Testing X-ray Measurements of Galaxy Cluster Outskirts with Cosmological Simulations
Camille Avestruz, Erwin T. Lau, Daisuke Nagai, Alexey Vikhlinin
aa r X i v : . [ a s t r o - ph . C O ] O c t T HE A STROPHYSICAL J OURNAL , SUBMITTED
Preprint typeset using L A TEX style emulateapj v. 8/13/10
TESTING X-RAY MEASUREMENTS OF GALAXY CLUSTER OUTSKIRTSWITH COSMOLOGICAL SIMULATIONS C AMILLE A VESTRUZ , E
RWIN
T. L AU , D AISUKE N AGAI , AND A LEXEY V IKHLININ Department of Physics, Yale University, New Haven, CT 06520, U.S.A.; [email protected] Yale Center for Astronomy & Astrophysics, Yale University, New Haven, CT 06520, U.S.A. Department of Astronomy, Yale University, New Haven, CT 06520, U.S.A. Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, U.S.A.
The Astrophysical Journal, submitted
ABSTRACTThe study of galaxy cluster outskirts has emerged as one of the new frontiers in extragalactic astrophysicsand cosmology with the advent of new observations in X-ray and microwave. However, the thermodynamicproperties and chemical enrichment of this diffuse and azimuthally asymmetric component of the intra-clustermedium are still not well understood. This work, for the first time, systematically explores potential observa-tional biases in these regions. To assess X-ray measurements of galaxy cluster properties at large radii ( > R c ),we use mock Chandra analyses of cosmological galaxy cluster simulations. The pipeline is identical to thatused for
Chandra observations, but the biases discussed in this paper are relevant for all X-ray observationsoutside of R c . We find the following from our analysis: (1) filament regions can contribute as much as 50%at R c to the emission measure, (2) X-ray temperatures and metal abundances from model fitted mock X-rayspectra in a multi-temperature ICM respectively vary to the level of 10% and 50%, (3) resulting density pro-files vary to within 10% out to R c , and gas mass, total mass, and baryon fractions vary all to within a fewpercent, (4) the bias from a metal abundance extrapolated a factor of 5 higher than the true metal abundanceresults in total mass measurements biased high by 20% and total gas measurements biased low by 10% and(5) differences in projection and dynamical state of a cluster can lead to gas density slope measurements thatdiffer by a factor of 15% and 30%, respectively. The presented results can partially account for some of therecent gas profile measurements in cluster outskirts by e.g., Suzaku . Our findings are pertinent to future X-raycosmological constraints with cluster outskirts, which are least affected by non-gravitational gas physics, aswell as to measurements probing gas properties in filamentary structures.
Subject headings: cosmology: theory — clusters: general — galaxies — methods : numerical — X-rays:galaxies:clusters INTRODUCTION
Clusters of galaxies are the largest gravitationally boundobjects in our universe. These objects are massive enoughto probe the tension between dark matter and dark energyin structure formation, making them powerful cosmologicaltools. Cluster-based cosmological tests use observed clus-ter number counts, the precision of which depends on howwell observable-mass relations can constrain cluster masses.Finely tuned observable-mass relations require an understand-ing of the thermal and chemical structure of galaxy clustersout to large radii, as well as an understanding of systematicbiases that may enter into observational measurements of theintracluster medium (ICM).A new area of study lies in studying the ICM in the out-skirts of clusters, a region where the understanding of ICMphysics remains incomplete (e.g., see Reiprich et al. 2013, foran overview). Cluster outskirts can be considered to be a cos-mic melting pot, where infalling substructure is undergoingvirialization and the associated infalling gas clumps are beingstripped due to ram pressure, depositing metal-rich gas in thecluster atmosphere, changing thermal and chemical structureof the ICM. Measurements at these regions will allow us toform a more complete assessment of the astrophysical pro-cesses, e.g. gas stripping and quenching of star formation ininfalling galaxies, that govern the formation and evolution ofgalaxy clusters and their galaxies.Gas properties in cluster outskirts are also more suitable for cosmological measurements; the effects of dissipative non-gravitational gas physics, such as radiative cooling and feed-back, are minimal in the outskirts compared with the innerregions. Gas density measurements in cluster outskirts willalso help constrain the baryon budget. While recent X-raycluster surveys have provided an independent confirmation ofcosmic acceleration and tightened constraints on the nature ofdark energy (Allen et al. 2008; Vikhlinin et al. 2009), the bestX-ray data, circa 2009, were limited to radii within half of thevirial radius, and cluster outskirts were largely unobserved.Recent measurements from the
Suzaku
X-ray satellite havepioneered the study of the X-ray emitting ICM in clusteroutskirts beyond R c (e.g., Bautz et al. 2009;Reiprich et al. 2009; Hoshino et al. 2010; Kawaharada et al.2010). These measurements had unexpected results in bothentropy and enclosed gas mass fraction at large radii. En-tropy profiles from Suzaku data were significantly flatterthan theoretical predictions from hydrodynamical simula-tions (George et al. 2009; Walker et al. 2012), and the en-closed gas mass fraction from gas mass measurements ofthe Perseus cluster exceeded the cosmic baryon fraction(Simionescu et al. 2011). These results suggest that gasesousinhomogeneities might cause an overestimate in gas densityat large radii.
Suzaku observations and subsequent studiesdemonstrated that measurements of the ICM in cluster out-skirts are complicated by contributions from the cosmic X-ray background, excess emission from gasesous substruc-tures (Nagai & Lau 2011; Zhuravleva et al. 2013; Vazza et al. Avestruz et al.2013; Roncarelli et al. 2013), and poorly understood ICMphysics in the outskirts. However, analyses of cluster outskirtswith both
Planck
Sunyaev-Zel’dovich and
ROSAT
X-ray mea-surements found otherwise (Eckert et al. 2013a,b). The mys-teries surrounding these X-ray measurements of cluster out-skirts are still unsettled due to a number of observational sys-tematic uncertainties, such as X-ray background subtraction.The latest deep
Chandra observations of the outskirts ofgalaxy cluster Abell 133, a visually relaxed cluster with X-ray temperature T X = 4 . R c .To address potential biases that could affect observedICM properties at large radii, we use mock Chandra ob-servations from high-resolution cosmological simulations ofgalaxy clusters. The goal of this paper is to characterize theproperties of diffuse components of the X-ray emitting ICM inthe outskirts of galaxy clusters, and to assess the implicationsfor X-ray measurements. Our analysis focuses on the effectsthat come from a combination of limited spectral resolutionand low photon counts in cluster outskirts, where spectral con-tributions from metal line emissions become more significantthan those from thermal bremsstrahlung emissions.Our approach is similar in spirit as in Rasia et al. (2008)which pioneered the study of systematics on abundance mea-surements on the ICM by analyzing the spectral propertiesof mock plasma spectra generated with XSPEC. We extendthe study of XSPEC generated mock spectra to the outskirtregions. Our results are valid for all observations of spectrafrom plasma with a multicomponent temperature.Specifically, we look at the accuracy of metal abundancemeasurements from spectral fitting and their consequential ef-fects on the measurements of projected X-ray temperature,emission measure, deprojected temperature, gas density, gasmass, and hydrostatic mass derived from these profiles. Weexplore potential biases by (1) measuring contributions ofclumps and filaments to emission measurements, (2) testingthe modeling of bremsstrahlung and metal line emissions inlow density regions, and (3) checking for line of sight depen-dence in the density slope in cluster outskirts and differencesin observed ICM profiles for an unrelaxed cluster.This work will be useful in assessing the robustness of X-ray measurements beyond R c , as we test both the spectralfitting of X-ray photons in low density regions, and the valid-ity of fitting formulae that are used to reconstruct ICM pro-files.Our paper is organized as follows: in Section 2 we de-scribe the simulations we used and the mock Chandra analy-sis pipeline. We present our results in Section 3, and give oursummary and discussion in Section 4. METHODOLOGY
Cosmological simulations
We use a sample of high-resolution cosmological simu-lations of cluster-sized systems from Nagai et al. (2007a,b,hereafter N07) that assumes a flat Λ CDM universe with Ω m =1 - Ω Λ = 0 . Ω b = 0 . h = 0 .
7, and σ = 0 .
9. The Hub- ble constant is defined as 100 h km s - Mpc - , and σ is themass variance within spheres of radius 8 h - Mpc. We simu-late these clusters with the Adaptive Refinement Tree (ART) N -body+gas-dynamics code (Kravtsov 1999; Kravtsov et al.2002; Rudd et al. 2008), an Eulerian code with spatial andtemporal adaptive refinement, and non-adaptive mass refine-ment (Klypin et al. 2001).Our simulations include radiative cooling, star formation,metal enrichment, stellar feedback, and energy feedback fromsupermassive black holes (see Avestruz et al., in prep. formore details). Briefly, we seed black hole particles with aseed mass of 10 h - M ⊙ in dark matter halos with M c > × h - M ⊙ , and allow the black holes to accrete gas ac-cording to a modified Bondi accretion model with a densitydependent boost factor (Booth & Schaye 2009). Black holesfeed back a fraction of the accreted rest mass energy in theform of thermal energy, ∆ E fb = ǫ r ǫ f ∆ M BH (1)where ǫ r = 0 . ǫ f = 0 . z = 0 blackhole mass.We focus on two X-ray luminous clusters from the N07sample to study the effects of metallicity, line of sight, and dy-namical state on X-ray measurements in cluster outskirts. Thecore-excised X-ray temperatures of CL6 and CL7 are T X =4 .
18 and T X = 4 .
11, respectively. The cluster mass and radiiare M c = 1 . × h - M ⊙ and R c = 626 . h - kpc forCL6, and M c = 1 . × h - M ⊙ and R c = 618 . h - kpcfor CL7. Each cluster is selected from simulation box withsize L box = 80 h - Mpc, run on a 128 uniform grid with8 levels of refinement, and has peak spatial resolution of ≈ . h - kpc. The corresponding dark matter particle massin our regions of interest is 2 . × h - M ⊙ . CL6 is an unre-laxed cluster that experienced a 1:1 major merger at z ≈ . z <
1. We refer the reader to N07 for furtherdetails.
Mock
Chandra
Pipeline
Using our simulated cluster sample, we create realisticmock X-ray maps of simulated clusters by (1) generating anX-ray flux map, and (2) convolving the flux map with the
Chandra response files to create photon maps. Below we out-line our mock
Chandra pipeline, where details can be foundin Nagai et al. (2007b).
X-ray Flux maps
We generate X-ray flux maps for each of the simulatedclusters along three orthogonal line-of-sight projections. Tocalculate the flux map, we project X-ray emission from hy-drodynamical cells along each line of sight within 3 × R vir ( ≈ × R c for the clusters in this work) of the cluster cen-ter. The X-ray emission as a function of energy from a singlehydrodynamical cell of volume ∆ V i is given as, j E , i = n e , i n p , i Λ E ( T i , Z i , z ) ∆ V i , (2)where n e , i , n p , i , T i , and Z i are respectively the electron num-ber density, proton number density, temperature, and metal-licity in that cell volume. We compute the X-ray emissiv-ity, Λ E ( T i , Z i , z ), using the MEKAL plasma code (Mewe et al.1985; Kaastra & Jansen 1993; Liedahl et al. 1995) and thesolar abundance table from Anders & Grevesse (1989). Wemultiply the plasma spectrum by the photoelectric absorp-tion corresponding to a hydrogen column density of N H =2 × cm - .We include results from four different flux maps: (1) Z sim ,(2) Z . , (3) Z . , and (4) Z . T X , . We generate the Z sim flux map from X-ray spectra using all hydrodynamic val-ues directly output from our simulation runs. To isolatethe effects of metallicity on our simulated spectra, we use Z i = 0 . Z ⊙ for all i -th cell elements in our Z . flux mapsand a constant Z i = 0 . ⊙ for all i -th cell elements in our Z . flux maps. Current observations do not yet probe metalabundance profiles in cluster outskirts, and often extrapo-late the abundance at radii beyond R c . Given the typicalrange of metallicity that observers have found near R c ( Z ≈ . ⊙ ) (e.g., Matsushita et al. 2007; Leccardi & Molendi2008; Komiyama et al. 2009; Werner et al. 2013), we choosethese two values for Z i to bracket a potential range of metal-licities in the outskirts. The order of magnitude differenceprobes how X-ray measurements of the ICM and inferredquantities might be affected by an incorrect extrapolation ormis-measurement of both a metal rich and metal poor envi-ronment.We remove the effects of a multi-temperature medium byfixing both the metal abundance to Z i = 0 . Z ⊙ , and the tem-perature to T X = 1 keV in the Z . T X , map, which is a char-acteristic temperature in the outskirts of a galaxy cluster in ourprobed mass range. Mock
Chandra
Photon Maps and Spectra
To simulate mock
Chandra data, we convolve the emissionspectrum with the response of the
Chandra front-illuminatedCCDs ACIS (both the
Chandra auxiliary response file (ARF)and redistribution matrix file (RMF)) and draw a number ofphotons at each position and spectral channel from the corre-sponding Poisson distribution. The photon maps for our mock
Chandra data has an exposure time of 2 Msec, comparable tothe 2 . Chandra observations of Abell 133. Sincewe are interested in probing physical causes for biased X-raymeasurements, we do not include any simulated backgrounds.From our mock photon maps, we generate images in the0 . - ∼ × - erg s - cm - in the 0 . - ∼ . × erg s - for clustersat z obs = 0 .
06, the observational redshift of our clusters. Anyremaining small-scale structures fall below our clump detec-tion limit. The total gas mass in excluded clumps is no morethan a few percent of the total enclosed gas mass.X-ray emission in cluster outskirts also receive azimuthallyasymmetric contributions from filamentary structures. Sim-ilar to the analysis of X-ray emission in A133 outskirts, af-ter clump removal, we identify filamentary regions as annularsections with an average flux that exceeds the average fluxwithin the entire annulus. We exclude the filamentary contri-bution from our final analysis. We include a sample imagemap with identified clumps and filaments in Figure 1.From the mock
Chandra photon map, we extract a grouped spectrum from concentric annular bins centered on the mainX-ray peak. Annular regions are defined with r out / r in = 1 . r out and r in are the outer and inner edges of the annularbin, respectively. The spectra exclude photons from regionscorresponding to clumps and filaments. For each spectrum,we perform a single-temperature fit in the 0 . -
10 keV bandusing XSPEC version 12.8 with the MEKAL model plasmacode and a C-statistic fit (Cash 1979) to extract the projectedX-ray temperature T X , proj ( r ) and abundance Z proj ( r ). We usethe same Chandra
ARF and RMF files to fit a model spectrumto the mock photon spectra. Temperature, metal abundance,and spectral normalization are the only free parameters in thefit.
Analysis of the ICM
To compare with observed
Chandra data, we reconstructthe emission measure profile and repeat the deprojection anal-ysis by Vikhlinin et al. (2006). This analysis reconstructs thespherically averaged density and temperature profiles of theintracluster medium with 3D analytic models. We also esti-mate the total cluster mass assuming hydrostatic equilibriumfrom the reconstructed 3D density and temperature profiles.To calculate the projected emission measure profile
EMM = R n e n p dl , we use two main input ingredients: (1) the az-imuthally averaged X-ray surface brightness profile XSB , and(2) the projected 2D X-ray temperature T X , proj and metallic-ity profiles Z proj described in Section 2.2.2. We can relate thedeprojected EMM to the two ingredients as follows,
XSB = Z dV n e n p Λ ( T , Z ) (3)We show the emission measure profile for CL7 along the y-axis, where filamentary features are most prominent in Fig-ure 2, and the corresponding effect of clump and filament re-moval from the analysis. The solid blue line corresponds tothe emission measure profile that include clumps and filamen-tary substructure, and dotted line corresponds to the emissionmeasure profile that excludes the substructure. The bottompanel shows the ratio between the profile with substructureto the profile without, and illustrates the extent of the excessemission in cluster outskirts. For this cluster, the excess of thebest-fit profile can be as much as a factor of 2 at R c to a fac-tor of 3 at 3 × R c . The errorbars correspond to the binneddata, and the excess in individual bins are as high as a fac-tor of 6 in the filamentary regions. Excess emission measurefrom a substructure is visible at R ≈ . - r out / r in = 1 .
1, giving us a count rate in each annularbin. We use the projected X-ray temperature and metal-licity to calculate the MEKAL model plasma emissivity, Λ ( T X , proj ( r proj ) , Z proj ( r proj ) , z ). With the relation given by Equa-tion (3), we use interpolated values of the model plasma emis-sivity to convert the count rate into a projected emission mea-sure.The deprojection process is fully described inVikhlinin et al. (2006), and can be summarized as fol-lows. The analytic form of our 3D gas density model is given Avestruz et al. F IG . 1.— Top: Mock Chandra maps of CL6 in 0 . - Chandra maps of CL7 in0 . - R [kpc]110 f E MM −5 −4 −3 −2 E MM [ c m k p c − ] R c R c R c R c R c R c allno clumps, no filaments F IG . 2.— We show the effects of clump and filament removal on the emis-sion measure profile for a sample projection for CL7 in the y direction. Pro-files with both clumps and filaments are shown in the solid lines, no clumpsin the dash-dotted lines, and no clumps nor filaments in the dotted lines. Theratio between solid (dash-dotted) and dotted is shown in the excess panel.Substructure in this cluster can boost the best fit emission measure by upto a factor of 3, and binned emission measure by up to a factor of 6 in thefilamentary regions. by, n p n e = n (cid:0) r / r c (cid:1) - α (cid:0) + r / r c (cid:1) β - α / (cid:0) + r γ / r γ s (cid:1) ǫ / γ + n (cid:0) + r / r c (cid:1) β (4)This is a modification of the β -model that allows for separatefitting of the gas density slope at small, intermediate, and largeradii.The analytic form of our 3D temperature profile is a prod-uct of two terms that describe key features of in observed pro-jected temperature profiles (Vikhlinin et al. 2005), T ( r ) = t ( r ) × t cool ( r ) (5)The first term takes into account the temperature decline atlarge radii with a broken power law with transition radius, r t , t ( r ) = T (cid:0) r / r t (cid:1) - a (cid:0) + ( r / r t ) b (cid:1) c / b (6)with a and b as free parameters. The second term modelsthe temperature decline in the central region due to radiativecooling, t cool = (cid:0) r / r cool (cid:1) a cool + T min / T (cid:0) r / r cool (cid:1) a cool + , (7)We project the 3D model of the temperature profile, weightingmultiple temperature components using the algorithm fromVikhlinin et al. (2006). We then fit the projected model to theprojected X-ray temperature from our mock data, fitting in theradial range between 0 . ≤ r / R c ≤ M HSE ( < r ) = - rT ( r ) µ m p G (cid:18) dln ρ g dln r + dln T dln r (cid:19) . (8)where µ ≈ .
59 is mean molecular weight for the fully ionizedICM, m p is the proton mass, and we have set the Boltzmannconstant to be unity such that the temperature T ( r ) is in unitof energy. RESULTS
To test the MEKAL modeling of bremsstrahlung and metalline emissions in low density regions, we perform the XSPECfit with a fixed metal abundance parameter: Z fix = 0 . Z ⊙ forthe Z . map, and Z fix = 0 . Z ⊙ for the Z . map. The metal-licity in the Z . map is close to the simulation metallicity incluster outskirts, but we use the Z . map to also check theeffects of a possible scenario where a higher abundance ofmetals sits in cluster outskirts. We then compare results fromeach Z fix case with the Z fit case, where we perform the fit forall three parameters.To eliminate the effects of a multi-temperature medium, wealso perform the XSPEC fit on our Z . T X , map. We gen-erate ICM profiles using a the best-fit metal abundance pa-rameter and fixed temperature, and compare with the profilesresulting from the known input metal abundance and temper-ature value. Recovered temperature and density profiles with knownabundance measurements
In this section, we check for consistency in gas density mea-surements from spectral fitting with the XSPEC fit performedusing only two free parameters: projected X-ray temperatureand normalization of fit. We use the Z . and Z . photonmaps for this experiment, and fix the abundance parameter tothe true abundance of the map.Figure 3 shows the best-fit value for the projected X-raytemperature for each map. In cluster outskirts, the best fit X-ray temperature for the Z . maps is systematically higherthan that of the Z . map, and the gas density differs by ∼ true temperature of the two mapsin each radial bin is identical, their measurements differ upto ∼ R ≈ × R c ) in Fig-ure 4. Compared to the spectrum from the Z . photon map(red points), the spectrum from the Z . map (blue points) hasa much lower emission from metal line cooling, and thereforeit has a suppressed peak in the energy spectrum. The solidlines show the best fit model spectrum for each map.With a metal abundance parameter fixed to the true valueof Z = 0 . ⊙ , the model spectrum overshoots the peak ofthe mock photon spectrum at the Fe-L complex between0.8 and 1.4 keV, and the best fit X-ray temperature is T X = 0 . ± . Z = 0 . ± . ⊙ ) and the X-ray temperature ( T X =0 . ± . Z . T X , map. This test isolates biases due to density inhomogeneities,lower photon counts, and instrumental response. For a mapwith uniform temperature, the fitting procedure well recoversthe input constant metallicity, and does not exhibit a system-atic decline. The best fit projected X-ray temperature is alsowell recovered to the fixed input value.The suppressed peak in the mock photon spectrum in Fig-ure 4 can be attributed to a high temperature phase of gas inthe outskirts. For gas with an average temperature of T X ≈ Z = 0 . Z ⊙ to better min-imize the residuals in the fitting procedure. If we were toincrease the X-ray temperature in the model spectrum, theresidual at the peak would improve, but at the expense of in-creasing the residual values at other energies.On the other hand, for the Z . spectra in Figure 4, thereare fewer photons from metal line cooling. Hence, most ofthe spectral features are hidden in Poisson noise. Becauseof the suppressed spectral features, the fitting procedure pro-duces a best-fit X-ray temperature and normalization that willbetter minimize the residuals near the peak of the spectrumas opposed to at energies with much fewer counts. To mini-mize the residuals near the peak, the fitted X-ray temperaturemust then be higher to harden the model spectrum, which iswhy the best fit projected X-ray temperature for the Z . mapis systematically higher in the outskirts, where photon countsare low. Testing metal abundance measurements in clusteroutskirts
The robustness of metal abundance measurements in thelow density outskirts is not well understood. In this section,we compare the abundance measurements from spectral fit-ting to the true abundance from our photon maps. We alsodiscuss the effects on the projected X-ray temperature, pro-jected emission measure and the reconstructed three dimen-sional gas density profile. We begin by comparing variationsin projected X-ray temperature T X , proj and abundance mea-surements Z proj from our Z . and Z . photon maps, wherewe have set the input metal abundance to constant values.Figure 6 compares ICM profiles from our Z . and Z . mock X-ray flux maps of CL7, projected along the z -axis.Results along other axis projections are qualitatively similar.The top left panel shows the metal abundance measurementsretrieved from our mock spectra, fitting the mock spectra toMEKAL model spectra using XSPEC. We define f Z fit / Z fix asthe ratio between the best fit abundance parameter to the trueabundance of our photon map. The f Z fit / Z fix profile shows thatthe while XSPEC fitting procedure recovers a Z fit value that iswithin a factor of 2 of the true input abundance of the photon Avestruz et al. R[kpc]0.91.01.11.2 f T x , p r o j T x , p r o j [ k e V ] R c R c Z :Z fix =Z true Z :Z fix =Z true −30 −29 −28 −27 −26 ρ g [ g c m − ] R c R c R [kpc]0.81.01.2 f ρ g F IG . 3.— (Left) The projected X-ray temperature from spectral fitting and (right) the corresponding deprojected 3-D density profile for CL7. Spectral fits useda fixed abundance parameter set to the true abundance of the photon map. Mock maps were generated with constant fixed metallicity ( Z . = 0 . Z ⊙ in red solidand Z . = 0 . Z ⊙ in blue dash-dotted). The black line in the f T X and f ρ g axes is the ratio between the profile obtained using a fixed Z . and the profile usinga fixed Z . . The black circles indicate the ratio between the T X , proj data points. Spectral fitting of two maps with identical gas density and temperature, butdiffering abundances, yield recovered density profiles that differ by up to 20%. map, the abundance recovery is significantly worse at largerradii.For the Z . map, the abundance measurements has a sys-tematic decline from the true value at large radii R & . R c .As discussed in Section 3.1, the metal abundance parame-ter adjusts to compensate for the multi-temperature mediumwhen we perform a single-temperture spectral fit . Thus, thebest fit abundance parameter will be smaller than the trueabundance in order to minimize the residual around the sup-pressed emission lines. However, if there are too few photonsfrom metal line cooling, the poisson errors become large, andtherefore this systematic effect is less apparent since the sta-tistical errors dominate. This is why the last two radial binsdo not show underestimation in abundance measurements.For the Z . map, the photon counts around the peak of thespectrum (0 . - Z . spectrum (See Figure 4). In addition to the intrinsi-cally less pronounced metal lines, the low photon count makesit more difficult for the fitting routine to discern between con-tributions from metal line cooling and thermal bremsstrahlungemissions, leading to more scatter in the abundance measure-ments shown in Figure 6. Note that similar scatter is seen inthe last two radial bins from the Z . map, which also has avery low photon count.The top right panel of Figure 6 shows the projected X-ray temperature profile (red and blue points for the respectiveabundances) from spectral fitting, and the corresponding bestfit projected 3D temperature model (thin lines of the samecolor). We measure the relative bias in T X , proj as the ratio ofthe projected X-ray temperature measured with abundance asa free parameter (the profile in Figure 6 compared with the X-ray temperature measured with the abundance parameter fixedto the true abundance of the map (see Figure 3). The X-raytemperature measurements from fits with abundance as a free parameter ( Z = Z fit ) vary up to ∼
10% of the measurementswith a fixed abundance parameter ( Z = Z true ). Note, the rela-tive bias in projected X-ray temperature from spectral fittingis in the same direction as that of the abundance measurement.The cause of the correlation in the relative bias of both theabundance and X-ray temperature becomes evident when wecompare model spectra (see Figure 7). When the best fit abun-dance is higher (lower) than the true abundance, the peak ofthe model spectrum will sit above (below) the mock spectrum,and a higher (lower) X-ray temperature is necessary to harden(soften) the spectrum to minimize the residuals. Note, whilefixing the metal abundance parameter to a value that is fivetimes the true value significantly biases the value of the bestfit X-ray temperature parameter, it does not significantly in-crease the error in the X-ray temperature parameter. The X-ray temperature fit is mostly sensitive to shape of the contin-uum, and most of the residual contributions to the error inX-ray temperature come from energy bins outside of the Fe-Lcomplex. Here, the model spectra are still within the errorbarsof the mock data.An interesting feature to note in the projected X-ray tem-perature profile is that the projected 3D temperature modelfits the data points well only out to R . R c . While the best-fit model profile is within the errorbars of the mock data, itdoes not capture the emerging trend of the decreasing slopeof the projected X-ray temperature at larger radii. The de-parture from the smooth temperature model profile is indica-tive of the complicated, inhomogeneous temperature structurein cluster outskirts. The smooth temperature model does nothave enough degrees of freedom to account for the bumpierfeatures of the T X , proj data points, indicating that perhaps amodified fitting formula would be appropriate for total massmeasurements outside of R c .Note, the errorbars at large radii for the metal abundance C o un t s [ k e V − ] Z =0.21 ±0.03, T x =0.760 ±0.016Z =0.50, T x =0.815 ±0.012Z =0.05, T x =0.944 ±0.027 R e s i d a l F IG . 4.— Energy spectra from the Z . (red) and Z . photon maps corre-sponding to the third to the last radial bin in Figure 3 ( R ≈ × R c ). Spectraare normalized to the size of the energy bin. The solid lines show the best fitmodel spectra with the abundance parameter fixed to the true abundance ofthe photon map. The dotted line corresponds to the best fit model spectrumallowing both the abundance and temperature values to vary. The legend la-bels the corresponding metallicity and best fit projected X-ray temperature.We show the residual value between the model and our mock data in thelower panel. The statistically significant difference between all of the best fitX-ray temperature values is due to a degeneracy in fitting a multi-temperaturemedium with a single temperature fit. The best fit abundance parameter forthe Z . map in this bin is lower than the true abundance to compensate forthe temperature that has been biased low. Note, the lower metal abundancein the Z . map produces fewer photons from metal line cooling, leading tosuppressed spectral features. There are 18 ,
800 counts in the bin from the Z . map, and 43 ,
000 counts in the bin from the Z . map. measurements are not as good as those of the X-ray tempera-ture measurements because the abundance fitting requires anorder of magnitude more photons to have the same statisticaluncertainty as the X-ray temperature. The metal abundance isonly affected by line emission contributions as opposed to theX-ray temperature, which also has contributions from thermalbremsstrahlung.From Equation (3), we can see that for the same X-ray sur-face brightness XSB at a given radius, a higher (lower) gasdensity is necessary to compensate for a lower (higher) X-ray emissivity Λ . The bottom two panels in Figure 6 showthe corresponding projected emission measure and gas den-sity for the same photon maps as in the top two panels. Theprojected emission measure varies to 20% at all radii, and thegas density to a level of 15%.We note that the apparent enhanced bias in projected emis-sion measure at intermediate radii for the Z . map, shown inred, is due to effects from integrating over the line of sight.The density profile measured with a fixed abundance param-eter is steeper at intermediate radii than the density profilemeasured with the best fit values for the abundance parame-ter. Hence, along an equal line of sight, there is a relativelylower emission measure in the fixed abundance case than inthe best fit abundance case.In summary, the relative bias in abundance and projected X-ray temperature are correlated, and vary up to 50% and 10%, −2 Z p r o j [ Z ⊙ ] R c R c R c R c R c R c xyz10 R[kpc]012 f Z f i t / Z f i x F IG . 5.— The best-fit projected metallicity along three different projectionsof CL7. Mock maps were generated with both a constant fixed metallicityof Z . = 0 . Z ⊙ and constant fixed temperature of T X = 1 keV. The loweraxis shows the ratio between the best-fit metal abundance and the true metalabundance. X-ray temperature and normalization are free parameters. In theabsence of a multi-temperature medium, the abundance is well recovered outto large radii. respectively. Both the resulting projected emission measureand gas density profiles vary to 10% at low and intermedi-ate radii, but deviate as much as 40% and 20% in emissionmeasure and gas density, respectively. Systematic effects of metal abundance measurements ongas profile measurements
As described in Section 3.2, a high (low) bias of the abun-dance and X-ray temperature measurements would lead to asuppressed (enhanced) density measurement in that radial bin.In this section, we examine the systematic effects on gasprofile measurements with a factor of 2 and 5 bias in the abun-dance fitting at all radii. For this test, we use the Z . mapwhere we take the metallicity to be constant at Z = 0 . Z ⊙ throughout the simulation. We choose this value since it iscloser to the metal abundance at large radii R & R c in ourself-consistent cluster simulation with metal enrichment andadvection. We run XSPEC fitting by fixing the abundance pa-rameter at metallicities a factor of 2 and 5 from the true valuefrom the map: Z fix / Z ⊙ = 0 . , . , . , .
25, and we com-pare the results to those fitted with the true value Z = 0 . Z ⊙ .To isolate the effects from all other quantities, we only allowthe projected X-ray temperature and the spectral normaliza-tion to be free parameters in the fitting .In Figure 8, we plotted the X-ray temperature, projectedemission measure, and gas density for each case of Z fix andcompare to the Z fix = Z true = 0 . Z ⊙ case (black solid line).Consistent with what we found in Section 3.2, an over (under)estimate of the abundance will lead to suppressed (enhanced)emission measurements and subsequently gas density mea-surements.If the abundance is measured to within a factor of two ofthe true metal abundance, the resulting density measurements Avestruz et al. −2 Z p r o j [ Z ⊙ ] R c R c R[kpc]0.41.01.62.2 f Z f i t / Z f i x R[kpc]0.91.01.11.2 f T x , p r o j T x , p r o j [ k e V ] R c R c Z :Z fit Z :Z fit R [kpc]012 f E MM −5 −4 −3 −2 E MM [ c m k p c − ] R c R c Z :Z fit Z :Z fit −30 −29 −28 −27 −26 ρ g [ g c m − ] R c R c R [kpc]0.81.01.2 f ρ g F IG . 6.— From top left panel in a clockwise direction, the projected metallicity, projected X-ray temperature, recovered 3-D density profiles, and projectedemission measure for CL7. Mock maps were generated with constant fixed metallicity ( Z . = 0 . Z ⊙ in red solid and Z . = 0 . Z ⊙ in blue dash-dotted). Thelower axes, labeled with an f , show the ratio between profiles recovered with metal abundance as a free parameter to profiles recovered with abundance fixedto the known input abundance. The systematic decline of the best-fit projected metallicity beyond R is due to the multi-temperature medium, as shown inFigure 4. The effects of low photon count from metal line cooling is visible in both the Z . best fit projected metallicity as well as in the last two bins of Z . .The bias in metal abundance measurement introduces variations of up to 15% in recovered density profiles. in the outskirts will be within 15%. However, for the mostextreme case of an abundance measurement biased by a factorof 5, the density measurements will have up to a 35% bias.The emission measure profile on the right panel of Figure 8shows that an abundance measurement that is higher than thetrue value can steepen the profile in cluster outskirts. For ob-servational measurements that extrapolate the abundance atlarge radii, this could introduce an even larger bias in theemission measure, density and temperature. Our tests that fix the abundance measurement to be a factor of two and fivetimes the true abundance show how severely this extrapola-tion can propagate to inferred quantities, even at R c . Mostnotably, for an assumed abundance that is a factor of fivehigher than the true abundance, the gas density is biased lowby ≈ Hydrostatic mass and gas mass fraction C o un t s [ k e V − ] Z =0.01, T x =0.865 ±0.030Z =0.01, T x =0.944Z =0.05, T x =0.944 ±0.027Z =0.25, T x =0.944Z =0.25, T x =1.134 ±0.025 R e s i d a l F IG . 7.— Grey errorbars show the energy spectrum from the Z . photonmap corresponding to the third to the last radial bin in Figure 3 ( R ≈ × R c ). Spectra are normalized to the size of the energy bin. The solid linesshow the best fit model spectra with the abundance parameter fixed (0 . Z ⊙ in blue, 0 . Z ⊙ in black, and 0 . Z ⊙ in cyan), and the best fit projectedX-ray temperature. The dotted lines correspond to model spectra with theprojected X-ray temperature fixed to T X , proj = 0 .
944 keV, which is the valueof the best-fit X-ray temperature with Z fix = Z true . We show the residual valuebetween the model and our mock data in the lower panel. The model spectrashow that the best-fit projected X-ray temperature will be biased in the samedirection as the best-fit abundance parameter, as shown in Figure 6. From Equation (8), we can see that the X-ray measurementof the hydrostatic mass of the cluster is sensitive to three quan-tities: the best fit 3D model of the temperature profile, thelogarithmic slope of the 3D temperature profile, and the log-arithmic slope of the gas density profile. The left panel ofFigure 9 illustrates how each of these quantities is affected bya systematic over (under) fit abundance.For the case of recovering the metal abundance to withina typical factor of two ( Z fix = 0 . Z ⊙ and Z fix = 0 . Z ⊙ ), allthree quantities are within <
10% at intermediate and largeradii. At large radii, the bias in the 3D temperature and log-arithmic density slope dominate to drive the hydrostatic massbias in the same direction. A lower temperature fit in the out-skirts corresponds to a shallower logarithmic density slope,and therefore a smaller hydrostatic mass (see dotted blue anddot-dashed green thick lines in the right panel of Figure 9).However, the lower fitted temperature and shallower logarith-mic slope correspond to a larger enclosed gas mass (see cor-responding thin lines in Figure 9), as underestimates in bothabundance and temperature lead to overestimates in emissionmeasure, therefore bias the gas density high.Figure 10 shows that the combined bias in enclosed gasmass and the hydrostatic mass drives the bias in gas fractionup for the low fixed abundance values, and down for the highfixed abundance values. This shows that a poor fit to the X-raytemperature and metallicity in cluster outskirts is potentiallycausing biases in the
Suzaku gas mass fraction measurementsat large radii. The same magnitude of errors can come from anincorrectly extrapolated metal abundance in cluster outskirts.For a standard factor of 2 error in the abundance measure- ment, the gas mass fraction is only affected by 10%, but anyunderestimate of the 3D temperature at large radii due to anoverly steep poor fit in the deprojection procedure could leadto an even higher gas mass fraction by driving the hydrostaticmass low.For an abundance that is measured 5 times lower than thetrue abundance, the measured gas mass fraction could be bi-ased high by 70% given a true abundance of Z = 0 . Z ⊙ .While the bias in enclosed gas mass and hydrostatic massat small radii is rather large, we do not consider this to berepresentative of biases present in observations; the error inabundance measurement is primarily an issue in the low den-sity outskirts where photon statistics are poor. Additionally,at smaller radii, the gas is much hotter, so the metal abun-dance and X-ray temperature parameters are less degeneratewith each other in the spectral fit. Potential issues for gas density slope measurements
We test two potential issues for density slope measurementsin cluster outskirts to compare with biases in measurementsfrom spectral fitting and metal abundance extrapolation: pro-jection bias from triaxiality and mass accretion history.To test the bias due to projection effects from traxiality, wecompute the inertia tensors of the gas mass at R c for ourclusters. For CL7, the semi-minor and semi-major axes of theinertia tensor lie almost parallel to the x - and y -axes respec-tively, allowing us to use the projections along these axes tomake a preliminary assessment of the effects of cluster shape.To test the effect of accretion history, we compare two clus-ters, CL6 and CL7 that have similar mass but differ in dy-namical states. CL6 has a recent major merger and CL7 hasbeen quiescent since z ∼
1. We compare these two clusters toidentify the extent of steepening that is mainly due to recentaccretion history.Figure 11 shows the logarithmic gas density slope for ourtwo test cases along various projections, CL6 and CL7. Inour relaxed cluster, CL7, the density slope projected alongeach axis varies by no more than 15%. We would naively ex-pect a significant difference between a line of sight along thesemi-major axis (y-axis) and along the semi-minor axis (x-axis). However, the difference in density slope between thesetwo line of sight projections amounts to only a few percent.Effects due to line of sight projection are not likely to signifi-cantly steepen observed density profiles.CL6 is a dynamically disturbed cluster that experienced a1:1 major merger at z ≈ .
6, and has a smaller infalling struc-ture at R & R c at z = 0. The x - and y -projections correspondto lines of sight that are perpendicular to the plane of accretingsubstructure. The effects of a merging substructure are visiblein the z -projection (dotted green), where the density slope isno longer monotonically increasing at larger radii. The out-skirts of CL6 cluster along the x - and y -projections have anobserved density profile that exceeds the density slope alongthe x -projection of CL7 by ≈ ≈ SUMMARY AND DISCUSSION
The study of galaxy cluster outskirts is the new frontier incluster-based cosmology. In this work, we used cosmologi-cal simulations of galaxy clusters to investigate potential bi-0 Avestruz et al. f E MM f T x , p r o j R [kpc]0.81.01.2 f ρ g R [kpc]10 −4 −3 −2 E MM [ c m k p c − ] R c R c Z fix =0.01Z ⊙ Z fix =0.025Z ⊙ Z fix =0.05Z ⊙ Z fix =0.1Z ⊙ Z fix =0.25Z ⊙ F IG . 8.— The above figures are profiles corresponding to the Z . flux maps. The plot on the left shows the ratio between profiles using various values for Z fix , and the corresponding profile with Z fix = Z true = 0 . ⊙ . From top to bottom panel, these respectively are the ratios in X-ray temperature, gas density, andprojected emission measure. The figure on the right shows the projected emission measure for each value of Z fix . At 3 × R c , an abundance measurement thatis biased by a factor of 2 results in gas densities that are biased by 15%. f T x , D f d l n T / d l n r R [kpc]0.81.01.2 f d l n ρ / d l n r R [kpc]10 M t o t , g a s ( < r ) R c R c Z fix =0.01Z ⊙ Z fix =0.025Z ⊙ Z fix =0.05Z ⊙ Z fix =0.1Z ⊙ Z fix =0.25Z ⊙ F IG . 9.— The above figures are profiles corresponding to the Z . flux maps, with various values of Z fix , identical to the data set from Figure 8. Left: Ratiosof quantities that contribute to the total mass measurement - the best fit 3D X-ray temperature, the logarithmic slope of the 3D temperature profile, and thelogarithmic slope of the density profile. Right: The total mass profile (thick lines), and the gas mass profile (thin lines). At 3 × R c , an abundance measurementthat is biased by a factor of 2 results a logarithmic gas density slope that is biased by < R [kpc]0.60.91.21.5 f f g f M g f M t o t R [kpc]0.00.10.20.30.40.5 f g a s [ Ω B / Ω M ] R c R c Z fix =0.01Z ⊙ Z fix =0.025Z ⊙ Z fix =0.05Z ⊙ Z fix =0.1Z ⊙ Z fix =0.25Z ⊙ F IG . 10.— The above figures are profiles corresponding to the Z . flux maps, with various values of Z fix , identical to the data set from Figure 8. The panelson the left are ratios of the gas mass, hydrostatic mass, and gas mass fraction, and the figure on the right is the gas mass fraction profile for each fixed value ofthe metal abundance. At 3 × R c , an abundance measurement that is biased by a factor of 2 results in a fractions that is biased by ∼ − d l n ρ / d l n r CL7,xCL7,yCL7,zCL6,xCL6,yCL6,z10 R [kpc]0.81.01.2 f d l n ρ / d l n r F IG . 11.— The logarithmic gas density slope profile of CL6 (dash-dottedlines) and CL7 (solid lines) along different projections. The z projection forCL6 is perpendicular to the plane of a merger. The bottom panel is the ratioof each profile to the profile of CL7 along the x-projection. The logarithmicgas density slope of CL6, a dynamically disturbed cluster, is steeper than thatof the CL7. ases in metal abundance measurements from X-ray spectra inthe low density cluster outskirts ( R > R ), where metal linecooling contributions become significant. Using X-ray mapswith a constant input metal abundance, we showed how sys-tematically biased metal abundance measurements affect ICMquantities such as the gas density and temperature profiles.In order to test other potential sources of systematic uncer-tainty in density slope measurements from Chandra observa-tions of Abell 133 (Vikhlinin et al., in prep.), we performedour mock X-ray analysis for projections along the semi-majorand semi-minor axes of a relaxed cluster, and for a dynami-cally disturbed cluster.To summarize our findings: • Metal abundance measurements in the outskirts of ourtest case of a relaxed cluster shows that the metal abun-dance is best recovered to within a factor of two. Dueto the presence of a multi-temperature medium, strongmetal line contributions will be systematically biasedlow. In the regime of few photon counts (either fromlow X-ray surface brightness or low abundance), accu-rate metal line contributions will be difficult to recoverfrom Poisson noise. • An incorrect measurement of the abundance in the lowdensity regions corresponds to fluctuations in measuredprojected X-ray temperature to within 10%. This tem-perature bias drives the projected emission measure ina direction opposite of the bias, thus affecting the gasdensity measurements as much as 15% in the outskirtsout to ≈ × R c . • We test the extent of biases that may arise from incor-rectly extrapolating the metal abundance measurementin galaxy cluster outskirts by fixing the metal abun-dance to a factor of 2 and a factor of 5 above and belowthe true abundance. An abundance assumed a factor of 5 too high biases the gas density measurement and theenclosed total mass by as much as 15% at R c . • In the cluster outskirts, an X-ray temperature that is bi-ased low by 10% corresponds to a decrease in the totalmass measurement, increase in the enclosed gas mass,and therefore a boosted gas mass fraction by ∼ • Two potential physical mechanisms for the derivedsharpening of density gradients in cluster outskirts areprojection and dynamical state of the cluster. Our testcases show that these contribute to no more than 15%and 30%, respectively, to the logarithmic gas densityslope. The bias due to an incorrect abundance measure-ment would affect the density slope by no more than5%.We note that fully testing the accuracy of X-ray measure-ments in cluster outskirts requires a comparison of measure-ments of ICM properties directly from the simulation with 3Dfilaments removed in a self-consistent manner with our pro-jected filament finder. While the test cases used in this studydo not include high mass clusters, any bias from incorrectabundance measurements would not exceed the bias found inthe lower mass counterparts. High mass clusters have largertemperatures at R c , resulting in a decreased relative contri-bution of metal line emissions to the X-ray emission measure.This analysis does not include any contributions from simu-lated backgrounds, and the effects of instrumental and X-raybackgrounds are additional sources of systematic biases thathave not been accounted for in our discussions.Furthermore, our analysis does not take into account non-equilibrium electrons at large radii, which have a lower tem-perature than the mean ICM gas temperature (Rudd & Nagai2009). The temperature bias from non-equilibrium electronsis small in clusters of the mass scales we tested, and weexpect non-equilibrium electrons to have less of an effecton the emission measure in the cluster outskirts than biasesfrom metal abundance measurements. Metal line contribu-tions have a significant contribution to the X-ray emission inthe low density cluster outskirts, and the X-ray emission hasa weak dependence on any biases in the temperature at allradii ( ∝ T / ). Alternatively, in higher mass clusters, non-equilibrium electrons have larger deviations from the meanICM gas temperature because the equilibration timescale islonger. The corresponding biases in X-ray measurements ofthe outskirts of high mass clusters is a topic for future work.Our findings emphasize the need for long exposure timesto have accurate measurements of metal abundance and X-ray temperature in the low density outskirts of galaxy clusters.Accurate X-ray measurements of the hydrostatic mass requirean additional term in the temperature modeling in cluster out-skirts to account for the shallow slope in the outermost radii.Additionally, accounting for errors in the metal abundancemeasurement will require more careful modeling of the ther-mal and chemical structure in cluster outskirts.In addition to gas inhomogeneities (e.g., Nagai & Lau2011), biased measurements of the metal abundance in clusteroutskirts may partially explain the enhanced gas mass fraction(e.g., Simionescu et al. 2011) and flattened entropy profiles(e.g., Walker et al. 2012) observed in cluster outskirts. Errorsin metal abundance could produce enhanced emission mea-surements and a steeper 3D temperature profile, leading tooverestimates in gas mass and underestimates in hydrostaticmass.3We have identified three potential mechanisms that con-tribute to density steepening in our mock data, and may affectthe observed density slope beyond R c : (1) overestimationof the abundance, (2) a line of sight corresponding to the semi-minor axis, and (3) rapid recent mass accretion. For (1), thebiases introduced by spectral fitting are not likely to exceed5%. The two physical mechanisms (2) and (3) had respectiveeffects of 15% and 30%.The mass accretion rate of a galaxy cluster can influenceits DM density (Cuesta et al. 2008; Diemer & Kravtsov 2014)and gas profiles (E.T. Lau et al., in prep.). Accurate mea-surements of gas density and temperature profiles from X-rayobservations of cluster outskirts will provide information onthe recent mass accretion history of a galaxy cluster, which isone of the dominant systematics in mass-observable relationsin cluster cosmology. Deep observations of cluster outskirtswill allow us to better understand the growth of galaxy clus-ters though mass accretion from the surrounding filamentarystructures.A careful consideration of all potential biases that enter into ICM measurements in galaxy cluster outskirts will enable usto fully use clusters as cosmological probes. Results from thiswork will also be useful for future missions, such as SMART-X and Athena+ , which will begin mapping the denser partsof the filamentary cosmic web.We thank the referee for the useful feedback, and ElenaRasia for comments on the manuscript. This work issupported by NSF grant AST-1009811, NASA ATP grantNNX11AE07G, NASA Chandra grants GO213004B andTM4-15007X, and by the facilities and staff of the Yale Uni-versity Faculty of Arts and Sciences High Performance Com-puting Center. CA acknowledges support from the NSF Grad-uate Student Research Fellowship and the Alan D. BromleyFellowship from Yale University. R c is the radius of the cluster enclosing an average matter density 200times the critical density of the Universe. http://smart-x.cfa.harvard.edu/3