CHARACTERIZING THE PROPERTIES OF CLUSTERS OF GALAXIES AS A FUNCTION OF LUMINOSITY AND REDSHIFT

The data were reduced using standard pipeline processing and the calibration implementation as of XMM Science Analysis Software (SAS) version 6.5. Background flares from soft protons were removed using light curve filtering in both soft (MOS: 0.3–10 keV, PN: 0.3–12 keV) and hard (MOS: 10–12 keV, PN: 12–14 keV) X-ray bands. For the soft band, light curves were binned in 10 s intervals while we used 100 s bins for the hard band. In both cases the data were not included during the time when the count rate exceeded 3σ above the quiescent count rate, indicating a proton flare.

The event files were also filtered for non-X-ray events by selecting only single and double pixel events for PN and single to quadruple events for MOS. Bad pixels and pixel columns were removed by applying the standard keywords in event selection; FLAG = 0 and #XMMEA_EM / #XMMEA_EP. The data reduction follows that described in Andersson et al. (2007).

All clusters are first analyzed using "standard" spectral analysis. This was conducted by extracting X-ray counts from a circular region centered on the peak of X-ray emission. The extraction radius was determined by estimating the radius at which a circle encompasses 90% of the background subtracted surface intensity.

The background was estimated using the surface intensity at the edge of the field. Background spectra were extracted in regions outside the source extraction region. Point sources were detected using SAS routine emldetect using only sources measured with a likelihood above 100.

The extracted spectra were fitted using XSPEC (Arnaud 1996) software, employing a MEKAL (Mewe et al. 1985, 1986; Kaastra 1992; Liedahl et al. 1995) thermal plasma model with solar abundances absorbed by a WABS (Morrison & McCammon 1983) model, which we allow to be fitted as a free parameter. In the fit, the redshift was fixed to the known optical value as listed in the NED database entry.

The results of these "standard" spectral fits including plasma temperature and metal abundances w.r.t. solar are listed in Table 1. The bolometric luminosities shown in the table are derived using the SPI analysis below. The redshift as given in the NED is also shown as well as the average effective exposure time after filtering. All observations had usable data for all three EPIC detectors with the exception of A665, A1413, A2261, A2597, and A3921 where only the two MOS detectors were available.

The cluster event files are modeled using SPI (Peterson et al. 2007) with a Monte Carlo model of the XMM-Newton EPIC Camera (Andersson et al. 2007). Within the SPI analysis, clusters are modeled as conglomerations of two-dimensional spatial Gaussians with individual spectral models. In this work, we use the MEKAL model to describe the thermal plasma. Model photons are simulated and propagated through the detector model adding background Monte Carlo events representing internal fluorescent lines, electronic noise and soft proton signals. Data and model photons are binned in three-dimensional adaptive bins and compared via a two-sample likelihood function. All parameters, spatial and spectral, are iterated in a Markov chain with an adaptive step length. All model samples in the converged part of the chain are used to reconstruct the cluster.

3.2.1. SPI Setup

3.2.2. Model Setup

3.2.3. Convergence

3.2.4. Simulated Clusters

In order to study the systematic effects induced by the X-ray mirror point-spread function (PSF) and limited statistics on faint sources, we simulate a massive, spherically symmetric, weak cooling core cluster at z = 0.1, 0.2, 0.4, and 0.8. The simulated cluster model consists of two superimposed beta models with core radii of 74 and 370 kpc, both with β = 0.7. The smaller, core component is assigned a spectral model with kT = 4 keV whereas the ambient component has kT = 9 keV. The normalization ratio of the cold to the hot component in terms of emission measure, ∫n_en_pdV, is 11/9 and the overall normalization is set so that the total bolometric luminosity of the cluster is L_Bol = 1.7 × 10⁴⁵ erg s⁻¹, a typical luminosity in our sample, present at all redshifts 0.1 ⩽ z < 0.8. The model is based on observations of the cool core clusters A2029 and A2241. Bolometric luminosities for the cluster sample are estimated using the 0.01–100 keV energy band as described in Section 4.1.

These simulated clusters all have the same background level (5 × 10⁻³ s⁻¹(')⁻²) and are assumed to have an effective exposure time of 20 ks in all three EPIC detectors. The assumed luminosity results in a total number of 850058, 209294, 46358, and 9353 cluster photons respectively for the z = 0.1, 0.2, 0.4, and 0.8 clusters in the adopted energy range (MOS:0.3–10 keV, pn:1.1–10 keV). As a reference, the same clusters are also simulated without background and reconstructed separately.

The apparent evolution with z in the derived properties for these identical clusters will be used to categorize the clusters in the sample. Figure 2 shows the original cluster model before propagated through the instrument model visualized with a luminosity and temperature map within the 1 Mpc radius. Luminosity per unit solid angle in this Figure is given in units of 10⁴⁴ erg s⁻¹ arcsec⁻² and temperature in keV.

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In order to follow the evolution of substructure in clusters we also simulate two sets of clusters with irregular morphology, one using a two "subcluster" model and the other, a three "subcluster" model. The two-subcluster model consists of two beta models, at 4 and 9 keV respectively, each with r_c = 74 kpc and β = 0.7 separated by 300 kpc. The total luminosity of the clusters is the same as for the original cool core cluster, L_Bol = 1.7 × 10⁴⁵ erg s⁻¹. The cluster model is shown in Figure 3. The three-subcluster model consists of three beta models, at 4, 9 and 9 keV respectively, with core radii of r_c = 74 kpc and β = 0.7. The clusters are separated by a triangle with sides of 300, 200, and 200 kpc. The total luminosity of the three clusters is L_Bol = 1.7 × 10⁴⁵ erg s⁻¹. This model is shown in Figure 4.

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We create 2 × 2 Mpc cluster luminosity and emission weighted temperature maps centered on the peak of cluster emission with 10 kpc bins. The method of creating median parameter maps is described in Andersson et al. (2007). Maps are created for each sample in the chain from iteration 750 to 2000 and are averaged by taking the median in each spatial point over the whole sample.

Point sources are removed by filtering out all cluster particles within a radius of −16.1 + log₁₀(L₂)15'', from the point source as detected by emldetect, where L₂ is the likelihood of detection. We find that this method successfully removes any point source contamination without removing the cluster flux in the region of the point source. This method is effective because particles that represent the point source emission are generally smaller in size whereas the cluster particles are larger and fill the region where the point source was removed. An example of such point source removal is shown in Figure 5, which shows maps of luminosity (top) and temperature (bottom) for Abell 3888 before (left) and after (right) point source removal.

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The reconstruction of the four simulated cool core clusters within 2 × 2 Mpc is shown in Figure 6 where luminosity per unit solid angle is given in units of 10⁴⁴ erg s⁻¹('')⁻² and temperature in units of keV. Here, the effects of the XMM PSF as well as the loss of photons with redshift is clearly seen as a distortion and flattening of the profile at high z.

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The maps for a subsample of clusters are shown in Figures 16–32 including the named clusters in Figure 1. We comment briefly on these maps in Section 5.4 and compare them to previously published results. Note that the temperatures in Table 1 do not necessarily agree with the scale shown in the figures. This is because these values are derived using two different methods. The tabulated values are determined using standard analysis (Section 3.1) requiring a single value of gas temperature for the whole cluster, whereas the parameter maps are created using a range of temperatures derived from the SPI runs (Section 3.2). The source of the disagreement has to do with the fact that SPI bases the spectral modeling on superpositions of thermal spectra with many different temperatures. The fact that the similarities between these spectra increase with higher temperature implies that the average temperature value of these also increases. It also becomes dependent on the prior range of temperatures (here: 0.5–15 keV). Whenever referring to the cluster average temperature, the tabulated values are used. A bolometric correction is applied to the cluster particles individually, where the temperature, abundance and emission measure of the particle are used to calculate a bolometric luminosity in the 0.01 eV to 100 keV range using the MEKAL spectral model. This is the luminosity that is listed in Table 1.

In all cases, except for the temperature and elemental abundance values used in Table 1, uncertainties are estimated by identifying five independent sets of samples in the Markov chain posterior. The samples are taken sequentially in ranges containing 250 iterations each, starting at iteration 750. The luminosity and temperature maps and the quantities derived from these are then evaluated separately and the uncertainty on these quantities is determined from the rms scatter within these five sets. This is a valid approximation of statistical uncertainty because each subset can be seen as an independent reconstruction of the cluster. The motions of particles in the parameter space are large and within a few iterations there is no memory about earlier states. This is largely due to the high dimensionality of the overall parameter space and the lack of local likelihood maxima. Of course, it would be more accurate to use more than five sets for this calculation but at least 250 iterations are needed in order to describe a cluster model and there is a limitation in CPU time to produce more iterations.

A common definition of a "cooling flow" cluster regards the central cooling time, t_cool = T/d(ln(T)), being much less than a "Hubble time," t_cool ≪ t_H. The calculation of the cooling time requires high-resolution spatially resolved spectroscopy which is not available for most of the clusters in our sample. Instead, we use the information about the steepness of the luminosity profile as well as the gradient of temperature in the cluster core to empirically select the clusters with cool cores.

The luminosity contrast is calculated as

$$\begin{equation} C_L = \frac{L\left(r \le 0.02 r_{500}\right)}{L\left(0.08 r_{500} \le r < 0.12 r_{500}\right)} \end{equation} \tag{ 1 }$$

where L is luminosity per solid angle. We use the luminosity based definition of r₅₀₀:

$$\begin{equation} r_{500} = 909 E(z) \left(\frac{L_{\rm bol}}{10^{44} \mathrm{erg s}^{-1}}\right)^{0.172} \mathrm{kpc}, \end{equation} \tag{ 2 }$$

where

$$\begin{eqnarray} &&E(z)=H(z)/H_0 = \nonumber \\ &&\sqrt{(1+z)^2 (1+\Omega _M z) - z (2+z) \Omega _\Lambda }. \end{eqnarray} \tag{ 3 }$$

This value of r₅₀₀ is derived by combining the M–T relation from Arnaud et al. (2005) and the L–T relation from Arnaud & Evrard (1999). The fractions of r₅₀₀ are used to account for the difference in size for clusters of various luminosities.

Similarly we define the temperature contrast as

$$\begin{equation} C_T = \frac{T\left(r \le 0.02 r_{500}\right) }{T\left(0.45 r_{500} \le r < 0.55 r_{500}\right)} \end{equation} \tag{ 4 }$$

where we choose to estimate the gradient farther from the core (0.5r₅₀₀) than for the luminosity contrast above because it is less sensitive to smearing by redshift-dependent effects. We find that the radius of 0.5r₅₀₀ is where we could get the most distinguishing power. We discuss the use of C_L and C_T toward identifying cooling clusters in Section 5.1.

One of the successful approaches toward assessing the amount of substructure in clusters is the power ratio method (Buote & Tsai 1995). It was used in Jeltema et al. (2005) for a sample of Chandra-observed clusters to establish that clusters are more dynamically active at high z. This method is based on the multipole expansion of the surface brightness Σ(R, ϕ) around the cluster centroid where the multipole moments a_m and b_m are

$$\begin{eqnarray} &&a_m(R) = \int _{R^{\prime } \le R} \Sigma (R,\phi) (R^{\prime })^m \cos m \phi ^{\prime } {d}^2 x^{\prime } \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&b_m(R) = \int _{R^{\prime } \le R} \Sigma (R,\phi) (R^{\prime })^m \sin m \phi ^{\prime } {d}^2 x^{\prime } \end{eqnarray} \tag{ 6 }$$

so that the powers in the multipole m can be written as

$$\begin{eqnarray} &&P_0 = \left(a_0 \mathrm{ln} (R) \right)^2 \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&P_m = \frac{1}{2 m^2 R^{2m}} \left(a^2_m + b^2_m \right). \end{eqnarray} \tag{ 8 }$$

Here, we define the location of the cluster centroid by the requirement that P₁ should be zero at this location. The above method is applied to the luminosity maps described in the previous section and the ratios P₂/P₀ and P₃/P₀ are calculated within a radius of 500 kpc. We expect that P₂/P₀ will be larger for elongated clusters whereas P₃/P₀ will be large for clusters with much substructure in the luminosity map.

We suggest that the variation of temperature inside the clusters can be estimated and characterized by taking a (non-standard) two point correlation of the temperature difference, weighted by products of luminosity so as to enhance temperature differences among the regions with the highest brightness. This is accomplished by taking the following sum over all 10 kpc × 10 kpc pixels in the generated maps:

$$\begin{equation} A(r_k)=\sqrt{ \sum ^{i,j} \sqrt{L_i L_j} (T_i - T_j)^2 / L_T } \\ \end{equation} \tag{ 9 }$$

where L_i is the luminosity of pixel i, T_i is the temperature and $L_T=\sum ^{i,j} \sqrt{L_i L_j}$. i and j are such that both (x_i, y_i) and (x_j, y_j) cover all points within 500 kpc of the centroid calculated in Section 4.4. A is binned based on the distance between i and j as 10k kpc<r_k ⩽ 10(k + 1) kpc, k = 0, ..., 99, where $r_k=\sqrt{(x_i-x_j)^2 + (y_i-y_j)^2}$ is the pixel distance and (x_i, y_i) is the spatial position of pixel i.

This statistic is designed to quantify the amount of distortion in the intra cluster medium, specifically regarding temperature features that can result from cluster mergers. Basically, it gives a "power spectrum" of strong temperature gradients over different scales, where the "strength" of the gradient is determined by $\sqrt{L_i L_j}$. In integrating this quantity over small or large distances it should be possible to distinguish small scale features, such as cooling cores, from larger scale disturbances. We discuss the application of this method in Section 5.3.1.

The first step is to identify clusters with cooling cores. We use the approach described in Section 4.3, and plot the luminosity contrast versus redshift in Figure 7 (left panel). Here, we also show the names for some well known clusters. Low L_X clusters (<4 × 10⁴⁴ erg s⁻¹) are shown as red circles, intermediate L_X clusters (4 × 10⁴⁴ erg s⁻¹ ⩽ L_X < 2 × 10⁴⁵ erg s⁻¹) as green stars and high L_X clusters (⩾2 × 10⁴⁵ erg s⁻¹) as blue squares. Likewise, the temperature contrast versus redshift is shown in Figure 7 (right panel).

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In Figure 7 we also show the calculated values for our simulated cool core cluster at redshift z = 0.1, 0.2, 0.4 and 0.8. It is easily seen that the effects of PSF smoothing and loss of photons due to increased distance severely affects both quantities. We show both the simulated clusters with (dark gray) and without (light gray) simulated background, totaling eight simulated clusters.

C_L is estimated at 0.1r₅₀₀ (∼10'' at z = 0.8) and it becomes increasingly difficult to estimate at high redshifts. This can be seen in the sharp drop in C_L in the trend of the simulated clusters above z = 0.4. At z = 0.8, the detection of cooling cores using C_L is no longer sensitive and thus we decide to rely on the C_T statistic alone for identification purposes above z = 0.6. We identify those clusters with a C_L above a line interpolated between the points set by the simulated clusters as cool core clusters. This cut corresponds to a linear interpolation of the points C_L = (7, 7, 5, 2.2) at z = (0.1, 0.2, 0.4, 0.6). This comfortably separates known cool core clusters (e.g., A2029) from disturbed and intermediate core clusters (e.g., A1689).

For C_T, the simulated clusters can be seen to have a trend that is more linear and not as dramatic as the trend in C_L. This is largely due to the larger radius (0.5r₅₀₀) used when calculating C_T. Using C_T to identify cool core clusters we make a cut corresponding to a straight line approximately following the trend of the simulated clusters so that clusters below the line, C_T ⩽ 0.85 + 0.15log z, are selected. These cuts select well known cooling clusters like A1068, A1835, and A2204 but fail to select clusters that are known not to have pronounced cool cores such as A1689 and A1413.

In subsequent plots we show the values for the reconstructions of the simulated clusters as filled circles and use this to measure how accurately a weak cooling core cluster at different redshifts can be resolved. The cuts are shown in Figure 7. In Figure 8 we also show C_T plotted against C_L, clearly showing the separation of cooling clusters like A1835 and A2204 (G, J, bottom right), nearly isothermal clusters such as A1413 and A1689 (B, K, center) and disturbed clusters like A2218 and A520 (H, E, top left). Out of 101 clusters, 31 are identified as cooling clusters.

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5.2.1. Power Ratios

In assessing the dynamical activity in the clusters in our sample we perform a multipole expansion of our luminosity maps as described in Section 4.4. The power ratios P₂/P₀ and P₃/P₀ are calculated to a radius of 500 kpc in order to quantify the amount of substructure in the sample. These are listed in Table 3.

Table 3. Power Ratios

Name	P2/P0(×10−7)	P3/P0(×10−7)
CIZAJ1645.4−7334	4.72 ± 0.95	0.41 ± 0.49
A1837	112 ± 7	2.37 ± 0.31
A3112	39.5 ± 1.6	0.085 ± 0.031
A1775	0.5 ± 0.33	3.86 ± 0.65
A399	62.0 ± 3.2	3.55 ± 0.31
A1589	298 ± 17	0.88 ± 0.51
A2065	208 ± 6	0.089 ± 0.032
A401	104 ± 3	0.92 ± 0.07
A2670	4.96 ± 0.91	4.07 ± 0.43
A2029	55.9 ± 2.1	0.051 ± 0.022
RXCJ1236.7−3354	21.6 ± 8.6	8.97 ± 2.72
RXCJ2129.8−5048	78.0 ± 7.8	27.1 ± 1.8
A2255	64.5 ± 5.0	0.58 ± 0.42
RXCJ0821.8+0112	106 ± 16	0.87 ± 0.58
RXCJ1302.8−0230	45.0 ± 11.7	0.59 ± 0.63
A1650	87.9 ± 2.3	0.24 ± 0.06
A1651	35.6 ± 3.1	0.12 ± 0.08
A2597	26.5 ± 0.6	0.058 ± 0.026
A1750	39.9 ± 5.7	1.84 ± 0.73
A478	46.4 ± 0.5	0.054 ± 0.004
A278	16.2 ± 2.1	1.23 ± 0.57
A2142	247 ± 9	0.15 ± 0.05
A3921	228 ± 10	2.2 ± 0.48
A13	88.2 ± 10.1	0.19 ± 0.12
A3911	344 ± 11	2.45 ± 0.51
RXCJ2319.6−7313	128 ± 18	0.22 ± 0.09
CL0852+1618	12.5 ± 9.6	22.9 ± 6.5
A3827	17.0 ± 1.7	0.96 ± 0.15
RXCJ0211.4−4017	18.3 ± 1.2	1.48 ± 0.55
A2241	7.01 ± 1.39	0.48 ± 0.36
PKS0745−19	36.0 ± 1.7	0.018 ± 0.01
RXCJ0645.4−5413	126 ± 6	2.11 ± 0.67
RXCJ0049.4−2931	13.3 ± 2.3	0.19 ± 0.14
A1302	28.9 ± 3.0	0.26 ± 0.23
RXCJ0616.8−4748	112 ± 18	3.0 ± 1.7
RXCJ2149.1−3041	29.3 ± 3.7	0.028 ± 0.015
RXCJ1516.3+0005	82.9 ± 4.0	0.63 ± 0.41
RXCJ1141.4−1216	19.4 ± 1.5	0.12 ± 0.11
RXCJ0020.7−2542	118 ± 6	1.36 ± 0.31
RXCJ1044.5−0704	47.0 ± 3.0	0.77 ± 0.09
RXCJ0145.0−5300	342 ± 29	2.63 ± 0.93
A1068	63.0 ± 4.4	1.21 ± 0.32
RXJ1416.4+2315	180 ± 20	3.37 ± 3.16
RXCJ0605.8−3518	47.0 ± 3.0	0.52 ± 0.17
A1413	192 ± 3	0.19 ± 0.1
RXCJ2048.1−1750	34.4 ± 4.4	3.92 ± 0.34
A3888	117 ± 7	6.33 ± 0.8
RXCJ2234.5−3744	119 ± 4	4.06 ± 0.49
A2034	84.5 ± 8.4	0.41 ± 0.21
A2204	3.81 ± 0.33	0.077 ± 0.045
RXCJ0958.3−1103	79.0 ± 9.1	0.14 ± 0.12
A868	105 ± 6	0.92 ± 0.98
RXCJ2014.8−2430	15.8 ± 1.2	0.37 ± 0.13
A2104	68.0 ± 16.7	2.83 ± 2.54
RXCJ0547.6−3152	29.4 ± 3.1	0.33 ± 0.14
A2218	72.8 ± 5.6	1.14 ± 0.36
A1914	32.4 ± 1.0	2.45 ± 0.26
A665	44.8 ± 3.5	7.75 ± 0.77
A1689	27.6 ± 2.0	0.64 ± 0.12
A383	1.63 ± 0.26	0.62 ± 0.17
A520	65.3 ± 11.2	3.87 ± 0.46
A2163	45.6 ± 5.9	10.7 ± 1.7
A209	98.8 ± 5.5	1.18 ± 0.83
A963	8.14 ± 1.32	1.34 ± 0.67
A773	83.2 ± 11.2	0.82 ± 0.39
A1763	218 ± 8	1.36 ± 1.3
A2261	12.9 ± 11.4	2.79 ± 1.7
A267	95.8 ± 9.2	0.92 ± 0.64
A2390	149 ± 7	2.97 ± 0.86
RXJ2129.6+0005	66.1 ± 1.4	0.21 ± 0.03
A1835	10.2 ± 0.4	0.36 ± 0.09
RXCJ0307.0−2840	14.6 ± 3.8	2.23 ± 0.51
E1455+2232	16.1 ± 0.9	0.12 ± 0.08
RXCJ2337.6+0016	282 ± 29	1.16 ± 0.42
RXCJ0303.8−7752	37.4 ± 13.7	2.01 ± 0.87
A1758	502 ± 13	0.78 ± 0.32
RXCJ0232.2−4420	50.6 ± 4.8	2.5 ± 0.62
ZW3146	11.6 ± 1.3	0.37 ± 0.08
RXCJ0043.4−2037	58.5 ± 12.9	2.31 ± 2.39
RXCJ0516.7−5430	296 ± 54	8.28 ± 5.55
RXJ0658−55	116 ± 3	8.73 ± 0.95
RXCJ2308.3−0211	23.3 ± 4.2	1.44 ± 0.74
RXJ2237.0−1516	198 ± 41	8.29 ± 7.19
RXCJ1131.9−1955	142 ± 18	2.03 ± 1.29
RXCJ0014.3−3022	5.54 ± 4.01	7.16 ± 1.1
MS2137−23	1.47 ± 0.96	0.37 ± 0.49
MS1208.7+3928	322 ± 293	23.4 ± 26.9
RXJ0256.5+0006	6.12 ± 2.42	64.0 ± 15.3
RXJ0318.2−0301	39.7 ± 20.2	4.11 ± 3.27
RXJ0426.1+1655	35.9 ± 37.5	4.66 ± 5.68
RXJ1241.5+3250	27.4 ± 17.9	12.6 ± 9.9
A851	304 ± 30	6.49 ± 2.24
RXCJ2228+2037	144 ± 27	8.23 ± 3.18
RXJ1347−1145	32.6 ± 1.0	0.87 ± 0.24
CL0016+16	114 ± 17	5.54 ± 2.87
MS0451.6−0305	82.3 ± 5.1	7.01 ± 2.19
RXJ1120.1+4318	137 ± 49	9.82 ± 7.75
MS1137.5+6625	11.5 ± 11.9	10.5 ± 7.8
MS1054.4−0321	220 ± 73	17.3 ± 14.6
WARPJ0152.7−1357	2880 ± 260	27.6 ± 19.4
CLJ1226.9+3332	11.1 ± 3.2	0.87 ± 0.4

Notes. Results of the power ratio analysis (Section 4.4) of the clusters.

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Here we divide the sample in three redshift bins; a local z bin (z <  0.1), a low-z bin (0.1 ⩽ z < 0.3) and a high-z bin (z > 0.3). The results of the multipole expansion are shown in Table 4 where we show the average values of P₂/P₀ and P₃/P₀ for the three samples. We also show the result obtained when using only the cooling core clusters identified in previous sections. These are not excluded from the other samples.

Table 4. Power Ratio Results

Sample	No.of objects	P2/P0(×10−7)	P3/P0(×10−7)
0.069 ⩽ z < 0.1	28	71 ± 3	1.7 ± 0.3
0.1 ⩽ z < 0.3	55	89 ± 5	2.2 ± 0.5
z ⩾ 0.3	18	65 ± 7	7.0 ± 2.8
Cooling clusters	31	39 ± 2	0.29 ± 0.07

Notes. Average values of the power ratios derived in different redshift bins.

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The values of P₂/P₀ and P₃/P₀ are plotted against z in Figures 9 and 10, respectively, where the simulated cool core clusters are shown as filled circles connected by black lines. In Figure 9 we also show the simulated two-component clusters as purple triangles connected by black lines and in Figure 10 we show the simulated three-component clusters as yellow triangles connected by black lines.

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The simulated cool core clusters represent the minimum amount of substructure that can be measured as they are simulated with perfect circular symmetry. This is true for clusters of similar luminosity and exposure. However, the P₃/P₀ values in Table 4 are higher than those of the simulated cool core clusters and above z = 0.3 most clusters have P₃/P₀ values similar to those of the simulated three-component clusters. This is most clearly seen around z = 0.4 where the data points cluster around the value of the three-component simulation. Toward z = 0.8 the simulated irregular cluster shows a decreasing value with redshift due to the smoothing effect introduced by PSF blurring and lack of photons. The data points around z = 0.6 and z = 0.8 show large P₃/P₀ values around 10⁻⁶, larger than for the simulated clusters.

To investigate further the proposed evolution of P₃/P₀ we restrict the analysis to clusters below z = 0.6 due to the apparent breakdown in the analysis as seen in the simulated clusters at z = 0.8. We also exclude the low-luminosity clusters (L_bol < 4 × 10⁴⁴ erg s⁻¹) from the sample since these are not present at all z. We bin the P₃/P₀ values for the remaining clusters in 5 bins of redshift (z = 0, 0.1, 0.2, 0.3, 0.4, 0.6) and fit these data points to a straight line in log z-log P₃/P₀ space. The binned data are shown along with the best-fit line in Figure 11. In order to distinguish between real and apparent evolution we interpolate the slope linearly for values of log P₃/P₀ from the simulated cool core clusters (gray circles) to the simulated clusters with irregular morphology (yellow stars). These interpolated lines are shown as dashed lines in Figure 11 where the red dashed line has the lowest χ² when compared to the data. Compared to the best line-fit (solid line) the dashed line models have Δχ² = 2.6, 1.1 and 3.3, respectively (from bottom to top). We conclude that the evolution in P₃/P₀ in addition to the redshift-dependent bias is only slightly more than 1σ significant.

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To compare our results with previous work we list our results for the clusters also found in Jeltema et al. (2005) along with their results in Table 5. These values were derived using Chandra data only. We find our values to be consistently about a factor 2 higher with a few exceptions but that they follow the same trend of low to high. This discrepancy could be related to the limited number of photons in the Chandra data causing structures to appear smoother but could also be due to artifacts in our modeling described further in following sections.

Table 5. Power Ratio Comparison

Name	P2/P0(×10−7)a	P2/P0(×10−7)	P3/P0(×10−7)a	P3/P0(×10−7)
A1413	58.0	192 ± 3	0.0525	0.19 ± 0.1
A2034	14.1	84.5 ± 8.4	0.476	0.41 ± 0.21
A2218	27.4	72.8 ± 5.6	0.233	1.14 ± 0.36
A1914	15.7	32.4 ± 1.0	0.709	2.45 ± 0.26
A665	19.4	44.8 ± 3.5	2.43	7.75 ± 0.77
A520	40.7	65.3 ± 11.2	2.43	3.87 ± 0.46
A963	5.03	8.14 ± 1.32	0.342	1.34 ± 0.67
A773	46.9	83.2 ± 11.2	−0.125	0.82 ± 0.39
A2261	4.57	12.9 ± 11.4	0.201	2.79 ± 1.7
A2390	58.0	149 ± 7	0.291	2.97 ± 0.86
A267	61.4	95.8 ± 9.2	−0.306	0.92 ± 0.64
RXJ2129.6+0005	17.6	66.1 ± 1.4	−0.0814	0.21 ± 0.03
A1758	188.0	502 ± 13	1.06	0.78 ± 0.32
ZW3146	4.42	11.6 ± 1.3	0.078	0.37 ± 0.08
MS2137−23	1.81	1.47 ± 0.96	0.00772	0.37 ± 0.49
CL0016+16	46.4	114 ± 17	0.316	5.54 ± 2.87
MS0451.6−0305	66.1	82.3 ± 5.1	2.19	7.01 ± 2.19
MS1137.5+6625	5.24	11.5 ± 11.9	0.115	10.5 ± 7.8
MS1054.4−0321	150.0	220 ± 73	10.3	17.3 ± 14.6
WARPJ0152.7−1357	264.0	2880 ± 260	12.4	27.6 ± 19.4
CLJ1226.9+3332	−0.821	11.1 ± 3.2	0.77	0.87 ± 0.4

Notes. Comparison of the power ratios in our analysis to the results using Chandra data by Jeltema et al. (2005). ^aDerived by Jeltema et al. (2005) from Chandra data.

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Using our definition of a cooling cluster, we treat these clusters separately in the subsequent analysis. The remaining clusters are divided into two redshift bins; 0.069 ⩽ z < 0.2 and z ⩾ 0.2. This results in 38 low-redshift clusters with $\bar{z}=0.11$ and 32 high-redshift clusters with $\bar{z}=0.39$.

The X-ray luminosity–temperature relations for these 3 samples are shown in Figure 12. We have used the bolometric luminosity as calculated within a 1 Mpc radius along with the temperatures given in Table 1. The low-redshift sample (red circles), the high-redshift sample (green stars), and the cooling cluster sample (blue squares) are shown along with their best-fit L_X–T relations (solid lines). The best-fit relations from the noncooling and cooling samples of Allen & Fabian (1998), at z = 0, are shown as dashed lines for comparison.

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Our best-fit values of L₆ and α_LT are shown in Table 6, where the bolometric X-ray luminosity is L_X = L₆(kT/6 keV$)^{\alpha _{LT}}$. We have added a systematic scatter in log(T) in order to achieve χ²/dof = 1.

It is obvious in this plot that it is of great importance to exclude cooling core clusters since these clusters have much higher L_X for a given T compared to noncooling clusters. It is of particular importance to select these clusters using a method unbiased in redshift. A selection bias at high redshift leading to the failure to identify cooling clusters properly could easily be interpreted as evolution. We have based our selection of cooling clusters on simulated identical clusters at various redshifts, propagating their photons through our detector model to best account for any distance-dependent systematic effect.

Fitting the data to a generalized L_X–T relation with a z dependence;

$$\begin{equation} L_X = L_{6} \left(\frac{T}{6\,\,\mathrm{keV}} \right) ^{\alpha _{LT}} (1+z)^{\beta _{LT}}, \end{equation} \tag{ 10 }$$

weak evolution is found with the best-fit values for the noncooling clusters shown in Table 7 along with fits for high luminosity (L_bol ⩾ 10⁴⁵ erg s⁻¹) noncooling clusters, high P₃/P₀ clusters (P₃/P₀ ⩾ 10⁻⁷) and for the cooling clusters separately.

The results for the noncooling clusters show weak evolution whereas the cooling cluster sample is consistent with no evolution. There is however large intrinsic scatter within the cooling sample. Interestingly, the sample containing the clusters with the highest luminosity substructure (high P₃/P₀, see Section 5.2.1) shows significant evolution with z. Slightly higher than that of the noncooling sample.

It is possible that this has to do with the incompleteness of the sample. A cut on P₃/P₀ selects a larger fraction of high-z clusters, since P₃/P₀ has a weak positive trend with z. High-redshift clusters tend to be more luminous by selection and this could cause a small bias in the observed evolution. We find that our results for the clusters without cool cores are in good agreement with theoretical models of cluster evolution that are based on balancing gas cooling with feedback at the entropy set by the cooling threshold (Voit et al. 2002; Voit 2005). This threshold K_c(T, t) is the entropy at which constant- entropy gas at temperature T radiates an energy equivalent to its thermal energy in time t. These models are also successful in explaining the observed L_X ∝ T³ behavior for clusters, contrary to early assumptions of pure gravitational self-similar collapse, where L_X ∝ T² was originally expected.

5.3.1. Temperature Correlation

Since the temperature two-point correlation function is new, we illustrate its power for four representative clusters that clearly show specific characteristics. A(r_k) is calculated for all spatial points, with 10 kpc resolution, within radius of 500 kpc of the cluster centroid.

First, we consider a well-known cooling core cluster, Abell 1835 (see Figure 13, the first panel). Here, A(r_k) is shown as a function of distance scale, r. As expected, there is considerable power at small spatial scales, implying a compact structure with temperature clearly different from the rest of the cluster. There is a sharp drop after 500 kpc where the core is no longer visible. Another extreme example is MS1054.4−0321 (Figure 13, the second panel), a highly substructured cluster consisting of multiple components. Here, the amplitude is high, distributed more uniformly over a large range of spatial scales, suggesting multiple high-luminosity components with different temperatures. As a third example, we also show the correlation function for a nearly isothermal cluster, A1689 (Figure 13, the third panel), where small temperature fluctuations lead to lower values across the range of spatial scales. Finally we show the correlation function for A2142 (Figure 13, fourth panel), where an offset of the cluster core causes the sharp drop seen in A1835 to be absent. The drop is gradual, implying an overall asymmetry in the temperature structure of the cluster.

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The temperature correlation function A(r_k) is integrated over 250–500 kpc (A₁) and over 750–1000 kpc (A₂) and this is displayed in Figure 14, plotted against redshift. This shows the magnitude of temperature gradients over small and large scales, respectively. These distance ranges are chosen since for the small scales, 250–500 kpc, the cooling cores dominate the statistic. To distinguish these fluctuations from larger scale, merger related disturbances we compare it to the 750–1000 kpc range where the core cannot be included (since we use a 500 kpc aperture).

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Strong cooling clusters such as A1835 and A2204 show large values of A₁ since the cool cores dominate the small scales. The same objects show low values of A₂ since these clusters are largely isothermal when the core is excluded. The same is true for our simulated cool core clusters, while the small scales cannot quite be resolved at high z. In contrast, disturbed clusters such as RXJ0658−55 or MS1054.4−0321 show generally lower values of A₁ and larger values of A₂ whereas nearly isothermal clusters such as A1689 or A1413 show low values for both. With the exception of a few low luminosity clusters at low-redshift, large-scale temperature structure (A₂) can be seen to increase slightly with redshift above z = 0.2 when compared to the simulated clusters. There is also an apparent drop in the temperature structure for smaller scales (A₁).

We note that the decrease in A₁ with redshift can be partly due to smoothing of the core caused by the PSF and loss of photons as seen in the evolution in the simulated clusters in Figure 14. However, it is unlikely that this effect is the cause of the increase in A₂ since temperature features are likely to be washed out by this effect. This is seen in the trend of A₂ for the simulated clusters.

In Figure 15 we plot A₂ against A₁ to show the separation of "relaxed" cool core clusters (lower right) from disturbed clusters (upper left).

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In this section, we describe briefly a few selected clusters in our sample, comparing our findings with previous results. The luminosity and temperature maps of these clusters are presented in Figures 16–32.

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5.4.1. Abell 1068

5.4.2. Abell 1413

5.4.3. Abell 1689

5.4.4. Abell 1750

5.4.5. Abell 1758

5.4.6. Abell 1775

5.4.7. Abell 1835

5.4.8. Abell 2029

5.4.9. Abell 2142

5.4.10. Abell 2204

5.4.11. Abell 520

5.4.12. MS0451.6−0305

5.4.13. MS1054.4−0321

5.4.14. MS2137−23

5.4.15. RXJ0658−55

5.4.16. RXJ1347−1145

5.4.17. ZW3146

Since the cluster sample was selected using all the public XMM data available at the time, it does not consist of a representative subset of clusters in general in the universe. However, many studies, and observation proposals, are based on either the most regular known clusters or the most complex. Hence, the sample should contain both these extremes of the cluster population which is what is compared in this work.

The X-ray background is modeled as opposed to subtracted which is the usual procedure in standard spectral modeling of extended X-ray sources. This introduces an uncertainty different from the unsubtracted background flux which is common when dealing with combinations of several sources of background with different spatial and spectral distributions. Here, the distortions of different backgrounds are modeled correctly, but instead, there is the possibility of cluster particles mimicking the background flux. This effect is small since the backgrounds are present across the entire field which is not the case for the majority of the cluster emission since all clusters are modeled out to at least 1 Mpc where the cluster emission tends to be faint. The low cluster flux at this radius does, however, impact on the precision of the temperature measurements at large radii. The nature of this uncertainty is seen in some temperature maps of median temperature as it reverts to the mean of the prior, which in this case is 7.75 keV.

The spectral shape of the instrumental background also means that it is not easily modeled by cluster particles. The particle background has a nearly flat spectral shape across the entire energy range whereas the X-ray photons are modulated by the energy-dependent effective area of the X-ray mirrors. This makes it statistically favorable to model these events using the background model which is not affected by mirrors as opposed to using the SPI particles.

In order to estimate the effect of the background modeling on the temperature at large radii we compare the temperature correlation results, A₁ and A₂, for the simulated regular clusters with and without background. These are shown as filled circles in dark and light gray respectively in Figures 14 and 15. The inclusion of background leads to an overestimation of approximately 25% in A₁ and 35% in A₂. This is comparable to the 1σ statistical errors and since the effect appears to be relatively independent of redshift we conclude that it does not affect the evolution of the temperature correlation.

Point sources can significantly alter maps of cluster luminosity and temperature. In the cases where we can identify the point sources from the original event file we remove the flux from these sources using the technique described above (Section 4.1, Figure 5). When a point source cannot be resolved it can have a large impact on the temperature maps even if the effect on the luminosity map is not visible. Ideally, Chandra data of the same cluster should be used in combination to identify the positions of point sources and to verify the accuracy of temperature features. Chandra data are, however, not available for all clusters in the sample and this type of analysis is beyond the scope of this work.

As mentioned earlier, the average temperature from the SPI modeling becomes slightly higher than when standard analysis routines are applied. This effect stems from the fact that we are using multiple particles in our modeling. Since the particles are extended and hundreds of particles are required to model a single cluster there will be multiple overlapping cluster particles at any given position. Each particle has a separate temperature and set of elemental abundances and this means that the model is inherently multiphase. There are many combinations of thermal spectra that can be constructed to agree with the observed data and the final model consists of a wide range of phases at each spatial position. This range is very much dependent on the prior range for the parameter in question. Here, these ranges are visualized by making use of the median of the range at each spatial coordinate and this value again depends on the prior range. Since the measured differences of thermal spectra in XMM become smaller with increasing temperature the median is naturally biased toward the higher part of the prior. This may lead to a potential luminosity bias in the bolometric correction since the luminosities are calculated separately for each smoothed particle which will cause a higher luminosity if the temperature is higher. We estimate this offset to be less than ∼20 % based on a comparison when using the temperature from the "standard" analysis in the bolometric correction.

Since the modeling itself is based on particles, even if between 100 and 1000 particles are required, there is a possibility for particle artifacts appearing in the luminosity and temperature maps. The cluster particles are constantly moving within the parameter space and interchanging positions with each other, and averaging over many Markov chain samples reduces this effect. However, since we are limited by CPU resources (see also Section 4.2) and only 250 samples are used in the error calculation these artifacts may still show up in the maps as lumpy structures and this may have a small effect increasing the power ratios. In comparing power ratios calculated from maps using 1500 samples with ones generated using 250 samples we establish that this effect corresponds to about a 20 % increase for ratios (P₂/P₀ or P₃/P₀) between 10⁻⁸ and 10⁻⁷ and less for larger values. However, this is not sufficient to explain the discrepancy in the comparison with the values from Jeltema et al. (2005).

In many of the plots, the trends in the relevant parameters with z for the simulated cool core clusters appear similar to the evolution for the sample as a whole. In most cases, the trend for the simulations only represents the minimum disturbance that can be measured and the similarity does not necessarily mean that the evolution is not real. In fact, we see from our simulations of irregular clusters that the trend for a cluster with high P₂/P₀ or P₃/P₀ is much more stable than that of the circularly symmetric clusters. Ideally, to make a more rigorous claim of evolution, clusters of many different morphologies should be simulated and the apparent evolution analyzed separately. This will be the subject of future work and we are limited here, again, by CPU constraints. Markov chains should also be run for several more iterations to limit model noise.

The derived luminosity–temperature relation for clusters clearly shows a large offset for cooling core clusters, as expected. This highlights the importance of applying a redshift-dependent criterion when excluding these sources. In contrast to previous work we do not attempt to correct for cooling cores by applying some standardized central profile. Instead we fit the relation for cooling clusters separately (see Allen & Fabian 1998) and find a similar slope of ∼3 but with a normalization ∼3 times higher than for the noncooling sample. The normalization of our L_X–T relation for cool core clusters is ∼50% higher for our sample compared to the findings of Allen & Fabian (1998). This is likely due to the use of ROSAT and Asca data to determine temperatures and luminosities, where cool core clusters would hard to model with the low spatial resolution and the limited energy band.

We also note that the scatter in the L_X–T relation is greater for cooling core clusters. We detect weak evolution in the noncooling L_X–T relation ∝(1 + z)^0.50±0.34 in good agreement with self-similar predictions and with an average slope of α_LT = 2.87 ± 0.14. This is in contrast to the findings of Vikhlinin et al. (2002) who detect strong evolution ∝(1 + z)^1.5±0.3. Due to the limited observable volume available at low z, high luminosity clusters are observed at mostly high redshifts. This could create a α_LT–β_LT bias. We note also that when studying β_LT for the cluster sample with P₃/P₀ > 10⁻⁷, the sample shows significant evolution. This may be an important effect to account for in large surveys relying on the L_X–T relation to measure the mass function of clusters over a range in z. Clearly, the L_X–T relation has to be measured in greater detail with larger samples of well exposed clusters in order to reach a consensus of its evolution. To measure the mass function of clusters to constrain cosmological parameters the knowledge of this evolution is crucial.

The power ratio analysis was applied uniformly for the 101 clusters within a 500 kpc radius. We find no significant increase in P₂/P₀ with z but detect an increase in P₃/P₀ implying a higher level of substructure at high redshift. Our sample of cooling core clusters shows significantly lower values of P₂/P₀ and P₃/P₀ than the noncooling sample, which is not surprising since cooling clusters generally are the most dynamically relaxed clusters.

Comparing this quantification of cluster structure with simulations of spherical clusters we show that all clusters in our sample are far from spherical, as determined by the value of P₂/P₀ and should not be modeled as such. This is easily realized when viewing the luminosity maps in Figures 16–32 which exhibit a large degree of ellipticity. The large amount of features visible in the temperature maps confirm that a large number of processes govern the properties of the intra-cluster medium, that cluster merging can give rise to highly complex thermal structure and that temperature seldom is a simple function of radius.

The SPI method inherently lends itself to a study of temperature structure of clusters, and to take advantage of it, we design a statistic, the "temperature two-point correlation," capable of distinguishing between cooling clusters, isothermal clusters and clusters with significant temperature differences over large distances, such as merging clusters. The analysis using this function hints at an increase in the large scale thermal structure of clusters with redshift, indicating more frequent merging in a denser universe. This indication could be confirmed using a joint approach utilizing both XMM-Newton and Chandra data sets.