Massive Halos in Millennium Gas Simulations: Multivariate Scaling Relations
aa r X i v : . [ a s t r o - ph . C O ] O c t Draft version October 29, 2018
Preprint typeset using L A TEX style emulateapj v. 03/07/07
MASSIVE HALOS IN MILLENNIUM GAS SIMULATIONS: MULTIVARIATE SCALING RELATIONS
R. Stanek
Department of Astronomy, University of Michigan, 500 Church St., Ann Arbor, MI 48109
E. Rasia
Chandra
Fellow, Department of Astronomy and Michigan Society of Fellows, University of Michigan, 500 Church St., Ann Arbor, MI48109
A. E. Evrard
Departments of Physics and Astronomy and Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109
F. Pearce
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
L. Gazzola
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
Draft version October 29, 2018
ABSTRACTThe joint likelihood of observable cluster signals reflects the astrophysical evolution of the coupledbaryonic and dark matter components in massive halos, and its knowledge will enhance cosmologi-cal parameter constraints in the coming era of large, multi-wavelength cluster surveys. We presenta computational study of intrinsic covariance in cluster properties using halo populations derivedfrom Millennium Gas Simulations (MGS). The MGS are re-simulations of the original 500 h − MpcMillennium Simulation performed with gas dynamics under two different physical treatments: shockheating driven by gravity only ( GO ) and a second treatment with cooling and preheating ( PH ). Weexamine relationships among structural properties and observable X-ray and Sunyaev-Zel’dovich (SZ)signals for samples of thousands of halos with M ≥ × h − M ⊙ and z <
2. While the X-rayscaling behavior of PH model halos at low-redshift offers a good match to local clusters, the modelexhibits non-standard features testable with larger surveys, including weakly running slopes in hot gasobservable–mass relations and ∼
10% departures from self-similar redshift evolution for 10 h − M ⊙ halos at redshift z ∼
1. We find that the form of the joint likelihood of signal pairs is generallywell-described by a multivariate, log-normal distribution, especially in the PH case which exhibits lesshalo substructure than the GO model. At fixed mass and epoch, joint deviations of signal pairs dis-play mainly positive correlations, especially the thermal SZ effect paired with either hot gas fraction( r = 0 . / .
69 for PH / GO at z = 0) or X-ray temperature ( r = 0 . / . PH model by combining thermal SZ and gas fraction measurements. Subject headings: galaxies:clusters – cosmology:theory INTRODUCTION
Accurate cosmology using surveys of clusters of galax-ies requires a robust description of the relations betweenobserved cluster signals and underlying halo mass. Evenwithout strong prior knowledge of the mass-signal rela-tion, cluster counts, in combination with other probes,add useful constraining power to cosmological parame-ters (Cunha et al. 2009). However, significant improve-ments can be realized when the error in mass variance isknown (Lima & Hu 2005; Cunha & Evrard 2009). Im-provements can also be gained by extending the modelto multiple observed signals (Cunha 2008), especiallywhen an underlying physical model can effectively reduce
Electronic address: [email protected] the dimensionality of the parameter sub-space associatedwith the model (Younger et al. 2006). The coming era ofmultiple observable signals from combined surveys in op-tical, sub-mm and X-ray wavebands invites a more holis-tic approach to modeling multi-wavelength signatures ofclusters.Signal covariance characterizes survey selection, interms of mass and additional observables. For the case ofX-ray selected samples, Nord et al. (2008) demonstratethat luminosity–temperature covariance can mimic ap-parent evolution in the luminosity–mass relation underanalysis that combines deep, X-ray-flux limited sampleswith local, shallow ones.Employing a selection observable with small mass vari-ance minimizes such errors. Recent work has shown Stanek et al.that the total gas thermal energy, Y , observable viaan integrated Sunyaev-Zeldovich (SZ) effect (Carlstrom2004) or via X-ray imaging and spectroscopy, is a sig-nal that scales as a power-law in mass with only ∼
15% scatter (White et al. 2002; Kravtsov et al. 2006;Maughan 2007; O’Hara et al. 2007; Zhang et al. 2008;Jeltema et al. 2008). However, unbiased estimates of themass selection function for Y or any other signal requiresaccurate knowledge of how the signal–mass scaling rela-tion evolves with redshift. The redshift behavior of sig-nals is generally not well known empirically, although re-cent work has begun to probe evolution in X-ray signalsto z ∼ ∼ ∼
15% scatter among individual systems (Rasia et al.2006; Nagai 2006; Jeltema et al. 2008). Cluster massescan also be measured by the shear induced on back-ground galaxies due to gravitational lensing. With thismethod, individual cluster masses have mass uncertain-ties ∼
20% due to cosmic web confusion (Hoekstra 2003;de Putter & White 2005), but large samples can reducethe uncertainty in the mean.The mean scaling behavior of samples binned in someselection signal offers another empirical path to measur-ing covariance. Non-zero covariance between the selec-tion and an independent, follow-up signal implies thatthe selection-binned scaling relation of the followup sig-nal with mass need not match that signal’s intrinsic massscaling. Comparison of scaling relations from differentlyselected samples thereby offers insight into covariance.Rykoff et al. (2008) offer a first attempt at this exer-cise for X-ray luminosity and optical richness using theoptically-selected SDSS maxbcg sample (Koester et al.2007). The sample contains ∼ ,
000 clusters for whichweak lensing mass estimates have been made by stackingthe shear of richness-binned sub-samples (Sheldon et al.2007; Johnston et al. 2007). Rykoff et al. (2008) stackRosat All-Sky Survey data (Voges et al. 1999) in thesame maxbcg richness bins, and find that the meanX-ray luminosity–mass relation derived with richnessbinning is consistent at the ∼ σ level with relationsderived solely from X–ray data (Reiprich & B¨ohringer2002; Stanek et al. 2006).A theoretical approach to studying cluster covari-ance is to realize populations via numerical simulation.While high resolution treatment of astrophysical pro-cesses, including star formation, supernova and AGNfeedback, galactic winds and thermal conduction havebeen included in recent simulations (Dolag et al. 2005;Borgani et al. 2006; Kravtsov et al. 2006; Sijacki et al.2008; Puchwein et al. 2008), the computational expensehas limited sample sizes to typically a few dozen ob-jects. A detailed study of population covariance requires larger sample sizes, as can be generated by lower reso-lution simulations of large cosmological volumes using amore limited physics treatment (Bryan & Norman 1998;Hallman et al. 2006; Gottl¨ober & Yepes 2007).We take the latter approach in this paper, focusing onthe bulk properties of massive halos identified the Millen-nium Gas Simulations (MGS), a pair of resimulations ofthe original 500 h − Mpc Millennium run (Springel et al.2005), each with 10 total particles, half representing gasand half dark matter. The pair of runs use different treat-ments for the gas physics — a gravity-only ( GO ) simula-tion, sometimes called “adiabatic”, in which entropy isincreased via shocks, and a simulation with cooling andpreheating ( PH ). The former ignores galaxies as both asink for baryons and a source of feedback for the hot intr-acluster medium (ICM). The latter also ignores the massfraction contribution of galaxies, but it approximates thefeedback effects of galaxy formation by a single param-eter, an entropy level imposed as a floor at high red-shift (Evrard & Henry 1991; Kaiser 1991; Bialek et al.2001; Kay et al. 2007; Gottl¨ober & Yepes 2007). Ourstudy focuses on samples of ∼ M > × h − M ⊙ examined at multiple epochs cov-ering the redshift range 0 < z < h − M ⊙ and halo mass is defined within a sphereencompassing a density contrast ∆ c = 200 times the crit-ical density. SIMULATIONS
Millennium Gas Simulations
The Millennium Gas Simulations (hereafter MGS)are a pair of resimulations of the original Millennium(Springel et al. 2005), a high-resolution, dark-matter-only simulation of a 500 h − Mpc volume. The sim-ulations were run with GADGET-2, treating the gasdynamics with smoothed particle hydrodynamics (SPH)(Springel 2005). As described in Hartley et al. (2008),the MGS use the initial conditions of the Millenniumsimulation, with 5 × dark matter particles, each ofmass 1 . × h − M ⊙ , and 5 × SPH gas par-ticles, each of mass 3 . × h − M ⊙ , resulting in amass resolution about 20 times coarser than the orig-inal Millennium N-body simulation. The gravitationalsoftening length is 25 h − kpc. The cosmological pa-rameters match the original: Ω m = 1 − Ω Λ = 0 . b = 0 . h = 0 .
73, and σ = 0 .
9. While some dif-ferences between the simulations are expected due to thedifference in mass resolution and gravitational softeninglength, Hartley et al. (2008) verify the positions of darkmatter halos to within 50 h − kpc between the originalMillennium and the MGS. The value of σ is higher thanthe WMAP3 value (Spergel et al. 2006), but we do notexpect it to strongly affect the results of this paper.In this paper, we consider two models of the MGS: a GO simulation where the only source of gas entropy changeis from shocks, and a PH simulation with preheating andcooling along with shock heating. The GO simulationis useful as a base model that can be easily comparedalos in Millennium Gas Simulations: Multivariate Scaling Relations 3to previous hydrodynamic simulations of galaxy clus-ters. Furthermore, comparing gravity-only simulationsto observations highlights the cluster properties that arestrongly affected by astrophysical processes beyond grav-itational heating.While adiabatic simulations match the self-similarprediction, L ∼ T , for the X-ray luminosity–temperature relation, observations show a steeper slope(Arnaud & Evrard 1999; Osmond & Ponman 2004).Preheating, the assumption of an elevated initialgas entropy at high redshift, was introduced byEvrard & Henry (1991) and Kaiser (1991) as a means toresolve the discrepancy in shape between the observedX-ray luminosity function and that expected from self-similar scaling of the cosmic mass function. The PH sim-ulation is tuned to match X-ray observations of clustersat redshift zero, particularly the luminosity-temperature(L-T) relation (Hartley et al. 2008). The preheating isachieved by boosting the entropy of every gas particle to200 keV cm at redshift z = 4. Although the preheat-ing dominates in the PH simulation, there is also coolingbased on the cooling function of Sutherland & Dopita(1993). Fewer than 2% of the baryons are converted tostars, however, and star formation is essentially haltedby the preheating at z = 4.This simple model certainly does not capture all of thecomplex astrophysical effects associated with star and su-permassive black hole formation in clusters. The centralentropy structure in local clusters is distinctly bimodal,with roughly half the population centered near the 200keV cm value used in the PH and the remainder cen-tered on an entropy level a factor of ten smaller. How-ever, the latter reflects potentially cyclical AGN feed-ing and feedback (Voit et al. 2002; Sijacki et al. 2007;Puchwein et al. 2008) that strongly influences only asmall mass fraction of the total cluster gas. Obser-vational evidence of ubiquitous galactic winds at highand moderate redshifts (Pettini et al. 2001; Weiner et al.2008) suggest that most of the heating of the ICM occursat high redshift. Further support for fast feedback comesfrom red sequence galaxies extending to redshift z = 1 . z ∼ − Halo Catalog
We identify halos in the simulation as spherical regions,centered on filtered density peaks, that encompass anaverage density of ∆ c ρ c ( z ), where ρ c ( z ) is the criticaldensity of the universe. Both dark matter and baryonsare included in the density measurement. We use ha-los identified at an overdensity of ∆ c = 200 for most ofour analysis. Halo centers were identified with an N th nearest neighbor approach, which approximates the localdensity by calculating the distance to the 32 nd nearestdark matter particle. The groupfinder begins with thedark matter particle with the highest local density, andworks outward in radius particle by particle (includinggas and dark matter) until the interior mean density is200 ρ c ( z ). The algorithm then identifies the densest darkmatter particle not already in a halo, and continues it-eratively until all overdense regions with more than 100particles have been identified. Overlapping halos are per-mitted; however, the center of mass of a halo may not be Fig. 1.—
Differential counts of halos versus total mass at redshiftzero for the PH simulations (filled, black points), the GO simulation(open, red points), and the prediction from the Tinker mass func-tion (Tinker et al. 2008) (solid, black line). in another halo.At redshift zero, in the PH simulation we have approxi-mately 220,000 halos with at least 100 particles, and 4474over a mass cut of M > × h − M ⊙ . These num-bers are higher in the GO simulation, with approximately370,000 halos with at least 100 particles, and 5612 overthe mass cut of M > × h − M ⊙ . In Figure 1, we plotdifferential halo counts as a function of total mass fromthe GO and PH simulations, as well as the prediction fromthe Tinker mass function (TMF) (Tinker et al. 2008).As discussed in Stanek et al. (2009), preheating causesa decrease in total halo mass of up to ∼
15% relativeto the GO treatment, with the largest effects at lowermasses and higher redshifts. While the GO halo spacedensity matches the TMF expectations well, the numberof halos in the PH case is lower, especially at lower mass.Among the halos over the mass cut of M > × h − M ⊙ , we identify overlapping halos. For a pairof overlapping halos, we denote the less massive halo asa “satellite” and the more massive halo as a “primary”halo. For the rest of the analysis in this paper, we ex-clude the satellite halos. At redshift zero we are left witha sample of 4404 halos in the PH simulation and 5498 ha-los in the GO simulation.We repeat the halo finding exercise at all redshiftsavailable for each model. In the case of PH, we employa total of 63 outputs extending to a redshift of two. Forthe GO, we analyze only a subset of outputs at redshifts, z = 0, 0 .
5, 1 . .
0. Unless otherwise noted, all of ouranalysis is for primary halos over the total mass limit of5 × h − M ⊙ . Bulk Halo Properties
With the primary halo samples identified, we calcu-late bulk properties for them that we roughly classifyinto “structural” and “observable” categories. The for-mer includes dark matter velocity dispersion, ICM massfraction, gas mass-weighted temperature, and halo con-centration while the latter includes X-ray luminosityand spectroscopic-like temperature, thermal Sunyaev- Stanek et al.Zel’dovich effect, and a dimensionless ICM emission mea-sure.As the MGS simulations are SPH treatments of thegas, integrals over volume map to summations over allparticles, via R dV ρ n → Σ i m i ρ n − i . We consider twomeasures of ICM temperature. First, we consider themass-weighted temperature. As GADGET-2 is a La-grangian simulation with equal mass gas particles, themass-weighted temperature is simply the average tem-perature of the particles in the halo: T m = 1 M Z V dV ρ T → N N X i T i . (1)We also calculate the spectroscopic-like temperature, T sl , as defined in Mazzotta et al. (2004), T sl = R n T α − / dV R n T α − / dV → P Ni ρ i T α − / i P Ni ρ i T α − / i , (2)with α = 0 .
75. The spectroscopic-like temperature offersa good match to the temperature derived from a one-component fit to an X-ray spectrum, but is far simplerto compute.We calculate X-ray luminosities, L = Z V dV ρ Λ( T ) → N X i ρ i ˜Λ( T i ) , (3)using MEKAL tables assuming fixed 0 . T ). We do this in energy bands of 0.7-2.0keV, 0.7-5.0 keV, and 0.7-7.0 keV, and also compute abolometric luminosity, L bol using a wide photon energyrange of [0.1-40.0] keV. These tables include both con-tinuum and emission lines derived from an assumptionof collisional ionization equilibrium (Mewe et al. 2003).We also calculate the global thermal Sunyaev-Zel’dovich signal, Y , parameter, following the conventionpresented in da Silva et al. (2000); Springel et al. (2001);Kravtsov et al. (2006). Y = (cid:18) k B σ T m e c A (cid:19) Z V dV n e T e → (cid:18) k B σ T m e c A M gas m p (cid:19) N X i T i . (4)With k B Boltzmann’s constant, σ T the Thomson cross-section, m e the electron mass, and m p the proton mass,and A the halo comoving area. The above expressionyields Y in units of h − Mpc . Radial Profile Measures
In addition to bulk cluster properties, we include twomeasures of radial structure in our analysis: halo concen-tration of the total mass and a dimensionless emissionmeasure for the hot gas.We measure the halo concentration, c , after fitting anNFW density profile (Navarro et al. 1997), ρ ( r ) ρ crit = δ c ( r/r s )(1 + r/r s ) , (5)to the radially-binned, total mass density profile of eachhalo. Following NFW, we take the concentration, c , tobe defined at a density threshold of 200 times the criti-cal density, c = r /r s . Halo concentration is a good proxy for halo formation epoch (Wechsler et al. 2002;Busha et al. 2007). We show below that baryon loss inthe PH case has a non-neglgible effect on halo concentra-tions.We also measure a dimensionless emission measure, orclumping factor, Q , that measures the contribution of gasdensity structure to the halo luminosity. Using a scaledradius, y = r/r , we write the radial gas density profileof a given halo as ρ ( yr ) = f ICM (200 ρ c ) g ( y ) , (6)where f ICM ≡ M gas ( < r ) /M is the halo’s ICM massfraction within r . With the overall ICM mass frac-tion factored out, g ( y ) becomes a dimensionless structurefunction normalized by 3 R dyy g ( y ) = 1.The second moment of g ( y ) defines the dimensionlessemission measureˆ Q ( T ) = (3 / π ) Z d yg ( y ) . (7)This definition allows the X-ray luminosity scaling to bewritten as L ∝ ρ c M Λ( T ) f ICM2 ˆ Q. (8)We show below how lower values of both f ICM and ˆ Q forthe PH case drive X-ray luminosities down by an order ofmagnitude at 10 h − M ⊙ relative to the GO treatment. MEAN SCALING RELATIONS
To characterize the mean behavior of halo properties,we perform a linear least squares fit to the natural log ofthe i th signal as a function of mass and redshift, usingthe form h s i i ( µ, a ) = s i, ( a ) + α i ( a ) µ, (9)where s i ≡ ln S i and µ ≡ ln( M/ h − M ⊙ ). Bracketsrepresent averaging in narrow mass bins at fixed epoch,so α i ( a ) is the slope and s i, ( a ) the normalization at10 h − M ⊙ of the i th signal at redshift, z = a − −
1. Inthe PH case, one expects the entropy floor to introducecurvature into the scaling relations (Voit et al. 2002),and so we extend this model to a quadratic with respectto µ for the ICM mass fraction and related measures forthis case.At each redshift, the mass scalings are derived fromthe sample of primary halos with M > × h − M ⊙ .We then fit the evolution of the normalization, presentedas a shift relative to the present value,∆ s i, ( a ) = s i, ( a ) − s i, (1) , (10)to the form ∆ s i, ( a ) = β ln( E ( a )) , (11)where E ( a ) = H ( a ) /H . For the PH model, this form isa poor fit to f ICM -related quantities. We find a better fitin terms of a quadratic in ln a ,∆ s i, ( a ) = ǫ + ǫ ln( a ) + ǫ (ln( a )) , (12)In this section, we present mean scaling relations anddiscuss deviations from self-similar expectations (Kaiser1986) for structural ( § § h ( s i − h s i i )( s j − h s j i ) i , arepresented in § Fig. 2.—
Scaling relations at redshift zero (left) and redshift evolution of fit parameters (right) for σ DM , T m , f ICM , and the NFWconcentration c (top to bottom). The PH model data are black, filled points and the GO are red, open points. Subsamples of halos are shownin the left panels, drawn in a manner that shows ∼
10% of the overall population with nearly uniform mass sampling. Solid lines show thebest-fit scalings derived from the entire sample. The right-hand panels present the evolution of the slope and the shifts in normalization,equation (10). Solid lines show fits to log-linear behavior, equation (11), except ficm which is quadratic in ln M , while the dotted linesgive the self-similar predictions listed in Table 3. Stanek et al.
Structural Quantities
The left panels of Figure 2 present scaling relations asa function of mass at z = 0 for sub-samples of the GO and PH halo samples. Four structural measures are pre-sented: dark matter velocity dispersion, σ DM ; intraclus-ter gas mass fraction, f ICM ; mass-weighted gas tempera-ture, T m ; and NFW concentration, c . Best-fit parametersto the mass scaling, equation (9), are presented in Table1 The right panels show redshift evolution of the slopesand the shifts in normalization, equation (10), for eachsignal. Error bars in the fit parameters are derived frombootstrapping resampling of the samples. In general, theuncertainties on the best fit parameters are very small, ∼ .
1% at redshift z = 0, and are much smaller than theintrinsic scatter at fixed mass about the median powerlaw relation. The errors grow larger at higher redshifts;the mass-limited sample size drops below 100 at z = 1 . PH case. Fits to the redshift evolution of the nor-malization, equation (11) are presented in Table 3. Dark matter velocity dispersion.
The velocity disper-sion of the dark matter particles is a fundamental mea-sure of the virial state of a halo. Observationally, thegalaxy velocity dispersion tracks ICM temperature in amanner consistent with virial expectations, but the pos-sibility of a ∼
10% bias relative the dark matter is stillallowed (Biviano & Katgert 2004; Becker et al. 2007). Inboth simulations, Figure 2a shows that the scaling withmass is slightly shallower than the self-similar predic-tion, σ DM ∝ M / . The dark matter velocity disper-sion at 10 h − M ⊙ is in good agreement with the1082 . ± . − value derived from a suite of N-body simulations by Evrard et al. (2008). As discussedin Evrard et al. (2008), a slight suppression of the slopeis expected from finite particle resolution when the lowmass halo cutoff corresponds to a few thousand particles,as is the case here. The fact that the PH and GO slopesare similar (0 .
341 and 0 . PH and GO cases, the slope remains constant tohigh redshift, as seen in the upper right panel of Figure2. The evolution of the normalization, σ DM, shownin Figure 2b, deviates slightly from the self-similar pre-diction, σ DM ∝ [ E ( z ) M ] / , with the velocity dispersionbeing 1 −
2% higher at z = 1 (values of β are given inTable 3). Despite strong baryon content differences dis-cussed below, the dark matter virial scaling under the PH and GO treatments remain remarkably consistent. Mass-weighted temperature.
The mass-weighed tem-perature, T m , a useful probe of the hydrodynamicstate of the ICM, is known to have small scatterwith respect to mass in simulations (Evrard et al. 1996;Bryan & Norman 1998; Borgani et al. 2004). Figure 2cshows that the slope of the GO scaling relation agreeswith the self-similar scaling, T ∼ M / , at the few per-cent level, whereas the PH slope is significantly less steep, α = 0 . ± . z ∼ PH , the resultant fractionalincrease in the characteristic entropy of a halo is largerfor low-mass halos than for high-mass halos. The ele-vated initial entropy drives the tilt as well as a ∼ T m − M relation in the PH simulation. But the effect diminishes at higher masses;the PH halos at 10 h − M ⊙ are only 10% hotter than Fig. 3.—
The normalized bulk kinetic energy in gas turbulentmotions, β gas = σ gas2 / ( kT m /µm p ), is shown as a function of massfor sub-samples of the PH (filled, black points) and GO (open, redpoints) halos. Lines show best-fit mean scalings. their GO counterparts.The slopes of the T m − M relation do not evolvestrongly with redshift in either model. The normaliza-tions, however, do not follow the self-similar expecta-tion, with the GO simulation lying ∼
10% low and the PH case 5% high at z = 1 relative to T m ∼ E ( a ) / scaling(Kaiser 1986). The self-similar prediction is for halos inperfect hydrostatic equilibrium, and mergers are knownto drive deviations from hydrostatic equilibrium. How-ever, the total kinetic energy of the gas — the sum ofthermal plus bulk kinetic, or turbulent components —should scale more closely to self-similar expectations.We define a dimensionless measure of the kinetic en-ergy content of the gas, β gas = σ gas2 / ( kT m /µm p ), where µ is the mean molecular weight of the gas. Figure 3.shows values of β gas for a subsample of halos at z = 0.The values differ strongly in the two simulations, withmean values at 10 h − M ⊙ of 0 .
16 in the GO simula-tion and 0 .
06 in the PH model. The positive scaling withmass reflects the later formation epoch of high-mass ha-los. The next generation of X-ray telescopes, such asIXO , will have high-resolution spectroscopy capable ofmeasuring σ gas , and thus be able to discriminate betweenthe model predictions in Figure 3.We measure the total gas kinetic energy, E tot = kT m µm p + σ gas2 , (13)and fit this to the standard form, equation (9). Figure4 shows the redshift evolution of the total gas kineticenergy in both models. The GO simulation respects theself-similar scaling, E tot ∼ [ M E ( a )] / at the few percentlevel to redshifts z = 1. At a given mass, the velocity dis-persion contribution to the total kinetic energy increaseswith redshift, and this enhanced turbulence is responsi-ble for driving the mass-weighted temperature away fromself-similar scaling, as seen in Figure 2. For the PH case, http://ixo.gsfc.nasa.gov/ alos in Millennium Gas Simulations: Multivariate Scaling Relations 7 TABLE 1Redshift Zero Mass Scalings a PH Simulation GO SimulationSignal s α s ασ DMb . ± . . ± .
001 6 . ± . . ± . kT m b . ± . . ± .
002 0 . ± .
001 0 . ± . kT slb . ± .
001 0 . ± .
002 0 . ± .
003 0 . ± . Y b − . ± .
002 1 . ± . − . ± .
002 1 . ± . L bolb − . ± .
003 1 . ± .
006 0 . ± .
005 1 . ± . c . ± .
005 0 . ± .
007 1 . ± . − . ± . Q . ± .
002 0 . ± .
003 1 . ± .
001 0 . ± . a Fit Parameters to equation (9) with uncertainties from Monte Carlo re-sampling. b Units: σ DM ( km s − ); kT m or kT sl (keV); L bol (10 erg s − cm − ). Fig. 4.—
The evolution of the normalization and slope of thetotal kinetic energy of the gas in the PH (filled, black points) andthe GO (open, red points) simulations. The dotted line is the self-similar prediction and the solid line is the measured evolution inthe PH simulation. the degree of turbulence is much lower, and the total en-ergy remains higher at high redshift due to the influenceof the initial entropy injection. Baryon fraction.
Figure 2e,f shows the scaling of thehot gas fraction, f ICM , with mass at z = 0 along with theredshift evolution of the slope and normalization. Forreasons discussed below, we show evolution in the PH normalization and slope at a mass of 5 × h − M ⊙ aswell as the fiducial mass of 10 h − M ⊙ .At redshift zero, the distribution of f ICM values within r in the GO simulation is simple: halos have a slightlydepleted baryon fraction, with mean f ICM = 0 .
90 , anda dispersion of 0 .
04. There is no trend with mass, asthe slope is 0 . ± . r by Crain et al. (2007) and Ettori et al.(2006). The simulation of the MareNostrum universeGottl¨ober & Yepes (2007) has twice our mass resolution,and a baryon fraction of f ICM = 0 .
92 at a virial radius, r vir , that encompasses a mean density of ∼
100 timesthe critical density. This small but significant increase isconsistent with the radial trend seen in our simulation.Our value of f ICM is lower than the 0 .
97 value measuredby Kravtsov et al. (2005) at r vir in their adiabatic AMR TABLE 2Redshift Zero PH Quadratic Mass Scalings a Signal s α α f ICM − . ± .
002 0 . ± . − . ± . Y b − . ± .
005 1 . ± . − . ± . L bolb − . ± .
012 1 . ± . − . ± . a Fits of the PH signal-mass relations at z = 0 to a quadratic,ln S = s + α ln M + α (ln M ) , with mass in units of10 h − M ⊙ . b Units: Y ( h − Mpc ); L bol (10 erg s − cm − ). TABLE 3 PH NormalizationEvolution in ln( E ( a )) a Signal β Self-Similar σ DM / T m / T sl / L bol / f ICM -0.44 0 Y / Q -0.13 N/A c -0.68 N/A a Equation 11. (Adaptive Mesh Refinement) code. That study includeda comparison of SPH and AMR simulations evolved un-der gravity only, and they note a statistically-significantoffset of ∼ r while the most massive halos are depleted by only ∼
10% relative to the GO case. A power-law form is apoor representation of the PH mean behavior of f ICM with mass, so we extend the model to a quadratic inln M and give best-fit parameters in Table 2. We presenta comparison with observed gas fractions in § GO simulation, the slope of the gas fractionremains consistent with zero at all redshifts, while themean gas fraction increases slightly, rising from f ICM = Stanek et al.
TABLE 4 PH Normalization Evolution in ln( a ) a Signal ǫ ǫ ǫ f ICM − . × − Y − . × − -0.308 -0.0141 L bol − . × − -0.786 0.477 a Equation 12. .
90 at z = 0 to 0 .
93 at z = 1. The latter effect isconsistent with the findings of Gottl¨ober & Yepes (2007)in the Marenostrum simulation, where the mean baryonfraction increases from 0 .
92 at z = 0 to f ICM = 0 .
94 at z = 1. While the mechanism responsible for this slightdrift is not fully understood, energy transfer from thedark matter to the gas during mergers may play a role(Pearce et al. 1994; McCarthy et al. 2007).The redshift evolution of the baryon fraction in the PH simulation depends on mass scale. Gas expelled fromlow-mass halos by preheating at z ∼ h − M ⊙ normalization scale, halos at z = 1 have a ∼
20% lowergas fraction compared to z = 0, and the local slope of the f ICM − M relation is also steeper at higher redshift. Theeffects of preheating dominate over the universal expan-sion at the 10 h − M ⊙ mass scale; hence the evolution ofthe normalization cannot be simply described as a powerof E ( z ). For the baryon fraction, and for other halo pa-rameters which depend strongly on the baryon fraction,we fit the evolution as a quadratic in ln( a ), with thebest-fit parameters presented in Table 4.At a higher mass scale of 5 × h − M ⊙ , the effects aremilder, and the local slope is very close to the slope of the GO model. The most massive halos are less affected bypreheating, and their baryon fractions are therefore moreappropriate to use as indicators of cosmology (Pen et al.2003; Allen et al. 2004, 2008). We note that the meangas fraction shift at 5 × h − M ⊙ in the PH modelis comparable to the 20 percent uniform prior on gasfraction applied by Allen et al. (2008) in their analysisof a Chandra sample of 42 clusters with kT > z = 1 .
1. The cosmological constraints fromthat work would thus not be strongly affected if a PH model prior on f ICM behavior were imposed.
NFW concentration.
Figure 2g,h show the behavior ofthe NFW concentration, c , derived from fits to the to-tal mass density profile of each halo. At 10 h − M ⊙ ,the mean concentration in the GO simulation, c = 3 .
3, issubstantially lower than the value of 5 . PH simulation, c = 2 .
6, is lower than the GO value. As discussed above, the PH halos have a lowerbaryon fraction than GO halos, and the increasing baryonloss at low masses has the effect of tilting the c − M re-lation so that the slope is positive rather than weaklynegative. X-ray and SZ Signals
We now turn to common bulk observed properties ofclusters: the SZ decrement, Y , and X-ray spectroscopic- like temperature, T sl , luminosity, L bol , and the emissionmeasure ˆ Q . Sunyaev-Zeldovich decrement.
From Figure 2c,e, weknow that preheated halos at fixed mass have, on aver-age, lower ICM mass fractions and higher mass-weightedtemperatures than their GO counterparts. Since the in-tegrated thermal SZ decrement, Y , is a product of thesetwo measures, we can anticipate some degree of cance-lation coming from this opposing behavior. Figure 5ashows that this cancellation is quite close to exact abovea mass scale of 2 × h − M ⊙ .The slope in the GO model is very close to the self-similar expectation of 5 /
3, but curvature in f ICM tiltsthe PH relation from a local slope of − . − . − h − M ⊙ . As with the f ICM − M relation, we fit Y to a quadratic in ln M for the PH casein order to account for the curvature and to get the bestpossible measure of the intrinsic scatter about the meanrelation. Parameters are given in Table 2.As discussed further in §
4, the dispersion about theintrinsic, mean Y − M relation is 12 ± Y − M normalization at high masses is notsensitive to cluster physics in our models, previouslyART simulations with cooling, star formation and su-pernova feedback (CSF) display a normalization drop by25% relative to the GO case Nagai (2006). At a basiclevel, the different behavior between the two studies sim-ply reflects the fact that the thermal pressure support, −∇ P/ρ , is insensitive to a multiplicative shift in gas den-sity normalization. Our models force all baryons into thehot phase, while a CSF treatment allows some fraction ofbaryons — 40% in the case of Nagai (2006) — to residein galactic sinks of stars and cold gas. Since observationsindicate that the ratio of stellar to hot gas mass declineswith increasing mass, from values near 0 . h − M ⊙ to 0 . h − M ⊙ (Giodini et al. 2009), one wouldanticipate that the observed Y − M relation to be steeperthan self-similar. A recent X-ray based estimate of the Y − M relation from a Chandra archival sample of groupsand clusters finds a slope of 1 .
75 at r over the massrange 10 − h − M ⊙ (Sun et al. 2009).Figure 5b compares the redshift evolution of the Y − M relation in the two simulations. The slope in the PH model steepens at higher redshift, and the normalizationdrifts below self-similar expectations by ∼
10% at z =1. The evolution at high mass ( ∼ × h − M ⊙ )in the PH model is similar to that of the GO model, inboth slope and normalization, since redshift z ∼
1. Atour chosen normalization mass of ∼ h − M ⊙ , theevolution departs from self-similar at z > .
5, drivenby the evolution of the baryon fraction at that scale.Like the baryon fraction, we fit the evolution of Y to aquadratic in ln( a ), with the best fit presented in Table4. We note that the PH behavior is mildly in conflictwith the CSF model evolution of Nagai (2006), whichdisplayed consistency with self-similar evolution at thealos in Millennium Gas Simulations: Multivariate Scaling Relations 9 ∼
20% level. With only 11 halos, that study could notaddress statistical differences at the level we do here.
Spectroscopic-like temperature.
We also consider thespectroscopic-like temperature, T sl , an analytic prescrip-tion derived by Mazzotta et al. (2004) to approximateobserved X-ray spectral temperatures. We find that T sl agrees well with the X-ray temperatures derived fromspectral fits to X-MAS2 mock observations (Rasia et al.2005). As seen in Figure 5c,d, the slopes in the PH and GO treatments agree well, α = 0 .
57, but at fixed mass the PH halos are ∼
40% hotter than the GO halos. The distribu-tion of the gas in the GO models contains cool, low entropycores of accreted sub-halos (Mathiesen & Evrard 2001),and these cool, dense clumps pull down the T sl measurerelative to the PH simulation. The cores in the latter casehave been effectively erased by the preheating.The redshift evolution of T sl, ( a ) in the PH simulationis very close to self-similar ( ∝ [ E ( a )] / ), and tracks wellthe mass-weighted behavior shown above (see Table 3).The evolution in the GO simulation departs dramaticallyfrom self-similar behavior at low redshifts. Bolometric luminosity and Emission measure.
The dif-ferent behaviors of the baryon fraction and temperaturein the PH and GO simulations drive X-ray luminosity dif-ferences, but the gas clumping, or emission measure, alsoplays a significant role. Figure 5e,g show the z = 0 massscalings of the bolometric luminosity, L bol , and dimen-sionless emission measure, ˆ Q . Because of the influenceof line emission, the GO scaling is less steep than the self-similar slope of L ∼ M / , and the emission measure isnearly constant with mass at a value ∼ .
5. In the PH case, the clumping factor is smaller by a factor of twoand displays a significant trend with mass. Combinedwith the f ICM behavior, the result is a suppression of L bol in the PH case by a factor of 10 at 10 h − M ⊙ , anda steepening of the slope in mass to 1 .
87, from 1 .
08 inthe GO case.Although the PH halos have a higher temperature atfixed mass, this effect is dwarfed by the decreases in f ICM and ˆ Q between the PH and GO treatments. The lower nor-malization of the ˆ Q − M relation reflects a shallower gasdensity profile in the PH case. In turn, the lower centralgas densities contribute to lowering the overall mass pro-file, driving the shift in concentrations to lower values forthe PH halos discussed above. As in the Y − M relation,the curvature in the f ICM − M relation in the PH modeldrives curvature in the L bol − M relation. Hartley et al.(2008) see this curvature in the L − T relation in the PH model, as well as in a large, local observed sample ofgalaxy clusters. We fit the L bol − M relation in the PH model to a quadratic in ln M , presenting the best fit inTable 2.The evolution of the normalization in PH is not a per-fect power of E ( a ), due to the complicated evolution of f ICM with redshift. We fit the evolution of L ( a ) to aquadratic in ln( a ) in the PH simulation, and note that itis weaker than the GO evolution. However, as shown inFigure 5f, this evolution is a function of mass. The largersolid points, for halos at 10 h − M ⊙ , show a weaker evo-lution than the evolution of the halos at 5 × h − M ⊙ ,shown by the smaller solid points. The latter halos evolvesimilarly to the GO halos, illustrating that the most mas-sive halos in the PH model are similar in structure and his- tory to the GO population. The evolution of the L bol − M relation in the PH simulation is driven mainly by the red-shift evolution of the hot gas fraction, with ˆ Q contribut-ing 10% of the decrease at z = 1. Comparison to Observations
As shown by Hartley et al. (2008), the PH simulationoffers a good match to the core-excised L − T relation oflocal clusters. Here, we briefly explore the level of agree-ment between the models and observations in this andother scaling relations. While not wishing to oversell thesimple physical treatment of the preheated model, whichis undoubtedly wrong in detail, we show below that itreproduces several scalings with quite high fidelity. Pre-heating appears to be a useful effective model. It isimportant to remember that precise comparison of ob-servations and simulation expectations requires carefulmodeling of survey selection and projection effects, andwe do not treat these effects here.Figure 6 compares the L bol − T sl relations with thecore-excised L bol − T X measurements of the local, rep-resentative REXCESS survey (Pratt et al. 2008). Theslope of the observed relation is somewhat shallower, andits scatter somewhat larger, than the PH model predic-tions, but these differences are at the level of a few tensof percent in luminosity, or less than ten percent in tem-perature. The small scatter in the core-excised L bol − T relation for REXCESS clusters (and PH halos) indicatesthat local galaxy clusters are well-behaved outside of thecore (Neumann & Arnaud 2001).At higher redshift, we consider the CCCP clusters, asubset of the 400 square degree survey which has beenfollowed-up with Chandra (Vikhlinin et al. 2008). Theseclusters range from approximately 0 . < z < .
8, so atfixed mass we scale, in a self-similar manner, the ob-served luminosities and temperatures to z = 0 . L bol − T sl relations. The latterare measured within ∆ c = 500 to be consistent with thetreatment of Vikhlinin et al. (2008). The comparison ispresented In Figure 7. The agreement is good, but themeasurement errors are larger than the REXCESS sam-ple. The scatter in the CCCP sample is larger, and thismay be due in part to the fact that cores have not beenexcised in the luminosity measurements of this sample.Figure 8 compares the mass scaling of ICM mass frac-tions at redshift zero in the models to XMM measure-ments for local clusters (Arnaud et al. 2007; Sun et al.2009). Values are measured within ∆ c = 500. To esti-mate observed cluster masses, hydrostatic estimates thatinclude a radial temperature gradient are employed. Wenote that Arnaud data agree with other observational de-terminations at high mass (Vikhlinin 2006; Giodini et al.2009), while the Sun et al. (2009) data extend to lowermass, ∼ h − M ⊙ . The PH model matches the ob-served f ICM − M relation well, with hot gas fractions afactor two less than the cosmic ratio, Ω b / Ω m at 10 anda strongly increasing trend toward higher masses. Largerobserved samples are needed to test the curvature anddegree of scatter.In Figure 9, we compare the T sl − M relation of themodels to that from Arnaud et al. (2007). The GO model,with its cool sub-halo cores, is strongly offset from thedata. The PH model relation is much closer, with a sim-ilar slope and scatter. There is, however, a consistent0 Stanek et al. Fig. 5.—
Scaling relations at redshift zero (left) and redshift evolution of fit parameters (right) for Y , T sl , L bol , and ˆ Q (top to bottom).Point and line styles are identical to Figure 2. alos in Millennium Gas Simulations: Multivariate Scaling Relations 11 Fig. 6.—
The redshift zero L bol − T sl relation for GO (open,red points), PH (filled, black points), measured within ∆ c = 500,and the core-excised L bol − T relation from the REXCESS survey(large, blue points) (Pratt et al. 2008). Fig. 7.—
The redshift z = 0 . L [0 . − . − T sl relation for PH (filled, black points), measured within ∆ c = 500, and the CCCP L [0 . − . − T relation from Vikhlinin et al. (2008) (large, bluepoints), scaled to redshift z = 0 . offset of ∼
15% in mass toward lower values in the ob-served sample. The magnitude of this offset is consistentwith the level of expected bias from hydrostatic mass es-timates, which simulations show tend to underestimatetrue masses by approximately 20% (Rasia et al. 2006;Nagai et al. 2007).Overall, the bulk X-ray properties of the PH simula-tion match the local scaling relations quite well. This isparticularly true after excising the core from the obser-vations, and after considering the mass bias introducedby hydrostatic mass estimates. Although the physics ofthe PH model is simple, it appears to provide useful rep-resentation for the behavior of the bulk of the hot ICMlying outside the cool core regions. COVARIANCE OF BULK PROPERTIES
Fig. 8.—
The redshift zero f ICM − M relation for GO (open, redpoints), and PH (filled, black points), measured within ∆ c = 500.The observations plotted are the data from Arnaud et al. (2007)(large, blue points) and from Sun et al. (2009) (green crosses). Fig. 9.—
The redshift zero T sl − M relation for GO (open, redpoints), PH (filled, black points), measured within ∆ c = 500, andthe data from Arnaud et al. (2007) (large, blue points). In this section, we explore the second moment of thehalo scaling relations. Understanding the variance atfixed mass is necessary for calculating the mass selectionproperties of signal-limited samples, and survey countstypically constrain only a linear combination of signalnormalization and variance (Stanek et al. 2006). In ad-dition, the signal covariance determines the precise struc-ture of scaling relations in signal-selected samples. Aworked example of the L − T relation expected for X-rayflux-limited samples is given by Nord et al. (2008).After defining terms, we begin by presenting the covari-ance of signals at fixed mass at the present epoch, thendemonstrate that redshift evolution in most elements isweak. We close with analysis of the mass selection prop-erties of signal pairs. Signal Covariance Matrix e µ at expansion epoch a , wedefine a symmetric covariance matrix, with elementsΨ ij ≡ h ( s i − ¯ s i ( µ, a ))( s j − ¯ s j ( µ, a )) i (14)where the mean values are determined by equation (9)and the brackets represent an ensemble average. The j th diagonal element of the covariance matrix is the j th signal variance, σ j ≡ Ψ jj . We present these measuresalong with the correlation matrix, C xy = Ψ xy / ( σ x σ y ),which expresses the covariance in normalized terms.Figure 10 presents a graphical representation of thedata comprising the covariance matrix for a subset ofsignals at redshift zero. The diagonal panels show thedistribution of signal deviations (in the natural log ofthe measured signal) about the mean for the PH (shaded)and GO (lines) treatments. Panels off the diagonal plotthe normalized deviations (( s j − ¯ s j ) /σ j vs . ( s i − ¯ s i ) /σ i )for signal pairs, with the lower and upper triangles show-ing PH and GO cases, respectively. The orientation andspread of halos in each panel determines the correlationcoefficient. For instance, the tight ellipse formed by thepopulation in the Y − f ICM panel indicates a high corre-lation coefficient between this pair of signals.The z = 0 values of the correlation coefficients, alongwith uncertainties from bootstrap resampling, are pre-sented in Table 6. As in the mean fit parameters, typicaluncertainties are on the order of ∼ Signal Variance at Fixed Mass
The assumption of log-normal variance is a common el-ement of the likelihood analysis used in cluster cosmologystudies (see references in § z = 0. To test the log-normal expec-tation, we list the normalized deviates at which each sig-nal’s ranked distribution reaches fixed percentile values,taken to be ± σ , ± σ and median/mean of a Gaussiandistribution. The difference between the listed valuesand their integer counterparts is a measure of the de-gree of local deformation from Gaussian in the frequencydistribution.While deviations from Gaussianity are apparent in es-sentially all measures, the typical percentile shifts in the PH model are only a few percent. Exceptions are a sig-nificant positive skew in the dark matter velocity dis-persion, with median location of − . σ and shifts inthe ± σ Gaussian tails to − . σ and 2 . σ , respectively.Most distributions are slightly leptokurtic, especially themass weighted temperature. Worth noting is the factthat the shapes of two important cluster selection ob-servables, Y and L bol , do not deviate by more than 0 . GO treatment, the shape of the dark mattervelocity dispersion is the same as in the PH case, showingpositive skew at the 10% level. However, the shapes ofthe hot gas properties of GO halos generally differ, to aslight degree, from the PH shapes; all measures tend tobe slightly skew negative. The differences can be sub- tle, as close inspection of the Y histograms in Figure 10confirms.Comparing the GO and PH distributions, we concludethat a log-normal approximation is a fairly accurate de-scription, but calculations demanding better than ∼ σ DM , we have preliminary evidence thatmergers drive the positive skew tail of the deviations;nearly all halos with a deviation of > σ have undergonea merger since redshift z = 0 . .
036 for f ICM to a high of 0 .
28 for L bol ,both under the GO treatment. For the PH case, L bol is highest, at 0 .
19 while σ DM is lowest at 0 . GO case, and both agree withthe scatter derived from the ensemble value presentedby Evrard et al. (2008). While the dark matter virialscaling is robust to simple physical treatments for thebaryons, (Lau et al. 2009) find that strongly dissipativebaryon physics depresses the slope of the σ DM − M re-lation by introducing mass- and redshift-dependent in-creases in halo velocity dispersion.By raising the halo sound speed, which drives the shockradius to larger values and lowers the Mach number ofinfalling material (Voit et al. 2002), preheating leads tomore thermally regular halo gas, with smaller scatterin mass-weighted temperature, T m , compared to the GO simulation. Naively, in a virialized cluster dominated byonly gravitational effects, one expects σ DM2 ∼ T m , and2 σσ DM ≃ σ T m . The values for the GO simulation are closeto this, with σ T m ∼ . σσ DM . However, the scatter in T m is much lower in the PH simulation, only 1 . σσ DM , or5 . f ICM shows the reverse behavior. In the GO simulation, the scatter is very small, 3 . PH model scatter of8 .
6% is more than twice that of the GO case. However,the scatter in this model is mass-dependent; splitting thesample at 3 × h − M ⊙ , the low-mass end has ∼ ∼
7% on the high-mass end.The opposing statistical shifts in f ICM and T m effec-tively cancel when combined to form the thermal SZsignal. Not only does the mean Y − M relation inFigure 5 agree well between the two simulations, butboth simulations have a similarly low scatter, ∼ Y and the steepslope of the Y − M relation make the thermal SZ effect, orits X-ray equivalent, an excellent mass proxy for clustersurveys. Furthermore, as discussed above, the distribu-tion of deviations in ln( Y ) at fixed mass is very close toa Gaussian distribution.In both models, the scatter in T sl at fixed mass is higheralos in Millennium Gas Simulations: Multivariate Scaling Relations 13 Fig. 10.—
Graphical representation of the data comprising the covariance matrix for the two simulations at z = 0. Diagonal panels plotthe distribution of deviations from mean mass scaling behavior, ( s i − ¯ s i ), where s i is the natural log of the i th signal. Shaded histogramsshow PH results and solid lines show GO data. Each off-diagonal panel plots the normalized deviations, ( s i − ¯ s i ) /σ i , for an ( i, j ) pair ofproperties. The lower triangle shows PH and the upper triangle shows GO behavior. than the scatter in T m at fixed mass. The difference isslight in PH , but over a factor of two in GO , a reflection ofthe larger amount of cool substructure in the latter treat-ment. The fact that T sl is sensitive to the gas physicstreatment means that opportunities exist for constrain-ing gas physics based on high-quality X-ray spectroscopyof large samples. Such a study could be provided by theproposed WFXT satellite .After NFW concentration, the X-ray luminosity hasthe highest level of scatter at fixed mass in both the PH ( σ ln L = 0 .
19) and GO ( σ ln L = 0 .
28) simulations.These values are consistent with the core extracted value, σ ln L = 0 . ± .
06, observed for REXCESS (Pratt et al.2008).
Off-Diagonal Elements of the Correlation Matrix http://wfxt.pha.jhu.edu/ There is much information about the physical processesdriving the ICM encoded in the correlation matrix ofTable 6. For cosmological studies, it would be usefulto identify pairs of cluster properties whose correlationcoefficient is insensitive to gas physics modeling. Highvalues of the correlation coefficient may point to pairs ofhalo properties that evolve on similar time scales duringmerger events. Below in § sec:imp, we show that highsignal correlations can improve halo mass selection byjoint signals considerably.Overall, the signal pair correlations tend to be higherin GO than in PH , but detailed differences warrant closerinspection. Considering the virial scaling as an anchor,we begin with consideration of correlations with σ DM .We then examine SZ and X-ray observables, and pointout extreme values for both cases.The covariance between σ DM and T m at fixed mass isan indicator of halo virialization. At redshift zero, thiscorrelation is much higher, C = 0 .
56 in GO than in PH ,4 Stanek et al. TABLE 5Scatter and Distribution of Deviates at Redshift Zero a Signal Scatter 2 . . . . PH σ DM . ± .
001 -1.66 -0.90 -0.11 0.85 2.43 PH f ICM . ± .
001 -2.14 -0.93 0.03 0.93 1.93 PH T m . ± .
002 -1.90 -0.84 -0.04 0.75 2.11 PH T sl . ± .
001 -2.17 -0.93 0.04 0.90 1.88 PH Y . ± .
002 -2.04 -0.97 0.04 0.90 1.95 PH L bol . ± .
002 -1.98 -0.98 0.004 1.00 1.94 PH ˆ Q . ± .
001 -1.89 -0.99 -0.04 0.97 2.19 PH c . ± .
008 -1.27 0.43 0.13 0.89 1.31 GO σ DM . ± .
001 -1.65 -0.88 -0.12 0.77 2.54 GO f ICM . ± .
001 -2.12 -0.98 0.03 0.99 1.90 GO T m . ± .
001 -2.23 -1.00 0.12 0.93 1.69 GO T sl . ± .
002 -2.32 -1.06 0.19 0.96 1.49 GO Y . ± .
001 -2.17 -1.00 0.10 0.95 1.75 GO L bol . ± .
003 -2.19 -1.03 0.10 0.97 1.70 GO ˆ Q . ± .
001 -2.59 -0.91 0.19 0.85 1.55 GO c . ± .
008 -2.42 -0.97 0.13 0.86 1.28 a The distribution of deviates shows locations, in terms of normalized deviates, δ/σ , at which the ranked distributions reaches the listed percentiles. Valuesfor a Gaussian distribution are listed in the first row. for each signal. Errorestimates on the scatter come from bootstrap resampling.
TABLE 6Correlation Coefficients at Redshift Zero a Signal σ DM T m T sl f ICM
Y L bol ˆ Q cσ DM − .
55 0.81 0.28 0.54 0.51 0.17 0.19 T m − T sl − f ICM -0.10 0.42 0.37 − Y − L bol − Q − c − a The redshift zero correlation coefficients, with the results fromthe PH simulation in the lower triangle and the results from the GO simulation in the upper, as in Figure 10. Uncertainties from boot-strapping resampling are on the order of 0 .
01 and are not shown. where it is 0 .
35. The halos in the GO simulation are gov-erned by gravitational effects only, so T m and σ DM excur-sions track each other closely. In PH , however, the pre-heating raises the sound speed in the halos, making thethermalization due to mergers less pronounced, therebydiminishing (though not eliminating) the coupling of T m and σ DM deviations.The ICM mass fraction behavior in the two simula-tions is quite different, as noted above, and this differ-ence is apparent in the covariance of σ DM and f ICM . Inthe PH simulation, the two properties are anti-correlated, C = − .
10 at redshift zero. In fact, this is the onlynegatively correlated pair of signal deviations exhibitedby the models. It is likely that this negative correla-tion is driven by the behavior of mergers. Halos in theearly stages of a merger will have a higher σ DM at fixedmass than the mean relation. If, during these mergerevents, the collisionless dark matter accretes faster thanthe baryons (since the extended PH gas envelopes of ac-creting satellites will be more easily ablated during themerger encounter), then the ICM mass fraction will be lo-cally depressed, driving an anti-correlation between σ DM and f ICM . We note that these effects are absent in the GO case, since the correlation of that model is positive, C = 0 .
28. The concentration and ICM emission measure aremeasures that are sensitive to formation history(Wechsler et al. 2002; Busha et al. 2007) and substruc-ture driven by merging. The correlations between theseparameters and σ DM are surprisingly modest, ∼ .
2, per-haps indicating that formation history is a more impor-tant driver, compared to recent mergers, for these mea-sures. There is not a strong sensitivity to gas physics inthe c − σ DM correlation, but the PH ˆ Q − σ DM correlation of0 .
32 is significantly larger than the GO value. In fact, theonly two signals with lower σ DM correlations in GO thanin PH are T sl and ˆ Q , both measures that are sensitive tosmall regions of cool, dense gas.Both Y and L bol have a weaker correlation with σ DM in PH than in GO . As Y ∝ f ICM T m , we expect that wecan describe the scatter in Y at fixed mass as σ Y = σ T + σ f + 2 C T f σ T σ f . (15)Examining Table 5, we see that, because the scatter in Y is largely dominated by T m in the GO model and f ICM in the PH case, the value of the correlation coefficient, C T f , is not an important factor. Our value for the scat-ter in Y is higher than that measured in the simulationsof Kravtsov et al. (2006), as they see an anti-correlationin the X-ray inferred deviations of gas mass and temper-alos in Millennium Gas Simulations: Multivariate Scaling Relations 15ature at fixed mass. In the intrinsic measures of Table 6,we find positive correlation between f ICM and T m withvalues of 0 .
42 ( PH ) and 0 .
48 (GO).It is worth noting the interesting case of C = 0 .
08 be-tween Y and σ DM in the PH simulation. Algebraically,this is unsurprising due to the positive correlation be-tween σ DM and T m and the negative correlation with f ICM . This lack of correlation suggests that SZ surveysshould produce cluster samples that unbiased with re-spect to dynamical state. Similarly, the low correlationbetween Y and ˆ Q , C = 0 .
12, shows that SZ surveysshould not be strongly biased by gas clumpiness.When metals are ignored, the bolometric luminosityscaling, L ∝ f ICM2 ˆ QT / , implies that the scatter inluminosity at fixed mass follows σ L = 4 σ f + σ Q + 14 σ T + 4Ψ fQ + 2Ψ fT + Ψ QT . (16)For the PH case, the scatter in ICM mass fraction is theprimary contributor, responsible for 90% of the variationin L bol . In contrast, f ICM variations account for only7% of the scatter in the GO case, where the majoritycontributors are variations in T sl and ˆ Q . Our results areconsistent with the work of Balogh et al. (2006), who usean analytic model to show that variation in halo structurecannot completely account for the observed variance in L bol at fixed mass.In both simulations, the correlation between T m and T sl at fixed mass is very high. Although a given halowill not have the same T m and T sl , the high correlationcoefficients indicate that the two temperature measuressimilarly trace the thermal state of a halo. The correla-tion is higher in the PH simulation than in the GO , andcool cores in the latter model also drive higher correla-tions between T sl and ˆ Q than between T m and ˆ Q . Thecorrelations between f ICM and temperature measures arenearly constant between the two simulations, with valuesin the range 0 . − T m and T sl .The highest measures of correlation are between T m and Y in the GO case (0 . f ICM and Y in the PH run (0 . Y and L bol is also large, ∼ .
7, in both treatments.This robust behavior is promising for cross-calibrationsof future, combined X-ray and SZ surveys Younger et al.(2006); Cunha (2008).Both simulations also have a significantly non-zero cor-relation coefficient between L bol and T sl , 0 .
73 in PH and0 .
67 in GO . As shown by Nord et al. (2008), the L − T scaling relation expected from X-ray flux-limited surveysis sensitive to the value of C T L , and studies of currentand future samples will be able to place limits on this cor-relation, using techniques similar to those employed byRozo et al. (2009) to constrain the correlation of massand X-ray luminosity at fixed optical richness for theSDSS maxbcg sample.
Redshift Evolution of the Signal Covariance
We plot the time evolution of the signal scatter in Fig-ure 11, with error bars come from bootstrap resampling.Going back to z = 2, few halo properties show any evo-lution with redshift. In the PH case, the scatter in baryonfraction slightly increases near z = 2, causing an increasein the scatter of Y and L . On the other hand, the scatter Fig. 11.—
Evolution of the scatter in eight bulk cluster proper-ties.
Fig. 12.—
Four pairs of signals which show little to no evolutionin the correlation coefficient. The black, solid line denotes the PH simulation, and the red, open points the GO simulation. in emission measure decreases at higher redshift. Notethat, at redshift two, there are only 62 halos in the PH simulation above our mass cut of 5 × h − M ⊙ . At red-shift zero, there is a higher scatter at the low-mass endof our mass range, as seen in Figure 2. Since the halos atredshift z = 2 are only slightly over the mass cut, theirhigher scatter may reflect this mass dependence ratherthan pure redshift evolution.Moving on to the off-diagonal elements, most pairs ofsignals show little evolution in the correlation coefficientwith redshift. Figure 12 shows four typical pairs withlittle evolution. Even as the physical density of haloschanges with redshift, we see that the interplay of Y and L or T sl and f ICM does not change. We see particularlylittle evolution in pairs of signals in the PH simulation.We do see evolution in a few signal pairs, notably be-tween σ DM and other signals in the GO simulation. Sev-eral pairs of signals – such as σ DM − T sl and f ICM − Y –6 Stanek et al. Fig. 13.—
Four pairs of signals which show some degree of evolu-tion in the correlation coefficient, particularly in the GO simulation.The black, solid line denotes the PH simulation, and the red, openpoints the GO simulation. evolve in the GO simulation, but not in the PH simulation.Furthermore, for σ DM − L bol , we see evolution in bothsimulations, but in opposite directions. The evolution ofthese pairs is shown in Figure 13.The correlation between σ DM and the SZ and X-raysignals increases towards high redshift in the GO simula-tion. We speculate that this is due to the increase in ˆ Q with time at fixed mass in the GO simulation, as shownin Figure 5. The gas in the GO simulation develops moresubstructure with time, and this tends to decouple theSZ and X-ray measurements from the dark matter ve-locity dispersion, σ DM . Although ˆ Q also increases withtime in the PH simulation, its normalization at all red-shifts is much lower than in the GO simulation. Hence,the substructure of the gas does not contribute much tothe T sl or L measures in the PH simulation, and there islittle evolution of their correlation with σ DM . Implications for Multi-Signal Mass Selection
At fixed signal, the scatter in halo mass is σ µi = σ i /α i ,where i labels the particular signal and α i is the slope ofthat signal–mass relation. From the analysis above, wecompute the mass scatter for a subset of z = 0 signalsand present the data in the first two columns of Table7. We see that Y provides the best mass selection un-der both physical treatments, with scatter ∼ PH case, the X-ray luminosity is quite good, with a10% scatter, while f ICM is the worst selector, with theweak f ICM − M slope producing a 28% scatter in mass.We will see below that this large scatter can actually beused to improve selection when paired with more precise,correlated signals.When selecting halos using multiple signals, the massvariance is Σ = ( α † Ψ − α ) − , where α is a vector ofslopes with elements α i . We consider the two-signal case,and let r ≡ C be the (traditional) correlation coefficientbetween signals 1 and 2.Given a pair of signal measurements, the mass selectionis log-normal with varianceΣ = (1 − r )( σ − µ + σ − µ − rσ − µ σ − µ ) − . (17) Fig. 14.—
For a range of correlation coefficients, we plot thedispersion in mass achieved by combining a pair of signals (1 and2) against the ratio of their dispersions. Different lines vary thecorrelation coefficient r between the pair of signals at fixed mass.Given a measurement of the first signal, a second measurementalways improves the mass scatter, except in the degenerate case, σ µ /σ µ = 1 and r = 1. While negatively correlated signals yieldmore improved mass selection than positively correlated pairs, theimprovement becomes independent of sign when the second signalis much noisier than the first, σ µ /σ µ ≫ The improvement in mass selection to be gained by apair of signals, relative to a single measurement, is dis-played in Figure 14. Here, we plot Σ /σ µ as a functionof the scatter ratio ( σ µ /σ µ ) for several values of thecorrelation coefficient. If the second signal is a bettermass proxy, σ µ ≪ σ µ , then it dominates the selec-tion. Combining signals with comparable mass selection, σ µ ∼ σ µ , can result in anything from no improvementin the degenerate case ( r →
1) to the √ r = 0) to dramatic improvementin the anti-correlated case ( r → − σ µ ≫ σ µ ,one still achieves significant improvement in mass selec-tion as long as the signal pair correlation is large in anabsolute sense.We evaluate equation (17) for a set of observable signalpairs and present the resultant mass scatter values in theright-hand columns of Table 7. In the PH simulation,the correlation coefficient between L and Y is r = 0 . . L bol (11%)or only Y (7 . L bol and Y , future multiwaveband surveys that join SZand X-ray detections should have a strong mass selectionproperties.Finally, we note that the best mass selection comes inthe PH case from combining Y with f ICM . The combina-tion of a strong correlation, r = 0 .
88, and, especially, therelatively large degree of scatter in mass at fixed f ICM (0 .
28) produce a scatter in mass of only 4 .
1% when mea-surements of these two signals are joined. We cautionthat this analysis does not take measurement errors intoaccount, and it is clear that high data quality will beimportant to realize this level of mass selection.alos in Millennium Gas Simulations: Multivariate Scaling Relations 17
TABLE 7Mass Scatter at Redshift Zero a Signal σ DM T sl f ICM
Y L bol σ DM . \ .
12 0 .
12 0 .
12 0 .
075 0 . T sl .
10 012 \ .
38 0 .
35 0 .
050 0 . f ICM .
11 0 .
12 0 . \ .
12 0 .
054 0 . Y .
062 0 .
069 0 .
041 0 . \ .
075 0 . L bol .
09 0 .
10 0 .
09 0 .
069 0 . \ . a The diagonal elements give the mass scatter for individual signals forthe PH \ GO cases. Off-diagonal elements give the halo mass scatter forsignal pairs, with the PH case in the lower triangle and the GO case inthe upper, as in Figure 10 . CONCLUSION
We analyze scaling relations for multiple properties ofmassive halos taken from a pair of gas dynamic simula-tions with different gas physics treatments. Our sam-ples contain tens of thousands of halos with masses M > × h − M ⊙ at redshifts z .
2. The phys-ical treatments of gravity only ( GO ) or preheating ( PH )are both highly idealized, but we show that the latter re-produces the scaling relation behavior of core-extractedX-ray measures of local clusters.The dark matter velocity dispersion scales with massand redshift according to self-similar expectations, indi-cating that the virial theorem is respected regardless ofgas treatment. However, the gas behavior in both treat-ments differs from self-similarity. The deviations in the GO case tend to be small and are related to the mass-limited sample definition.The entropy injection of the PH model drives more sub-stantial deviations from self-similar scaling. At z = 0,the ICM mass fraction varies with mass in a mannerroughly consistent with observed measurements of localclusters. We find that f ICM requires a quadratic fit inlog-mass, and mild curvature in the logarithmic scalingsof Y and L bol as a function of mass is also evident. Whilethe ICM in PH halos is lower in mass compared to the GO case, it is also slightly hotter. The effects on f ICM and T m nearly cancel when combined to form the thermal SZsignal, leading to nearly identical Y – M scaling relationsabove 3 × for both PH and GO cases at low redshift.The ICM mass fraction at fixed mass declines weaklywith redshift, by 10% in 5 × h − M ⊙ halos at z = 1and with larger reductions at lower masses. While Chan-dra observations of optically-selected clusters in the RCSsurvey show evidence of reduced gas fractions at z = 1(Hicks et al. 2006), further work is needed to address thequantitive level of agreement between the observationsand PH model expectations. The PH baryon fraction evo-lution drives departures from self-similar predictions inthe Y – M and L bol – M relations; slopes tend to steepenslightly and the normalization at 10 h − M ⊙ is lowerthan self-similar expectations at high redshift.We present the first systematic investigation of prop-erty covariance in the computational samples of massivehalos. The data generally support a multivariate log-normal form for the joint distribution of signals at fixedmass and redshift. All measures depart somewhat froman exact gaussian form, but the deviations in gas mea-sures are smaller for the PH model due to the suppressionof substructure caused by the preheating. Most signalpairs exhibit positive correlations, with the lone excep- tion of − . σ DM and f ICM in the PH case. Thethermal SZ signal displays a robust 13% scatter that isstrongly correlated with variations in both ICM gas massand temperature, with f ICM dominating in the PH caseand T m being more important in the GO treatment.Combining multiwavelength observations offers an op-portunity to improve selection of clusters by their intrin-sic mass. We derive the mass variance of signal pairs andshow that combining strongly correlated signals alwaysimproves mass selection, even when one of the signals byitself is a comparatively poor mass proxy. The combina-tion of thermal SZ and ICM mass fraction in the PH caseselects halo mass with just 4% intrinsic rms scatter.Identifying the root causes behind the terms inthe covariance matrix is a considerable task thatwe leave for future work. Mergers (Roettiger et al.1997; Ricker & Sarazin 2001) and assembly bias(Boylan-Kolchin et al. 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