AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2018October 26, 2018
Probing the dynamical state of galaxy clusters
Ewald Puchwein and Matthias Bartelmann
Zentrum f¨ur Astronomie der Universit¨at Heidelberg, ITA, Albert- ¨Uberle-Str. 2, 69120 Heidelberg, Germany
Astronomy & Astrophysics, submitted
ABSTRACT
We show how hydrostatic equilibrium in galaxy clusters can be quantitatively probed combining X-ray, SZ, and gravitational-lensingdata. Our previously published method for recovering three-dimensional cluster gas distributions avoids the assumption of hydrostaticequilibrium. Independent reconstructions of cumulative total-mass profiles can then be obtained from the gas distribution, assuminghydrostatic equilibrium, and from gravitational lensing, neglecting it. Hydrostatic equilibrium can then be quantified comparing thetwo. We describe this procedure in detail and show that it performs well on progressively realistic synthetic data. An application to acluster merger demonstrates how hydrostatic equilibrium is violated and restored as the merger proceeds.
Key words. galaxies: clusters: general – x-rays: galaxies: clusters – submillimeter – gravitational lensing
1. Introduction
Numerous observations show that galaxy clusters frequently ex-hibit irregular shapes and violent dynamics. On the theoret-ical front, simulations indicate that cluster-sized dark matterhalos are often well described as triaxial ellipsoids, but notas spheres (Jing & Suto 2002). Nevertheless, clusters are of-ten interpreted as spherically-symmetric objects in hydrostaticequilibrium, which is a potential source of error. For example,Hallman et al. (2006) show that intrinsic variations in clusterslimit the accuracy of cluster gas mass estimates to about 10%when using such simple assumptions.Several authors tried to relax the restricting assumption ofspherical symmetry and aimed at a joint analysis of differenttypes of cluster data. Zaroubi et al. (1998) suggested to basethe reconstruction of axisymmetric, three-dimensional gravita-tional cluster potentials on the Fourier slice theorem, extrapolat-ing Fourier modes into the “cone of ignorance”. They appliedtheir technique to simulated data and showed that it performswell (Zaroubi et al. 2001). Dor´e et al. (2001) followed a per-turbative approach, and Lee & Suto (2004) proposed to adaptparameters of triaxial halo models, all by combining differentdata sets such as X-ray, (thermal) Sunyaev-Zel’dovich (SZ) andgravitational-lensing maps. A similar method was applied to databy De Filippis et al. (2005).An alternative approach based on the iterative Richardson-Lucy deconvolution was suggested by Reblinsky (2000) andReblinsky & Bartelmann (2001). It aims at the gravitationalpotential, assumes only axial symmetry of the main clusterbody, avoids extrapolations in Fourier space, and can easilybe extended to include additional data sets. In Puchwein &Bartelmann (2006) (hereafter Paper I) we developed the latteralgorithm further. However, instead of aiming at the gravita-tional potential, which would require us to assume a relationbetween the gas distribution and the gravitational field, we pro-posed methods to reconstruct the three-dimensional cluster gasdensity and temperature distribution from X-ray and thermal SZeffect observations. These methods do not require any equilib-rium assumption other than local thermal equilibrium and againassume only axial symmetry with respect to an arbitrarily in- clined axis. Using synthetic observations of analytically mod-elled and numerically simulated galaxy clusters we showed thatthese reconstruction methods perform very well, even in thepresence of observational noise, deviations from axial symme-try and cluster substructure.In this work we use this gas reconstruction algorithm to-gether with novel methods to reconstruct the three-dimensionalgravitational potential from lensing data in order to probe hy-drostatic equilibrium in galaxy clusters and to quantify the ac-curacy of mass estimates based on the assumption of hydro-static equilibrium. We will also introduce methods to find three-dimensional reconstructions of the gravitational potential and ofthe mass profiles of relaxed galaxy clusters from X-ray and ther-mal SZ observations alone. All these methods are tested withsynthetic observations of analytically modelled and numericallysimulated galaxy clusters. Mass estimates based on the gas re-construction and the assumption of hydrostatic equilibrium arecompared to lensing mass estimates and to the original analyticor simulated masses.
2. Three-dimensional cluster reconstructiontechniques
In Paper I a novel technique to reconstruct the intra-clustermedium in three dimensions by a combined analysis of X-rayand thermal Sunyaev-Zel’dovich effect observations was intro-duced (hereafter also called XSZ reconstructions). It assumesonly axial symmetry of the cluster halo with respect to an ar-bitrarily inclined axis and does not require any equilibrium as-sumption other than local thermal equilibrium. The iterativemethod is based on Richardson-Lucy deconvolution (Lucy 1974,1994) and is a generalisation of employing it to reconstructionsof three-dimensional axisymmetric quantities form their projec-tion along the line-of-sight (Binney et al. 1990). Due to the as-sumed axial symmetry the reconstructed gas densities and tem-peratures depend only on the distance R from the symmetry axisand the coordinate Z along the axis. a r X i v : . [ a s t r o - ph ] J un E. Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters
In this study we employ the same three-dimensional gas re-construction method as in Paper I except for one modification.We now use a more realistic model for the X-ray emission of theintra-cluster medium that also includes line emission. The syn-thetic X-ray observations of the reconstructed cluster halo thatare performed during the iterative deprojection (see Paper I) arenow calculated with the MEKAL emission model (see Liedahlet al. (1995); Kaastra & Mewe (1993)) and the WABS model forgalactic absorption (Morrison & McCammon 1983). More pre-cisely we use the X-ray spectral fitting software package XSPEC(Arnaud 1996) to create a table of the cooling function with themodels mentioned above, assuming a constant metallicity of 0.3times the Solar value and an equivalent hydrogen column den-sity of 5 × atoms cm − . We then use this table to producethe synthetic X-ray maps needed during the reconstruction andwe also employed it to create the synthetic observations of an-alytically modelled and numerically simulated clusters that arediscussed in sections 3 and 4. Note that it is not necessary tochange the equations for the iterative corrections of the gas den-sity and temperature (Eq. (22) and (23) in Paper I), which werederived considering only thermal bremsstrahlung, because smallerrors introduced by using these equations are corrected in sub-sequent iteration steps. It also does not significantly affect thenumber of iterations needed to achieve a good reconstruction. We want to use these three-dimensional cluster gas reconstruc-tions to find the gravitational potentials and total mass distribu-tions of relaxed galaxy clusters and to probe the dynamical stateof potentially unrelaxed clusters by comparing such gas recon-structions to an analysis of lensing data.To find the gravitational potential of a cluster from the distri-bution of the cluster gas we assume that the gas is in hydrostaticequilibrium. Then the gas density ρ , the gas pressure p and thegravitational potential φ satisfy ∇ φ = − ∇ p ρ . (1)In principle this equation can be used to find the gravitationalpotential of relaxed clusters from three-dimensional reconstruc-tions of their intra-cluster medium. However due to deviationsfrom hydrostatic equilibrium, and the presence of observationalnoise and cluster substructure violating axial symmetry, the curlof − ∇ p / ρ will not vanish exactly for the reconstructed gas dis-tributions. Thus one cannot obtain a unique solution for φ di-rectly from Eq. (1).To get a unique solution we first derive − ∇ p / ρ on the gridin R and Z space on which the gas reconstruction was calculated(see also Paper I). Then we aim to determine the potential φ forwhich ∇ φ is closest to − ∇ p / ρ . We do that by finding the valuesof the potential φ at all grid points which minimise the deviation ∑ neighbours i , j (cid:0) φ j − φ i + p j − p i ( ρ j + ρ i ) (cid:1) , (2)between these two vector fields. Here p i , p j are the gas pres-sures, ρ i , ρ j the gas densities and φ i , φ j the gravitational poten-tials at the R and Z coordinates of grid points i and j . The sumextends only over such pairs of grid points i and j that are near-est neighbours. Conjugate gradient minimisation starting with aguess φ i = φ i . However, to reduce noise in the potential it turned out to befavourable to add a penalty function to (2) that requires the sec-ond derivatives of the potential to be small. We also multiplyeach term in the sum in Eq. (2) and the penalty function by aweight factor. So the function we end up minimising is, ∑ neighbours i , j w ( r i , j ) (cid:0) φ j − φ i + p j − p i ( ρ j + ρ i ) (cid:1) + w p ∑ i (cid:0) ( φ iR > + φ iR < − φ i ) + ( φ iZ > + φ iZ < − φ i ) (cid:1) , (3)where r i , j is the distance from the cluster centre to the midpointof the line connecting grid points i and j . iR > , iR < and iZ > , iZ < are the indices of the neighbouring grid points of point i in the R and Z directions respectively. The weighting function w ( r ) is chosen equal to one in the central region of the cluster, for r < . l , then it smoothly goes to zero, and vanishes for r > . l ,close to the perimeter of the box with side length l that is used forthe gas reconstruction (see Paper I). This is necessary becausethere are significant artefacts in the gas reconstruction close tothe perimeter (see also Paper I). When using the potential recon-struction algorithm proposed here they would have a non-localeffect on the potential reconstruction and would thereby reduceits quality also near the cluster centre. The weight factor w p = ∇ p / ρ . This allows us todefine a cumulative mass M < r , XSZ as a function of radius r fromthe cluster centre by M < r , XSZ ≡ π G (cid:90) − ∇ p ρ dA , (4)where G is Newton’s constant and the integral extends overthe surface of a sphere with radius r around the cluster centre.The numerical evaluation of the integral is done using 128 sam-pling points which are equally spaced in the angular coordinate θ ≡ arctan ( Z / R ) . For each point the component of ∇ p / ρ per-pendicular to the surface is calculated from the gas reconstruc-tion and multiplied by the area of the corresponding ring. Lensing observations allow reconstructions of the lensing poten-tial (see e.g. Cacciato et al. (2006)), which is simply the suitablyrescaled projection of the lens gravitational potential along theline-of-sight. Once the lensing potential is found, Richardson-Lucy deconvolution can be applied to deproject it in order toobtain the three-dimensional gravitational potential. Again axialsymmetry with respect to an arbitrarily inclined axis needs to beassumed.We employ the deprojection algorithm discussed in sections2.1 and 2.2 of Paper I to obtain such three-dimensional recon-structions of the gravitational potential.In Paper I the optimal number of iterations for three-dimensional gas reconstructions was studied. Richardson-Lucy . Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters 3 deconvolution reproduces large scale structure quickly, while itconverges slowly to small scale structure. It turned out that forgas reconstructions based on X-ray and SZ data it is best to useabout five iterations. For a smaller number of iterations the clus-ter structure is not recovered sufficiently well, while for a largernumber of iterations the reconstruction algorithm tries to repro-duce small-scale observational noise which can reduce the re-construction quality again. However as the lensing potential is amuch smoother quantity than the X-ray surface brightness or theSZ temperature decrement it is favourable to use a larger numberof iterations for deprojections of the lensing potential.However even when using a large number of iterations, prob-lems with the gravitational potential reconstruction arise forsmall inclination angles i between the line-of-sight and the sym-metry axis, because then the assumption of axial symmetry con-tains least information (see also Paper I) and a reconstructionthat, compared to the original halo, is stretched along the sym-metry axis can still reproduce the lensing observations ratherwell. For a cluster with a roughly spherical gravitational poten-tial and for a small inclination angle one gets too large correctionfactors close to the symmetry axis during the first few iterationsteps when starting from a flat guess and thus the reconstructionafter a few iterations is overly extended along that axis. As thepower to determine the halo elongation along the symmetry axisis limited for small inclination angles the reconstruction algo-rithm takes very long to recover from this. To avoid this problem,it is thus favourable to start with a guess that has already moreor less the right shape. We can get such a guess by doing a grav-itational potential reconstruction from a flat guess with a smallnumber of iterations and by then making the obtained potentialspherically symmetric while preserving its profile. This spheri-cally symmetrised potential can then be used as a first guess forthe actual reconstruction with a larger number of iterations. 10iterations were used to produce spherically symmetrised guessesfor reconstructions from synthetic lensing data in sections 3 and4. The actual reconstructions use 30 iterations and start eitherfrom such a spherically symmetrised guess or from a flat guessas specified there.The lensing three-dimensional gravitational potential recon-structions can then be used to find the total mass distribution andcan be compared to reconstructions from X-ray and SZ data. Inorder to have a quantity that can be directly compared to M < r , XSZ we define in analogy to Eq. (4) a lensing cumulative mass M < r , lensing ≡ π G (cid:90) ∇ φ dA , (5)where φ is the three-dimensional gravitational potential obtainedby deprojecting the lensing potential. The numerical evaluationof the integral is done in the same way as for M < r , XSZ . Hydrostatic equilibrium in galaxy clusters can be probed bycomparing cluster reconstructions based on X-ray and SZ datato lensing reconstructions. In principle this could be done bycomparing the gravitational potential obtained by minimisingEq. (3) to the one found by deprojecting the lensing potential.However as the gravitational potential is not uniquely defined itis more favourable to compare the cumulative masses defined inEqs. (4) and (5). If the cluster is exactly in hydrostatic equilib-rium, so that Eq. (1) is satisfied, the masses should be identicalfor all distances r from the cluster centre except for small de-viations caused by reconstruction errors. Otherwise differencesbetween the masses directly reflect the differences between the gravitational field and ∇ p / ρ . In the next two sections we testthis method to probe hydrostatic equilibrium in galaxy clustersby performing such a comparison using synthetic observation ofanalytically modelled and numerically simulated clusters.
3. Probing hydrostatic equilibrium in analyticallymodelled clusters
We use an analytic halo model with a NFW total (gas+DM) den-sity profile to test the methods introduced in section 2. Thus thetotal matter density ρ m and the gravitational potential are givenby ρ m = c rr s ( + rr s ) , (6) φ = π Gcr s r ln (cid:0) r s r + r s (cid:1) , (7)where r s is the NFW scaling radius and c = ρ ( r s ) fixes the nor-malisation of the density profile. For the cluster gas we assumein this toy model that the ratio f of ∇ p / ρ to − ∇ φ is constant butcan be different from 1. So Eq. (1) generalises to f ∇ φ = − ∇ p ρ . (8)We further assume a polytropic equation of state T ∝ ρ γ − forthe cluster gas, where T is the gas temperature and γ the poly-tropic index. Then the gas density ρ and temperature T satisfy ρ ∝ (cid:20) ( − γ ) φγ (cid:21) γ − , (9) kT = f ( − γ ) φγ ¯ m , (10)where k is Boltzmann’s constant and ¯ m is the mean gas particlemass. In the following, we adopt γ = .
2, which is consistentwith X-ray temperature profiles of nearby clusters (Markevitchet al. 1998), and fix the normalisation of ρ by requiring abaryon fraction of 0.12 at the scale radius, which we set to r s = h − kpc. Note that the lengths here and below are givenin comoving units. A reduced Hubble parameter of h = . c is chosen to be 1 . × h − M (cid:12) / ( h − kpc ) . Thesechoices for r s and c correspond to a massive galaxy cluster. Totest the reconstruction methods we put this analytically modelledcluster at a redshift of z = . For the X-ray observations we use the table of the cooling func-tion discussed in section 2.1 and a 128 × ×
128 grid with1 . h − Mpc side length to project the gas distribution and geta map of the X-ray surface brightness in a 0.25-7.0 keV band.Then, except for reconstructions we specifically characterise asdone without observational noise, we add photon noise corre-sponding to 10 observed source photons to these maps usingthe same method as in Paper I.The thermal SZ maps are generated like in Paper I by project-ing the product of cluster gas density and temperature along theline-of-sight onto a 128 ×
128 grid with 1 . h − Mpc side length.
E. Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters
Then the result is appropriately rescaled and, unless stated other-wise, noise corresponding to future ALMA Band 3 observationsis added. In Band 3 (84-116 GHz) and in its compact configura-tion, ALMA will be able to achieve a temperature sensitivity of50 µ K at a spatial resolution of ∼ h − Mpc side length which is centred on thecluster along the line-of-sight and calculate the convergence.The grid we use for this purpose is chosen such that each pixelcorresponds to roughly 1 / n g = / arcmin . For instance for a cluster at redshift z = . ×
44 convergence map covers the projection of the 6 h − Mpccube on the sky.For lensing reconstructions with observational noise, nor-mally distributed noise with variance σ κ = σ σ ε π n g a (cid:16) − exp (cid:0) − a σ (cid:1) − (cid:114) π a σ erf (cid:0) a √ σ (cid:1)(cid:17) , (11)is added to each pixel of the convergence map. Here σ κ is thevariance expected for a weak lensing reconstruction of the con-vergence for a density n g of background galaxies with an intrin-sic ellipticity dispersion σ ε , and for an angular pixel size a ofthe convergence map (see van Waerbeke 2000). It is assumedthat the galaxy ellipticities are smoothed with a Gaussian of an-gular standard deviation σ before the reconstruction. We choose σ = a and σ ε = . × We applied these methods to produce synthetic X-ray, SZ andlensing observation for the analytic halo described above. Wethen generate three-dimensional reconstructions of the clustergas based on X-ray and SZ data and reconstructions of the grav-itational potential based on lensing data using the methods de-tailed in section 2 and in Paper I.Figure 1 shows the cumulative mass profiles M < r , XSZ ( r ) ,obtained from the XSZ reconstructions, and M < r , lensing ( r ) , ob-tained from the lensing reconstructions, and compares them tothe original analytic profile. The profiles are shown for recon-structions based on data without observational noise and forreconstructions based on noisy data and for inclination angles i = ◦ and i = ◦ between the symmetry axis and the line-of-sight. The inclination angles were assumed to be known forthe reconstructions. See Paper I for methods to determine themfrom the observations. For the gas reconstructions ratios f of ∇ p / ρ to − ∇ φ of f = . f = . f = .
0) and when using datawithout noise and an inclination i = ◦ (see upper right panel M < r [ h − M (cid:12) ] distance r from cluster centre [ h − kpc]original analytic mass i = ◦ , lensing reconstr., flat prior i = ◦ , lensing reconstr., spher. prior i = ◦ , lensing reconstr., flat prior i = ◦ , lensing reconstr., spher. prior Fig. 2.
Cumulative mass profiles M < r ( r ) of an ellipsoidal an-alytic halo and its reconstructions from lensing maps withoutnoise. Inclination angles of i = ◦ and i = ◦ were used forthe synthetic observations and assumed to be known for the re-constructions. Lensing reconstructions are shown for a flat andfor the spherically symmetrised prior.of Fig. 1). They also excellently match the original analytic pro-file. The only significant difference between the profiles is thatthe lensing mass is too small very close to the cluster centre.However, this is completely expected because the lensing ob-servations lack the resolution required to accurately resolve thisregion. When perturbing the hydrostatic equilibrium by 20%,in other words when assuming f = .
8, the XSZ reconstructedmass profile is essentially 0.8 times the original analytic pro-file as theoretically expected. For such a halo one can easily seea significant ( ∼ i = ◦ (left panels) the accuracy of the re-constructions is somewhat lower and one can also see significantdifferences between the lensing reconstructions based on a flatprior and a spherically symmetrised prior. The latter reproducethe original profiles much better. Thus, for such small inclina-tions one can detect deviations from hydrostatic equilibrium bycomparing lensing reconstructions based on a spherically sym-metrised prior to XSZ reconstructions. Also note that for a ran-domly oriented cluster sample only about 13% of the clustershave inclination angles smaller than 30 ◦ . Paper I contains a moredetailed discussion of the dependence of the reconstruction ac-curacy on the inclination angle.Above we used spherically symmetrised priors in the lensingreconstruction of spherically symmetric halos. It is reassuring,but not really surprising that this works well. We thus need tocheck whether or not a spherically symmetrised prior also im-proves the lensing reconstruction quality of elliptical halos forsmall inclination angles. In Figure 2 we show lensing recon-structions of the cumulative mass profile of an elliptic analytichalo with an NFW density profile but isodensity surfaces thatare prolate spheroids with a major to minor axis ratio of 2 to 1.The lensing reconstructions with a spherically symmetrised priorreproduce the original analytic profile well, both for small andfor large inclination angles. On the other hand when using a flatprior we again obtain too small lensing masses for small inclina-tion angles. It is thus favourable to use a spherically symmetrisedprior for the iterative deprojection of the lensing potential. . Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters 5 i = ◦ , no noise original analytic masslensing reconstr., flat priorlensing reconstr., spher. priorf=1.0 XSZ reconstr.0.8 x original analytic massf=0.8 XSZ reconstr. M < r [ h − M (cid:12) ] i = ◦ , no noise 0 100 200 300 400 500 600distance r from cluster centre [ h − kpc] i = ◦ , noise3.532.521.510.50 0 100 200 300 400 500 600 M < r [ h − M (cid:12) ] distance r from cluster centre [ h − kpc] i = ◦ , noise Fig. 1.
Cumulative mass profiles M < r ( r ) of an analytic halo and its reconstructions from X-ray and SZ maps with and withoutobservational noise as well as from lensing maps with and without noise. The upper panels show the results obtained from mapswithout noise, while the lower panels show the profiles found from noisy maps. For the reconstructions shown in the left panels aninclination angle of 30 ◦ was used and assumed to be known, while the right panels show the corresponding results for an inclinationangle of 70 ◦ . Lensing reconstructions are shown for a flat and for the spherically symmetrised prior. The XSZ reconstructions weredone for halos with ratios f of ∇ p / ρ to − ∇ φ of 1 . In Figure 3 we show three-dimensional reconstructions of thegravitational potential of the analytic halo from X-ray and SZdata and from lensing data as well as the original analytic grav-itational potential described by Eq. (7). Reconstructions that arebased on idealised observations without noise and on more re-alistic noisy observations are shown. The XSZ potential recon-structions were obtained from the X-ray, SZ cluster gas recon-structions by assuming hydrostatic equilibrium and by using theminimisation method described in section 2.2. The lensing re-constructions were obtained directly by deprojecting the lensingpotential. The XSZ reconstructions reproduce the inner region ofthe cluster well, while the lensing reconstructions lack the res-olution to accurately resolve this innermost part. Between dis-tances r from the cluster centre of 150 h − kpc and 450 h − kpcboth reconstruction methods yield very good results. Fartheroutside the lensing reconstruction is still accurate, while theXSZ reconstruction becomes more and more unrealistic. Thisis partly due to different noise properties. But in the exampleshown in Figure 3 it is also due to the smaller box size usedfor the XSZ reconstructions and reconstruction artefacts that de-velop close to the perimeter of this box. The weighting func-tion w ( r ) , which was introduced to prevent non-local effects ofthese artefacts, was chosen to decrease from unity to zero be-tween r = h − kpc and r = h − kpc in these XSZ recon-structions. Thus they become unrealistic farther outside.
4. Probing the dynamical state of numericallysimulated clusters
In the previous section, novel methods to probe hydrostatic equi-librium in clusters of galaxies were tested with synthetic obser-vations of analytically modelled clusters. For a more realistictest we now apply these methods to a sample of four numeri-cally simulated galaxy clusters. The same sample was also usedin Paper I. The simulations were carried out by Klaus Dolagwith the GADGET-2 code (Springel 2005), a new version ofthe parallel TreeSPH simulation code GADGET (Springel et al.2001). The cluster regions were extracted from a dissipation-less(dark matter only) simulation with a box size of 479 h − Mpcof a flat Λ CDM model with Ω m = . h = . σ = . Ω b = .
04. The final mass resolutionwas m DM = . × h − M (cid:12) and m gas = . × h − M (cid:12) fordark-matter and gas particles within the high-resolution region,respectively. The simulations we use follow the dynamics of the E. Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters analytic_potRZ.fit_0 potRZ_xray_sz_recon_shifted_to_analytic.fit_0 potRZ_lens_recon_shifted_to_analytic.fit_0 analytic_potRZ.fit_0 potRZ_xray_sz_recon_shifted_to_analytic.fit_0 potRZ_lens_recon_shifted_to_analytic.fit_0 analytic_potRZ.fit_0 potRZ_xray_sz_recon_shifted_to_analytic.fit_0 potRZ_lens_recon_shifted_to_analytic.fit_0 analytic_potRZ.fit_0 potRZ_xray_sz_recon_shifted_to_analytic.fit_0 potRZ_lens_recon_shifted_to_analytic.fit_0 Z [ h − kp c ] Z [ h − kp c ] Z [ h − kp c ] Z [ h − kp c ] Z [ h − kp c ] R [ h − kpc] R [ h − kpc] R [ h − kpc] R [ h − kpc] R [ h − kpc]analytic XSZ reconstr., no noise lensing reconstr., no noise XSZ reconstr., noise lensing reconstr., noise Fig. 3.
Gravitational potential of an analytic halo and its reconstructions from synthetic X-ray and SZ observations and from syn-thetic lensing observations each with and without observational noise. The XSZ reconstructions work well even close to the clustercentre, where lensing observations lack the resolution to accurately resolve the central peak. However, farther outside the lensingreconstructions perform better, due to the different noise properties, but in the example shown here also because of the smaller boxsize used for the XSZ reconstructions.dark matter and the adiabatic evolution of the cluster gas, butthey ignore radiative cooling. They are described in more detailin Puchwein et al. (2005) and Dolag et al. (2005).
We produced synthetic X-ray, thermal SZ and lensing observa-tions of these four simulated clusters for 28 simulation snapshotsbetween redshifts 0.58 and 0.1 and three lines-of-sight using es-sentially the same methods as in section 3.2 for the analytic halo.The only difference is that we did not use a three-dimensionalgrid for projections along the line-of-sight. For the X-ray andSZ maps the X-ray luminosities and integrated Compton y pa-rameters of the gas particles are projected directly onto a two-dimensional 128 ×
128 grid by using the particles’ projectedSPH smoothing kernel. The convergence of the simulated clus-ters is found in a similar way by projecting the masses of bothgas and dark matter particles onto a two-dimensional grid, whosedimensions are again chosen such that one pixel corresponds toroughly 1 / M < r , XSZ ( r ) and M < r , lensing ( r ) .In Figure 4 we show these profiles for two clusters that didnot experience a major merger recently. For comparison we alsoshow the original simulated mass profile and the profile thatwould be expected from the original simulated gas distributionby assuming hydrostatic equilibrium. The latter is calculated like M < r , XSZ ( r ) , however directly from the simulated gas distributionrather than the reconstructed one. We again use 128 rings whichare equally spaced in the polar angle θ to numerically evaluatethe surface integral in Eq. (4), but as the simulated gas distribu- tion is not perfectly axisymmetric we use 128 sampling pointsequally spaced in the longitude angle for each of these rings.The gas density ρ and the pressure gradient ∇ p are calculated ateach sampling point using the SPH formalism, i.e. by summingup the contributions from all nearby particles using their SPHsmoothing kernels and the gradients thereof. For these relaxedclusters the XSZ reconstructed profiles and the lensing recon-structed profiles agree well with each other, with the originalmass profile and the profile obtained from the original gas dis-tribution. This shows that for such relaxed clusters this methodallows accurate and consistent lensing and XSZ mass estimates.The results also confirm that these clusters are close to hydro-static equilibrium.It is reassuring that this novel method to probe hydrostaticequilibrium works well for clusters that do not have a record ofrecent mergers. However clusters that do experience such vio-lent events may be even more interesting to study. In Figure 5we show a cluster at four different times during a merger. Foreach of these snapshots we show reconstructions of the cumula-tive mass profile from synthetic X-ray, SZ and lensing observa-tions, as well as the original mass profile and the profile obtainedfrom the simulated gas distribution. Again observational noisewas added to the synthetic maps used for the reconstruction. Wealso show X-ray maps of the cluster for each of the four snap-shots. These are however idealised noise-free versions and justmeant to illustrate what is going on in the cluster. To facilitatefollowing the merger we also show the approximate trajectoryof the relevant infalling subhalo in the X-ray maps.In the first snapshot (upper left panel) the main cluster halo isstill close to hydrostatic equilibrium. The lensing and XSZ massestimates still agree well for radii r smaller than the distance tothe infalling subhalo. In the second snapshot (upper right panel),after the subhalo has passed the main halo, shocked gas causes atoo large XSZ mass estimate from roughly the subhalo distanceoutwards. The mass profile obtained directly from the simulatedgas distribution shows the same behaviour and thus confirms thatthis is not an artefact of the reconstruction but a real, signifi-cant deviation from hydrostatic equilibrium, which is recoveredby the reconstruction or in this example even somewhat over-estimated. The lensing reconstruction still reproduces the origi-nal simulated mass profile well. Thus by comparing lensing and . Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters 7012345 0 100 200 300 400 500 600 M < r [ h − M (cid:12) ] distance r from cluster centre [ h − kpc]cluster g1 z=0.25 0 100 200 300 400 500 600distance r from cluster centre [ h − kpc]cluster g51 z=0.3original simulated massfrom simulated gas distributionlensing reconstr.XSZ reconstr. Fig. 4.
Cumulative mass profiles M < r ( r ) of relaxed simulated clusters g1 at redshift z = .
25 and g51 at redshift z = .
3. Profilesof the original simulated mass distribution, of the lensing and of the XSZ reconstructions are shown, as well as the profile obtaineddirectly from the simulated gas distribution by assuming hydrostatic equilibrium. The lensing and XSZ reconstructions are basedon synthetic observation that contain observational noise. For such relaxed clusters both the lensing and the XSZ reconstructionsagree very well with the original mass profile.XSZ cumulative mass profiles one can directly see the devia-tions from hydrostatic equilibrium. The third snapshot (lowerleft panel) shows that when the bow shock moves outward onecan also obtain too low cluster masses by assuming hydrostaticequilibrium during a merger. Again the effect can be seen in boththe mass profiles obtained directly from the simulated gas distri-bution and obtained from the three-dimensional XSZ gas recon-struction. In the fourth snapshot (lower right panel) hydrostaticequilibrium is already largely restored, even if one can still seethe pronounced bow shock in the X-ray map.These simulations show that deviations from hydrostaticequilibrium during mergers can be faithfully recovered by thecluster reconstructions methods introduced above.
To determine the typical scatter in cumulative mass profile re-constructions and quantify the significance of detections of devi-ations from hydrostatic equilibrium we repeated the reconstruc-tion of the merging simulated cluster shown in the upper rightpanel of Fig. 5 with different noise realisations and for differentlines-of-sight.For the left panel of Fig. 6 we used the same line-of-sight asin Fig. 5 but 50 different noise realisations for the synthetic X-ray, thermal SZ and lensing observations. The noise realisationswere obtained using different seeds for the random number gen-erator employed for adding noise to the synthetic observations.The mean XSZ and lensing reconstructed profiles and the 1- σ errors are shown as well as the profile of the original simulatedmass distribution and the profile obtained directly from the sim-ulated gas distribution by assuming hydrostatic equilibrium. Thedeviations from hydrostatic equilibrium are reliably detected. Asexpected for a cluster that contains substructure that violates ax-ial symmetry there are also some systematic deviations such thatthe mean profiles are not centred exactly on the simulated pro-files.For the right panel we started with a sample of synthetic lens-ing, X-ray and SZ observations along 50 different randomly ori- ented lines-of-sight. All contain realistic observational noise. Itturned out that for projections for which the merging subhaloresponsible for perturbing hydrostatic equilibrium is almost di-rectly in front of or behind the main halo detecting deviationsfrom hydrostatic equilibrium is less reliable. This is not surpris-ing as the signal from the region where hydrostatic equilibrium isstrongly perturbed is superimposed with a larger signal from themain halo, so that the contributions to such projections are diffi-cult to separate. For the right panel of Fig. 6 we thus decided toreject all 16 lines-of-sight for which the projected distance of therelevant subhalo from the main halo centre is less than 200 h − kpc, as well as one line-of-sight which happened to be inclinedby only 2 ◦ with respect to the cluster’s symmetry axis whichis to small for a faithful reconstruction. The mean and the 1- σ errors of the reconstructions that were based on the 33 remain-ing lines-of-sight are shown. Again deviations from hydrostaticequilibrium can be reliably detected.As discussed in Paper I reconstruction artefacts can appearclose to the perimeter of the box used for the reconstructions.As we can see in the right panel of Fig. 6 they can dominate theXSZ reconstructed cumulative mass profiles errors from roughly r = h − kpc outwards for some lines-of-sight, when using a1 . h − Mpc sidelength box for the XSZ reconstruction. Thuswhen the quality of the observations allows studying a largerregion one should also use an appropriately larger box for theXSZ reconstruction to avoid this problem.
5. Summary and discussion
We proposed a novel method to obtain three-dimensional recon-structions of a galaxy cluster’s gravitational potential and cumu-lative mass profile from X-ray and thermal SZ observations un-der the assumption of hydrostatic equilibrium and independentlydropping this assumption from lensing data. If only X-ray andthermal SZ data is available accurate reconstructions of relaxedclusters can be obtained. If, however, lensing data is available aswell, hydrostatic equilibrium can be probed, also in dynamicallyactive clusters, by comparing these independent reconstructions.
E. Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters
Fig. 5.
Cumulative mass profiles M < r ( r ) and X-ray surface brightness maps of simulated cluster g51at four different redshifts duringa merger. The approximate trajectory of the infalling subhalo is illustrated in the X-ray maps. Profiles of the original simulated massdistribution, of the lensing and of the XSZ reconstructions are shown, as well as the profile obtained directly from the simulatedgas distribution by assuming hydrostatic equilibrium. The lensing and XSZ reconstructions are based on synthetic observation thatcontain observational noise. The X-ray maps shown above are however idealised noise-free versions and were rotated such as to allhave the same orientation in space. . Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters 901234567 100 200 300 400 500 600 M < r [ h − M (cid:12) ] distance r from cluster centre [ h − kpc] 100 200 300 400 500 600distance r from cluster centre [ h − kpc]original simulated massfrom simulated gas distributionlensing reconstr.XSZ reconstr.different noise realisations different lines-of-sight Fig. 6.
Mean XSZ and lensing reconstructed cumulative mass profiles M < r ( r ) and their 1- σ errors of merging simulated clusterg51 obtained for different noise realisations (left panel) and different lines-of-sight (right panel). Profiles of the original simulatedmass distribution and the profiles obtained directly from the simulated gas distribution by assuming hydrostatic equilibrium areshown for reference. X-ray, SZ and lensing observations along one line-of-sight but with 50 different noise realisations were usedfor the reconstructions whose mean and 1- σ errors are shown in the left panel. For the right panel we started with a sample of noisysynthetic observations along 50 different randomly oriented lines-of-sight. However we rejected projections for which the mergingsubhalo is almost directly behind or in front of the main halo (projected distance < h − kpc) as well as one line-of-sight withan inclination of only 2 ◦ with respect to the clusters symmetry axis which is to small for a faithful reconstruction. The mean and the1- σ errors of the profiles reconstructed from the observations along the remaining 33 lines-of-sight are shown. For both the differentnoise realisations and the different lines-of-sight deviations from hydrostatic equilibrium can be reliably detected.The three-dimensional reconstructions are based on iterativeRichardson-Lucy deconvolution and assume only axial symme-try of the cluster halo with respect to an arbitrarily inclined axis.The X-ray and thermal SZ data are used to first reconstruct thethree-dimensional cluster gas density and temperature distribu-tion. No equilibrium assumption except local thermal equilib-rium is needed for that. Then the gravitational potential and thecumulative mass profile can be obtained from these reconstruc-tions under the assumption of hydrostatic equilibrium. For thelensing reconstructions we deproject the lensing potential ob-tained by a weak lensing or a combined weak and strong lensinganalysis. This yields the three-dimensional gravitational poten-tial, from which we can get independent cumulative mass pro-files by exploiting Gauss’s law. The X-ray and thermal SZ anal-ysis (abbreviated by XSZ throughout this work) and the lensinganalysis are then compared in order to probe hydrostatic equi-librium and to test the accuracy of mass estimates based on theassumption of hydrostatic equilibrium.These methods were tested with synthetic X-ray, thermal SZand lensing observations of analytically modelled and numeri-cally simulated galaxy clusters. Except where specifically notedrealistic observational noise was added to the synthetic observa-tions.For analytically modelled clusters in hydrostatic equilibriumwe found: – Consistent and accurate lensing and X-ray, SZ based cumu-lative mass profiles M < r , lensing ( r ) and M < r , XSZ ( r ) can be ob-tained. – The accuracy somewhat decreases for very small inclinationangle between the line-of-sight and the cluster’s symmetryaxis. – Higher accuracy for the iterative deprojection of the lens-ing potential for small inclination angles are achieved withspherically symmetrised priors. – Faithful three-dimensional reconstructions of the gravita-tional potential can be obtained from both lensing observa-tions and from an XSZ analysis.For analytically modelled clusters that are not in hydrostaticequilibrium, we showed that the deviations from equilibrium canbe effectively probed by a comparison of lensing and XSZ recon-structions even when realistic observational noise is present.From reconstructions based on synthetic observations of asample of numerically simulated galaxy clusters we conclude: – For clusters that did not experience recent mergers consistentand accurate lensing and XSZ cumulative mass profiles arefound. – Although these clusters are not perfectly axisymmetric andnoise is added to the synthetic data, the accuracy of recon-structed cumulative mass profiles is typically better than 10to 15% for both the X-ray, SZ and the lensing reconstruc-tions. – On the other hand in clusters in the process of merging devia-tions from hydrostatic equilibrium can be accurately probed,except for cases where the relevant merging subhalo appearsdirectly in front of or behind the main halo’s centre.
Acknowledgements.
We are deeply indebted to Klaus Dolag, who generouslyprovided us access to the numerical simulations of the cluster sample that wasused in this work. E. P. was supported by the German Science Foundation undergrant number BA 1369/6-1within the framework program SPP 1177.
References0 E. Puchwein & M. Bartelmann: Probing the dynamical state of galaxy clusters