On the Cluster Physics of Sunyaev-Zel'dovich Surveys I: The Influence of Feedback, Non-thermal Pressure and Cluster Shapes on Y-M Scaling Relations
aa r X i v : . [ a s t r o - ph . C O ] S e p S UBMITTED TO A P J Preprint typeset using L A TEX style emulateapj v. 2/16/10
ON THE CLUSTER PHYSICS OF SUNYAEV-ZEL’DOVICH SURVEYS I:THE INFLUENCE OF FEEDBACK, NON-THERMAL PRESSURE AND CLUSTER SHAPES ON Y - M SCALINGRELATIONS
N. B
ATTAGLIA , J. R. B
OND , C. P FROMMER , J. L. S
IEVERS
Department of Astronomy and Astrophysics, University of Toronto, 50 St George, Toronto ON, Canada, M5S 3H4 Canadian Institute for Theoretical Astrophysics, 60 St George, Toronto ON, Canada, M5S 3H8 McWilliams Center for Cosmology, Carnegie Mellon University, Department of Physics, 5000 Forbes Ave., Pittsburgh PA, USA, 15213 Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, D-69118 Heidelberg, Germany Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton NJ, USA, 08544
Submitted to ApJ
ABSTRACTThe utility of large Sunyaev Zel’dovich (SZ) surveys for determining cosmological parameters from clusterabundances is limited by the theoretical uncertainties in the integrated SZ-flux-to-mass relation, Y - M . We explorehow non-thermal pressure and the anisotropic shape of the gas distribution of the intracluster medium (ICM)impacts Y - M scaling using a suite of hydrodynamical TreePM-SPH simulations of the cosmic web in large periodicboxes. We contrast results for models with different treatments of entropy injection and transport, varying radiativecooling, star formation and accompanying supernova feedback, cosmic rays, and energetic feedback from activegalactic nuclei (AGN) and/or starbursts. We find that the gas kinetic-to-thermal pressure ratio from internal bulkmotions depends on the cluster mass, and increases in the outer-cluster due to enhanced substructure, as does theasphericity of the ICM gas (which is substantially more pronounced for the dark matter). The asphericity is lessdependent on the mass and on variations in the simulated physics. With only a ∼ -
10% correction to projected(observable) ellipticities, we can infer the 3D ellipticities. We find radii around R – within which the meandensity is 500 times the critical density – are the most robust for studying virial properties of clusters, being farenough out to avoid the complex “short-distance” physics of the cluster core, having a relatively low non-thermalto thermal pressure ratio ( ∼ Y - M -slope roughly follows the self-similar Y ∼ M / prediction, except for a steepeningdue to a deficit of gas in lower mass clusters at low redshift in our AGN-feedback simulations. The overall Y - M amplitudes with AGN feedback and radiative cooling are lower than for the shock-heating-only case, by ∼ Y - M -scatter, from ∼
11% to ∼ ∼
11% if we select clusters with lower kinetic pressure.If we split the cluster system into lower, middle and upper bands of P kin / P th , we find a ∼
10% effect on Y - M .A 3-split on asymmetry as measured by the long-to-short axis ratio has a <
10% effect on Y - M , but using 3D-sphericalized estimates instead of projected (cylindrical) has a a ∼
30% effect. Identifying observable secondparameters related to internal bulk flows and anisotropy for cluster-selection to minimize Y - M -scatter in a (fuzzy)“fundamental plane” would allow tighter cosmological parameter constraints. Subject headings:
Cosmic Microwave Background — Cosmology: Theory — Galaxies: Clusters: General —Large-Scale Structure of Universe — Methods: Numerical INTRODUCTION
Clusters are the largest gravitationally-collapsed objects inthe universe, forming at sites of constructive interference oflong waves in the primordial density fluctuations, the coher-ent peak-patches (Bardeen et al. 1986; Bond & Myers 1996).The interiors are separated from the Hubble-flow, but main-tain contact with the nearby cosmic web through ongoingaccretion and mergers as they evolve. Although there is astrong internal baryon-to-dark-matter density bias, a conse-quence of collisional-to-collisionless physics, when cluster-scale-averaged the smoothed densities are nearly in the univer-sal Hubble-volume-smoothed proportion. Clusters have provento be useful cosmological probes as the rarest collapsed-eventtracers of the growth of structure in the universe, with a well-defined number count that steeply falls as mass and redshift in-crease. The number density tail is very sensitive to changes incosmological parameters and primordial non-Gaussianity. Inclusters most of the baryons are in the form of a hot diffuseplasma, the intracluster medium (ICM). The remaining baryons are housed in the cluster’s numerous stars and galaxies. Obser-vations of gas in the cluster system not only reveal the detailedastrophysical processes at work in the ICM, but the counts de-rived as a function of the global-cluster-observables such asthermal energy content can allow for a high precision probe ofthe cosmological parameters defining the count density shapeand amplitude.This scheme of using the cluster system for cosmology mustrely on simulations capturing the physics at work: the clus-ters have been revealed to be too complex for simple spherical-ized analytical modelling as the observations have been pro-gressively refined and resolutions improved. We hope that thebasic global observables will be sufficiently robust to the highresolution complexities that the cluster/group system can becosmologically useful, but that must be demonstrated by de-tailed theoretical work with a necessarily heavy computationalcomponent. Direct observation of mass or gravitational energyor overall binding energy of clusters would be ideal, but weare stuck with what can be observed, in the optical, X-ray andmicrowave/radio/sub-mm. Each derived observable from these BATTAGLIA, BOND, PFROMMER, SIEVERSwindows into clusters is fraught with complication that requiresa computational understanding. A thermal Sunyaev-Zel’dovich(SZ) (Sunyaev & Zeldovich 1970) probe directly observes theintegrated Compton- y parameter which is a measure of thecluster’s global gas heat-energy content, a volume-average ofthe thermal gas pressure, and this is related to gravitationalenergy through the virial relation, so it might be expected toprovide a robust probe. In this paper, and the following se-quence, BBPS2,3,4 (Battaglia et al. 2011a,b,c), a follow-on to(Battaglia et al. 2010), we focus on the SZ effect, the Comptonup-scattering of cosmic microwave background (CMB) pho-tons by hot electrons with its unique signature of a spatially-varying distortion of the CMB spectrum, a decrement in ther-modynamic temperature at frequencies below ∼
220 GHz, andan excess above. The SZ signal is proportional to the integratedelectron pressure, so the hot gas of the ICM dominates the ef-fect. The SZ surface brightness is independent of the redshifta specific cluster is at. Hence SZ surveys have a different se-lection function in redshift and in mass than X-ray and opticalcluster surveys do, being generically more sensitive to higherredshift clusters. The combination of the three probes can pro-vide more robust, tighter constraints on cosmological parame-ters than any one can alone.In a large cluster survey there is a wealth of informationcontained on cosmology and structure formation. The abun-dance of clusters, their distribution in redshift, and their spa-tial clustering should be determined purely by the geometryof the universe, the power spectrum of initial density fluctu-ations, and cosmological parameters such as the rms ampli-tude of the (linear) density power spectrum on cluster-massscales, σ , the mass-energy density in baryons, dark matter,and dark energy, and the equation of state of the latter. In SZsurveys, the number counts as a function of the total SZ flux(integrated Compton- y parameter Y ) and redshift and the an-gular power spectrum are two complementary probes of cos-mology (e.g., Birkinshaw 1999; Carlstrom et al. 2002). Iden-tifying clusters through blind SZ surveys and measuring theirintegrated power spectrum have been long term goals in CMBresearch, and are reaching fruition through, e.g. , the SouthPole Telescope, SPT (e.g., Lueker et al. 2010; Shirokoff et al.2010; Keisler et al. 2011; Vanderlinde et al. 2010), the Ata-cama Cosmology Telescope, ACT (e.g., Fowler et al. 2010;Dunkley et al. 2010; Marriage et al. 2010), and the Planck satellite (e.g., Planck Collaboration et al. 2011b,c,a). To de-termine cosmological parameters from number counts requiresunderstanding the relationship of SZ observables such as totalSZ flux to fundamental cluster properties such as mass M . Andto determine them from the SZ power spectrum requires know-ing the sum of the squares of pressure profiles of unresolvedgroups and clusters, as well as of resolved ones, weighted bythe counts. Both depend sensitively on σ , hence can providean independent measure of it. Such extraction from the SZprobes is inevitably entangled with the uncertainties in the as-trophysical properties of the ICM. This paper deconstructs theinfluence of various physical processes on the Y - M scaling re-lation. BBPS2 does the same for the SZ power spectrum.Previous work attempted to calibrate the Y - M scal-ing relation through observations (e.g., Benson et al. 2004;Bonamente et al. 2008; Marrone et al. 2009; Andersson et al.2010; Sayers et al. 2011; Marrone et al. 2011), self-calibrationtechniques (e.g., Majumdar & Mohr 2003, 2004; Lima & Hu2004; Chaudhuri & Majumdar 2011; Nath & Majumdar 2011),simulations (e.g., da Silva et al. 2004; Motl et al. 2005;Schäfer et al. 2006a,b; Bonaldi et al. 2007) and analytical ap- proaches (Bode et al. 2007; Shaw et al. 2008; Mroczkowski2011). Combining Y - M scaling relations so determined withthe survey selection function and marginalizing over asso-ciated statistical and systematic uncertainties can enable ac-curate determination of cosmological parameters. Using asmall sample of SZ-clusters, SPT (Vanderlinde et al. 2010) andACT (Sehgal et al. 2011) determined some cosmological con-straints, e.g., on σ . However, the errors on σ are dominatedby systematic uncertainties in the underlying cluster physics,making this approach not competitive with other cosmologicalprobes. Hence in order to improve upon the determination ofcosmological parameters, a better understanding of the massproxies and their scatter is needed (Nagai 2006; Shaw et al.2008; Stanek et al. 2010; Yang et al. 2010; Krause et al. 2011). ICM processes
Clusters have been increasingly revealed to be complex sys-tems as the data has progressively improved, necessitating arevision of the simplified pictures popular in the eighties forinterpreting the data. For example, pioneering work by Kaiser(1986) assumed that clusters were self-similar systems with themass determining their ICM thermodynamic properties. Asshown by subsequent X-ray observations, this self-similar de-scription is broken, especially on group scales; low-mass sys-tems are less luminous in comparison to the self-similar expec-tation (see Voit 2005, for a review). Studying how non-thermalprocesses such as magnetic fields, cosmic rays, active galac-tic nuclei (AGN), star formation, radiative cooling and bulkmotions contribute to the energy balance and thermodynamicstability within clusters is a very active research field. It re-mains unclear how these processes vary with cluster radius ordynamical state. State-of-the-art simulations are about the onlytool available for building a consistent picture of clusters. Herewe contribute to these non-thermal studies by using our simu-lations to explore the three effects that influence the Y - M scal-ing relation. These are the feedback processes that appear tobe necessary to explain the thermodynamic characteristics ofthe ICM and avoid a cooling catastrophe leading to too muchstar formation, non-thermal pressure support from bulk mo-tions internal to the clusters that are a natural consequence of adynamically evolving structure formation hierarchy, and devi-ations from spherical symmetry. Energetic feedback
In many clusters the ICM cooling times are much shorterthan a Hubble time (Fabian 1994; Cavagnolo et al. 2009),which should cause extremely high star formation rates thatare well beyond what is observed. However, current simula-tions with only radiative cooling and star formation excessivelyover-cool cluster centers (e.g., Suginohara & Ostriker 1998;Lewis et al. 2000; Pearce et al. 2000), even with the addition ofsupernova feedback. This leads to too many stars in the clus-ter cores, an unphysical rearrangement of the thermal and hy-drodynamic structure, and creates problems when comparingsimulations to observations, in particular for the entropy andpressure profiles. Self-regulated, inhomogeneous energy feed-back mechanisms by, e.g., AGN are very successful in globallystabilizing the group and cluster "atmospheres", and, in partic-ular, preventing the cooling catastrophe (Churazov et al. 2001).Observations of cool core galaxy clusters show evidence forAGN-moderation of the cooling and AGN feedback can po-tentially heat the surrounding ICM from kpc sized bubblesto hundreds of kpc sized outbursts (McNamara et al. 2005).N THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 3In hydrodynamical simulations, it has been shown that incor-porating a sub-grid for AGN feedback can resolve the over-cooling problem (e.g., Sijacki et al. 2007, 2008; Battaglia et al.2010; McCarthy et al. 2011). The effects of AGN feedback onthe ICM will mainly alter the cluster and group cores, wherethe actual physics is poorly resolved and understood. Theseeffects can seen to be dramatic in X-ray observations (e.g.,Fabian et al. 2003), since the emission is proportional to gasdensity squared. Since the SZ signal is proportional to thegas pressure, these effects are smaller. Hence, AGN feedbackshould only perturb the integrated thermal SZ signal, with anamplitude that is not yet known.
Non-thermal pressure support
Studying non-thermal pressure support from bulk motion inclusters has a long history and was first noticed in simulationsby Evrard (1990), who showed that estimates for the bind-ing mass of a cluster using a hydrostatic isothermal β -modelin comparison to a fit to the surface brightness profile dif-fered by 15%. They found that inclusion of velocity disper-sion in the hydrostatic isothermal β -model reconciled this dif-ference between binding masses. Including the support fromresidual gas motions in the hydrostatic cluster mass estima-tor improved the match with the true cluster mass (Rasia et al.2004), with increasing kinetic pressure at larger cluster radii(Lau et al. 2009). The amount of energy in these bulk motionsare of the order of 20% to 30% at radii of interest for cosmol-ogy (Battaglia et al. 2010; Burns et al. 2010). However, kineticpressure support has only recently been included in analyticaland semi-analytical templates for the thermal SZ power spec-trum (Shaw et al. 2010; Trac et al. 2011). Of course cosmolog-ical hydrodynamical simulations fully include this contributionand thus do not require additional modeling of kinetic pressureeffects. While of importance for correctly interpreting SZ mea-surements, the X-ray observations of clusters have been cali-brated to partly take this effect into account when determiningmass from the X-ray inferred total thermal energy (e.g., usingthe Y X - M relation, Kravtsov et al. 2006).In this paper, our focus is on the effects of bulk motionswithin clusters. These dominate the total kinetic pressure bud-get since there is generally a smaller fraction of energy in ahydrodynamical turbulent cascade compared to the energy onthe injection scale which is well resolved for the relevant largescale motions. Quantifying turbulence in clusters is becomingfeasible with simulations that include the modeling of sub-gridturbulence (e.g., Iapichino & Niemeyer 2008) as well as sim-ulations with magnetohydrodynamics and anisotropic thermalconduction (Parrish et al. 2011).The method of smoothed particle hydrodynamics (SPH) thatwe are using for solving the inviscid Euler equations in thiswork is perfectly suited for studying large-scale bulk motionswhich dominate the kinetic pressure support due to its La-grangian and conservative nature. However, it is known thatSPH in its standard implementation poorly resolves hydrody-namical instabilities, such as of Kelvin-Helmholtz or Rayleigh-Taylor type (Agertz et al. 2007). In non-radiative simulationsof cluster formation, adaptively-refined mesh codes generatea larger core entropy level in comparison to SPH simulationswhich is presumably due to the difference in the amount ofmixing in SPH and mesh codes and possibly related to a differ-ent treatment of vorticity in the simulations (e.g., Frenk et al.1999; Mitchell et al. 2009; Vazza et al. 2011). A recent com-parison of a galaxy formation simulation with the SPH tech-nique and the recently developed moving mesh code AREPO (Springel 2010) enabled – for the first time – to test the na-ture of the hydrodynamic solver with otherwise identical im-plementations of the gravity solver, the sub-resolution physics,and the detailed form of the initial conditions. The movingmesh calculations resulted in more disk-like galaxy morpholo-gies in comparison to SPH. This difference originated froman artificially high heating rate with SPH in the outer partsof haloes, caused by viscous dissipation of inherent sonic ve-locity noise of neighboring SPH particles, an efficient damp-ing of subsonic turbulence injected in the halo infall regionpotentially by the artificial viscosity employed by SPH, andbecause of a higher efficiency of gas stripping in AREPO(Vogelsberger et al. 2011). Based on these results it may bequestionable to analyse small-scale velocity power spectra and2-point structure functions in SPH simulations; however, thedominating large-scale bulk motions which is our primary in-terest should be followed accurately. This theoretical expec-tation is confirmed by analyzing otherwise identical clustersimulations run with AREPO and GADGET/SPH that showan equal kinetic pressure contribution outside the core region( r > . R ) for the two numerical techniques (Puchwein &Springel, priv. comm., 2011). ICM shapes
DM halo shapes have been studied extensively. For SZ ob-servations, the shape of the gas distribution of the ICM is im-portant, especially in the far field of the intracluster mediumwhich contributes substantially to the total integrated SZ flux(Battaglia et al. 2010). The assumption of spherical symmetryis often made when calculating cluster properties from observa-tions and in analytical prescriptions so we would like to assessits validity. Semi-analytic models that employ the full three-dimensional information of a dissipationless dark-matter-onlysimulation use the shape of the resulting gravitational clusterpotentials, so it is important to study how such shapes com-pare with those that include the dissipational gas component.Recent numerical work has shown the impact of cooling andstar-formation on the properties of ICM shape for a sample sizeof 16 clusters (Lau et al. 2011), however, such a study has notbeen extended to a larger sample. Furthermore, the question ofhow energetic feedback in the cluster cores affects ICM shapeshas not been addressed.
Overview
In this work we explore a large statistical sample of simu-lated clusters with identical initial conditions but employingdifferent models for sub-grid physics. We quantify the impor-tance of non-thermal pressure support and ICM shapes on theSZ Y - M scaling relation. In Section 2, we briefly describe thesimulations and sub-grid physics used. We present our resultsfor non-thermal pressure support from bulk motions and ICMshapes in Sections 3 and 4, respectively. The impact of theseprocesses and changes in the simulated physics on the Y - M scaling relation is presented in Section 5. In Section 6 we sum-marize our results and conclude. COSMOLOGICAL SIMULATIONS AND CLUSTER DATA SET
We described the basic suite of hydrodynamical modelsused in Battaglia et al. (2010). We simulate tens of large-scale boxes of the cosmic web in order to improve our statis-tics of the number of objects while simultaneously aiming fora sufficiently high mass resolution to map out the core re-gions of those clusters and groups which are the target of cur- BATTAGLIA, BOND, PFROMMER, SIEVERSrent SZ cluster surveys and which dominate the SZ powerspectrum signal on scales larger than 1’. Here we charac-terize the average behaviour of the properties of the ICMover a large mass and redshift range using a modified ver-sion of the GADGET-2 (Springel 2005) code which employsSPH and treePM for the gravity solver. For each modeledphysics, we simulate a sequence of 10 boxes of side length165 h - Mpc = 200 Mpc with periodic boundary conditions, en-compassing N DM = N gas = 256 DM and gas particles. Thisgives an initial gas particle mass of m gas = 3 . × h - M ⊙ and a DM particle mass of m DM = 1 . × h - M ⊙ . Weadopt a minimum comoving (Plummer) gravitational smooth-ing length of ε s = 20 h - kpc. Our SPH densities are computedwith 32 neighbours. For our standard calculations, we adopt atilted Λ CDM cosmology, with total matter density (in units ofthe critical) Ω m = Ω DM + Ω b = 0 .
25, baryon density Ω b = 0.043,cosmological constant Ω Λ = 0.75, a present day Hubble con-stant of H = 100 h km s - Mpc - with h = 0 .
7, a spectral indexof the primordial power-spectrum n s = 0.96 and σ = 0.8.We compare results for three variants of simulated physics:(1) the classic non-radiative ‘adiabatic’ case with only grav-itational formation shock heating ; (2) an extended radiativecooling case with star formation, supernova (SN) energy feed-back and cosmic rays (CRs) from structure formation shocks(for more information on CRs, see Pfrommer et al. 2006a,2007; Enßlin et al. 2007; Jubelgas et al. 2008); (3) AGN feed-back in addition to radiative cooling, star formation, and SNfeedback. Radiative cooling and heating were computed as-suming an optically thin gas of primordial composition ina time-dependent, spatially uniform ultraviolet background.Star formation and supernovae feedback were modelled us-ing the hybrid multiphase model for the interstellar mediumof Springel & Hernquist (2003). The CR population is mod-elled as a relativistic population of protons described by anisotropic power-law distribution function in momentum spacewith a spectral index of α = 2 .
3, following Enßlin et al. (2007).With those parameters, the CR pressure causes a small reduc-tion in the integrated Compton- y parameter (Pfrommer et al.2007), but can result in interesting modifications of the localintracluster y -map.The AGN feedback prescription we adopt for our standardsimulations (for more details see Battaglia et al. 2010) allowsfor lower resolution and hence can be applied to large-scalestructure simulations. It couples the black hole accretion rate tothe global star formation rate (SFR) of the cluster, as suggestedby Thompson et al. (2005). If the SFR is larger than an ob-servationally motivated threshold, ˙ M ∗ > ⊙ yr - , the thermalenergy is injected into the ICM at a rate which is proportionalto the SFR within a given spherical region. The AGN feedbackin these box simulations injects approximately one third of totalinjected energy in the cluster formation phases at z > < z <
2, and the final third below z = 1 (analogous tojet/bubble like feedback). These fractions depend moderatelyon the the numerical resolution; increasing the resolution en-ables to resolve the growth of smaller halos at earlier times andcauses a higher fraction of energy injection at higher redshifts(see Battaglia et al. 2010, for a discussion).We define the virial radius of a cluster, R ∆ , as the radiusat which the mean interior density equals ∆ times the criti-cal density , ρ cr ( z ) (e.g., for ∆ = 200 or 500). For comparison,we will use an alternative definition of the virial radius, R ∆ , m ,where the mean interior density is compared to the mean mat- ter density , ¯ ρ m ( z ). For clarity the critical density and the meanmatter density are, ρ cr ( z ) = 3 H π G (cid:2) Ω m (1 + z ) + Ω Λ (cid:3) , (1) ¯ ρ m ( z ) = 3 H π G Ω m (1 + z ) . (2)Here we have assumed a flat universe ( Ω m + Ω Λ = 1 ) and areonly interested in times after the matter-radiation equality, i.e.,the radiation term with Ω r is negligible. We chose to define thevirial radius with respect to the critical density in continuitywith recent cluster measurements. The merits and utilities ofboth these definitions are discussed later in Appendix C.We apply the following two-step algorithm to compute thevirial mass of a cluster in our simulations. First, we find allclusters in a given snapshot using a friends-of-friends (FOF) al-gorithm (Huchra & Geller 1982). Then, using a spherical over-density method with the FOF values as starting estimates, werecursively calculate the center of mass, the virial radius, R ∆ ,and mass, M ∆ , contained within R ∆ , and compute the radiallyaveraged profiles of a given quantity with radii scaled by R ∆ .We then form a weighted average of these profiles for the entiresample of clusters at a given redshift unless stated otherwise.We use the integrated Compton y -parameter as our weightingfunction, Y ∆ = σ T m e c Z R ∆ P e ( r )4 π r d r ∝ E th ( < R ∆ ) , (3)where σ T is the Thompson cross-section, m e is the elec-tron mass and P e is electron pressure. For a fully ionizedmedium of primordial abundance, the thermal pressure P = P e (5 X H + / X H +
1) = 1 . P e , where X H = 0 .
76 is the pri-mordial hydrogen mass fraction. NON-THERMAL CLUSTER PROFILES
Several simulations (Evrard 1990; Rasia et al. 2004;Lau et al. 2009) showed that the kinetic pressure from bulkmotions contributes a small but still significant amount of en-ergy within R and this importance increases for larger clusterradii (Lau et al. 2009; Battaglia et al. 2010; Burns et al. 2010).Thus, it is important to accurately quantify the kinetic pres-sure contribution as it biases the hydrostatic cluster masses andis significant to the total energy budget within clusters. Thereare two kinetic pressure contributions, namely large-scale, un-virialized bulk motions and subsonic turbulence. For a Kol-mogorov power spectrum of turbulence, the energy is domi-nated by the largest scales which we resolved and characterizein our simulations. Hence we believe that our approach cap-tures the majority of the kinetic pressure contribution. Kinetic pressure support
The internal bulk motions in the medium can be quantified bythe mass-averaged velocity fluctuation tensor, h δ V i δ V j i , whichis associate with the kinetic pressure (stress) tensor, P kin , i j = ρ h δ V i δ V j i , P kin ≡ Tr P kin / ρ h δ V · δ V i / ,δ V = a ( υ - ¯ υ ) + a H ( z ) ( x - ¯ x ) . (4)In this paper we focus on the trace, P kin , which we refer toas the kinetic pressure. Issues associated with the anisotropicstress tensor will be made explicit where they appear, and areN THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 5 r / R P k i n / P t h AGN feedback, z = 01.1 x 10 M O • < M < 1.7 x 10 M O • M O • < M < 2.7 x 10 M O • M O • < M < 4.2 x 10 M O • M O • < M < 6.5 x 10 M O • M O • < M < 1.01 x 10 M O • M O • < M < 1.57 x 10 M O • Shaw et al. 2010Trac et al. 2010 R R vir r / R P k i n / P t h AGN feedback, 1.7 x 10 M O • < M < 2.7 x 10 M O • z = 0 z = 0.3 z = 0.5 z = 0.7 z = 1.0 z = 1.5Shaw et al. 2010, z = 0Shaw et al. 2010, z = 1 R R vir Figure 1.
The ratio of kinetic and thermal pressure support, P kin / P th , depends on mass and redshift. We show the median of P kin / P th as a function of radius forthe AGN feedback simulations for various mass bins at z = 0 (left) and as a function of redshift for a fixed mass bin (right). We additionally show the 25 th and 75 th percentile values for the lowest mass bin at z = 0 (dotted). In both panels we illustrate the 1 and 2 σ contributions to Y ∆ centered on the median for the feedbacksimulation by horizontal purple and pink error bars which extends out to 4 R (Battaglia et al. 2010). Two analytical models for the P kin by Shaw et al. (2010) andTrac et al. (2011) are shown with the dash dot and dashed lines, respectively. The Shaw et al. (2010) model matches our result in the mass bin 2 . × M ⊙ ≤ M ≤ . × M ⊙ at intermediate cluster radii (this mass bin best represents the mean mass of their sample at redshift zero), but also illustrates the need for amass dependence in future analytical models. The dependence of P kin / P th on cluster mass is driven by the variation of P kin with mass (see Fig. 2 below). r / R P t h / P f it , P k i n / P f it P th / P fit P kin / P fit AGN feedback, z = 01.1 x 10 M O • < M < 1.7 x 10 M O • M O • < M < 2.7 x 10 M O • M O • < M < 4.2 x 10 M O • M O • < M < 6.5 x 10 M O • M O • < M < 1.01 x 10 M O • M O • < M < 1.57 x 10 M O • R R vir r / R P t h / P f it , P k i n / P f it P th / P fit P kin / P fit AGN feedback, 1.7 x 10 M O • < M < 2.7 x 10 M O • z = 0 z = 0.3 z = 0.5 z = 0.7 z = 1.0 z = 1.5 R R vir Figure 2.
Mass and redshift dependence of P th and P kin , normalized to an empirical fit from BBPS2, P fit , to the scaled thermal pressure, P th / P ∆ . We show the meanscaled thermal and kinetic pressure profiles at z = 0 as a function of radius for the AGN feedback simulations in various mass bins (left), and for various redshifts atfixed mass bin (right). also explored in more detail in BBPS3. The code uses comov-ing peculiar velocities, which are translated into the internalcluster velocities relative to the overall mean cluster velocity inthe Hubble flow by the relation given, where H ( z ) is the Hub-ble function, a is the scale factor, υ (= d x / d t ) is the peculiarvelocity and x is the comoving position of each particle. Thegas-particle-averaged cluster bulk flow within R is ¯ υ and thecenter of mass within R is ¯ x .The radial profiles of the kinetic-to-thermal pressure, P kin / P th , shown in Fig. 1 for various mass bins demonstratean overall mass dependence at all cluster radii, predominatelydriven by the variation of P kin and not P th with mass. We showthis explicitly in Fig. 2 where we scale P kin with the virial ana- logue of the thermal pressure, P ∆ ≡ GM ∆ ∆ ρ cr ( z ) f b / (2 R ∆ ) , f b = Ω b / Ω m . (5)This behavior reflects the average formation history of galaxygroups and clusters which, according to the hierarchical pic-ture of structure formation, sit atop the mass hierarchy, withthe most massive clusters forming and virializing near to thepresent time. In contrast, the median galaxy group ( M =10 M ⊙ ) has stopped forming today as can be seen by thedramatically decreasing mass accretion rates implying that theassociated virializing shocks have dissipated the energy asso-ciated with the growth of these objects and hence decreasingthe kinetic pressure support (Wechsler et al. 2002; Zhao et al. BATTAGLIA, BOND, PFROMMER, SIEVERS2009; Pfrommer et al. 2011). The semi-analytic model fornon-thermal pressure support by Shaw et al. (2010) falls inthe middle of the mass bins chosen since this model resultsfrom a sample of 16 high resolutions adaptive mesh refinement(AMR) simulations of individual galaxy clusters (Lau et al.2009) which have a similar mass range. We provide a simplefit for the mass dependence of P kin / P th in Appendix A. We findthat the radius at which P kin = P th is just beyond the sphericalcollapse definition for R vir from Bryan & Norman (1998), R vir = (cid:18) M vir π ∆ cr ( z ) ρ cr ( z ) (cid:19) / , (6)where ∆ cr ( z ) = 18 π + Ω ( z ) - - Ω ( z ) - and Ω ( z ) = Ω m (1 + z ) (cid:2) Ω m (1 + z ) + Ω Λ (cid:3) - . Hence, this radius represents apossible physical definition for the virialized boundary of clus-ters.The redshift evolution of P kin / P th is dramatic. At higher red-shift, P kin is increasing faster than P th over all radii (cf. Fig.2), such that at z = 1, P kin / P th is approximately twice that at z = 0. In the picture of hierarchical structure formation, at anygiven redshift the most massive objects are currently assem-bled and hence show the largest kinetic pressure contribution incomparison to smaller objects that formed on average earlier.Or equivalently, at fixed cluster mass, the relative contributionfrom kinetic pressure and the relative amount of substructureincreases with redshift. In particular, the relative mass accre-tion rates increase from z = 0 to z = 2 by a factor 3 for clus-ters ( M = 10 M ⊙ ) and 10 for groups ( M = 10 M ⊙ ) (seePfrommer et al. 2011; Gottlöber et al. 2001). This strong evo-lution in P kin / P th is lessened by a different choice of scalingradius, i.e., if we normalize by R , m instead of R (cf. Ap-pendix C). Although this ratio cannot be observed, we will useit as an indicator for the dynamical state of clusters in our simu-lations. Results from Lau et al. (2009) find a similar correlationbetween P kin and the X-ray definition of dynamical state, froma smaller sample of 16 clusters. At z = 1, the Shaw et al. (2010)semi-analytic model for non-thermal pressure support does notmatch our simulations as well as it does at redshift zero. The formation of clusters and the associated accretion of sub-structure are driven by the depth of the cluster gravitational po-tential. Therefore, it is not surprising that we find kinetic pres-sure support to be ubiquitous in the three differently simulatedphysics cases (cf. Fig. 3). Looking at the median of this non-thermal pressure support we find similar radial profiles withinthe 25 th and 75 th percentiles of the complete distribution ofclusters. In the AGN feedback simulations we find marginallylower values for P kin / P th . These differences are well withinthe 25 th and 75 th percentiles implying consistency across dif-ferently modeled physics. Thus, our model of AGN feedbackdoes not significantly alter the kinetic pressure support at lowredshift although there seems to be a hint that this may be thecase at larger radii at redshifts z ∼ Our kinetic pressure contribution is larger at the center compared to thatin the model by Shaw et al. (2010). This discrepancy is probably a manifes-tation of the well-known core entropy problem in numerical simulations. In(adaptive) grid codes there is a larger level of core entropy generated in com-parison to SPH codes implying that the enhanced entropy (which results fromdissipating gas motions) is accompanied by a smaller amount of kinetic pres-sure. This is presumably due to the difference in the amount of mixing in SPHand mesh codes and possibly related to a different treatment of vorticity in thesimulations (e.g., Frenk et al. 1999; Mitchell et al. 2009; Vazza et al. 2011). r / R P k i n / P t h M O • < M < 2.7 x 10 M O • AGN feedbackShock heatingRadiative cooling z = 0 z = 1 R R vir Figure 3.
Shown is the median of P kin / P th as a function of radius for differ-ent physics models at z = 0 (solid) and z = 1 (dashed) with the 25 th and 75 th percentile values shown for the AGN feedback simulations at z = 0 (dotted).Results are shown for the mass bin 1 . × M ⊙ ≤ M ≤ . × M ⊙ totake out the dependence on mass of P kin / P th . The kinetic pressure contributionis similar for our differently simulated physics, suggesting that gravitationalprocesses dictate that contribution (while AGN feedback slightly decreases thekinetic pressure contribution, especially for higher redshifts). The horizontalpurple and pink error bars have the same meaning as in Fig. 1. r / R M H S E ( < r ) / M t o t ( < r ) AGN feedback, z = 01.1 x 10 M O • < M < 1.7 x 10 M O • M O • < M < 2.7 x 10 M O • M O • < M < 4.2 x 10 M O • M O • < M < 6.5 x 10 M O • M O • < M < 1.01 x 10 M O • M O • < M < 1.57 x 10 M O • R R vir Figure 4.
Assessing the bias in hydrostatic masses, M HSE , due to the kineticpressure. The median of M HSE / M tot as a function of radius for AGN feedbacksimulations for various mass bins, with the 25 th and 75 th percentile valuesshown for the smallest mass bin (dotted). Assuming hydrostatic equilibriumfor all clusters of a given mass will bias the mass values low by 20 to 25%.The scatter about the median – represented by the 25 th and 75 th percentiles– amounts to approximately 5%. This bias is not representative for a relaxedcluster sample which will likely have a smaller bias since the calibration ofsuch a sample against numerical cluster simulations shows (Kravtsov et al.2006). tensor. We treat in detail the velocity anisotropy in BBPS3.The main results are that the core regions near the center, thevelocity distribution starts to become isotropic for the gas ingroups and (to a lesser extent) for the DM and gas in largerclusters. The positive values of velocity anisotropy around thevirial radius indicate (radial) infall, whereas the strong decreaseat even larger radii (very noticeably in the DM) is caused by theturn-around of earlier collapsed shells, which minimizes theradial velocity component such that the tangential componentsN THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 7dominate the velocity. Hydrostatic Masses
Even if clusters are in hydrostatic equilibrium, balancing thegravitational force to the pressure gradient yields ∇ P = ρ g → - ρ GM ( < r ) ˆ r / r for spherical symmetry , (7)and using this relation to estimate cluster masses will give thewrong result if one does not include non-thermal pressure, inparticular the kinetic pressure.As others have shown (Evrard 1990; Rasia et al. 2004;Lau et al. 2009), assuming that all the pressure in Eq. (7) isthermal ( P = P th ) is incorrect; for clarity we define M HSE to bethe mass derived using P = P th . Comparing M HSE to the truemass inside a given radius, M tot , we find that M HSE on aver-age underestimates M tot by 20–25% depending on the radius(cf. Fig 4). This bias is almost independent of cluster mass outto R , the current maximum radius typically observed by theX-ray telescopes Chandra and
XMM-Newton . We can under-stand this weak mass dependence by rewriting the total pres-sure P = P ( P th / P + P kin / P ). Since P kin / P ∝ M / (see Fig. 19),the hydrostatic mass estimates inherit a similarly weak depen-dence on mass. Thus, an overall correction to the hydrostaticmass is reasonable for these measurements.Individual clusters can stray from this generalization, sinceeach cluster has a unique dynamical state and formation his-tory. These deviations are suggested by the scatter of ∼
5% be-tween the 25 th and 75 th percentiles of the complete distribution.For cluster samples that are selected against major mergers (forwhich the assumption of spherical symmetry will also be ques-tionable), the correction factor will necessarily be smaller, e.g.,for quality X-ray data of a Chandra sample, the hydrostaticmass correction was found to be of the order M HSE ∼ CLUSTER SHAPES
Generally, we expect clusters to be triaxial since they growby accretion and through merging along filamentary structuresthat impose tidal gravitational forces upon the forming clusters.Following Dubinski & Carlberg (1991), we estimate this non-sphericity of cluster gas and dark matter (DM) by computingthe normalized moment-of-inertia tensor, h δ x δ x i i j ( r < R | w ) = P α w α ( x i , α - ¯ x i )( x j , α - ¯ x j ) P α w α , (8)for several weightings w α of the contribution of particle α thatlies within a given radius R . The tensor measures the variancein the spatial fluctuations within R , with δ x i = x i - ¯ x i the devia-tion of the particle position x i from the centre-of-mass ¯ x i ( < R )of the region. Using mass weighting, w α = m α , for the darkmatter or the SPH gas particles gives the moment-of-inertia inits usual form. It has an effective ρ ( x ) x d ln x reach in its probeof the unit vector combination ˆ δ x i ˆ δ x j , hence preferentially feelsthe outskirts, near R . In the Appendix B, we explore how ourresults are modified with a weight w α = m α / x α less sensitiveto the outskirts: this weight just mass-averages the unit vectorproduct, hence emphasizes the more isotropic interior. Sincethe tSZ signal is of primary interest to us, we also considerthermal-energy weighting, with w α = m α T α the product of themass and temperature.We quantify the asphericity of cluster gas and DM by twoparametrizations: the axis ratios (in particular the ratio of the largest-to-smallest main axis, c / a ); and the three-dimensionalasymmetry parameters for symmetric tensors introduced byBBKS (Bardeen et al. 1986). Both use the eigenvalues λ i ofthe moment-of-inertia tensor at a prescribed radius, ordered by λ < λ < λ . The ellipsoid associated with the tensor has axislengths a = √ λ , b = √ λ , and c = √ λ . (We note that Lau et al.(2011) express their results in terms of eigenvalue ratios, defin-ing a ′ = λ , b ′ = λ , and c ′ = λ ). The eigenvectors E i associatedwith λ i are also used to rotate the clusters to their principal axesand to explore alignment variations with radius.The BBKS-style asymmetry parameters are defined by e = λ - λ ¯ λ , p = λ - λ + λ ¯ λ , ¯ λ ≡ λ + λ + λ . (9)Following BBKS (Bardeen et al. 1986), we refer to e as the el-lipticity and p as the prolaticity. When p is positive the clustersare prolate, and when p is negative the clusters are oblate. Thuswe also define the oblateness o ≡ - p θ ( - p ), equal to | p | for neg-ative p and zero for positive p . Although e is the most strik-ing indicator of cluster elongation, the degree of prolateness oroblateness are also necessary to specify the general triaxialityof the morphological configuration on a given smoothing scale. Overall shapes and their profiles
We rotate all clusters into the moment-of-inertia tensor frameusing the eigenvector matrix E , so x ′ = E x . The output order-ing is arbitrary; we choose the convention that the major axis isaligned with the x-axis and the minor axis is aligned with the z-axis. In Figs. 5 and 6 we show the results for DM and gas whichhave been obtained by computing the weighted moment-of-inertia tensor within 3 R , rotating into the moment-of-inertiatensor frame, and stacking the respective distributions, i.e., gasdensity and pressure as well as DM density. The rotated con-tours show obvious elongations along the major axis; with theellipticity being larger at z = 1 in comparison to z = 0. The elon-gation is larger for the DM distribution in comparison to thegas density and pressure which show very similar behaviour.Even in the rotated stacked distributions, the innermost contourlines become more spherical because they are intrinsically lesselliptical (see below) and because the main axes of the innerdistributions are twisted relative to those at 3 R so that theirellipticity partially averages out to become more spherical (seeSect. 4.5).In order to quantify these results, we show the mass depen-dence and redshift evolution of the ellipticity within R inFig. 7. Due to the dissipationless nature of DM, its elliptic-ity is larger (smaller ratio of c / a ) in comparison to that ofthe gas. This is because the kinetic energy of the accretedgas is dissipated at cluster accretion shocks – a process thaterases part of the memory of the geometry of the surroundinglarge scale structures and their tidal force field. Those accre-tion shocks are typically forming at radii > R as suggestedby numerical simulations (Miniati et al. 2000; Ryu et al. 2003;Pfrommer et al. 2006b; Skillman et al. 2008; Vazza et al. 2009)or indirectly by the action of shock waves on radio plasmabubbles, which represents a novel method of finding forma-tion shocks (e.g., Ensslin et al. 2001; Pfrommer & Jones 2011).Following these qualitative considerations, it is not surprisingthat the ellipticity of the gas distribution does not show anymass dependence while the DM distribution of more massive BATTAGLIA, BOND, PFROMMER, SIEVERS -3 -2 -1 0 1 2 3 x / R -3-2-10123 z / R AGN feedback, z = 0Stacked gasRotated stacked gasRotated stacked pressure -3 -2 -1 0 1 2 3 x / R -3-2-10123 z / R AGN feedback, z = 0Stacked DMRotated stacked DM Figure 5.
Stacked density and pressure distributions with and without rotations into the principle axis frame of the correspondingly weighted moment-of-inertiatensors at z = 0. Left: We compare rotated distributions of the gas density (red) and pressure (blue) to the non-rotated stacked gas density (black) at z = 0. Right: Shownis the same as on the left for DM. The non-rotated clusters average out to form spherical iso-density contours, while the rotated clusters clearly show elongationsalong the major axis (defined here as the x -axis). The thicker lines approximately show the radii R , R and R from the inside out. These contours have beensmoothed to a pixel size of 0.09 R . The horizontal purple and pink error bars have the same meaning as in Fig. 1. -3 -2 -1 0 1 2 3 x / R -3-2-10123 z / R AGN feedback, z = 1Stacked gasRotated stacked gasRotated stacked pressure -3 -2 -1 0 1 2 3 x / R -3-2-10123 z / R AGN feedback, z = 1Stacked DMRotated stacked DM Figure 6.
Same as in Fig. 5, but at z = 1. N THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 9 M [ h -1 M O • ]0.60.70.80.91.0 c / a c / a | r ,gas c / a | r ,DM Dissipationless KE05 z c / a c / a | r ,gas c / a | r ,DM Dissipationless KE05
Figure 7.
We show axis ratios of clusters that are obtained by computing the moment-of-inertia tensor of the gas (red) and DM mass distributions (blue) within R and stacking those in bins of cluster mass and redshift. The resulting mean and standard deviation of the axis ratio c / a is shown as a function of M at z = 0 (leftpanel) and at fixed average mass bin of M = 2 . × h - M ⊙ as a function z (right panel). See Table 1 for fit values; here, we have chosen to quote h - M ⊙ to compare directly with the dissipationless simulations by Kasun & Evrard (2005, KE05). Shocks dissipate the kinetic energy of the gas which causes larger axisratios/smaller ellipticities, this couples through gravity to the DM distribution and sphericalize their axis ratios, resulting in smaller ellipticities in comparison todissipationless simulations alone, e.g., by KE05 (that do not follow the hydrodynamics of the gas). c / a c / a | r ,gas c / a | p c / a | r ,DM AGN feedbackRadiative coolingShock heating D (Pressure - Gas density) R R vir r / R -50510 D c / a [ % ] e , p , o AGN feedback, z = 0 e , p , o r ,gas e , p , o p e , p , o r ,DM epo R R vir r / R -40-2002040 D ( e , p , o ) [ % ] Figure 8.
Average cluster axis ratios and ellipticities for the DM mass (blue), gas mass (red) and pressure (green) distributions within a scaled radius r / R . Left:Shown is the axis ratio c / a as a function of scaled radius for all simulated physics models. In the bottom panel we show the relative differences of the shockheating (long-dashed) and the radiative cooling simulations (short-dashed) with respect to AGN feedback simulations. Right: We show the ellipticity, prolaticity andoblaticity as a function of scaled radius. The bottom panel shows the relative differences between the pressure and gas density weightings of the moment-of-inertiatensor. The axis ratio c / a and ellipticity show the same trends. We find clusters to be more prolate then oblate. In the regions beyond R vir the sudden decrease inthe axis ratios can be attributed to other nearby groups and collapsed objects (that is also seen as an enhanced density clumping at these radii, Nagai & Lau 2011;Battaglia et al. 2011c). The pressure-weighted shapes tightly track the density-weighted shapes with deviations of less than 5%. The horizontal purple and pink errorbars have the same meaning as in Fig. 1. Table 1
Axis ratio fits for cluster as a function of mass and redshift. B M A M B z A z DM 0 . ± . - . ± .
006 0 . ± . - . ± . . ± . - . ± .
002 0 . a ± . - . ± . . ± . - . ± .
01 0 . ± . - . ± . a We use a re-normalized value from M = 1 × h - M ⊙ to M = 2 . × h - M ⊙ . clusters shows a larger ratio of c / a in comparison to smallersystems. However, the ellipticity of the gas and DM distribu-tion are increasing as a function of redshift, at about the samerate. This can be understood by the fact that 1) a given massrange of clusters shows a larger degree of morphological dis-turbances/merging at higher redshifts which probe on averagedynamically younger objects and 2) the redshift evolution ofthe velocity anisotropy (cf. BBPS3) which shows that the av-erage location of accretion shocks moves to smaller radii (ifscaled by R ). Hence at larger redshifts, also the gas distribu-tion probes the infall/pre-accretion shock region that is shapedby the tides exerted by the far-field of clusters.We compare the results from our simulations directly withthose of Kasun & Evrard (2005) in Figure 7 and Table 1.Other work (Allgood et al. 2006; Gottlöber & Yepes 2007;Macciò et al. 2008; Lau et al. 2011) on DM and gas shapeshave used different mass definitions, axis definitions and cos-mologies than we do, making quantitative comparisons diffi-cult, but we note that these various results were shown to beconsistent with Kasun & Evrard (2005). For the mass and red-shift functional fits, Kasun & Evrard (2005) define c / a ( M ) = B M (1 + A M ln[ M / h - M ⊙ ]) and c / a ( z ) = B z (1 + z ) A z , respec-tively. The axis ratios that we find for the DM mass de-pendence and redshift evolution have slopes consistent withKasun & Evrard (2005), but as an overall trend, our axis ratiosare more spherical than theirs. Some of this may be traced todiffering cosmological parameters, but some may be becausethe less aspherical baryons may have an impact on the DM el-lipticity. The effect of baryons on DM has been explored in,e.g., Rudd et al. (2008).In the following, we will show radial profiles of the axis ra-tios and asymmetric parameters that are obtained by comput-ing the moment-of-inertia tensor at 30 different radii for eachcluster. In Figure 8, we report on the overall radial distribu-tion of c / a and ellipticity in the gas and DM distributions.Within R , the ellipticities of gas density and pressure arerather flat at a level of c / a ≃ . - .
9. As laid out above,this is because dissipation effects at the accretion shocks causean effective sphericalization and erase the memory of large-scale tidal fields. In contrast, ellipticities are increasing for theDM as a function of radius due to the dissipationless nature ofDM, i.e. c / a decreases from values around 0.8 in the centerto 0.7 at R . The radial behaviour may be due to increasedtidal effects on DM substructures at small radii which causesa dramatic drop of their central mass density (Springel et al.2008a,b; Pinzke et al. 2011). Effectively this causes a redistri-bution of a clumped (elliptical) to a smooth distribution thatis able to couple more efficiently to the (more spherical) gasdistribution. Studying the asymmetric parameters, we find thatif a cluster is prolate, it is on average more elliptical than anoblate one that is always close to spherically symmetric.We find that the average axis ratios and ellipticities have a c / a , b / a , r s AGN feedback, z = 0 c / a | r ,gas c / a | p c / a | r ,DM Full 3D, c / a Random 2D, b / a Correlation coefficient r s R R vir r / R D ( b / a - c / a ) [ % ] Figure 9.
Top: We compare the average of the 2D axis ratios (dashed) of threerandom orthogonal projections to the 3D axis ratios (solid) for the DM mass(blue), gas mass (red) and pressure (green) from the clusters in the AGN feed-back simulations. Additionally, we show the linear correlation coefficient, r s (dotted line), between the projected 2D and the 3D axis ratios. The horizon-tal purple and pink error bars have the same meaning as in Fig. 1. Bottom:Shown is the relative differences between the projected 2D and 3D axis ratios.While the relative difference between the projected 2D and 3D axis ratios varybetween 15–20% for the DM mass distribution (with the 2D axis ratios beingmore spherical), the relative differences are smaller for the mass and pressuredistribution of the gas, with values between 5–10%. As expected, the projected2D and the 3D axis ratios are correlated with an increasing correlation coeffi-cient at larger radii which suggests that the substructure distribution that drivesthe asphericity also causes this correlation. pronounced break in their slopes at r ∼ . R . The break inthe ellipticity arises from substructure in the cluster outskirts.Recent X-ray observations of the Perseus cluster find a strongsignature of clumping in gas density (Simionescu et al. 2011);qualitatively consistent with the findings in simulations but notquantitatively (Nagai & Lau 2011). This gas density clump-ing is a direct tracer of substructure and becomes important atroughly the same radius where we find the break in the elliptic-ity. Interestingly, this effect is not only seen in DM and gas butalso in pressure, which suggests that the pressure is clumpedin a similar fashion as the gas density. In order to accuratelymodel the outskirts of clusters, semi-analytic models will needto properly deal with the substructure. In the Appendix B, weshow that one can attempt to counteract or lessen the impact ofsubstructure on the gas, pressure and DM shapes by includingan r - -weighting when calculating the moment-of-inertia ten-sor (cf. Eq. (8)). In future work, we will further explore theissue of substructure. How shape profiles depend on modeled physics
We also address the influence that changes in the simulatedphysics has on cluster shapes in Fig. 8. While the ellipticityof the gas is slightly larger in non-radiative models, it is verysimilar for the gas distribution in our radiative models (radia-tive cooling and star formation with and without AGN feed-back). Dissipating accretion shocks seem to explain the over-all behavior rather well and the different physical models onlyN THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 11marginally change the cluster shapes in the gas. In the DM,however, there is still a pronounced difference among our tworadiative physics models with the ellipticities of the AGN feed-back model being larger that in our pure radiative model. Thissmall ellipticity is a remnant of overcooling that our pure radia-tive model suffers with an associated star formation rate that isunphysically high. Most of these stars form out of the cold,dense gas in the core region which causes a decreasing centralpressure support so that gas at larger radii moves in adiabat-ically and causes a deeper potential which in turn causes theDM to adiabatically contract. Enhanced dissipation processesin the gas sphericalize the potential which is then communi-cated to the DM during this central settling. We find that in-cluding AGN feedback counteracts the overcooling issue andmodifies the DM shapes on the level of 5% in comparison toour pure radiative simulations (cf. Fig 8).Our general trends are similar to those reported by Lau et al.(2011) who also find that the DM distribution is more spher-ical for radiative simulations in comparison to non-radiativemodels. However, the differences between radiative and non-radiative simulations are not as extreme as those found inLau et al. (2011), since our radiative simulations do not havethe level of (catastrophic) cooling in the central regions, be-cause their simulations have higher resolution and includecooling from metals which we do not. The AGN feedbackstabilizes the cooling and, thereby, lessens this sphericalizingeffect on the DM ellipticity.
Projected and intrinsic shapes
In order to tie the underlying 3-dimensional structure of clus-ters to observable 2-dimensional projections, we compare theintrinsic 3D axis ratios to axis ratios of random 2D projections,i.e. we project the DM density and gas density/pressure distri-butions along a randomly chosen direction and then computethe 2D moment-of-inertia tensor. The results are shown in Fig-ure 9. We find that the 2D axis ratios ( b / a ) for both, the gasdensity and pressure are systematically closer to unity than the3D ratios c / a by ∼ ∼
15% underestimate of theintrinsic (3D) axis ratios. Using the linear correlation coeffi-cient statistic ( r s ), we find that the random 2D axis ratios arestrongly correlated with the intrinsic 3D axis ratios with a ra-dially increasing correlation strength (cf. Fig 9). As expected, c / a serves as a limit to the observable projected (2D) axis ratio.We find that the mean 2D axis ratio, as a function of cluster ra-dius, closely tracks c / a , modulo a roughly constant ∼ - Mass and redshift dependence of shape profiles
Both the density- and pressure-weighted ellipticities showthe same general trends with radius and cluster mass. The ellip-ticity increases with increasing cluster mass by ∼
50% over themass ranges shown (cf. Fig 10). On the right-hand side of Fig.10, we show the redshift evolution of the cluster shapes andfind that the ellipticity is a stronger function of redshift than themass. For increasing redshift, the break in the ellipticity profilemoves to smaller radii (when scaled to R ). Both behaviors,the mass and redshift dependence can be understood in the hi-erarchical picture for structure formation, where clusters showincreased mass accretion rates and hence an increased level ofsubstructure for larger clusters (at a given redshift) or, equiv-alently, for a cluster of given mass at higher redshifts which probe on average systematically younger systems. Similar tothe non-thermal pressure support, the redshift evolution foundin the ellipticities are lessened by a different choice of scalingradius (cf. Appendix C). This result suggests that using a sin-gle (constant) ellipticity profile for clusters is not sufficient forpercent level accuracy.Pressure-weighted ellipticities are marginally more sphericalthan the density-weighted ellipticities for r < R (cf. Figs. 8and 10). However, between R and 2 R the behavior is re-versed. This is because the core region shows a smaller kineticpressure support implying that hydrostatic forces had time toact and to smooth out the pressure distribution whereas at largerradii, pressure-weighted ellipticities are affected more by in-fall caused by a noticeable pressure clumping at these radii (cf.BBPS4). We have shown that scales around R are the mostrobust for studying the virial properties of clusters, since allellipticities show only ∼
10% redshift evolution and mass de-pendence on these scales. Taken together with the relativelymodest degree of non-thermal pressure support on these scalesfound in § 3 and that these scales are far enough out to avoid thecomplications of the intricate “short-distance” physics of thecluster core region, we can give further justification for whatis already the practice in the X-ray cluster community, drivenby the nature of the X -ray data, namely a focus on R andenvirons for "global" cluster properties. Alignment variations and semi-analytical models
Semi-analytic models for the baryon distribution in clustersinclude an underlying assumption that baryons will arrangethemselves along equipotential surfaces (or in some cases theDM density-weighted surfaces). Given the importance of thisassumption, we test its validity in our simulations. In Figure 7,we plot the ratio c / a for both dark matter and gas as function ofcluster mass and redshift. While c / a for DM haloes decreaseswith halo mass as expected (Jing & Suto 2002), we find that c / a is constant for the gas distribution. This is potentially aproblem for semi-analytic models of ICM gas (Ostriker et al.2005; Bode et al. 2009), which solve for the resulting gas dis-tribution in a DM-dominated gravitational potential as obtainedfrom dissipationless simulations while allowing for a constantnon-thermal pressure contribution (in the latter case). How-ever, the gravitational potential from the DM is more sphericalthan the underlying matter distribution (e.g. Lau et al. 2011),and so the semi-analytic shape estimates are not as discrepantas one might expect from Figure 7.A more important issue is the alignment of the gas or pres-sure with respect to the DM. We calculate the angular differ-ence between the major axes of the DM and those of the gasand pressure major axes at a given radius, using the moment-of-inertia tensor eigenvectors E , DM ( r ) · E , gas ( r ). When calcu-lating misalignment, the major axes of nearly spherical objectsare poorly defined quantities. To avoid this problem we cal-culate, in each radial bin, a weighted average using 1 - c / a asthe weight. Furthermore, we exclude the region inside 0 . R since the gas and pressure shapes are nearly spherical, with c / a | DM > .
75. On average at a given radius, the cluster gasand pressure are 20 -
30 degrees misaligned from the major axisof the DM (cf. Fig. 11). In the next section we show SZ mea-surements of the total thermal energy in clusters, Y , stronglydepend on the projection axis through the cluster. Thus, mis-alignment between the semi-analytic baryon distortion and the“true” distribution may cause biases when using semi-analyticmodels to, e.g. , tie weak-lensing and SZ observations together.2 BATTAGLIA, BOND, PFROMMER, SIEVERS e M O • < M < 1.7 x 10 M O • M O • < M < 2.7 x 10 M O • M O • < M < 4.2 x 10 M O • M O • < M < 6.5 x 10 M O • M O • < M < 1.01 x 10 M O • M O • < M < 1.57 x 10 M O • e r ,gas e p e r ,DM AGN feedback, z = 0 R R vir r / R -50050 D e [ % ] e AGN feedback z = 0 z = 0.3 z = 0.5 z = 0.7 z = 1.0 z = 1.5 e r ,gas e p e r ,DM R R vir r / R D e [ % ] Figure 10.
The dependence on mass and the redshift evolution of the cumulative ellipticity profile as a function of r / R . Left: Shown is the ellipticity profile at z = 0 for various mass bins. Bottom left: Shown are the relative differences in ellipticity to the lowest mass bin (1 . × M ⊙ < M < . × M ⊙ ). Overthis mass range, the cluster ellipticities show a noticeable but not substantial mass dependence within R , in contrast to the stronger dependence on P kin / P th . Right:Shown is the ellipticity profile for various redshift bins. The horizontal purple and pink error bars have the same meaning as in Fig. 1. Bottom right: Shown is therelative difference of ellipticity at a given redshift to z = 0. The redshift evolution of the ellipticity (especially at large radii, r > R ) is driven by the larger amountof substructures at higher redshifts due to the increased mass accretion rate of group/cluster halos at these redshifts. The pressure-weighted ellipticities track thedensity-weighted ellipticities well and show the same trends with redshift. r / R Dq [ D e g ] AGN feedbackRadiative coolingShock heating Dq r ,gas Dq p R R vir Figure 11.
The weighted median angles between the DM major axis and gas(red) and pressure (green) axes at each radius for all simulated physics models:AGN feedback (solid), radiative cooling (short-dashed), and shock heating-only (long-dashed). The 25 th and 75 th percentile values are shown for the gasdensity in the AGN feedback model (dotted). On average the gas and pressureaxes are misaligned by 20 to 30 degrees to the DM principle axis, independentof our simulated physics models. However, both simulations with radiativecooling show more misalignment in the inner regions than the non-radiativesimulations. The light colors and lines represent the region where the averagecluster shape is close to spherical ( c / a | DM > .
75) such that the major axes arenot well defined and their angles are approaching a random distribution. Notethat we have weighted the average angles by 1 - c / a to down-weight the anglesfrom the spherical ICM shapes and the cluster interiors. The horizontal purpleand pink error bars have the same meaning as in Fig. 1. SZ SCALING RELATION
In this section we explore the impact of AGN feedback, clus-ter shapes and kinetic pressure support on the SZ-flux-to-massrelation, Y - M , using our large sample of clusters. We computethe SZ flux for all clusters for both, spherical boundaries andcylindrical apertures ( Y sph and Y cyl ). For the cylindrical aper-ture calculations the total fluxes are computed along each axisof the moment-of-inertia frame, measured at R , and addi-tionally along each axis of another randomly-oriented frame.We choose the line of sight boundaries for the cylindrical in-tegrations to be three times the radius of the aperture. Thisprocedure enables quantifying the importance of substructure,which we have already shown in Sections 3 and 4 to be signifi-cant at radii beyond R . From the calculated Y ∆ values we fitan average scaling relation, Y ∆ = 10 B (cid:18) M ∆ × h - M ⊙ (cid:19) A h - Mpc , (10)where A and B are the fit parameters for the slope and normal-ization, respectively. We weight each cluster by its Y ∆ whenfitting for A and B to keep the low-mass clusters from com-pletely dominating the fit. Self-similar Y -M scaling relation
We review the expectations for Y in the idealized case ofa cluster in virial equilibrium to help understand how possi-ble deviations from the self-similar Y - M relation and the scat-ter about it may arise. Starting with Eq. (3), which has beenN THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 13 M [ M O • ]10 -6 -5 -4 Y , s ph [ M p c ] E ( z ) - / D A z = 0, HSE correctedAGN feedbackArnaud et. al. 2010 (REXCESS)Marriage et al. 2011 (ACT)Andersson et el. 2010 (SPT)Planck Collaboration 2011 Figure 12.
The Y - M scaling relation for the AGN feedback simulationscompared to recent X-ray results from Arnaud et al. (2010) and SZ resultsfrom ACT (Marriage et al. 2010), SPT (Andersson et al. 2010), and Planck (Planck Collaboration et al. 2011a). We have applied the 15% correction tothe X-ray M HSE from Kravtsov et al. (2006). rewritten as, Y = σ T m e c Z R d V P e = ( γ - σ T m e c x e X H µ E gas , (11)where x e is the electron fraction defined as the ratio of electronand hydrogen number densities x e = n e / n H = ( X H + / (2 X H ) =1 . γ = 5 / µ = 4 / (3 X H + + X H x e ) =0 .
588 denotes the mean molecular weight for a fully ionizedmedium of primordial abundance, and we assume equilibriumbetween the electron and ion temperatures. Next, we definethe characteristic temperature of the halo (Komatsu & Seljak2002) as kT = GM µ m p R = µ m p G H M E ( z )] / , (12)so we can write the total thermal energy of the halo withEq. (12) as E gas = 32 N gas kT = (1 - f ∗ ) f b f c GM R = (1 - f ∗ ) f b f c G ρ cr ( z )] / M / . (13)Here f ∗ . M ∗ / M b is the stellar mass fraction within the haloand f c is the correction factor for the fraction of missingbaryons at a given overdensity. Then we insert Eq. (13) intoEq. (11) to get the integrated Compton- y parameter within R , Y = ( γ - σ T m e c x e X H µ (1 - f ∗ ) f b f c G h π ρ cr ( z ) i / M / = 97 . h - kpc E ( z ) / (cid:18) M h - M ⊙ (cid:19) / Ω b .
043 0 . Ω m (14)For Eq. (14), we set f ∗ = 0, f c = 0 .
93 (as calculated from ourshock heating simulations at R ) and adopted the cosmolog-ical parameters of our simulation. This simple analytical ex-pression for the Y - M scaling relation allows one to explore theassumptions underlying its derivation. More specifically, wetest the assumptions of spherical gravitational potential, zero non-thermal pressure support, and constant f b (and for simu-lation with star formation, constant f ∗ ) at R ∆ , independent ofcluster mass. Comparison to data
In Figure 12, we compare Y sph for our simulated clusters tothe X-ray results from Arnaud et al. (2010), and the SZ re-sults from ACT (Marriage et al. 2010), SPT (Andersson et al.2010), and Planck (Planck Collaboration et al. 2011a). Weadopt the 15% correction to the X-ray M HSE estimates fromKravtsov et al. (2006) which is valid for the respective obser-vational sample selection criterion. Our Y sph -M relation withAGN feedback is consistent with the current data from X-rayand SZ observations. However, at group scales, our simulationsslightly overpredict the SZ flux due to the too high gas frac-tions, f gas , in our simulations compared to X-ray observations(cf. BBPS4). Potentially our simulations are missing some ofthe relevant physics that governs f gas (see, e.g., Pfrommer et al.2011) or underestimate the action of AGN feedback on thesemass scales.The Y sph reported by SZ surveys for known clusters use an X-ray-derived estimate of the aperture size. This is useful becausethe cluster radii are typically poorly measured in SZ, and so theX-ray aperture fixes the SZ measurement along the otherwisedegenerate aperture flux/aperture radius relation. However, thisprior introduces correlations between the X-ray and SZ obser-vations, which makes comparisons between these observationsdifficult to interpret. Physics dependence of the Y -M relation
In Figure 13 we show the dependence of the Y - M relation onour three simulated physics models, i.e., shock heating, radia-tive cooling and star formation, and AGN feedback. The starkdifferences between the shock heating simulation and the tworadiative simulation models arise from the loss of baryons inthe ICM to star formation. The radiative cooling simulationsshow a constant normalization offset of ∼ f ∗ values for these simulations. In Table 2 we showthat the self-similar expectation of Eq. (14) almost completelycaptures the cluster thermodynamics in our simulations whenintegrated over cluster-sized apertures. Including more physi-cally motivated sub-grid models in the simulations, we find thatboth, the shock heating and radiative cooling slopes are consis-tent with this self-similar derivation for the Y - M relation, whilethe AGN feedback simulations have a steeper, mass-dependentslope. This break from self-similarity in the AGN simulationsarises from the suppression of star formation in the higher massclusters and a feedback-induced deficit of gas inside the lowermass clusters. Over the redshift ranges we explore ( z = 0 to z = 1 .
5) and for all simulated physics models, the Y - M scalingrelation normalization changes as predicted by self-similar evo-lution and the slopes remain essentially constant (cf. Fig. 14).So, the Y - M relations from AGN simulations are different at z = 0, but evolve as predicted by self-similar evolution. Thisresult is independent of the two aperture sizes chosen, whichcorrespond to over-densities of 200 and 500 times the criticaldensity (Fig. 14). As we have repeatedly found in previoussections, the clusters interior to the radii R and even R are relatively well behaved, with only modest impact of clusterellipticities and kinetic pressure.To quantify the scatter, we compute the relative devia-tion of each cluster from the mean relation, δ Y ∆ / Y = ( Y ∆ - Y ∆ , fit ) / Y ∆ , fit , and then fit this distribution with a Gaussian prob-4 BATTAGLIA, BOND, PFROMMER, SIEVERS Table 2 Y - M scaling relation fits for different simulated physics, sub-sampling in kinetic-to-thermal energy and ellipticity (of the density and pressure distribution), andalong different projected axes yielding Y cyl . ∆ = 200 z = 0 z = 0 . z = 1 B A σ Y B A σ Y B A σ Y Simulated physicsTheory, Eq. (14) -4.88 1.67 - -4.81 1.67 - -4.74 1.67 -Shock heating -4.87 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± K / U a Lower 3 rd -4.88 ± ± ± ± ± ± ± ± ± rd -4.92 ± ± ± ± ± ± ± ± ± rd -4.94 ± ± ± ± ± ± ± ± ± c / a (gas) a Lower 3 rd -4.94 ± ± ± ± ± ± ± ± ± rd -4.92 ± ± ± ± ± ± ± ± ± rd -4.90 ± ± ± ± ± ± ± ± ± c / a (pressure) a Lower 3 rd -4.94 ± ± ± ± ± ± ± ± ± rd -4.92 ± ± ± ± ± ± ± ± ± rd -4.90 ± ± ± ± ± ± ± ± ± b / a (gas) a Lower 3 rd -4.93 ± ± ± ± ± ± ± ± ± rd -4.92 ± ± ± ± ± ± ± ± ± rd -4.91 ± ± ± ± ± ± ± ± ± Y cyl rotated a Minor axis -4.87 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Y cyl random a axis 1 -4.86 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± a For fits to all sub-samples/projections, we use our AGN feedback model. Fit parameters are defined in Eq. (10). b For the b / a sub-sampling of Y cyl , we chose random axis 1. M [ M O • ]10 -6 -5 Y , s ph / ( M / M O • ) / [ M p c ] z = 0AGN feedbackRadiative coolingShock heating -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) z = 0 AGN feedbackRadiative coolingShock heatingGaussian fit Figure 13.
The normalization, slope and scatter of the Y - M scaling relations all depend on the simulated physics. Left: The Y - M scaling relations at z = 0 for allsimulated physics models: shock heating (green), radiative cooling (blue), and AGN feedback (red). The y -axis has been scaled by M / to highlight the deviationsfrom self-similarity. Right: The probability distributions for the relative deviation, δ Y ∆ / Y , with respect to the best fits for all three physics models. We also showGaussian fits (dotted lines) and include Poisson deviations for the AGN feedback simulations (grey band). We find that the AGN feedback simulations have thelargest scatter and a steeper slope compared to the other simulations. N THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 15 z -5.0-4.9-4.8-4.7-4.6-4.5-4.4 B Y Y AGN feedbackRadiative coolingShock heatingSelf-similar (Eq. 14) z A Y Y AGN feedbackRadiative coolingShock heatingSelf-similar (Eq. 14)
Figure 14.
All simulated Y - M scaling relations evolve self-similarly with redshift according to Eq. (14). We show the Y - M scaling relation fits for the normalization, B , (left panel) and slope, A , (right panel) as a function of redshift and for two different cluster masses M and M , and compare those to the self similar predictionfor M (dotted black). The Y - M relation of AGN feedback simulations has a different slope, but shows no anomalous redshift evolution relative to self-similarevolution. M [ M O • ]10 -6 -5 Y [ M p c ] / ( M / M O • ) / AGN feedback, z Y Y major axis < 3 R Y middle axis < 3 R Y minor axis < 3 R -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) AGN feedback, z = 0 Y Y major axis < 3 R Y middle axis < 3 R Y minor axis < 3 R Gaussian fit
Figure 15.
Rotating the clusters into their major, middle and minor axes and calculating projected (cylindrical) Y - M relations shows the effect of infalling substruc-ture. Left: The cylindrical Y - M scaling relations from the AGN simulations for clusters that have been rotated into their major, middle, and minor axes defined bycomputing the (3D) moment-of-inertia tensor within R . Right: The probability distributions for the scatter, δ Y ∆ , relative to the best fits for all three distributions,each representing a distinctive rotation as well as the spherical distribution (black). We include the Gaussian fits (dotted lines) and the Poisson deviations for themajor axis rotation (grey band). Rotating the clusters such that integration happens along the major axis increases the total Y values, while further distorting andincreasing the scatter (due to the large cluster-to-cluster variance in the infall regions). The Y cyl values are integrated along the given axis from - R to 3 R ;hence for any given cluster Y cyl ≥ Y sph . M [ M O • ]10 -6 -5 Y , s ph [ M p c ] / ( M / M O • ) / AGN feedback, z = 0 K / U upper 3 rd K / U middle 3 rd K / U lower 3 rd -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) AGN feedback, z = 0 K / U upper 3 rd K / U middle 3 rd K / U lower 3 rd Gaussian fit
Figure 16.
Sub-sampling the Y - M relations by the kinetic-to-thermal energy ratio ( K / U ) for the AGN simulations. Left: The Y - M scaling relation for the three K / U sub-samples, upper 3 rd (red), middle 3 rd (green) and lower 3 rd (blue), with the corresponding slope fitted to those points. The y -axis has been scaled by M / to highlight the deviations from self-similarity. Right: The probability distributions for the relative deviation, δ Y ∆ / Y , with respect to the best fits for the threesub-samples and the total distribution (black), including the Gaussian fits (dotted lines) and the Poisson deviations for the upper 3 rd sub-sample (grey band). Thesub-sample of K / U with the largest kinetic pressure support (upper 3 rd ) shows systematically lower total Y values for a given mass as well as larger scatter, while thelower K / U sub-sample has the lowest scatter of ∼ Y . ability distribution function (PDF), G ( δ Y ∆ / Y ) = A exp (cid:20) - ( δ Y ∆ / Y ) σ Y (cid:21) . (15)Here the parameter A is the normalization and σ Y is the vari-ance, which we will refer to as the scatter. Here we have chosento model the variation about the mean as a Gaussian, while pre-vious work by Stanek et al. (2010) showed that a log-normaldistribution is also a reasonable description of the scatter. InAppendix D, we show that within the (Poisson) uncertainties,the scatter is clearly Gaussian distributed and only approxi-mately log-normal. Forcing a log-normal distribution intro-duces higher-order moments such as skewness and kurtosis ascan be seen by the tails in the distributions and their asymmet-ric shapes.We find that the scatter, σ Y , for the entire sample of clustersis between 11 % and 13 % (cf. Fig. 13 and Table 2), which isconsistent with previous work (Nagai 2006; Shaw et al. 2008;Stanek et al. 2010; Yang et al. 2010). In the simplest simula-tions with only shock heating the source for this scatter in the Y - M relation has been proposed to arise from the formationtime, the concentration, and the dynamical state of the cluster(Yang et al. 2010). As our simulations include more sub-gridphysics models the scatter increase from ∼
11% to ∼
13% at z = 0 and changes further from ∼
11% to ∼
15% at z = 1. Of thethree different physics models, the simulations with AGN feed-back gives the largest scatter, which is consistent with semi-analytic results (Shaw et al. 2008). This model for AGN feed-back is self-regulated (Battaglia et al. 2010) and injects ∼ / z = 1 when the average cluster mass is sig-nificantly smaller and the associated potentials are shallower sothat a fixed energy injection by AGNs may in principle have astronger impact . Thus, the increased scatter in the Y - M rela-tion from the AGN feedback simulations compared to the sim- Similar results were found by McCarthy et al. (2011) in simulations witha more detailed feedback prescription. ulations without feedback is a result of the energy injection,which heats and disturbs the ICM. This statement is in accor-dance with previous results from Battaglia et al. (2010), wherethey showed the impact of AGN feedback on the pressure pro-files of clusters and found that simulations with feedback had ashallower asymptotic pressure profile slopes than those withoutfeedback. Thus, the intermittent nature of energy injection intothe group system early-on results in a larger scatter in the Y - M relation compared to simulations without energetic feedback. Cylindrical Apertures
For pointed SZ observations of clusters and SZ surveys, anatural, model-independent observable is the projected flux, Y cyl (Mroczkowski et al. 2009; Sayers et al. 2011). We find Y cyl > Y sph in all cases, whether we chose the projection alonga principal or a random axis. This is due to the assumed ex-tension along the line-of-sight integration which we choose tobe three times the aperture radius; in observations, structurebeyond this scale may additionally contribute in some cases.In fact, a projection integral out to 3 R decreases the Y - M slope for the AGN feedback simulations such that it becomesconsistent with the self-similar slope (cf. Table 2). We find nodifference between the random 2D projections and the integra-tion along the middle or minor axes with respect to the nor-malization and slope (cf. Fig. 15 and Table 2). The scatter forthe random 2D projections is marginally larger than the projec-tions along middle and minor axes. Our results show that theintegration along the major axis yields dramatically differentresults, both, for the normalization and scatter in comparisonto projections along the other axes. This has its origin in themore extended tails of the PDF (cf. Fig 15). The normaliza-tion and scatter between the major axis and the other axes in-crease by ∼
7% and ∼ ∼
12% increasein the normalization and an increase in scatter by ∼ M [ M O • ]10 -6 -5 Y , s ph [ M p c ] / ( M / M O • ) / AGN feedback, z = 0 c / a upper 3 rd c / a middle 3 rd c / a lower 3 rd -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) AGN feedback, z = 0 c / a upper 3 rd c / a middle 3 rd c / a lower 3 rd Gaussian fit M [ M O • ]10 -6 -5 Y , s ph [ M p c ] / ( M / M O • ) / AGN feedback, z = 0 b / a upper 3 rd b / a middle 3 rd b / a lower 3 rd -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) AGN feedback, z = 0 b / a upper 3 rd b / a middle 3 rd b / a lower 3 rd Gaussian fit
Figure 17.
Sub-sampling the Y - M relations by the gas c / a axis ratio (upper panels) gas b / a axis ratio (lower panels) for the AGN simulations. Left: The Y - M scaling relation for the three c / a sub-samples, upper 3 rd (red), middle 3 rd (green) and lower 3 rd (blue), with the corresponding slope fitted to those points. The y-axishas been scaled by M / to highlight the deviations from self-similarity. Right: The probability distributions for the relative deviation, δ Y ∆ / Y , with respect to thebest fits for the three sub-samples and the total distribution (black), including the Gaussian fits (dotted lines) and the Poisson deviations for the upper 3 rd sub-sample(grey band). The sub-sample of c / a containing the lowest values (largest ellipticities) shows systematically lower total Y values for a given mass and larger scatter,while the more spherical high c / a sub-sample shows a lower scatter of ∼ c / a axis ratio sub-sample has similar results(cf.Table 2). The sub-sample of b / a containing the lowest values (largest projected ellipticities) shows larger scatter than the sub-sample with the highest values by ∼ Y - M relation at z = 0, however, smaller in comparison to the c / a sub-sampling. moment-of-inertia tensor beyond R . Toward a fundamental plane of Y -M
After quantifying the scatter of the entire sample, we aim atunderstanding its origin. This may enable us to either constructa linear combination of physically motivated observables thatminimizes the scatter or to employ sub-sampling of the full dis-tribution according to some parameter so that the resulting dis-tribution exhibits a smaller intrinsic scatter and potentially al-lows for tighter cosmological constraints (e.g., Afshordi 2008).In the previous sections we explored the average radial trendsfor kinetic pressure support from bulk motions and gas den-sity/pressure shapes of the ICM. Utilizing this information, werank order clusters according to their kinetic pressure supportand intrinsic shape information. We follow the same fittingprocedure as above for subsets of the lower 3 rd , middle 3 rd , andupper 3 rd of the correspondingly sorted distributions in order to demonstrate the impact of kinetic pressure support and as-phericity on the Y - M relation fits and scatter. For the rest of thissection we concentrate our analysis on the Y - M relations of theAGN feedback simulations, since they show the largest scatter(this will provide an upper limit on the scatter) and are mostlikely our best representation of “real” clusters in comparisonto the other simulated physics models. We compute the ratio ofkinetic-to-thermal energy, K / U , within radial bins and use thisratio as a measure of dynamical state for the galaxy clusters.We define the internal kinetic energy, K , and thermal energy, U , of a cluster as K ( < r ) ≡ X i m gas , i P kin , i ρ i , (16) U ( < r ) ≡ X i m gas , i P th , i ρ i , (17)8 BATTAGLIA, BOND, PFROMMER, SIEVERS c / a K / U Figure 18.
The correlations between the K / U ratio and 1 - c / a . Here the redpoints represent each cluster in the simulations and the blue crosses are averagequantities. The linear correlation coefficient is 0.575 where m and ρ are the gas mass and the SPH density, respec-tively for all particles i less than radius r . The ratio K / U is thevolume integrated analog of the ratio P kin / P th shown in § 3 andhence is also an indicator of formation history and substruc-ture. For the sub-sample with the highest ratio of K / U , we finda smaller normalization (cf. Fig. 16 and Table 2) where the dif-ference between this upper 3 rd of the distribution and lower 3 rd is ∼ Y -values.More massive clusters are typically in the high K / U samplerather then the other two samples. We find that the sub-samplewith the smallest K / U values shows the lowest scatter, ∼ K / U values (e.g., the lowest 6 th ) does not decrease thescatter, which is limited to ∼ c / a and b / a respec-tively, as defined in Sec. 4. Following the same procedure asfor the K / U sub-sample and restricting ourselves to the AGNfeedback simulations, we find that splitting the clusters up byellipticity, c / a , gives similar results in comparison to K / U -splitting. The galaxy clusters with smaller ellipticities havelarger total Y values and less scatter, while the more triaxialclusters have lower total Y and large scatter (cf. Fig. 17). Thesetrends are reflected in the fit parameters of the sub-sample Y - M relation shown in Table 2, where the differences betweenthe upper 3 rd and lower 3 rd sub-samples normalization param-eters is ∼ b / a statistics has the greatest impact on the scatter. The sub-sample of clusters that appear to be elongated in the plane ofthe sky have larger scatter than the more spherical clusters (cf.Fig. 17 and Table 2). Also, the b / a sub-sampling causes asimilar bias in the Y - M relation in comparison to the c / a sub-sampling, but not as significant.The results from sub-sampling clusters according K / U and c / a indicate that there are correlations between these physical We show one projection for the b / a sub-sampling. The other projectionsyield similar results. properties and the scatter in the Y - M relation. In Figure 18we show that larger K / U ratios correlate with larger 1 - c / a ,i.e. larger triaxiality, with a linear coefficient value, r s = 0 . Y - M relation. Simi-lar results were found by Rasia et al. (2011) and Krause et al.(2011) Previous work by Yang et al. (2010) found mass trendsin the measured scatter, which is consistent with our findingsafter extrapolating their lower mass range to our larger masses.However, their conclusion is different from ours, since theyclaim that the scatter is most sensitive to the DM concentration;a finding that may partially be due to the insufficient resolutionin their simulations. DISCUSSION AND CONCLUSIONS
In this paper we demonstrate that the spatial distribution ofthe ICM, kinetic pressure support from bulk motions, and self-regulated thermal energy feedback in clusters cores (that werefer to as AGN feedback) all play very important roles forthe thermal properties of galaxy clusters. In particular, the ob-servables for large SZ cluster surveys, such as ACT, SPT and
Planck , will be modified by these processes. Below we high-light and expand on our main results.
Non-thermal pressure support and cluster shapes:
The con-tribution to the overall pressure support in clusters from bulkmotions, P kin , increases substantially for larger radii and is astrong function of both, cluster mass and redshift. Includ-ing AGN feedback marginally decreases P kin / P th in compari-son to the other (more simplified) simulation models, namelyour shock heating-only model and that which additionally in-cludes radiative cooling, star formation, supernova feedback,and CRs. However, the difference is not substantial enough tobe statistically inconsistent with the variance around the me-dian of P kin / P th . The mass dependence and redshift evolutionof P kin / P th is governed by P kin and a direct result of the hi-erarchical growth of structure. Semi-analytic approaches arejust beginning to model P kin . The full dependence on radius,mass and redshift of this component is, by definition, self-consistently included in hydrodynamic simulations.We find that the distribution of gas density and pressure areweak functions of the simulated physics models within R (excluding the cluster core) and that AGN feedback mildlymodifies the average gas shapes. The cluster mass dependenceof the ellipticity is more moderate in comparison to P kin / P th .The ellipticity is small within R with little redshift evolution.In combination with the comparably small non-thermal pres-sure support at these scales (which rises dramatically beyondthis characteristic radius), the small clumping factor measuredin our simulations (cf. BBPS4), and the small modificationof our simulated cluster physics at these radii (in particular ofour implementation of AGN feedback), this result is reassur-ing for X-ray observations of clusters which use R to char-acterize clusters with high-quality Chandra and
XMM Newton observations. Hence, our analysis theoretically supports thischoice of radius (which was initially motivated by the simu-lations in Evrard et al. (1996)) and justifies some of the mainassumptions such as spherical symmetry and an almost radius-independent hydrostatic mass bias of ∼ -
25% when using afair sample of clusters without morphological selection whichN THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 19may be applicable for the future eROSITA sample.We find substantial redshift evolution in different dynamicalquantities, e.g., P kin / P th , the velocity anisotropy, and anisotropyparameters such as ellipticities. This is in particular the case forthe changes in the power-law behaviors of the radial profile ofthese quantities such as the sudden break in ellipticities whichmoves to smaller radii as the redshift increases (when scaledto R ). The break and the more pronounced ellipticities and P kin / P th outside a characteristic radius are a direct result of in-creased level of substructure predicted by hierarchical struc-ture formation and the associated higher mass accretion rate athigher redshift. We explicitly show (in the Appendix C) thatmost of this redshift evolution is somewhat artificial and canbe absorbed in a re-definition of the virial radius: scaling withthe radius that contains a mean density of 200 times the aver-age mass density rather than the critical density of the universeconsiderably weakens the observed trends with redshift. Thisalso suggests a physical definition of the virial radius in termsof dynamical quantities (that, however, remain poorly definedobservationally), e.g., the equipartition radius of thermal andkinetic pressure, the region where the velocity anisotropy be-comes strongly radial, or the radius at which the ellipticity orsubstructure level increases dramatically. These seemingly dif-ferent criteria all select a rather similar radius around R , m ;almost independent of redshift.On scales > R , stacking analyses of projected SZ clusterimages can be done with data from SZ experiments such asACT and SPT, and, provided there is a suitable sample size,one may be able to detect projected gas pressure shapes, po-tentially even in bins of redshift. The results on the randomlyprojected 2D axis ratios represent the theoretical expectations.Any statistics from the intrinsic 3D distribution is highly corre-lated with the projected 2D distribution; we find that the (moreelliptical) intrinsic cluster shapes can on average be inferredfrom their projected analogues by applying a ∼ -
10% cor-rection on the ellipticity. Another interesting outcome fromour shape analysis is that there is no direct and simple map-ping of shapes and alignments for DM spatial distribution tothe gas and pressure distributions possible mostly due to thedifference in substructure distribution and dissipational natureof the gas. This result is troublesome for semi-analytic mod-els which use dissipationless simulations as a template to solvefor the gas distributions and pressure shapes. Such a methodwill produce additional triaxiality and misalignment for sucha semi-analytical model of the ICM. The overall magnitude ofthe shape is reconciled by using the gravitational potential (e.g.,Ostriker et al. 2005; Bode et al. 2009; Trac et al. 2011) whichhas been shown to be less triaxial (Lau et al. 2011) than theDM. However, providing an algorithm to re-alignment thesepseudo gas distributions is a non-trivial task.
Y -M scaling relations:
Our simulations are in good agree-ment with the current Y - M scaling relations from both X-rayobservations and SZ surveys. However, to properly predict the Y - M scaling relations for an SZ experiment such as ACT, SPTor Planck without any prior knowledge of cluster masses, care-ful mock observations are needed. Those would have to in-clude a simulation of the CMB sky with associated experimentnoise and adopt the relevant cluster selection pipelines for thegiven experiment that employs the same cluster profile used formatched filtering in order to include all the systematics and po-tential biases that are intrinsic to the data analysis, e.g., X-raypriors on the aperture size.We find that the inclusion of AGN feedback causes a de-viation from the predictions of self-similar evolution for both the normalization and slope of the Y - M relation (as measuredwithin R ). However, we recover the self-similar slope againin our projected Y - M scaling relations (where we integratealong a cylinder of half-height 3 R ), suggesting that AGNfeedback pushes a fraction of its gas beyond the virial radiusand a larger aperture/projection radius is able to recover thethermal energy from this larger reservoir of gas.Including AGN feedback also increases scatter in the Y - M relation compared to simulations that include shock heatingalone, from ∼
11 % to ∼
13 %. Interestingly, sorting the clus-ters into sub-samples of K / U and c / a will reduce this scat-ter; e.g., K / U sub-sampling reduces the scatter from ∼ ∼ Y - M relation. The scatter ultimately orig-inates from the merging history with its redshift and mass de-pendent accretion rates; those determine the non-thermal pres-sure support, the level of substructure, and the ellipticity. Whilesub-sampling on one of these secondary tracers may decreasethe scatter, it is unlikely to decrease much more if more tracersare used (as they probe the same underlying process, albeit witha different weighting). Conversely, our band-splitting analysison the Y - M relations suggests that large outliers from the meanrelations would be interesting candidates for follow up withhigh resolution SZ observations, since they are more likely tohave larger kinetic pressure support and ellipticities.A fundamental point to take away is that all results at largerradii ( > R ) for the kinetic pressure support and ICM shapesare dominated by substructure. We also see the impact of sub-structure on the cylindrical Y - M scaling relation when integrat-ing along the major axis with which substructure is preferen-tially aligned. Quantifying substructure statistically is difficultbecause of the problem of double-counting: the large volumecontained within the radius that contains 95% of the total SZflux, 4 R , necessarily leads to overlapping volumes of neigh-boring clusters, especially at high-redshift. Thus, this propertyremains challenging to model phenomenologically or analyti-cally.As discussed previously in the literature (e.g. Battaglia et al.2010; Sun et al. 2011) SZ galaxy cluster may provide furtherinsight into the interesting astrophysics associated with theICM of clusters. This however may significantly complicatecosmological analyses in producing competitive constraints.However, these are exciting prospects for studies of feedbackand other energy injection processes within clusters especiallyat higher redshift since the selection function of SZ cluster sur-veys probes clusters which populate the massive and high red-shift end of the distribution.We thank Mike Nolta, Norm Murray, Hy Trac, Gus Evrard,Alexey Vikhlinin, Andrey Kravtsov, Laurie Shaw, Doug Rudd,and Diasuke Nagai for useful discussions. Research in Canadais supported by NSERC and CIFAR. Simulations were run onSCINET and CITA’s Sunnyvale high-performance computingclusters. SCINET is funded and supported by CFI, NSERC,Ontario, ORF-RE and UofT deans. C.P. gratefully acknowl-edges financial support of the Klaus Tschira Foundation. Wealso thank KITP for their hospitality during the 2011 galaxycluster workshop. KITP is supported by National Science0 BATTAGLIA, BOND, PFROMMER, SIEVERSFoundation under Grant No. NSF PHY05-51164. REFERENCESAfshordi, N. 2008, ApJ, 686, 201Agertz, O. et al. 2007, MNRAS, 380, 963Allgood, B., Flores, R. A., Primack, J. R., Kravtsov, A. V., Wechsler, R. H.,Faltenbacher, A., & Bullock, J. S. 2006, MNRAS, 367, 1781Andersson, K. et al. 2010, arXiv:1006.3068Arnaud, M., Pratt, G. W., Piffaretti, R., Böhringer, H., Croston, J. H., &Pointecouteau, E. 2010, A&A, 517, A92Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, ApJ, 304, 15Battaglia, N., Bond, J. R., Pfrommer, C., & Sievers, J. L. 2011a, in prep.—. 2011b, in prep.—. 2011c, in prep.Battaglia, N., Bond, J. R., Pfrommer, C., Sievers, J. L., & Sijacki, D. 2010,ApJ, 725, 91Benson, B. A., Church, S. E., Ade, P. A. R., Bock, J. J., Ganga, K. M., Henson,C. N., & Thompson, K. L. 2004, ApJ, 617, 829Birkinshaw, M. 1999, Phys. Rep., 310, 97Bode, P., Ostriker, J. P., & Vikhlinin, A. 2009, ApJ, 700, 989Bode, P., Ostriker, J. P., Weller, J., & Shaw, L. 2007, ApJ, 663, 139Bonaldi, A., Tormen, G., Dolag, K., & Moscardini, L. 2007, MNRAS, 378,1248Bonamente, M., Joy, M., LaRoque, S. J., Carlstrom, J. E., Nagai, D., &Marrone, D. P. 2008, ApJ, 675, 106Bond, J. R., & Myers, S. T. 1996, ApJS, 103, 1Bryan, G. L., & Norman, M. L. 1998, ApJ, 495, 80Burns, J. O., Skillman, S. W., & O’Shea, B. W. 2010, ApJ, 721, 1105Carlstrom, J. E., Holder, G. P., & Reese, E. D. 2002, ARA&A, 40, 643Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2009, ApJS, 182, 12Chaudhuri, A., & Majumdar, S. 2011, ApJ, 728, L41Churazov, E., Brüggen, M., Kaiser, C. R., Böhringer, H., & Forman, W. 2001,ApJ, 554, 261da Silva, A. C., Kay, S. T., Liddle, A. R., & Thomas, P. A. 2004, MNRAS, 348,1401Dubinski, J., & Carlberg, R. G. 1991, ApJ, 378, 496Dunkley, J. et al. 2010, arXiv:1009.0866Enßlin, T. A., Pfrommer, C., Springel, V., & Jubelgas, M. 2007, A&A, 473, 41Ensslin, T. A., Simon, P., Biermann, P. L., Klein, U., Kohle, S., Kronberg, P. P.,& Mack, K.-H. 2001, ApJ, 549, L39Evrard, A. E. 1990, ApJ, 363, 349Evrard, A. E., Metzler, C. A., & Navarro, J. F. 1996, ApJ, 469, 494Fabian, A. C. 1994, ARA&A, 32, 277Fabian, A. C., Sanders, J. S., Allen, S. W., Crawford, C. S., Iwasawa, K.,Johnstone, R. M., Schmidt, R. W., & Taylor, G. B. 2003, MNRAS, 344,L43Fowler, J. W. et al. 2010, arXiv:1001.2934Frenk, C. S. et al. 1999, ApJ, 525, 554Gottlöber, S., Klypin, A., & Kravtsov, A. V. 2001, ApJ, 546, 223Gottlöber, S., & Yepes, G. 2007, ApJ, 664, 117Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423Iapichino, L., & Niemeyer, J. C. 2008, MNRAS, 388, 1089Jing, Y. P., & Suto, Y. 2002, ApJ, 574, 538Jubelgas, M., Springel, V., Enßlin, T., & Pfrommer, C. 2008, A&A, 481, 33Kaiser, N. 1986, MNRAS, 222, 323Kasun, S. F., & Evrard, A. E. 2005, ApJ, 629, 781Keisler, R. et al. 2011, arXiv:1105.3182Komatsu, E., & Seljak, U. 2002, MNRAS, 336, 1256Krause, E., Pierpaoli, E., Dolag, K., & Borgani, S. 2011, arXiv:1107.5740Kravtsov, A. V., Vikhlinin, A., & Nagai, D. 2006, ApJ, 650, 128Lau, E. T., Kravtsov, A. V., & Nagai, D. 2009, ApJ, 705, 1129Lau, E. T., Nagai, D., Kravtsov, A. V., & Zentner, A. R. 2011, ApJ, 734, 93Lewis, G. F., Babul, A., Katz, N., Quinn, T., Hernquist, L., & Weinberg, D. H.2000, ApJ, 536, 623Lima, M., & Hu, W. 2004, Phys. Rev. D, 70, 043504Lueker, M. et al. 2010, ApJ, 719, 1045Macciò, A. V., Dutton, A. A., & van den Bosch, F. C. 2008, MNRAS, 391,1940Majumdar, S., & Mohr, J. J. 2003, ApJ, 585, 603—. 2004, ApJ, 613, 41Marriage, T. A. et al. 2010, aXiv:1010.1065Marrone, D. P. et al. 2011, arXiv:1107.5115—. 2009, ApJ, 701, L114McCarthy, I. G., Schaye, J., Bower, R. G., Ponman, T. J., Booth, C. M.,Vecchia, C. D., & Springel, V. 2011, MNRAS, 412, 1965McNamara, B. R., Nulsen, P. E. J., Wise, M. W., Rafferty, D. A., Carilli, C.,Sarazin, C. L., & Blanton, E. L. 2005, Nature, 433, 45 Miniati, F., Ryu, D., Kang, H., Jones, T. W., Cen, R., & Ostriker, J. P. 2000,ApJ, 542, 608Mitchell, N. L., McCarthy, I. G., Bower, R. G., Theuns, T., & Crain, R. A.2009, MNRAS, 395, 180Motl, P. M., Hallman, E. J., Burns, J. O., & Norman, M. L. 2005, ApJ, 623,L63Mroczkowski, T. 2011, ApJ, 728, L35Mroczkowski, T. et al. 2009, ApJ, 694, 1034Nagai, D. 2006, ApJ, 650, 538Nagai, D., & Lau, E. T. 2011, ApJ, 731, L10Nath, B. B., & Majumdar, S. 2011, arXiv:1105.2826Ostriker, J. P., Bode, P., & Babul, A. 2005, ApJ, 634, 964Parrish, I. J., McCourt, M., Quataert, E., & Sharma, P. 2011, arXiv:1109.1285Pearce, F. R., Thomas, P. A., Couchman, H. M. P., & Edge, A. C. 2000,MNRAS, 317, 1029Pfrommer, C., Chang, P., & Broderick, A. E. 2011, arXiv:1106.5505Pfrommer, C., Enßlin, T. A., Springel, V., Jubelgas, M., & Dolag, K. 2007,MNRAS, 378, 385Pfrommer, C., & Jones, T. W. 2011, ApJ, 730, 22Pfrommer, C., Springel, V., Enßlin, T. A., & Jubelgas, M. 2006a, MNRAS,367, 113—. 2006b, MNRAS, 367, 113Pinzke, A., Pfrommer, C., & Bergstrom, L. 2011, arXiv:1105.3240Planck Collaboration et al. 2011a, arXiv:1101.2026—. 2011b, arXiv:1101.2024—. 2011c, arXiv:1101.2043Rasia, E., Mazzotta, P., Evrard, A., Markevitch, M., Dolag, K., & Meneghetti,M. 2011, ApJ, 729, 45Rasia, E., Tormen, G., & Moscardini, L. 2004, MNRAS, 351, 237Rudd, D. H., Zentner, A. R., & Kravtsov, A. V. 2008, ApJ, 672, 19Ryu, D., Kang, H., Hallman, E., & Jones, T. W. 2003, ApJ, 593, 599Sayers, J., Golwala, S. R., Ameglio, S., & Pierpaoli, E. 2011, ApJ, 728, 39Schäfer, B. M., Pfrommer, C., Bartelmann, M., Springel, V., & Hernquist, L.2006a, MNRAS, 370, 1309Schäfer, B. M., Pfrommer, C., Hell, R. M., & Bartelmann, M. 2006b, MNRAS,370, 1713Sehgal, N. et al. 2011, ApJ, 732, 44Shaw, L. D., Holder, G. P., & Bode, P. 2008, ApJ, 686, 206Shaw, L. D., Nagai, D., Bhattacharya, S., & Lau, E. T. 2010, ApJ, 725, 1452Shirokoff, E. et al. 2010, arXiv:1012.4788Sijacki, D., Pfrommer, C., Springel, V., & Enßlin, T. A. 2008, MNRAS, 387,1403Sijacki, D., Springel, V., Di Matteo, T., & Hernquist, L. 2007, MNRAS, 380,877Simionescu, A. et al. 2011, Science, 331, 1576Skillman, S. W., O’Shea, B. W., Hallman, E. J., Burns, J. O., & Norman, M. L.2008, ApJ, 689, 1063Springel, V. 2005, MNRAS, 364, 1105—. 2010, MNRAS, 401, 791Springel, V., & Hernquist, L. 2003, MNRAS, 339, 289Springel, V. et al. 2008a, MNRAS, 391, 1685—. 2008b, Nature, 456, 73Stanek, R., Rasia, E., Evrard, A. E., Pearce, F., & Gazzola, L. 2010, ApJ, 715,1508Suginohara, T., & Ostriker, J. P. 1998, ApJ, 507, 16Sun, M., Sehgal, N., Voit, G. M., Donahue, M., Jones, C., Forman, W.,Vikhlinin, A., & Sarazin, C. 2011, ApJ, 727, L49Sunyaev, R. A., & Zeldovich, Y. B. 1970, Ap&SS, 7, 3Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ, 630, 167Trac, H., Bode, P., & Ostriker, J. P. 2011, ApJ, 727, 94Vanderlinde, K. et al. 2010, ApJ, 722, 1180Vazza, F., Brunetti, G., & Gheller, C. 2009, MNRAS, 395, 1333Vazza, F., Dolag, K., Ryu, D., Brunetti, G., Gheller, C., Kang, H., & Pfrommer,C. 2011, arXiv:1106.2159Vogelsberger, M., Sijacki, D., Keres, D., Springel, V., & Hernquist, L. 2011,arXiv:1109.1281Voit, G. M. 2005, Reviews of Modern Physics, 77, 207Wechsler, R. H., Bullock, J. S., Primack, J. R., Kravtsov, A. V., & Dekel, A.2002, ApJ, 568, 52White, M. 2002, ApJS, 143, 241Yang, H., Bhattacharya, S., & Ricker, P. M. 2010, ApJ, 725, 1124Zemp, M., Gnedin, O. Y., Gnedin, N. Y., & Kravtsov, A. V. 2011,arXiv:1108.5384Zhao, D. H., Jing, Y. P., Mo, H. J., & Börner, G. 2009, ApJ, 707, 354
N THE CLUSTER PHYSICS OF SUNYAEV ZEL’DOVICH SURVEYS I 21 r / R P k i n / P t o t ( M / x M O • ) / AGN feedback, z = 01.1 x 10 M O • < M < 1.7 x 10 M O • M O • < M < 2.7 x 10 M O • M O • < M < 4.2 x 10 M O • M O • < M < 6.5 x 10 M O • M O • < M < 1.01 x 10 M O • M O • < M < 1.57 x 10 M O • Shaw et al. 2010 R R vir Figure 19.
The kinetic pressure-to-total pressure is weakly mass-dependent, P kin / P tot ∝ M / as indicated by the scaling of the y -axis. Shown is the medianof P kin / P tot as a function of radius for the AGN feedback simulations for vari-ous mass bins with the 25 th and 75 th percentile values illustrated by the dottedlines for the lowest mass bin at z = 0. For comparison, we also show the modelfor P kin / P tot by Shaw et al. (2010), which has been fit to match AMR simula-tions (dash-dotted). We illustrate the 1 and 2 σ contributions to Y ∆ centeredon the median for the feedback simulation by horizontal purple and pink errorbars. Therefore, ignoring this mass dependence results in a 60 % difference inthis ratio for an order of magnitude change in the cluster mass. The median of P kin / P th scales as M / , which results in a larger difference. r / R -20-15-10-50510 D c / a [ % ] AGN feedback, z = 0 c / a | r ,gas c / a | p c / a | r ,DM R R vir Figure 20.
Shown is the relative difference between axis ratios with and with-out the r - weighting for the gas-density (red line), DM-density (blue line) andgas-pressure (green line) weightings. Additionally including the r - weightingin the definition of the moment-of-inertia tensor down-weights the contributionat larger radii by . A. FITTING FUNCTION FOR P KIN / P TOT
In Section 3 we show that the ratio P kin / P th is a functionof mass. However, the previous empirical fitting function for P kin / P tot (Shaw et al. 2010) does not include a mass depen-dence, P kin P tot ( r , z ) = α ( z ) (cid:18) rR (cid:19) n nt (cid:18) M × M ⊙ (cid:19) n M , (A1) where α ( z ) ≡ α (1 + z ) β for low redshifts ( z .
1) and the fitparameters are α = 0 . ± . β = 0 . n nt = 0 . ± .
25, andby construction, n M = 0. In Fig. 19 we compare the fittingfunction for Eq. (A1) and P kin / P tot , split by different mass binwhich have been scaled by M / , i.e. n M = 1 /
5, that minimizesour χ . We chose a normalization of 3 × M ⊙ to matchthe fitting function of Shaw et al. (2010). Thus, the mediandifference between P kin / P tot of a 10 M ⊙ and a 10 M ⊙ clusteris ∼ P kin / P th depends more sensitively on mass. We findthat the mass dependence for this ratio amounts to M / . B. DOWN-WEIGHTING THE SUBSTRUCTURE IN THEMOMENT-OF-INERTIA TENSOR
For both the gas density- and pressure-weighting of themoment-of-inertia tensor, the inclusion of an additional x - -weighting has a relatively minor influence on cluster shapes(cf. Fig. 20) and we do not see large differences in the axisratios at the larger radii. The x - -weighting does lessen the in-fluence of substructure which we have seen to be important atradii beyond R , but it does not remove it or isolate its signal.This would be a non-trivial task for any stacking analysis as itwas recently suggested by Zemp et al. (2011). C. CLUSTERS IN VELOCITY SPACE AND A DYNAMICALRADIUS DEFINITION
The radial trends over redshift seen in Figures 1 and 10 callfor re-examination of the choice for the working definition ofradius, which is directly related to the definition of the clus-ter mass (see White 2002, for a more thorough discussion ofcluster mass definitions in dissipationless simulations). It hasbeen the common choice by both observers and theorists todefine the mass within an radii where the average overden-sity is greater than a large multiple of a given backgrounddensity, such as ρ cr ( z ) and ¯ ρ m ( z ). For low redshift observa-tions, the more popular definition has been the ρ cr ( z ) as the iso-density surface, since no prior knowledge of Ω m is required.The question remains what definition is physically more intu-itive when comparing across various redshifts. At late times( z < R ∆ definition will result in the clus-ter radius shrinking as time approaches present day. Using the R ∆ , m scaling we find that the radial regions at which kineticpressure is in equipartition with thermal pressure and the sharpbreak found in the ICM ellipticity align at ∼ R ∆ , m (cf. Fig.21). In BBPS3 we show that the velocity anisotropy has thesame radial trends as in Figures 1 and 10 and that 200 R ∆ , m traces a distinct dynamical region of clusters, the splash-backradius, i.e., is caused by the turn-around of earlier collapsedshells which minimizes the radial velocity component such thatthe tangential components dominate the velocity. D. GAUSSIAN OR LOG-NORMAL SCATTER?
Previous approaches quantified the scatter around the best-fit Y - M scaling relation with a log-normal distribution, i.e. theycharacterized the distribution of δ log Y ∆ = log Y ∆ - log Y ∆ , fit with a Gaussian. Deviations from this log-normal distributionwere computed with the Edgeworth expansion, introducingsubstantial higher order moments, such as skewness and kur-2 BATTAGLIA, BOND, PFROMMER, SIEVERS r / R P k i n / P t h AGN feedback, 1.7 x 10 M O • < M < 2.7 x 10 M O • z = 0 z = 0.3 z = 0.5 z = 0.7 z = 1.0 z = 1.5Shaw et al. 2010, z = 0Shaw et al. 2010, z = 1 r / R e AGN feedback z = 0 z = 0.3 z = 0.5 z = 0.7 z = 1.0 z = 1.5 e r ,gas e p e r ,DM Figure 21.
The choice for our working definition of virial radius has an impact on the redshift evolution of both, the kinetic pressure support (left) and ellipticity(right) of clusters. The figures shown here are the same as Fig. 1 and Fig. 10 except that the dimensionless radius has been scaled by R , m instead of R . Withthis definition of virial radius, the redshift evolution of both, kinetic pressure support and ellipticity is weakened, especially in the outer regions. -0.4 -0.2 0.0 0.2 0.4 d Y D / Y P ( d Y D / Y ) AGN feedback, z = 0Gaussian fitLog-Normal fit Figure 22.
Comparison of Gaussian and log-normal scatter relative to the best-fit Y - M scaling relation at z = 0. We show the distribution of the relative linear deviation from the mean relation δ Y ∆ / Y (cf. Eq. 15) with the solid blue lineand compare it to a Gaussian fit (blue dotted line) and log-normal fit ( δ log Y ∆ ,red dashed line). The Poisson deviations are shown with the grey band. Herewe transformed the fit to the δ log Y ∆ distribution into δ Y ∆ / Y so they couldbe shown together. The δ Y ∆ / Y distribution is fit by a Gaussian better than the δ log Y ∆ distribution, with χ ( δ Y ∆ / Y ) ∼ χ ( δ log Y ∆ ) ∼
7. Forcing alog-normal distribution introduces higher-order moments such as skewness andkurtosis as can be seen by the asymmetric shapes of the tails in the log-normalfit. tosis (e.g., Yang et al. 2010). Using a non-linear least squaresapproach we fit a Gaussian to both the δ log Y ∆ and δ Y ∆ / Y dis-tributions. In Fig. 22 we show that δ Y ∆ / Y distribution is abetter fit by a Gaussian within the (Poisson) uncertainties thanthe δ log Y ∆ distribution, with χ ∼ χ ∼