How does our choice of observable influence our estimation of the centre of a galaxy cluster? Insights from cosmological simulations
Weiguang Cui, Chris Power, Veronica Biffi, Stefano Borgani, Alexander Knebe, Giuseppe Murante, Dunja Fabjan, Geraint F. Lewis, Greg B. Poole
aa r X i v : . [ a s t r o - ph . GA ] D ec Mon. Not. R. Astron. Soc. , 1–11 (2015) Printed 7 August 2018 (MN L A TEX style file v2.2)
How does our choice of observable influence our estimationof the centre of a galaxy cluster? Insights fromcosmological simulations.
Weiguang Cui, , ⋆ Chris Power , , Veronica Biffi , Stefano Borgani , , , Alexan-der Knebe , , Giuseppe Murante , Dunja Fabjan , , Geraint F. Lewis & Greg B.Poole ICRAR, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia ARC Centre of Excellence for All-Sky Astrophysics (CAASTRO) Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy INAF – Astronomical Observatory of Trieste, via Tiepolo 11, I-34131 Trieste, Italy INFN – Sezione di Trieste, I-34100 Trieste, Italy Departamento de F´ısica Te´orica, M´odulo 15, Facultad de Ciencias, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Astro-UAM, UAM, Unidad Asociada CSIC Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia Sydney Institute for Astronomy School of Physics, A28, The University of Sydney NSW 2006, Australia School of Physics, University of Melbourne, Parksville, VIC 3010, Australia
ABSTRACT
Galaxy clusters are an established and powerful test-bed for theories of both galaxyevolution and cosmology. Accurate interpretation of cluster observations often requiresrobust identification of the location of the centre. Using a statistical sample of clustersdrawn from a suite of cosmological simulations in which we have explored a rangeof galaxy formation models, we investigate how the location of this centre is affectedby the choice of observable – stars, hot gas, or the full mass distribution as can beprobed by the gravitational potential. We explore several measures of cluster centre:the minimum of the gravitational potential, which would expect to define the centreif the cluster is in dynamical equilibrium; the peak of the density; the centre of BCG;and the peak and centroid of X-ray luminosity. We find that the centre of BCG cor-relates more strongly with the minimum of the gravitational potential than the X-raydefined centres, while AGN feedback acts to significantly enhance the offset betweenthe peak X-ray luminosity and minimum gravitational potential. These results high-light the importance of centre identification when interpreting clusters observations,in particular when comparing theoretical predictions and observational data.
Key words: cosmology: theory – galaxies: clusters: general – galaxies: formation
Currently favoured models of cosmological structure forma-tion are hierarchical – lower mass systems merge progres-sively to form more massive structures, with galaxy clustersrepresenting the final state of this process. They are widelyused as cosmological probes (e.g von der Linden et al. 2014;Mantz et al. 2015), but they are also unique laboratories fortesting models of gravitational structure formation, galaxyevolution, thermodynamics of the intergalactic medium, andplasma physics (e.g. Kravtsov & Borgani 2012). ⋆ E-mail: [email protected]
Observationally, galaxy clusters are usually identi-fied through optical images (e.g. Postman et al. 1996;Gladders & Yee 2000; Ramella et al. 2001; Koester et al.2007; Robotham et al. 2011), X-ray observations (e.g.Ebeling et al. 1998; B¨ohringer et al. 2004; Liu et al. 2013),the Sunyaev-Zel’dovich effect (e.g. Vanderlinde et al.2010; Planck Collaboration et al. 2011; Williamson et al.2011), and weak and strong gravitational lensing (e.g.Johnston et al. 2007; Mandelbaum et al. 2008; Zitrin et al.2012). A fundamental step in any of these procedures isidentification of the cluster centre. For example, it is nat-ural to adopt the optical/X-ray luminosity peak/centroid orbrightest cluster galaxy (BCG) position as the centre of an c (cid:13) Weiguang Cui, et al. optically or X-ray selected cluster respectively, whereas thelocation of the minimum of the lensing potential is morenatural when considering strong and weak lensing.It is interesting to ask how observational estimates ofthe cluster centre relate to assumptions about the under-lying physical mass distribution. This can have importantconsequences for our interpretation of observations, poten-tially biasing recovery of properties such as mass and con-centration (e.g. Shan et al. 2010b; Du & Fan 2014). Theo-retically, it is natural to select the location of the minimumof the gravitational potential as the cluster centre, providedthe cluster is dynamically relaxed. If the hot X-ray emit-ting intra-cluster gas is in hydrostatic equilibrium withinthe cluster potential and orbiting stars are in dynamicalequilibrium, then we should expect good agreement betweenthese different observable centre tracers and the potentialminimum. However, typical clusters are not in dynamicalequilibrium – they form relatively recently and have under-gone or are undergoing significant merging activity, result-ing in disturbed mass distributions (e.g. Thomas et al. 1998;Power et al. 2012) – and so we might anticipate systematicoffsets between optical, X-ray and potential centres.The goal of this paper is to estimate the size of offsetthat we might expect by using a statistical sample ofsimulated galaxy clusters to measure cluster centres asdetermined by different observables (e.g. centre of BCG,X-ray emitting hot gas) and the minimum of the gravita-tional potential. We also assess how these measurementsare affected by AGN feedback, which we would expectto influence the distribution of hot gas, but could alsoinfluence when and where stars form. Before we presentthe results of our analysis, we review briefly results fromobservations.We argued that typical clusters are not in dynamicalequilibrium, and so we should expect offsets between centresestimated using different tracers. This is borne out by ob-servations, which suggest that where one locates a cluster’scentre will depend on the choice of the tracer. Lin & Mohr(2004) looked at the offsets between BCGs and X-ray peaks(or centroids) and found that about 75 per cent of identi-fied clusters had offsets within 0 . r (where r is theradius within which the enclosed mean matter overdensityis 200 times the critical density of the Universe), 90 per centwithin 0 . r , and ∼
10 per cent contamination level ofpossibly misidentified BCGs. Mann & Ebeling (2012) foundthat the offsets between BCGs and X-ray peaks are well ap-proximated by a log-normal distribution, centred at ∼ . ∼
21 kpc. Rozo & Rykoff (2014) foundthat ∼
80 per cent of their clusters have a perfect agree-ment ( < ∼
50 kpc) between the X-ray centroid and the cen-tral galaxy position; interestingly, the remaining clusterswere undergoing ongoing mergers and had offsets .
300 kpc(see also von der Linden et al. 2014, for similar findings).Zitrin et al. (2012) found that the offset between a BCG’slocation and the peak of smoothed dark matter density iswell described by a log-normal distribution centred around ∼ . h − kpc, and the size of this offset increases with red-shift, while Shan et al. (2010a) characterized the offsets be-tween the X-ray peaks and strong lensing centres and found that about 45 per cent of their clusters show offsets of order40 − h − kpc.Identifying cluster centres observationally is notstraightforward, however. For example, Oguri et al. (2010)found that the distribution of separations between the lo-cation of the BCG and the lensing centre has a long tail,and that the typical error on the mass centroid measure-ment in weak lensing is ∼ h − kpc. George et al. (2012)found BCGs are one of the best tracers of a cluster’s centre-of-mass, with offsets typically less than 75 h − kpc, butthese measurements are susceptible to how the centre is de-fined (e.g. intensity centroids vs intensity peaks) and this cancause a 5 −
30 per cent bias in stacked weak lensing anal-yses. Also, evidence of recent or ongoing merging activitycorrelates with increased offsets, as revealed by, for example,the Rozo & Rykoff (2014) result mentioned already. Inter-estingly, the centroid shift (offsets of a system’s X-ray sur-face brightness peak from its centroid) is usually a good in-dicator of a cluster’s dynamical state and recent merging ac-tivity (e.g. Mohr et al. 1993; Poole et al. 2006). Large offsetsbetween the centre of mass and the minimum of the grav-itational potential have been shown to be good indicatorsof recent merging activity and systems that are out of dy-namical equilibrium (e.g. Thomas et al. 1998; Power et al.2012).We note briefly that measurements of velocity offsets ingroups and clusters also imply spatial offsets. For example,van den Bosch et al. (2005) estimated that central galaxiesoscillate about the potential minimum with an offset of ∼ ∼ . R vir , or ∼ − h − kpc at halo mass of log M = 13 − . AHF,Knollmann & Knebe 2009; Gill et al. 2004a) , or via an SPH-style density evaluation (e.g.
PIAO, Cui et al. 2014b) . In thispaper, we use the SO method to identify halos and com-pute the location of the density peak using an SPH ker-nel approach. If we can better understand the astrophys- c (cid:13) , 1–11 n the Estimation of Galaxy Clusters Centres ical origin of observed centre offsets, then we can recovermore accurate measurements of cluster mass profiles (e.g.Shan et al. 2010b), reconstruction of assembly histories (e.g.Mann & Ebeling 2012), and tests of cosmological modelswith cases such as bullet clusters (e.g. Forero-Romero et al.2010).In the following sections, we describe how we have usedcosmological hydro-simulations with different baryon mod-els (see also Cui et al. 2012, 2014b) to select the statis-tical sample of clusters ( § § §
5, and comment on their significance for interpretation ofobservations of galaxy clusters.
We use three large–volume cosmological simulations, namelytwo hydrodynamical simulations in which we include differ-ent feedback processes, and one dark matter only N-bodysimulation. All these simulations are described in Cui et al.(2012, 2014b); here we summarise the relevant details.We assume a flat ΛCDM cosmology, with cosmologicalparameters of Ω m = 0 .
24 for the matter density parameter,Ω b = 0 . σ = 0 . n s = 0 .
96 for the primordialspectral index, and h = 0 .
73 for the Hubble parameter inunits of 100 h km s − Mpc − . The three simulations wereset up using the same realisation of the initial matter powerspectrum, and reproduce the same large-scale structures. Werefer to the dark matter only simulation as the DM run. Bothhydrodynamical simulations include radiative cooling, starformation and kinetic feedback from supernovae; in one casewe ignore feedback from AGN (which is referred as the CSFrun), while in the other we include it (which is referred asthe AGN run).We use the TreePM-SPH code GADGET-3 , an improvedversion of the public
GADGET-2 code (Springel 2005),which includes a range of prescriptions for galaxy forma-tion physics (e.g. cooling, star formation, feedback). Grav-itational forces are computed using a Plummer equivalentsoftening fixed to ǫ Pl = 7 . h − kpc from z = 0 to 2 and fixedin comoving units at a higher redshift. As we will see, oursoftening length 7 . h − kpc is comparable to – and in caseslarger than – the offsets between the minimum potential andmaximum SPH density positions, centre of BCGs and X-rayemission-weighted centres. However, the minimum potentialposition is determined by the whole cluster, which should beless affected by the softening length. Thus, we expect thatthese offsets are accurate to within a softening length.Haloes are identified using the spherical overdensity(SO) algorithm PIAO (Cui et al. 2014b), assuming an over-density criterion of ∆ c = 200 . Densities are computed In the following, the overdensity value ∆ c is expressed inunits of the cosmic critical density at a given redshift, ρ c ( z ) =3 H ( z ) / (8 πG ). using a SPH kernel smoothed over the nearest 128 neigh-bours; this allows us to determine the maximum density inthe halo, which we also identify as the density-weighted cen-tre of the halo. All of the particle types (dark matter, gas,stars) contribute equally to the density computation.We select our cluster sample from the DM run SOhalo catalogue, with the requirement that M > . × h − M ⊙ ; this gives a total of 184 halos in our sample,with a maximum mass of ∼ . × h − M ⊙ . The corre-sponding SO halos in the AGN and CSF runs are identifiedby cross-matching the dark matter components using theunique particle IDs (see more details in Cui et al. 2014b);we find no systems less massive than 1 . × h − M ⊙ inthis cross-matched catalogue. In this paper, we only focuson the clusters at redshift z = 0.Examples of a visually and dynamically disturbed andundisturbed clusters (lower and upper panels respectively)at z = 0 are shown in Fig. 1, where we show qualitativeprojected density distributions in the AGN, CSF and DMruns (from left to right). In the case of the dark matter maps(rightmost panels), only the dark matter contributes to theRGB value of a pixel. The projected density of dark matterwithin a pixel lies in the range (0,255), and this is used toset the “B” of the RGB value of the pixel; if this densityexceeds a threshold, we set the RGB value to white. Whencombining dark matter, gas and stars (leftmost and middlepanels), both the dark matter and gas contribute to theRGB value. As before, the projected density of dark matteris scaled to the range (0,255), but without a threshold, andit is used to set the “B” of the RGB value; the projecteddensity of has is scaled to the range (0,255) and is used toset the “R” of the RGB value; and the RGB value of starsis set to white, with a transparency of 0.5. By constructingthe projected density maps in this way, we can get a sensefor the relative projected densities of dark matter and gasin the systems; the projected dark matter density dominatesthe hot gas density at larger radii in both systems, but isdominated by the hot gas density at smaller radii. In this paper, we focus on 4 different definitions of the clustercentre. We quote centres of potential and density, which arereadily measured in the simulation data, by their 3D values,while we use projected (2D) values for centres derived frommock observational data.
Minimum of the Gravitational Potential:
This is thephysically intuitive definition of the cluster centre, and isexpected to correspond to the lensing centre. For all parti-cles within the r radius, we select the one with the mostnegative value of the potential as the cluster centre. Theparticle’s potential is directly coming from the simulations.We will take this minimum potential position as the baseline for comparison in this paper. Maximum of the SPH Density:
In constructing our halocatalogue using the SO algorithm implemented in
PIAO , weestimate the densities of particles by smoothing over nearest c (cid:13) , 1–11 Weiguang Cui, et al.
Figure 1.
Examples of one visually and dynamically disturbed (upper panel) and one undisturbed (lower panel) galaxy cluster at z = 0from our suite of simulations (AGN, CSF, DM, from left to right). For the hydrodynamical simulated clusters, we use blue and redcolours to represent dark matter and gas particle (SPH) densities, white represents optical stellar luminosity with a surface brightnessof µ > . mag/arcsec in the SDSS r band; the DM only equivalent is shown in the rightmost panel. The symbols (+,x, ◦ , (cid:3) ) identifythe location of the cluster centre of mass (+); minimum of the gravitational potential (x); maximum of SPH kernel weighted density ( ◦ );and the iterative centre of mass ( (cid:3) ). For the two hydrodynamical runs, we show also the BCG position in SDSS r band using red filledcircles. The open and filled black star symbols indicate the X-ray peak and centroid positions, respectively. We refer to section § neighbours using the SPH kernel, and identify the particlewith the highest density as the halo centre. Optical Centres of the BCG:
Our hydrodynamical CSFand AGN runs include star formation. Using the methodapplied in Cui et al. (2011), we assign luminosities to eachof the star particles that form by assuming that they con-stitute single stellar populations with ages, metallicities andmasses given the corresponding particle’s properties in therun. Adopting the same initial mass function as the simula-tion, the spectral energy distribution of each particle is com-puted by interpolating the simple stellar population tem-plates of Bruzual & Charlot (2003). We consider the threestandard SDSS r , g , and u bands in this paper. The lu-minosity of each star particle is smoothed to a 2-D map(projected to the xy-plane), with each pixel having a sizeof 5 h − kpc. We adopt the same spline kernel used for the Although we employ this particular density estimate in thispaper, we note that there are several methods to locate the centrewhen using the SO algorithm; in appendix A, we show how threedifferent density peak estimators differ.
SPH calculations with 49 SPH neighbours, which is equiva-lent to 30 h − kpc (see Cui et al. 2014a, for more details).Note that the minimum offset cut for later relevant plotswill be set to half the image pixel size, 2 . h − kpc.The centre of BCG is identified as the most lumi-nous image pixel of each band within the BCG. To se-lect the BCG, we first separate the intra-cluster light fromgalaxies. As shown in Fig. 1, the surface brightness cut( µ > . mag/arcsec ) employed observationally is notsuitable for our simulated data because it would include toomuch intra-cluster light. Cui et al. (2014a) has shown thatthe physical intra-cluster light identification method (basedon the star’s velocity information Dolag et al. 2010) impliesmuch higher surface brightness threshold values. For thisreason, we adopt the surface brightness threshold values, µ = 23 , . mag/arcsec for the CSF and AGN runs, re-spectively. Although these two values are for V-band lumi-nosity in Cui et al. (2014a), we apply them here to the threeSDSS bands without further corrections. This is because weare only interested in position of the brightest pixel insidethe BCG in this paper; corrections should not affect ourfinal results. Pixels above the surface brightness threshold c (cid:13) , 1–11 n the Estimation of Galaxy Clusters Centres are grouped together to form a galaxy by linking all neigh-bouring pixels, starting from the brightest pixel. The mostluminous galaxy is selected as the BCG. In each band, weselect the centre of the most luminous pixel inside the BCGas the centre. Centres of X-ray Emission:
We estimate the X-ray emis-sion from each of the simulated clusters using the
PHOX code (see Biffi et al. 2012, 2013, for a more detailed de-scription). Specifically, we simulate the X-ray emission ofthe intra-cluster medium (ICM) by adopting an absorbedAPEC model (Smith et al. 2001), where the WABS absorp-tion model (Morrison & McCammon 1983) is used to mimicthe Galactic absorption and the main contribution from thehot ICM comes in the form of bremsstrahlung continuumplus metal emission lines. The latter is obtained from the im-plementation of the APEC model for a collisionally-ionizedplasma comprised within the XSPEC package (v.12.8.0).For any gas element in the simulation output, the modelspectrum predicts the expected number of photons, withwhich we statistically sample the spectral energy distribu-tion.In the approach followed by PHOX, the synthetic X-ray photons are obtained from the ideal emission spectrumcalculated for every gas particle belonging to the clusterICM, depending on its density, temperature, metallicity and redshift (we assume z = 0 .
05 for the X-ray luminosityand angular-diameter distances). We consider only the posi-tion of the X-ray centre in this work, and do not expect theparticular choice of redshift or metallicity to affect it signif-icantly. To obtain the photon maps, we assume a realisticexposure time of 50 ks and convolve the ideal photon-list ofevery cluster with the response matrices of Chandra (ACIS-S detector); this accounts for the instrument characteristicsand sensitivity to the incoming photon energies. In this pro-cess, the maps (i) are originally centered on the cluster po-tential centre, (ii) cover a circular region of R radius, and(iii) have a the same pixel size of 5 h − kpc as the opticalimage.In this work, we consider the x - y projection and the fullenergy band of the detector. In addition, we also apply thesame SPH smoothing procedure as used for the optical im-age, but using each pixel’s photon counts from the PHOXX-ray maps instead of stellar luminosity. The X-ray peakposition is identified as the pixel with the maximum valueof photon counts. We note here that using this simple X-ray peak position as the X-ray centre can be biased by thesatellites (see Mantz et al. 2015, for more discussions aboutdifferent X-ray centre tracers). The centroid of the X-raymap is computed basing on the method of B¨ohringer et al.(2010); Rasia et al. (2013), modified to take the X-ray peakposition as the initial centre and reset to the centre of massfrom photon counts within the shrinking radius after eachiteration. We reduce the radius to 85 per cent of the previousiteration, starting at an initial radius of R in projection,until a fixed inner radius R is reached. The X-ray cen- http://heasarc.gsfc.nasa.gov/xanadu/xspec/. In this work, a fiducial average metallicity of Z = 0 . Z ⊙ is assumed, for simplicity, with solar abundances according toAnders & Grevesse (1989). Gas Dark matter StarCSF 36 38 110AGN 50 18 116 Table 1.
Numbers of clusters in which the densest particle be-longs to a given particle type (i.e. gas, dark matter or stars). troid is the centre of mass position at the final step. We usethis iterative method to locate the centroid, because thereare many un-relaxed clusters in our sample. Note that theminimum offset cut for later relevant plots is also set to thesize of half a pixel, 2 . h − kpc. In Fig. 2, we investigate the offset between the maximum ofSPH density and the minimum of the gravitational potentialpositions in the DM, CSF and AGN runs (upper, middle andlower panels respectively). We reset offsets R off < h − kpcto 1 h − kpc, for an easier visualization. • In the DM run, we find typical offsets of ∼ h − kpc,which is comparable with the simulation softening length asindicated by the horizontal dashed line in all panels. Thoseclusters with large offsets contain massive compact substruc-tures that are in the process of merging and the systemshows obvious signs of disturbance. • In the CSF run, the typical offsets are smaller thanthe softening length of the simulation ( < ∼ h − kpc), butin some cases there are offsets as large as > ∼ h − kpc.Close inspection shows that star and dark matter particlestend to be the particles defining the maximum SPH densitywithin these systems; we indicate this explicitly by markingthe particles that trace the maximum of the density withsymbols defined in the legend. • In common with the CSF run, the majority of clusters inthe AGN run have offsets smaller than the softening length, < ∼ h − kpc. As in the CSF run, and as shown in table 1,star particles tend to define the location of the density peak.We have visually inspected those clusters that have largeoffsets in Fig. 2 and find, unsurprisingly, that the densitypeak is associated with a massive satellite galaxy (e.g. thedisturbed cluster in the upper row of Fig. 1). This indicatesthat these clusters with large offsets are normally undergoingmajor mergers and are visually disturbed.We did not differentiate between the material that con-tributes to the estimate of the maximum SPH density posi-tion (i.e. gas, star and dark matter particles are given equalweight) in Fig. 2; we now show this in Fig. 3. Here the maxi-mum SPH density positions computed from each of the threeparticle types are offset with respect to the potential centreof the cluster in the CSF and AGN runs (left and right panelsrespectively). In this calculation, we include only particles ofthe same species (i.e. dark matter, gas, stars) when calculat-ing densities. The particle with the maximum SPH densityis selected as the density peak for the given component. c (cid:13)000
Numbers of clusters in which the densest particle be-longs to a given particle type (i.e. gas, dark matter or stars). troid is the centre of mass position at the final step. We usethis iterative method to locate the centroid, because thereare many un-relaxed clusters in our sample. Note that theminimum offset cut for later relevant plots is also set to thesize of half a pixel, 2 . h − kpc. In Fig. 2, we investigate the offset between the maximum ofSPH density and the minimum of the gravitational potentialpositions in the DM, CSF and AGN runs (upper, middle andlower panels respectively). We reset offsets R off < h − kpcto 1 h − kpc, for an easier visualization. • In the DM run, we find typical offsets of ∼ h − kpc,which is comparable with the simulation softening length asindicated by the horizontal dashed line in all panels. Thoseclusters with large offsets contain massive compact substruc-tures that are in the process of merging and the systemshows obvious signs of disturbance. • In the CSF run, the typical offsets are smaller thanthe softening length of the simulation ( < ∼ h − kpc), butin some cases there are offsets as large as > ∼ h − kpc.Close inspection shows that star and dark matter particlestend to be the particles defining the maximum SPH densitywithin these systems; we indicate this explicitly by markingthe particles that trace the maximum of the density withsymbols defined in the legend. • In common with the CSF run, the majority of clusters inthe AGN run have offsets smaller than the softening length, < ∼ h − kpc. As in the CSF run, and as shown in table 1,star particles tend to define the location of the density peak.We have visually inspected those clusters that have largeoffsets in Fig. 2 and find, unsurprisingly, that the densitypeak is associated with a massive satellite galaxy (e.g. thedisturbed cluster in the upper row of Fig. 1). This indicatesthat these clusters with large offsets are normally undergoingmajor mergers and are visually disturbed.We did not differentiate between the material that con-tributes to the estimate of the maximum SPH density posi-tion (i.e. gas, star and dark matter particles are given equalweight) in Fig. 2; we now show this in Fig. 3. Here the maxi-mum SPH density positions computed from each of the threeparticle types are offset with respect to the potential centreof the cluster in the CSF and AGN runs (left and right panelsrespectively). In this calculation, we include only particles ofthe same species (i.e. dark matter, gas, stars) when calculat-ing densities. The particle with the maximum SPH densityis selected as the density peak for the given component. c (cid:13)000 , 1–11 Weiguang Cui, et al. R off [kpc/h] N CSF
DM CenterGAS CenterSTAR Center R off [kpc/h] AGN
DM CenterGAS CenterSTAR Center
Figure 3.
As in Fig. 2, we show the offset between the density and potential centres in the CSF (left panel) and AGN (right panel)runs, but now we split according to particle type, where solid, dotted and dashed histograms correspond to dark matter, gas and starsrespectively. Note that offsets R off < h − kpc are reset to 1 h − kpc. In the CSF run, there is broad agreement between themaximum SPH density and minimum potential position off-sets computed for each of the particle types; these offsetsare within ∼ h − kpc, while those systems with offsets & h − kpc are visually idenitified as disturbed. In starkcontrast to the CSF run and also to the result from Fig. 2,in the AGN run there is a clear separation in the maximumSPH density and minimu potential position offsets computedfrom dark matter particles on the one hand and star and gasparticles on the other. The dark matter particles have off-sets similar to those found in the CSF run, clustering within ∼ h − kpc, but the star particles have two offset peaksat ∼ ∼ h − kpc, while gas particles particles haveoffsets spread between ∼ − h − kpc. The large offsetswe see in the stellar component arise because the identifiedcentres are located in satellite galaxies, which are compact,rather than in the BCG. This is also linked to the largeoffsets we find in the gas component, which arise becausestrong AGN feedback can expel gas to a large cluster-centricradius and helps to suppress star formation over much ofthe lifetime of the BCG by inhibiting the accumulation ofdense gas at small radii. Similar trends arising from AGNhave been reported in Ragone-Figueroa et al. (2012, 2013);Cui et al. (2014a). Note that this figure is primarily of the-oretical interest; it shows how the centre of density changesas we sample the different components in the simulation,something that would be challenging to do observationally! We now consider the relationship between the centre ofBCGs and minimum potential positions, where we employthe method of Cui et al. (2011) as described in § u , g and r bands. Note that we do not include the effects of dust whencalculating luminosities, and so we potentially omit band-dependent dust attenuation that could, in principle, biasour conclusion. To compare with observations, we focus on 2D x - y projections here. The minimum offset is set to halfof the pixel size 2 . h − kpc.In Fig. 4 we show how the distribution of offsets betweenthe centre of BCGs and minimum potential positions. Theresults for both the CSF and AGN runs (left and right panelsrespectively) are in broad agreement, and similar to thoseshown in Fig. 2 for the offset between density and potentialpeaks; most of the offsets are within the softening lengthfor both CSF and AGN runs. We find no dependence onmeasured (i.e. u , g or r ) band. In Fig. 5, we show the distribution of offsets between the X-ray peak, centroid positions and cluster potential centre inthe CSF and AGN runs (left and right panels respectively).Here we note some interesting differences. • In the CSF run, the offset distributions of peak posi-tions show a peak at ∼ h − kpc, with a second peak at ∼ h − kpc; this is larger than for the offset between thecentre of BCGs and minimum potential positions. While,the X-ray centroid offset show a wide spread distributionfrom ∼ h − kpc towards ∼ h − kpc. • In the AGN run, the offset distributions for both X-ray peak and centroid have a peak at ∼ h − kpc. Thisis slightly less than the offsets between the gas componentdensity and cluster potential centres from Fig. 3.Compared with the X-ray peak centre, the centroid is morestable for both CSF and AGN runs. They tend to havesimilar distributions, despite the AGN feedback model.However, the centroid offsets from CSF runs have noclear peak compared to the AGN runs. There is no strongevidence of the secondary peak for the centroid offsets. TheX-ray peak offsets for CSF run are smaller than the AGNrun, which indicates that the AGN feedback has strongereffects on the X-ray peak position. c (cid:13) , 1–11 n the Estimation of Galaxy Clusters Centres R off [kpc/h] N CSF rgu R off [kpc/h] AGN rgu
Figure 4.
The histogram of the offsets between centre of BCGs and cluster potential centre. Left panel is the results from CSF clusters,while the right panel is for AGN clusters. Three optical luminosity bands u , g , r are indicated in the upper-right legend. The verticaldashed lines are the softening length in the simulations. R off [kpc/h] N CSF
X-ray PeakX-ray Centroid R off [kpc/h] AGN
X-ray PeakX-ray Centroid
Figure 5.
Similar to Fig. 4, the histogram of the offsets between X-ray centres and the cluster potential centre. Left panel is the resultsfrom CSF clusters, while the right panel is for AGN clusters. The peak and centroid indicators are shown in the legend. The verticaldashed lines are the softening length in the simulations.
These results suggest that the centre of BCGs should be amore reliable and precise tracer of the underlying gravita-tional potential, and is also less likely to be influenced bythe AGN feedback.
Using a suite of cosmological N -body and hydrodynami-cal simulations, we have constructed a mass- and volume-complete simulated galaxy cluster catalogue. We have con-sidered a pure dark matter (i.e. N -body only) model and twogalaxy formation models that include cooling, star formationand supernova feedback, with and without AGN feedback(the CSF and AGN runs respectively); this allows us to ex-plore in a systematic fashion the impact of these two baryonmodels on the properties of galaxy clusters. In this paper,we have assessed how estimates of galaxy cluster centres are influenced by the mode of measurement – using X-ray emit-ting hot gas, the centre of BCGs, or the total mass distri-bution, which is accessible via gravitational lensing, say. Inall cases we compare to the location of the minimum of thegravitational potential of the system, which we would ex-pect to define a physically reasonable centre of the system,assuming that it is in dynamical equilibrium.The main results of our analysis are summarised as fol-lows. • We find that the maximum local density, computed us-ing an SPH kernel smoothing over 128 nearest neighbours,is in good agreement with the minimum of the gravitationalpotential regardless of the assumed galaxy formation model, provided we include all particles – dark matter, gas and stars– in the calculation . In the CSF runs, we find offsets betweenthe maximum SPH density and minimum potential positionsat < ∼ h − kpc; in the AGN runs, these offsets are evensmaller than the CSF runs. However, both runs have a small c (cid:13)000
Using a suite of cosmological N -body and hydrodynami-cal simulations, we have constructed a mass- and volume-complete simulated galaxy cluster catalogue. We have con-sidered a pure dark matter (i.e. N -body only) model and twogalaxy formation models that include cooling, star formationand supernova feedback, with and without AGN feedback(the CSF and AGN runs respectively); this allows us to ex-plore in a systematic fashion the impact of these two baryonmodels on the properties of galaxy clusters. In this paper,we have assessed how estimates of galaxy cluster centres are influenced by the mode of measurement – using X-ray emit-ting hot gas, the centre of BCGs, or the total mass distri-bution, which is accessible via gravitational lensing, say. Inall cases we compare to the location of the minimum of thegravitational potential of the system, which we would ex-pect to define a physically reasonable centre of the system,assuming that it is in dynamical equilibrium.The main results of our analysis are summarised as fol-lows. • We find that the maximum local density, computed us-ing an SPH kernel smoothing over 128 nearest neighbours,is in good agreement with the minimum of the gravitationalpotential regardless of the assumed galaxy formation model, provided we include all particles – dark matter, gas and stars– in the calculation . In the CSF runs, we find offsets betweenthe maximum SPH density and minimum potential positionsat < ∼ h − kpc; in the AGN runs, these offsets are evensmaller than the CSF runs. However, both runs have a small c (cid:13)000 , 1–11 Weiguang Cui, et al. log (M halo [h −1 M ⊙ ]) R o ff [ k p c / h ] DM log (M halo [h −1 M ⊙ ]) R o ff [ k p c / h ] CSF
GasDark matterStar log (M halo [h −1 M ⊙ ]) R o ff [ k p c / h ] AGN
GasDark matterStar
Figure 2.
The offset between the maximum SPH density andminimum potential positions as a function of halo mass. From topto bottom, these panels are for DM, CSF, AGN runs, respectively;On the right hand of each panel, we show a histogram distributionof the offsets. The horizontal dashed lines are the softening lengthin the simulations. As indicated in the legends of middle andbottom panels, the different color symbols represent the type ofthe highest density particle, i.e. cluster centre. amount of clusters with very large offsets ( > ∼ h − kpc).This is because the density peak is associated with a satellitegalaxy.If we compute the maximum local density for individualparticle types, we find differences that depend on the as-sumed galaxy formation model. The offsets for different par-ticles in CSF run are within the simulation softening length.However, many clusters in AGN run have very large offsetsbetween the density peak evaluated from both stellar andgas particles and the potential centre. The strong feedbackfrom the AGN not only expels gas particles, which have theoffset at ∼ h − kpc, but also reduces the stellar den-sity within the central galaxy, in which case the peak of thestellar density is more likely to be associated with a satellitegalaxy. • Using projected optical luminosities in SDSS r , g and u bands, we identify the centre of BCG from star particlesin the CSF and AGN runs. We find that centre of BCGs areclose to the potential centre, within the softening length inboth runs and independent of the assumed band. A smallfraction of the clusters have large offsets in both CSF andAGN runs; these belong to disturbed clusters, in which theidentified BCG is offset from the centre of the potential byvisually checking. • Identifying the location of both the peak and centroidpositions of X-ray emission from realistic maps, we findslightly larger peak offsets ∼ h − kpc in the CSF run(with a second peak at ∼ h − kpc); ∼ h − kpc in theAGN run. The X-ray centroid offset seem more stable thanX-ray peak, which have less effect from the AGN feedback.It has a wide spread from ∼ h − kpc to ∼ h − kpc.There is no clear peak in the CSF run; while the AGN runhas a similar peak as its X-ray peak offset.It is interesting to ask how well our simulations matchobservations, which has a bearing on the general applicabil-ity of our results. We note that we have already used thesame cosmological simulation data to compare baryon andstellar mass fractions with observations in Cui et al. (2014b)(see their Fig. B1 for details). There it was shown that bothof these fractions computed from our AGN simulation areconsistent with observations, whereas the CSF runs predictvalues that are larger than observed; this is to be expected,arising because of overcooling. In the nIFTy cluster compar-ison project Sembolini et al. 2015a,b, a single galaxy clusterhas been simulated in a cosmological context with a range ofstate-of-the-art astrophysics codes, and in the runs that em-ploy the physics of galaxy formation (e.g. radiative cooling,star formation, feedback from supernovae and AGN) it hasbeen shown the results from the model used in this paper isconsistent with the results of other codes (Sembolini et al.2015b; Cui et al., In Preparation) in global cluster proper-ties. However, galaxies inside this cluster show striking code-to-code variations in stellar and gas mass (Elahi et al. 2015),which implies different spatial distributions for the gas andstellar components. Thus, we caution that the choice of in-put physics in simulations of this kind can have a strongquantitative influence on the results.We find that the distribution of offsets between the cen-tre of BCG and X-ray emission centres with respect to thepotential centre is smaller than is found observationally; thiscould be due to in part to observational inaccuracies (image c (cid:13) , 1–11 n the Estimation of Galaxy Clusters Centres resolution, identification of lensing centre) and in part to ourassumption that the potential centre, calculated from the 3Ddistribution of matter within the cluster, is well-matched tothe lensing centre. However, our results agree with observa-tions that centre of BCG is a better tracer of the cluster cen-tre than the X-ray emission weighted centre (George et al.2012). However, the claim that the BCG is a better tracerrequires identifying BCGs correctly in the first place in ob-servations, which is not straightforward. Our simulation re-sults suggest that the simple grouping method after ICLextraction in Section 3 does a good job. Offsets betweenX-ray and lensing centres are in fact observed at a level of100 kpc (e.g. Allen 1998; Shan et al. 2010a; George et al.2012). However, the observed offsets between lensing andBCGs are usually smaller. For example, Oguri et al. (2010)found that the offsets between weak lensing and BCG areat ∼ h − kpc, while the strong lensing has even closerposition to BCG (Oguri et al. 2009). With large statisticalsamples, Zitrin et al. (2012) also suggested smaller offsetsbetween the weak lensing and BCG position. These supportthat the BCG traces the minimum gravitational potentialposition better than the X-ray data.The large offset tail found in clusters from both the cen-tre of BCG and X-ray center are basically consistent with thesecondary peak found by Johnston et al. (2007); Zitrin et al.(2012). These large offsets should be caused by dynami-cally unrelaxed clusters undergoing mergers, in which theoptical luminosity and X-ray centres can be located at amassive satellite galaxy, which is away from the cluster po-tential centre. Using a set of hydrodynamical simulationsof mergers of two galaxy clusters, Zhang et al. (2014) findthat significantly large SZ-X-ray peak offsets ( >
100 kpc) canbe produced during the major mergers of galaxy clusters.This finding is basically agreed to the second peak for X-raypeak-potential offsets from our CSF runs. These large offsetsindicate these clusters are not relaxed. This highlights theimportance of dynamical state in the centre determination,something we will address in a follow-up paper.Finally, we have considered only spatial offsets in thisstudy, the first of a series. We expect to find dynamical off-sets within clusters. Subhaloes or satellite galaxies in N-bodyand hydrodynamic simulations are found to have velocitiesdiffering from the dark matter halos (e.g. Diemand et al.2004; Gao et al. 2004; Gill et al. 2004b; Munari et al. 2013;Wu et al. 2013). These velocity offsets are closely con-nected to the cluster center offsets. Gao & White (2006);Behroozi et al. (2013) demonstrated that dark matter halocores are not at rest relative to the halo bulk or substructureaverage velocities and have coherent velocity offsets acrossa wide range of halo masses and redshifts. We revisit thisusing our cluster sample in our next paper, surveying notonly the dark matter but also gas and stars, and considerits implications for turbulence and accretion onto AGN.
ACKNOWLEDGEMENTS
We thank the referee for their thorough and thoughtful re-view of our paper. All the figures in this paper are plot-ted using the python matplotlib package (Hunter 2007).Simulations have been carried out at the CINECA super-computing Centre in Bologna, with CPU time assigned through ISCRA proposals and through an agreement withthe University of Trieste. WC acknowledges the supportsfrom University of Western Australia Research Collabora-tion Awards PG12105017, PG12105026, from the SurveySimulation Pipeline (SSimPL; )and from iVEC’s Magnus supercomputer under NationalComputational Merit Allocation Scheme (NCMAS) projectgc6. WC, CP, AK, GFL, and GP acknowledge support ofARC DP130100117. CP, AK, and GFL acknowledge sup-port of ARC DP140100198. CP acknowledges support ofARC FT130100041. VB, SB and GM acknowledge supportfrom the PRIN-INAF12 grant ’The Universe in a Box: Multi-scale Simulations of Cosmic Structures’, the PRIN- MIUR01278X4FL grant ’Evolution of Cosmic Baryons’, the IN-DARK INFN grant and ’Consorzio per la Fisica di Trieste’.AK is supported by the
Ministerio de Econom´ıa y Compet-itividad (MINECO) in Spain through grant AYA2012-31101as well as the Consolider-Ingenio 2010 Programme of the
Spanish Ministerio de Ciencia e Innovaci´on (MICINN) un-der grant MultiDark CSD2009-00064. He further thanks TheCure for faith. GP acknowledges support from the ARC Lau-reate program of Stuart Wyithe.
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APPENDIX A: IDENTIFYING DENSITYPEAKS
We considered a number of approaches to estimating thelocation of the maximum density of the cluster. Here webriefly review three – one that was used in the study, andtwo others from the literature. • The Smoothed Particle Hydrodynamics (SPH) methodadopts the kernel smoothing approach that is commonly c (cid:13) , 1–11 n the Estimation of Galaxy Clusters Centres used in hydrodynamics; we have implemented and testedthis method in Cui et al. (2014b) using 128 neighbours whencalculating densities. This is the method used in the PIAO halo finder and the one used in this study. • The Iterative Centre of Mass (ICM) method estimatesthe mass-weighted centre in an iterative fashion, using allparticles within a shrinking spherical volume until conver-gence in the estimated centre is achieved (cf. Power et al.2003); we define convergence when consecutive centres agreeto within 1 h − kpc. • The Voronoi Tessellation Density (VTD) method par-titions the volume into cells using the distance between ad-jacent points to define cell boundaries, and uses the inversevolume of the cell to estimate the local density at the po-sition of each particle; it requires no free parameters. Weuse the publicly available convex hulls program (Clarkson1992) implemented in python . We note that this approach issensitive to the finite resolution of the simulation.In Fig. A1, we show the offsets between the three estimates(SPH, ICM, and VTD) of the maximum density positionand the location of the minimum of the gravitational poten-tial (red circles, blue diamonds and green inverted triangles,respectively) for each of the clusters in our DM sample. Thehistograms in the right hand panel are the correspondingto projected distributions of cluster offsets. Fig. A1 showsthat the performance of the three estimators, as measuredby the typical size of offset with respect to the location ofthe minimum of the gravitational potential, is comparable,although the SPH method – implemented in
PIAO and usedin this study – should be favoured – 87 . h − kpc.This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000