Planck/SDSS Cluster Mass and Gas Scaling Relations for a Volume-Complete redMaPPer Sample
Pablo Jimeno, Jose-Maria Diego, Tom Broadhurst, Ivan De Martino, Ruth Lazkoz
MMon. Not. R. Astron. Soc. , 1–13 (0000) Printed 2 June 2017 (MN L A TEX style file v2.2)
Planck /SDSS Cluster Mass and Gas Scaling Relations fora Volume-Complete redMaPPer Sample
Pablo Jimeno (cid:63) , Jose M. Diego , Tom Broadhurst , , I. De Martino , Ruth Lazkoz Department of Theoretical Physics and History of Science, University of the Basque Country UPV-EHU, 48040 Bilbao, Spain Instituto de F´ısica de Cantabria (CSIC-UC), Avda. Los Castros s/n, 39005 Santander, Spain IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36-5 48008 Bilbao, Spain
Draft version 2 June 2017
ABSTRACT
Using
Planck satellite data, we construct SZ gas pressure profiles for a large, volume-complete sample of optically selected clusters. We have defined a sample of over 8,000redMaPPer clusters from the Sloan Digital Sky Survey (SDSS), within the volume-complete redshift region 0 . < z < . Planck data over the full range of richness. Dividing thesample into richness bins we simultaneously solve for the mean cluster mass in eachbin together with the corresponding radial pressure profile parameters, employing anMCMC analysis. These profiles are well detected over a much wider range of clustermass and radius than previous work, showing a clear trend towards larger break radiuswith increasing cluster mass.Our SZ-based masses fall ∼
24% below the mass–richness relations from weak lens-ing, in a similar fashion as the “hydrostatic bias” related with X-ray derived masses.We correct for this bias to derive an optimal mass–richness relation finding a slope1 . ± .
04 and a pivot mass log ( M /M (cid:12) ) = 14 . ± . λ = 60. Finally, we derive a tight Y – M relation over a wide range of cluster mass,with a power law slope equal to 1 . ± .
07, that agrees well with the independentslope obtained by the
Planck team with an SZ-selected cluster sample, but extends tolower masses with higher precision.
Key words: cosmology: observations — dark matter — galaxies: clusters: general— galaxies: clusters: intracluster medium
Galaxy clusters are powerful cosmological probes that pro-vide complementary constraints in the era of “Precision Cos-mology”. They contribute accurate consistency checks andunique new competitive constraints because of the well un-derstood cosmological sensitivity of their numbers and clus-tering (Jain et al. 2013; Huterer et al. 2015; Dodelson & Park2014; Pouri et al. 2014; Pan & Knox 2015). The growth ofstructures has led the observational evidence to support darkenergy dominance today, in combination with complemen-tary constraints (Efstathiou et al. 1990; Lahav et al. 1991;Bahcall 2000; Allen et al. 2011; Carroll et al. 1992; Ostriker& Steinhardt 1995; Bahcall & Fan 1998). To realize their fullcosmological potential, large, homogeneous samples of clus-ters are now being constructed out to z (cid:39) (cid:63) E-mail: [email protected] surveys (Oguri et al. 2017; Benitez et al. 2014; Jimeno et al.2015, 2017).Currently the best direct lensing masses are limited torelatively small subsamples of X-ray and Sunyaev-Zel’dovich(SZ) selected clusters, totalling ∼
100 clusters (Umetsu et al.2014; Zitrin et al. 2015; Okabe & Smith 2016). One of themain efforts is focused on defining scaling relations betweenclusters with such weak lensing masses and the more widelyavailable X-ray, SZ and/or optical richnesses with the rea-sonable expectation that these relations may provide massproxies in the absence of direct lensing masses. Such prox-ies have a physical basis for clusters that appear to be viri-alised, so that X-ray temperature and emissivity profiles canprovide virial masses under hydrostatic equilibrium. Inde-pendently, the SZ distortion of the cosmic microwave back-ground (CMB) spectrum relates the density and temper-ature of cluster gas through inverse Compton scattering,and hence naturally anticipated to scale approximately withcluster mass. The cleanest mass proxy is arguably provided c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J un Jimeno et al. by the number of member galaxies, the so called richness,implicit in the assumption that the dominant cluster darkmatter is collisionless like galaxies, and indeed the mass andthe richness are found to be closely proportional (Bahcall1977; Girardi et al. 2000).The cluster mass–richness relation, crucial in any at-tempt to use large number of clusters detected in the op-tical to constrain cosmological parameters, has been esti-mated in the past decade using cluster catalogues derivedfrom the Sloan Digital Sky Survey (SDSS, Gunn et al. 2006)data, like the MaxBCG (Koester et al. 2007), the GMBCG(Hao et al. 2010), the WHL12 (Wen et al. 2012), or themore sophisticated redMaPPer cluster catalogue, both inits SDSS (Rykoff et al. 2014) and DES (Rykoff et al. 2016)versions. This relation can be estimated directly obtainingcluster masses from X-rays, weak lensing, SZ effect or ve-locity dispersion measurements in clusters (Johnston et al.2007; Andreon & Hurn 2010; Saro et al. 2015; Sereno et al.2015; Saro et al. 2016; Simet et al. 2016; Mantz et al. 2016;Melchior et al. 2016), or indirectly, using numerical simula-tions (Angulo et al. 2012; Campa et al. 2017) or compar-ing the observed abundances or clustering amplitudes withmodel predictions (Rykoff et al. 2012; Baxter et al. 2016;Jimeno et al. 2017).The dynamical evolution and growth of galaxy clustersare driven by the dominant dark matter, but the relevantobservables depend on the physical state of the baryons.Hence scaling relations between clusters observables andmass are not direct, but have been predicted to follow physi-cally self-similar relations (Kaiser 1986; Kravtsov & Borgani2012) that have been tested observationally and with hydro-dynamical N-body simulations. Specifically, the integratedthermal SZ effect (Sunyaev & Zeldovich 1972), the X-rayluminosity, and the temperature are predicted to scale withthe mass of the galaxy clusters as Y ∝ M / , L X ∝ M / and T ∝ M / , respectively. While simulations agree withthe self-similar model (White et al. 2002; da Silva et al. 2004;Motl et al. 2005; Nagai 2006; Wik et al. 2008; Aghanim et al.2009), X-ray and SZ observations have uncovered departuresfrom self-similarity that may be explained by complicationsdue to cluster mergers, including shocked gas, cool gas cores,and energy injection from active galactic nuclei (AGN)(Voit2005; Arnaud et al. 2005, 2007; Pratt et al. 2009; Vikhlininet al. 2009; De Martino & Atrio-Barandela 2016). Differ-ences between the observations and purely gravitationallypredicted scaling relations then provide insights into theinteresting physics of the intracluster medium (Bonamenteet al. 2008; Marrone et al. 2009; Arnaud et al. 2010; Melinet al. 2011; Andersson et al. 2011; Comis et al. 2011; Czakonet al. 2015).In practice, samples of strong SZ-selected clusters thatare also bright X-ray sources are currently being used tocalibrate the SZ–mass relation (Arnaud et al. 2010; PlanckCollaboration et al. 2013a, 2014d; Saro et al. 2015), but,since such clusters are often out of hydrostatic equilibriumfor the reasons mentioned above, an SZ–mass scaling rela-tion requires a correction for “hydrostatic mass bias” (Nagaiet al. 2007b; Zhang et al. 2010; Shi & Komatsu 2014; Sayerset al. 2016). This hot gas related bias can be broadened byother systematics like object selection process or by tem-perature inhomogeneities in X-ray measurements. Anotherapproach to calibrate the masses of the cluster sample is to stack clusters in terms of richness and measure the SZ sig-nal as a function of richness. This was done first by PlanckCollaboration et al. (2011) using the MaxBCG catalogueand stacking the Planck data, and more recently by Saroet al. (2016) using an initial sample of 719 DES clusters withSouth Pole Telescope (SPT) SZ data and assuming variouspriors to extract the SZ signal.In this work, we extract the
Planck
SZ signal from ∼ . < z < . Planck multi-frequency data over a wide rangeof cluster richness. We only need to assume a weak priorfor the global gas fraction using X-ray measurements to si-multaneously derive a more “self-sufficient” method to de-rive SZ pressure profiles and the corresponding mean clustermasses binned by richness. Comparing masses derived thisway with those expected from weak lensing mass–richnessrelations found in the literature, we derive the level of in-trinsic bias for our sample and we then derive both a debi-ased mass–richness and a Y – M relation describing ourobservational results.This paper is organised as follows. In Sec. 2 we describethe data that we use in our analysis, namely the redMaPPercluster catalogue and Planck
HFI maps. We present the ba-sic theory associated to the SZ effect in Sec. 3, as well as themass–richness relation and the different models that we con-sider. In Sec. 4 we process the
Planck data to obtain SZ mapsgiven in terms of the Compton parameter y and use them toconstrain, through a joint likelihood analysis, the universalpressure profile parameters and the mean masses of the clus-ter subsamples considered. In Sec. 5 we make an estimationof the value of the mass bias and obtain the optimal mass–richness relation able to describe our bias-corrected masses.Finally, we use all the results obtained in the previous sec-tions to derive a Y – M relation in Sec. 6, and presentour conclusions in Sec. 7.Throughout this paper we adopt a fiducial flat ΛCDMcosmology with a matter density Ω m = 0 . H = 70 km s − Mpc − .We also consider h ≡ H / (100 km s − Mpc − ) and h ≡ H / (70 km s − Mpc − ). Cluster masses are givenin terms of an spherical overdensity ∆ c ( m ) with re-spect to the critical (mean) density of the Universe, M ∆ c ( m ) = (4 / π ∆ ρ c ( m ) ( z ) r . Unless stated otherwise, werefer to M c as M . The pioneering Sloan Digital Sky Survey (SDSS, Gunn et al.2006) combines photometric and spectroscopic observationsand has mapped the largest volume of the Universe in theoptical to date, covering around 14,000 deg of the sky. Theinformation obtained has been made public periodically viaData Releases (DR), and many different cluster catalogueshave been carefully constructed using very different cluster-finder algorithms (Koester et al. 2007; Hao et al. 2010; Wen c (cid:13) , 1–13 luster Scaling Relations et al. 2012). We focus our analysis on one of these SDSS-based catalogues, the red-sequence Matched-filter Proba-bilistic Percolation cluster catalogue (redMaPPer, Rykoffet al. 2014), based on SDSS DR8 photometric data, as givenin the public 6.3 version (Rykoff et al. 2016). We use this cat-alogue because it offers a very low rate of projection effects( < z λ , and the richness λ ofeach cluster. In this catalogue, the richness estimations re-lies on a multi-colour self-training procedure that calibratesthe red-sequence as a function of redshift. The richness of acluster is defined as: λ = (cid:88) p mem θ L θ R , (1)where p mem is the membership probability of each galaxyfound near the cluster, and θ L and θ R are luminosity andradius-dependent weights. A more in-depth explanation ofthe algorithm features can be found in Rykoff et al. (2014)and Rozo et al. (2015b).The resulting redMaPPer catalogue covers an effectivearea of 10,401 deg , and contains 26,111 clusters in the0 . (cid:54) z photo (cid:54) .
55 redshift range. Finally, it should bementioned that this catalogue is volume-complete up to z (cid:46) .
33, and has a richness cutoff of λ/S ( z ) >
20, where S ( z ) is the “scale factor” that relates the richness with thenumber of observed galaxies above the magnitude limit ofthe survey, λ/S ( z ). This detection threshold corresponds toa mass limit of approximately M (cid:62) . × M (cid:12) . We combine the above optically selected cluster sample fromSDSS with the all-sky temperature maps derived by the
Planck space mission (Planck Collaboration et al. 2014a).Although these maps have already been used to constructcatalogues of SZ sources (Planck Collaboration et al. 2014b)and an all-sky Compton y parameter map (Planck Collabo-ration et al. 2014e), we reprocess them for our own purposes.To obtain the Compton parameter maps required inour analysis, we use the Planck full mission High-FrequencyInstrument maps (HFI, Planck Collaboration et al. 2014c)at 100, 143, 217, and 353 GHz. These maps are providedin
HEALPix format (G´orski et al. 2005), with a pixelisationof N side = 2,048, which correspond to a pixel resolution of ∼ . Planck effective beams for each of the100, 143, 217, and 353 GHz channels can be approximatedby circular Gaussians with FWHM values of 9.66, 7.27, 5.01and 4.86 arcmin, respectively. To compute the contribution of the SZ signal in the
Planck temperature maps, we alsomake use of the spectral transmission information of each ofthese frequency channels, as given in Planck Collaborationet al. (2014c).
Here we briefly introduce the equations that describe thethermal Sunyaev-Zel’dovich (SZ) effect. For a derivation,we refer the reader to the papers of Sunyaev & Zel’dovich(1980); Rephaeli (1995), or the more recent work by Birkin-shaw (1999), Carlstrom et al. (2002) and Diego et al. (2002).Ignoring relativistic corrections, the SZ spectral distortionof the CMB, expressed as temperature change, is:∆ T SZ T CMB = g ( x ) y , (2)where x = ( h ν ) / ( k B T CMB ) is the dimensionless frequency, y is the Compton parameter, and g ( x ) = (cid:18) x e x + 1 e x − − (cid:19) . (3)The Compton parameter y is equal to the optical depth, τ e , times the fractional energy gain per scattering, and isgiven by: y = σ T m e c (cid:90) ∞ P ( l ) d l , (4)where σ T is the Thomson cross section and P is the in-tracluster pressure produced by free electrons. Integratingover the solid angle of the cluster one obtains the integratedCompton parameter: Y = (cid:90) Ω y dΩ = D − A σ T m e c (cid:90) ∞ d l (cid:90) A clu P ( l ) d A , (5)where A clu is the area of the cluster in the plane of the sky,and D A ( z ) is the angular diameter distance at redshift z .Assuming an spherical model for the cluster, we havethat the Compton parameter y at a distance r from thecenter of the cluster is equal to: y ( r ) = σ T m e c (cid:90) ∞−∞ P (cid:16)(cid:112) r (cid:48) + r (cid:17) d r (cid:48) , (6)and thus the integrated Compton parameter Y , obtainedintegrating to a distance R from the center of the cluster, isgiven by: Y ( R ) = (cid:90) R π y ( r ) r d r , (7)which has units of Mpc . It should be noted that, as y is aprojected along the line of sight quantity, Y is the so called“cylindrical” integrated Compton parameter Y cyl , and notthe “spherical” integrated Compton parameter, which wouldbe obtained directly from the pressure profile doing: Y sph ( R ) = σ T m e c (cid:90) R π P ( r ) r d r . (8)In practice, we work with Y cyl when dealing with obser-vations, as this is the quantity that can be measured fromthe data, and we use Y sph when dealing with models. Once c (cid:13)000
Here we briefly introduce the equations that describe thethermal Sunyaev-Zel’dovich (SZ) effect. For a derivation,we refer the reader to the papers of Sunyaev & Zel’dovich(1980); Rephaeli (1995), or the more recent work by Birkin-shaw (1999), Carlstrom et al. (2002) and Diego et al. (2002).Ignoring relativistic corrections, the SZ spectral distortionof the CMB, expressed as temperature change, is:∆ T SZ T CMB = g ( x ) y , (2)where x = ( h ν ) / ( k B T CMB ) is the dimensionless frequency, y is the Compton parameter, and g ( x ) = (cid:18) x e x + 1 e x − − (cid:19) . (3)The Compton parameter y is equal to the optical depth, τ e , times the fractional energy gain per scattering, and isgiven by: y = σ T m e c (cid:90) ∞ P ( l ) d l , (4)where σ T is the Thomson cross section and P is the in-tracluster pressure produced by free electrons. Integratingover the solid angle of the cluster one obtains the integratedCompton parameter: Y = (cid:90) Ω y dΩ = D − A σ T m e c (cid:90) ∞ d l (cid:90) A clu P ( l ) d A , (5)where A clu is the area of the cluster in the plane of the sky,and D A ( z ) is the angular diameter distance at redshift z .Assuming an spherical model for the cluster, we havethat the Compton parameter y at a distance r from thecenter of the cluster is equal to: y ( r ) = σ T m e c (cid:90) ∞−∞ P (cid:16)(cid:112) r (cid:48) + r (cid:17) d r (cid:48) , (6)and thus the integrated Compton parameter Y , obtainedintegrating to a distance R from the center of the cluster, isgiven by: Y ( R ) = (cid:90) R π y ( r ) r d r , (7)which has units of Mpc . It should be noted that, as y is aprojected along the line of sight quantity, Y is the so called“cylindrical” integrated Compton parameter Y cyl , and notthe “spherical” integrated Compton parameter, which wouldbe obtained directly from the pressure profile doing: Y sph ( R ) = σ T m e c (cid:90) R π P ( r ) r d r . (8)In practice, we work with Y cyl when dealing with obser-vations, as this is the quantity that can be measured fromthe data, and we use Y sph when dealing with models. Once c (cid:13)000 , 1–13 Jimeno et al. a pressure profile has been adopted, any measurement of Y cyl ( n r ) can be straightforwardly converted in terms of Y cyl ( r ), and the latter to Y sph ( r ). We refer the readerto appendix A of Melin et al. (2011) for a more detailedexplanation of how to convert between definitions.Finally, as y is dimensionless, Y can also be expressedin units of arcmin : Y [arcmin ] = D A ( z ) − (cid:18) × π (cid:19) Y [Mpc ] (9)From now on, we refer to Y sph ( r ) as Y , given inMpc units. In this work we adopt the generalised NFW (GNFW) “uni-versal pressure profile” proposed by Nagai et al. (2007b),that has a flexible double power-law form: P ( x ) = P ( c x ) γ [1 + ( c x ) α ] ( β − γ ) /α , (10)where x = r/r is the scaled dimensionless physical radius.The physical pressure is given by: P ( x ) = P (cid:18) M × h − M (cid:12) (cid:19) α p P ( x ) , (11)where: P =1 . × − E ( z ) / × (cid:18) M × h − M (cid:12) (cid:19) / h keV cm − , (12)and α p = 0 .
12 accounts for the deviation from the self-similar scaling model (Arnaud et al. 2010). The value of r is given by: 43 π ρ c ( z ) r = M , (13)where ρ c ( z ) is the critical density of the Universe at redshift z , defined as: ρ c ( z ) = 3 H E ( z ) πG , (14)and E ( z ) = Ω m (1 + z ) + (1 − Ω m ).From Eq. 10, it is clear that the slopes of the pressureprofile are given, at different r -scaled distances, by γ for x (cid:28) /c , α for x ∼ /c , and β for x (cid:29) /c . In ouranalysis and following the approach by Planck Collaborationet al. (2013a), we leave P , c , α , and β as free parameters.The low resolution of the Planck data does not have thepower to constrain γ , so we fix it to γ = 0 .
31, value obtainedby Arnaud et al. (2010) from a sample of 33
XMM-Newton
X-ray local clusters in the r < r range.
To improve our analysis, we use established results regardingthe global gas fraction f gas in clusters, particularly, those byPratt et al. (2009), who derived a mass–gas fraction relationusing precise hydrostatic mass measurements of 41 Chandra and
XMM-Newton clusters (Vikhlinin et al. 2006; Arnaudet al. 2007; Sun et al. 2009), which is also in good agree-ment with the results obtained from the
REXCESS sample (B¨ohringer et al. 2007). According to their analysis, theseclusters, whose masses range from 10 M (cid:12) to 10 M (cid:12) , fol-low the mean mass–gas fraction relation:ln (cid:16) f gas , E ( z ) − / (cid:17) = ( − . ± . . ± .
03) ln (cid:18) M × M (cid:12) (cid:19) . (15)To compute the gas fraction we first need to compute thegas mass: M gas , = (cid:90) r µ e m u n e ( r ) 4 π r d r , (16)where µ e = 1 .
15 is the mean molecular weight per free elec-tron, m u is the atomic mass unit, and n e ( r ) is the elec-tron density. Because the intra-cluster pressure is given by P ( r ) = n e ( r ) k B T , assuming an isothermal model for thecluster one can directly derive M gas , from the adoptedpressure profile (Eq. 11).For the temperature, we use the mean mass–temperature relation given by Lieu et al. (2016):log (cid:18) M E ( z ) h − M (cid:12) (cid:19) = (cid:0) . +0 . − . (cid:1) + (cid:0) . +0 . − . (cid:1) log (cid:18) k B T keV (cid:19) , (17)which was obtained combining weak lensing mass estimateswith Chandra and
XMM-Newton temperature data of 38clusters from the XXL survey (Pacaud et al. 2016), 10 clus-ters from the COSMOS survey (Kettula et al. 2013), and48 from the Canadian Cluster Comparison Project (CCCP,Mahdavi et al. 2013; Hoekstra et al. 2015), spanning a tem-perature range T (cid:39) −
10 keV.It is worth mentioning that if an isothermal model isassumed and we consider that M gas ∝ f b M , where M is thetotal cluster mass and f b is the baryon gas fraction, fromEq. 5 we have that the integrated Compton parameter scalesas Y ∝ f b M T D − A . However, even clusters in hydrostaticequilibrium are not strictly isothermal, and temperaturesare commonly observed to drop by a factor of ∼ r (cid:46) −
200 kpc because of strong radiative cool-ing, described best by a broken power law with a transitionregion (Vikhlinin et al. 2006). In any case, these scales arenot resolved by
Planck and in our analysis the assumptionthat the temperature is constant is a good approximationfor the radial scales considered in this work.
In the redMaPPer catalogue, for each cluster the 5 mostprobable central galaxies (CGs) are provided with their cor-responding centering probabilities. Usually, there is one CGwith a much higher probability of being the real CG thanthe other 4, so we consider the most probable CG to bethe center of the cluster. In any case, it is now known that,because clusters are still evolving systems, CGs do not al-ways reside at the deepest part of the DM halo potentialwell (von der Linden et al. 2007), but sometimes have highpeculiar velocities, are displaced with respect to the peak ofthe X-ray emission (Rozo & Rykoff 2014), or are wronglyidentified satellite galaxies (Skibba et al. 2011). c (cid:13) , 1–13 luster Scaling Relations In stacked measurements on clusters, miscentering isone of the main sources of noise, and should be taken intoaccount. When modelling the SZ signal coming from stackedsamples of clusters, we introduce this effect considering theresults obtained by Johnston et al. (2007), who found a CG-center offset distribution that could be fitted by a 2D Gaus-sian with a standard deviation of σ = 0 . h − Mpc for theCGs that were not accurately centered, that occurs between20 and 40 per cent of the time as a function of cluster rich-ness, with a probability p mc ( λ ) = (2 .
13 + 0 . λ ) − .However, it should be noted that this value of0 . h − Mpc is about 2 arcmin at z ∼ .
2, scale well be-low the resolution of the
Planck data we work with, so wedo not expect this miscentering to introduce a high level ofnoise in our stacked measurements of the SZ effect.
Usually referred to as hydrostatic equilibrium (HE) masses,in their derivation there is an implicit assumption that thepressure is purely thermal. However, we may expect a non-negligible contribution to the total pressure from bulk andturbulent gas motions related to structure formation his-tory, magnetic fields, and AGN feedback (Shi & Komatsu2014; Planck Collaboration et al. 2014d). Such non-thermalcontributions to the total pressure would therefore causemasses estimated using X-ray or SZ observations to be bi-ased low with uncertain estimates ranging between 5% to20% (Nelson et al. 2014; Nagai et al. 2007a; Rasia et al.2006; Sembolini et al. 2013).We simply relate the HE mass estimates M HE , ob-tained from our SZ observations to true masses M through a simple mass independent bias: M HE , = (1 − b ) M , (18)where (1 − b ) is the bias factor. This term can include notonly the bias coming from departures from HE, but fromobservational systematics or sample selection effects. In order to explore the connection between the mass andthe optical richness λ in clusters, i.e., the number of galax-ies contained within them, one needs to assume a form todescribe the relation between cluster richness and mass. Weconsider the standard power law cluster mass–richness meanrelation: (cid:104) M | λ (cid:105) = M (cid:18) λλ (cid:19) α M | λ , (19)where M is a reference mass at a given value of λ = λ ,and α M | λ is the slope of the mass–richness relation. In ourcase, we consider λ = 60.To compute the mean masses of our cluster subsamples,we first consider the probability P ( M | λ obs ) of having a givenvalue of the mass M for a cluster with λ obs : P ( M | λ obs ) = (cid:90) P ( M | λ ) P ( λ | λ obs ) d λ , (20)with P ( M | λ ) a delta function, as the relation between massand richness is given by Eq. 19. Following the usual approach (Lima & Hu 2005), we consider that P ( λ | λ obs ) follows a log-normal distribution: P ( λ | λ obs ) = 1 (cid:113) πσ λ | λ obs exp[ − x ( λ )] , (21)where: x ( λ ) = ln λ − ln λ obs (cid:113) σ λ | λ obs , (22)and σ ln λ | λ obs is the fractional scatter on the halo richness atfixed observed richness, which is assumed to be constantwith cluster redshift and richness. Because P ( M | λ ) is adelta function, and considering Eq. 19, we also have that σ λ | λ obs = σ M | λ obs . So, the mean mass (cid:104) M (cid:105) of one of therichness bins considered, with λ ∈ [ λ obs i , λ obs i +1 ] and contain-ing N i clusters, is given by: (cid:104) M (cid:105) = 1 N i N i (cid:88) j =1 (cid:90) M P ( M | λ obs j ) d M . (23)Finally, when dealing with mean values we can workin terms of (cid:104) ln M | λ (cid:105) instead of (cid:104) M | λ (cid:105) , with (cid:104) ln M | λ (cid:105) =ln (cid:104) M | λ (cid:105) − . σ M | λ obs . We refer the reader to Rozo et al.(2009a) and Simet et al. (2016) for a discussion of this trans-formation. We choose our parametrisation (Eq. 19) becausethe resulting mean relation is less affected by the uncertaintyin σ ln M | λ obs .Since it was made public, there have been multiple at-tempts to constrain in different ways the parameters of thisrelation using the redMaPPer cluster catalogue (Rykoff et al.2012; Baxter et al. 2016; Farahi et al. 2016; Li et al. 2016;Melchior et al. 2016; Miyatake et al. 2016; Saro et al. 2015,2016; Simet et al. 2016; Jimeno et al. 2017). Although someof these works introduced a redshift dependence in the mass–richness relation, it was weakly constrained in all cases, andcompatible with no redshift evolution at all. Given the smallredshift range in which we work, redshift evolution is not im-portant for our analysis and we refer our result to the meanredshift of our sample, z = 0 . We first divide the redMaPPer cluster catalogue in 6 inde-pendent log-spaced richness bins, and take all those clustersthat reside within the 0 . < z < .
325 volume-completeredshift region. This leaves a total of 8,030 clusters, dis-tributed in number and mean richness as shown in Table 2.Then, for each cluster subsample, we produce and stackthe ν = 100, 143, 217 and 353 GHz 2.5 deg × Planck maps associated to the clusters in each subsample and pro-duce the corresponding SZ maps following a technique sim-ilar to the one used in Planck Collaboration et al. (2016),based on internal linear combinations (ILC) of the four dif-ferent HFI maps. In our case we do not use the 70 GHzSZ map, as we prefer to smooth all the maps to a com-mon higher 10 arcmin resolution instead. We also use the M − M combination to clean the M map, where M ν is the Planck map at frequency ν . To convert from ∆ T SZ to c (cid:13)000
325 volume-completeredshift region. This leaves a total of 8,030 clusters, dis-tributed in number and mean richness as shown in Table 2.Then, for each cluster subsample, we produce and stackthe ν = 100, 143, 217 and 353 GHz 2.5 deg × Planck maps associated to the clusters in each subsample and pro-duce the corresponding SZ maps following a technique sim-ilar to the one used in Planck Collaboration et al. (2016),based on internal linear combinations (ILC) of the four dif-ferent HFI maps. In our case we do not use the 70 GHzSZ map, as we prefer to smooth all the maps to a com-mon higher 10 arcmin resolution instead. We also use the M − M combination to clean the M map, where M ν is the Planck map at frequency ν . To convert from ∆ T SZ to c (cid:13)000 , 1–13 Jimeno et al. λ : [20, 29) λ : [29, 43) λ : [43, 63) λ : [63, 93) λ : [93, 136) λ : [136, 200) Figure 1. × y maps in 6 richness bins (with the richness ranges indicated oneach stamp), obtained through the process described in Sec. 4.1,showing the SZ effect is strongly detected in these cluster stacksover the full richness range. These subsamples are defined in thevolume-complete redshift region 0 . < z < . y units (Eq. 2), we compute the different effective spectralresponses integrating the expected SZ spectrum (Eq. 3) overeach Planck bandpass.Our final SZ maps of the cluster subsamples considered,given in terms of the Compton parameter y and shown inFig. 1, are produced as a combination of the SZ and SZ maps, weighting them by the inverse of the variance of eachmap. This particular combination has been proposed by the Planck team to maximise the signal-to-noise of the SZ com-ponent whilst minimising the contamination from Galacticemission and extragalactic infrared emission within clusters(Planck Collaboration et al. 2016).
We now combine the data from
Planck and the constraintsimposed by the gas fractions to perform a likelihood anal-ysis that enables us to constrain the pressure profile pa-rameters and the mean masses of the 6 cluster subsam-ples. We explore the values of our 4+6 dimensional model φ = ( P , c , α , β, M , ... , M ) through a Monte CarloMarkov Chain (MCMC) analysis.For a given value of r ( M , z ) we measure theCompton parameter y within a disk of radius x = 0 .
35 and in6 annulus given by radii x i and x i +1 , where the x i = r i /r values are log-spaced between 0.35 and 3.5. This results in a y vector of 7 values. Then, to account for the background wesubtract the mean value of the signal obtained from an an-nulus of radii x out and x out + x FWHM , where x out = 3 .
5, and x FWHM = θ FWHM D A ( z ) /r corresponds in x -space to the θ FWHM = 10 arcmin FWHM effective resolution of the SZmaps. The values of r used to both model the signal andmeasure it from the data are obtained from M throughEq. 13.The log-likelihood employed has the form:ln L ( φ ) = (cid:88) k =1 ln L k ( y k | φ k ) , (24) r/r r / r Figure 2.
Normalized covariance matrix used in the likelihoodanalysis for a mass of M = 5 × M (cid:12) , corresponding to r = 1 .
11 Mpc. At the mean redshift of the sample and for thesame mass, r/r = 3 . where φ k = ( P , c , α , β, M k ), y k is the data vectorobtained from the cluster subsample k , andln L k ( y k | φ k ) ∝ − χ ( y k , φ k , C ) , (25)where C is the covariance matrix, and χ ( y k , φ k , C ) = (cid:16) y k − µ ( φ k ) (cid:17) C − (cid:16) y k − µ ( φ k ) (cid:17) T , (26)with µ the model values drawn from φ k . To model the signal,for each φ k configuration we produce mock maps of theCompton parameter y as a function of redshift. Then, wemimic miscentering effects adding to the mock maps thesame maps smoothed with a 2D Gaussian of width σ mc =0 . h − Mpc, and weighted by p mc ( λ ) = (2 .
13 + 0 . λ ) − .Finally, we produce a weighted map integrating over theredshift distribution of the subsample considered, convolveit with a 10 arcmin FWHM Gaussian, and perform the samemeasurements made in the Planck data maps.It should be noted that the mock y maps that we createto fit the observed signal are generated from the pressureprofile as given by Eq. 11 and for a given total model mass M k . Hence, to compute ln L k ( y k | φ k ) we do not rely on thegas mass or f gas .The covariance matrixes C are estimated from N R =1,000 patches randomly chosen within the redMaPPer foot-print, where the same measurement described above is done.As this measurement depends on the M value proposed,the 7 × C ij ( M ) = 1 N R − N R (cid:88) n =1 ( y ni − (cid:104) y i (cid:105) )( y nj − (cid:104) y j (cid:105) ) . (27)As an example, the covariance matrix obtained consideringa mass of M = 5 × M (cid:12) is shown in Fig. 2.Finally, we consider the gas fraction constraints intro-ducing a Gaussian prior ∼ N ( f gas , σ f gas ), where f gas is es-timated from the results of Pratt et al. (2009), as explained c (cid:13) , 1–13 luster Scaling Relations Table 1.
The recovered values of the GNFW universal pressureprofile parameters, together with the best fit values.
Parameter Mean value Best fit P . ± .
00 5 . c . ± .
95 1 . α . ± .
25 0 . β . ± .
66 3 . Table 2.
Richness range, number of clusters, mean richness andmean masses recovered from the SZ signal of the cluster subsam-ples studied.
Richness range N (cid:104) λ (cid:105) (cid:10) M HE , (cid:11) [10 M (cid:12) ] [136 , . ± . , . ± . ,
93) 293 74.3 2 . ± . ,
63) 902 50.8 1 . ± . ,
43) 2308 34.6 1 . ± . ,
29) 4440 23.8 0 . ± . above. In each MCMC step we compute, following the pro-cedure described in Sec. 3.3, the 6 gas fractions associatedto a given set φ of pressure profile parameters and masses,and then use Eq. 15 to model the expected gas fraction f gas for each value of M k , which we use for the prior. To es-timate σ f gas , we add in quadrature the errors derived fromthe uncertainties on both the mass–gas fraction (Eq. 15)and the mass–temperature (Eq. 17) relations, which are ob-tained propagating through a Monte Carlo (MC) method.Because we have decided to be as conservative as possibleon the relations employed and the resulting uncertainties inthis analysis are large, we notice that the contribution thatthis prior has in the final estimated values of the pressureprofile is small, only limiting those models where the gasfraction f gas , takes values below 0.05 or above 0.20 formasses in the ∼ − M (cid:12) range.For all P , c , α , β , M , ... , M , we consider flatuninformative priors ∼ U ( ∞ , −∞ ), allowing for a widerange of different model-masses configurations. The derived posterior probabilities of the GNFW universalpressure profile parameters P , c , α and β are displayedin Fig. 3. To compute the center (mean) and the scale (dis-persion) of the marginalised posterior distributions, we usethe robust estimators described in Beers et al. (1990). Thevalues obtained with this method, together with the best fitvalues, are listed in Table 1.The y radial profiles recovered for the cluster subsam-ples considered are displayed in Fig. 4, together with thejoint best fit y profile model obtained, as shown in Table 1.Finally, the mean masses recovered for the cluster sub-samples are listed in Table 2. For this model ([ P , c , γ, α, β ]), and using the XMM-Newton
X-ray data up to r/r (cid:46) z < .
2) clusters covering the 10 M (cid:12) XMM-Newton data, the bestfit values [6.41, 1.81, 0.31, 1.33, 4.13], where the Arnaudet al. (2010) value of γ = 0 . 31 had been previously fixed, aswe have done in our analysis. By comparison with pure esti-mates, it can be seen in Fig. 5 of Planck Collaboration et al.(2013a), that a large uncertainty is present in the estimationof these parameters, with a high degeneracy between them,which we attribute to the larger radial extent of our SZ pro-files and the wider cluster mass coverage of our sample.We also notice that the external slope, β , derived inthis work points to shallower profiles in the outer part of theclusters. This is in agreement with the results derived in theComa and Virgo clusters (Planck Collaboration et al. 2013b,2016) based on Planck data where the SZ signal extendsto beyond the virial radius in those clusters. Like in thosepapers, we can reach similar distances from the virial radiiand be sensitive to the external slope of clusters where thesignal from neighbouring merging filaments is expected toflatten the SZ profile. We now obtain the value of the mass bias comparing ourresults with recent weak lensing mass derivations that makeuse of stacked subsamples of the redMaPPer cluster cata-logue. We consider the results by Simet et al. (2016, S16from now on), who, making use of SDSS data for the weaklensing mass estimates of the redMaPPer clusters in the0 . < z < . 33 redshift region, obtained one of the mostprecise mass–richness relations to date, and those by Mel-chior et al. (2016, M16 hereafter), which, using DES SV datato make stacked measurements of the weak lensing shear asa function of mean cluster richness and mean cluster red-shift, measured a redshift-dependent mass–richness relationof comparable precision to that of S16. Because both of themare given in terms of M m , we consider a NFW profileNavarro et al. (1996) and the mass–concentration relationof Bhattacharya et al. (2013) to convert between differentmass definitions.At the same time, we derive the optimal mass–richnessrelation able to describe our data, considering the probabil-ity distribution of the bias obtained from the comparisonwith S16 and M16 results. To do so, we first compare ourmasses with the masses estimated by S16 and M16 with ajoint likelihood:ln L bias = ln L S16 + ln L M16 , (28)where L S16 and L M16 are computed comparing our 6 masses c (cid:13)000 33 redshift region, obtained one of the mostprecise mass–richness relations to date, and those by Mel-chior et al. (2016, M16 hereafter), which, using DES SV datato make stacked measurements of the weak lensing shear asa function of mean cluster richness and mean cluster red-shift, measured a redshift-dependent mass–richness relationof comparable precision to that of S16. Because both of themare given in terms of M m , we consider a NFW profileNavarro et al. (1996) and the mass–concentration relationof Bhattacharya et al. (2013) to convert between differentmass definitions.At the same time, we derive the optimal mass–richnessrelation able to describe our data, considering the probabil-ity distribution of the bias obtained from the comparisonwith S16 and M16 results. To do so, we first compare ourmasses with the masses estimated by S16 and M16 with ajoint likelihood:ln L bias = ln L S16 + ln L M16 , (28)where L S16 and L M16 are computed comparing our 6 masses c (cid:13)000 , 1–13 Jimeno et al. β c α P β c α Figure 3. Marginalised posterior distributions of the universal pressure profile parameters P , c , α and β , as obtained in ourMCMC analysis. Contours represent 68% and 95% confidence levels. The best fit values, corresponding to [ P , c , γ, α, β ] =[5 . , . , . , . , . h − / , 1.18, 0.31,1.05, 5.49], is marked with a red triangle, and Planck Collaboration et al. (2013) best fit, [6.41, 1.81, 0.31, 1.33, 4.13], with a greendiamond. with the masses predicted by the S16 and the M16 mass–richness relations, with ln L ∝ − χ / 2, and: χ [(1 − b )] = (cid:0) M − M model (cid:1) C − (cid:0) M − M model (cid:1) T , (29)where the bias-corrected masses are computed as M = M HE , / (1 − b ) , following Eq. 18, and the S16 and M16 M model model masses are obtained evaluating the corre-sponding S16 or M16 mass–richness relation at the 6 clustersubsamples mean richnesses. To obtain the resulting bias-dependent mass–richnessrelation, we compare these bias-corrected masses with themasses predicted by a generic mass–richness relation, whichwe compute following the steps described in Sec. 3.6 andconsidering a given set of free parameters log M , α M | λ and σ M | λ , which we constrain. The global likelihood has theform: ln L = ln L bias + ln L M | λ , (30)where L M | λ is computed now through another chi-square c (cid:13) , 1–13 luster Scaling Relations r/r λ : [20 , , h λ i : 23 , N : 4440 M HE , = 0 . × M fl r/r λ : [29 , , h λ i : 34 , N : 2308 M HE , = 0 . × M fl r/r -7 -6 -5 y E ( z ) − / λ : [43 , , h λ i : 50 , N : 902 M HE , = 1 . × M fl λ : [63 , , h λ i : 74 , N : 293 M HE , = 2 . × M fl λ : [93 , , h λ i : 109 , N : 68 M HE , = 4 . × M fl -7 -6 -5 y E ( z ) − / λ : [136 , , h λ i : 157 , N : 19 M HE , = 6 . × M fl Figure 4. The values of y recovered at different scaled radii for the cluster subsets divided by richness, and ordered as in Fig. 1. Theerror bars are the square root of the diagonal elements of the covariance matrix. The red lines represent the prediction from the universalpressure profile with the best fit parameters, [ P , c , γ, α, β ] = [5 . , . , . , . , . Planck beam with respect to r for the 6 cluster subsamples. function similar to that of Eq. 29, but considering both thevalue of the bias and the values of the mass–richness relationparameters for the model masses.Because the values of the SZ-estimated masses are cor-related, this new covariance matrix C is obtained directlyfrom the MCMC analysis performed in Sec. 4, and used inthis new MCMC likelihood calculation. We also include theerrors coming from the S16 and M16 model uncertainties,adding them in quadrature to the diagonal of the covariancematrix.For log M , α M | λ and (1 − b ) we assume flat unin-formative priors, meanwhile for σ M | λ we assume the inversegamma distribution prior ∼ IG ( (cid:15), (cid:15) ), with (cid:15) = 10 − (An-dreon & Hurn 2010). We notice that this analysis, based on mean masses ratherthan individual measurements, is not able to constrain thevalue of σ M | λ . The value of the scatter, difficult to constrainin general, is usually found to be between 0.15 and 0.30. Saroet al. (2015) find σ ln M | λ obs = 0 . +0 . − . , meanwhile Rozo &Rykoff (2014) and Rozo et al. (2015a), comparing individualredMaPPer clusters with X-ray and SZ mass estimates, finda value σ ln M | λ obs ≈ . ± . M , α M | λ and (1 − b ) recovered from the MCMC is shown in Fig. 5. Asin the previous section, we use the Beers et al. (1990) esti-mator to obtain the values from the marginalised posteriordistributions. The results are shown in Table 5.2. Table 3. The confidence values of the mass–richness relationparameters and of the bias factor (1 − b ), together with the bestfit. α M | λ is the slope of the mass–richness relation, and log M is the pivot mass, evaluated at λ = 60 and given in terms of M c . The stacking analysis performed is not able to constrainthe value of σ M | λ , which is largely determined by the prior, andthus not shown. Parameter Mean value Best fit log ( M /M (cid:12) ) 14 . ± . 040 14 . α M | λ . ± . 04 1 . − b ) 0 . ± . 05 0 . σ M | λ - - We now compare the value of the bias that we recoverwith those found in the literature. We also compare the as-sociated bias-dependent mass–richness relation with thosemass–richness relations used to obtain it, and with the oneobtained recently by Baxter et al. (2016). We would like toremind the reader that our estimations of both the bias andthe mass–richness relation depend on the results of S16 andM16, and should not be considered independent of them.Comparing “observed” with “true” masses coming fromseveral numerical simulations, Planck Collaboration et al.(2014d) derived a mass-dependent bias with a mean valueof (1 − b ) = 0 . +0 . − . . However, a bias of the order of 40per cent was needed in order to reconcile Planck clusterscounts with CMB observations, and its value is still the fo-cus of intense research. von der Linden et al. (2014), compar- c (cid:13)000 05 0 . σ M | λ - - We now compare the value of the bias that we recoverwith those found in the literature. We also compare the as-sociated bias-dependent mass–richness relation with thosemass–richness relations used to obtain it, and with the oneobtained recently by Baxter et al. (2016). We would like toremind the reader that our estimations of both the bias andthe mass–richness relation depend on the results of S16 andM16, and should not be considered independent of them.Comparing “observed” with “true” masses coming fromseveral numerical simulations, Planck Collaboration et al.(2014d) derived a mass-dependent bias with a mean valueof (1 − b ) = 0 . +0 . − . . However, a bias of the order of 40per cent was needed in order to reconcile Planck clusterscounts with CMB observations, and its value is still the fo-cus of intense research. von der Linden et al. (2014), compar- c (cid:13)000 , 1–13 Jimeno et al. (1 − b ) α M | λ log ( M / M fl ) ( − b ) α M | λ Figure 5. Posterior probability distributions of the mass–richness parameters log M and α M | λ , and of the bias factor(1 − b ). The best fit location is marked with a white star. Themass–richness relation scatter, σ M | λ , cannot be constrained byour data and its posterior probability distribution is greatly in-fluenced by the posterior, and thus not shown. ing Planck cluster mass estimates with weak lensing massesfrom the Weighing the Giants (WtG) project, observed ahigher bias of (1 − b ) = 0 . ± . 06. In a similar way butusing 50 clusters from the Canadian Cluster ComparisonProject (CCCP), Hoekstra et al. (2015) found a value of(1 − b ) = 0 . ± . ± . 06. On the other hand, Smith et al.(2016) obtained a smaller bias of the order of the 5 per centfrom a sample of 50 clusters with X-ray and weak lensingmasses in the 0 . < z < . z > . 3. AlthoughSereno & Ettori (2016) found a value of the bias of the orderof 25 per cent for Planck masses with respect to their weaklensing masses, it was claimed that this value was stronglydependent on redshift, following Smith et al. (2016) results.Saro et al. (2016), following an approach similar to ours butusing stacked measurements of the SZ signal of 719 DESredMaPPer clusters with South Pole Telescope (SPT) data,needed a bias as high as (1 − b ) = 0 . ± . 05 to follow themodel predictions when the Arnaud et al. (2010) pressureprofile was assumed. However, this bias was reduced to therange (1 − b ) = 0 . − . λ > 80 were taken into account, or when other scaling re-lations were considered. More recently, and using a sampleof 35 Planck clusters that were within the area covered bythe CFHTLenS and RCSLenS photometric surveys, Serenoet al. (2017) found that the Planck estimated masses werebiased low by ∼ ± 11 per cent with respect to weak lensingmasses, consistent with our findings here.Regarding the mass–richness relation, in M16 a redshift-dependent mass–richness relation was derived with a slope 20 30 50 100 150 200 λ M [ M fl ] This workMelchior et al. (2016)Simet et al. (2016)Baxter et al. (2016) Figure 6. The blue shaded region represents our 1 σ confidenceinterval of the mean mass–richness relation with the parametri-sation shown in Table 5.2, derived from the mean bias-correctedmasses (black circles) of the cluster subsamples considered, dis-played as a function of the original sample mean richness. Theerrors on the M masses include the uncertainty on the valueof the bias. The two mass–richness relations that we have used toobtain an estimate of the bias, namely those derived by Melchioret al. (2016, M16) (red solid line) and Simet et al. (2016, S16)(black dashed line), are shown, together with Baxter et al. (2016)mass–richness relation (green dotted line). of 1 . ± . ± . 06 for a pivot mass log ( M m /M (cid:12) ) =14 . ± . ± . 022 at a pivot richness λ = 30 and z = 0 . 5. Evaluated at our mean redshift z = 0 . 245 andtheir pivot richness λ = 30, their mass–richness relationgives a mass log ( M m /M (cid:12) ) = 14 . ± . M m , we find at λ = 30 a valueof log ( M m /M (cid:12) ) = 14 . ± . . +0 . − . for a pivot mass of log ( M m /h − M (cid:12) ) =14 . ± . ± . 023 at λ = 40. Converting to theirunits, we find that our mass–richness relation at their pivotrichness yields a mass equal to log ( M m /h − M (cid:12) ) =14 . ± . . < z < . 33 redshift region, found a value ofthe slope of the mass–richness relation of 1 . ± . 16 anda pivot mass of ln( M m /M (cid:12) ) = 33 . ± . 18 at λ = 35and z = 0 . 25. In our mass–richness relation at their pivotrichness, we find ln( M m /M (cid:12) ) = 33 . ± . c (cid:13) , 1–13 luster Scaling Relations Y – M RELATION We now straightforwardly derive the Y – M relation byconsidering the values of P , c , α , β and (1 − b ) obtained,and assuming again a value of γ = 0 . 31. Following the usualnotation we have: E − / ( z ) (cid:20) Y − Mpc (cid:21) = 10 A (cid:20) (1 − b ) M × M (cid:12) (cid:21) B , (31)in which the redshift evolution is considered to be self-similar(Kaiser 1986). From the results previously obtained, we find: A = − . ± . ,B = 1 . ± . . The value of the slope is actually in good agreement withthe expectation from self-similarity, B ∼ / 3. This valueof the slope is within 1 σ from the results of B = 1 . ± . 08 obtained in Planck Collaboration et al. (2014d), witha normalisation factor of A = 0 . ± . 02, different fromour value of A but compensated by the difference in the(1 − b ) factor. Our result is also compatible with those ofSereno et al. (2015), who obtained values for the slope of 1.4- 1.9, albeit a relatively wide range of slope, by consideringsamples of clusters with weak lensing mass estimates.It should be noted that this is the mean Y – M rela-tion, so, although small in the SZ case, a scatter σ Y | M shouldbe considered when applying it to individual clusters, justas when converting observed richness into mass (Eqs. 20 and21). In Planck Collaboration et al. (2014d) a value of of thescatter of ∼ 15 per cent is considered, meanwhile in Serenoet al. (2015) a scatter of the the 15 - 30 per cent order isestimated.Finally, as a consistency check on our results, we mea-sure Y cyl (3 . r ) in the 6 y maps corresponding to the6 cluster subsamples, deriving the values of r from themean masses obtained in the MCMC analysis, listed in Ta-ble 2, and converting to Y using the universal pressureprofile as given by the P , c , α and β parametrisation ofTable 1. We derive the errors from 1,000 random patcheswithin the redMaPPer footprint in a similar way as we didin Sec. 4.2. We propagate the errors coming from the un-certainty in the pressure profile parameters, the masses andthe bias factor drawing a large enough number of samplesfrom the posterior probability distribution of the 4 + 6 + 1dimensional parameter space, repeating the measurements,and adding in quadrature the variance of the Y cyl (3 . r )values obtained.Our mean Y – M relation and the measured valuesof Y are shown in Fig. 7, where we also plot the rela-tion considered in Planck Collaboration et al. (2014d), withwhich good agreement is found. In comparison our work fol-lows a tighter relation, which is attributed to the relativelylarge size of our cluster sample. In this chapter we have presented and employed a newmethod to constrain simultaneously the GNFW universalpressure profile parameters and the masses of 6 differentrichness subsamples of clusters using stacked measurementsof the SZ effect. We then estimated both a mass–richness M [ M fl ] -6 -5 -4 Y E ( z ) − / [ M p c ] This workPlanck Collaboration et al. (2014d) Figure 7. The blue shaded region shows our 1 σ confidence in-terval of the scaling relation between Y and M , as given bythe posterior probability distribution of the pressure profile pa-rameters P , c , α and β , the masses, and the bias factor. Theblack circles dots are the observed Y cyl (3 . r ) values for thecluster subsamples, converted to Y using the pressure profileparameters of Table 1 and the masses of Table 2. The errors onthe M masses include the uncertainty coming from the bias.As the red shaded region, we plot the Planck Collaboration et al.(2014d) Y – M relation for comparison, where we have con-sidered not only the errors in the slope and the normalisation,but also the uncertainty coming from the bias factor correction. and a Y – M relation using weak lensing mass estimatesfound in the literature.Using the positions in the sky of ∼ . < z < . 325 volume-complete redshiftregion, we have produced and stacked the Planck full mis-sion SZ maps of 6 richness subsamples of clusters, and con-strained the common GNFW universal pressure profile pa-rameters and the mean masses of each subsample througha MCMC analysis, obtaining better constraints than previ-ous works, with P = 6 . ± . c = 1 . ± . α =0 . ± . 25 and β = 3 . ± . 66 for a fixed value of γ = 0 . . × M (cid:12) (cid:46) M (cid:46) × M (cid:12) range. The universal pressure profile parameters best fit isfound at [ P , c , γ, α, β ] = [5 . , . , . , . , . Chandra and XMM-Newton clusters.We also notice that the external slope, β , derived inthis work points to shallower profiles in the outer part of theclusters. This is in agreement with the results derived in theComa and Virgo clusters (Planck Collaboration et al. 2013b,2016) based on Planck data where the SZ signal extendsto beyond the virial radius in those clusters. Like in thosepapers, we can reach similar distances from the virial radiiand be sensitive to the external slope of clusters where thesignal from neighbouring merging filaments is expected toflatten the SZ profile.Then we have compared our SZ-based masses with inde-pendent weak lensing mass estimates from the mass–richnessrelations of Simet et al. (2016, S16) and Melchior et al. (2016,M16), finding a mean mass bias of (1 − b ) = 0 . ± . 05. This c (cid:13)000 05. This c (cid:13)000 , 1–13 Jimeno et al. value does not depend strongly on the richness subsampleconsidered, and is in good agreement with other independentrecent estimates of the HE bias. Because we are working inthe 0 . < z < . 325 redshift region it may not be surpris-ing that we differ with the higher redshift base estimate ofSmith et al. (2016), but this may be understood given theirclaim that this bias may be enhanced in the z > . α M | λ = 1 . ± . 04 for a pivot mass log ( M c /M (cid:12) ) =14 . ± . 041 evaluated at λ = 60 at the mean redshift of thesample, z = 0 . Y – M relation E − / ( z ) (cid:104) Y (cid:105) ∝ A (cid:104) (1 − b ) M (cid:105) B with a normalisationfactor A = − . ± . 04 and a slope B = 1 . ± . 07. 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