X-RAY CAVITIES IN A SAMPLE OF 83 SPT-SELECTED CLUSTERS OF GALAXIES: TRACING THE EVOLUTION OF AGN FEEDBACK IN CLUSTERS OF GALAXIES OUT TO z = 1.2

To identify systems with X-ray cavities, we visually search the central 100 kpc of each cluster for circular or ellipsoidal surface brightness depressions in the Chandra X-ray images. Past studies on local clusters have found that the majority of X-ray cavities are located within this region (e.g., McNamara & Nulsen 2007, HL12).

As a first indicator, we create unsharp-masked images for each cluster to enhance deviations in the original Chandra X-ray image. This method consists of subtracting a strongly smoothed image from a lightly smoothed image and has been used extensively in the literature for X-ray cavity studies (e.g., Blanton et al. 2009; Sanders et al. 2009; Machacek et al. 2011). For the strongly smoothed image, we use Gaussian smoothing scales on the order of 40–80 kpc to match the underlying large-scale cluster emission, and for the lightly smoothed image, we use Gaussian smoothing scales on the order of 5–10 kpc to match the typical length scale of a cavity. Next, we build a best-fitting King model, centered on the X-ray peak, for the surface brightness distribution of all clusters showing hints of depressions. We use the lightly smoothed Chandra X-ray image to build the King model and subtracted the resulting King model image from the original Chandra X-ray image. We refer to these as the "King-subtracted" images. The King model is defined as

$$\begin{eqnarray}&&I(r)={{I}_{0}}{{\left( 1+{{\left( \frac{r}{{{r}_{0}}} \right)}^{2}} \right)}^{-\beta }}+{{C}_{0}},\end{eqnarray} \tag{ 1 }$$

where ${{I}_{0}}$ is the normalization, ${{r}_{0}}$ is the core radius, and ${{C}_{0}}$ is a constant. We allow all three of these parameters to vary while building the King model.

We try several binning and smoothing factors for all clusters and also test the robustness of the King-subtracted method by shifting the position of the central peak by $\pm 2-3$ pixels. We examine three energy bands: $0.5-7\ {\rm keV}$, $0.3-2\ {\rm keV}$, and $0.6-3\ {\rm keV}$. We only consider cavities to be real if they are visually identifiable in all X-ray images, including the original X-ray images (excessive binning and smoothing eventually removes any indication of a depression).

Out of the 83 SPT–SZ clusters with Chandra X-ray observations, we find that 10 clusters have surface brightness depressions in their Chandra X-ray images. However, to minimize false identification, three co-authors (Hlavacek-Larrondo, McDonald, and Allen) independently classified each cluster based on the visual significance of its X-ray cavities, clusters were classified as having either convincing X-ray cavities, somewhat convincing X-ray cavities, or unconvincing X-ray cavities. We then tabulated the classifications and only kept those that were classified as having convincing or somewhat convincing X-ray cavities by at least two co-authors. This resulted in the rejection of 2 of the original 10 clusters: SPT-CLJ0334–4659 (z = 0.45) and SPT-CLJ2043–5035 (z = 0.7234).

The final list of clusters with X-ray cavities is shown in Table 1. We note that six of these clusters were unanimously classified as having convincing or somewhat convincing X-ray cavities by all three co-authors. From now on, we refer to these as the clusters with "convincing" X-ray cavities. These are shown in the top portion of Table 1. The remaining two clusters, SPT-CLJ0156–5541 (z = 1.2) and SPT-CLJ2342–5411 (z = 1.075), were classified as having unconvincing X-ray cavities by one co-author. To emphasize the uncertainty related to these two particular systems, we highlight them differently in all tables (see bottom portion) and figures (square symbols instead of triangles).

Table 1.  SPT–SZ Clusters with Candidate X-ray Cavities

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Name	z	${{L}_{{\rm cool}\;\,7.7{\rm Gyrs}}}$	${{L}_{{\rm cool}\;\,{\rm r}\,=\,50\,{\rm kpc}}}$	${{L}_{{\rm cool}\;\,{\rm z}\,=\,2}}$	N	${{R}_{{\rm l}}}$	${{R}_{{\rm w}}}$	R	PA	PV	${{P}_{{\rm cav}}}$	Radio ...
		(${{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$)	(${{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$)	(${{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$)		(kpc)	(kpc)	(kpc)	(o)	(1059 erg)	(${{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$)
SPT-CLJ0000–5748	0.7019	7.9	5.2	5.0	#1P	18.6	10.7	37	140	1.9 ± 0.8	9.9 ± 5.0	✓
					#2P	15.7	15.7	32	322	3.5 ± 1.6	19.2 ± 9.7	...
SPT-CLJ0033–6326	0.59712	0.7	0.7	0.3	#1P	33.3	20.0	62	100	3.3 ± 1.5	7.7 ± 3.9	✓a
					#2P	30.7	23.3	73	280	3.2 ± 1.4	5.6 ± 2.8
SPT-CLJ0509–5342	0.4607	1.3	1.5	0.9	#1C	16.9	16.9	32	42	1.9 ± 0.8	6.8 ± 3.4	×b
					#2C	24.9	13.4	35	223	1.8 ± 0.8	7.1 ± 3.6	...
SPT-CLJ0616–5227	0.6838	1.9	1.6	0.2	#1C	24.0	17.0	62	82	1.7 ± 0.8	2.4 ± 1.2	✓
					#2P	24.0	17.0	37	234	2.2 ± 1.0	5.2 ± 2.6	...
SPT-CLJ2331–5051	0.576	5.5	3.6	3.8	#1P	13.7	10.5	27	59	1.1 ± 0.5	6.4 ± 3.2	✓
					#2P	12.4	9.2	20	222	1.6 ± 0.7	21.3 ± 10.7	...
SPT-CLJ2344–4243c	0.596	157.0	117.7	144.1	#1P	9.3	9.3	20	115	6.3 ± 2.9	84 ± 42	✓
					#2P	10.0	10.0	20	356	7.8 ± 3.6	107 ± 54	...
SPT-CLJ0156–5541	1.2	$\lt 0.02$	1.2	$\lt 0.02$	#1P	20.9	20.9	40	139	2.5 ± 1.1	3.8 ± 1.9	✓a
					#2P	21.2	39.0	40	335	4.6 ± 2.1	9.8 ± 4.9	...
SPT-CLJ2342–5411	1.075	3.1	1.8	0.7	#1P	13.9	13.9	26	196	0.9 ± 0.4	2.8 ± 1.4	×

Note. The first division contains the six systems with visually convincing cavities, while the second contains the two systems with marginally visually convincing cavities; see paragraph 4 in Section 3.2 for details. (1) Name; (2) redshift; (3) bolometric cooling luminosity within which ${{t}_{{\rm cool}}}$ is equal to $7.7\;{\rm Gyr}$; (4) same but defined within r = 50 kpc; (5) same but defined within which ${{t}_{{\rm cool}}}$ is equal to the look-back time since z = 2; (6) cavity number (N) with the depth of the cavity compared to the surrounding X-ray emission: C for "clear" and P for "Potential"; see Section 3.1; (7) cavity radius along the jet (errors are ±20%); (8) cavity radius perpendicular to the jet (errors are ±20%); (9) distance from the central AGN to the center of the cavity; (10) position angle of the cavity for north to east; (11) cavity enthalpy; (12) cavity power; (13) radio emission associated with AGNs in the BCG.

^aRadio source not centered on the BCG. ^bDynamic range limited by the presence of a nearby, un-associated, bright radio source. ^cPhoenix cluster (e.g., McDonald et al. 2012).

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The raw, unsharp-masked, and King-subtracted images of the final list of clusters with X-ray cavities are shown in Figure 2. The smoothing and binning factors are indicated in the lower-left corner of each image. "BX" refers to the image binning factor: "B1" means that the image was not binned, whereas "B2" means that each pixel corresponds to 4 pixels in the original image. The smoothing factor is denoted as "SX," where "X" corresponds to the sigma of a Gaussian in units of pixels once the image was binned. For each unsharp-masked image, we show the two smoothing scales used to create the image, and for each King-subtracted image we show the smoothing scale adopted for the original image before creating and subtracting a King model. Those with brackets were additionally smoothed for illustrative purposes. The optical or near-infrared images obtained as part of the SPT 2500 deg² follow-up campaign for photometric redshifts are also shown (High et al. 2010; Song et al. 2012; Bleem et al. 2015). These were taken with the Spitzer Space Telescope, the Swope 1 m telescope, the CTIO Blanco 4 m telescope, or the Baade Magellan 6.5 m telescope (see Bleem et al. 2015, for details).

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The depth of each cavity was estimated in the X-ray data using an azimuthal averaged surface brightness (see Figure 3). We measure azimuthal surface brightness profiles from the X-ray data (left panels of Figure 2), using an annulus that encompassed the cavities, centered on the X-ray peak (middle-right panels of Figure 2). Each annulus was divided into eight azimuthal sectors, containing roughly 50 counts per sector. On average, the depressions lie ∼1σ–3σ below the surrounding X-ray emission and contain 10%–40% less counts, roughly consistent with X-ray cavities seen in local clusters. Based on these results, we refer to the cavities lying 1σ–2σ below the surrounding X-ray emission as "potential" cavities (see "P" in Column 5 of Table 1), requiring deeper Chandra data to confirm them, while we refer to those lying 2σ–3σ (or more) below the surrounding X-ray emission as "clear" cavities (see "C" in Column 5 of Table 1). We note that the latter are only found among the six SPT–SZ clusters with visually "convincing" X-ray cavities, as explained earlier in this section.

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In Table 1, we also highlight the SPT–SZ clusters with X-ray cavities that have a radio source associated with the central regions. Since all of the low-redshift clusters with X-ray cavities have a radio-active BCG, we expect this to be similarly the case for the SPT–SZ clusters with X-ray cavities. We use the 843 MHz Sydney University Molonglo Sky Survey (SUMSS, synthesized beam width of ∼40$^{\prime\prime }$; Bock et al. 1999; Mauch et al. 2003), but owing to the large beam size, we cannot resolve any extended jet-like emission. We therefore use the radio data simply to verify that the clusters in our sample with X-ray cavities have radio-loud BCGs. We note that the radio sources in SPT-CLJ0033–6326 and SPT-CLJ0156–5541 are not centered on the BCG, but that the BCG lies well within the beam size of SUMSS. While the position uncertainty of sources detected in SUMSS is only $\sim {{10}^{\prime\prime }},$ these BCGs may still contain a radio source that is contributing to the overall emission in the beam, since they lie well within the beam size. We therefore consider that these sources have a radio counterpart for the purposes of this paper. SPT-CLJ0509–5342 lies within two beam sizes of a nearby 120 mJy radio source, making its detection by SUMSS uncertain since SUMSS is dynamic range limited by 1:100 on average (Bock et al. 1999). This bright radio source is most likely not associated with the BCG since, at its redshift, it would be located some 700 kpc from the galaxy. Finally, we find no evidence from the SUMSS maps of a radio source in SPT-CLJ2342–5411. Considering that this source is the second most distant cluster among our candidates (z = 1.075), the non-detection may simply be due to the limited sensitivity of the survey. In summary, and as expected, the majority of clusters with clear X-ray cavities (Table 1) have a detected radio counterpart.

To study the evolution of AGN feedback in BCGs, the energetics of the X-ray cavities in the SPT–SZ sample must be computed. These are estimated using the standard techniques, which we describe below (see Bîrzan et al. 2004, and references therein). Assuming that the cavity is filled with a relativistic fluid, the total enthalpy is

$$\begin{eqnarray}&&{{E}_{{\rm bubble}}}=4pV,\end{eqnarray} \tag{ 2 }$$

where p is the thermal pressure of the ICM at the radius of the bubble (estimated from X-ray data, assuming $p={{n}_{e}}kT$) and V is the volume of the cavity. We assume that the cavities are of prolate shape. The volume is then given by $V=4\pi R_{{\rm w}}^{2}{{R}_{{\rm l}}}/3$, where ${{R}_{{\rm l}}}$ is the projected semimajor axis along the direction of the jet and ${{R}_{{\rm w}}}$ is the projected semimajor axis perpendicular to the direction of the jet. Errors on the radii are assumed to be ±20%, and the jet is defined as the line that connects the central AGN to the middle of the cavity. The position of the central AGN is chosen to coincide with the position of the BCG as seen from the optical or infrared images (Figure 2). If two central dominant galaxies were present in the optical images, we chose the brightest one as the BCG. There is only one cluster where this applies: SPT-CLJ0616–5227. Modifying the location of the BCG to coincide with the second dominant galaxies modifies the cavity energetics by a factor of $\leqslant 2$, which is not significant for the purposes of this study.

In Table 1, we give the constraints on the X-ray cavity radii (${{R}_{{\rm l}}}$ and ${{R}_{{\rm w}}}$), enthalpy (pV), and cavity power (${{P}_{{\rm cav}}}$). Cavity powers are determined by dividing the total enthalpy of the X-ray cavity ($4pV$) by its age. The latter is given by the buoyancy rise time (Churazov et al. 2001):

$$\begin{eqnarray}&&{{t}_{{\rm buoyancy}}}=R\sqrt{\frac{S{{C}_{{\rm D}}}}{2gV}}.\end{eqnarray} \tag{ 3 }$$

Here, R is the projected distance from the central AGN to the middle of the cavity, S is the cross-sectional area of the bubble ($S=\pi R_{{\rm w}}^{2}$), ${{C}_{{\rm D}}}$ is the drag coefficient (assumed to be 0.75; Churazov et al. 2001), and g is the local gravitational potential. We use the values of g derived by M13 (see resulting values in Table 1) but stress that the typical uncertainties are larger than those reported in M13: they are on the order of ±50%, as mentioned in Section 2. The enthalpy and powers of the X-ray cavities in the SPT–SZ sample are therefore known, at best, to within a factor of a few.

In Figure 4, we plot the cooling luminosity of the 83 SPT–SZ clusters with Chandra observations using the standard 7.7 Gyrs cooling time definition. This plot shows that the majority of the SPT–SZ clusters with X-ray cavities lie in the strongest cool-core clusters. This is expected in part since the majority of the counts in the Chandra X-ray images will be concentrated toward the central regions for cool-core clusters, due to highly peaked X-ray surface brightness distributions. This higher concentration of counts in the central regions makes it easier to identify depressions.

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AGN feedback in clusters of galaxies is known to be finely tuned to the energy needed to offset cooling of the hot ICM out to $z\sim 0.6$ (HL12). We illustrate this in Figure 5, where we plot data points from Nulsen et al. (2009, black circles), Nulsen et al. (2009, black stars) and HL12 (blue circles). In green, we show the SPT–SZ clusters with X-ray cavities. Here, we adopt the first definition of the cooling luminosity, defined as the bolometric X-ray luminosity within which the cooling time is equal to 7.7 Gyrs. Figure 5 shows that, on average, X-ray cavities can offset cooling of the hot ICM even in the highest-redshift sources. We note that this result remains true if we use the other two definitions of the cooling luminosity (see Section 4). We discuss these results in Section 6.4.1.

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In Figure 6, we address whether AGN feedback in clusters of galaxies, as probed by the presence and properties of X-ray cavities, is evolving significantly with redshift. In this figure, we plot the mechanical power of the X-ray cavities (${{P}_{{\rm cav}}}$), the enthalpy of the X-ray cavities ($P{{V}_{{\rm total}}}$), and the cooling luminosity defined with the 7.7 Gyrs threshold (${{L}_{{\rm cool}}}$) for each cluster, as a function of redshift. In the bottom right panel, we also plot the average X-ray cavity radius ${{R}_{{\rm average}}}$ defined as ${{({{R}_{l}}\times {{R}_{{\rm w}}})}^{1/2}}$ for each X-ray cavity. Since we plot the radius for each cavity, there are more data points in this panel. We only include the objects in Rafferty et al. (2006, black circles) and HL12 (blue circles) since Nulsen et al. (2009) do not provide any redshift information. In these plots and for the remainder of the paper, we focus only on the systems with convincing cavities (i.e., we exclude SPT-CLJ0156–5541 and SPT-CLJ2342–5411). Figure 6 shows that there is no significant evolution in any of the cavity properties for the largest and most powerful outflows. To provide a first-order correction for the dependency of mass on the cavity powers, we also plot in Figure 6 the ratio between the mechanical power and the cooling luminosity as a function of redshift. As we will discuss in Section 6.1, we are strongly biased against finding small cavities ($r\lesssim 10$ kpc) in the XVP Chandra data. Removing these small cavities from Figure 7 for the lower-redshift samples does not affect the results. We further discuss Figures 5 and 6 in Section 6.4.1.

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While the results presented in Section 5 are interesting, there may be several biases present in the cavity selection method and the analysis. Below we address these potential biases.

First, we will clearly miss cavities below the resolution limit (${{R}_{{\rm average}}}\lesssim \;10$ kpc; see Table 1), as well as those that lie on a jet axis parallel to our line of sight. Furthermore, we limited the search to the central regions of each cluster, where most of the X-ray counts are located, and disregarded any X-ray depression identified at $r\gt 100$ kpc. Future, deeper X-ray observations would be necessary to detect these more extended cavities. We are likely therefore to miss any extremely large cavities like those in MS 0735.6+7421. Although these are expected to be energetically very important, they are expected to be rare with only two such systems currently known (McNamara & Nulsen 2007).

In Figure 8, we show the Chandra X-ray images of the massive galaxy cluster 4C+55.16 (z = 0.2412, Iwasawa et al. 2001; Hlavacek-Larrondo et al. 2011). The left panels show the deep Chandra images (74 ks), which reveal a large southern X-ray cavity. The right panels show the same image but for a reduced exposure time of only 4 ks and binned by a factor of 1.7 (pixel size of $0\buildrel{\prime\prime}\over{.} 836\times 0\buildrel{\prime\prime}\over{.} 836$) to mimic the appearance of the cluster as if it were located at the average redshift of the SPT–SZ clusters with X-ray cavities ($z\sim 0.7$). We reduce the exposure time to 4 ks since this reduces the total number of cluster counts to ∼2000, similar to the counts in the Chandra X-ray images of the SPT–SZ clusters. Figure 8 shows that the cavity in 4C+55.16 remains visible in all panels. According to our criteria described in Section 3.1, we would identify this X-ray cavity as a "potential" cavity since the cavity emission lies only 1σ–2σ below the surrounding X-ray emission. We note that adding an additional 75 ks blank field exposure to the images, mimicking the increase in background from the long exposure needed to get 2000 counts for high-redshift clusters, does not change our results.

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Even though Figure 8 demonstrates that we can identify cavities with as few as 2000 counts, we are likely missing several cavities due to limited data quality of the 83 SPT–SZ clusters with Chandra observations. To test the probability of identifying cavities in the SPT–SZ cluster survey, we applied the same imaging processes as in Figure 8 to the 13 MACS clusters with X-ray cavities: we reduced the exposure times such that each cluster contained only ∼2000 counts and binned the images to mimic the appearance of each cluster as if it were located at $z\sim 0.7$. Interestingly, we found that only ∼60% of these X-ray cavities would then have been detected using the same criteria as those in Section 3.1, and that an even smaller fraction (∼20%) would have had "clear" cavities with a cavity emission 2σ–3σ (or more) below the surrounding X-ray emission, as defined in Section 3.1. This demonstrates that we are likely missing cavities in the SPT–SZ sample due to the limited X-ray depth of the survey (see also Enßlin & Heinz 2002; Diehl et al. 2008; Bîrzan et al. 2012). Moreover, the strongly peaked X-ray surface brightness distributions of strong cool-core clusters also imply that the X-ray counts will be highly concentrated toward the central regions in these clusters. We may therefore be preferentially selecting X-ray cavities in strong cool-core clusters.

Simulations have shown that the increased SZ signal due to the presence of a cool core is not significant (Motl et al. 2005; Pipino & Pierpaoli 2010), and that, similarly, star formation does not fill in the SZ decrement significantly (e.g., McDonald et al. 2012). In particular, Lin et al. (2009) showed that at z = 0.6, less than 2% of clusters with a mass exceeding ${{10}^{14}}\;{{M}_{\odot }}$ will host a radio AGN that contributes to more than 20% of the SZ signal, even when accounting for the fraction of systems with flat or inverted spectra. This fraction is even smaller at higher redshift due to the decreasing flux. These biases should therefore not be significant.

Finally, we recall that the majority of the cavities presented here lie only $1\sigma -2\sigma$ below the surrounding X-ray emission. These depressions could therefore be caused by other activity in the cluster such as merger-induced asymmetries in the X-ray gas distribution. This is especially true for SPT–SZ clusters, since cluster merger rates are expected to increase with redshift (e.g., Cohn & White 2005; Ettori & Brighenti 2008). It is also important to note that cluster mergers can provide an additional source of heating to the X-ray gas. At high redshift, AGN feedback may therefore no longer be the only heating source of the ICM.

Out of the 83 SPT–SZ clusters with Chandra observations, we find that six clusters have visually convincing X-ray cavities. For these six systems, all of the cavities are found in pairs, most of which are similar in size and symmetrically located on either side of the AGN in the BCG. In Section 3.1, we also showed that these cavities lie $1\sigma -3\sigma$ below the surrounding X-ray emission. While the probability of having one $1\sigma -3\sigma$ random fluctuation in the Chandra X-ray images is high, due to the Poisson nature of the observations, the probability of having two such fluctuations in an eight-element azimuthal array, as in Figure 3, is substantially lower: ∼37% for two ∼1σ fluctuations within r = 100 kpc. Moreover, the probability of having the two fluctuations at 180° ± 30° from one another in PA is only ∼10%. If we also require them to have matching radii to within ±50 kpc in radius from one another, as observed in Table 1, then the number drops to ∼7.5% and even further for two ∼2σ (≲0.2%) and two ∼3σ (≲0.001%) fluctuations. Overall, the probability that the cavities presented here are caused by random fluctuations, given the PA and radii properties observed in Table 1, is very small. Instead, the depressions may well be cavities being carved out by twin radio lobes.

We note that all of these calculations were computed using a Monte Carlo approach, assuming a random normal distribution for the cavity flux and a random uniform distribution for the PAs and radii of the cavities. The statistics are also based on the annuli shown in Figure 3, which do not cover the full radial extent ($r\lt 100$ kpc) considered in the cavity selection. By adding additional spatial elements, we would increase the chance of a false detection (by adding more draws) but would also increase the significance of the individual detections (by improving the measurement of the "background"). Thus, we expect that these probabilities are within a factor of ∼2 of what one would get by doing a more rigorous analysis. In principle, it should also be possible to derive an approach, similar to the one used here, but focusing on the probability of finding $1\sigma -3\sigma$ fluctuations by eye within the entire region of $r\lt 100$ kpc. However, it is very difficult to quantify this selection function, as visual identification can vary significantly from one person to another. We therefore choose to focus only on reporting approximate probabilities in this paper, based on the annuli statistics shown in Figure 3.

In this section, we discuss various properties of the detected X-ray cavities in the SPT–SZ sample (e.g., Figure 9). First, we note that the average power, enthalpy, and radius of the cavities in the six SPT–SZ clusters with convincing X-ray cavities are $\sim 5\times {{10}^{45}}\ {\rm erg}\ {{{\rm s}}^{-1}}$, $\sim 6\times {{10}^{59}}\ {\rm erg}$, and ∼16 kpc, respectively, whereas the average cooling luminosity for these same clusters is $\sim (2-3)\times {{10}^{45}}\ {\rm erg}\ {{{\rm s}}^{-1}}$ (depending on which definition we adopt; see Section 4). The X-ray cavities in these six SPT–SZ clusters therefore provide, on average, enough energy to offset cooling of the hot ICM. This statement remains true even if (1) we include SPT-CLJ0156–5541 and SPT-CLJ2342–5411 in the calculations and (2) we only consider the two SPT–SZ clusters with "clear" cavities (SPT-CLJ0509–5342 and SPT-CLJ0616–5227) where the cavity emission lies $2\sigma -3\sigma$ below the surrounding emission. For the latter, the average cavity power, enthalpy, and radius would be $\sim 0.8\times {{10}^{45}}\ {\rm erg}\ {{{\rm s}}^{-1}}$, $\sim 3\times {{10}^{59}}\ {\rm erg}$, and ∼18 kpc, respectively, whereas the average cooling luminosity would be $\sim 0.1\times {{10}^{45}}\ {\rm erg}\ {{{\rm s}}^{-1}}$.

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We also note that the detection fraction in the sample is 7% since 6 of the 83 SPT–SZ clusters with Chandra observations have visually convincing X-ray cavities. If we only consider the two clusters with "clear" cavities, then the fraction drops to 2%. This is a factor of several less than the fraction observed in local clusters, where the fraction with X-ray cavities is 20%–30% (Dunn & Fabian 2006; Rafferty et al. 2006; Bîrzan et al. 2012; Fabian 2012; HL12). However, our result is more likely a lower limit, due to selection effects (see Section 6.1), which will need to be quantified with deeper observations and studies of other cluster samples at high redshifts.

6.4.1. Implications for AGN Feedback

6.4.2. Implications for Supermassive Black Hole Growth

Our results show that powerful radio mode feedback may be operating in massive clusters of galaxies for over half of the age of the universe (>7 Gyrs, corresponding to the look-back time since $z\sim 0.8$). If we assume once more that the duty cycles of the AGNs in BCGs remain high, as they do for local BCGs, then this implies that the supermassive black holes in BCGs may have accreted a substantial amount of mass to power the X-ray cavities. To determine this, we use the following equation relating the jet (or cavity) power (${{P}_{{\rm cav}}}$) to the black hole accretion rate ($\dot{M}$):

$$\begin{eqnarray}&&{{P}_{{\rm cav}}}=\eta \dot{M}{{c}^{2}},\end{eqnarray} \tag{ 5 }$$

where η is the efficiency and $c$ is the speed of light. We assume that the conversion efficiency between accreted mass and jet power is $\eta =0.1$ (Churazov et al. 2005; Merloni & Heinz 2008; Gaspari et al. 2012) but stress that if it were lower, then the black holes would need to accrete even more mass. Assuming that the average jet power of SPT–SZ clusters with X-ray cavities is a representative value for massive cool core clusters ($(0.8-5)\times {{10}^{45}}\ {\rm erg}\ {{{\rm s}}^{-1}}$), and integrating Equation (5) over 7 Gyrs while assuming nearly constant duty cycles, we find that the supermassive black holes in these BCGs must have accreted $(1-6)\times {{10}^{9}}\;{{M}_{\odot }}$ in mass to power the radio jets responsible for carving out the observed X-ray cavities. If correct, this would imply that supermassive black hole growth in BCGs may be important not only at earlier times during quasar mode feedback, when the black holes are accreting at rates near the Eddington limit (see Alexander & Hickox 2012, for a review), but also at later times when the black holes are accreting at low rates and driving powerful jetted outflows (see also HL12; Ma et al. 2013). We note that if we only consider the lower limit to the duty cycle (11% for cool core clusters), then the accreted mass is $(0.1-1)\;\times \;{{10}^{9}}\;{{M}_{\odot }}$ and remains substantial.