# The SPT-Chandra BCG Spectroscopic Survey I: Evolution of the Entropy
Threshold
for Cooling and Feedback in Galaxy Clusters Over the Last 10 Gyr

Michael S. Calzadilla Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Michael
McDonald Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Bradford A. Benson Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
Avenue, Chicago, IL 60637, USA  Fermi National Accelerator Laboratory, MS209,
P.O. Box 500, Batavia, IL 60510, USA  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637, USA  Lindsey E. Bleem High Energy Physics Division, Argonne National
Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL 60637, USA  Judith H. Croston School of Physical Sciences, The Open
University, Walton Hall, Milton Keynes, MK7 6AA, UK  Megan Donahue Michigan
State University, Physics and Astronomy Dept., East Lansing, MI 48824-2320,
USA  Alastair C. Edge Department of Physics, University of Durham, South Road,
Durham, DH1 3LE, UK  Benjamin Floyd Faculty of Physics and Astronomy,
University of Missouri – Kansas City, 5110 Rockhill Road, Kansas City, MO
64110, USA  Gordon P. Garmire Huntingdon Institute for X-ray Astronomy, LLC,
Huntingdon, PA 16652, USA  Julie Hlavacek-Larrondo Département de Physique,
Université de Montréal, Succ. Centre-Ville, Montréal, Québec, H3C 3J7, Canada
Minh T. Huynh CSIRO Space and Astronomy, PO Box 1130, Bentley WA 6102,
Australia  Gourav Khullar Department of Physics and Astronomy, and PITT PACC,
University of Pittsburgh, Pittsburgh, PA 15260, USA  Ralph P. Kraft Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Brian R. McNamara Department of Physics and Astronomy, University of
Waterloo, Waterloo, ON N2L 3G1, Canada  Waterloo Centre for Astrophysics,
University of Waterloo, Waterloo, ON N2L 3G1, Canada  Perimeter Institute for
Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
Allison G. Noble School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA  Charles E. Romero Center for Astrophysics —
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Florian
Ruppin Univ. Lyon, Univ. Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon,
F-69622, Villeurbanne, France  Taweewat Somboonpanyakul Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall,
Stanford, CA 94305, USA  G. Mark Voit Michigan State University, Physics and
Astronomy Dept., East Lansing, MI 48824-2320, USA

###### Abstract

We present a multi-wavelength study of the brightest cluster galaxies (BCGs)
in a sample of the 95 most massive galaxy clusters selected from South Pole
Telescope (SPT) Sunyaev-Zeldovich (SZ) survey. Our sample spans a redshift
range of $0.3<z<1.7$, and is complete with optical spectroscopy from various
ground-based observatories, as well as ground and space-based imaging from
optical, X-ray and radio wavebands. At $z\sim 0$, previous studies have shown
a strong correlation between the presence of a low-entropy cool core and the
presence of star-formation and a radio-loud AGN in the central BCG. We show
for the first time that a central entropy threshold for star formation
persists out to $z{\sim}1$. The central entropy (measured in this work at a
radius of 10 kpc) below which clusters harbor star-forming BCGs is found to be
as low as $K_{\mathrm{10~{}kpc}}=35\pm 4$ keV cm2 at $z<0.15$ and as high as
$K_{\mathrm{10~{}kpc}}=52\pm 11$ keV cm2 at $z\sim 1$. We find only marginal
(${\sim}1\sigma$) evidence for evolution in this threshold. In contrast, we do
not find a similar high-$z$ analog for an entropy threshold for feedback, but
instead measure a strong evolution in the fraction of radio-loud BCGs in high-
entropy cores as a function of redshift. This could imply that the cooling-
feedback loop was not as tight in the past, or that some other fuel source
like mergers are fueling the radio sources more often with increasing
redshift, making the radio luminosity an increasingly unreliable proxy for
radio jet power. We also find that our SZ-based sample is missing a small
(${\sim}$4%) population of the most luminous radio sources ($\nu
L_{\nu}>10^{42}$ erg s-1), likely due to radio contamination suppressing the
SZ signal with which these clusters are detected.

High-redshift galaxy clusters (2007) — Intracluster medium (858) — Cooling
flows (2028) — Star formation (1569) — Active galactic nuclei (16)

††facilities:  CXO, WISE. NSF/US Department of Energy 10m South Pole Telescope
(SPTpol). Magellan 6.5m Telescopes (Clay/LDSS3C, Clay/PISCO, Baade/IMACS).
Gemini South Telescope (GMOS-S). Blanco 4m Telescope (DECam, MOSAIC-II). Swope
1m Telescope (SITe3). ††software: astropy (Astropy Collaboration et al.,
2013), numpy (Harris et al., 2020), scipy (Virtanen et al., 2020), pandas
(Team, 2023), CIAO (Fruscione et al., 2006), XSPEC (Arnaud, 1996). Prospector
(Johnson et al., 2021), python-FSPS (Foreman-Mackey et al., 2014), SEDpy
(Johnson, 2021), matplotlib (Hunter, 2007), seaborn (Waskom, 2021),
jupyter/IPython Notebooks (Kluyver et al., 2016), SAOImage DS9 (Joye & Mandel,
2003), pyRAF (Science Software Branch at STScI, 2012), Source Extractor
(Bertin & Arnouts, 1996)

##  1 Introduction

Galaxy clusters, the largest gravitationally bound structures in the Universe,
are intricate ecosystems that allow us to investigate numerous astrophysical
processes. Central to our understanding of these systems is the hot ($T\sim
10^{7}$ K) intracluster medium (ICM), a diffuse gas that permeates the space
between the member galaxies and emits X-rays via radiative cooling. In the
central regions of many clusters, where this reservoir of gas is relatively
colder and denser, the more frequent interactions of ICM particles increases
their rate of cooling and X-ray production, and eventually creates an inward
flow of material called a “cooling flow,” where the characteristic cooling
time is much shorter than the age of the universe (e.g. Fabian, 1994). Based
on the standard cooling flow model, this should lead to a substantial
accumulation of cool gas over time in the cluster core, subsequently fueling
prodigious star formation rates in the central dominant brightest cluster
galaxy (BCG) and mass accretion onto the BCG’s supermassive black hole (SMBH).
Despite these theoretical expectations, actual observations reveal that the
amount of cooling implied by weak soft X-ray line strengths (Peterson &
Fabian, 2006) or in the form of stars or cold molecular gas reservoirs is
orders of magnitude smaller than predicted (e.g. Johnstone et al., 1987;
McNamara & O’Connell, 1989; Allen, 1995; Crawford et al., 1999; Rafferty et
al., 2006; O’Dea et al., 2008; Donahue et al., 2015; McDonald et al., 2018),
culminating in what we know as the cooling flow problem.

The leading theory for which process must be counteracting this expected
cooling is feedback from active galactic nuclei (AGN; see reviews by McNamara
& Nulsen 2007, 2012; Donahue & Voit 2022). AGN are actively accreting SMBHs in
the centers of galaxies, which can emit enormous amounts of energy in the form
of radiation (“radiative” or “quasar mode” feedback) or jetted outflows
(“mechanical” or “radio mode” feedback). In this self-limiting feedback
mechanism, the very precipitation out of the hot ICM that forms stars in the
BCG eventually feeds the central SMBH which channels its accretion energy
toward heating its surroundings and preventing further cooling. Radio mode
feedback is especially effective at suppressing runaway cooling on large
scales as it drives powerful jets, shocks, sound waves, and turbulence (e.g.
Churazov et al., 2001; Li et al., 2017; Reynolds et al., 2002; Yang &
Reynolds, 2016; Soker et al., 2001; Zhuravleva et al., 2014; Hillel & Soker,
2017; Yang et al., 2019; Gaspari, 2015). Evidence for such a tightly-regulated
feedback loop can be found, for instance, in multi-wavelength observations
that show a strong correlation between the work $p\Delta V$ done by radio jets
as they expand against the surrounding ICM and that of the cooling luminosity
of the ICM due to radiative losses (e.g. Bîrzan et al., 2004). Observations
also seem to imply that every cool core (CC) cluster – clusters whose central
cooling times are short compared to the age of the Universe – hosts a radio
loud AGN (Sun, 2009). Finally, studies like Cavagnolo et al. (2008) also
provide compelling evidence that once the central entropy of the ICM falls
below a critical threshold, it becomes locally unstable to multiphase cooling
and triggers both star formation and AGN activity (see also Nulsen, 1986;
Pizzolato & Soker, 2005; Rafferty et al., 2008; Main et al., 2017; Hogan et
al., 2017; Pulido et al., 2018).

While there is a wealth of evidence to support the AGN feedback mechanism (see
Fabian, 2012, for a review), many details remain to be worked out. Of
particular relevance to this study, how the balance between cooling and
feedback was established and has evolved with time is still a largely
unexplored area of research. Only recently has a window into earlier cosmic
epochs been opened, with the advent of Sunyaev-Zeldovich-based (SZ)
cosmological surveys discovering thousands of distant galaxy clusters in the
past decade up to redshifts of $z\lesssim 2$ (e.g. Vanderlinde et al., 2010;
Bleem et al., 2015; Planck Collaboration et al., 2016; Hilton et al., 2021),
allowing for studies of their cooling and heating properties, among others.
Past flux-limited studies have also enabled evolutionary studies, but are
often limited by their biased selection of rarer, intrinsically brighter
objects with increasing redshift, which the mass-limited selection of SZ
surveys does not suffer from. Using these large SZ samples, we have so far
learned that over the past 9 Gyr (i.e. $z\lesssim 1$) neither the CC fraction,
nor the distributions of central cooling times and entropies of the ICM have
had any significant evolution (see e.g. McDonald et al., 2013; Ruppin et al.,
2021). The mass, size, and metallicity of cool cores have similarly
experienced no evolution (McDonald et al., 2016a, b, 2017). Examples of
extreme ICM cooling at high redshifts have also provided insight into how fast
some clusters and BCGs can grow, as in the case of SPT2215 (Calzadilla et al.,
2023) and SpARCS1049 (Webb et al., 2015; Hlavacek-Larrondo et al., 2020). On
the heating side, we have also learned that AGN feedback has been operating
since at least $z\sim 1$, so it must have been established at earlier times,
and that the ratio of AGN heating power to cooling luminosity has also
remained relatively constant at a gentle
$P_{\mathrm{cav}}/L_{\mathrm{cool}}<1$ (see Rafferty et al., 2006; Nulsen et
al., 2009; Hlavacek-Larrondo et al., 2012, 2015; Calzadilla et al., 2019;
Ruppin et al., 2023).

In this study, we use one such sample of SZ-selected galaxy clusters from the
South Pole Telescope (SPT) SPT-SZ survey (Bleem et al., 2015). We focus on the
higher-mass subsample that has been followed up with Chandra, defined in
McDonald et al. (2013, 2014), all of which have high-angular resolution X-ray
observations in hand. The X-ray properties of this sample have been studied in
detail in works like McDonald et al. (2013), Mantz et al. (2016), and Sanders
et al. (2018), and we re-analyze these data here to measure their central
entropies. Unique to this study, however, is optical spectroscopy for every
BCG in the sample, to look at where the cooling out of the hot X-ray emitting
ICM has concentrated into forming stars. We also utilize high-resolution radio
data to determine whether the cooling flows have ultimately accreted onto the
central SMBH in these BCGs and are triggering mechanical AGN feedback. This
one of a kind cluster survey is currently the best and only dataset capable of
answering how the largest galaxies, their central supermassive black holes,
and their large-scale environments have grown and co-evolved over the past
$\sim 10$ Gyr. In this first paper of a series, we introduce the sample and
the new data, and address whether the well-established threshold for ICM
cooling instabilities that we see in local systems (e.g. Cavagnolo et al.,
2008) is still present at higher redshifts and whether it has evolved with
time. In other words, we seek to address whether the conditions for cooling
have evolved. In $\S\\!\\!$ 2 we describe our sample in more detail, explain
our BCG selection process, and how we reduced all of our multi-wavelength
data. Section $\S\\!\\!$ 3 connects our new star formation measurements to the
central ICM entropies and discusses whether an entropy threshold exists at
higher redshifts and if it has evolved with time, and compare to previous
efforts to study this effect. We further discuss the connection between the
entropy threshold and feedback in $\S\\!\\!$ 4, as well as discuss potential
sources of bias. Finally, we summarize our takeaway results and describe
future work in $\S\\!\\!$ 5. Throughout this paper, we assume a flat
$\Lambda$CDM cosmology with $H_{0}=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$, and
$\Omega_{\Lambda}=0.7$. All measurement errors are $1\sigma$ unless noted
otherwise.

##  2 Sample & Observations

Our sample was chosen based on the ${\sim}100$ SPT-SZ selected clusters that
have been followed up with Chandra, ATCA, and Magellan observing campaigns,
described further in Bleem et al. (2015) and in the following sections. After
performing spectroscopic followup of the BCGs (with selection described in
$\S\\!\\!$ 2.2.1), some of these resulted in mis-identifications, resulting in
a total remaining sample of 95 clusters. Additionally, we compare our results
to the low-$z$ ACCEPT cluster sample (Cavagnolo et al., 2009), which we will
describe in more detail in $\S\\!\\!$ 4. The mass-redshift distribution for
both of these samples can be seen in Figure 1.

Figure 1: Mass vs redshift distribution of the data considered in this study.
In blue are the clusters from our sample, which were drawn from the SPT-SZ
survey, with a median mass and redshift of $\left<M_{500}\right>=4.2\times
10^{14}$ M⊙ and $\left<z\right>=0.6$. The lower mass limit of our sample
(${\sim}2\times 10^{14}$ M⊙) is shown as a horizontal dashed line. To look for
evolutionary trends, we also compare our sample to the nearby ($0<z<0.3$)
ACCEPT sample of galaxy clusters (Cavagnolo et al., 2009), shown here as grey
points. For these comparisons, we enforce a minimum mass of $2\times 10^{14}$
M⊙ on the ACCEPT sample. Figure 2: Multi-wavelength images for one of the
clusters in our sample (SPT-CLJ0000-5748), showing smoothed X-ray data from
Chandra (left panel), optical data from DES (middle panel), and radio data
from ATCA (right panel), with beam size shown in the bottom left. From the
X-ray data, we can quantify how much hot ($T\sim 10^{7}$ K) gas is cooling out
of the ICM to subsequently fuel both star formation onto the BCG that ionizes
warm ($T\sim 10^{4}$ K) gas at optical wavelengths, as well as accretion onto
the BCG’s central AGN which we can see at radio wavelengths. The BCG (selected
as described in $\S\\!\\!$ 2.2.1) is marked in all panels with a cross ($+$).
ATCA contours are shown at $[6,50,90,130,170]\times\sigma_{\mathrm{rms}}$,
where $\sigma_{\mathrm{rms}}\sim 32\,\mu$Jy beam-1.

###  2.1 X-ray (Chandra)

As the crucial fuel source for eventual star formation and SMBH accretion, we
must first examine the X-ray emitting ICM in each of the galaxy clusters in
our sample. All of our systems have been observed with the Chandra X-ray
Observatory, with the bulk of the observations coming from the multi-cycle
Chandra X-ray Visionary Project (XVP; PI: B. Benson) which observed the 80
most massive clusters from SPT-SZ sample above $z>0.3$, or a Cycle 16 Large
Program that observed 10 SPT-selected clusters at $z>1.2$ (PI: McDonald).
X-ray data for the remaining clusters in our sample were obtained either
through archival data, or various smaller Guest Observer (GO; PIs: McDonald,
Hlavacek-Larrondo, Mohr) or Guaranteed Time Observer (GTO; PIs: Garmire,
Murray) programs. More details on these observations can be found in McDonald
et al. (2013, 2014). An example X-ray image of one of the clusters in our
sample can be found in Figure 2 (left panel).

To reduce and analyze these X-ray data, we used the Chandra Interactive
Analysis of Observations (CIAO) v4.14.0 software with CALDB v4.9.8, in a
standard fashion similar to McDonald et al. 2013; Calzadilla et al. 2019,
2023; Ruppin et al. 2021, 2023. All observations were made with the ACIS-I
instrument. We applied the latest gain and charge-transfer inefficiency
corrections using the chandra_repro script, as well as improved background
screening for observations taken in the VFAINT telemetry mode. Flare removal
from the lightcurves is done using the lc_clean script, then point sources are
identified with a wavelet filter decomposition using wavdetect (e.g. Vikhlinin
et al., 1998) and subsequently masked to produce a clean event file. Surface
brightness profiles are extracted from 20 concentric circular annuli following
the uniform binning scheme of McDonald et al. (2017) and Ruppin et al. (2021),
with the outer radius of the $i^{\mathrm{th}}$ annulus defined as:

$r_{\mathrm{out,i}}=(a+bi+ci^{2}+di^{3})\times R_{500}$ (1)

where $(a,b,c,d)=(13.779,-8.8148,7.2829,-0.15633)\times 10^{-3}$, and
$R_{500}$ is the radius where the mean density is 500 times the critical
density of the Universe at that redshift and is obtained from Bleem et al.
(2015). The annuli are centered on the X-ray peak positions – which are found
according to the procedure described in Ruppin et al. (2021) – so as to trace
where the ICM is cooling most rapidly. The surface brightness profile is then
extracted at these annuli using the dmextract tool over the $0.7-2.0$ keV
energy range.

####  2.1.1 Spectral Fitting

To measure a spectroscopic temperature profile for each object, we extract
spectra using specextract along a much coarser binning scheme with outer radii
defined by $(0.1,0.3,0.8)\times R_{500}$ and centered on the same X-ray peak
as before. Our data quality varies considerably across the sample, and this
coarse binning scheme is meant to optimally sample the worst cases where
observations resulted in a few hundred counts per cluster. Many of our more
deeply-observed clusters have sufficient quality to do a finer radial
sampling, but we opted to have a uniform binning scheme in order to avoid any
resolution bias in comparing spectroscopic profiles and central quantities
later on in our analysis. Each of the spectra in these three bins were modeled
over the $0.7-7.0$ keV range as an optically-thin, X-ray emitting plasma in
XSPEC v3.0.9 with the APEC/AtomDB thermal spectral model (Smith et al., 2001),
in addition to PHABS for photoelectric absorption (Morrison & McCammon, 1983).
We adopt Anders & Grevesse (1989) abundances for consistency with previous
literature, and Hydrogen column densities from the Leiden-Argentine-Bonn
survey (Kalberla et al., 2005). Redshifts are fixed to the updated values from
Bocquet et al. (2019). The metallicity is fixed to a value of $0.3Z_{\odot}$.
The background is assumed to be a combination of instrumental and
astrophysical backgrounds. The instrumental background is obtained by
normalizing the unscaled stowed background to the count rate of the
observations in the $9-12$ keV band and subtracting from the source spectra.
The remaining astrophysical background is measured from an off-source region
across the ACIS-I chips, and modeled as a second APEC component (fixed at
$kT=0.18$ keV, $Z=Z_{\odot},z=0$) to model soft Galactic X-ray emission, as
well as a BREMSS model (fixed at $kT=40$ keV) to model unresolved point
sources (e.g. McDonald et al., 2019), with normalizations scaled by the ratio
of areas of the background to source extraction regions. The resulting
temperature profile was fit with the analytical model from Vikhlinin et al.
(2006), which was first projected along the line of sight:

$kT(x)=kT_{0}\frac{\left(\frac{x}{0.045}\right)^{1.9}+\frac{T_{\mathrm{min}}}{T_{0}}}{\left(\frac{x}{0.045}\right)^{1.9}+1}\frac{1}{\left[1+\left(\frac{x}{0.6}\right)^{2}\right]^{0.45}}$
(2)

where $x=r/R_{500}$, and $T_{0}$ and $T_{\mathrm{min}}$ are free parameters
representing the mean and core ICM temperatures respectively.

To model the ICM density, we use the surface brightness profile extracted from
the fine annuli defined above, which can be expressed in terms of the emission
measure ${\rm EM}(r)=\int n_{e}(r)n_{p}(r)dl$:

$S_{X}(\theta)=\frac{1}{4\pi(1+z)^{4}}\epsilon(T,z)\times EM(r)$ (3)

where $z$ is the redshift, $\epsilon(T,z)$ is the temperature and redshift-
dependent ICM emissivity, and $\theta=r/D_{A}(z)$ is the radius scaled by
angular diameter distance. At each annulus, we calculate a $S_{X}(\theta)$ to
EM conversion factor by extracting a spectrum, and modelling it with an APEC
model, fixing the temperature to the interpolated best-fit Vikhlinin et al.
(2006) model at this radius, allowing only normalization to vary. The APEC
normalization is defined as

$N=\frac{10^{-14}}{4\pi[D_{A}(z)(1+z)]^{2}}\int n_{e}n_{p}dld\Omega$ (4)

where $D_{A}(z)$ is the angular diameter distance, $dl$ is the line of sight
differential, $d\Omega$ is the solid angle differential. Because the APEC
normalization contains the EM term, we can use the surface brightness profile
$S_{X}(\theta)$ to compute an EM profile. We can arrive at the electron
density ($n_{e}$) profile by projecting the analytic density profile from
Vikhlinin et al. (2006) onto the plane of the sky to fit this EM profile:

$n_{e}(r)=n_{e0}\frac{\left(r/r_{c}\right)^{-\frac{\alpha}{2}}}{\left[1+\left(r/r_{c}\right)^{2}\right]^{\frac{3\beta}{2}-\frac{\alpha}{4}}}\frac{1}{\left[1+\left(r/r_{s}\right)^{\gamma}\right]^{\frac{\epsilon}{2\gamma}}}.$
(5)

which is a modified beta model with a cusp, rather than a flat core (defined
by $n_{e0}$, $r_{\rm c}$, $\alpha$, $\beta$), a steeper outer profile slope
(defined by $r_{\rm s}$, $\gamma$, and $\epsilon$). The fitting procedure is
done with the emcee (Foreman-Mackey et al., 2013) MCMC ensemble sampler python
package, as described in Ruppin et al. (2021).

Following the calculation of our temperature and density profiles (see Figure
3), it is straightforward to compute additional thermodynamic profiles. Of
particular relevance for our analysis, we compute the pseudo-entropy profile
as $K(r)=n_{e}^{-2/3}kT$. As a central quantity, we quote the entropy at a
radius of 10 kpc for all of our cluster X-ray profiles in Table 1. We have
compared the results of this analysis to the thermodynamic profiles published
for the same clusters in McDonald et al. (2013) and Sanders et al. (2018), and
find good agreement overall. In particular, the agreement with Sanders et al.
(2018) is excellent at all radii, while our profiles disagree somewhat with
McDonald et al. (2013) due, largely, to a different choice of center –
McDonald et al. (2013) use the large-scale centroid as the center, while here
we use the X-ray peak. We will return to this difference in $\S\\!\\!$ 3.2.

Figure 3: Thermodynamic profiles for one of the clusters in our sample (SPT-
CLJ0000-5748). Top panel: electron density ($n_{e}$) profile. Middle panel:
temperature ($kT$) profile. Bottom: entropy ($K=n_{e}^{-2/3}kT$) profile. In
all panels, the dark and light blue bands show the 1$\sigma$ and 2$\sigma$
confidence interval, respectively, while the solid line shows the best fit.
The density and entropy profiles are showing the three dimensional model,
while the temperature panel is showing the projected model fit to the data.

###  2.2 Optical/IR

####  2.2.1 BCG Selection

To trace whether the cooling that we measure from the X-ray-emitting ICM is
fueling star formation in the central BCG, we first have to identify which
member galaxy is indeed the BCG. In the local universe, this may typically be
done by eye after a galaxy cluster has had sufficient time to relax and lead
to a dominant BCG. However, at higher redshifts, when clusters are still
disturbed and galaxies are still colliding and accreting smaller galaxies, it
is often not obvious. Thus, we employ a more algorithmic approach, beginning
with a probability-based BCG assignment. In summary, this analysis uses the
optical and near-infrared spectral energy distribution (SED), in combination
with the EAZY (Brammer et al., 2008) and FAST (Kriek et al., 2009) codes to
produce redshift ($p(z)$) and stellar mass ($p(M_{*})$) distributions for each
galaxy in each cluster. These distributions are used to compute the
probability that each galaxy is the most massive, based on $p(M_{*})$, in the
three-dimensional cluster volume based on $p(z)$. For high quality data this
methodology typically assigns nearly 100% probability to a single massive
galaxy, but this probability decreases as the data quality decreases, as it
should. More details on this BCG probability assignment can be found in
Somboonpanyakul et al. (2022) and will be provided in Noble et al. (in prep).

After assigning these BCG probabilities to member galaxies in each cluster, we
then consider a hierarchy of conditions before confirming a final BCG
candidate: (1) If a candidate had a BCG probability of $p\gtrsim 90\%$, we
considered this sufficient after additional visual inspection. (2) If there
were no obviously dominant galaxies with $p\gtrsim 90\%$ or there were two or
more galaxies with roughly equal BCG likelihoods, then we chose the galaxy at
the center of gravitational lensing arcs, assuming these arcs are part of a
larger lensing ring. These arcs are visually identified and are present
throughout much of our sample, and have been the subject of many papers and
follow-up observations (e.g. Bleem et al., 2015). (3) If there were no signs
of lensing, we chose the brightest galaxy near the X-ray peak positions
measured in McDonald et al. (2013). X-ray centroids were also available, but
the X-ray peak will more closely follow where most of the cooling occurs, and
can move relative to the underlying gravitational potential just as the BCG
should. With this heuristic, we publish these new BCG coordinates for the
entire sample in Table 1, and proceed to collect follow-up spectroscopy for
each BCG.

####  2.2.2 Spectroscopy

To determine whether the ICM is forming stars as it cools and deposits onto
the central BCG, we gather spectroscopic data to look for signatures of young
stars ionizing their surroundings. For a large portion of the objects in our
sample, archival optical spectroscopy was available from the SPT
collaboration’s follow-up campaigns. Most of this spectroscopy (${\sim}60\%$)
was collected prior to 2017, with the GMOS spectrograph at the Gemini Southern
Telescope, as well as the FORS2 spectrograph on the Very Large Telescope.
Additional details on these observations and their reduction and calibration
can be found in Ruel et al. (2014) and Bayliss et al. (2016) where they were
originally presented. Lastly, we used the spectroscopic data on the two
intermediate redshift systems presented in Sifón et al. (2013).

Beyond these archival observations, we present here new spectroscopy for one-
third of the objects in our sample, including a few re-observations of
previously observed targets due to [O ii] falling in a chip gap, insufficient
depth, or poorly calibrated data. For these new observations, we used the
IMACS spectrograph on the Magellan/Baade telescope for 13 systems in October
2019, 7 systems in October 2021, and 3 systems in November 2021. We also used
the LDSS3 spectrograph to observe 2 systems in January 2020, 2 systems in
October 2020, 1 system in November 2020, 4 systems in September 2021, and 1
system in November 2021. Filters and gratings for our observations were chosen
to search for the presence of redshifted [O ii] emission from young stars
ionizing their surrounding gas. Exposure times were calculated to reach a
sensitivity limit of $L_{\mathrm{[O\,\textsc{ii}]}}>1.2\times 10^{40}$ erg s-1
to enable a SFR measurement down to a sensitivity of at least 1 M⊙ yr-1.

To reduce these spectroscopic data, we used the standard
pyRAF/IRAF111https://iraf-community.github.io/pyraf.html tools. Science and
calibration arc exposures were bias and flat-field corrected using the
imred.ccdred package. We also use the response task from the twodspec.longslit
to fit for the shape of the lamp spectrum in the dispersion direction before
flat-fielding. For multi-object spectra, the one dimensional (1D) BCG spectrum
was identified and traced in the flat-fielded science frames using the
twodspec package apall with background subtraction. 1D spectra were extracted
from the arc frames using the same traced apertures, after which a calibration
solution was calculated with the identify task. This wavelength solution was
applied to the science frames with the onedspec refspec and dispcor tasks. The
procedure for long-slit spectra was similar, except wavelength calibration was
done first with identify, reidentify, and fitcoords. This wavelength solution
was then applied to the two-dimensional spectrum along with a rectilinear
transformation with the transform task, with background subtraction performed
with the background task. Finally, a 1D spectrum was then extracted with the
apall task.

####  2.2.3 Photometry

Optical/IR imaging for every cluster in the SPT sample has been obtained for
the purpose of optical/IR confirmation and the assessment of photometric
redshifts. The acquisition and reduction of these data, which come from a wide
variety of ground-based optical telescopes, are presented in full detail in
Bleem et al. (2015). This work also describes the full details of our aperture
photometry, star-galaxy separation, and photometric calibration.

Infrared photometry was obtained from the WISE satellite using the W1 ($3.4\mu
m$) and W2 ($4.6\mu m$) bands. We cross-matched our BCG coordinates with the
AllWISE source catalog (Cutri et al., 2021), and extracted the photometric
values where there was a match within 2″.

####  2.2.4 BCG Star Formation Rates

Figure 4: Spectral energy distribution (SED) fit of one of our clusters, SPT-
CLJ0000-5748. Using the calibrated optical and infrared photometry of each of
our clusters, we can correct for the shape of our initially uncalibrated
spectra and flux calibrate using the SED fitting code prospector. Photometric
filter bandpasses are plotted in gray and labeled. Model photometry and
spectroscopy are shown in red, while observed data are shown in blue. In the
inset we show our fit to the residual spectrum after subtracting the best-fit
model SED from the calibrated spectrum. A linear plus Gaussian model is used
to look for the presence of [O ii]$\lambda\lambda 3726,3729~{}\mathrm{\AA}$
which signals star formation and is covered over our sample’s entire redshift
range ($0.3<z<1.7$).

The wavelength-calibrated spectra obtained from the reduction steps outlined
in $\S\\!\\!$ 2.2.2 were fit in combination with optical (griz/VRI), as well
as IR WISE (W1, W2) photometry, using the prospector (Johnson et al., 2021)
SED fitting code, which is especially helpful in providing spectrophotometric
calibration of these uncalibrated spectra (based on calibrated photometry),
constraining the underlying stellar continuum, and providing a rough estimate
of the amount of intrinsic attenuation. Prospector uses Markov Chain Monte
Carlo (MCMC) methods to perform stellar population synthesis (SPS) based on
the Python-FSPS framework (Conroy et al., 2009; Foreman-Mackey et al., 2014),
incorporating the MILES stellar spectral library (Falcón-Barroso et al., 2011)
and MIST isochrones (Choi et al., 2016). Magnitudes are all converted to the
AB system, and the appropriate filter throughputs corresponding to each
different telescope and instrument were loaded using the python package sedpy
(Johnson, 2021).

In fitting our spectrophotometry, we attempt only to model the stellar
continuum, and mask out telluric absorption lines as well as bright emission
lines like [O ii], [O iii] $\lambda 5007$, and the Hydrogen Balmer series
lines H$\alpha~{}\lambda 6563$ \+ [N II] $\lambda\lambda 6549,6585$,
H$\beta~{}\lambda 4861$, H$\gamma~{}\lambda 4340$, and H$\delta~{}\lambda
4102$, which are typically associated with ionization from star formation and
AGN (e.g. Kennicutt, 1998; Kewley et al., 2004). For a given BCG, once we
obtain our best-fit model, we subtract it from our flux-calibrated spectrum
and look for the presence of emission lines in the residual spectrum. To model
the line-free stellar continuum, we employ a simple delayed-tau parametric
star formation history of the form ${\rm
SFR}(t,\tau)\propto\left(t/\tau\right)e^{-t/\tau}$, as in Khullar et al.
(2022), with a Salpeter initial mass function (Salpeter, 1955), dust
attenuation following Calzetti et al. (2000), and additional free parameters
to capture a burst of recent star formation as well. For the spectrum shape
calibration, a multiplicative calibration vector describing the ratio between
observed and model spectra was defined using a third order Chebyshev
polynomial with Prospector’s PolySpecModel class. We also define a noise model
using a multiplicative noise inflation (i.e. “jitter”) term and a pixel
outlier mixture model. Bayesian priors for all of the parameters described
above, as well as as the free parameters of stellar mass, metallicity, age,
and redshift, are reported in Table 2.

An example SED fit to spectrophotometry for one of the BCGs in our sample,
SPT-CLJ0000-5748, can be found in Figure 4. In this figure we show the
observed photometry and the now flux- and shape-calibrated observed spectrum,
along with the best-fit model to the stellar continuum and photometry from
prospector. After subtracting the best-fit stellar continuum from the observed
spectrum, we then proceed with fitting this residual spectrum with a linear
background to model any excess continuum, plus a Gaussian to model possible [O
ii] emission, shown in the inset in Figure 4. The spectra are converted to the
rest-frame using the best-fit BCG redshift. We fit a region spanning
$3727\mathrm{\AA}\pm 100\mathrm{\AA}$. A variable slope $m$ and intercept $b$,
allowed to vary uniformly over the ranges $m\in[-1,1]\times 10^{-16}$ erg s-1
cm-2 and $b\in[-5,5]\times 10^{-14}$ erg s-1 cm-2 Å-1, respectively, are used
to model out any residual structure to the continuum resulting from an
insufficiently accurate SED fit. For the Gaussian component meant to capture
potential [O ii] emission, the position is allowed to vary uniformly over the
range of $3727\mathrm{\AA}\pm 31\mathrm{\AA}$ (i.e. $\pm 2500$ km s-1), the
amplitude between ${\pm}1\times 10^{-15}$ erg s-1 , and a velocity dispersion
between $0.6<\sigma<6.0$ (i.e. $50-500$ km s-1). These fits are performed
using the ensemble MCMC sampler python package emcee (Foreman-Mackey et al.,
2013) with a Gaussian likelihood, 128 walkers, 1000 burn-in steps, and 10000
production steps. To allow for convergence, we let the chains run until they
were at least 50 times the autocorrelation length, guaranteeing a sufficient
number of functionally independent samples. An example of one of these fits
performed to the residual spectrum of SPT-CLJ0000-5748 is shown in the inset
of Figure 4. At each iteration in the MCMC fits, a flux was calculated from
integrating that specific model Gaussian. In the case of non-detections, the
flux upper limit we quote is the 84th percentile of the MCMC model fluxes. A
fit was considered a non-detection of [O ii] emission if the 0.15 percentiles
(i.e. $-3\sigma$) of all these integrated fluxes was negative, or if the
median peak model flux density was not at least $2\sigma$ above the continuum
flux density adjacent to the line center. With these stringent criteria, we
find that 22/95 spectra have [O ii] detections. Some potential detections are
missed, but we opted for purity over completeness for the purposes of our
analysis in the sections that follow.

The line fluxes we measure can all be found in Table 1. The fluxes are
converted to luminosities and then extinction-corrected using $E(B-V)=0.32\pm
0.13$ based on the distribution of BCG reddening measurements from Crawford et
al. (1999), and using the extinction law of Calzetti et al. (2000), assuming
$R_{V}=3.1$. Luminosities are converted to SFRs using the
SFR-L${}_{\mathrm{[O\,\textsc{ii}]}}$ relation and scatter from Kewley et al.
(2004, eqn. 4).

###  2.3 Radio

To connect the cooling, X-ray-emitting ICM to synchrotron radiation resulting
from black hole feeding in the BCG, we also gathered radio data for each of
our systems. A targeted followup campaign of XVP clusters was made with the
Australia Telescope Compact Array (ATCA). These clusters were initially
observed at 2 GHz in January 2015, reaching an rms noise of 28-55 $\mathrm{\mu
Jy}$ beam-1. Additional, higher-resolution observations at 5 and 9 GHz were
made in August 2016 to followup those systems where a strong detection was
found at 2 GHz. At these frequencies, the radio maps reached an rms noise of
30-67 $\mathrm{\mu Jy}$ beam-1 (5 GHz), and 19-77 $\mathrm{\mu Jy}$ beam-1 (9
GHz). All ATCA observations were reduced with the 21 February 2015 release of
the Miriad software package (Sault et al., 1995). 60 out of the 95 clusters in
our sample had available ATCA data at any observing frequency. For the
remainder of our sample, we used 887.5 MHz radio data from the Rapid ASKAP
Continuum Survey222https://data.csiro.au/collection/csiro:52217 (RACS) first
data release (Hale et al., 2021; McConnell et al., 2020). RACS has worse
angular resolution ($\theta\sim 25\arcsec$) than ATCA ($\theta\sim
3-6\arcsec$), but the RACS maps in Hale et al. (2021) have uniform sensitivity
(${\sim}0.3$ mJy rms) and coverage of the entire SPT-SZ footprint with a
declination range of $\delta\in[-80^{\circ},+30^{\circ}]$.

To consider a radio source a detection associated with our BCG, we required
that the source position was within 5″of the BCG coordinates. For the ATCA
data, the angular resolution was good enough to visually confirm these
associations (see right panel in Figure 2). These ATCA data provided 18/95
confirmed radio detections. The ASKAP data matching contained 27/95
detections, confirming 15 of the 18 ATCA detections, and found an additional
12 unique detections only for the systems that had no ATCA data (i.e. the
ASKAP did not misidentify any detections where there was also higher-
resolution ATCA data). Thus we find a total of 30/95 (32%) radio detections.
The ATCA data maps were modeled interactively with a variable combination of
2D Gaussians, with major and minor axes and position angles fixed to the point
spread function (psf) model parameters in each observing frequency. The sky
position and amplitudes of the Gaussians were free to vary, and were estimated
using an MCMC analysis. More details on this fitting procedure can be found in
Ruppin et al. (2023). Radio fluxes and uncertainties were estimated by
sampling the posterior distributions of the model parameters at each
frequency. For non-detections, we instead quoted an upper limit based on
3$\times$ the rms of the radio maps. Similarly for the ASKAP data, we quoted
the integrated flux of the RACS DR1 “continuum_component” source closest to
our BCG within 5″, and for the non-detections we again quoted 3$\times$ the
rms of the radio maps. Fluxes were converted to k-corrected 1.4 GHz rest-frame
radio power assuming $L\propto\nu^{\alpha}$, and using

$L_{1.4~{}\mathrm{GHz},{\rm rest}}=\frac{4\pi
D_{L}^{2}(z)}{(1+z)^{1+\alpha}}\left(\frac{1.4~{}\mathrm{GHz}}{\nu}\right)^{\alpha}F_{\nu,{\rm
obs}}$ (6)

where $z$ is the redshift, $D_{L}(z)$ is the luminosity distance, and $\alpha$
is the power-law spectral index for the radio source. For ATCA data where
multiple frequency bands were available (2, 5, and/or 9 GHz), the $\alpha$ was
measured directly, while for the rest of the ATCA and all of the ASKAP data,
we set $\alpha=-1.12\pm 0.06$ from the analysis of SPT radio sources from
Ruppin et al. (2023).

##  3 Redshift Dependence of the Entropy Threshold for Multiphase Cooling

In the decade after the launch of Chandra, the new era of high-resolution
X-ray imaging allowed us to investigate and trace in extraordinary detail the
conditions that lead to rapid cooling of the ICM. The seminal work done by
Cavagnolo et al. (2008, 2009), for instance, demonstrated that below a
threshold of $K_{0}<30$ keV cm2 in central ICM entropy, rapid cooling is
simultaneously associated with high levels of H$\alpha$ emission (see also
Nulsen, 1986; Pizzolato & Soker, 2005; Rafferty et al., 2008; Main et al.,
2017; Hogan et al., 2017; Pulido et al., 2018), which probes multiphase
cooling and is a good indicator of star formation and AGN activity (e.g.
Kennicutt, 1998; Kewley et al., 2004). This threshold is also seen in cooling
time, where multiphase cooling and feedback ensues below a threshold of
$t_{\rm cool}<1$ Gyr. McDonald et al. (2010) also showed that ICM gas
coincident with star forming filaments observed in H$\alpha$ has cooling times
that are shorter than their surroundings by roughly a factor of 4. This is
strong evidence that we can link ICM cooling to the fueling of star formation,
and that this same process initiates a self-regulating feedback loop in the
form of feedback triggered by AGN feeding. However, given the flux-limited
nature of many previous cluster surveys, these links could only be confidently
claimed for nearby, low-$z$ systems for the most part. In this work, we make
the first attempt to extend this line of investigation to determine whether
the entropy threshold has evolved and had the same influence in the past, or
in other words, whether the conditions for ICM cooling & AGN feedback, have
evolved with time.

Figure 5: Central entropy of the ICM measured at a radius of 10 kpc
($K_{\mathrm{10~{}kpc}}$) vs luminosity of the [O ii] lines
($L_{\mathrm{[O\,\textsc{ii}]}}$) measured as in Figure 4. Detections are
shown in blue, and upper limits are shown in red. For the first time, we show
that the entropy threshold (shown here as a vertical dashed line) below which
the hot ICM can condense into multiphase gas and trigger star formation
persists to high ($z{\sim}1$) redshifts. Figure 6: Central entropy
($K_{\mathrm{10~{}kpc}}$) vs H$\alpha$ or $\mathrm{[O\,\textsc{ii}]}$ emission
line luminosity (indicators of star formation). For low-$z$ clusters from the
ACCEPT sample (left two panels), we show H$\alpha$ luminosities that are more
readily available at low-$z$, while for high-$z$ clusters from this work (next
three panels) we show $\mathrm{[O\,\textsc{ii}]}$ luminosities which are
available for our higher-$z$ sample. Uncertainties are omitted for clarity.
The two samples are split into various redshift bins to look for evolutionary
trends in the central entropy threshold originally observed at low-$z$
(vertical dashed line). The median redshift in each bin is noted in the top
left of each panel. We see that the entropy threshold of
$K_{\mathrm{10~{}kpc}}\approx 60$ keV cm2 (vertical dashed line) persists out
to our highest redshift bin.

In Figure 5 we show the central ICM entropies as described in $\S\\!\\!$ 2.1
and measured at a radius of 10 kpc, plotted against the BCG [O ii]
luminosities ($\S\\!\\!$ 2.2.4) for our sample, which like H$\alpha$, probes
thermally unstable multiphase cooling. We see that the vast majority of the [O
ii] detections (and all of the strongest detections) lie to the left of a
central entropy value of $K_{\mathrm{10~{}kpc}}=60$ keV cm2, indicating that
an entropy threshold for cooling persists in this sample spanning $0.3<z<1.7$
(i.e. 10 Gyr in evolution). This persistence suggests that multiphase
condensation of the ICM are already feeding star formation in central BCGs
when the universe was only a few Gyr old.

###  3.1 Has the Entropy Threshold Changed With Time?

To directly compare our analysis results with low-$z$ systems to look for
evolutionary trends, we also use the same reference sample where this central
entropy threshold was first measured. To do so, we re-analyzed the Archive of
Chandra Cluster Entropy Profile Tables (ACCEPT) dataset from Cavagnolo et al.
(2009) in a manner more consistent with the data reduction and analysis steps
described in $\S\\!\\!$ 2.1 to eliminate biases and allow for a more fair
comparison of these low-$z$ systems with our higher-$z$ sample. The central
entropy threshold $K_{0}$ of Cavagnolo et al. (2008) is somewhat model-
dependent, and cannot be reliably estimated with the quality of X-ray data
available for our entire sample. Instead, we choose to measure a central
entropy at a projected radius of 10 kpc, which can be measured directly and
coincides with a typical cooling radius for most clusters. In general, we find
that $K_{0}$ and $K_{\mathrm{10~{}kpc}}$ agree well, but the latter is far
less sensitive to observation depth. In addition to this, we filter the ACCEPT
sample to only include the systems whose masses are $M_{\mathrm{500}}>2\times
10^{14}$ M⊙, which is roughly the minimum mass of our sample (see Figure 1).

While Figure 5 suggests that an ICM entropy threshold persists at higher
redshifts, we would also like to assess if it has evolved with time. To that
end, we separate our sample into three separate redshift bins with roughly
equal number of clusters per bin: $0.278<z<0.50$, $0.50<z<0.75$, and
$0.75<z<1.70$. We again plot $L_{\mathrm{[O\,\textsc{ii}]}}$ vs
$K_{\mathrm{10kpc}}$ for these separate redshift bins in Figure 6. In addition
to this, we plot the $L_{\mathrm{H\alpha}}$ data for the ACCEPT clusters,
splitting into two different redshift bins of $0.003<z<0.15$ and
$0.15<z<0.296$. We see that with increasing redshift, the
$K_{\mathrm{10kpc}}=40-60$ keV cm2 central entropy threshold identified in
low-$z$ systems separates [O ii] detections from non-detections reasonably
well as we go to higher redshifts. To quantify this, we use a support vector
classifier that maximizes the margin between two different classes ([O ii]
detections vs non-detections in this case), with the scikit-learn python
package (Pedregosa et al., 2011). Here, the margin is the distance between the
threshold value and the observations closest to that threshold (i.e. the
support vectors). In each redshift bin, we sample the $K_{\mathrm{10~{}kpc}}$
values for each cluster within their uncertainties for 100 bootstrap
iterations, each time solving for the threshold value of entropy
$K_{\mathrm{thresh}}$ that maximizes the margin between detections and non-
detections on a one-dimensional grid. Class weights are balanced proportional
to the frequency of each class. The resulting median values and their 16th and
84th percentiles are sampled from the bootstrap distributions and plotted in
Figure 7. Plotting these threshold entropy values versus redshift for the same
redshift bins as in Figure 6, we can see that the threshold entropies increase
slightly with redshift, from a minimum of $K_{\mathrm{10~{}kpc}}=35\pm 4$ keV
cm2 in the lowest redshift bin ($z<0.15$) to $K_{\mathrm{10~{}kpc}}=52\pm 11$
keV cm2 in the highest redshift bin ($0.75<z<1.7$). Ultimately, these values
are all consistent at the ${\sim}1.5\sigma$ level, indicating that there is no
strong redshift evolution in the entropy threshold for ICM cooling.

Figure 7: Bootstrapped threshold values of entropy that separate [O ii]
detections from non-detections in each redshift bin. For each bootstrap
iteration, we use a support vector classifier to find the threshold value that
maximizes the margin between the two classes. We see that, while there is a
slight increase in the measured threshold value with redshift, there is no
strong statistical evidence for evolution.

###  3.2 Comparison to McDonald et al. (2016)

This work builds on the work of McDonald et al. (2016a, hereafter M16), who
looked at central entropies and SFRs for most of the BCGs in this sample. In
M16, however, only 36 BCGs had optical spectroscopy, so their SFR measurements
were supplemented with UV- and IR-based SFR measurements and upper limits.
Such a heterogeneous dataset may introduce some systematics that are difficult
to account for. Our sample of uniform and consistently measured [O ii]
detections and upper limits allow for a more straightforward analysis and
interpretation of some of the topics tackled in M16, and our measurements
benefit from tighter measurement uncertainties and better statistics from a
complete spectroscopic followup of our targets. One of the findings reported
from M16 was that a significant number of their BCGs had SFRs $>10$ M⊙ yr-1
(31/90 = 34%), consistent with findings by Webb et al. (2015) and Bonaventura
et al. (2017), and in contrast with low-$z$ systems where the occurrence rate
is closer to a few percent (e.g. Donahue et al., 2010; Fraser-McKelvie et al.,
2014). In our study of purely [O ii] SFRs, we find that only 22/95 clusters
have [O ii] detections, and only 9/95 (${\sim}9\%$) have SFRs $\gtrsim 10$ M⊙
yr-1. This occurrence rate is in closer agreement with the $z\sim 0$ studies.
Another result from M16 was that in contrast to $z\sim 0$ studies, they
observed no significant correlation between BCG star formation signatures and
the central ICM entropy of the host cluster. As we showed in Figure 5 and
Figure 6, however, we do observe a clear correlation.

There may be many reasons for our disagreements with McDonald et al. (2016a)
in the number of star-forming BCGs, the most likely of which being AGN
contamination in the mid-IR SFRs and misidentification of BCGs based on lower
quality data. The BCG selection of McDonald et al. (2016a) prioritized sample
completeness at the expense of purity, with some BCG candidates later
identified as foreground stars or galaxies. M16 also used large-scale
centroids instead of X-ray peaks for their thermodynamic profile centers,
which would lead to different (often higher) central entropy measurements. If
at higher redshifts it is more common for the X-ray peak and centroid
positions to differ significantly, then these “sloshing” or offset cool
coreswould be mislabeled as high-entropy cores in McDonald et al. (2016a).
Using the offset values from Sanders et al. (2018), we find that the average
angular scale difference between X-ray peak and centroid positions is about
$10\arcsec$. Given the evolution in the angular diameter distance with
redshift, this angular offset corresponds to a physical offset of ${\sim}80$
kpc, which can certainly make the difference in calling a system a cool core
or not. Alternatively, if mid-IR AGN were more common at high-$z$, which we
know is the case from Somboonpanyakul et al. (2022), then the mid-IR SFRs used
in McDonald et al. (2016a) in the absence of optical SFRs would again wash out
any sign of an entropy threshold. All of these factors likely contribute to
some extent to the disagreement with our results.

##  4 Redshift Dependence of the Entropy Threshold for AGN Feedback

###  4.1 A Disappearing Dichotomy

Figure 8: Central entropy of the ICM measured at a radius of 10 kpc
($K_{\mathrm{10~{}kpc}}$) vs 1.4 GHz rest luminosity of the central radio
source ($\nu L_{\nu}$) whenever one was associated with the BCG (i.e. a
detection, in blue) or of the radio map rms noise (i.e. upper limits, shown in
red). In contrast to low-$z$ systems, or like the case for star formation
(Figure 5), in our sample we do not see a correlation between a cluster’s
central entropy and AGN activity. This could be due to an actual evolution in
the entropy threshold, due to a less tight feedback-cooling cycle or higher
mergers at high-$z$ competing with cooling flows to drive more gas towards the
AGN, for instance; or, an artificial evolution due to some biases we discuss
in $\S\\!\\!$ 4.2.

As mentioned above, the work by Cavagnolo et al. (2008) showed that the ICM
becomes thermally unstable to cooling when its central entropy drops below
some threshold $K_{0}$. This localized cooling not only fuels star formation,
but it also triggers AGN activity by feeding the central BCG’s SMBH. Sun
(2009) also showed that at low-$z$, all strong radio BCGs (defined as having
$\nu L_{\nu}>10^{39.5}$ erg s-1) have short central cooling times (${<}4$ Gyr)
making the connection between cooling and feedback quite clear. With our
sample, we can investigate whether this link is still present at higher
redshifts. In Figure 8, we show our central entropies measured at a radius of
10 kpc ($K_{\mathrm{10~{}kpc}}$) now versus the 1.4 GHz rest-frame
luminosities of the central radio sources (see $\S\\!\\!$ 2.3). One of the
striking features of these new data is that there is no apparent threshold
between radio source detections and non-detections, in contrast to the optical
luminosities associated with star formation of Figure 5. On aggregate, the
connection between thermally unstable cooling and AGN feedback seen at lower
redshifts appear to be gone at higher redshifts. This finding is consistent
with that of Bîrzan et al. (2017), who found no evidence for a separation
between cooling flow (CF) clusters and non-CF clusters based on the central
radio source’s luminosity at $z>0.6$.

Figure 9: Same as in Figure 8, but separated into redshift bins and compared
to ACCEPT clusters with $M_{500}\gtrsim 2\times 10^{14}$ M⊙ as before in
Figure 6. Error bars are omitted for clarity. Just as before, there is no
apparent correlation between a central entropy threshold and radio AGN
activity. In the low-z ACCEPT clusters, the demarcation between radio
detection and non-detection is clear at $K_{\mathrm{10~{}kpc}}=60$ keV cm2 in
the lowest redshift bin. A higher proportion of radio detections is above the
threshold in the $0.15<z<0.295$ redshift bin, but it is clear that the entropy
distribution is still bimodal. In our SZ-based sample, the correlation
vanishes abruptly. One possible source of bias is contamination to the SZ
signal by the radio source, potentially making our sample deficient in high
power radio sources. In the right three panels, a horizontal dashed line shows
the level at which radio emission from a $4.2\times 10^{14}$ M⊙ cluster at
each median redshift (shown in the upper left of each panel) would suppress
100% of the SZ signal. The varying strength of this effect of this is explored
in more detail in $\S\\!\\!$ 4.2 and Figure 10.

To see whether the disappearance of the entropy threshold for feedback is a
gradual one or not, we again split our sample into individual redshift bins
and compare to the ACCEPT data from Cavagnolo et al. (2009) in Figure 9. The
ACCEPT sample used radio luminosities from the NRAO VLA Sky Survey (NVSS;
Condon et al., 1998) and the Sydney University Molonglo Sky Survey (SUMSS;
Mauch et al., 2003). In the ACCEPT data, there is a clear dichotomy between
radio detections and non-detections on either side of the
$K_{\mathrm{10~{}kpc}}=60$ keV cm2 boundary. However, this dichotomy seems to
grow weaker with redshift, as roughly ${\sim}$10% of the detections shown in
the lowest redshift bin of $0.006<z<0.15$ (2 out of 19) have an entropy above
this threshold, while ${\sim}20$% (5/24) of the detections are above the same
threshold in the $0.15<z<0.295$ redshift bin. The proportion of radio
detections above the threshold is approximately 50% in the next redshift bins
associated with our higher redshift SPT sample, abruptly washing out any sign
of an entropy threshold.

This strong redshift evolution in the proportion of radio detections above and
below the central entropy threshold seen in Figure 9 could be interpreted in a
number of ways. First, if this is a real astrophysical effect, it could imply
that the cooling-feedback cycle observed in nearby systems was not as tightly-
regulated in the past. For instance, in some cases, higher-redshift systems
may often exhibit cooling that is offset from a BCG, without the opportunity
for AGN accretion and feedback (e.g. Hlavacek-Larrondo et al., 2020; Trudeau
et al., 2019). The more chaotic environments of high-redshift galaxy clusters
that are still assembling via mergers could naturally result in more chaotic
pathways of directing gas towards accreting central black holes. As the
fraction of galaxies participating in mergers increases with redshift (e.g.
Brodwin et al., 2013; Martini et al., 2013), this activity is capable of
driving gas towards the AGN and causing them to accrete, which would confound
the specific correlation between cooling flows and AGN accretion that would
otherwise result in a clean dichotomy in central entropy. Another related
explanation could be that the lack of a dichotomy is an effect of
misinterpreting the source of radio emission we measure. Recent results are
making it clear that radio power becomes a progressively worse indicator of
jet power at high-$z$, and an object’s radio loudness is not necessarily due
to the presence of jets. For instance, Bîrzan et al. (2017) find an evolution
in the radio luminosity function with redshift (see also Main et al., 2017;
Butler et al., 2019), suggesting that higher redshift radio sources are more
typically associated with high-excitation radio galaxies which accrete more
efficiently. In other words, at higher redshifts, a radio source is more often
in quasar mode than in jet mode, which is consistent with recent studies (e.g.
Hlavacek-Larrondo et al., 2013; Russell et al., 2013; Somboonpanyakul et al.,
2022), and we must be more thoughtful about the measurements we use for
comparison to account for this. Studying a radio source’s SED with multiple
observing frequency bands would also allow for the decomposition of different
contributing components like radio cores and jets. These studies will be made
possible for the southern hemisphere SPT sources with upcoming ASKAP data
releases, for instance (see Duchesne et al., 2023). All of these sources of
confusion mentioned above are probably contributing to the lack of an observed
entropy threshold for feedback to some degree.

###  4.2 A Lack of Extremely Luminous Radio Galaxies in SPT Clusters

In addition to the lack of an observed higher-$z$ entropy threshold predicting
the occurrence of radio sources as in low-$z$ systems, we also find a lack of
very strong radio sources. This is qualitatively consistent with earlier
findings based on ROSAT cluster surveys using a more limited redshift range
(Samuele et al., 2011). One possible explanation for this dearth is that high
mass clusters simply do not host high radio power sources as often as low-mass
ones. In the ACCEPT data shown in the left two panels of Figure 9, we note
that the highest radio powers are associated with lower-mass clusters (e.g.
Hydra A, 3C388, Abell 2597), which are removed from the comparison sample when
applying our mass cut of $2\times 10^{14}$ M⊙. However, even after applying a
mass cut to the ACCEPT data, there remain clusters at $z<0.3$ that have
central AGN with $\nu L_{\nu}>10^{42}$ erg s-1, including Cygnus A, Hercules
A, 4C55.16, Abell 1942, PKS0745-191, Abell 2390, Abell 1361, and
RXCJ0331.1-2100. These 8 clusters represent roughly 7% of the population in
this mass and redshift regime. However, it is worth noting that the ACCEPT
survey is highly biased: 51% of the clusters in the ACCEPT sample are cool
cores, compared to $\sim$30% in SZ-selected samples (e.g., Andrade-Santos et
al., 2017; Ruppin et al., 2021). Correcting for this, we would expect the
underlying population of high-radio-luminosity BCGs to be closer to $\sim$4%
of the total population in high mass clusters.

Why are these extremely high radio luminosity systems missing in our sample?
One potential concern that we consider is that the SZ surveys are biased
against BCGs with extremely high radio luminosities. With the broadband
spectrum from the synchrotron radiation associated with accreting SMBHs, the
flux from a higher power radio source is capable of contaminating the SZ
signal at the same frequencies used to detect these clusters. The SZ decrement
could in effect be “filled in” by excess radio emission, making the SZ
detections less significant. How much of an effect radio emission from a BCG
has on the observed SZ signal has been quantified in various studies (e.g.
Sayers et al., 2013; Lin et al., 2015; Hogan et al., 2015; Dicker et al.,
2021; Rose et al., 2022). In the horizontal dashed lines plotted in Figure 9,
we use equation 8 from Lin et al. (2015) to show the level at which radio bias
($\delta\zeta/\zeta$) is 100% of the SZ signal for a $M_{500}=4.2\times
10^{14}$ M⊙ cluster (the median mass of our cluster sample) at the median
redshift in each column. Above this bias threshold we do not expect to see any
SZ-selected clusters. We do not plot these thresholds for the ACCEPT data as
these clusters are not SZ-selected and would not suffer from the same type of
selection bias. To investigate this further, we also plot what the radio bias
at various levels ($\delta\zeta/\zeta=\\{1\%,~{}10\%,~{}30\%,~{}100\%\\}$)
looks like as a finer function of redshift in Figure 10, again at a fixed
median cluster mass of $4.2\times 10^{14}$ M⊙. At all redshifts, we find a
lack of clusters above the line indicating a bias of 30%. Above a luminosity
of $\nu L_{\nu}=10^{42}$ erg s-1, this should only result in an absence of
${\sim}$4% of systems in our sample selection, as we estimated above from the
low-$z$ ACCEPT data. We seem to also have a scarcity of clusters at $z>1$ and
$\nu L_{\nu}\gtrsim 10^{41}$ erg s-1, an effect that cannot be attributed to a
bias in the SZ survey.

It is important to stress that this is only an attempt to quantify our lack of
extremely powerful radio sources, and that this radio contamination is not
capable of contributing to our lack of an observed dichotomy between radio
detections and non-detections discussed in $\S\\!\\!$ 4.1. Even restricting
the ACCEPT data in Figure 9 to systems with $\nu L_{\nu}<10^{42}$ erg s-1,
there is still a very clear contrast above and below the central entropy
threshold, unlike in the higher-$z$ data. The radio contamination bias to the
SZ signal is not dependent on entropy, thus we should still expect to see a
dichotomy in Figure 8 and Figure 9 if one existed, assuming only that the 1.4
GHz radio luminosity is a good proxy for the AGN jet power.

Figure 10: 1.4 GHz radio luminosity vs redshift. Lines of constant flux are
overlaid, depicting the amount of bias ($\delta\zeta/\zeta$) by which the
radio flux suppresses the SZ signal by 1%, 10%, and 100% (Lin et al., 2015).
We see that our sample contains no clusters for which the SZ signal in a
M${}_{500}=4.2\times 10^{14}$ M⊙ cluster would be biased by ${>}$30%, which
could be indicating a significant source of selection bias for SZ-based
surveys.

Alternatively, our sample may just be missing clusters with extremely high
radio luminosities because our roughly constant mass versus redshift selection
(see Figure 1) singles out systems at higher redshifts that are progenitors of
increasingly rarer systems with no low-$z$ equivalents. An evolution in the
entropy threshold for feedback may look different when considering a sample of
clusters along a common evolutionary track, such as the one considered in
Ruppin et al. (2021, 2023). Spectroscopic followup on such a descendent-
antecedent sample would be a promising path forward.

##  5 Summary & Future Work

We present a multi-wavelength study of the most massive SZ-selected clusters
from the SPT-SZ survey, spanning ${\sim}$10 Gyr in cosmic evolution
($0.3<z<1.7$). Our X-ray imaging, optical spectroscopy, and radio imaging of
these clusters allows us to connect the cooling out of the hot ICM to star
formation and AGN activity in the central BCG. Our results can be summarized
as follows:

  1. 1.

For the first time, we show that a central ICM entropy threshold for
multiphase condensation persists over the entire redshift range of our sample
($0.3<z<1.7$). This entropy threshold has only previously been demonstrated
for nearby ($z\lesssim 0.3$) systems.

  2. 2.

The threshold for ICM cooling is measured at a radius of 10 kpc for each
cluster and is found to increase slightly with redshift, from a minimum of
$K_{\mathrm{10~{}kpc}}=35\pm 4$ keV cm2 in the lowest redshift bin ($z<0.15$)
to $K_{\mathrm{10~{}kpc}}=52\pm 11$ keV cm2 in the highest redshift bin
($0.75<z<1.7$). Ultimately, these values are roughly consistent with no
statistically significant evolution.

  3. 3.

In contrast, the same entropy threshold that defines which systems should have
cooling that triggers star formation and AGN feedback at low-$z$ does not
predict which clusters will have radio mode feedback at higher redshifts. We
find a roughly equal number of radio detections above and below the
$K_{\mathrm{10~{}kpc}}\sim 60$ keV cm2 entropy threshold.

  4. 4.

This lack of a radio detection dichotomy with core entropy at high-$z$ could
imply that the ICM cooling-AGN feedback cycle was not as tightly-regulated in
the past as it is today, or perhaps a competing source of AGN fueling like
mergers, which are more prevalent at higher redshifts as well as quasars.
Radio luminosities may also be an increasingly poor proxy for AGN jet power at
higher redshifts, where the quasar occurrence rate is higher.

  5. 5.

We also find an absence of extremely high radio luminosity ($\nu
L_{\nu}>10^{42}$ erg s-1) sources in our sample. This is possibly due to an SZ
selection bias, as we find that our sample is missing clusters where the radio
suppresses the SZ signal by as little as $\gtrsim$30%. Importantly, this
effect cannot explain the absence of the radio detection dichotomy with core
entropy at high-$z$.

This rich dataset provides many exciting possibilities for studying the
evolution of the ICM cooling and AGN feedback cycle in clusters. This study is
the first in a series of such possible studies. In upcoming papers, we plan to
explore whether the effectiveness of AGN feedback has changed with time, by
studying the ratio of SFR to ICM cooling rate in these clusters, as has been
hinted at in Calzadilla et al. (2022), McDonald et al. (2018), Fogarty et al.
(2017), etc. We also plan to use this dataset to investigate whether AGN
feedback was more or less bursty in the past, by studying the correlation
between AGN radio power and cooling luminosity as in Ruppin et al. (2023),
Hlavacek-Larrondo et al. (2012, 2015), etc. Finally, we will examine in
further detail the growth of BCGs using our higher quality optical
spectroscopy, and see whether specific SFRs of BCGs can tell us if they are
fueled preferentially via mergers or residual cooling flows, as in McDonald et
al. (2016a).

MSC acknowledges support from the NASA Headquarters under the Future
Investigators in NASA Earth and Space Science and Technology (FINESST) award
20-Astro20-0037. MSC and MM acknowledge additional financial support for this
work provided by the National Aeronautics and Space Administration through
Chandra Award Number GO0-21124A issued by the Chandra X-ray Center, which is
operated by the Smithsonian Astrophysical Observatory for and on behalf of the
National Aeronautics Space Administration under contract NAS8-03060. The
scientific results reported in this article are based on observations made by
the Chandra X-ray Observatory, and this research has made use of software
provided by the Chandra X-ray Center (CXC) in the application package, CIAO.

The South Pole Telescope program is supported by the National Science
Foundation (NSF) through award OPP- 1852617. Partial support is also provided
by the Kavli Institute of Cosmological Physics at the University of Chicago.
Work at Argonne National Lab is supported by UChicago Argonne LLC, Operator of
Argonne National Laboratory (Argonne). Argonne, a U.S. Department of Energy
Office of Science Laboratory, is operated under contract no. DE-
AC02-06CH11357. Work at Fermi National Accelerator Laboratory, a DOE-OS, HEP
User Facility managed by the Fermi Research Alliance, LLC, was supported under
Contract No. DE-AC02-07CH11359. This paper used data gathered with the 6.5 m
Magellan Telescopes located at Las Campanas Observatory, Chile. We thank the
staff of Las Campanas for their dedicated service, which has made these
observations possible. PISCO observations are supported by NSF AST-1814719.

## References

  * Allen (1995) Allen, S. W. 1995, MNRAS, 276, 947, doi: 10.1093/mnras/276.3.947
  * Anders & Grevesse (1989) Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197, doi: 10.1016/0016-7037(89)90286-X
  * Andrade-Santos et al. (2017) Andrade-Santos, F., Jones, C., Forman, W. R., et al. 2017, ApJ, 843, 76, doi: 10.3847/1538-4357/aa7461
  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Bayliss et al. (2016) Bayliss, M. B., Ruel, J., Stubbs, C. W., et al. 2016, ApJS, 227, 3, doi: 10.3847/0067-0049/227/1/3
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393, doi: 10.1051/aas:1996164
  * Bîrzan et al. (2017) Bîrzan, L., Rafferty, D. A., Brüggen, M., & Intema, H. T. 2017, MNRAS, 471, 1766, doi: 10.1093/mnras/stx1505
  * Bîrzan et al. (2004) Bîrzan, L., Rafferty, D. A., McNamara, B. R., Wise, M. W., & Nulsen, P. E. J. 2004, ApJ, 607, 800, doi: 10.1086/383519
  * Bleem et al. (2015) Bleem, L. E., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27, doi: 10.1088/0067-0049/216/2/27
  * Bleem et al. (2020) Bleem, L. E., Bocquet, S., Stalder, B., et al. 2020, ApJS, 247, 25, doi: 10.3847/1538-4365/ab6993
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55, doi: 10.3847/1538-4357/ab1f10
  * Bonaventura et al. (2017) Bonaventura, N. R., Webb, T. M. A., Muzzin, A., et al. 2017, MNRAS, 469, 1259, doi: 10.1093/mnras/stx722
  * Brammer et al. (2008) Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503, doi: 10.1086/591786
  * Brodwin et al. (2013) Brodwin, M., Stanford, S. A., Gonzalez, A. H., et al. 2013, ApJ, 779, 138, doi: 10.1088/0004-637X/779/2/138
  * Butler et al. (2019) Butler, A., Huynh, M., Kapińska, A., et al. 2019, A&A, 625, A111, doi: 10.1051/0004-6361/201834581
  * Calzadilla et al. (2019) Calzadilla, M. S., McDonald, M., Bayliss, M., et al. 2019, ApJ, 887, L17, doi: 10.3847/2041-8213/ab5b07
  * Calzadilla et al. (2022) Calzadilla, M. S., McDonald, M., Donahue, M., et al. 2022, ApJ, 940, 140, doi: 10.3847/1538-4357/ac9790
  * Calzadilla et al. (2023) Calzadilla, M. S., Bleem, L. E., McDonald, M., et al. 2023, ApJ, 947, 44, doi: 10.3847/1538-4357/acc6c2
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682, doi: 10.1086/308692
  * Cavagnolo et al. (2008) Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2008, ApJ, 683, L107, doi: 10.1086/591665
  * Cavagnolo et al. (2009) —. 2009, ApJS, 182, 12, doi: 10.1088/0067-0049/182/1/12
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102
  * Churazov et al. (2001) Churazov, E., Brüggen, M., Kaiser, C. R., Böhringer, H., & Forman, W. 2001, ApJ, 554, 261, doi: 10.1086/321357
  * Condon et al. (1998) Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693, doi: 10.1086/300337
  * Conroy et al. (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486, doi: 10.1088/0004-637X/699/1/486
  * Crawford et al. (1999) Crawford, C. S., Allen, S. W., Ebeling, H., Edge, A. C., & Fabian, A. C. 1999, MNRAS, 306, 857, doi: 10.1046/j.1365-8711.1999.02583.x
  * Cutri et al. (2021) Cutri, R. M., Wright, E. L., Conrow, T., et al. 2021, VizieR Online Data Catalog, II/328
  * Dicker et al. (2021) Dicker, S. R., Battistelli, E. S., Bhandarkar, T., et al. 2021, MNRAS, 508, 2600, doi: 10.1093/mnras/stab2679
  * Donahue & Voit (2022) Donahue, M., & Voit, G. M. 2022, Phys. Rep., 973, 1, doi: 10.1016/j.physrep.2022.04.005
  * Donahue et al. (2010) Donahue, M., Bruch, S., Wang, E., et al. 2010, ApJ, 715, 881, doi: 10.1088/0004-637X/715/2/881
  * Donahue et al. (2015) Donahue, M., Connor, T., Fogarty, K., et al. 2015, ApJ, 805, 177, doi: 10.1088/0004-637X/805/2/177
  * Duchesne et al. (2023) Duchesne, S. W., Thomson, A. J. M., Pritchard, J., et al. 2023, PASA, 40, e034, doi: 10.1017/pasa.2023.31
  * Fabian (1994) Fabian, A. C. 1994, ARA&A, 32, 277, doi: 10.1146/annurev.aa.32.090194.001425
  * Fabian (2012) —. 2012, ARA&A, 50, 455, doi: 10.1146/annurev-astro-081811-125521
  * Falcón-Barroso et al. (2011) Falcón-Barroso, J., Sánchez-Blázquez, P., Vazdekis, A., et al. 2011, A&A, 532, A95, doi: 10.1051/0004-6361/201116842
  * Fogarty et al. (2017) Fogarty, K., Postman, M., Larson, R., Donahue, M., & Moustakas, J. 2017, ApJ, 846, 103, doi: 10.3847/1538-4357/aa82b9
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Foreman-Mackey et al. (2014) Foreman-Mackey, D., Sick, J., & Johnson, B. 2014, python-fsps: Python bindings to FSPS (v0.1.1), v0.1.1, Zenodo, Zenodo, doi: 10.5281/zenodo.12157
  * Fraser-McKelvie et al. (2014) Fraser-McKelvie, A., Brown, M. J. I., & Pimbblet, K. A. 2014, MNRAS, 444, L63, doi: 10.1093/mnrasl/slu117
  * Fruscione et al. (2006) Fruscione, A., McDowell, J. C., Allen, G. E., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. D. R. Silva & R. E. Doxsey, 62701V, doi: 10.1117/12.671760
  * Gaspari (2015) Gaspari, M. 2015, MNRAS, 451, L60, doi: 10.1093/mnrasl/slv067
  * Hale et al. (2021) Hale, C. L., McConnell, D., Thomson, A. J. M., et al. 2021, PASA, 38, e058, doi: 10.1017/pasa.2021.47
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hillel & Soker (2017) Hillel, S., & Soker, N. 2017, ApJ, 845, 91, doi: 10.3847/1538-4357/aa81c5
  * Hilton et al. (2021) Hilton, M., Sifón, C., Naess, S., et al. 2021, ApJS, 253, 3, doi: 10.3847/1538-4365/abd023
  * Hlavacek-Larrondo et al. (2013) Hlavacek-Larrondo, J., Fabian, A. C., Edge, A. C., et al. 2013, MNRAS, 431, 1638, doi: 10.1093/mnras/stt283
  * Hlavacek-Larrondo et al. (2012) —. 2012, MNRAS, 421, 1360, doi: 10.1111/j.1365-2966.2011.20405.x
  * Hlavacek-Larrondo et al. (2015) Hlavacek-Larrondo, J., McDonald, M., Benson, B. A., et al. 2015, ApJ, 805, 35, doi: 10.1088/0004-637X/805/1/35
  * Hlavacek-Larrondo et al. (2020) Hlavacek-Larrondo, J., Rhea, C. L., Webb, T., et al. 2020, ApJ, 898, L50, doi: 10.3847/2041-8213/ab9ca5
  * Hogan et al. (2015) Hogan, M. T., Edge, A. C., Geach, J. E., et al. 2015, MNRAS, 453, 1223, doi: 10.1093/mnras/stv1518
  * Hogan et al. (2017) Hogan, M. T., McNamara, B. R., Pulido, F. A., et al. 2017, ApJ, 851, 66, doi: 10.3847/1538-4357/aa9af3
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Johnson (2021) Johnson, B. D. 2021, bd-j/sedpy: sedpy v0.2.0, v0.2.0, Zenodo, Zenodo, doi: 10.5281/zenodo.4582723
  * Johnson et al. (2021) Johnson, B. D., Leja, J., Conroy, C., & Speagle, J. S. 2021, ApJS, 254, 22, doi: 10.3847/1538-4365/abef67
  * Johnstone et al. (1987) Johnstone, R. M., Fabian, A. C., & Nulsen, P. E. J. 1987, MNRAS, 224, 75, doi: 10.1093/mnras/224.1.75
  * Joye & Mandel (2003) Joye, W. A., & Mandel, E. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 295, Astronomical Data Analysis Software and Systems XII, ed. H. E. Payne, R. I. Jedrzejewski, & R. N. Hook, 489
  * Kalberla et al. (2005) Kalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005, A&A, 440, 775, doi: 10.1051/0004-6361:20041864
  * Kennicutt (1998) Kennicutt, Robert C., J. 1998, ARA&A, 36, 189, doi: 10.1146/annurev.astro.36.1.189
  * Kewley et al. (2004) Kewley, L. J., Geller, M. J., & Jansen, R. A. 2004, AJ, 127, 2002, doi: 10.1086/382723
  * Khullar et al. (2022) Khullar, G., Bayliss, M. B., Gladders, M. D., et al. 2022, ApJ, 934, 177, doi: 10.3847/1538-4357/ac7c0c
  * Kluyver et al. (2016) Kluyver, T., Ragan-Kelley, B., Pérez, F., et al. 2016, in IOS Press, 87–90, doi: 10.3233/978-1-61499-649-1-87
  * Kriek et al. (2009) Kriek, M., van Dokkum, P. G., Labbé, I., et al. 2009, ApJ, 700, 221, doi: 10.1088/0004-637X/700/1/221
  * Li et al. (2017) Li, Y., Ruszkowski, M., & Bryan, G. L. 2017, ApJ, 847, 106, doi: 10.3847/1538-4357/aa88c1
  * Lin et al. (2015) Lin, H. W., McDonald, M., Benson, B., & Miller, E. 2015, ApJ, 802, 34, doi: 10.1088/0004-637X/802/1/34
  * Main et al. (2017) Main, R. A., McNamara, B. R., Nulsen, P. E. J., Russell, H. R., & Vantyghem, A. N. 2017, MNRAS, 464, 4360, doi: 10.1093/mnras/stw2644
  * Mantz et al. (2016) Mantz, A. B., Allen, S. W., Morris, R. G., & Schmidt, R. W. 2016, MNRAS, 456, 4020, doi: 10.1093/mnras/stv2899
  * Martini et al. (2013) Martini, P., Miller, E. D., Brodwin, M., et al. 2013, ApJ, 768, 1, doi: 10.1088/0004-637X/768/1/1
  * Mauch et al. (2003) Mauch, T., Murphy, T., Buttery, H. J., et al. 2003, MNRAS, 342, 1117, doi: 10.1046/j.1365-8711.2003.06605.x
  * McConnell et al. (2020) McConnell, D., Hale, C. L., Lenc, E., et al. 2020, PASA, 37, e048, doi: 10.1017/pasa.2020.41
  * McDonald et al. (2018) McDonald, M., Gaspari, M., McNamara, B. R., & Tremblay, G. R. 2018, ApJ, 858, 45, doi: 10.3847/1538-4357/aabace
  * McDonald et al. (2010) McDonald, M., Veilleux, S., Rupke, D. S. N., & Mushotzky, R. 2010, ApJ, 721, 1262, doi: 10.1088/0004-637X/721/2/1262
  * McDonald et al. (2013) McDonald, M., Benson, B. A., Vikhlinin, A., et al. 2013, ApJ, 774, 23, doi: 10.1088/0004-637X/774/1/23
  * McDonald et al. (2014) —. 2014, ApJ, 794, 67, doi: 10.1088/0004-637X/794/1/67
  * McDonald et al. (2016a) McDonald, M., Stalder, B., Bayliss, M., et al. 2016a, ApJ, 817, 86, doi: 10.3847/0004-637X/817/2/86
  * McDonald et al. (2016b) McDonald, M., Bulbul, E., de Haan, T., et al. 2016b, ApJ, 826, 124, doi: 10.3847/0004-637X/826/2/124
  * McDonald et al. (2017) McDonald, M., Allen, S. W., Bayliss, M., et al. 2017, ApJ, 843, 28, doi: 10.3847/1538-4357/aa7740
  * McDonald et al. (2019) McDonald, M., McNamara, B. R., Voit, G. M., et al. 2019, ApJ, 885, 63, doi: 10.3847/1538-4357/ab464c
  * McNamara & Nulsen (2007) McNamara, B. R., & Nulsen, P. E. J. 2007, ARA&A, 45, 117, doi: 10.1146/annurev.astro.45.051806.110625
  * McNamara & Nulsen (2012) —. 2012, New Journal of Physics, 14, 055023, doi: 10.1088/1367-2630/14/5/055023
  * McNamara & O’Connell (1989) McNamara, B. R., & O’Connell, R. W. 1989, AJ, 98, 2018, doi: 10.1086/115275
  * Morrison & McCammon (1983) Morrison, R., & McCammon, D. 1983, ApJ, 270, 119, doi: 10.1086/161102
  * Nulsen et al. (2009) Nulsen, P., Jones, C., Forman, W., et al. 2009, in American Institute of Physics Conference Series, Vol. 1201, The Monster’s Fiery Breath: Feedback in Galaxies, Groups, and Clusters, ed. S. Heinz & E. Wilcots, 198–201, doi: 10.1063/1.3293033
  * Nulsen (1986) Nulsen, P. E. J. 1986, MNRAS, 221, 377, doi: 10.1093/mnras/221.2.377
  * O’Dea et al. (2008) O’Dea, C. P., Baum, S. A., Privon, G., et al. 2008, ApJ, 681, 1035, doi: 10.1086/588212
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825, doi: 10.48550/arXiv.1201.0490
  * Peterson & Fabian (2006) Peterson, J. R., & Fabian, A. C. 2006, Phys. Rep., 427, 1, doi: 10.1016/j.physrep.2005.12.007
  * Pizzolato & Soker (2005) Pizzolato, F., & Soker, N. 2005, ApJ, 632, 821, doi: 10.1086/444344
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A27, doi: 10.1051/0004-6361/201525823
  * Pulido et al. (2018) Pulido, F. A., McNamara, B. R., Edge, A. C., et al. 2018, ApJ, 853, 177, doi: 10.3847/1538-4357/aaa54b
  * Rafferty et al. (2008) Rafferty, D. A., McNamara, B. R., & Nulsen, P. E. J. 2008, ApJ, 687, 899, doi: 10.1086/591240
  * Rafferty et al. (2006) Rafferty, D. A., McNamara, B. R., Nulsen, P. E. J., & Wise, M. W. 2006, ApJ, 652, 216, doi: 10.1086/507672
  * Reynolds et al. (2002) Reynolds, C. S., Heinz, S., & Begelman, M. C. 2002, MNRAS, 332, 271, doi: 10.1046/j.1365-8711.2002.04724.x
  * Rose et al. (2022) Rose, T., Edge, A., Kiehlmann, S., et al. 2022, MNRAS, 509, 2869, doi: 10.1093/mnras/stab3217
  * Ruel et al. (2014) Ruel, J., Bazin, G., Bayliss, M., et al. 2014, ApJ, 792, 45, doi: 10.1088/0004-637X/792/1/45
  * Ruppin et al. (2021) Ruppin, F., McDonald, M., Bleem, L. E., et al. 2021, ApJ, 918, 43, doi: 10.3847/1538-4357/ac0bba
  * Ruppin et al. (2023) Ruppin, F., McDonald, M., Hlavacek-Larrondo, J., et al. 2023, ApJ, 948, 49, doi: 10.3847/1538-4357/acc38d
  * Russell et al. (2013) Russell, H. R., McNamara, B. R., Edge, A. C., et al. 2013, MNRAS, 432, 530, doi: 10.1093/mnras/stt490
  * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161, doi: 10.1086/145971
  * Samuele et al. (2011) Samuele, R., McNamara, B. R., Vikhlinin, A., & Mullis, C. R. 2011, ApJ, 731, 31, doi: 10.1088/0004-637X/731/1/31
  * Sanders et al. (2018) Sanders, J. S., Fabian, A. C., Russell, H. R., & Walker, S. A. 2018, MNRAS, 474, 1065, doi: 10.1093/mnras/stx2796
  * Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 433, doi: 10.48550/arXiv.astro-ph/0612759
  * Sayers et al. (2013) Sayers, J., Mroczkowski, T., Czakon, N. G., et al. 2013, ApJ, 764, 152, doi: 10.1088/0004-637X/764/2/152
  * Science Software Branch at STScI (2012) Science Software Branch at STScI. 2012, PyRAF: Python alternative for IRAF, Astrophysics Source Code Library, record ascl:1207.011.  http://ascl.net/1207.011
  * Sifón et al. (2013) Sifón, C., Menanteau, F., Hasselfield, M., et al. 2013, ApJ, 772, 25, doi: 10.1088/0004-637X/772/1/25
  * Smith et al. (2001) Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91, doi: 10.1086/322992
  * Soker et al. (2001) Soker, N., White, Raymond E., I., David, L. P., & McNamara, B. R. 2001, ApJ, 549, 832, doi: 10.1086/319433
  * Somboonpanyakul et al. (2022) Somboonpanyakul, T., McDonald, M., Noble, A., et al. 2022, AJ, 163, 146, doi: 10.3847/1538-3881/ac5030
  * Sun (2009) Sun, M. 2009, ApJ, 704, 1586, doi: 10.1088/0004-637X/704/2/1586
  * Team (2023) Team, T. P. D. 2023, pandas-dev/pandas: Pandas, v2.1.1, Zenodo, Zenodo, doi: 10.5281/zenodo.3509134
  * Trudeau et al. (2019) Trudeau, A., Webb, T., Hlavacek-Larrondo, J., et al. 2019, MNRAS, 487, 1210, doi: 10.1093/mnras/stz1364
  * Vanderlinde et al. (2010) Vanderlinde, K., Crawford, T. M., de Haan, T., et al. 2010, ApJ, 722, 1180, doi: 10.1088/0004-637X/722/2/1180
  * Vikhlinin et al. (2006) Vikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, ApJ, 640, 691, doi: 10.1086/500288
  * Vikhlinin et al. (1998) Vikhlinin, A., McNamara, B. R., Forman, W., et al. 1998, ApJ, 502, 558, doi: 10.1086/305951
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Waskom (2021) Waskom, M. 2021, The Journal of Open Source Software, 6, 3021, doi: 10.21105/joss.03021
  * Webb et al. (2015) Webb, T., Noble, A., DeGroot, A., et al. 2015, ApJ, 809, 173, doi: 10.1088/0004-637X/809/2/173
  * Yang et al. (2019) Yang, H. Y. K., Gaspari, M., & Marlow, C. 2019, ApJ, 871, 6, doi: 10.3847/1538-4357/aaf4bd
  * Yang & Reynolds (2016) Yang, H. Y. K., & Reynolds, C. S. 2016, ApJ, 829, 90, doi: 10.3847/0004-637X/829/2/90
  * Zhuravleva et al. (2014) Zhuravleva, I., Churazov, E., Schekochihin, A. A., et al. 2014, Nature, 515, 85, doi: 10.1038/nature13830

##  Appendix A Tables

Table 1: BCG Data

Name | $z_{\mathrm{cluster}}$ | $\alpha_{\mathrm{BCG}}$ | $\delta_{\mathrm{BCG}}$ | $z_{\mathrm{BCG}}$ | $K_{\mathrm{10~{}kpc}}$ | $L_{\mathrm{1.4~{}GHz}}$ | $f_{\mathrm{[O\,\textsc{ii}]}}$ | $L_{\mathrm{[O\,\textsc{ii}]}}$ | SFR
---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [keV cm2] | [$10^{40}$ erg s-1] | [$10^{-16}$ erg s-1 cm-2] | [$10^{40}$ erg s-1] | [M⊙ y-1]
SPT-CLJ0000-5748 | 0.702 | 0.25015 | -57.8093 | 0.701 | $20^{+5}_{-4}$ | $32\pm 3$ | $8.9\pm 0.4$ | $840\pm 500$ | $53^{+44}_{-25}$
SPT-CLJ0013-4906 | 0.408 | 3.33048 | -49.1105 | 0.410 | $55^{+20}_{-6}$ | ${<}0.19$ | ${<}0.18$ | ${<}8.2$ | ${<}0.29$
SPT-CLJ0014-4952 | 0.752 | 3.70417 | -49.8852 | 0.744 | $120^{+60}_{-20}$ | ${<}0.59$ | ${<}0.3$ | ${<}59$ | ${<}2.1$
SPT-CLJ0033-6326 | 0.597 | 8.471 | -63.4449 | 0.597 | $57^{+20}_{-10}$ | ${<}0.88$ | $0.51\pm 0.2$ | $32\pm 20$ | $2.1^{+1.7}_{-0.98}$
SPT-CLJ0037-5047 | 1.026 | 9.44771 | -50.789 | 1.029 | $57^{+40}_{-20}$ | ${<}3.4$ | ${<}0.19$ | ${<}80$ | ${<}2.9$
SPT-CLJ0040-4407 | 0.350 | 10.20821 | -44.1307 | 0.350 | $110^{+50}_{-10}$ | ${<}0.89$ | ${<}2.1$ | ${<}65$ | ${<}2.4$
SPT-CLJ0058-6145 | 0.885 | 14.58777 | -61.7671 | 0.830 | $88^{+40}_{-10}$ | $19\pm 1$ | ${<}0.46$ | ${<}130$ | ${<}4.9$
SPT-CLJ0102-4603 | 0.841 | 15.67791 | -46.071 | 0.839 | $53^{+20}_{-20}$ | ${<}0.71$ | ${<}0.018$ | ${<}4.8$ | ${<}0.17$
SPT-CLJ0102-4915 | 0.870 | 15.74073 | -49.272 | 0.869 | $18^{+0.5}_{-5}$ | ${<}1.3$ | $2.9\pm 0.07$ | $460\pm 300$ | $29^{+23}_{-14}$
SPT-CLJ0106-5943 | 0.348 | 16.61974 | -59.7202 | 0.351 | $72^{+30}_{-8}$ | $1.2\pm 0.05$ | ${<}3$ | ${<}89$ | ${<}3.3$
SPT-CLJ0123-4821 | 0.655 | 20.79565 | -48.3563 | 0.655 | $160^{+80}_{-20}$ | ${<}0.52$ | ${<}0.19$ | ${<}27$ | ${<}0.99$
SPT-CLJ0151-5954 | 1.034 | 27.84222 | -59.9054 | 1.008 | $110^{+60}_{-20}$ | ${<}0.92$ | ${<}0.14$ | ${<}59$ | ${<}2.2$
SPT-CLJ0156-5541 | 1.288 | 29.03803 | -55.703 | 1.293 | $120^{+50}_{-10}$ | ${<}3.6$ | ${<}0.22$ | ${<}170$ | ${<}5.9$
SPT-CLJ0200-4852 | 0.498 | 30.14204 | -48.8713 | 0.499 | $260^{+40}_{-10}$ | ${<}0.29$ | ${<}1.2$ | ${<}86$ | ${<}3.2$
SPT-CLJ0205-5829 | 1.322 | 31.44879 | -58.4821 | 1.320 | $72^{+10}_{-50}$ | ${<}1.2$ | ${<}0.35$ | ${<}280$ | ${<}9.9$
SPT-CLJ0212-4657 | 0.654 | 33.12022 | -46.9488 | 0.651 | $46^{+20}_{-9}$ | $3.5\pm 0.2$ | ${<}1.1$ | ${<}150$ | ${<}5.5$
SPT-CLJ0217-5245 | 0.343 | 34.31195 | -52.7603 | 0.341 | $67^{+20}_{-20}$ | ${<}0.073$ | ${<}2.6$ | ${<}77$ | ${<}2.8$
SPT-CLJ0232-4421 | 0.284 | 38.07726 | -44.3467 | 0.289 | $26^{+6}_{-8}$ | $3.7\pm 0.07$ | $6.5\pm 0.6$ | $71\pm 40$ | $4.5^{+3.9}_{-2.1}$
SPT-CLJ0232-5257 | 0.556 | 38.20587 | -52.9532 | 0.560 | $36^{+2}_{-40}$ | ${<}0.9$ | ${<}0.89$ | ${<}83$ | ${<}3$
SPT-CLJ0234-5831 | 0.415 | 38.67609 | -58.5236 | 0.414 | $13^{+0.6}_{-4}$ | $2.2\pm 0.3$ | $30\pm 0.3$ | $760\pm 400$ | $49^{+44}_{-23}$
SPT-CLJ0235-5121 | 0.278 | 38.93863 | -51.3513 | 0.279 | $120^{+40}_{-20}$ | ${<}0.096$ | ${<}2.2$ | ${<}40$ | ${<}1.4$
SPT-CLJ0236-4938 | 0.334 | 39.25696 | -49.636 | 0.336 | $97^{+50}_{-20}$ | $1.3\pm 0.07$ | ${<}1.5$ | ${<}40$ | ${<}1.4$
SPT-CLJ0243-5930 | 0.635 | 40.86283 | -59.5173 | 0.633 | $100^{+30}_{-20}$ | ${<}0.42$ | $0.63\pm 0.2$ | $44\pm 30$ | $2.8^{+2.7}_{-1.3}$
SPT-CLJ0252-4824 | 0.421 | 43.20825 | -48.4162 | 0.421 | $69^{+10}_{-30}$ | ${<}0.26$ | ${<}0.81$ | ${<}40$ | ${<}1.4$
SPT-CLJ0256-5617 | 0.580 | 44.12593 | -56.2974 | 0.609 | $160^{+50}_{-50}$ | ${<}0.4$ | ${<}0.22$ | ${<}22$ | ${<}0.82$
SPT-CLJ0304-4401 | 0.458 | 46.07024 | -44.0254 | 0.455 | $100^{+30}_{-10}$ | ${<}0.29$ | ${<}1.2$ | ${<}69$ | ${<}2.6$
SPT-CLJ0304-4921 | 0.392 | 46.06729 | -49.3573 | 0.393 | $35^{+10}_{-6}$ | ${<}0.2$ | ${<}1.5$ | ${<}59$ | ${<}2.3$
SPT-CLJ0307-5042 | 0.550 | 46.96054 | -50.7012 | 0.555 | $110^{+40}_{-20}$ | ${<}0.23$ | ${<}2.2$ | ${<}200$ | ${<}7.2$
SPT-CLJ0307-6225 | 0.580 | 46.81977 | -62.4465 | 0.578 | $20^{+2}_{-20}$ | $20\pm 2$ | $1.8\pm 0.4$ | $110\pm 60$ | $6.8^{+5.8}_{-3.2}$
SPT-CLJ0310-4647 | 0.707 | 47.63545 | -46.7857 | 0.707 | $73^{+30}_{-10}$ | ${<}0.78$ | ${<}0.011$ | ${<}1.7$ | ${<}0.061$
SPT-CLJ0313-5334 | 1.474 | 48.48543 | -53.5708 | 1.477 | $73^{+9}_{-50}$ | ${<}1.3$ | ${<}0.27$ | ${<}280$ | ${<}9.9$
SPT-CLJ0324-6236 | 0.750 | 51.05101 | -62.5988 | 0.746 | $79^{+30}_{-10}$ | ${<}0.45$ | ${<}1.4$ | ${<}270$ | ${<}9.9$
SPT-CLJ0330-5228 | 0.442 | 52.73718 | -52.4703 | 0.440 | $150^{+60}_{-10}$ | $17\pm 2$ | ${<}1.8$ | ${<}100$ | ${<}3.5$
SPT-CLJ0334-4659 | 0.485 | 53.54573 | -46.9959 | 0.485 | $30^{+10}_{-3}$ | $10\pm 1$ | $26\pm 0.2$ | $990\pm 600$ | $63^{+53}_{-29}$
SPT-CLJ0346-5439 | 0.530 | 56.7309 | -54.6486 | 0.532 | $51^{+20}_{-8}$ | $21\pm 2$ | ${<}0.55$ | ${<}45$ | ${<}1.7$
SPT-CLJ0348-4515 | 0.359 | 57.07136 | -45.2498 | 0.363 | $95^{+40}_{-10}$ | ${<}0.19$ | ${<}1.3$ | ${<}40$ | ${<}1.5$
SPT-CLJ0352-5647 | 0.649 | 58.23958 | -56.7977 | 0.649 | $55^{+20}_{-8}$ | ${<}0.39$ | ${<}0.59$ | ${<}81$ | ${<}2.9$
SPT-CLJ0406-4805 | 0.737 | 61.73024 | -48.0825 | 0.736 | $100^{+50}_{-20}$ | ${<}1.1$ | ${<}0.026$ | ${<}4.6$ | ${<}0.17$
SPT-CLJ0411-4819 | 0.424 | 62.81791 | -48.315 | 0.429 | $29^{+8}_{-8}$ | ${<}0.31$ | ${<}1.1$ | ${<}56$ | ${<}2$
SPT-CLJ0417-4748 | 0.579 | 64.34616 | -47.8132 | 0.581 | $31^{+10}_{-3}$ | ${<}0.2$ | ${<}2$ | ${<}200$ | ${<}7.1$
SPT-CLJ0426-5455 | 0.642 | 66.51718 | -54.9253 | 0.635 | $230^{+100}_{-40}$ | ${<}0.82$ | ${<}0.67$ | ${<}89$ | ${<}3.3$
SPT-CLJ0438-5419 | 0.421 | 69.57358 | -54.3223 | 0.420 | $72^{+20}_{-20}$ | ${<}0.19$ | ${<}3.6$ | ${<}160$ | ${<}6.3$
SPT-CLJ0441-4855 | 0.843 | 70.44958 | -48.9234 | 0.808 | $47^{+10}_{-7}$ | ${<}1.5$ | ${<}0.068$ | ${<}17$ | ${<}0.63$
SPT-CLJ0449-4901 | 0.792 | 72.28175 | -49.0213 | 0.786 | $150^{+50}_{-30}$ | $32\pm 2$ | ${<}0.2$ | ${<}45$ | ${<}1.7$
SPT-CLJ0456-5116 | 0.562 | 74.11716 | -51.2764 | 0.562 | $150^{+80}_{-10}$ | ${<}0.28$ | ${<}0.33$ | ${<}30$ | ${<}1.1$
SPT-CLJ0509-5342 | 0.461 | 77.33914 | -53.7035 | 0.461 | $52^{+20}_{-4}$ | $2\pm 0.2$ | $15\pm 0.2$ | $510\pm 300$ | $33^{+27}_{-15}$
SPT-CLJ0516-5430 | 0.295 | 79.15568 | -54.5005 | 0.297 | $150^{+20}_{-80}$ | ${<}0.084$ | ${<}4.8$ | ${<}97$ | ${<}3.7$
SPT-CLJ0522-4818 | 0.296 | 80.56489 | -48.3048 | 0.299 | $31^{+10}_{-6}$ | ${<}0.19$ | ${<}6.7$ | ${<}140$ | ${<}5$
SPT-CLJ0528-5300 | 0.768 | 82.02214 | -52.9981 | 0.766 | $60^{+20}_{-20}$ | $140\pm 10$ | $0.79\pm 0.04$ | $94\pm 60$ | $6.1^{+4.8}_{-2.9}$
SPT-CLJ0533-5005 | 0.881 | 83.40337 | -50.0958 | 0.880 | $83^{+30}_{-20}$ | ${<}0.77$ | ${<}0.069$ | ${<}19$ | ${<}0.72$
SPT-CLJ0542-4100 | 0.640 | 85.70855 | -41.0001 | 0.642 | $140^{+60}_{-10}$ | $62\pm 5$ | ${<}0.97$ | ${<}130$ | ${<}4.6$
SPT-CLJ0546-5345 | 1.066 | 86.65741 | -53.7588 | 1.064 | $84^{+40}_{-9}$ | ${<}7.1$ | ${<}0.014$ | ${<}6.5$ | ${<}0.24$
SPT-CLJ0551-5709 | 0.423 | 87.89828 | -57.1412 | 0.423 | $98^{+40}_{-20}$ | ${<}0.31$ | ${<}0.23$ | ${<}11$ | ${<}0.38$
SPT-CLJ0555-6406 | 0.345 | 88.85376 | -64.1057 | 0.345 | $100^{+40}_{-20}$ | ${<}0.13$ | $1.8\pm 0.4$ | $30\pm 20$ | $1.9^{+1.7}_{-0.88}$
SPT-CLJ0559-5249 | 0.609 | 89.93006 | -52.8242 | 0.610 | $130^{+40}_{-20}$ | $37\pm 0.1$ | ${<}0.62$ | ${<}70$ | ${<}2.7$
SPT-CLJ0607-4448 | 1.401 | 91.89507 | -44.8041 | 1.401 | $29^{+10}_{-5}$ | $6.1\pm 0.6$ | $0.29\pm 0.03$ | $150\pm 90$ | $9.6^{+8}_{-4.6}$
SPT-CLJ0615-5746 | 0.972 | 93.96551 | -57.7802 | 0.972 | $51^{+10}_{-7}$ | ${<}2.3$ | $0.65\pm 0.2$ | $130\pm 80$ | $8.6^{+7.7}_{-4.1}$
SPT-CLJ0616-5227 | 0.684 | 94.14205 | -52.4525 | 0.688 | $31^{+3}_{-20}$ | ${<}1.1$ | $0.89\pm 0.1$ | $77\pm 50$ | $4.9^{+4.5}_{-2.2}$
SPT-CLJ0640-5113 | 1.316 | 100.0725 | -51.2178 | 1.317 | $85^{+40}_{-9}$ | ${<}1.5$ | ${<}0.011$ | ${<}8.4$ | ${<}0.32$
SPT-CLJ0655-5234 | 0.470 | 103.9698 | -52.568 | 0.473 | $200^{+100}_{-30}$ | ${<}0.17$ | ${<}0.22$ | ${<}13$ | ${<}0.51$
SPT-CLJ2011-5725 | 0.279 | 302.8624 | -57.4197 | 0.278 | $37^{+10}_{-4}$ | ${<}0.096$ | ${<}0.56$ | ${<}10$ | ${<}0.37$
SPT-CLJ2031-4037 | 0.342 | 307.9719 | -40.6252 | 0.339 | $82^{+30}_{-10}$ | $17\pm 0.2$ | ${<}3.3$ | ${<}90$ | ${<}3.5$
SPT-CLJ2035-5251 | 0.528 | 308.7946 | -52.8564 | 0.534 | $150^{+50}_{-60}$ | ${<}0.36$ | ${<}0.46$ | ${<}38$ | ${<}1.4$
SPT-CLJ2040-4451 | 1.478 | 310.2384 | -44.8594 | 1.469 | $66^{+30}_{-20}$ | ${<}1.4$ | ${<}0.086$ | ${<}89$ | ${<}3.3$
SPT-CLJ2043-5035 | 0.723 | 310.8231 | -50.5923 | 0.723 | $14^{+3}_{-4}$ | $9\pm 0.7$ | $20\pm 0.6$ | $2000\pm 1000$ | $130^{+110}_{-61}$
SPT-CLJ2106-5844 | 1.132 | 316.5192 | -58.7412 | 1.131 | $26^{+0.7}_{-20}$ | $36\pm 0.8$ | $5.2\pm 0.2$ | $1600\pm 900$ | $100^{+87}_{-49}$
SPT-CLJ2135-5726 | 0.427 | 323.9147 | -57.4376 | 0.429 | $56^{+20}_{-7}$ | ${<}0.18$ | ${<}0.14$ | ${<}6.8$ | ${<}0.24$
SPT-CLJ2145-5644 | 0.480 | 326.4666 | -56.7481 | 0.481 | $53^{+20}_{-8}$ | ${<}0.3$ | ${<}1.9$ | ${<}130$ | ${<}4.6$
SPT-CLJ2146-4633 | 0.933 | 326.6473 | -46.5504 | 0.928 | $98^{+40}_{-20}$ | $170\pm 10$ | ${<}0.09$ | ${<}29$ | ${<}1.1$
SPT-CLJ2148-6116 | 0.571 | 327.1784 | -61.2795 | 0.572 | $73^{+20}_{-40}$ | $7.7\pm 0.7$ | ${<}0.31$ | ${<}30$ | ${<}1.1$
SPT-CLJ2218-4519 | 0.636 | 334.7467 | -45.3144 | 0.635 | $53^{+4}_{-50}$ | ${<}0.16$ | ${<}0.23$ | ${<}29$ | ${<}1.1$
SPT-CLJ2222-4834 | 0.652 | 335.7112 | -48.5764 | 0.651 | $32^{+7}_{-10}$ | $4.1\pm 1$ | $2.7\pm 0.2$ | $210\pm 100$ | $14^{+11}_{-6.3}$
SPT-CLJ2232-5959 | 0.595 | 338.1409 | -59.9981 | 0.594 | $35^{+10}_{-4}$ | ${<}0.33$ | ${<}0.1$ | ${<}11$ | ${<}0.42$
SPT-CLJ2233-5339 | 0.440 | 338.315 | -53.6526 | 0.439 | $120^{+60}_{-10}$ | ${<}0.6$ | $2\pm 0.5$ | $60\pm 40$ | $3.9^{+3.3}_{-1.8}$
SPT-CLJ2236-4555 | 1.170 | 339.2176 | -45.9305 | 1.180 | $63^{+30}_{-8}$ | ${<}2.3$ | ${<}0.006$ | ${<}3.5$ | ${<}0.13$
SPT-CLJ2245-6206 | 0.586 | 341.2587 | -62.1267 | 0.560 | $65^{+20}_{-30}$ | ${<}2.2$ | ${<}0.37$ | ${<}39$ | ${<}1.4$
SPT-CLJ2248-4431 | 0.351 | 342.1832 | -44.5308 | 0.347 | $85^{+20}_{-3}$ | $0.98\pm 0.02$ | ${<}3$ | ${<}93$ | ${<}3.4$
SPT-CLJ2258-4044 | 0.897 | 344.7011 | -40.7418 | 0.897 | $100^{+50}_{-20}$ | ${<}1.7$ | ${<}0.1$ | ${<}30$ | ${<}1.1$
SPT-CLJ2259-6057 | 0.855 | 344.7541 | -60.9595 | 0.788 | $79^{+30}_{-7}$ | $100\pm 8$ | ${<}0.044$ | ${<}12$ | ${<}0.43$
SPT-CLJ2301-4023 | 0.835 | 345.4708 | -40.3876 | 0.857 | $54^{+20}_{-6}$ | ${<}0.96$ | ${<}0.14$ | ${<}35$ | ${<}1.3$
SPT-CLJ2306-6505 | 0.530 | 346.7231 | -65.0882 | 0.529 | $48^{+5}_{-30}$ | ${<}0.55$ | ${<}0.6$ | ${<}48$ | ${<}1.8$
SPT-CLJ2325-4111 | 0.358 | 351.2988 | -41.2037 | 0.362 | $170^{+80}_{-20}$ | ${<}0.43$ | ${<}1.2$ | ${<}40$ | ${<}1.4$
SPT-CLJ2331-5051 | 0.576 | 352.9631 | -50.865 | 0.578 | $23^{+6}_{-6}$ | $14\pm 1$ | $1.7\pm 0.4$ | $97\pm 50$ | $6.1^{+4.7}_{-2.8}$
SPT-CLJ2332-5053 | 0.560 | 353.0249 | -50.8849 | 0.579 | $65^{+30}_{-9}$ | ${<}0.56$ | ${<}0.87$ | ${<}84$ | ${<}3$
SPT-CLJ2335-4544 | 0.547 | 353.7854 | -45.7391 | 0.546 | $130^{+40}_{-40}$ | ${<}0.28$ | ${<}0.051$ | ${<}4.5$ | ${<}0.16$
SPT-CLJ2337-5942 | 0.775 | 354.3651 | -59.7013 | 0.779 | $130^{+60}_{-10}$ | ${<}1.7$ | ${<}0.53$ | ${<}110$ | ${<}4$
SPT-CLJ2341-5119 | 1.003 | 355.3015 | -51.3291 | 1.003 | $71^{+30}_{-6}$ | $23\pm 2$ | ${<}0.46$ | ${<}180$ | ${<}6.5$
SPT-CLJ2341-5724 | 1.259 | 355.3533 | -57.417 | 1.257 | $42^{+20}_{-10}$ | $14\pm 0.2$ | $0.19\pm 0.06$ | $78\pm 40$ | $4.9^{+3.9}_{-2.2}$
SPT-CLJ2342-5411 | 1.075 | 355.6913 | -54.1848 | 1.081 | $37^{+10}_{-6}$ | ${<}1.9$ | ${<}0.51$ | ${<}250$ | ${<}9.1$
SPT-CLJ2344-4243 | 0.596 | 356.1829 | -42.7201 | 0.596 | $30^{+9}_{-2}$ | $53\pm 0.2$ | $170\pm 2$ | $10000\pm 6000$ | $670^{+550}_{-320}$
SPT-CLJ2345-6405 | 1.000 | 356.2376 | -64.0972 | 1.127 | $120^{+60}_{-20}$ | ${<}1.6$ | ${<}0.17$ | ${<}69$ | ${<}2.5$
SPT-CLJ2352-4657 | 0.902 | 358.0678 | -46.9602 | 0.908 | $25^{+3}_{-20}$ | ${<}0.83$ | ${<}0.25$ | ${<}76$ | ${<}2.6$
SPT-CLJ2355-5055 | 0.320 | 358.9478 | -50.928 | 0.318 | $21^{+1}_{-10}$ | $0.22\pm 0.006$ | $13\pm 2$ | $190\pm 100$ | $12^{+10}_{-5.6}$
SPT-CLJ2359-5009 | 0.775 | 359.9324 | -50.1722 | 0.775 | $150^{+80}_{-20}$ | ${<}0.32$ | ${<}0.66$ | ${<}140$ | ${<}4.8$

Note. — Column 1: SPT cluster name. Column 2: redshift from SPT-SZ catalog.
Column 3, 4: R.A. and Declination of BCG. Column 5: BCG redshift as determined
from SED fitting of spectrophotometry. Column 6: ICM pseudo-entropy measured
at a radius of 10 kpc. Column 7: k-corrected 1.4 GHz luminosity measured from
radio data. Column 8: Raw measured [O ii] flux that has not been corrected by
extinction. Column 9: [O ii] luminosity measured from spectroscopic line
fitting. Column 10: [O ii] SFR estimate, using Kewley et al. (2004)

Table 2:  Free parameters used in Prospector SED fitting  Parameter | Description | Priors
---|---|---
$z_{\rm obs}$ |  Observed redshift, initalized to mean $z$ from Bleem et al. (2020) |  TopHat: [$z-0.05$, $z+0.05$]
MBCG (M⊙) |  Total stellar mass formed |  Log10 uniform: [$10^{9},10^{13}$]
$\log(Z/Z_{\odot})$ |  Stellar metallicity in log solar units |  Clipped normal: $\mu=0.0$, $\sigma=0.3$, range=[$-2.0,1.0$]
$t_{\rm age}$ |  Age of galaxy |  TopHat: [$0$, age of universe at $z_{\rm obs}$]
$\tau$ |  e-folding time of star formation history in Gyr |  Log10 uniform: [$0.01,3.0$]
$D$ |  Optical depth for stellar light attenuation by dust for old stars using extinction curve from Calzetti et al. (2000), where observed flux $I=I_{0}e^{-D}$ |  TopHat: [$0,1$]
$f_{\rm burst}$ |  Fraction of total stellar mass formed in a recent star formation burst |  TopHat: [$0,0.5$]
$f_{\rm age,burst}$ |  Time at which burst happens, as a fraction of $t_{\rm age}$ |  TopHat: [$0.1,1$]
$\sigma_{v}$ |  Velocity smoothing in km s-1 |  TopHat: [$150,800$]
specnorm |  Spectrum normalization factor to match photometry |  Log10 uniform: [$1\times 10^{-6},10.0$]
specjitter |  Multiplicative noise inflation term, which inflates the noise in all spectroscopic pixels as necessary to get a statistically acceptable fit |  TopHat: [$1,10$]
specoutlier |  Pixel outlier mixture model, to marginalize over poorly modeled noise like residual sky lines or missing absorption lines |  TopHat: [$0.0001,1$]
($p_{1}$, $p_{2}$, $p_{3}$) |  Continuum calibration (Chebyshev) polynomial |  TopHat: n=3: [$-0.2/(n+1),0.2/(n+1)$]