THE GROWTH OF COOL CORES AND EVOLUTION OF COOLING PROPERTIES IN A SAMPLE OF 83 GALAXY CLUSTERS AT 0.3 < z < 1.2 SELECTED FROM THE SPT-SZ SURVEY

The clusters used in this work were selected based on their SZ signature in the 2500 deg² SPT-SZ survey. The SPT-SZ survey was completed in 2011 November, producing maps in three frequency bands (95, 150, and 220 GHz), with a key science goal of discovering clusters via the SZ effect (Staniszewski et al. 2009; Vanderlinde et al. 2010; Williamson et al. 2011; Benson et al. 2013; Reichardt et al. 2013).

The clusters considered in this work have additionally been observed with the Chandra X-Ray Observatory, with exposures typically sufficient to obtain ∼2000 X-ray source counts. The majority of the clusters have been observed through a Chandra X-Ray Visionary Project to obtain X-ray imaging of the 80 clusters detected with the highest SZ significance (ξ) in the first 2000 deg² of the 2500 deg² SPT-SZ survey at z > 0.4 (B. A. Benson et al., in preparation, hereafter B13). While B13 analyze the full XVP sample, we exclude 6 of the 80 clusters only observed with XMM-Newton, which does not have sufficient angular resolution to resolve the cool cores in typical high-redshift clusters. In addition, we include nine clusters at z > 0.3 also detected in the SPT-SZ survey that were observed by Chandra, a sub-sample that primarily consists of clusters observed either in previous Chandra GO and GTO proposals from the SPT-SZ collaboration, or in other proposals to observe SZ-selected clusters from the Atacama Cosmology Telescope (Marriage et al. 2011) and Planck (Planck Collaboration et al. 2011) collaborations. We note that every SPT-selected cluster that was targeted with Chandra yielded an X-ray detection—perhaps unsurprising due to the dependence of both techniques on a rich ICM.

The final sample used in this work, referred to hereafter as SPT-XVP, is summarized in Table 1. The sample consists of 83 clusters, spanning a redshift range of 0.3 < z < 1.2 and a mass range of ∼2 × 10¹⁴ < M₅₀₀ < 20 × 10¹⁴ M_☉/h₇₀. The clusters were all identified in the SPT-SZ survey maps with an SPT detection significance, ξ, spanning a range from 5.7 < ξ < 43. As was done in Vanderlinde et al. (2010), we predict the SPT survey completeness using cosmological and scaling relation constraints of the ξ-mass relation. We assume the ΛCDM cosmological constraints from Reichardt et al. (2013) when using a cosmic microwave background data set and the SPT cluster catalog. At our median redshift of z ∼ 0.7, the SPT-XVP sample is expected to be ∼50% complete at M₅₀₀ = 4 × 10¹⁴ M_☉/h₇₀ and nearly 100% complete at 6 × 10¹⁴ M_☉/h₇₀. These completeness thresholds are nearly redshift independent, varying by ≲15% over the redshift range of the sample.

Our basic data reduction and analysis follows closely that outlined in Vikhlinin et al. (2005) and Andersson et al. (2011). Briefly, this procedure includes filtering for background flares, applying the latest calibration corrections, and determining the appropriate blank sky background. In addition to using blank-sky backgrounds, we simultaneously model additional background components from Galactic sources as well as unresolved cosmic X-ray background (CXB) sources in off-source regions. Point sources were identified using an automated routine following a wavelet decomposition technique (Vikhlinin et al. 1998), and then visually inspected. Clumpy, asymmetric substructure was masked by hand, and excluded in calculations of the global temperature. The center of the cluster was chosen by iteratively measuring the centroid in a 250–500 kpc annulus. This choice, rather than the peak of emission, can play a significant role in whether or not the cluster is ultimately classified as a cool core or not—a subject we will return to in Section 4.

Global cluster properties (L_{X, 500}, M₅₀₀, T_{X, 500}, M_{g, 500}) used in this work are derived in B13, following closely the procedures described in Andersson et al. (2011). For each of these quantities, the subscript refers to the quantity measured within R₅₀₀—the radius within which the average enclosed density is 500 times the critical density. We estimate R₅₀₀ by requiring the measured quantities (T_X, M_g, Y_X) to satisfy a set of scaling relations between T_{X, 500}, M_{g, 500}, and Y_{X, 500} and M₅₀₀ (Vikhlinin et al. 2009). Each of these three scaling relations are individually satisfied by iteratively adjusting R₅₀₀. In this paper, we use R₅₀₀ from the Y_X–M relation only. Further details on both the data reduction and the derivation of global X-ray properties can be found in Vikhlinin et al. (2005) and Andersson et al. (2011), respectively.

The surface brightness profile for each cluster, extracted in the energy range 0.7–2.0 keV, is measured in a series of 20 annuli, with the outer radii for each annulus defined as

$$\begin{equation} r_i=1.5\,R_{500}\left(\frac{i}{20}\right)^{1.5} i=1\ldots n. \end{equation} \tag{ 1 }$$

Following the techniques described in Vikhlinin et al. (2006), we correct these surface brightness profiles for spatial variations in temperature, metallicity, and the telescope effective area. Calibrated surface brightness profiles (see Figure 16) are expressed as a projected emission measure integral, ∫n_en_pdl, where n_e and n_p are the electron and proton densities, respectively. We model the calibrated surface brightness profile with a modified beta model:

$$\begin{equation} n_en_p = n_0^2\frac{(r/r_c)^{-\alpha }}{(1+r^2/r_c^2)^{3\beta -\alpha /2}}\frac{1}{(1+r^3/r_s^3)^{\epsilon /3}}, \end{equation} \tag{ 2 }$$

where n₀ is the core density, and r_c and r_s are scaling radii of the core and extended components, following Vikhlinin et al. (2006). This three-dimensional (3D) model is numerically projected along the line of sight, yielding a model emission measure profile that is fit to the data. We estimate the 3D gas density assuming n_e = Zn_p and ρ_g = m_pn_eA/Z, where A = 1.397 and Z = 1.199 are the average nuclear charge and mass, respectively, for a plasma with metal abundance 30% of solar (0.3 Z_☉). The calibrated surface brightness profiles and best-fit projected gas density models for the full sample are shown in Figure 16.

In recent studies of high-redshift cool core clusters (e.g., Vikhlinin et al. 2007; Santos et al. 2008, 2010; Semler et al. 2012), the presence of a cool core has been quantified solely by the central cuspiness of the surface brightness profile. While measuring the central deprojected temperature and cooling time typically requires >10,000 X-ray counts, the surface brightness profile can often be constrained in the central region with as few as ∼500 counts, making this an inexpensive method of classifying cool cores. To classify a sample of high-redshift clusters as cool core or non-cool core, Vikhlinin et al. (2007) defined a "cuspiness" parameter,

$$\begin{equation} \alpha \equiv \left. \frac{d\log \rho _g}{d\log r}\right|_{r=0.04\,R_{500}}. \end{equation} \tag{ 3 }$$

This parameter has been shown to correlate well with the central cooling time for galaxy clusters at z ∼ 0 (Vikhlinin et al. 2007; Hudson et al. 2010). Vikhlinin et al. (2007) showed that α was typically higher for a low-redshift sample of clusters (z < 0.5), suggesting a rapid evolution in cool core strength (Vikhlinin et al. 2007). While easily measurable, this parameter has the drawback that it assumes that the cool core radius evolves at the same rate as the cluster radius (R₅₀₀).

An alternative measure of the surface brightness cuspiness is the "concentration" parameter, as defined by Santos et al. (2008):

$$\begin{equation} c_{{\rm SB}} \equiv \frac{F_{0.5\hbox{--}5.0\,\rm {keV}}(r <40\, \rm {kpc})}{F_{0.5\hbox{--}5.0\,\rm {keV}}(r<400 \,\rm {kpc})}, \end{equation} \tag{ 4 }$$

where F_{0.5–5.0 keV} is the X-ray flux in the energy bandpass 0.5–5.0 keV. This value can range from ∼0 (no flux peak), to 1 (all flux in central 40 kpc). This choice of parameter is relatively insensitive to redshift effects, such as worsening spatial resolution, reduced counts, and k-corrections (Santos et al. 2008), but has the potential drawback that it assumes no evolution in the cooling radius.

We will use both the full 3D density profile (n_e(r)), as well as the commonly-used single-parameter estimates of profile peakedness (α, c_SB) to trace the evolution of cool cores in this unique sample.

2.4.1. ρ_g(r), Φ(r), T_X(r)

Many recent works have verified the presence, or lack, of high-redshift cool cores via surface brightness quantities, as discussed in the previous section. We wish to extend this analysis further and quantify the cooling properties of the ICM. With only ∼2000 X-ray counts per cluster, we cannot perform a full temperature and density deprojection analysis, as is typically done at low redshift (e.g., Vikhlinin et al. 2006; Sun et al. 2009). Instead, motivated by earlier studies with the Einstein X-ray observatory (White et al. 1997; Peres et al. 1998), we combine our knowledge of the X-ray surface brightness and a coarse temperature profile with assumptions about the underlying dark matter distribution to produce best-guess 3D temperature profiles. This procedure, which will be described in complete detail in an upcoming paper (M. McDonald et al., in preparation), is summarized below.

First, a three-bin temperature profile is derived by extracting X-ray spectra in logarithmically-spaced annuli over the range 0 < r < R₅₀₀. These spectra were fit in xspec (Arnaud 1996) with a combined phabs(mekal) model.³⁶ In cases requiring additional background components, as determined from off-source regions of the field (see B13 for more details), an additional mekal (Galactic) or bremss (CXB) component was used, with temperatures fixed to 0.18 keV and 40 keV, respectively. For the source model, we fix the abundance to 0.3 Z_☉ and the hydrogen absorbing column, n_H, to the average from the Leiden-Argentine-Bonn survey (Kalberla et al. 2005). The resulting temperature profiles for the full sample are shown in Figure 16.

In order to model the underlying dark matter potential, we use a generalized Navarro–Frenk–White (GNFW; Zhao 1996; Wyithe et al. 2001) profile:

$$\begin{equation} \rho _{D}=\frac{\rho _{D,0}}{(r/r_s)^{\beta _{D}}(1+r/r_s)^{3-\beta _{D}}}, \end{equation} \tag{ 5 }$$

where ρ_{D, 0} is the central dark matter density, r_s is a scale radius related to the halo concentration by C = R₂₀₀/r_s, and β_D is the inner slope of the dark matter profile. This model is similar to the NFW (Navarro et al. 1997) profile at large radii, but has a free "cuspiness" parameter at small radii. We estimate the initial values of ρ_{D, 0} and r_s using the measured M₅₀₀ and the mass–concentration relation (Duffy et al. 2008).

Given an assumed 3D functional form of both the dark matter (Equation (5)) and gas density profiles (Section 2.3, Equation (2)), and further assuming a negligible contribution from stars to the total mass, we can derive the 3D temperature profile by combining hydrostatic equilibrium,

$$\begin{equation} \frac{dP}{dr} = -\frac{GM(r)\rho (r)}{r^2}, \end{equation} \tag{ 6 }$$

with the ideal gas law (P = n_TkT, where n_T = n_e + n_p). This temperature profile is projected along the line of sight (weighted by $n_e^2 T^{1/2}$), producing a two-dimensional (2D) temperature profile which is compared to the data, allowing a calculation of χ². We repeat this process, varying both the normalization of the GNFW halo (ρ_{D, 0}), and thus the total dark matter mass, as well as the inner slope (β_D), while requiring that the mass–concentration (Duffy et al. 2008) and M₅₀₀–P₅₀₀ (Nagai et al. 2007) relations are always satisfied (removing r_s and P₅₀₀ as free parameters), until a stable minimum in the χ²(ρ_{D, 0}, β_D) plane is found. The net result of this process is a 3D model of the gas density, gas temperature, and gravitational potential for each cluster (see Figure 16).

2.4.2. t_cool(r), K(r), (r)

While a centrally concentrated surface brightness profile is an excellent indicator that the ICM is cooling rapidly (e.g., Hudson et al. 2010), we ultimately would like to quantify, in an absolute sense, the strength of this cooling. Classically, clusters have been identified as "cooling flows" if the cooling time in the central region is less than the age of the universe, with the cooling time defined as:

$$\begin{equation} t_{{\rm cool}} = \frac{3}{2}\frac{n_TkT}{n_e n_{{\rm H}}\Lambda (T,Z)}, \end{equation} \tag{ 7 }$$

where Λ(T, Z) is the cooling function for an optically thin plasma (Sutherland & Dopita 1993). Similarly, the specific entropy of the gas is defined as:

$$\begin{equation} K = \frac{kT}{n_e^{2/3}}. \end{equation} \tag{ 8 }$$

Both the central cooling time and central entropy are smallest in the centers of cool core clusters, and are distributed bimodally over the full cluster population (e.g., Cavagnolo et al. 2009; Hudson et al. 2010). In nearby, well-studied clusters, the central cooling time and entropy are well-defined, as both of these functions tend to flatten at radii less than ∼10 kpc. However, for lower signal-to-noise data, the measurement of central cooling time is strongly dependent on the choice of bin size (see e.g., White et al. 1997; Peres et al. 1998), and inward extrapolation can be risky if a flattening of the surface brightness profile is not observed due to poor spatial sampling. For this work, we choose as our central bin r < 0.012 R₅₀₀ (r ≲ 10 kpc), which roughly corresponds to the first data point in our surface brightness profiles. While these quantities are not truly "central," this choice allows us to avoid the increasingly large uncertainty associated with extrapolating our temperature and density fits as r → 0.

Following White et al. (1997), we estimate the classical mass deposition rate, $\dot{M}(r)$, using the following formula:

$$\begin{equation} \frac{dM}{dt}(r_i)=\frac{L_X(r_i)-(\Delta \phi (r_i)+\Delta h(r_i)){\dot{M}}({<}r_{i-1})}{h(r_i)+f(r_i)\Delta \phi (r_i)}, \end{equation} \tag{ 9 }$$

where L_X(r_i) is the X-ray luminosity in shell i, Δϕ(r_i) is the change in the gravitational potential across shell i, h(r_i) is the temperature in units of energy per particle mass, h(r_i) = (5/2)(kT(i)/μm_p), and f(r_i) is the fraction of the shell that the gas crosses before dropping out of the flow. This equation calculates the cooling rate due to X-ray radiation (dM/dt∝L_X/kT) corrected for the gravitational work done on the gas as it falls inward toward the center of the cluster's gravitational potential. There are currently no constraints on what f(r_i) should be, so we choose the mid-point (f(r_i) = 0.5). We note that varying f(r_i) from 0 to 1 typically alters the estimate of dM/dt by only ∼5%. We integrate the mass deposition rate out to the radius at which the cooling time equals the age of the universe at the epoch of the cluster. The resulting dM/dt(r < r_cool) represents the time-averaged cooling rate if the cluster as we currently observe it has been in equilibrium for all time. We note that, by this definition, our sample ought to have overall smaller cooling radii due to the fact that these high-redshift clusters have had less time to cool than their low-redshift counterparts.

In previous studies (e.g., Hudson et al. 2010), the central entropy and cooling time are calculated from a combination of 3D, central electron density, n_{e, 0}, and a 2D, aperture temperature measured in some small aperture (e.g., r ≲ 0.05 R₅₀₀). For clusters with only ∼2000 X-ray counts, this central aperture may only contain ∼100 counts, making the estimate of a central temperature complicated. However, in cool core clusters, where a significant fraction of the flux originates from this small aperture, we can measure a reliable temperature and compare to our 3D models described above.

In Figure 1, we show the measured spectroscopic temperature (kT_{0, 2D}) in an aperture of r < 0.1 R₅₀₀ (with AGN masked), where the outer radius was chosen to maximize the number of X-ray counts, while still capturing the central temperature drop in cool core clusters, following the universal profile shown in Vikhlinin et al. (2006). The modeled 3D temperature profile was projected onto this same aperture (kT_{0, 3D}) to enable a fair comparison of the two quantities. While the uncertainty in the 2D temperature for these small X-ray apertures is high, there appears to be good agreement between the models and data (reduced χ² = 87.2/83), suggesting that this technique is able to recover the "true" central temperature of the cluster.

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Based on this agreement, we feel confident extrapolating inward into a regime without sufficient X-ray counts to measure the spectroscopic temperature and proceed throughout Section 3 utilizing the central (r < 0.012 R₅₀₀), deprojected model temperatures to calculate t_{cool, 0} and K₀. In Section 4 we will return to the comparison between 2D and 3D quantities to determine the dependence of our results on this extrapolation and our choice of models. In Table 2 we provide the results of both the 2D and 3D analyses for all 83 clusters in the SPT-XVP sample.

Table 2. Cooling Properties of 83 Clusters in the SPT-XVP Sample

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Cluster Name	ne, 0	kT0	cSB	α	K0	tcool, 0	dM/dt7.7	dM/dtUniv
	(cm−3)	(keV)			(keV cm2)	(Gyr)	(M☉ yr−1)	(M☉ yr−1)
SPT-CLJ0000-5748	0.164$^{+0.004}_{-0.004}$	4.2 ± 0.6	0.29$^{+0.03}_{-0.04}$	1.00 ± 0.04	14.1$^{+ 2.2}_{- 2.1}$	0.18$^{+ 0.03}_{- 0.03}$	437.3$^{+ 68.1}_{- 51.9}$	437.3$^{+ 68.1}_{- 51.9}$
SPT-CLJ0013-4906	0.007$^{+0.000}_{-0.000}$	8.0 ± 1.6	0.03$^{+0.00}_{-0.02}$	0.09 ± 0.01	229.8$^{+ 50.9}_{- 49.5}$	6.83$^{+ 1.57}_{- 1.51}$	...	33.3$^{+ 8.5}_{- 5.6}$
SPT-CLJ0014-4952	0.004$^{+0.000}_{-0.000}$	7.4 ± 0.9	0.03$^{+0.00}_{-0.01}$	0.03 ± 0.00	309.6$^{+ 42.1}_{- 41.2}$	11.44$^{+ 1.63}_{- 1.58}$	...	...
SPT-CLJ0033-6326	0.042$^{+0.002}_{-0.002}$	2.3 ± 1.4	0.08$^{+0.01}_{-0.03}$	0.77 ± 0.06	19.2$^{+ 12.4}_{- 11.8}$	0.47$^{+ 0.31}_{- 0.29}$	57.1$^{+ 87.8}_{- 21.5}$	83.1$^{+ 127.8}_{- 31.4}$
SPT-CLJ0037-5047	0.009$^{+0.001}_{-0.001}$	5.1 ± 7.3	0.08$^{+0.03}_{-0.04}$	0.41 ± 0.14	115.7$^{+ 183.8}_{- 161.9}$	3.67$^{+ 6.13}_{- 5.08}$	0.0$^{+ -0.0}_{- 0.0}$	0.0$^{+ -0.0}_{- 0.0}$
SPT-CLJ0040-4407	0.018$^{+0.000}_{-0.000}$	7.5 ± 1.9	0.07$^{+0.00}_{-0.02}$	0.38 ± 0.04	108.0$^{+ 29.9}_{- 28.9}$	2.34$^{+ 0.67}_{- 0.64}$	86.2$^{+ 29.6}_{- 17.6}$	159.8$^{+ 54.9}_{- 32.5}$
SPT-CLJ0058-6145	0.031$^{+0.001}_{-0.001}$	2.4 ± 1.7	0.06$^{+0.00}_{-0.02}$	0.63 ± 0.06	24.0$^{+ 18.5}_{- 17.8}$	0.64$^{+ 0.51}_{- 0.48}$	25.4$^{+ 70.3}_{- 10.8}$	15.2$^{+ 42.1}_{- 6.4}$
SPT-CLJ0102-4603	0.025$^{+0.001}_{-0.001}$	0.9 ± 0.5	0.07$^{+0.01}_{-0.02}$	0.66 ± 0.09	10.3$^{+ 6.1}_{- 5.8}$	0.29$^{+ 0.18}_{- 0.16}$	58.7$^{+ 74.1}_{- 21.0}$	58.7$^{+ 74.1}_{- 21.0}$
SPT-CLJ0102-4915	0.009$^{+0.000}_{-0.000}$	16.2 ± 2.3	0.03$^{+0.01}_{-0.01}$	0.06 ± 0.00	364.6$^{+ 54.5}_{- 53.8}$	7.10$^{+ 1.09}_{- 1.07}$	...	...
SPT-CLJ0123-4821	0.004$^{+0.000}_{-0.000}$	7.3 ± 1.5	0.04$^{+0.00}_{-0.02}$	0.09 ± 0.04	276.3$^{+ 61.9}_{- 60.1}$	9.77$^{+ 2.28}_{- 2.18}$	...	...
SPT-CLJ0142-5032	0.005$^{+0.000}_{-0.000}$	8.5 ± 1.7	0.03$^{+0.00}_{-0.02}$	0.06 ± 0.01	307.7$^{+ 70.5}_{- 67.7}$	10.03$^{+ 2.43}_{- 2.29}$	...	...
SPT-CLJ0151-5954	0.004$^{+0.000}_{-0.000}$	6.2 ± 2.8	0.04$^{+0.00}_{-0.02}$	0.02 ± 0.02	264.9$^{+ 130.5}_{- 123.3}$	10.62$^{+ 5.46}_{- 5.02}$	...	...
SPT-CLJ0156-5541	0.008$^{+0.000}_{-0.000}$	9.7 ± 2.5	0.06$^{+0.00}_{-0.02}$	0.07 ± 0.06	238.6$^{+ 70.9}_{- 66.9}$	6.06$^{+ 1.92}_{- 1.76}$	...	...
SPT-CLJ0200-4852	0.017$^{+0.000}_{-0.000}$	7.0 ± 3.0	0.05$^{+0.00}_{-0.02}$	0.52 ± 0.05	105.5$^{+ 48.6}_{- 47.0}$	2.42$^{+ 1.14}_{- 1.09}$	...	2.4$^{+ 1.9}_{- 0.7}$
SPT-CLJ0212-4657	0.004$^{+0.000}_{-0.000}$	8.2 ± 2.4	0.04$^{+0.00}_{-0.02}$	0.04 ± 0.01	340.7$^{+ 108.6}_{- 104.5}$	12.14$^{+ 4.03}_{- 3.81}$	...	...
SPT-CLJ0217-5245	0.019$^{+0.000}_{-0.000}$	1.6 ± 1.5	0.06$^{+0.00}_{-0.02}$	0.75 ± 0.04	21.8$^{+ 21.3}_{- 20.7}$	0.73$^{+ 0.72}_{- 0.69}$	5.8$^{+ 100.7}_{- 2.8}$	5.8$^{+ 100.7}_{- 2.8}$
SPT-CLJ0232-5257	0.020$^{+0.001}_{-0.001}$	1.6 ± 0.8	0.04$^{+0.00}_{-0.02}$	0.64 ± 0.05	21.3$^{+ 12.2}_{- 11.6}$	0.70$^{+ 0.41}_{- 0.38}$	40.4$^{+ 46.6}_{- 14.1}$	40.4$^{+ 46.6}_{- 14.1}$
SPT-CLJ0234-5831	0.058$^{+0.001}_{-0.001}$	5.7 ± 0.6	0.18$^{+0.01}_{-0.03}$	0.80 ± 0.05	38.1$^{+ 4.4}_{- 4.3}$	0.63$^{+ 0.08}_{- 0.08}$	427.8$^{+ 46.7}_{- 38.4}$	427.8$^{+ 46.7}_{- 38.4}$
SPT-CLJ0236-4938	0.011$^{+0.000}_{-0.000}$	3.1 ± 1.7	0.06$^{+0.00}_{-0.02}$	0.54 ± 0.04	61.8$^{+ 35.9}_{- 35.0}$	2.20$^{+ 1.30}_{- 1.25}$	2.2$^{+ 2.8}_{- 0.8}$	2.2$^{+ 2.8}_{- 0.8}$
SPT-CLJ0243-5930	0.015$^{+0.000}_{-0.000}$	6.4 ± 1.9	0.05$^{+0.00}_{-0.02}$	0.35 ± 0.05	104.3$^{+ 31.9}_{- 31.1}$	2.55$^{+ 0.80}_{- 0.77}$	15.2$^{+ 6.2}_{- 3.4}$	15.2$^{+ 6.2}_{- 3.4}$
SPT-CLJ0252-4824	0.008$^{+0.000}_{-0.000}$	2.3 ± 1.0	0.03$^{+0.00}_{-0.02}$	0.44 ± 0.06	58.6$^{+ 26.5}_{- 25.7}$	2.47$^{+ 1.15}_{- 1.09}$	2.3$^{+ 1.7}_{- 0.7}$	2.3$^{+ 1.7}_{- 0.7}$
SPT-CLJ0256-5617	0.004$^{+0.000}_{-0.000}$	14.2 ± 3.1	0.04$^{+0.00}_{-0.02}$	0.04 ± 0.01	547.9$^{+ 130.1}_{- 125.9}$	14.71$^{+ 3.64}_{- 3.47}$	...	...
SPT-CLJ0304-4401	0.003$^{+0.000}_{-0.000}$	10.1 ± 2.4	0.02$^{+0.01}_{-0.01}$	0.04 ± 0.01	441.8$^{+ 112.6}_{- 109.6}$	14.72$^{+ 3.88}_{- 3.73}$	...	...
SPT-CLJ0304-4921	0.055$^{+0.001}_{-0.001}$	4.0 ± 0.8	0.13$^{+0.01}_{-0.02}$	0.76 ± 0.03	28.0$^{+ 5.6}_{- 5.5}$	0.53$^{+ 0.11}_{- 0.11}$	139.0$^{+ 31.9}_{- 21.9}$	194.7$^{+ 44.7}_{- 30.7}$
SPT-CLJ0307-5042	0.007$^{+0.000}_{-0.000}$	7.2 ± 2.2	0.05$^{+0.00}_{-0.02}$	0.16 ± 0.05	204.9$^{+ 67.4}_{- 65.6}$	6.31$^{+ 2.13}_{- 2.05}$	...	...
SPT-CLJ0307-6225	0.003$^{+0.000}_{-0.000}$	6.4 ± 1.2	0.03$^{+0.00}_{-0.02}$	0.04 ± 0.01	322.7$^{+ 66.6}_{- 64.1}$	13.91$^{+ 3.04}_{- 2.87}$	...	...
SPT-CLJ0310-4647	0.025$^{+0.001}_{-0.001}$	3.1 ± 1.1	0.06$^{+0.00}_{-0.02}$	0.51 ± 0.05	35.9$^{+ 14.4}_{- 13.7}$	0.97$^{+ 0.40}_{- 0.38}$	26.3$^{+ 15.3}_{- 7.1}$	13.9$^{+ 8.1}_{- 3.7}$
SPT-CLJ0324-6236	0.025$^{+0.001}_{-0.001}$	3.5 ± 1.7	0.09$^{+0.02}_{-0.03}$	0.44 ± 0.07	40.8$^{+ 21.6}_{- 20.7}$	1.05$^{+ 0.57}_{- 0.54}$	49.3$^{+ 48.8}_{- 16.4}$	49.3$^{+ 48.8}_{- 16.4}$
SPT-CLJ0330-5228	0.008$^{+0.000}_{-0.000}$	1.5 ± 0.7	0.03$^{+0.00}_{-0.02}$	0.47 ± 0.05	37.1$^{+ 16.2}_{- 15.9}$	1.63$^{+ 0.72}_{- 0.71}$	20.6$^{+ 15.2}_{- 6.1}$	20.6$^{+ 15.2}_{- 6.1}$
SPT-CLJ0334-4659	0.057$^{+0.001}_{-0.001}$	4.3 ± 0.6	0.18$^{+0.01}_{-0.03}$	0.84 ± 0.03	29.4$^{+ 4.4}_{- 4.2}$	0.53$^{+ 0.08}_{- 0.08}$	73.9$^{+ 11.4}_{- 8.7}$	73.9$^{+ 11.4}_{- 8.7}$
SPT-CLJ0346-5439	0.037$^{+0.001}_{-0.001}$	1.5 ± 0.8	0.08$^{+0.01}_{-0.02}$	0.71 ± 0.04	13.7$^{+ 7.3}_{- 7.0}$	0.37$^{+ 0.20}_{- 0.19}$	104.0$^{+ 105.5}_{- 34.8}$	104.0$^{+ 105.5}_{- 34.8}$
SPT-CLJ0348-4515	0.015$^{+0.000}_{-0.000}$	2.4 ± 1.5	0.06$^{+0.00}_{-0.02}$	0.50 ± 0.05	40.7$^{+ 26.0}_{- 25.0}$	1.39$^{+ 0.91}_{- 0.86}$	20.1$^{+ 31.2}_{- 7.6}$	20.1$^{+ 31.2}_{- 7.6}$
SPT-CLJ0352-5647	0.015$^{+0.000}_{-0.000}$	2.7 ± 1.5	0.07$^{+0.02}_{-0.03}$	0.43 ± 0.05	44.2$^{+ 25.4}_{- 24.5}$	1.46$^{+ 0.86}_{- 0.81}$	23.8$^{+ 28.5}_{- 8.4}$	23.8$^{+ 28.5}_{- 8.4}$
SPT-CLJ0406-4805	0.025$^{+0.001}_{-0.001}$	3.1 ± 2.5	0.09$^{+0.01}_{-0.03}$	0.66 ± 0.07	36.5$^{+ 30.4}_{- 28.9}$	0.98$^{+ 0.84}_{- 0.78}$	9.9$^{+ 36.2}_{- 4.3}$	9.9$^{+ 36.2}_{- 4.3}$
SPT-CLJ0411-4819	0.005$^{+0.000}_{-0.000}$	7.8 ± 1.9	0.03$^{+0.00}_{-0.02}$	0.05 ± 0.01	274.3$^{+ 74.5}_{- 71.9}$	9.18$^{+ 2.60}_{- 2.47}$	...	...
SPT-CLJ0417-4748	0.086$^{+0.002}_{-0.002}$	6.3 ± 0.8	0.21$^{+0.02}_{-0.03}$	0.80 ± 0.03	32.5$^{+ 4.5}_{- 4.4}$	0.45$^{+ 0.07}_{- 0.06}$	517.0$^{+ 74.2}_{- 57.7}$	517.0$^{+ 74.2}_{- 57.7}$
SPT-CLJ0426-5455	0.003$^{+0.000}_{-0.000}$	8.9 ± 15.1	0.03$^{+0.00}_{-0.02}$	0.04 ± 0.01	407.6$^{+ 718.4}_{- 688.2}$	14.77$^{+26.46}_{-24.83}$	0.0$^{+ -0.0}_{- 0.0}$	0.0$^{+ -0.0}_{- 0.0}$
SPT-CLJ0438-5419	0.023$^{+0.000}_{-0.000}$	8.9 ± 1.5	0.07$^{+0.00}_{-0.02}$	0.44 ± 0.03	110.3$^{+ 19.4}_{- 19.1}$	2.08$^{+ 0.38}_{- 0.37}$	91.7$^{+ 18.3}_{- 13.1}$	207.4$^{+ 41.3}_{- 29.5}$
SPT-CLJ0441-4855	0.047$^{+0.001}_{-0.001}$	3.4 ± 1.0	0.13$^{+0.02}_{-0.03}$	0.62 ± 0.03	26.1$^{+ 7.7}_{- 7.5}$	0.55$^{+ 0.17}_{- 0.16}$	230.1$^{+ 89.1}_{- 50.2}$	154.7$^{+ 59.9}_{- 33.7}$
SPT-CLJ0446-5849	0.004$^{+0.001}_{-0.001}$	7.6 ± 2.1	0.02$^{+0.00}_{-0.02}$	0.03 ± 0.02	286.9$^{+ 110.5}_{- 95.5}$	9.95$^{+ 4.41}_{- 3.58}$	...	...
SPT-CLJ0449-4901	0.004$^{+0.000}_{-0.000}$	9.8 ± 5.8	0.03$^{+0.00}_{-0.02}$	0.12 ± 0.06	358.9$^{+ 223.3}_{- 215.3}$	11.14$^{+ 7.09}_{- 6.72}$	...	...
SPT-CLJ0456-5116	0.008$^{+0.000}_{-0.000}$	10.8 ± 5.7	0.04$^{+0.00}_{-0.02}$	0.24 ± 0.05	264.5$^{+ 146.4}_{- 142.4}$	6.32$^{+ 3.57}_{- 3.42}$	...	...
SPT-CLJ0509-5342	0.056$^{+0.001}_{-0.001}$	3.7 ± 1.1	0.13$^{+0.01}_{-0.03}$	0.86 ± 0.02	25.0$^{+ 7.7}_{- 7.6}$	0.48$^{+ 0.15}_{- 0.15}$	39.2$^{+ 16.3}_{- 8.9}$	39.2$^{+ 16.3}_{- 8.9}$
SPT-CLJ0528-5300	0.018$^{+0.001}_{-0.001}$	2.0 ± 1.3	0.07$^{+0.02}_{-0.03}$	0.50 ± 0.06	29.2$^{+ 19.3}_{- 18.5}$	0.97$^{+ 0.65}_{- 0.62}$	9.6$^{+ 16.2}_{- 3.7}$	9.6$^{+ 16.2}_{- 3.7}$
SPT-CLJ0533-5005	0.020$^{+0.001}_{-0.001}$	1.0 ± 1.1	0.04$^{+0.01}_{-0.03}$	0.57 ± 0.12	13.5$^{+ 16.2}_{- 15.0}$	0.41$^{+ 0.51}_{- 0.46}$	15.2$^{+ 146.9}_{- 8.0}$	9.1$^{+ 87.8}_{- 4.8}$
SPT-CLJ0542-4100	0.025$^{+0.000}_{-0.000}$	11.3 ± 4.4	0.01$^{+0.01}_{-0.01}$	0.61 ± 0.04	133.8$^{+ 54.3}_{- 52.9}$	2.17$^{+ 0.90}_{- 0.87}$	2.0$^{+ 1.2}_{- 0.5}$	2.0$^{+ 1.2}_{- 0.5}$
SPT-CLJ0546-5345	0.020$^{+0.001}_{-0.001}$	5.3 ± 2.6	0.07$^{+0.01}_{-0.02}$	0.33 ± 0.05	70.4$^{+ 37.6}_{- 36.1}$	1.69$^{+ 0.93}_{- 0.88}$	168.2$^{+ 169.4}_{- 56.2}$	37.3$^{+ 37.6}_{- 12.5}$
SPT-CLJ0551-5709	0.005$^{+0.000}_{-0.000}$	9.3 ± 2.5	0.03$^{+0.00}_{-0.02}$	0.17 ± 0.05	329.4$^{+ 91.8}_{- 89.7}$	10.26$^{+ 2.93}_{- 2.84}$	...	...
SPT-CLJ0555-6406	0.009$^{+0.000}_{-0.000}$	5.4 ± 1.9	0.04$^{+0.00}_{-0.02}$	0.31 ± 0.05	128.3$^{+ 48.7}_{- 47.2}$	4.06$^{+ 1.58}_{- 1.51}$	...	4.8$^{+ 2.7}_{- 1.3}$
SPT-CLJ0559-5249	0.008$^{+0.000}_{-0.000}$	3.5 ± 1.0	0.03$^{+0.00}_{-0.01}$	0.30 ± 0.04	90.1$^{+ 25.7}_{- 25.2}$	3.48$^{+ 1.01}_{- 0.99}$	4.2$^{+ 1.6}_{- 0.9}$	4.2$^{+ 1.6}_{- 0.9}$
SPT-CLJ0616-5227	0.024$^{+0.000}_{-0.000}$	4.6 ± 1.1	0.09$^{+0.01}_{-0.03}$	0.44 ± 0.05	56.6$^{+ 14.6}_{- 14.2}$	1.35$^{+ 0.36}_{- 0.35}$	247.1$^{+ 78.3}_{- 47.9}$	158.2$^{+ 50.2}_{- 30.7}$
SPT-CLJ0655-5234	0.005$^{+0.000}_{-0.000}$	18.2 ± 11.9	0.04$^{+0.00}_{-0.02}$	0.09 ± 0.06	655.7$^{+ 458.4}_{- 434.8}$	15.41$^{+11.13}_{-10.29}$	...	...
SPT-CLJ2031-4037	0.016$^{+0.000}_{-0.000}$	12.2 ± 2.4	0.05$^{+0.00}_{-0.02}$	0.37 ± 0.04	189.8$^{+ 39.9}_{- 38.9}$	3.43$^{+ 0.75}_{- 0.72}$	35.8$^{+ 8.7}_{- 5.9}$	97.6$^{+ 23.7}_{- 15.9}$
SPT-CLJ2034-5936	0.026$^{+0.001}_{-0.001}$	3.7 ± 3.6	0.06$^{+0.00}_{-0.02}$	0.51 ± 0.05	41.9$^{+ 41.9}_{- 40.2}$	1.04$^{+ 1.06}_{- 1.00}$	56.2$^{+1323.3}_{- 27.5}$	28.0$^{+ 658.7}_{- 13.7}$
SPT-CLJ2035-5251	0.002$^{+0.000}_{-0.000}$	6.2 ± 1.4	0.05$^{+0.00}_{-0.02}$	0.03 ± 0.02	362.8$^{+ 93.4}_{- 88.8}$	17.13$^{+ 4.70}_{- 4.36}$	...	...
SPT-CLJ2043-5035	0.143$^{+0.002}_{-0.002}$	3.5 ± 0.2	0.31$^{+0.02}_{-0.04}$	1.00 ± 0.03	12.7$^{+ 1.0}_{- 1.0}$	0.18$^{+ 0.02}_{- 0.01}$	749.9$^{+ 53.6}_{- 46.9}$	749.9$^{+ 53.6}_{- 46.9}$
SPT-CLJ2106-5844	0.012$^{+0.000}_{-0.000}$	7.2 ± 1.3	0.04$^{+0.00}_{-0.02}$	0.12 ± 0.05	135.8$^{+ 26.8}_{- 25.9}$	3.42$^{+ 0.71}_{- 0.68}$	381.5$^{+ 81.5}_{- 57.1}$	28.2$^{+ 6.0}_{- 4.2}$
SPT-CLJ2135-5726	0.012$^{+0.000}_{-0.000}$	4.2 ± 1.5	0.06$^{+0.00}_{-0.02}$	0.31 ± 0.05	79.4$^{+ 30.2}_{- 29.3}$	2.43$^{+ 0.95}_{- 0.91}$	27.5$^{+ 15.4}_{- 7.3}$	50.1$^{+ 28.1}_{- 13.2}$
SPT-CLJ2145-5644	0.032$^{+0.001}_{-0.001}$	15.7 ± 8.1	0.10$^{+0.01}_{-0.02}$	0.71 ± 0.04	155.6$^{+ 84.1}_{- 81.4}$	2.03$^{+ 1.12}_{- 1.07}$	5.9$^{+ 6.3}_{- 2.0}$	5.9$^{+ 6.3}_{- 2.0}$
SPT-CLJ2146-4632	0.012$^{+0.000}_{-0.000}$	2.1 ± 1.3	0.05$^{+0.00}_{-0.02}$	0.42 ± 0.07	40.3$^{+ 26.6}_{- 25.6}$	1.54$^{+ 1.04}_{- 0.98}$	11.5$^{+ 19.5}_{- 4.4}$	4.4$^{+ 7.5}_{- 1.7}$
SPT-CLJ2148-6116	0.007$^{+0.000}_{-0.000}$	3.7 ± 1.3	0.04$^{+0.00}_{-0.02}$	0.26 ± 0.06	98.5$^{+ 36.2}_{- 35.1}$	3.78$^{+ 1.43}_{- 1.36}$	1.1$^{+ 0.6}_{- 0.3}$	1.1$^{+ 0.6}_{- 0.3}$
SPT-CLJ2218-4519	0.010$^{+0.000}_{-0.000}$	3.0 ± 1.2	0.05$^{+0.00}_{-0.02}$	0.37 ± 0.06	65.0$^{+ 28.6}_{- 27.5}$	2.41$^{+ 1.09}_{- 1.03}$	2.3$^{+ 1.6}_{- 0.7}$	2.3$^{+ 1.6}_{- 0.7}$
SPT-CLJ2222-4834	0.060$^{+0.002}_{-0.002}$	2.4 ± 1.0	0.13$^{+0.01}_{-0.03}$	0.77 ± 0.04	15.8$^{+ 7.2}_{- 6.9}$	0.34$^{+ 0.16}_{- 0.15}$	165.4$^{+ 122.4}_{- 49.4}$	165.4$^{+ 122.4}_{- 49.4}$
SPT-CLJ2232-5959	0.054$^{+0.001}_{-0.001}$	3.8 ± 0.8	0.15$^{+0.01}_{-0.03}$	0.70 ± 0.05	26.8$^{+ 6.0}_{- 5.8}$	0.51$^{+ 0.12}_{- 0.12}$	228.3$^{+ 59.3}_{- 39.0}$	228.3$^{+ 59.3}_{- 39.0}$
SPT-CLJ2233-5339	0.013$^{+0.000}_{-0.000}$	4.3 ± 1.4	0.06$^{+0.00}_{-0.02}$	0.40 ± 0.05	77.0$^{+ 26.8}_{- 25.9}$	2.26$^{+ 0.82}_{- 0.78}$	21.7$^{+ 10.5}_{- 5.3}$	48.2$^{+ 23.2}_{- 11.8}$
SPT-CLJ2236-4555	0.026$^{+0.001}_{-0.001}$	2.0 ± 0.8	0.08$^{+0.01}_{-0.03}$	0.47 ± 0.06	22.7$^{+ 9.9}_{- 9.5}$	0.67$^{+ 0.30}_{- 0.28}$	68.7$^{+ 46.6}_{- 19.8}$	68.7$^{+ 46.6}_{- 19.8}$
SPT-CLJ2245-6206	0.002$^{+0.000}_{-0.000}$	11.3 ± 22.5	0.02$^{+0.01}_{-0.01}$	0.03 ± 0.01	680.8$^{+1385.5}_{-1345.6}$	25.04$^{+51.52}_{-49.32}$	0.0$^{+ -0.0}_{- 0.0}$	0.0$^{+ -0.0}_{- 0.0}$
SPT-CLJ2248-4431	0.033$^{+0.000}_{-0.000}$	13.0 ± 0.9	0.07$^{+0.00}_{-0.02}$	0.48 ± 0.02	128.0$^{+ 9.5}_{- 9.4}$	1.79$^{+ 0.14}_{- 0.14}$	652.4$^{+ 48.2}_{- 42.0}$	997.1$^{+ 73.6}_{- 64.1}$
SPT-CLJ2258-4044	0.007$^{+0.000}_{-0.000}$	7.9 ± 2.0	0.04$^{+0.00}_{-0.02}$	0.06 ± 0.05	207.9$^{+ 56.4}_{- 54.7}$	5.98$^{+ 1.68}_{- 1.61}$	10.9$^{+ 3.7}_{- 2.2}$	...
SPT-CLJ2259-6057	0.037$^{+0.001}_{-0.001}$	5.1 ± 1.2	0.09$^{+0.00}_{-0.02}$	0.58 ± 0.03	45.8$^{+ 11.4}_{- 11.2}$	0.91$^{+ 0.23}_{- 0.23}$	68.6$^{+ 21.2}_{- 13.1}$	68.6$^{+ 21.2}_{- 13.1}$
SPT-CLJ2301-4023	0.017$^{+0.000}_{-0.000}$	9.6 ± 1.6	0.10$^{+0.01}_{-0.02}$	0.25 ± 0.05	147.9$^{+ 27.8}_{- 27.0}$	3.00$^{+ 0.59}_{- 0.57}$	96.2$^{+ 19.6}_{- 13.9}$	52.3$^{+ 10.6}_{- 7.6}$
SPT-CLJ2306-6505	0.004$^{+0.000}_{-0.000}$	3.8 ± 2.7	0.03$^{+0.00}_{-0.02}$	0.19 ± 0.08	158.8$^{+ 115.7}_{- 111.6}$	7.45$^{+ 5.55}_{- 5.26}$	...	...
SPT-CLJ2325-4111	0.004$^{+0.000}_{-0.000}$	10.4 ± 5.0	0.03$^{+0.00}_{-0.02}$	0.10 ± 0.04	417.4$^{+ 210.9}_{- 204.0}$	13.06$^{+ 6.76}_{- 6.44}$	...	...
SPT-CLJ2331-5051	0.102$^{+0.002}_{-0.002}$	3.9 ± 0.6	0.22$^{+0.02}_{-0.04}$	0.91 ± 0.03	18.0$^{+ 2.9}_{- 2.8}$	0.28$^{+ 0.05}_{- 0.04}$	236.0$^{+ 40.0}_{- 29.9}$	236.0$^{+ 40.0}_{- 29.9}$
SPT-CLJ2335-4544	0.017$^{+0.000}_{-0.000}$	5.5 ± 4.6	0.06$^{+0.00}_{-0.02}$	0.50 ± 0.05	83.6$^{+ 72.4}_{- 70.1}$	2.10$^{+ 1.85}_{- 1.77}$	3.5$^{+ 17.6}_{- 1.6}$	3.5$^{+ 17.6}_{- 1.6}$
SPT-CLJ2337-5942	0.013$^{+0.000}_{-0.000}$	15.6 ± 3.3	0.05$^{+0.00}_{-0.02}$	0.14 ± 0.05	281.7$^{+ 65.8}_{- 63.9}$	4.98$^{+ 1.21}_{- 1.16}$	84.6$^{+ 23.2}_{- 15.0}$	32.9$^{+ 9.0}_{- 5.8}$
SPT-CLJ2341-5119	0.046$^{+0.001}_{-0.001}$	5.2 ± 2.0	0.08$^{+0.00}_{-0.02}$	0.55 ± 0.04	40.4$^{+ 16.3}_{- 15.8}$	0.74$^{+ 0.31}_{- 0.29}$	253.8$^{+ 156.1}_{- 70.0}$	152.8$^{+ 94.0}_{- 42.1}$
SPT-CLJ2342-5411	0.077$^{+0.002}_{-0.002}$	1.9 ± 0.4	0.17$^{+0.03}_{-0.04}$	0.81 ± 0.08	10.3$^{+ 2.4}_{- 2.4}$	0.21$^{+ 0.05}_{- 0.05}$	235.7$^{+ 65.2}_{- 42.0}$	198.4$^{+ 54.9}_{- 35.3}$
SPT-CLJ2344-4243	0.286$^{+0.004}_{-0.004}$	9.5 ± 0.5	0.38$^{+0.03}_{-0.05}$	1.29 ± 0.03	22.0$^{+ 1.4}_{- 1.4}$	0.17$^{+ 0.01}_{- 0.01}$	1881.1$^{+ 113.5}_{- 101.3}$	1881.1$^{+ 113.5}_{- 101.3}$
SPT-CLJ2345-6405	0.007$^{+0.000}_{-0.000}$	6.9 ± 1.7	0.04$^{+0.00}_{-0.02}$	0.14 ± 0.07	186.0$^{+ 51.6}_{- 49.5}$	5.67$^{+ 1.65}_{- 1.55}$	...	...
SPT-CLJ2352-4657	0.030$^{+0.001}_{-0.001}$	1.6 ± 1.0	0.07$^{+0.00}_{-0.02}$	0.60 ± 0.05	17.1$^{+ 11.3}_{- 10.9}$	0.49$^{+ 0.33}_{- 0.32}$	49.6$^{+ 84.8}_{- 19.2}$	49.6$^{+ 84.8}_{- 19.2}$
SPT-CLJ2355-5055	0.046$^{+0.002}_{-0.002}$	1.4 ± 0.5	0.13$^{+0.01}_{-0.03}$	0.84 ± 0.04	10.7$^{+ 4.2}_{- 4.0}$	0.27$^{+ 0.11}_{- 0.10}$	105.4$^{+ 60.1}_{- 28.1}$	105.4$^{+ 60.1}_{- 28.1}$
SPT-CLJ2359-5009	0.009$^{+0.000}_{-0.000}$	3.0 ± 1.1	0.06$^{+0.00}_{-0.02}$	0.30 ± 0.05	71.0$^{+ 28.6}_{- 27.6}$	2.79$^{+ 1.15}_{- 1.10}$	...	...

Notes. This table provides cool core and cooling flow properties for the full sample of 83 galaxy clusters at 0.3 < z < 1.2. A complete description of how these values were measured is presented in Section 2. As in Sections 2–4, we use the subscript "0" to represent the measured quantity in our smallest annulus: r < 0.012 R₅₀₀. All of these data are deprojected, using our mass-modeling technique described in Section 3. (1) Cluster name; (2) central (r < 0.012 R₅₀₀) electron density; (3) central (r < 0.012 R₅₀₀) temperature; (4) surface brightness concentration (Santos et al. 2008); (5) electron density slope at 0.04 R₅₀₀; (6) central (r < 0.012 R₅₀₀) entropy; (7) central (r < 0.012 R₅₀₀) cooling time; (8) mass deposition rate within r(t_cool < 7.7 Gyr); (9) mass deposition rate within r(t_cool < t_Univ).

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In Figure 2, we present the radial entropy and cooling time profiles for our full sample, based on the deprojection procedures described in Section 2.4. For comparison, we show the average entropy profiles for low-redshift cool core and non-cool core clusters from Cavagnolo et al. (2009). Perhaps surprisingly, there is no qualitative difference in the entropy profiles between this sample of high-redshift, massive, SZ-selected clusters and the low-redshift sample of groups and clusters presented in Cavagnolo et al. (2009).

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Naively, one might expect the mean central entropy to decrease with time, as clusters have had more time to cool. However, the similarity between low-redshift clusters and this sample, which has a median redshift of 0.63 and a median age nearly half of the z ∼ 0 sample, indicates that the entropy and cooling time profiles are unchanging. This suggests that the characteristic entropy and cooling time profiles, having minimal core entropies of ∼10 keV cm² and cooling radii of ∼100 kpc, were established at earlier times than we are probing with this sample (z ≳ 1).

In order to look for evolution in the entropy profile, we plot in Figure 3 the average entropy profile for cool core (K₀ < 30 keV cm²) and non-cool core (K₀ > 30 keV cm²) clusters in the SPT-XVP sample, divided into two redshift bins corresponding to z < 0.75 and z > 0.75. These are compared to the average profiles from Cavagnolo et al. (2009), for clusters at z ≲ 0.1. In general, the average profiles are indistinguishable in the inner few hundred kiloparsecs, with high-redshift clusters having slightly higher entropy at small radii (r ≲ 200 kpc) than their low-redshift counterparts. At larger radii, the profiles vary according to the self-similar E(z)^4/3 scaling (Pratt et al. 2010). These results suggest that the outer entropy profile is following the gravitational collapse of the cluster, while the inner profile has some additional physics governing its evolution, most likely baryonic cooling. The mild central entropy evolution in the right panel of Figure 3 could be thought of as the effect of "forcing" an evolutionary scaling in a regime where the profile is unevolving.

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The combination of Figures 2 and 3 suggests that the cooling properties of the ICM in the inner ∼200 kpc have remained relatively constant over timescales of ∼8 Gyr. The short cooling times at these radii imply that the core entropy profile should change on short timescales. The fact that this is not observed suggests that some form of feedback has offset cooling on these exceptionally long timescales, keeping the central entropy at a constant value. There is evidence that mechanical feedback from AGN is stable over such long periods of time (Hlavacek-Larrondo et al. 2012), perhaps maintaining the observed entropy floor of 10 keV cm² since z ∼ 1.2.

In Figure 4, we compare the derived central entropy and cooling time (see Section 2.4 for details on deriving central quantities) for the SPT-XVP sample to those for the low-redshift clusters in the Chandra Cluster Cosmology Project (hereafter CCCP; Vikhlinin et al. 2006, 2009). Overall, we find excellent agreement between the two samples. While it is unsurprising that these two quantities are correlated, due to their similar dependence on both T_X and n_e, the normalization and distribution of points along the sequence is reassuring. Both the SPT-XVP and the CCCP clusters have a slightly higher normalization than found by Hudson et al. (2010), which can be accounted for by the fact that both of these samples target more massive clusters. Indeed, Hudson et al. (2010) showed that the scatter about the t_{cool, 0}–K₀ correlates with the cluster temperature, with high-T_X clusters lying above the relation and low-T_X groups lying below the relation. Similar to previous low-redshift studies (e.g., Cavagnolo et al. 2009; Hudson et al. 2010), we see hints of multiple peaks in both t_{cool, 0} and K₀, with minima at ∼1 Gyr and 50 keV cm², respectively (e.g., Cavagnolo et al. 2009; Hudson et al. 2010). This threshold, separating cool core from non-cool core clusters, appears to be unchanged between the low-redshift and high-redshift samples. We will return to this point in Section 3.3.

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In Figure 5, we plot the distribution of mass deposition rates, dM/dt, for our SPT-selected sample. We show the integrated cooling rate within two radii: r(t_cool = t_Univ) and r(t_cool = 7.7 Gyr). The former is more physically motivated, representing the amount of gas that has had time to cool since the cluster formed. The latter is motivated by the desire to have the definition of the cooling radius be independent of redshift—the choice of a 7.7 Gyr timescale is arbitrary and was chosen simply to conform with the literature (e.g., O'Dea et al. 2008).

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We find that clusters at high-z have overall smaller time-averaged cooling rates, which is unsurprising given that they have had less time to cool. If we remove this factor by instead computing the cooling rate within a non-evolving aperture (r[t_cool = 7.7 Gyr]), we find no significant difference between the mass deposition rates measured in intermediate-redshift (0.4 < z < 0.75) and high-redshift (0.75 < z < 1.1) clusters. These sub-samples have median mass deposition rates of 49 M_☉ yr⁻¹ and 57 M_☉ yr⁻¹ (excluding non-cooling systems), respectively. For comparison, we also show the distribution of cooling rates for nearby clusters from White et al. (1997), for which the distribution is cut off at dM/dt ≲ 50 M_☉ yr⁻¹ due to poorer sampling and, thus, reduced sensitivity to modest cooling rates. However, as evidenced by the cumulative distribution, at dM/dt > 50 M_☉ yr⁻¹ the three samples are nearly identical, suggesting very little evolution in the rate of cooling in the ICM over timescales of ∼8 Gyr.

Figures 3 and 5 suggest that the cooling properties of the ICM vary little from z ∼ 0 to z ∼ 1. In order to directly quantify this, we show in Figure 6 the evolution of dM/dt, K₀, and t_{cool, 0} with redshift. For each quantity, we separate cool core and non-cool core clusters using the following thresholds: dM/dt > 0 M_☉ yr⁻¹, K₀ < 30 keV cm², and t_{cool, 0} < 1 Gyr. This figure more clearly shows that there is very little, if any, evolution in the cooling properties of SPT-selected clusters over the range 0.3 < z < 1.2. We compare the range of dM/dt, K₀, and t_{c, 0} observed for these clusters to samples of nearby clusters from White et al. (1997) and the CCCP and find no appreciable change. The entropy floor, at K₀ ∼ 10 keV cm², is constant over the full redshift range of our sample, consistent with earlier work by Cavagnolo et al. (2009) which covered clusters at 0 < z ≲ 0.5. The data are also consistent with the self-similar expectation (E(z)^4/3; Pratt et al. 2010), which predicts only a factor of ∼1.6 change in central entropy from z = 1 to z = 0. This self-similar evolution is based on gravity alone—the fact that it is an adequate representation of the data in the central ∼100 kpc of clusters, where cooling processes should be responsible for shaping the entropy profile, suggests that cooling is offset exceptionally well over very long over the past ∼7 Gyr. In the absence of feedback, cool core clusters at z ∼ 1 should have K₀ → 0 in <1 Gyr.

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Figures 2–6 suggest that there is little change in the ICM cooling properties in the cores of X-ray- and SZ-selected clusters since z ∼ 1.2, in agreement with earlier studies at lower redshift (e.g., Cavagnolo et al. 2009). However, several recent studies have argued that high-redshift clusters are less centrally concentrated than their low-redshift counterparts (Vikhlinin et al. 2007; Santos et al. 2008, 2010) based on measurements of surface brightness concentration, suggesting that "cool cores" and "cooling flows" may not have the same evolution and, thus, are not necessarily coupled at high redshift as they are now.

In Figure 7, we duplicate the analyses of Vikhlinin et al. (2007), Santos et al. (2008), and Semler et al. (2012) in order to look for evolution in the cool core properties. We find that the number of galaxy clusters classified as "cool core" by both α and c_SB (see Section 2.3) decreases with redshift, from ∼40% at z ∼ 0 to ∼10% at z ≳ 0.75. These results confirm the evolution in cool core strength reported by Vikhlinin et al. (2007) and Santos et al. (2008) for X-ray selected samples; however, the evolution appears to be a bit slower for the SZ-selected sample, with several strong cool cores in the range 0.5 < z < 0.75 (Semler et al. 2012). The higher fraction of "moderate" cool cores in Figure 7 at higher redshift in consistent with recent work by Santos et al. (2010). All samples agree that there is a lack of strong, classical cool cores at z > 0.75, which seems to be in opposition to the results presented in Figures 2–6 which suggest no evolution in the cooling properties. One possible explanation for this discrepancy is that both the concentration parameter, c_SB, and the cuspiness parameter, α, (see Equations (3) and (4)) assume no evolution in the scale radius of the cool core: c_SB assumes a radius of 40 kpc, while α uses 0.04 R₅₀₀.

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To further investigate the surface brightness evolution, we move away from single-parameter measures of surface brightness concentration and, instead, directly compare the X-ray surface brightness profiles for low- and high-redshift clusters in Figure 8. At z < 0.75, we confirm that, overall, clusters with low central entropy (K₀ < 30 keV cm²) have more centrally concentrated surface brightness profiles than those with high central entropy (K₀ > 30 keV cm²). However, at high redshift (z > 0.75) we find a lack of strongly concentrated clusters, with only a weak increase in concentration for the clusters with low central entropy, consistent with earlier studies of distant X-ray-selected clusters (e.g., Vikhlinin et al. 2007; Santos et al. 2008, 2010). This difference in surface brightness concentration is not a result of increased spatial resolution for low-redshift clusters—the difference in spatial resolution between the centers of these redshift bins is only ∼30%.

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This result becomes even more dramatic when the data are deprojected into gas density, rather than surface brightness, and including z ∼ 0 clusters for comparison. In Figure 9, we compare the ICM gas density profiles for the sample presented in this work to a low-redshift sample (CCCP; Vikhlinin et al. 2009). We restrict the low-redshift sample to clusters with M₅₀₀ > 3 × 10¹⁴ M_☉, in order to approximate the SPT-selection cut. This comparison is particularly appropriate since our reduction and analysis pipeline is identical to that used by Vikhlinin et al. (2009). Figure 9 shows that the 3D gas density profiles become more centrally concentrated with decreasing redshift, with nearly an order of magnitude difference in central gas density between cool core clusters at z ∼ 0.1 and z ∼ 0.9. On the contrary, non-cooling clusters (K₀ > 30 keV cm²) experience no appreciable evolution in the central physical density over the same timescale. The combination of Figures 8 and 9 seem to suggest that the dense cores which are associated with cooling flows have built up slowly over the past ∼8 Gyr. This scenario would explain the lack of centrally concentrated, or "cuspy," clusters at high redshift.

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The results presented thus far suggest that the dense cores associated with cooling flow clusters were not as pronounced ∼8 Gyr ago, despite the fact that clusters at these early times had similar cooling rates and central entropy (Figure 6). We propose that these cores have grown over time as a direct result of cooling flows being halted by feedback. In this scenario, gas from larger radii cools and flows inward, but it hits a "cooling floor" at ∼10 keV cm², below which cooling is less efficient. This floor is likely a result of some form of feedback, with the most promising explanation currently being mechanical energy injection from the central AGN (e.g., Fabian 2012; McNamara & Nulsen 2012). This concept of a cooling floor is supported by observations. Peterson & Fabian (2006) show, using high resolution X-ray spectroscopy of nearby galaxy clusters, that gas at temperatures less than ∼1/3 of the ambient ICM temperature is cooling orders of magnitude less effectively than predicted. Further, in agreement with Cavagnolo et al. (2009), we show in Figure 6 that there is a lower entropy limit in the cores of galaxy clusters of ∼10 keV cm² which is roughly constant over the range 0 < z < 1.2. The fact that the ICM appears unable to cool efficiently below ∼1/3 of the ambient temperature, or ∼10 keV cm², implies that, if material is indeed flowing into the cluster core, then we should observe a build up of low-entropy gas in the core.

To test this hypothesis, we measure the "cool core mass" for all clusters in our sample with K₀ < 30 keV cm². The cool core mass is defined as

$$\begin{equation} M_{{\rm cool}} = 4\pi \int _0^{0.1\,R_{500}} (\rho _{g}-\langle \rho _{g,\,{\rm NCC}}\rangle)r^2dr, \end{equation} \tag{ 10 }$$

where 〈ρ_g, NCC〉 represents the median non-cool core (K₀ > 150 keV cm²) density profile (Figure 9) and the outer radius of 0.1 R₅₀₀ is roughly where the uncertainty in ρ_g is similar in scale to the difference between the median cool core and non-cool core profiles. In Figure 10, we plot the cool core mass, M_cool, versus redshift for the full sample of SPT-XVP clusters with K₀ < 30 keV cm², including five nearby (z < 0.1) clusters from Vikhlinin et al. (2006). We find a rapid evolution in the total cool core mass, with an order of magnitude increase in M_cool between z ∼ 1 and z ∼ 0.5. As we show in Figure 10, the median growth is fully consistent with a constant cooling flow since z = 1 with dM/dt = 150 M_☉ yr⁻¹. The range of cool core masses is consistent with cooing flowing initiating at 0.8 ≲ z ≲ 2, providing the first constraints on the onset of cooling in galaxy clusters.

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Figure 10 suggests that cool cores at z ∼ 0 are a direct result of long-standing cooling flows (Figure 5) coupled with a constant entropy floor (Figure 6)—most likely the result of AGN feedback. The long-standing balance between cooling and feedback prevents gas from cooling completely and, instead, leads to an accumulation of cool gas in the core of the cluster. Such a build-up of low-entropy, cool material is demonstrated in simulations of galaxy clusters for which the energy loss by cooling is matched by AGN feedback over timescales of several Gyr (e.g., Gaspari et al. 2011).

This simple evolutionary scenario, which we offer as an explanation for the increase in central gas density in clusters from z ∼ 1 to z ∼ 0.5, is based on a sample of massive (M₅₀₀ > 2 × 10¹⁴ M_☉), rich galaxy clusters. While the cooling rate (dM/dt) is proportional to cluster mass (White et al. 1997), there is no evidence that the presence of a cool core is dependent on whether the host is a rich galaxy cluster or a poor group (McDonald et al. 2011a). Thus, if these trends hold at z ≫ 0.1, we would expect future surveys of low-mass, high-redshift clusters, via optical (e.g., LSST; Tyson 2002) or X-ray (e.g., eRosita; Predehl et al. 2007) detection, to see a similar decline in the cuspiness of cool cores at high redshift. At this point, however, such an extrapolation to lower masses is purely speculative.

In Figure 6, we show that there is no measurable evolution in the minimum central entropy, K₀, over the range 0.3 < z < 1.2. Coupled with the apparent increase in central density, this would imply that cool cores today are warmer than their high-z counterparts. Since a detailed spatial comparison, such as we did for density, is more challenging with a spectroscopically measured quantity, we reduce the problem to a single measurement of "central" temperature. Here we define the central temperature as the spectroscopically measured temperature within 0.05 R₅₀₀, with central point sources masked. We consider here only systems with K₀ < 30 keV cm², which are the most centrally concentrated systems in our sample, allowing us to use a smaller aperture than in Figure 1.

In Figure 11, we see that, indeed, for clusters classified as cool core on the basis of their central entropy (K₀ < 30 keV cm²), central temperature increases with decreasing redshift. The slope of this relation is equally consistent with the expected self-similar evolution, as well as what is required to have no evolution in K₀ over this redshift interval (dashed line, see also Figure 6). This figure seems to suggest that there may be some additional heating in low-redshift cluster cores above the self-similar expectation, perhaps resulting from AGN feedback. However, we stress that these data are insufficient to distinguish between these two scenarios, and so we defer any further speculation on the evolution of the central temperature to an upcoming paper which will perform a careful stacking analysis of these clusters.

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The combined evidence presented in Sections 4.1 and 4.2 support the scenario that cooling material has been gradually building up in cluster cores over the last ∼8 Gyr, but has been prevented from completely cooling by an almost perfectly balanced heating source. While at first glance it would appear that the amount of energy injected by feedback must increase rapidly to offset increased cooling ($L_{{\rm cool}} \propto n_e^2 T^{1/2}$), much of this is offset by the fact that the gravitational potential in the core is increasing, leading to more heating as cooling material falls into the potential well. So, while cool cores are becoming denser with decreasing redshift, the actual cooling rates, and by extension the energy needed for feedback to offset cooling, have remained nearly constant since z ∼ 1.

Recent observations of the "Phoenix Cluster" (McDonald et al. 2012a, 2013) suggest that some clusters may undergo episodes of runaway cooling, perhaps before the feedback responsible for establishing the cooling flow was fully established, or that the feedback mechanism is strongly episodic.

Much effort has recently focused on determining the fraction of high-redshift clusters which harbor a cool core (Vikhlinin et al. 2007; Santos et al. 2008, 2010; Samuele et al. 2011; McDonald 2011; Semler et al. 2012). However, based on the results of this paper, we now know that the inferred evolution in the cool core fraction depends strongly on the criteria that are used to classify cool cores. In Figure 12 we demonstrate this point, showing that the measured fraction of high-z cool cores is drastically different if the classification of cool cores is based on the presence of cooling (K₀, t_{cool, 0}, dM/dt; ∼35%) or cooled (α, c_SB; ∼5%) gas. This figure shows that, at high redshift, it is important to differentiate between "cooling flows" and "cool cores" when classifying galaxy clusters as cooling or not—a distinction which is unnecessary in nearby clusters.

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Figure 12 shows that the fraction of strongly-cooling clusters (dM/dt > 50 M_☉ yr⁻¹, K₀ < 35 keV cm², t_{cool, 0} < 1 Gyr) undergoes little evolution over the range 0.3 < z < 1.2. This is qualitatively apparent in Figure 6. The fraction of strongly-cooling clusters increases from 25$^{+11}_{-5}$% to 35$^{+10}_{-10}$%, consistent at the 1σ level with no change. This figure shows that, not only is the rate at which the ICM is cooling roughly constant since z ∼ 1 (Figures 5 and 6), but the fraction of clusters that experience strong cooling is also nearly constant over similar timescales.

It is also worth noting here the overall agreement in Figure 12 between the evolution of ICM cooling inferred from X-ray properties (this work) and optical properties (McDonald 2011) at z ⩽ 0.5. The latter sample was drawn from optically selected catalogues, and used emission-line nebulae as a probe of ICM cooling. This overall agreement suggests that the evolution of cooling properties is relatively independent of how the clusters are selected (optical versus SZ). The steep rise in the cooling fraction at z < 0.5 was interpreted by McDonald (2011) as being due to timing—we are seeing clusters transitioning from "weak" to "strong" cool cores as the central cooling time drops over time.

While the observed cool core evolution presented here is interesting, it may suffer from a combination of several biases in both the sample selection and in the analysis. Below we address four biases and quantify their effects on the observed cool core evolution.

4.4.1. Three-Dimensional Mass Modeling

The results presented thus far rely on our ability to estimate the central temperature based on assumptions about the dark matter halo and hydrostatic equilibrium (see Section 2.3). While we established in Section 2.5 that these estimates are reliable, it is worthwhile to investigate whether any of the observed evolutionary trends are due to this approach. In Figure 13, we compare the central entropy calculated using a 2D, spectroscopically measured temperature (see Section 2.5) to our 3D model calculation. We find very good one-to-one agreement between these two quantities, with the scatter being uncorrelated with redshift. This suggests that the observed lack of evolution in K₀ is not a result of our modeling technique. We further demonstrate this in Figure 14, showing the evolution of the central entropy based on the 2D central temperature. Comparing to low-redshift clusters from the CCCP, with K_{0, 2D} calculated in the same way, we confirm the lack of evolution in the central entropy from Figure 6 over the range 0 < z < 1.2.

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4.4.2. Increased SZ Signal in Cool Cores

4.4.3. X-Ray Centroid Determination

In Section 2.2 we describe our method of determining the cluster center, which is based on the X-ray emission in an annulus of 250–500 kpc. This method, which reduces scatter in X-ray scaling relations by finding the large-scale center rather than what may be the displaced core, will yield less centrally concentrated density profiles than if we chose the X-ray peak as the cluster center. Regardless of how the center is defined, we can investigate how our choice can affect the resulting density profile.

In Figure 15, we compare the measured surface brightness concentration (c_SB; Santos et al. 2008) based on our large-scale centroid and the X-ray peak. This figure confirms that, for relaxed strong cool cores, the X-ray peak and the large-scale centroid are nearly equivalent. By switching to c_{SB, peak}, the number of high-z strong cool cores (c_SB > 0.155) would remain constant at one, while the number of low-z strong cool cores would increase from five to seven. Repeating this exercise for moderate cool cores (0.075 < c_SB < 0.155) we find increases of five (from one to six) and two (from three to five) for high- and low-redshift clusters, respectively. This large difference is primarily due to the arbitrary definition of moderate cool cores—if we switched to a threshold of 0.07 rather than 0.075, the number of high-z clusters which would be re-classified as moderate cool cores by re-defining the center would decrease from five to two. Perhaps most importantly, the increase in c_SB resulting from changing the center to the X-ray peak appears to have no dependence on redshift, as the lower panel of Figure 15 demonstrates. Thus, while the choice of center certainly affects the shape of the density profile, there is no evidence that this could result in low-z clusters being measured to be more centrally concentrated than their high-z counterparts.

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4.4.4. Radio-loud and Star-forming BCGs

4.4.5. X-Ray Underluminous Clusters