lines) obtained with a Monte Carlo simulation.Note that the intensity of the background components and their relative contribution vary significantly from cluster to cluster. ( Right)
Example of the PSF–corrected background–subtracted surface brightness profile as obtained after the analysis outlined in Sect. 2.2.This example refers to Abell1835, one of the objects with the largest smearing effect due to the combination of the telescope’s PSFand the centrally peaked intrinsic profile.On the other hand, no significant effect is noticed when thevalues of the normalization of the thermal model K obtainedfrom the two different instruments are compared. The combinedprofile is then the direct result of the weighted mean of the twoestimates.Unlike in LM08, where the focus was on the measure of the T gas profile in outer regions, here we need to recover a detaileddescription of both the T gas and surface brightness S b profilesat large and small radii. A significant improvement comparedto the treatment by LM08 has been the correction of the spec-tral mixing between different annuli caused by the finite PSF ofthe MOS instruments. We adopted the cross-talk modificationof the ancillary region file (ARF) generation software (using the crossregionarf parameter of the argen task of SAS), treatingthe cross-talk contribution to the spectrum of a given annulusfrom a nearby annulus as an additional model component (seeSnowden et al. 2008). This is a thermal model with parameterslinked to the thermal spectrum fitted to the nearby annulus andassociated to the appropriate ARF file of that region (i.e., theusual ARF familiar to X-ray astronomers). We found the cor-rection to be important in particular to the first two annuli usedin the analysis. The annuli have been therefore fitted jointly inXSPEC version 12 (Arnaud 1996), which allows to associate dif-ferent models to different RMF and ARF files. A comparison ofthe values obtained with this modelling and the values quoted in Snowden et al. (2008) for the 16 clusters in common with oursample give results in agreement within the errors. We extend the spectral analysis presented in LM08 with a spatialanalysis of the combined exposure–corrected M1-M2 images.We extract surface brightness profiles from MOS imagesin the energy band . − . keV, in order to keep the back-ground as low as possible with respect to the source. For thisreason, we avoid the intense fluorescent instrumental lines of Al( ∼ . keV) and Si ( ∼ . keV) (LM08). To correct for thevignetting, we divide the images by the corresponding exposuremaps. From the surface brightness profiles, we subtract the back-ground that is estimated starting from the spectral modelling ofthe background components in the external ring 10–12 arcmin(see LM08 for details on the adopted models). We recall herethat in the procedure of LM08 the normalizations of the back-ground components are the only free parameters of the fit andthat the galactic foreground emission, the cosmic X-ray back-ground and the cosmic ray induced continuum give a significantcontribution in the . − . keV energy range. The intensitiesof the background components in the annulus 10–12 arcmin aregiven by the count rates predicted by the best fit spectral modelin this region. In order to associate errors to these count rates,
4. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 2.
Example of the results of the two analyses adopted for the mass reconstruction. (Top and middle panels, left)
Gas densityprofile as obtained from the deprojection of the surface brightness profile compared to the one recovered from the deprojection ofthe normalizations of the thermal model in the spectral analysis; observed temperature profile with overplotted the best-fit model(from
Method 1 ). (Top and middle panels, right) Data (diamonds) and models (dashed lines) of the projected gas density squaredand temperature (from
Method 2 ). (Bottom, left) Constraints in the r s − c plane with the prediction (in green) obtained by imposingthe relation c = 4 . / (1 + z ) × (cid:0) M / h − M ⊙ (cid:1) − . from M08. (Bottom, right) Gas mass fraction profile obtainedfrom
Method 1 (gray) and
Method 2 (red).we perform a simulation within XSPEC: we allow the normal-izations of the background components to vary randomly withintheir errors, we obtain the count rates associated to this fakemodel and we iterate this procedure. The error on the level of thebackground components is the width of the distribution of thesimulated count rates. Using these values in the outer annulus, we reconstruct the background profile at all radii. The “photon”components (CXB and galactic foreground) are affected by vi-gnetting in the same way as the source photons and, therefore, di-viding by the exposure map effectively corrects also these back-ground components for the vignetting. In order to reconstructthe “photon” background profile, it is thus sufficient to rescale
5. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters the count rate for the mean vignetting in the outer annulus (con-stant blue profile in Fig. 1). On the contrary, the instrumentalbackground does not suffer from vignetting and, therefore, di-viding the image by the exposure map “mis-corrects” this com-ponent. In order to consider this effect, we divide the correspond-ing count rate by the vignetting profile (that we derive from theexposure map in the . − . keV), obtaining the growing greencurve in Fig. 1. The total background profile (red line in Fig. 1)is the sum of the photon (blue) and instrumental (green) profiles.The surface brightness profiles S b ( r ) have been first ex-tracted from the combined images and binned by requiring afixed number of 200 counts in each radial bin to preserve all thespatial information available. After the background subtraction,they have been corrected for the PSF smearing. For this purpose,a sum of a cusped β − model and of a β − model (Cavaliere &Fusco-Femiano 1978) with seven free parameters, f m ( r ) = a × h x − a × (cid:0) x (cid:1) . − a + a / + a (cid:0) x (cid:1) . − a i (with x = r/a and x = r/a ), is convolved with thepredicted PSF (Ghizzardi 2001) and fitted to the observedprofile background–subtracted S b ( r ) to obtain the best-fitconvolved model f c ( ¯ a i ; r ) . Finally, to correct S b ( r ) for thePSF-convolution, we apply a correction at each radius ˆ r where S b ( r ) is measured equal to the ratio f m ( ¯ a i ; ˆ r ) /f c ( ¯ a i ; ˆ r ) . Anexample of the results of the procedure is shown in Fig. 1. Thesecorrected profiles are, finally, used in the following analysis upto the radial limit, R sp , beyond which the ratio between theprofile and the error on it (including the estimated uncertaintyon the measurement of the background) is below 2.
3. Estimates of the mass profiles
We use the profiles of the spectroscopically determined ICMtemperature and of the PSF–corrected surface brightness esti-mated, as described in the previous section, to recover the X-raygas, the dark and the total mass profiles, under the assumptionsof the spherical geometry distribution of the Intracluster Medium(ICM) and that the hydrostatic equilibrium holds between ICMand the underlying gravitational potential. We apply the two fol-lowing different methods: – ( Method 1 ) This technique is described in Ettori et al. (2002)and has been widely used to recover the mass profiles in re-cent X-ray studies of both observational (e.g. Morandi et al.2007; Donnarumma et al. 2009, 2010) and simulated datasets(e.g. Rasia et al. 2006; Meneghetti et al. 2010) against whichit has been thoroughly tested.We summarize here the algorithm adopted and how it usesthe observed measurements. Starting from the X-ray surfacebrightness profile and the radially resolved spectroscopictemperature measurements, this method puts constraints onthe parameters of the functional form describing the darkmatter M DM , defined as the total mass minus the gas mass(we neglect the marginal contribution from the mass in starsthat amounts to about 10-15 % of the gas mass in mas-sive systems –see, e.g., discussion in Ettori et al. 2009 andAndreon 2010–, and is here formally included in the M DM term). In the present work, we adopt a NFW profile: M DM ( < r ) = M tot ( < r ) − M gas ( < r ) = 4 π r ρ s f ( x ) ,ρ s = ρ c,z c ln(1 + c ) − c/ (1 + c ) ,f ( x ) = ln(1 + x ) − x x, (1) where x = r/r s , ρ c,z = 3 H z / πG is the critical density atthe cluster’s redshift z , H z = H × (cid:2) Ω Λ + Ω m (1 + z ) (cid:3) / is the Hubble constant at redshift z for an assumed flatUniverse ( Ω m + Ω Λ = 1 ), and the relation R = c × r s holds.The two parameters ( r s , c ) are constrained by minimizinga χ statistic defined as χ T = X i ( T data , i − T model , i ) ǫ T,i (2)where the sum is done over the annuli of the spectral anal-ysis; T data are the either deprojected or observed tempera-ture measurements obtained in the spectral analysis; T model are either the three-dimensional or projected values of theestimates of T gas recovered from the inversion of the hy-drostatic equilibrium equation (see below) for a given gasdensity and total mass profiles; ǫ T is the error on the spec-tral measurements. The gas density profile, n gas , is estimatedfrom the geometrical deprojection (Fabian et al. 1981, Krisset al. 1983, McLaughlin 1999, Buote 2000, Ettori et al. 2002)of either the measured X-ray surface brightness or the esti-mated normalization of the thermal model fitted in the spec-tral analysis (see Fig. 2). In the present study, we considerthe observed spectral values of the temperature and evaluate T model by projecting the estimates of T gas over the annuliadopted in the spectral analysis accordingly to the recipe inMazzotta et al. (2004) and using the gas density profile ob-tained from the deprojection of the PSF–deconvolved surfacebrightness profile (see Sect. 2.2). We exclude the deprojecteddata of the gas density within a cutoff radius of 50 kpc be-cause the influence of the central galaxy is expected to be notnegligible, in particular for strong low-entropy core systems.The values of T gas are then obtained from − Gµm a n gas M tot ( < r ) r = d ( n gas × T gas ) dr , (3)where G is the universal gravitational constant, m a is theatomic mass unit and µ =0.61 is the mean molecular weightin atomic mass unit. To solve this differential equation, weneed to define a boundary condition that is here fixed to thevalue of the pressure measured in the outermost point ofthe gas density profile, P out = P gas ( R sp ) = n gas ( R sp ) × T gas ( R sp ) , where T gas ( R sp ) is estimated by linear extrapo-lation in the logarithmic space, if required. The systematicuncertainties introduced by this assumption on P out are dis-cussed in the next section. Note that by applying Method 1 the errors on the gas density do not propagate into the esti-mates of the parameters of the mass profile and are used bothto define the range of the accepted values of P out and to eval-uate the uncertainties on the gas mass profiles. The allowedrange at σ of the two interesting parameters, r s and c , isdefined from the minimum and the maximum of the valuesthat permit χ T to be lower or equal to min ( χ T ) + 1 . Theaverage error on the mass is then the mean of the upper andlower limit obtained at each radius from the allowed rangesat σ of r s and c . Only for the purpose of estimating theprofile of M gas ( < r ) , and eventually to provide the extrapo-lated values, the deprojected gas density profile is fitted withthe generic functional form described in Ettori et al. (2009)and adapted from the one described in Vikhlinin et al. (2006), n gas = n gas , ( r/r c , ) − α × (cid:0) r/r c , ) (cid:1) − . α + α / × (1 + ( r/r c , ) α ) − α /α .
6. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 3. ( First 2 panels on the left ) Relative errors on M (black) and c (red) estimated by the two methods. The median valuesare indicated by a dashed line. ( ) Ratios between the best-fit result on the scale radius r s ( ) andon R ( ) and outermost radius reached with the spatial analysis. – ( Method 2 ) The second method follows the approach de-scribed in Humphrey et al. (2006) and Gastaldello et al.(2007) where further details of this technique ( in particularin Appendix B of Gastaldello et al. 2007) are provided. Weassume parametrizations for the gas density and mass pro-files to calculate the gas temperature assuming hydrostaticequilibrium, T ( r ) = T n gas , n gas ( r ) − µm a Gk B n gas ( r ) Z rr n gas M tot drr , (4)where n gas is the gas density, n gas , and T are densityand temperature at some “reference” radius r and k B isBoltzmann’s constant. The n gas and T ( r ) profiles are fittedsimultaneously to the data to constrain the parameters of thegas density and mass models. The parameters of the massmodel are obtained from fitting the gas density and temper-ature data and goodness-of-fit for any mass model can beassessed directly from the residuals of the fit. The quality ofthe data, in particular of the temperature profile, motivatedthe use of this approach rather than the default approachof parametrizing the temperature and mass profiles to cal-culate the gas density used in Gastaldello et al. (2007). Weprojected the parametrized models of the three-dimensionalquantities, n and T , and fitted these projected emission-weighted models to the results obtained from our analysisof the data projected on the sky. With respect to the papercited above, the XSPEC normalization have been derivedconverting the XMM surface brightness in the . − . keVband using the effective area and observed projected tem-perature and metallicity obtained in the wider radial binsused for spectral extraction. The models have been integratedover each radial bin (rather than only evaluating at a sin-gle point within the bin) to provide a consistent comparison.We considered an NFW profile of eq. 1 for fitting the to-tal mass and two models for fitting the gas density profile:the β model (Cavaliere & Fusco-Femiano 1978) a double β model in which a common value of beta is assumed, and acusped β model (Pratt & Arnaud 2002; Lewis et al. 2003).The last two models have been introduced to account for thesharply peaked surface brightness in the centers of relaxedX-ray systems and they provide the necessary flexibility toparametrize adequately the shape of the gas density profilesof the objects in our sample when the traditional β modelfails in fitting the data. Hereafter, we define M ∆ = M DM ( < R ∆ ) (i.e. M is thedark matter enclosed within a sphere where the mean clusteroverdensity in dark matter only is 200 times the critical densityat the cluster’s redshift) and f gas ( < R ∆ ) is the ratio between thegas mass, M gas , and the total mass, M tot = M DM + M gas , es-timated within R ∆ , where the overdensity is here estimated byusing the total (i.e. dark + gas) mass profile.The best-fit values obtained for an assumed NFW dark mat-ter mass profiles are quoted in Table 2. In Table 3, we presentour estimates of R , R and the gas mass fraction f gas = M gas /M tot , that is hereafter considered within R to avoida problematic extrapolation of the data up to R . In Fig. 3,we show the relative errors provided from the two methods onthe estimates of c and M . The distribution of the statis-tical uncertainties is comparable, with median values of 15–20% on both c and M with Method 1 and
Method 2 .Also the distributions of the measurements of c and M are very similar, with 1st–3rd quartile range of . − . and . − . × M ⊙ with Method 1 and . − . and . − . × M ⊙ with Method 2 .Moreover, the two methods show a good agreement betweenthe two estimates of the gas mass fraction f gas ( < R ) , asshown in Fig. 5. We measure a median (1st, 3rd quartile) of . . , . , and a median relative error of 12%, with Method 1 and . . , . , and a relative error of 10%,with Method 2 .As shown in the last two panels of Fig. 3, we note that thelarge majority of our data is able to define a scale radius r s wellwithin the radial range investigated in the spectral and spatialanalysis, allowing a quite robust constraints of the fitted param-eters.To rely on the best estimates of the concentration and mass,we define in the following analysis a further subsample by col-lecting the clusters that satisfy the criterion that the upper valueat σ of the scale radius, as estimated from the 2 methods, islower than the upper limit of the spatial extension of the detectedX-ray emission, i.e. ( r s + ǫ r s ) < R sp . Imposing this condition,we select the 26 clusters where a more robust (i.e. with welldefined and constrained free parameters) mass reconstruction isachievable.
4. Systematics in the measurements of c , M and f gas The derived quantities c , M and f gas ( < R ) are mea-sured with a relative statistical error of about 20, 15 and 10 %,
7. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 4.
Best-fit values of r s , c , gas mass fraction f gas at R and dark mass M DM within R as obtained from Method 1 (diamonds;the ones including red points indicate the objects where the condition ( r s + ǫ r s ) < R sp is satisfied by both methods. R sp here plottedas horizontal line in the upper panel).respectively (see Section 3 and Figures 3 and 5). Here, we inves-tigate the main uncertainties affecting our techniques that will betreated as systematic effects in the following analysis.We consider two main sources of systematic errors: (i) theanalysis of our dataset, both for what concerns the estimates ofthe gas temperature and the reconstructed gas density profile; (ii)the limitations and assumptions in the techniques adopted for themass reconstruction.In Table 4, we summarize our findings tabulated as rela-tive median difference with respect to the estimates obtainedwith Method 1 . Overall, we register systematic uncertainties of ( − , +1) % on c , ( − , +3) % on M and ( − , +4) % on f gas ( < R ) , where these ranges represent the minimum andmaximum estimated in the dataset investigated and quoted inTable 4. The ICM properties of the present dataset have been studiedthrough spatially resolved spectroscopic measurements of thegas temperature profile and deprojected, PSF–corrected surfacebrightness profile as accessible to
XMM-Newton (see Section 2).
8. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 5.
Estimates of M ( left ), c ( center ) and f gas ( < R ) ( right ) with the two methods. ( Upper panels ) The color codeindicates the objects at z < . (blue), in the range . < z < . (green) and at z > . (red). ( Lower panels ) Distribution ofLow (LEC), Medium (MEC), High (HEC) Entropy Core systems.To assess the systematics propagated through the temper-ature measurements, we present the results obtained with M2 only, i.e. before any correction introduced from the harder spec-tra observed with M1 (see Sect. 2.1). Overall, the systematics arein the order of a few per cent, with the largest offset of about per cent on the concentration and mass measurements at R and beyond.When the deprojected spectral values of the gas temperature,instead of the projected ones, are compared with the predictionsfrom the model, we measure differences below % (see datasetlabelled “ T ”).On the gas density profile, we investigate the role playedfrom the use of a functional form instead of the values obtaineddirectly from deprojection. To this purpose, we use a revisedform of the one introduced from Vikhlinin et al. (2006) to fitthe gas density profile and, then, we adopt it as representative ofthe gas density profile to be put in hydrostatic equilibrium withthe gravitational potential in equation 3. The measurements ob-tained are labelled “fit n gas ” and show discrepancies in the orderof 1 per cent or less. With the intention to assess the the bias affecting the recon-structed mass values, we make use of the gas temperature anddensity profiles through two independent techniques (labelled
Method 1 and
Method 2 ), as described in Section 3. With respect to
Method 1 , Method 2 provides differences on M DM that arelower than 10 per cent, increasing from about 1 per cent at R up to 7 per cent at R (see Table 4). The bias on f gas remainsstable around 3–4 per cent, suggesting that some systematics af-fect also the estimate of M gas . This is due to the application oftwo different functional forms in Method 1 and
Method 2 overa radial range that extends beyond the observational limit (see,e.g., Fig. 3).The mass reconstruction of
Method 1 depends upon theboundary condition on the gas pressure profile. In particular, tosolve the differential equation 3, an outer value on the pressureis fixed to the product of the observed estimate of the gas densityprofile at the outermost radius and an extrapolated measurementof the gas temperature. Using a grid of values for the pressureobtained from the best-fit results of the gas density and tempera-ture profiles, we evaluate a systematic bias on the mass of about3 per cent, on the gas mass fraction of 1 per cent, and on c ofabout 1 per cent (see dataset labelled “ P out ”).
5. The c − M relation In this section, we investigate the c − M relation. We notethat our sample has not been selected to be representative ofthe cluster population in the given redshift range and, in themean time, does not include only relaxed systems. Therefore,the results here presented on the c − M relation have to bejust considered for a qualitative comparison with the predictions
9. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 6.
Data in the plane ( c , M ) used to constrain the cosmological parameters (Ω m , σ ) . The dotted lines show the predictedrelations from Eke et al. (2001) for a given Λ CDM cosmological model at z = 0 (from top to bottom: σ = 0 . and σ = 0 . ).The shaded regions show the predictions in the redshift range . − . for an assumed cosmological model in agreement withWMAP-1, 5 and 3 years (from the top to the bottom, respectively) from Bullock et al. (2001; after Macci`o et al. 2008). The dashedlines indicate the best-fit range at σ obtained for relaxed halos in a WMAP-5 years cosmology from Duffy et al. (2008; thin lines: z = 0 . , thick lines: z = 0 . ). Color codes and symbols as in Fig. 5.from numerical simulations and to assess differences or simili-tude with previous work on this topic.As we show in Fig. 6 using the measurements obtained with Method 1 , the median relation between concentration and totalmasses for CDM halos as function of redshift is represented wellfrom the analytic algorithms, as in N97, E01 and B01. Thesemodels relate the halo properties to the physical mechanism ofhalo formation. Considering the weak dependence of the haloconcentrations on the mass and redshift, Dolag et al. (2004) in-troduced a two-parameter functional form, c = c M B / (1 + z ) .We consider this relation in its logarithmic form and fit linearlyto our data the expression: log ( c × (1 + z )) = A + B × log (cid:18) M M ⊙ (cid:19) . (5)A minimum in the χ distribution is looked for by taking intoaccount the errors on both the coordinates (we use the routineFITEXY in IDL). The errors are assumed to be Gaussian inthe logarithmic space, although they are properly measured asGaussian in the linear space.We also express our results in term of the concentration c expected for a dark matter halo of h − M ⊙ and equal to A once the parameters in equation 5 are used. We convert to c even the results from literature obtained, for instance, at differentoverdensity, as described in the Appendix.We measure A ≈ . and B systematically lower than − . ,with the best-fit results obtained through Method 1 that prefer,with respect to
Method 2 , a relation with slightly higher normal-ization (by ∼ per cent) and flatter (by − per cent) distri-bution in mass. In both cases, a total scatter of σ log c ≈ . ismeasured both in the whole sample of 44 objects, where the sta- tistical scatter related to the observed uncertainties is still domi-nant, and in the subsample of 26 selected clusters.When a slope B = − . is assumed, as measured in numer-ical simulations over one order of magnitude in mass almost in-dependently from the underlying cosmological model (see e.g.Dolag et al. 2004, Macci`o et al. 2008), the measured normal-izations of the c − M relation fall into the range of theestimated values for samples of simulated clusters (see Table 5).All the values of normalization and slope are confirmed,within the estimated errors, with both the BCESbisector method(as described in Akritas & Bershady 1996 and implementedin the routines made available from M.A. Bershady) and aBayesian method that accounts for measurement errors in linearregression, as implemented in the IDL routine LINMIX ERRbyB.C. Kelly (see Kelly 2007). As we quote in Table 5, with theselinear regression methods (and after bootstrap resampling ofthe data in BCES), we measure a typical error that is larger bya factor − in normalization and up to in the slope than thecorresponding values obtained through the covariance matrix ofthe FITEXY method.These values compare well with the measurements obtainedfrom numerical simulations of DM-only galaxy clusters, al-though these simulations sample, on average, mass ranges lowerthan the ones investigated here. Recent work from Shaw et al.(2006) and Macci`o et al. (2008) summarize the findings. Theslope of the relation, as previously obtained from B01 and D04,lies in the range ( − . , − . , with a preferred value ofabout − . . The normalizations for low-density Universe witha relatively higher σ , as from WMAP-1, are more in agree-ment with the observed constraints on, e.g., c . For instance,M08 find c = 4 . , . , . for relaxed objects in a back-ground cosmology that matches WMAP-1, 3 and 5 year data,
10. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters respectively . Shaw et al. (2006) measure c = 4 . using a flatUniverse with Ω m = 0 . and σ = 0 . . D04 for a Λ CDM with σ = 0 . require c = 4 . . All these values show the sensitiv-ity of the normalization to the assumed cosmology, that is furtherdiscussed in the section where constraints on the cosmologicalparameters (Ω m − σ ) will be obtained through the measured c − M relation. Neto et al. (2007) study the statistics of the haloconcentrations at z = 0 in the Millennium Simulation (with anunderlying cosmology of Ω m = 1 − Ω Λ = 0 . , σ = 0 . ) andfind that a power-law with B = − . and c = 4 . fits fairlywell the relation for relaxed objects, with an intrinsic logarithmicscatter for the most massive objects of 0.092 (see their Fig. 7).We note, however, that, while the normalizations we mea-sure for a fixed slope B = − . are well in agreement withthe results from numerical simulations, a systematic lower valueof the slope is measured, when it is left to vary. To test the ro-bustness of this evidence, we have implemented Monte-Carloruns using the best-fit central values estimated in N-body simu-lations (see Appendix B for details). With almost no dependenceupon the input values from numerical simulations and using theFITEXY technique that provides the results with the most sig-nificant deviations from B ≈ − . , we measure in the 3 sam-ples here considered (i.e. all 44 objects, the selected 26 objects,and the only 11 LEC objects) a probability of about 0.5 (1), 20(42) and 26 (46) per cent, respectively, to obtain a slope lowerthan the measured σ upper limit. These result confirm thatthe systematic uncertainties present in the measurements of theconcentration and dark mass within R are still affecting thesample of 44 objects, whereas they are significantly reduced inthe selected subsamples.Our best-fit results are in good agreement also with previ-ous constraints obtained from X-ray measurements in the samecosmology. Pointecouteau et al. (2005) measure c ≈ . and B = − . ± . in a sample of ten nearby ( z < . ) andrelaxed objects observed with XMM-Newton in the temperaturerange − keV. Zhang et al. (2006) measure a steeper slope of − . ± . , probably affected from few outliers, in the REFLEX-DXL sample of 13 X-ray luminous and distant ( z ∼ . ) clus-ters observed with XMM-Newton , that, they claim, are how-ever not well reproduced from a NFW profile. Voigt & Fabian(2006) show a good agreement with B01 results and B ≈ − . for their estimates of 12 mass profiles of X-ray luminous ob-jects observed with Chandra in the redshift range . − . .A good match with the results in D04, and within the scatterfound in simulations, is obtained with 13 low-redshift relaxedsystems with T gas in the range . − keV as measured with Chandra in Vikhlinin et al. (2006). Schmidt & Allen (2007), us-ing
Chandra observations of 34 massive relaxed galaxy clusters,measure B = − . ± . (95% c.l.), significantly steeperthan the value predicted from CDM simulations. Leaving freethe redshift dependence that they estimate to be consistent withthe (1 + z ) − expected evolution, they measure a normaliza-tion c ≈ . ± . (95% c.l.), definitely higher than ourbest-fit parameter. Buote et al. (2007) fit the c − M relationfrom 39 systems in the mass range . − × M ⊙ se-lected from Chandra and
XMM-Newton archives to be relaxed.Analysing the tabulated values of the 20 galaxy clusters with M > M ⊙ , that include the most massive systems fromthe XMM-Newton study of Pointecouteau et al. (2005) and the
Chandra analysis in Vikhlinin et al. (2006), we measure B = − . ± . and c ≈ . ± . . We refer to Appendix A for a detailed discussion of the conversionsadopted
Overall, we conclude however that the slope of the c − M re-lation cannot be reliably determined from the fitting over a nar-row mass range as the one considered in the present work andthat, once the slope is fixed to the expected value of B = − . ,the normalization, with estimates of c in the range . − . ,agrees with results of previous observations and simulations fora calculations in a low density Universe. Following Leccardi et al. (2010), we have employed the pseudo-entropy ratio ( σ ≡ ( T IN /T OUT ) × ( EM IN /EM OUT ) − / , where IN and OUT define regions within ≈ R and encircled inthe annulus with bounding radii 0.05-0.20 R , respectively,and T and EM are the cluster temperature and emission mea-sure) to classify our sample of 44 galaxy clusters accordingly totheir core properties. We identify 17 High-Entropy-Core (HEC),11 Medium-Entropy-Core (MEC) and 16 Low-Entropy-Core(LEC; see Table 1) systems. While the MEC and HEC objectsare progressively more disturbed (about 85 per cent of the merg-ing clusters are HEC) and with a core that presents less evidencein the literature of a temperature decrement and a peaked surfacebrightness profile (intermediate, ICC, and no cool core, NCC,systems), the LEC objects represent the prototype of a relaxedcluster with a well defined cool core (CC in Table 1) at low en-tropy (see also Cavagnolo et al. 2009). These systems are pre-dicted from numerical simulations to have higher concentrationsfor given mass, by about 10 per cent, and lower scatter, by about15-20 per cent, in the c − M relation (e.g. M08, Duffy et al.2008).Out of 16, eleven LEC objects are selected under the condi-tion that their scale radius is within the radial coverage of ourdata. We measure their c − M relation to have slightly lowernormalization ( A ≈ . − . , c ≈ . − . ) and flatter dis-tribution ( B = − . ± . ) than the one observed in the selectedsubsample of 26 objects, with a dispersion around the logarith-mic value of the concentration of 0.08, that is about 40 per centlower than the similar value observed in the latter sample. Thisis consistent in a scenario where disturbed systems have an es-timated concentration through the hydrostatic equilibrium equa-tion that is biased higher (and with larger scatter) than in relaxedobjects up to a factor of 2 due to the action of the ICM motions(mainly the rotational term in the inner regions and the randomgas term above R ), as discussed in Lau et al. (2009; see alsoFang et al. 2009, Meneghetti et al. 2010) for galaxy clusters ex-tracted from high-resolution Eulerian cosmological simulations.
6. Cosmological constraints from themeasurements of c , M , and f gas N − body simulations have provided theoretical fitting functionsthat are able to reproduce the distribution of the concentrationparameter of the NFW density profile as function of halo massand redshift (e.g. NFW, E01, B01, N07). Basically, all thesesemi-empirical prescriptions provide the expected values of theconcentration parameter for a given set of cosmological parame-ters (essentially, the cosmic matter density, Ω m , and the normal-ization of the power spectrum on clusters scale, σ ) for a givenmass (the estimated cluster dark mass, M , in our case) at themeasured redshift of the analyzed object. They assume that theconcentration reflects the background density of the Universeat the formation time of a given halo. The cosmological modelinfluences the concentration and virial mass because of the cos-
11. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters mic background density and the evolution of structure formation.For instance, the NFW model uses two free parameters, ( f, C ) ,to define the collapse redshift at which half of the final mass M is contained in progenitors of mass ≥ f M , with C repre-senting the ratio between the characteristic overdensity and themean density of the Universe at the collapse redshift. We use ( f, C ) = (0 . , .B01 assume, instead, an alternative model to improve theagreement between the predicted redshift dependence of the con-centrations and the results of the numerical simulations by usingtwo free parameters, F and K , where F is still a fixed fraction( . in our study) of a halo mass at given redshift and K in-dicates the concentration of the halo at the collapse redshift. K has to be calibrated with numerical simulations and is fixed hereto be equal to (see also Buote et al. 2007 for a detailed dis-cussion on the role played from the parameters F and K on theprediction of the concentrations as function of the backgroundcosmology and halo masses). M08 have revised this model byassuming that the characteristic density of the halo, that in B01scales as (1 + z ) , is independent of redshift. This correctionpropagates into the growth factor of the concentration parame-ter that becomes shallower with respect to the mass dependenceat masses higher than h − M ⊙ , permitting larger concentra-tions at the high-mass end than the original B01 formulation.The prescription in E01 defines with the only parameter C σ (equal to 28, in our analysis, as suggested in their original work)the collapse redshift z c through the relation D ( z c ) σ eff ( M s ) = C − σ , where D ( z ) is the linear growth factor, σ eff is the effectiveamplitude of the linear power spectrum at z = 0 and M s is thetotal mass within the radius at which the circular velocity of anNFW halo reaches its maximum and that is equal to 2.17 timesthe scale radius, r s .As tested in high-resolution numerical simulations (see, e.g.,N07, M08, Duffy et al. 2008), these 3 formulations provide dif-ferent predictions over different mass range and redshift: formassive systems a z < , as the ones under investigation inthe present analysis, the original B01 tends to underestimate theconcentration at fixed halo mass; its revised version after M08partially compensate for this difference but still shows some ten-sion with numerically simulated objects (see, e.g., figure 5 inM08); NFW overestimates the concentration, whereas E01 pro-vide good estimates (see, e.g. figure 2 in Duffy et al. 2008) alsoconsidering its simpler and more robust formulation, being de-pendent upon a single parameter that does not need an indepen-dent calibration from simulations evolved with a given back-ground cosmology (note, indeed, that as pointed out in M08,both NFW and B01 models have normalizations that, ideally,have to be determined empirically for each assumed cosmologywith a dedicated numerical simulation).Hereafter, we consider E01 as the model of reference and usethe other prescriptions as estimate of the systematics affectingour constraints.In particular, to constrain the cosmological parameters of in-terest, σ and Ω m , we calculate first the concentration c ,ijk = c ( M i , Ω m , j , σ ,k ) predicted from the model investigated ateach cluster redshift for a given grid of values in mass, M i , cos-mic density parameter, Ω m , j , and power spectrum normalization, σ ,k .Then, we proceed with the following analysis:1. a new mass M ,j and concentration c ,j are estimatedfrom the X-ray data for given Ω m , j ; 2. we perform a linear interpolation on the theoretical predic-tion of c ,ijk to associate a concentration ˆ c ,jk to the newmass M ,j for given Ω m , j and σ ,k ;3. we evaluate the merit function χ c χ c = χ c (Ω m , j , σ ,k ) = X data ,i ( c ,i − ˆ c ,jk ) ǫ ,i + σ c , (6)where ǫ ,i is the σ uncertainty related to the measured c ,i and σ c is the scatter intrinsic to the mean predictedvalue ˆ c ,jk as evaluated in Neto et al. (2007; see theirfig. 7 and relative discussion). They estimate in the massbin . − . h − M ⊙ a logarithmic mean value ofthe concentration parameter of . , with a dispersion of . , corresponding to a relative uncertainty of . . Wetake into account these estimates by associating to the ex-pectation of ˆ c ,jk a scatter equals to log ˆ c ,jk ± ǫ c , where ǫ c = 0 . × log ˆ c ,jk ;4. a minimum in the χ c distribution, χ c, min , is evaluated andthe regions encompassing χ c, min + (2 . , . , . are es-timated to constrain the best-fit values and the , , σ in-tervals in the (Ω m , σ ) plane shown in Fig. 7. To representthe observed degeneracy in the σ − Ω m plane, we quote inTable 6 (and show with a dashed line in Fig. 7) the best-fitvalues of the power-law fit σ Ω γ m = Γ , obtained by fittingthis function on a grid of values estimated, at each assigned Ω m , the best-fit result, and associated σ error, of σ .5. A further constraint on the Ω m parameter that allows us tobreak the degeneracy in the σ − Ω m plane (as highlightedfrom the banana-shape of the likelihood contours plotted inFig. 7) is provided from the gas mass fraction distribution.We use our estimates of f gas ( < R ) = f from Method1 quoted in Table 3. We follow the procedure described inEttori et al. (2009) and assume: (i) Ω b h = 0 . ± . and H = 70 . ± . from the best-fit results of the jointanalysis in Komatsu et al. (2008), (ii) a depletion parameterat R b = 0 . ± . , (iii) a contribution of coldbaryons to the total budget f cold = 0 . ± . f gas . All thequoted errors are at σ level. Then, we look for a minimumin the function χ f = χ f (Ω m , j ) χ f = X data ,i h f ,i (1 + f cold ) /b − ˆ f bar ,j i ǫ f,i , (7)where ˆ f bar ,j = Ω b / Ω m ,j and ǫ f,i is given from thesum in quadrature of all the statistical errors, namely, on f , f cold , H , b and Ω b .6. We combine the two χ distribution, χ = χ c + χ f , andplot in Fig. 7 the constraints obtained from both χ c only and χ , quoting the best-fit results in Table 6.7. The effect of the systematic uncertainties, assumed to be nor-mally distributed, is also considered by propagating them inquadrature to the measurements of c and f , as obtainedfrom the analysis summarized in Table 4. The constraints ob-tained after this further correction are indicated with label“ + syst” in Table 6.The cosmological constraints we obtain with 3 different ana-lytic models (E01, B01+M08, NFW) are summarized in Table 6and likelihood contours for the model of reference E01 are plot-ted in Fig. 7. To represent the observed degeneracy, we con-strain the parameters of the power-law fit σ Ω γ m = Γ . As ex-pected from the properties of the prescriptions, E01 provides
12. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. 7.
Cosmological constraints in the (Ω m , σ ) plane obtained from equations 6 and 7 by using predictions from the model byEke et al. (2001). The confidence contours at , , σ on 2 parameters (solid contours) are displayed. The combined likelihood withthe probability distribution provided from the cluster gas mass fraction method is shown in red. The dashed green line indicatesthe power-law fit σ Ω γ m = Γ . The best-fit results are quoted in Table 6. A relative logarithmic scatter of 0.139 (see Sect. 6) isconsidered in the models. Systematic uncertainties on c and f gas ( < R ) as quoted in Table 4 are also propagated. ( Left ) Fromthe subsample of 26 clusters satisfying the condition ( r s + ǫ r s ) < R sp ; ( center ) from the subsample of the LEC objects; ( right ) fromall the 44 clusters.constraints on σ , for given Ω m , that lie between the other two,with γ = 0 . ± . and Γ = 0 . ± . (at σ level; sta-tistical only). We break the degeneracy of the best-fit values inthe ( σ , Ω m ) plane by assuming that the cluster baryon fractionrepresents the cosmic value well. We obtain that σ = 1 . ± . and Ω m = 0 . ± . (at σ level). When the subsample of11 LEC clusters, that are expected to be more relaxed and witha well-formed central cooling core, is considered, we measure γ = 0 . ± . , Γ = 0 . ± . σ = 0 . ± . and Ω m = 0 . ± . (at σ level).We confirm that, assumed correct the ones measured withE01, NFW tends to overestimate the predicted concentrationsand, therefore, requires lower normalization σ of the powerspectrum, whereas B01+M08 compensate with larger values of σ the underestimate of c with respect to E01.We assess the systematics affecting our results by compar-ing the cosmological constraints obtained by assuming (i) differ-ent algorithms to relate the cosmological models to the derived c − M relation, (ii) biases both in the concentration parameter( b c = 0 . ), from the evidence in numerical simulations that re-laxed halos have an higher concentration by about 10 per cent(e.g. Duffy et al. 2008), and in the dark matter ( b M = 1 . ) mea-surements from the evidence provided from hydrodynamial sim-ulations that the hydrostatic equilibrium might underestimate thetrue mass by 5-20 per cent (e.g. see recent work in Meneghetti etal. 2010). As expected, lower concentrations and higher massespush the best-fit values to lower normalizations of the powerspectrum at fixed Ω m , with an offset of about 10 per cent with b c = 0 . and of few per cent b M = 1 . and M tot .
7. Summary and Conclusions
We present the reconstruction of the dark and gas mass fromthe
XMM-Newton observations of 44 massive X-ray luminousgalaxy clusters in the redshift range . − . . We estimate adark ( M tot − M gas ) mass within R in the range (1st and 3rdquartile) − × M ⊙ , with a concentration c between2.7 and 5.3, and a gas mass fraction within R between 0.11and 0.16. By applying the equation of the hydrostatic equilibrium tothe spatially resolved estimates of the spectral temperature andnormalization, we recover the underlying gravitational potentialof the dark matter halo, assumed to be well described from aNFW functional form, with two independent techniques.Our dataset is able to resolve the temperature profiles up toabout . − . R and the gas density profile, obtained fromthe geometrical deprojection of the PSF–deconvolved surfacebrightness, up to a median radius of . R . Beyond this radialend, our estimates are the results of an extrapolation obtained byimposing a NFW profile for the total mass and different func-tional forms for M gas .We estimate, with a relative statistical uncertainty of − , the concentration c and the mass M of the dark mat-ter (i.e. total − gas mass) halo. We constrain the c − M relation to have a normalization c = c × (1 + z ) × (cid:0) M / M ⊙ (cid:1) − B of about . − . and a slope B between − . and − . (depending on the methods used to recover thecluster parameters and to fit the linear correlation in the logarith-mic space), with a relative error of about 5% and 15%, respec-tively. Once the slope is fixed to the expected value of B = − . ,the normalization, with estimates of c in the range . − . ,agrees with results of previous observations and simulations forcalculations done assuming a low density Universe. We concludethus that the slope of the c − M relation cannot be reliablydetermined from the fitting over a narrow mass range as the oneconsidered in the present work, altough the steeper values mea-sured are not significantly in tension with the results for simu-lated halos when the subsamples of the most robust estimatesare considered (see Sect. 5 and Appendix B). We measure a to-tal scatter in the logarithmic space of about 0.15 at fixed mass.This value decreases to 0.08 when the subsample of LEC clustersis considered, where a slightly lower normalization and flatterdistribution is measured. This is consistent in a scenario wheredisturbed systems have an estimated concentration through thehydrostatic equilibrium equation that is biased higher (and withlarger scatter) than in relaxed objects up to a factor of 2 due tothe action of the ICM motions (see e.g. Lau et al. 2009).
13. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters We put constraints on the cosmological parameters ( σ , Ω m )by using the measurements of c and M and by comparingthe estimated values with the predictions tuned from numericalsimulations of CDM universes. In doing that, we propagate thestatistical errors (with a relative value of about − % at σ level) and consider the systematic uncertainties present both inthe simulated datasets ( ∼ %) and in our measurements ( ∼ %; see Table 4). To represent the observed degeneracy, weconstrain the parameters of the power-law fit σ Ω γ m = Γ andobtain γ = 0 . ± . and Γ = 0 . ± . (at σ level)when the E01 formalism is adopted. Different formalisms (likethe ones in B01, revised after M08, and NFW) induce variationsin the best-fit parameters in the order of 20 per cent. A furthervariation of about 10 per cent occurs if a bias of the order of 10per cent is considered on the estimates of c and M .We break the degeneracy of the best-fit values in the( σ , Ω m ) plane by assuming that the cluster baryon fraction rep-resents the cosmic value well. We obtain that σ = 1 . ± . and Ω m = 0 . ± . (at σ level; statistical only).When the subsample of 11 LEC clusters, that are expectedto be more relaxed and with a well-formed central cooling core,is considered, we measure γ = 0 . ± . , Γ = 0 . ± . σ = 0 . ± . and Ω m = 0 . ± . (at σ level).All these estimates agree well with similar constraints ob-tained for an assumed low-density Universe in Buote et al.(2007; . < σ < . at 99% confidence for a Λ CDM modelwith Ω m = 0 . ) and with the results obtained by analysing themass function of rich galaxy clusters [see, e.g., Wen, Han & Liu(2010) that summarizes recent results obtained by this cosmo-logical tool], showing that the study of the distribution of themeasurements in the c − M DM − f gas plane provides a validtechnique already mature and competitive in the present era ofprecision cosmology.However, we highlight the net dependence of our results onthe models adopted to relate the properties of a DM halo to thebackground cosmology. In this context, we urge the N − bodycommunity to generate cosmological simulations over a largebox to properly predict the expected concentration associated tothe massive ( > M ⊙ ) DM halos as function of σ , Ω m andredshift. The detailed analysis of the outputs of these datasetswill provide the needed calibration to make this technique morereliable and robust. ACKNOWLEDGEMENTS
The anonymous referee is thanked for suggestions that haveimproved the presentation of the work. We acknowledge thefinancial contribution from contracts ASI-INAF I/023/05/0and I/088/06/0. This research has made use of the X-RaysClusters Database (BAX) which is operated by the Laboratoired’Astrophysique de Tarbes-Toulouse (LATT), under contractwith the Centre National d’Etudes Spatiales (CNES).
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Appendix A: Conversion between differentoverdensity and c − M relations The total mass within a given overdensity ∆ is defined in thepresent work as M ∆ = 43 πR ∆ ρ c,z , (A.1)where ρ c,z = 3 H z / (8 πG ) is the critical density of the Universeat the cluster’s redshift z , R ∆ = c ∆ r s is the radius within whichthe mean cluster overdensity is ∆ times ρ c,z and the relation withthe concentration c ∆ and the scale radius r s holds by definitionof the NFW mass profile. We assume ∆ = 200 . Hereafter, werefer to ∆ as any other assumed overdensity. In case it is referredto the background density of the Universe, ρ b = Ω m , z ρ c,z , it isstraightforward to correct ∆ by Ω m , z = Ω m (1 + z ) /H z torecover the definition in equation A.1.To convert the tabulated values to our definition of the c − M relation, c = c z (cid:18) M M (cid:19) B , (A.2)where M = 10 h − M ⊙ , we proceed as follows:1. by definition, M ∆ / ( R ∆) is constant and R ∆ /c ∆ is fixedfrom the measurement of the scale radius. Therefore, we canwrite M ∆ c = M c ∆200 (A.3)2. c ∆ and c are related through the assumed NFW mass den-sity profile (cid:18) c c ∆ (cid:19) ln(1 + c ∆ ) − c ∆ / (1 + c ∆ )ln(1 + c ) − c / (1 + c ) = ∆200 . (A.4)This function is monotonic and easily to resolve numericallyto estimate C = c ∆ /c , that is a quantity that dependsmostly on ∆ and only marginally on the guessed c , asshown in Fig. A.1. For instance, for ∆ = 178Ω . , z , whichestimate the virial overdensity predicted from the sphericalcollapse model in a flat Universe with a contribution fromdark energy (Eke et al. 2001), C = 1 . and . at z = 0 and z = 0 . , respectively, for Ω m = 0 . , with deviationswithin 2% in the range c = 3 − .3. for a given relation c ∆ = c (1 + z ) − ( M ∆ /M ∗ ) B , we sub-stitute the above relations to obtain after simple algebricaloperations: c = c C B − z (cid:18) ∆200 M M ∗ M M (cid:19) B , (A.5)or c = c C B − (cid:18) ∆200 M M ∗ (cid:19) B . (A.6) Fig. A.1.
Numerical solution to equation A.4 for an assumed c in the range − (from the thinnest to the thickest line). Appendix B: Monte-Carlo realizations of the c − M relation We have run Monte-Carlo (MC) simulations to test the robust-ness of the observed deviations in the c − M relation de-scribed in Section 5. We have used as input values the best-fitresults (defined in the following analysis as ¯ c and ¯ B ) obtainedin the numerical simulations from Neto et al. (2007; see theirequations 4 and 5) and Macci`o et al. (2008; see Table A1 andA2) and listed in Tab. 5. We have considered the results for boththe complete sample and the relaxed objects only. To each clus-ter in our sample with measured mass M ,i and redshift z i , weassign the concentration c ,i defined as c ,i = 10 l i (B.1) l i = log h ¯ c × (cid:0) M ,i / (cid:1) ¯ B / (1 + z i ) i + R × ǫ log c , where R is a random value extracted from a Gaussian distribu-tion and ǫ log c is the scatter in the log-Normal distribution mea-sured in the numerical simulations ( ∼ . for samples includ-ing all the simulated objects and ∼ . for the sample of the re-laxed ones; the actual values are quoted in N07 and in Table A1and A2 of M08). We assume that (1) our LEC objects followthe distribution obtained for relaxed simulated clusters; (2) allthe remaining clusters follow the distribution estimated for thecomplete simulated halo sample; (3) the 3 samples consideredin our analysis with 44, 26 and 11 clusters, respectively, are builtconsidering whether each object is a LEC and/or has an upperlimit at σ on the scale radius lower than R sp , as discussed inSection 5. To be conservative in our approach, we fit equation 5to the distribution in the c − M with the FITEXYtechniquethat is the one that provides the most significant deviations fromthe results obtained in numerical simulations. We repeat this pro-cess 10,000 times and obtain the plots shown in Fig. B.1 for eachof the case investigated. In Table B.1, we summarize our findingfrom which we conclude that, while the best-fit values estimated
15. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Fig. B.1.
Distribution of the best-fit values of the normalization A and slope B after 10,000 MC realizations. The input values(in these plots from M08 / WMAP-5) are indicated with verticalsolid (for all the simulated objects) and dashed (for the relaxedones) lines. The red solid line represents the central value for thecorresponding sample as quoted in Table 5. The red dotted linesshow the σ uncertainties.for the samples of 26 and 11 clusters are within the overall dis-tribution expected in numerical simulations, the sample of 44clusters provides results on the slope B that lie on the lower endof the distribution, as probable consequence of the uncertaintiespresent both on the estimates of c and M for the 18 clus-ters that are indeed not selected for the further analysis and onthe residual bias affecting the measurements of c under thehypothesis of the hydrostatic equilibrium (see Section 5.1 and,e.g., Lau et al. 2009).
16. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 2.
Results on the mass reconstruction.
Method 1 Method 2
Cluster R sp R xsp r s c M χ T (N) r s c M χ T (N) χ n (N)kpc kpc kpc M ⊙ kpc M ⊙ RXCJ0003.8+0203
605 414 143 +36 − . +1 . − . . ± .
23 2 . ±
77 5 . ± .
49 2 . ± .
76 2 . . Abell3911
836 754 261 +108 − . +1 . − . . ± .
50 10 . ±
211 3 . ± .
79 5 . ± .
73 10 . . Abell3827
792 767 390 +89 − . +0 . − . . ± .
73 3 . ±
66 4 . ± .
94 5 . ± .
86 2 . . RXCJ0049.4-2931
386 371 71 +30 − . +3 . − . . ± .
16 5 . ±
17 9 . ± .
84 1 . ± .
16 6 . . Abell2034
690 866 979 +7 − . +0 . − . . ± .
17 7 . ±
110 4 . ± .
68 7 . ± .
47 9 . . RXCJ1516.5-0056
411 502 563 +0 − . +0 . − . . ± .
42 8 . ±
139 4 . ± .
00 2 . ± .
95 4 . . RXCJ2149.1-3041
663 513 251 +41 − . +0 . − . . ± .
21 24 . ±
22 4 . ± .
26 3 . ± .
24 8 . . RXCJ1516.3+0005
624 514 185 +67 − . +1 . − . . ± .
41 2 . ±
92 5 . ± .
05 3 . ± .
98 2 . . RXCJ1141.4-1216
676 519 496 +60 − . +0 . − . . ± .
37 13 . ±
19 3 . ± .
13 4 . ± .
22 3 . . RXCJ1044.5-0704
847 566 286 +23 − . +0 . − . . ± .
18 13 . ±
63 3 . ± .
42 3 . ± .
58 8 . . Abell1068
998 1026 564 +66 − . +0 . − . . ± .
48 8 . ± . ± .
05 4 . ± .
13 5 . . RXCJ2218.6-3853
764 587 597 +184 − . +0 . − . . ± .
62 9 . ±
175 3 . ± .
78 7 . ± .
29 7 . . RXCJ0605.8-3518
893 598 369 +47 − . +0 . − . . ± .
36 17 . ±
43 4 . ± .
32 4 . ± .
46 9 . . RXCJ0020.7-2542
695 603 473 +245 − . +1 . − . . ± .
67 17 . ±
205 3 . ± .
82 12 . ± .
79 5 . . Abell1413 +23 − . +0 . − . . ± .
32 6 . ±
40 5 . ± .
60 5 . ± .
65 5 . . RXCJ2048.1-1750
806 619 742 +80 − . +1 . − . . ± .
12 2 . ±
578 1 . ± .
40 8 . ± .
01 3 . . RXCJ0547.6-3152
847 624 443 +253 − . +0 . − . . ± .
51 4 . ±
158 3 . ± .
95 7 . ± .
20 2 . . Abell2204
858 837 816 +137 − . +0 . − . . ± .
19 58 . ±
30 3 . ± .
09 13 . ± .
64 33 . . RXCJ0958.3-1103 +260 − . +0 . − . . ± .
02 3 . ±
223 2 . ± .
37 12 . ± .
28 1 . . RXCJ2234.5-3744
745 640 506 +261 − . +2 . − . . ± .
15 1 . ±
271 3 . ± .
23 15 . ± .
11 1 . . RXCJ2014.8-2430
999 878 462 +59 − . +0 . − . . ± .
53 28 . ±
70 3 . ± .
44 9 . ± .
02 13 . . RXCJ0645.4-5413 +135 − . +1 . − . . ± .
12 7 . ±
174 5 . ± .
76 5 . ± .
14 12 . . Abell2218 +95 − . +2 . − . . ± .
74 11 . ±
129 4 . ± .
62 6 . ± .
57 10 . . Abell1689
999 974 211 +22 − . +0 . − . . ± .
44 16 . ±
28 7 . ± .
66 7 . ± .
75 9 . . Abell383
740 589 435 +95 − . +0 . − . . ± .
37 27 . ±
81 3 . ± .
68 5 . ± .
91 11 . . Abell209 +272 − . +0 . − . . ± .
23 8 . ±
311 3 . ± .
92 6 . ± .
87 8 . . Abell963
995 813 377 +107 − . +0 . − . . ± .
83 5 . ±
58 6 . ± .
06 4 . ± .
72 4 . . Abell773
977 846 605 +408 − . +1 . − . . ± .
12 6 . ±
188 3 . ± .
61 8 . ± .
88 5 . . Abell1763 +194 − . +2 . − . . ± .
74 8 . ±
91 8 . ± .
76 3 . ± .
77 3 . . Abell2390 +0 − . +0 . − . . ± .
16 11 . ±
13 1 . ± .
03 53 . ± .
48 5 . . Abell2667
966 885 993 +0 − . +0 . − . . ± .
45 4 . ±
116 2 . ± .
27 13 . ± .
92 2 . . RXCJ2129.6+0005
883 702 418 +68 − . +0 . − . . ± .
44 3 . ±
55 4 . ± .
46 4 . ± .
61 2 . . Abell1835 +46 − . +0 . − . . ± .
41 10 . ±
157 2 . ± .
16 20 . ± .
89 9 . . RXCJ0307.0-2840
691 951 611 +297 − . +0 . − . . ± .
39 5 . ±
69 5 . ± .
66 5 . ± .
94 5 . . Abell68
634 746 834 +0 − . +0 . − . . ± .
97 4 . ±
234 1 . ± .
25 21 . ± .
20 5 . . E1455+2232
946 752 214 +26 − . +0 . − . . ± .
29 2 . ±
35 6 . ± .
54 3 . ± .
54 1 . . RXCJ2337.6+0016
803 1004 332 +342 − . +3 . − . . ± .
91 1 . ±
490 3 . ± .
11 8 . ± .
31 2 . . RXCJ0303.8-7752
906 1007 1115 +14 − . +1 . − . . ± .
33 5 . ±
525 3 . ± .
58 7 . ± .
26 4 . . RXCJ0532.9-3701
781 787 278 +170 − . +2 . − . . ± .
83 3 . ±
195 5 . ± .
36 7 . ± .
17 3 . . RXCJ0232.2-4420 +0 − . +0 . − . . ± .
90 12 . ±
257 3 . ± .
64 6 . ± .
71 13 . . ZW3146 +61 − . +0 . − . . ± .
49 27 . ±
86 2 . ± .
26 10 . ± .
30 18 . . RXCJ0043.4-2037
940 823 186 +196 − . +5 . − . . ± .
24 10 . ±
85 9 . ± .
67 3 . ± .
88 7 . . RXCJ0516.7-5430
821 1061 785 +405 − . +2 . − . . ± .
88 1 . ±
261 3 . ± .
84 6 . ± .
45 2 . . RXCJ1131.9-1955 +494 − . +1 . − . . ± .
50 6 . ±
629 1 . ± .
84 19 . ± .
26 3 . . Notes.
We quote the name of the object, the upper limit of the radial range investigated in the spatial ( R sp ) and spectral analysis ( R xsp ), thebest-fit values of the scale radius, the concentration parameters, M and minimum χ with the corresponding degrees of freedom. In the case of Method 2 , we quote two minimum χ , corresponding to the minima obtained from the simultaneous fits of the temperature ( χ T ) and gas density( χ n ) profiles. 17. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 3.
Estimates of R , R and the gas mass fraction. Method 1 Method 2
Cluster R R f gas R R f gas kpc kpc < R kpc kpc < R RXCJ0003.8+0203 ±
65 824 ±
38 0 . ± .
049 1360 ±
122 899 ±
50 0 . ± . Abell3911 ±
88 1044 ±
41 0 . ± .
017 1773 ±
155 1130 ±
75 0 . ± . Abell3827 ±
84 1228 ±
45 0 . ± .
012 1823 ±
87 1184 ±
40 0 . ± . RXCJ0049.4-2931 ±
59 666 ±
37 0 . ± .
020 1071 ±
39 721 ±
29 0 . ± . Abell2034 ±
140 1569 ±
75 0 . ± .
007 1957 ±
108 1267 ±
114 0 . ± . RXCJ1516.5-0056 ±
65 1039 ±
38 0 . ± .
009 1309 ±
159 845 ±
98 0 . ± . RXCJ2149.1-3041 ±
52 846 ±
30 0 . ± .
027 1452 ±
36 942 ±
26 0 . ± . RXCJ1516.3+0005 ±
95 940 ±
54 0 . ± .
074 1502 ±
107 991 ±
81 0 . ± . RXCJ1141.4-1216 ±
55 1047 ±
31 0 . ± .
012 1551 ±
27 1003 ±
23 0 . ± . RXCJ1044.5-0704 ±
35 923 ±
19 0 . ± .
009 1531 ±
72 996 ±
21 0 . ± . Abell1068 ±
57 1140 ±
31 0 . ± .
007 1645 ±
13 1061 ± . ± . RXCJ2218.6-3853 ±
159 1275 ±
84 0 . ± .
018 1900 ±
167 1222 ±
132 0 . ± . RXCJ0605.8-3518 ±
64 1057 ±
29 0 . ± .
012 1643 ±
49 1071 ±
25 0 . ± . RXCJ0020.7-2542 ±
228 1329 ±
124 0 . ± .
016 2182 ±
200 1415 ±
82 0 . ± . Abell1413 ±
64 1207 ±
21 0 . ± .
010 1809 ±
58 1188 ±
28 0 . ± . RXCJ2048.1-1750 ±
155 1110 ±
80 0 . ± .
044 2008 ±
269 1187 ±
109 0 . ± . RXCJ0547.6-3152 ±
161 1251 ±
85 0 . ± .
057 1882 ±
168 1219 ±
81 0 . ± . Abell2204 ±
79 1549 ±
44 0 . ± .
008 2319 ±
33 1477 ±
47 0 . ± . RXCJ0958.3-1103 ±
174 1366 ±
87 0 . ± .
013 2191 ±
174 1363 ±
106 0 . ± . RXCJ2234.5-3744 ±
293 1474 ±
164 0 . ± .
067 2377 ±
294 1542 ±
159 0 . ± . RXCJ2014.8-2430 ±
56 1245 ±
32 0 . ± .
014 2067 ±
70 1323 ±
16 0 . ± . RXCJ0645.4-5413 ±
133 1243 ±
65 0 . ± .
020 1811 ±
183 1174 ±
83 0 . ± . Abell2218 ±
120 1100 ±
53 0 . ± .
019 1820 ±
120 1122 ±
66 0 . ± . Abell1689 ±
40 1279 ±
24 0 . ± .
008 1946 ±
54 1304 ±
21 0 . ± . Abell383 ±
79 1015 ±
39 0 . ± .
042 1697 ±
100 1090 ±
17 0 . ± . Abell209 ±
125 1267 ±
57 0 . ± .
015 1873 ±
197 1196 ±
54 0 . ± . Abell963 ±
95 1153 ±
50 0 . ± .
015 1586 ±
74 1049 ±
36 0 . ± . Abell773 ±
257 1350 ±
130 0 . ± .
041 1959 ±
170 1140 ±
92 0 . ± . Abell1763 ±
105 1079 ±
52 0 . ± .
025 1575 ±
88 1028 ±
39 0 . ± . Abell2390 ±
63 1695 ±
36 0 . ± .
013 3484 ±
67 2026 ±
57 0 . ± . Abell2667 ±
36 1478 ±
22 0 . ± .
018 2259 ±
103 1417 ±
72 0 . ± . RXCJ2129.6+0005 ±
60 1099 ±
30 0 . ± .
012 1619 ±
63 1042 ±
16 0 . ± . Abell1835 ±
86 1540 ±
46 0 . ± .
012 2539 ±
100 1583 ±
34 0 . ± . RXCJ0307.0-2840 ±
199 1302 ±
103 0 . ± .
017 1695 ±
78 1114 ±
59 0 . ± . Abell68 ±
127 1457 ±
71 0 . ± .
008 2549 ±
165 1489 ±
155 0 . ± . E1455+2232 ±
46 980 ±
26 0 . ± .
013 1445 ±
59 954 ±
14 0 . ± . RXCJ2337.6+0016 ±
192 1178 ±
96 0 . ± .
027 1894 ±
278 1225 ±
173 0 . ± . RXCJ0303.8-7752 ±
179 1347 ±
93 0 . ± .
016 1888 ±
301 1203 ±
122 0 . ± . RXCJ0532.9-3701 ±
179 1186 ±
102 0 . ± .
027 1835 ±
233 1207 ±
105 0 . ± . RXCJ0232.2-4420 ±
141 1380 ±
71 0 . ± .
013 1798 ±
167 1152 ±
137 0 . ± . ZW3146 ±
49 1206 ±
26 0 . ± .
010 2040 ±
77 1293 ±
22 0 . ± . RXCJ0043.4-2037 ±
157 1068 ±
82 0 . ± .
032 1472 ±
95 982 ±
133 0 . ± . RXCJ0516.7-5430 ±
246 1273 ±
114 0 . ± .
022 1767 ±
112 1135 ±
57 0 . ± . RXCJ1131.9-1955 ±
206 1325 ±
93 0 . ± .
023 2513 ±
271 1475 ±
97 0 . ± . Notes.
These estimates refer to the mass models obtained with two different methods (see Table 2) and are evaluated at the ovedensities determinedfrom the total (i.e. dark + gas) mass profiles. All the quoted errors are at σ level.18. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 4.
Median deviations measured in the distribution of c , M DM and f gas . Dataset (ˆ c − c ) /c ( ˆ M DM − M DM ) /M DM ( ˆ f gas − f gas ) /f gas Method 2 − .
013 +0 .
008 +0 . M2 +0 . − .
017 +0 . T − . − .
036 +0 . fit n gas +0 .
001 +0 .
011 +0 . P out − .
011 +0 . − . at R ( − . , +0 . − . , +0 . − . , +0 . Method 2 − − .
015 +0 . M2 − − .
018 +0 . T − − .
046 +0 . fit n gas − +0 . − . P out − +0 . − . at R − ( − . , +0 . − . , +0 . Method 2 − − .
073 +0 . M2 − − .
013 +0 . T − − .
059 +0 . fit n gas − +0 .
004 +0 . P out − +0 . − . at R − ( − . , +0 . − . , +0 . Notes.
The deviations are measured with respect to the estimates obtained from the combined M1 + M2 profile with the Method 1 for the wholesample of 44 clusters. Dataset: (
Method 2 ) Method 2 is used for mass reconstruction; (M2) only the T ( r ) profile from M2 is used; ( T D ) thedeprojected spectral measurements of T ( r ) are used in Method 1 instead of the projected estimates of T model (see Sect. 3); (fit n gas ) a model fittedto the gas density profile is used in Method 1 ; ( P out ) the outer value of the pressure is not fixed. 19. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 5.
Best-fit values of the c − M relation. Dataset c A B σ log c All objects (44 clusters)
Method 1 - WeightedMean . +0 . − . . ± . − . . / . Method 1 - FITEXY . ± .
07 0 . ± . − . ± .
023 0 . / . Method 1 - BCES . ± .
18 0 . ± . − . ± .
071 0 . / . Method 1 - LINMIX . ± .
21 0 . ± . − . ± .
077 0 . / . Method 2 - WeightedMean . +0 . − . . ± . − . . / . Method 2 - FITEXY . ± .
05 0 . ± . − . ± .
015 0 . / . Method 2 - BCES . ± .
16 0 . ± . − . ± .
084 0 . / . Method 2 - LINMIX . ± .
20 0 . ± . − . ± .
067 0 . / . Selected objects (26 clusters)
Method 1 - WeightedMean . +0 . − . . ± . − . . / . Method 1 - FITEXY . ± .
17 0 . ± . − . ± .
050 0 . / . Method 1 - BCES . ± .
38 0 . ± . − . ± .
116 0 . / . Method 1 - LINMIX . ± .
47 0 . ± . − . ± .
125 0 . / . Method 2 - WeightedMean . +0 . − . . ± . − . . / . Method 2 - FITEXY . ± .
13 0 . ± . − . ± .
063 0 . / . Method 2 - BCES . ± .
57 0 . ± . − . ± .
259 0 . / . Method 2 - LINMIX . ± .
45 0 . ± . − . ± .
131 0 . / . only LEC objects (11 clusters) Method 1 - WeightedMean . +0 . − . . ± . − . . / . Method 1 - FITEXY . ± .
15 0 . ± . − . ± .
051 0 . / . Method 1 - BCES . ± .
52 0 . ± . − . ± .
229 0 . / . Method 1 - LINMIX . ± .
45 0 . ± . − . ± .
150 0 . / . Method 2 - WeightedMean . +0 . − . . ± . − . . / . Method 2 - FITEXY . ± .
11 0 . ± . − . ± .
054 0 . / . Method 2 - BCES . ± .
75 0 . ± . − . ± .
303 0 . / . Method 2 - LINMIX . ± .
36 0 . ± . − . ± .
133 0 . / . SimulationsB01 .
29 0 . − . D04 .
01 0 . − . S06 – all, relaxed . , .
86 0 . , . − . , − . N07 – all, relaxed . , .
33 0 . , . − . , − . M08 / WMAP-1 – all, relaxed . , .
18 0 . , . − . , − . M08 / WMAP-3 – all, relaxed . , .
41 0 . , . − . , − . M08 / WMAP-5 – all, relaxed . , .
56 0 . , . − . , − . Notes.
The best-fit values refer to equation 5 and are obtained by using (i) the linear least–squares fitting with errors in both variables (FITEXY),(ii) the linear regression method BCES, (iii) a Bayesian linear regression method (LINMIX). In the last column, the total ( σ tot = P Ni ( y i − A − Bx i ) /N ) and statistical ( σ stat = P Ni ǫ y i /N ) scatters are quoted, where y i = log ( c (1 + z )) , x i = log M , ǫ y i is the statistical erroron y i and N is the number of objects.20. Ettori et al.: Mass profiles and c − M DM relation in X-ray luminous galaxy clusters Table 6.
Cosmological constraints on σ and Ω m . Model N data γ Γ χ c σ Ω m χ adding f gas E01 26 . ± .
030 0 . ± . . +0 . − . . +0 . − . . ± .
042 0 . ± . . +0 . − . . +0 . − . . ± .
026 0 . ± . . +0 . − . . +0 . − . . ± .
040 0 . ± . . +0 . − . . +0 . − . . ± .
086 0 . ± . . +0 . − . . +0 . − . b c ) 26 . ± .
032 0 . ± . . +0 . − . . +0 . − . b M ) 26 . ± .
030 0 . ± . . +0 . − . . +0 . − . b c , b M ) 26 . ± .
032 0 . ± . . +0 . − . . +0 . − . M tot ) 26 . ± .
030 0 . ± . . +0 . − . . +0 . − . Notes.
These cosmological contraints are obtained from equations 6 and 7 corresponding to the confidence contours shown in Fig. 7. To representthe observed degeneracy, we quote the best-fit values of the power-law σ Ω γ m = Γ . Errors at σ (95.4%) level of confidence are indicated. Table B.1.
Results of the 10,000 MC runs of the c − M relation fitted using the expression in equation 5. Model N obj mean (rms) A mean (rms) B B obs ± σ P σ ( B obs ) P σ ( B obs ) N07 44 . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . . . − . . − . ± . Notes. B obs is the best-fit result quoted in Table 5. P σ ( B obs ) and P σ ( B obs ) indicate the percentage of MC runs that provides an estimate of B lower than B obs + 1 σ and B obs + 3 σσ