Hydrodynamic Simulation of Non-thermal Pressure Profiles of Galaxy Clusters
AA CCEPTED TO A P J: J
ULY
14, 2014
Preprint typeset using L A TEX style emulateapj v. 5/2/11
HYDRODYNAMIC SIMULATION OF NON-THERMAL PRESSURE PROFILES OF GALAXY CLUSTERS K AYLEA N ELSON , E
RWIN
T. L AU , AND D AISUKE N AGAI
Department of Astronomy, Yale University, New Haven, CT 06520, U.S.A.; [email protected] Department of Physics, Yale University, New Haven, CT 06520, U.S.A. Yale Center for Astronomy & Astrophysics, Yale University, New Haven, CT 06520, U.S.A.A
CCEPTED TO A P J: July 14, 2014
ABSTRACTCosmological constraints from X-ray and microwave observations of galaxy clusters are subjected to sys-tematic uncertainties. Non-thermal pressure support due to internal gas motions in galaxy clusters is one ofthe major sources of astrophysical uncertainties. Using a mass-limited sample of galaxy clusters from a high-resolution hydrodynamical cosmological simulation, we characterize the non-thermal pressure fraction profileand study its dependence on redshift, mass, and mass accretion rate. We find that the non-thermal pressure frac-tion profile is universal across redshift when galaxy cluster radii are defined with respect to the mean matterdensity of the universe instead of the commonly used critical density. We also find that the non-thermal pres-sure is predominantly radial, and the gas velocity anisotropy profile exhibits strong universality when galaxycluster radii are defined with respect to the mean matter density of the universe. However, we find that the non-thermal pressure fraction is strongly dependent on the mass accretion rate of the galaxy cluster. We providefitting formulae for the universal non-thermal pressure fraction and velocity anisotropy profiles of gas in galaxyclusters, which should be useful in modeling astrophysical uncertainties pertinent to using galaxy clusters ascosmological probes.
Subject headings: cosmology: theory – galaxies: clusters: general – methods: numerical INTRODUCTION
Clusters of galaxies are the largest gravitationally boundobjects in the universe and therefore trace the growth of largescale structure. In recent years, X-ray and microwave obser-vations have enabled detailed studies of the structure and evo-lution of galaxy clusters and significantly improved the use ofthese systems as powerful cosmological probes (Allen et al.2011, for a review). However, current cluster-based cosmo-logical constraints are limited by systematic uncertainties as-sociated with cluster astrophysics. Controlling these astro-physical uncertainties is therefore critical for exploiting thefull statistical power of ongoing and upcoming cluster sur-veys, such as
Planck and eROSITA .One of the main challenges in using clusters as cosmolog-ical probes lies in the accurate determination of their masses.Cluster mass estimates from X-ray and Sunyaev-Zel’dovich(SZ) observations are based on the assumption that cluster gasis in hydrostatic equilibrium with their gravitational poten-tial, but there have been inconsistencies between the hydro-static mass and the mass estimated from gravitational lensing(e.g., Zhang et al. 2010; Mahdavi et al. 2013; von der Lin-den et al. 2014; Applegate et al. 2014). Hydrodynamical sim-ulations suggest that this hydrostatic mass bias arises fromnon-thermal pressure support in clusters that is not accountedfor in current X-ray and SZ cluster mass measurements (e.g.,Evrard et al. 1996; Rasia et al. 2006; Nagai et al. 2007b; Pif-faretti & Valdarnini 2008). Simulations also suggest that ac-counting for the non-thermal pressure support can recover thecluster mass to within a few percent (e.g., Rasia et al. 2004;Lau et al. 2009; Nelson et al. 2012, 2014). To date, it has beenwidely assumed that the bias in hydrostatic mass is constantwith redshift and mass, but it is unclear whether this assump- tion is valid. For upcoming cluster surveys, which will detectclusters out to z ≈ .
5, it is necessary to characterize the massand redshift dependence of the mass bias and its impact oncosmological inferences.Non-thermal pressure can also affects the interpretation ofthe angular power spectrum of the thermal SZ signal, origi-nated from the inverse Compton scattering of the CMB pho-tons off hot electrons in galaxy clusters. The amplitude ofthe angular power spectrum of the thermal SZ signal ( C (cid:96) ) isvery sensitive to the amplitude of matter density fluctuations( σ ) as C (cid:96) ∝ σ − (Komatsu & Seljak 2002). Non-thermalpressure is one of the main astrophysical uncertainties sincemost of the thermal SZ signal comes from integrated ther-mal pressure from the hot gas in the intracluster and intra-group medium at large radii, where the level of non-thermalpressure is comparable to that of thermal pressure, and wherethe energy injection from stars and active galactic nuclei areexpected to be subdominant. The inclusion of non-thermalpressure support can change the amplitude of the thermalSZ power spectrum by as much as 60% (Shaw et al. 2010;Battaglia et al. 2010; Trac et al. 2011), significantly affect-ing its constraint on σ . Since the thermal SZ angular powerspectrum gets contributions from galaxy groups and clustersin a wide range of redshifts and mass, a proper understandingof the mass and redshift dependence of the non-thermal pres-sure support is critical for using the SZ power spectrum andits high-order moment counterparts (Bhattacharya et al. 2012;Hill & Sherwin 2013) as cosmological probes.The upcoming ASTRO-H mission, equipped with high-resolution X-ray spectrometer, will measure internal gas mo-tions in galaxy clusters from doppler broadening of emissionlines (Takahashi et al. 2010). However, due to its limitedsensitivity, the
ASTRO-H measurements of the non-thermalpressure will be limited to only the inner regions of nearbymassive clusters, and it will be difficult to extend these mea- a r X i v : . [ a s t r o - ph . C O ] A ug Nelson, Lau, & Nagaisurements to the outer regions or high-redshift clusters wherethe effects of non-thermal pressure are expected to be moresignificant.In the absence of observational constraints, hydrodynam-ical cosmological simulations can serve as guides for char-acterizing the effects of non-thermal pressure, particularly atlarge cluster radii and at high redshifts. In this paper we buildupon previous works (Shaw et al. 2010; Battaglia et al. 2012)by examining the non-thermal pressure fraction for a mass-limited sample of highly resolved massive galaxy clusters ina wide range of mass, redshifts and dynamical states. Weshow that the mean non-thermal pressure fraction as well asthe gas velocity anisotropy profiles exhibit remarkable univer-sality with redshift and mass, when the cluster mass is definedwith respect to the mean mass density of the universe, insteadof the critical density. We also find that these profiles showcluster-to-cluster scatter which depends primarily on the massaccretion rate of the clusters, which only affects the normal-ization of the profiles. We present fitting formulae for theseuniversal profiles. These formulae should useful for charac-terizing the effects of non-thermal pressure on the hydrostaticmass bias, incorporating their effects in semi-analytic mod-els of thermal SZ power spectrum, and calibrating analyticalmodels of the non-thermal pressure profiles of clusters (e.g.,Shi & Komatsu 2014).The paper is organized as follows. In Section 2 we givean overview of the different mass definitions and describe ourdynamical state proxy; in Section 3 we describe our simula-tions of galaxy cluster formation; in Section 4 we present ourfindings; and finally we offer our conclusions and discussionsin Section 5. THEORETICAL OVERVIEW
Cluster Mass Definitions
Galaxy clusters form at the intersections of large-scale fil-amentary structures in the universe. As such, they have nowell-defined physical edge, and we must adopt some conven-tion for determining a boundary for the systems and their en-closed mass. The common approach is to define the boundaryof a cluster as a sphere enclosing an average matter densityequal to some chosen reference overdensity, ∆ ref , times somereference background density, ρ ref . The mass of the cluster isthen M ∆ ref ≡ π ∆ ref ρ ref ( z ) r ∆ ref (1)where r ∆ ref is the cluster radius. The two common choices ofthe background density ρ ref are the critical density, ρ c ( z ), andthe mean matter density, ρ m ( z ), ρ c ( z ) = 3 H π G (cid:0) Ω m (1 + z ) + Ω Λ (cid:1) , (2) ρ m ( z ) = 3 H π G Ω m (1 + z ) . (3)in the standard Λ CDM spatially flat cosmological model. Thereference overdensity, ∆ ref , is usually chosen to be a numberclose to 18 π ≈ Ω m = 1 − Ω Λ = 1. In the more realistic flat Λ CDMuniverse, the virial overdensity is not constant and varies withredshift (e.g., Bryan & Norman 1998).Conventionally ρ c ( z ) has been adopted as the referencebackground density for the cluster mass definition: ρ ref = ρ c ( z ), with ∆ ref = ∆ c = 500 or 200. This is a convenient choicebecause it depends only on the critical density and does not re-quire additional prior knowledge of Ω m . ∆ c = 500 has beenmost widely used in analyzing Chandra and
XMM-Newton clusters, since it corresponds to the radius where these ob-servatories can reliably measure gas density and temperatureprofiles of the intracluster medium (ICM). ∆ c = 200 is alsowidely used since it is close to the virial overdensity in thespherical collapse model at z = 0.A recent work by Diemer & Kravtsov (2014) demonstratesthat while dark matter density profiles with halos definedwith respect to the critical density ρ c ( z ) exhibit more self-similar behavior in their inner regions, adopting the refer-ence density to be the mean matter density ρ ref = ρ m ( z ) resultsin a more self-similar density profile in the outskirts, wherethe clusters are more sensitive to the recent mass accretion.Since gas motions are driven by mass accretion and are moresignificant in the outer regions of clusters, the non-thermalpressure fraction profile is expected to be more self-similarwhen scaled with r ∆ m . In this paper, we compare the non-thermal pressure fraction profile as well as the gas velocityanisotropy profile using two different mass definitions basedon ∆ c = 200 and ∆ m = 200. For reference, r c ≈ . r m and r c ≈ . r m for our sample at z = 0. Mass Accretion Rate
We use the mass accretion rate of galaxy clusters to identifytheir dynamical state at z = 0. Following Diemer & Kravtsov(2014) we use the quantity Γ as a proxy of mass accretionrate, such that Γ m ≡ ∆ log( M m ( a )) / ∆ log( a ) , (4)where M m ( a ) is the mass of the cluster or its most massiveprogenitor measured at expansion factor a . Γ is computedfrom the difference of each respective quantity between z = 0and z = 0 .
5. The most massive progenitor of each cluster istracked and identified using merger trees as described in Nel-son et al. (2012). A higher Γ means the halo experiences agreater physical mass accretion between z = 0 and z = 0 .
5. Wenote that this definition of mass accretion naturally accountsfor physical mass accretion and isolates effects of pseudo evo-lution in cluster mass (due to evolution of the background ref-erence density ρ ref ( z ), see Diemer et al. 2013), since we arecomparing the increase in halo mass between two fixed red-shifts.Also note that the choice of the redshift z = 0 . Γ is not the only (or necessarily the best) way ofcharacterizing the mass accretion rate. We will explore al-ternative methods for quantifying the mass accretion rate infuture work. For now, we adopt this definition to aid compar-ison between our results and the N -body results of Diemer &Kravtsov (2014). SIMULATIONS
Hydrodynamical Simulations of Galaxy Clusters
We analyze simulated massive galaxy clusters presentedpreviously in Nelson et al. (2014). We refer the reader to thatpaper for more details. We briefly summarize the relevant pa-rameters below.In this work we analyze a high-resolution cosmologicalsimulation of 65 galaxy clusters in a flat Λ CDM model withWMAP five-year (
WMAP5 ) cosmological parameters: Ω m =1 − Ω Λ = 0 . Ω b = 0 . h = 0 . σ = 0 .
82, where theon-thermal Pressure Profiles of Galaxy Clusters 3 r/r c P r a nd / P t o t z=0.0z=0.5z=1.0z=1.5 r/r m P r a nd / P t o t our fit (eq 5) F IG . 1.— Redshift dependence of the profile of non-thermal pressure fraction P rand / P total , with radius scaled with respect to r c ( left ) and r m ( right ). Theshaded regions denote the 1- σ scatter around mean at z = 0. Our fitting formula is over plotted in the dashed line. Hubble constant is defined as 100 h km s − Mpc − and σ is themass variance within spheres of radius 8 h − Mpc. The simula-tion is performed using the Adaptive Refinement Tree (ART) N -body+gas-dynamics code (Kravtsov 1999; Kravtsov et al.2002; Rudd et al. 2008), which is an Eulerian code that usesadaptive refinement in space and time and non-adaptive re-finement in mass (Klypin et al. 2001) to achieve the dynamicranges necessary to resolve the cores of halos formed in self-consistent cosmological simulations. The simulation volumehas a comoving box length of 500 h − Mpc, resolved using auniform 512 grid and 8 levels of mesh refinement, implyinga maximum comoving spatial resolution of 3 . h − kpc.Galaxy clusters are identified in the simulation using a vari-ant of the method described in Tinker et al. (2008) (see Nelsonet al. (2014) for a more detailed description of this method).We selected clusters with M c ≥ × h − M (cid:12) and re-simulated the regions, defined as 5 × r vir , surrounding the se-lected clusters with higher resolution. The resulting simula-tion has effective mass resolution of 2048 surrounding theselected clusters, allowing a corresponding dark matter par-ticle mass of 1 . × h − M (cid:12) . The current simulation onlymodels gravitational physics and non-radiative hydrodynam-ics. As shown in Lau et al. (2009), the exclusion of cool-ing and star formation have negligible effect (less than a fewpercent) on the total contribution of gas motions to the non-thermal pressure support for r ≥ . r c , the radial range wefocus on in this work. We show that our results are insensitiveto dissipative physics in Appendix B.To study the evolution of the non-thermal pressure frac-tion, we extract halos from four redshift outputs: z =0 . , . , . , .
5. At each redshift we apply an additionalmass-cut to ensure mass-limited samples at all epochs. Themass-cuts and resulting sample sizes are as follows: 65clusters with M m ≥ × h − M (cid:12) at z = 0, 48 clusterswith M m ≥ . × h − M (cid:12) at z = 0 .
5, 42 clusters with M m ≥ . × h − M (cid:12) at z = 1 .
0, and 42 clusters with M m ≥ × h − M (cid:12) at z = 1 . Computing the Non-thermal Pressure Fraction
The non-thermal pressure fraction is defined as P rand P total = P rand P rand + P therm = σ σ + (3 kT /µ m p ) (5)where k is the Boltzmann constant, m p is the proton mass, µ =0 .
59 is the mean molecular weight for the fully ionized ICM, σ gas and T are the mass-weighted 3D velocity dispersion andmass-weighted temperature of the gas averaged over sphericalradial shells, respectively.To compute spherically averaged profiles of the mass-weighted 3D velocity dispersion and mass-weighted temper-ature we divide the analysis region for each cluster into 99spherical logarithmic bins from 10 h − kpc to 10 h − Mpc in theradial direction from the cluster center, which is defined asthe position with the maximum binding energy. Our resultsare insensitive to the exact choice of binning. We follow theprocedure presented in Zhuravleva et al. (2013), and we referthe reader to it for the details of the procedure and its impacton the non-thermal pressure measurements. Briefly, for eachradial bin we exclude contribution from gas that lies in thehigh-density tail in the gas distribution, which remove small-scale fluctuations in the non-thermal pressure due to gas sub-structures while preserving the profiles of the global ICM. Inaddition, we smooth the profiles by applying the Savitzky-Golay filter used in Lau et al. (2009).In computing the gas velocity dispersion, we first choosethe rest frame of the system to be the center-of-mass velocityof the total mass interior to each radial bin. We rotate the co-ordinate system for each radial bin such that the z -axis alignswith the axis of the total gas angular momentum of that bin.We then compute mean (cid:104) v i (cid:105) and mean-square gas velocities (cid:104) v i (cid:105) weighted by the mass of each gas cell. The velocity dis-persion is computed as σ i = (cid:112) (cid:104) v i (cid:105) − (cid:104) v i (cid:105) for both the radial Nelson, Lau, & Nagaiand tangential velocity components σ r and σ t . The 3D veloc-ity dispersion is simply σ gas = (cid:112) ( σ r + σ t ) / RESULTS
Universality with Redshift
In Figure 1 we present the non-thermal pressure fractionprofile, P rand / P total ( r ), averaged for our cluster sample at fourredshifts z = 0 . , . , . , .
5. In the two panels we show theevolution of the profiles normalized using two different haloradii r c and r m on the left and right, respectively. Theshaded regions depict the 1- σ scatter around the mean for z = 0. The scatter is comparable at all redshifts and has beenomitted for clarity. For r c , there is a strong redshift evo-lution of both the shape and normalization of P rand / P total . Atlow redshift, the non-thermal pressure fraction is ≈
10% inthe inner regions of the clusters, increasing up to >
40% out-side of r > r c . At higher redshift, the non-thermal pressuresystematically constitutes a larger fraction of the total pres-sure support in the systems. Moreover, the profile increaseswith radius more rapidly, reaching up to ≈
45% of the totalpressure at r c at z = 1 .
5, twice the fraction at z = 0.In the right panel, we again show the redshift evolution of P rand / P total , however, this time normalizing the profiles usingthe halo radius r m . When the halo radius is defined with themean background density of the universe, the radial depen-dence of P rand / P total remains, with ≈
10% non-thermal pres-sure support in the core rising steadily to r = r m where thepressure fraction flattens out to ≈
50% pressure support inthe cluster outskirts. However, the very strong redshift depen-dence seen using r c as the halo radius has completely disap-peared. The universality we find in our non-thermal pressurefraction is in agreement with the very weak redshift depen-dence Battaglia et al. (2012) show in their non-thermal pres-sure fraction profile, also scaled with r m .The simple reason behind this universality with r m is thatthe non-thermal pressure fraction is sensitive to the mass ac-cretion rate of the clusters. The rate at which mass is accret-ing depends on the physical density contrast between the haloand the mean background density, precisely the quantity usedin the definition of r m . We will provide a more detailed ex-planation of this redshift universality in our follow-up paper. Dependence on Mass and Mass Accretion Rate
In order to characterize the scatter in the non-thermal pres-sure fraction profile, we examine the mass and dynamicalstate dependence of the profile. In Figure 2, we divide thesample into three equal sized mass bins at z = 0, shown inred, green and blue from lowest mass to highest mass sub-samples, respectively. We find no systematic dependence onmass in the profile. However, it is worth noting that our sam-ple only encompasses the massive end of the cluster popula-tion with M m ≥ × h − M (cid:12) or M c ≥ × h − M (cid:12) correspondingly.In the left panel of Figure 3, we investigate the dynamicalstate dependence of the non-thermal pressure fraction at z = 0.The mean profiles are shown for three equal sized Γ bins,shown in red, green, and blue lines from the most slowly tomost rapidly accreting systems, respectively. The high Γ sub-sample has systematically higher non-thermal pressure frac-tion at all radii by 5% −
15% than the most slowly accretingsubsample. This is consistent with previous works which alsofind significantly larger fractions of non-thermal pressure inmerging or unrelaxed systems (Vazza et al. 2011; Nelson et al. r/r m P r a nd / P t o t z =0 . × In our definition of P rand in Equation (5), we assume the gasvelocities in the galaxy clusters are isotropic. In this sectionwe explore the relative importance of the radial and tangentialcomponents in the non-thermal pressure fraction by examin-ing the velocity anisotropy parameter β , β ( r ) = 1 − σ t ( r )2 σ r ( r ) , (6)where a positive (negative) value of β indicates more radial(tangential) motion. In Figure 4, we examine the redshift evo-lution of the anisotropy with the cluster radius scaled withrespect to r c ( left panel) and r m ( right panel). Similar tothe P rand / P total profile, the profile scaled with respect to criti-cal density shows significant redshift dependence. The profilescaled with respect to mean matter density, on the other hand,shows universality across the radial range of 0 . (cid:46) r / r m (cid:46) . 5. The gas velocities are slightly radial in the inner regions(e.g., β ≈ . r = 0 . r m ) and becomes increasingly radial,reaching β ≈ . r = r m .In the cluster outskirts, the anisotropy decreases as the gasmotions become more isotropic again. The apparent drop in β is partly due to our definition of the velocity anisotropy inEquation (6), in which we neglected the contributions fromcoherent motions in both radial and tangential directions. Atlarge radii, there is a coherent radial motion toward the clustercenter (e.g., see Figure 1 in Lau et al. 2009), causing σ r (rep-on-thermal Pressure Profiles of Galaxy Clusters 5 r/r m P r a nd / P t o t z =0 Γ < 1.691.69 < Γ < 2.682.68 < Γ < 4.24 r/r m P r a nd / P t o t renormalized using ¯Γ F IG . 3.— Dynamical state dependence of the P rand / P total profiles at z = 0. The sample has been divided into three equal sized mass accretion rate bins as denotedin the legend, shown in red, green and blue from most slowly to most rapidly accreting clusters. In the right panel the profiles have been renormalized by the ratioof Eq 7 to Eq 8 for the mean Γ value in each bin, respectively, in order to remove the dynamical state dependence. The shaded regions denote the 1- σ scatter inthe most relaxed bin. r/r c β z=0.0z=0.5z=1.0z=1.5 r/r m β our fit F IG . 4.— Redshift dependence of the profile of velocity anisotropy, with radius scaled with respect to r c ( left ) and r m ( right ). The shaded regions denotethe 1- σ scatter around the mean at z = 0. resenting the random radial motions) and hence β to decreasewith radius. Fitting Formulae In this section, we provide fitting formulae for the univer-sal non-thermal pressure fraction and gas velocity anisotropyprofiles discussed in the previous sections. By using r m forthe halo radius, our universal non-thermal pressure profile is well-described by the following fitting formula, P rand P total ( r ) = 1 − A (cid:26) + exp (cid:20) − (cid:18) r / r m B (cid:19) γ (cid:21)(cid:27) (7)where the best fit parameters are A = 0 . ± . B =0 . ± . γ = 1 . ± . . ≤ r / r m ≤ . z = 1 . 5. The best fit line is plotted in the right panel of Fig-ure 1. In the Appendix, we also supply the adjusted fittingformula for the scaling of radii with respect to critical. De-spite using r ∆ c , we are still able to provide an accurate fit outto z = 1, since the formula is based on the universal r ∆ m pro-file.As shown in the left panel of Figure 3, the varied mass ac-cretion histories of the clusters in our sample is a major sourceof scatter in the non-thermal pressure fraction profile. To ac-count for the mass accretion rate, we provide a correction fac-tor for the normalization of our fitting formula at z = 0 using Γ as a parameter, P rand P total ( r ; z = 0)= 1 − (0 . − . Γ m ) (cid:26) + exp (cid:20) − (cid:18) r / r m B (cid:19) γ (cid:21)(cid:27) , (8)where we keep our best fit parameters B = 0 . 841 and γ =1 . 628 obtained earlier by fitting in the radial range of 0 . ≤ r / r m ≤ . 5. We use the best fit A (Equation (7)) values forthe profiles of each of the 65 clusters in our sample, with B and γ fixed at our best fit values, to determine the function of Γ . Inthe right panel of Figure 3, we show the same three Γ bins asin the left panel but renormalized using Equation (8), fit usingthe mean Γ value in each bin respectively. While there is someresidual variation in the slope of the profile for different massaccretion bins, including this normalization correction factordecreases the scatter at r m from 0 . 067 to 0 . σ agreement between the adjusted profiles.Previous attempts to characterize the non-thermal pressurefraction have used the r c (or r c ) as the cluster radius andtherefore have included an additional factors to account forits strong redshift evolution (Shaw et al. 2010) and mass de-pendence (Battaglia et al. 2012). We have confirmed that the r c profiles for our data are well fit by their fitting formulaeat z = 0 . β ( r ) = ( r / r t ) − a (1 + ( r / r t ) b ) c / b , (9)fit between 0 . ≤ r / r m ≤ . 5, where the best fit param-eters are r t = 1 . ± . a = 2 . ± . b = − . ± . 183 and c = − . ± . . ≤ r / r m ≤ . z = 1 . 5, with the exception of z = 0 . 5, which is only a robustfit for r ≤ r m . SUMMARY AND DISCUSSION In this work we presented the redshift and mass inde-pendent non-thermal pressure fraction profile using a mass-limited, cosmologically representative sample of 65 massivegalaxy clusters from a high resolution hydrodynamical cos-mological simulation. This result is relevant in accounting forthe systematic effects of non-thermal pressure on X-ray andmicrowave measurements of galaxy clusters and cosmologi-cal inferences based on these measurements.We found that the mean non-thermal pressure fraction pro-file exhibits remarkable universality in redshift and masswhen we define the size of cluster halos using the mean matter density of the universe, instead of the critical density. How-ever, we also showed that there is strong dependence in thenon-thermal pressure fraction profile on the halo’s mass ac-cretion rate: clusters that are rapidly accreting have an overallhigher non-thermal pressure fraction. As such, the mass ac-cretion rate is a major source of systematic scatter in the meannon-thermal pressure fraction profile.A robust and quantitative proxy for measuring mass accre-tion rate is therefore needed to account for this effect, es-pecially with the upcoming multi-wavelength cluster surveyswhere statistical errors will be considerably smaller than sys-tematic uncertainties arising from our ignorance of cluster as-trophysics. We note that the current method of characterizingthe mass accretion rate using the fractional mass increase be-tween z = 0 and z = 0 . z = 0 clusters. Future work should focus on de-veloping quantitative measures of the mass accretion rate ofhalos that can be applied to halos across a wide range of red-shifts, and relate these measures to observable proxies of thedynamical states of clusters.We found no systematic mass dependence in the univer-sal non-thermal pressure fraction profile. But given that oursample contains only massive clusters, the independence inmass should be checked with sample of lower mass halos.Since slowly accreting halos have smaller non-thermal pres-sure fraction, we expect that lower mass groups and galaxies,which should experience less physical mass accretion thanhigh mass clusters, to have lower non-thermal pressure frac-tion profile. However, we note that smaller mass halos aremore susceptible to non-gravitational physics (such as gascooling and energy injections from stars and active galacticnuclei) which can influence the net accretion rate into andwithin the halos in a non-trivial way.We found that the gas velocity is predominantly radial, withthe velocity anisotropy parameter increasing from ≈ . ≈ . . r m to r m . Moreover, we found that thevelocity anisotropy profile is also universal across redshiftwhen halos are defined using the mean matter density, indi-cating that gas velocity anisotropy is also a self-similar quan-tity. This result can be useful since the velocity anisotropycannot be easily measured in observations, as we can onlymeasure line-of-sight velocities from Doppler width measure-ments, e.g., with the upcoming ASTRO-H in the near future.Measurements of gas motions tangential to the line-of-sightare possible with resonant scattering but difficult (e.g., Zhu-ravleva et al. 2011).We provided fitting formulae for the universal non-thermalpressure fraction and gas velocity anisotropy profiles thatwork remarkably well within r m and out to redshift z = 1 . . (cid:46) r / r m (cid:46) . 5. We note, however, that our simulation does not modelplasma effects which can amplify gas turbulence and pro-vide extra non-thermal pressure support (e.g., Parrish et al.2012). Physical viscosity, on the other hand, can decrease thelevel of gas turbulence which lowers the non-thermal pres-sure support. The results presented in this paper based on hy-drodynamical simulations serve as baseline for further studiesof these effects. Magnetic fields and cosmic rays can alsoprovide additional non-thermal pressure (e.g., Laganá et al.2010). However, their contributions are expected to be small.The typical magnetic field strength of (cid:46) µ G in the ICMcorresponds to magnetic pressure fraction of (cid:46) (cid:46) γ -ray observations of Fermi -LAT ( Fermi -LAT Collaboration 2013). It is important to note,however, that the constraints on the contribution from cosmicrays assume that the cosmic ray distribution follows that of thethermal ICM and, therefore, a flattened distribution of cosmicrays could result in an increased contribution to the ICM en-ergy (e.g., Zandanel & Ando 2014)The upcoming ASTRO-H mission will measure gas mo-tions in galaxy clusters and should provide observational con- straints on the level of the non-thermal pressure fraction inthese systems. However, the observational constraints will belimited to inner regions ( (cid:46) r c ≈ . r m ) of nearby mas-sive clusters, due to the lack of sensitivities in low-densityregions in cluster outskirts. Extending these measurementsto the outskirts or high-redshift clusters must await the nextgeneration of X-ray missions, such as SMART-X and/or Athena+ . Alternatively, kinematic SZ effect can probe in-ternal gas motions of electrons in galaxy clusters (Inogamov& Sunyaev 2003; Nagai et al. 2003). Since the SZ signalis independent of redshift and linearly proportional to gasdensity (unlike X-ray emission which is proportional to gasdensity squared), measurements of the kSZ effect with high-resolution, multifrequency radio telescopes, such as CCAT ,might enable characterization of the non-thermal pressure inthe outer regions of high-redshift clusters.Previous works have used the definition of cluster mass nor-malized with respect to the critical density of the universe. Inthis work, we argue that an alternative definition based on themean mass density of the universe is a preferred choice for thenon-thermal pressure profile as well as velocity anisotropy ofgas in clusters. It would be interesting to check whether othergas properties exhibit similar universality when the clusterprofiles are normalized with respect to the mean mass density.We will investigate these issues in our next paper and exploretheir implications for understanding the evolution cluster gasstructure and their application to cosmology.We thank Nick Battaglia, Eiichiro Komatsu, and the anony-mous referee for comments on the manuscript. This work wassupported in part by NSF grant AST-1009811, NASA ATPgrant NNX11AE07G, NASA Chandra grants GO213004Band TM4-15007X, the Research Corporation, and by the fa-cilities and staff of the Yale University Faculty of Arts andSciences High Performance Computing Center. REFERENCESAllen, S. W., Evrard, A. E., & Mantz, A. 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(2012) (dashed) fitting formulas with our data at z = 0 and z = 1, blueand red lines respectively. The shaded regions denote the 1- σ scatter around mean at each redshift.APPENDIXA. FITTING THE NON-THERMAL PRESSURE FRACTION PROFILE WITH RESPECT TO CRITICAL DENSITY To date, it has been common to use the critical density to define the radius and mass of galaxy clusters. We therefore provide aconversion for our fitting formula for the universal non-thermal pressure profile defined respect to the mean mass density of theuniverse, so that it can easily be used with alternative definitions of halos’ radius and mass based on the critical density. Thisconversion method is adapted from Appendix C in Hu & Kravtsov (2003). The converted fitting formula is expressed as P rand P total = 1 − A (cid:26) + exp (cid:20) − (cid:18) η r / r m B (cid:19) γ (cid:21)(cid:27) (A1)where A , B , and γ are the best fit parameters given in Section 4.4. The conversion factor η is given by η = c m (cid:32) (cid:112) a f p + (3 / + f (cid:33) (A2)where c m ≡ r m / r s is the concentration parameter, and f = ln(1 + c m ) − c m / (1 + c m ) c m (cid:18) ∆ c E ( z ) Ω m (1 + z ) (cid:19) , (A3) p = a + a ln f + a (ln f ) . (A4)Here ∆ c is the chosen overdensity with respect to the critical density of the universe, E ( z ) ≡ H ( z ) / H ( z = 0) = (cid:112) Ω m (1 + z ) + Ω Λ isthe normalized Hubble parameter for a flat cosmology, and a = 0 . a = − . / a = − . × − and a = − . × − .In left panel of Figure 5, we show how our fitting formula recovers the non-thermal pressure fraction measured from simulationsat four different redshifts, z = 0 . , . , . , . 5. To convert between r m and r c we use the mean concentration at each redshiftbin as calculated from concentration-mass relation in Bhattacharya et al. (2013). We find excellent agreement between our fit andour data in the redshift range of 0 ≤ z ≤ 1. At z = 1 . r c .Previous attempts to describe the non-thermal pressure profile used the scale radius r c . In right panel of Figure 5 we compareour sample to the fitting formula from Battaglia et al. (2012) (dashed line) and Shaw et al. (2010) (dotted line) at two redshifts, z = 0 . z = 1 . 0, shown in the blue and red lines, respectively. The Shaw et al. (2010) fitting formula slightly underestimatesour results at z = 0. At z = 1 . 0, the fit has approximately the same slope as ours between r = 0 . r c and r = 1 . r c , butunderestimates the pressure fraction slightly at r (cid:46) . r c and overestimates the fraction at the larger radii. The deviation inon-thermal Pressure Profiles of Galaxy Clusters 9 . . . . . . . r/r m . . . . . . . P r a nd / P t o t NRCSF . . . . . . . r/r m − . − . . . . . . . β ( r ) NRCSF F IG . 6.— Left: P rand / P total profiles for the 16 clusters taken from Nagai et al. (2007a). Right: The gas velocity anisotropy profiles β for the same set of clusters.The red lines correspond to clusters simulated with non-radiative (NR) physics, and blue lines correspond to the same clusters simulated with radiative cooling,star formation and supernova feedback (CSF). The shaded regions indicate 1- σ scatter around the mean. Shaw et al. (2010) is likely due to the fact that their cluster sample was small (only 16) and heterogeneous. The Battaglia et al.(2012) fitting formula adopts the redshift dependence from Shaw et al. (2010) and adds an additional factor to account for themass dependence which they calibrate using a larger sample of simulated clusters at z = 0. At z = 0, this additional factor results ina slight overestimate of the non-thermal pressure fraction out to r = 2 r c . At z = 1, their fit shows the same redshift dependenceas Shaw et al. (2010), but with a lower normalization due to the mass dependence factor, which was only calibrated at z = 0. B. EFFECTS OF DISSIPATIVE PHYSICS ON THE NON-THERMAL PRESSURE FRACTION AND GAS VELOCITY ANISOTROPY In this section we examine the effect of dissipative physics on the non-thermal pressure fraction and gas velocity anisotropyin group and cluster sized systems. We compare profiles for the set of 16 clusters from Nagai et al. (2007a), simulated with twodifferent gas physics: non-radiative (NR) physics; and with radiative cooling, star formation, and supernova feedback (CSF). InFigure 6 we show the mean profiles of the non-thermal pressure fraction P rand / P total and the gas velocity anisotropy β for theclusters at z = 0. There is no systematic dependence on gas physics in the radial range of 0 . (cid:46) r / r m (cid:46) . 5. The profilesbetween the two runs are consistent between each other within 1- σσ