Deeply cooled core of the Phoenix galaxy cluster imaged by ALMA with the Sunyaev-Zel'dovich effect

We present measurements of the thermal Sunyaev-Zel'dovich effect (SZE) toward SPT-CL J2334-4243 (the Phoenix galaxy cluster) at z=0.597 by the Atacama Large Millimeter/submillimeter Array (ALMA) in Band 3. The SZE is imaged at 5"resolution (corresponding to the physical scale of 23kpc/h) within 200kpc/h from the central galaxy with the peak signal-to-noise ratio exceeding 11. Combined with the Chandra X-ray image, the ALMA SZE data further allow for non-parametric deprojection of electron temperature, density, and entropy. Our method can minimize contamination by the central AGN and the X-ray absorbing gas within the cluster, both of which largely affect the X-ray spectrum. We find no significant asymmetry or disturbance in the SZE image within the current measurement errors. The detected SZE signal agrees well with the average pressure profile of local cool-core clusters. Unlike any other known clusters, however, gas temperature drops by at least a factor of 5 toward the center. We identify ~6x10^{11} M_sun cool gas with temperature ~3keV in the inner 20kpc/h. Taken together, our results imply that the gas is indeed cooling efficiently and nearly isobarically down to this radius in the Phoenix cluster.


Introduction
It has long been argued that the density of thermal gas particles in cores of galaxy clusters is large enough for these particles to cool radiatively, leading to a runaway "cooling flow" toward the cluster center (e.g., Fabian 1994 for review). Such rapid gas cooling or associated star formation in central galaxies, however, has not been observed in galaxy clusters in the local Universe. Radiative gas cooling must then be suppressed, e.g., by feedback from an Active Galactic Nucleus (AGN) often hosted by a central galaxy (e.g., McNamara & Nulsen 2007;Fabian 2012), while the exact mechanism is yet uncertain.
The Phoenix galaxy cluster, SPT-CL J2344-4243, at redshift z = 0.597 possibly provides a unique counter example to suppressed gas cooling mentioned above. It is the most X-ray luminous galaxy cluster known to date with exceptionally high concentration of thermal gas within the central 100 kpc; the predicted cooling (or mass deposition) rate amounts toṀ cool = 2000 − 4000 M⊙ yr −1 (McDonald et al. 2012;McDonald et al. 2013;Ueda et al. 2013). The central galaxy of the Phoenix cluster is also unique in that its star formation rateṀSF = 400 − 900 M⊙ yr −1 (McDonald et al. 2012;McDonald et al. 2013;Mittal et al. 2017) is among the largest in any galaxy at z < 1, it hosts a dusty type II quasar (Ueda et al. 2013), and it is associated with extended filaments of warm (∼ 10 4 K) ionized gas (McDonald et al. 2014a) and cold molecular gas (Russell et al. 2017).
One of the key questions on the nature of the Phoenix cluster is how much thermal gas is in fact cooling to sufficiently low temperatures. While the presence of cool plasma with temperature kT < ∼ 3 keV, where k is the Boltzmann constant, has been reported in the literature (Ueda et al. 2013;Pinto et al. 2018;McDonald et al. 2019), the deposition rate of such gas is often inferred to be lower than the values quoted above (Tozzi et al. 2015;Pinto et al. 2018;McDonald et al. 2019). This may imply that radiative cooling is still suppressed at kT > 3 keV in this cluster. A major challenge in the X-ray analysis of this cluster is the presence of a bright AGN and possibly cold neutral gas in the central galaxy; the former dominates the X-ray emission at energies E > 2 keV and the latter modifies the spectrum at E < 2 keV via absorption (McDonald et al. 2019; see also section 4.1). It is hence crucial to explore the physical states of the gas by other independent methods.
In this paper, we report on measurements of the thermal Sunyaev-Zel'dovich effect (SZE: Sunyaev & Zel'dovich 1970;Sunyaev & Zel'dovich 1972) toward the Phoenix cluster by the Atacama Large Millimeter/submillimeter Array (ALMA). While the cluster was first identified via the SZE by the South Pole Telescope (SPT) (Carlstrom et al. 2011;Williamson et al. 2011), its internal structures were not explored because of moderate angular resolution (> 1 ′ ) of the SPT. We present the first spatially-resolved SZE image of the Phoenix cluster at angular resolution of 5 ′′ using ALMA Band 3 (section 3). We further explore thermodynamic properties of the hot gas in conjunction with the X-ray data taken by Chandra (section 4). The high resolution SZE image by ALMA provides an independent and complementary probe of the intracluster medium (ICM) to X-rays.
Throughout the paper, we adopt a standard set of cosmological density parameters, ΩM = 0.3 and ΩΛ = 0.7. We use the dimensionless Hubble constant h ≡ H0/(100km/s/Mpc); given controversial results on the value of h (e.g., Verde, Treu & Riess 2019) we do not fix it unless otherwise stated. In this cosmology, the angular size of 1 ′′ corresponds to the physical size of 4.67 h −1 kpc at the source redshift z = 0.597. The errors are given in 1σ and the coordinates are given in J2000. and 15 separate blocks for the 12-m and 7-m arrays, respectively. Each execution block lasted less than 80 minutes including overheads. The number of antennas and the calibrators slightly varied among the execution blocks as listed in table 1. All the data were taken at four continuum bands centered at 85, 87, 97, and 99 GHz, yielding the overall central frequency of 92 GHz with an effective bandwidth of 7.5 GHz. Compact configurations were adopted to cover the baseline ranges of 3.7-145 kλ and 2.1-15.6 kλ for the 12-m and 7-m arrays, respectively, where λ is the observed wavelength. They yielded the angular resolution and the maximum recoverable scale of ∼ 2 ′′ and ∼ 60 ′′ , respectively.
The target field, centered at (23 h 44 m 43. s 90, −42 • 43 ′ 12. ′′ 00), had a diameter of about 1.5 ′ covered with 7 hexagonal mosaic pointings by both arrays. An equal spacing of 34.2 ′′ between the pointings was adopted, yielding approximately the Nyquist sampling for the 12-m array and much denser sampling for the 7-m array.
Throughout this paper, we used the visibility data produced by the second stage of ALMA's Quality Assurance process (QA2).
Imaging was done with the Common Astronomy Software Applications package (CASA: McMullin et al. 2007) version 5.6.1. The procedure was similar to Kitayama et al. (2016). We adopted the multi-frequency synthesis mode in joint mosaic imaging. Natural weighting was used and all the images presented were corrected for primary beam attenuation. Table 2 lists the parameters of the synthesized beams as well as the 1σ noise levels of the synthesized image within 45 ′′ from the field center. The noise levels were measured on a difference map created after subtracting the compact sources as described in section 3.1, dividing the data set in half, taking a difference between their dirty images, and dividing it by a factor 2 to correct for the reduction of the integration time.

Compact sources
There are four compact sources detected by emission above the 5σ significance level in our target field. As described below, we determined their positions and flux densities by fitting with the CASA task uvmodelfit the 12-m array visibility data at baselines longer than 30kλ (corresponding to  the spatial scale smaller than λ/(30kλ) ∼ 7 ′′ ) to eliminate contamination by the extended SZE. Source identification and subtraction are done in the uv plane and fully independent of the image synthesis method. The results are summarized in table 3. These sources are also detected at 18 GHz by ATCA and their properties are discussed in detail in a separate paper (Akahori et al. 2019). The assigned source names are the same as in Akahori et al. (2019); C1, C2, and C3 are central sources in descending order of their 18GHz flux, whereas W is a western off-center source.
The brightest central source, C1, is an AGN hosted by the central galaxy. Figure 1a shows a dirty image toward source C1 produced by the visibility data at > 30kλ. Throughout this paper, we denote the projected angular distance and the deprojected physical distance from source C1 by θ and r, respectively.
When the best-fit point source model for source C1 (table 3) is subtracted from the visibility data, two weaker sources, C2 and C3, become apparent near the cluster center (figure 1b). In addition, there is another off-center source, W, at θ ∼ 25 ′′ to the west from source C1 (figure 1c). As the shapes of sources C3 and W are consistent with the synthesized beam, they are modeled by point sources. We checked that the results in table 3 are essentially unchanged when these sources are modeled by a Gaussian varying its full width at half maximum (FWHM) as a free parameter. On the other hand, source C2 appears to be elongated from north-west to south-east. We thus modeled this source by an elliptical Gaussian, fixing the major-to-minor axis ratio at 0.05, to obtain the major axis FWHM of 1.3 ′′ ± 0.5 ′′ and the position angle of −9.4 • ± 21.8 • , with the centroid position and flux density listed in table 3. We checked that the results are insensitive to the assumed value of the axis ratio as long as the minor axis remains negligible compared to the synthesized beam size. For comparison, if source C2 is modeled by a point source, the fitted flux is lower by about 25% (table  3). Figure 1d shows a long baseline (> 30kλ) image after all the detected sources are removed from the visibility data. The residuals have the root mean square (rms) value of 0.0146 mJy/beam and are consistent with noise. This confirms that identification and subtraction of the sources are properly done.

The Sunyaev-Zel'dovich effect
The source-subtracted visibility data were deconvolved with the Multi-Scale CLEAN algorithm (Cornwell 2008;Rich et al. 2008;Steeb & Rau 2019) using the CASA task tclean. We adopted [0, 4 ′′ , 8 ′′ , 16 ′′ , 32 ′′ , 64 ′′ ] as the FWHMs of the Gaussian components used in Multi-Scale CLEAN implemented in CASA version 5.6.1 1 . As illustrated in figure 2, this is an optimal choice for the present target and assure maximal recovered flux as well as minimal residuals. We used a circular mask region with radius θ = 42 ′′ , a flux threshold of 0.024 mJy, and a loop gain of 0.05. Figure 3 shows the deconvolved SZE image created from the visibility data taken by the 12-m and 7-m arrays after the compact sources described in section 3.1 are removed. The image has been smoothed to an effective beam size of 5 ′′ FWHM for display purposes; unless otherwise stated, quantitative analysis in this paper was done on the unsmoothed image with 2.25 ′′ × 1.92 ′′ FWHMs. The rms noise level measured on the difference map smoothed to the 5 ′′ resolution is 0.025 mJy/beam at θ < 45 ′′ . The SZE decrement is detected at more than 11σ statistical significance. The SZE peak is located at 3.6 ′′ south from the central AGN (source C1), with the SZE intensities at the two positions of −0.294 ± 0.025 mJy/beam (SZE peak) and −0.280 ± 0.025 mJy/beam (source C1 location), respectively, on the smoothed image. The offset is hence not statistically significant given the noise level of the ALMA data. The integrated SZE flux density within θ = 40 ′′ is −11.9 ± 0.4 mJy. The mean signal within the annulus at 40 ′′ < θ < 45 ′′ is consistent with zero and −0.13 ± 0.17 µJy/arcsec 2 .
We plot in figure 4 azimuthally averaged intensity profiles in four quadrants with position angles of 315 The statistical error in each bin is computed using equation (1) of Kitayama et al. (2016). The SZE intensities in four quadrants are consistent with one another within the error, while the signal in the north quadrant tends to be weaker than the other directions at θ < 15 ′′ .

Comparison to X-ray data
To compare with the ALMA SZE data, we extracted the Xray data of the Phoenix cluster taken by Chandra ACIS-I (ObsID: 13401, 16135, 16545, 19581, 19582, 19583, 20630, 20631, 20634, 20635, 20636, and 2079). The total exposure time is 551.5 ksec. The data were processed using CIAO   (Fruscione et al. 2006) and the Calibration database (CALDB) version 4.8.2. The backgrounds were estimated from the off-center region at 3 ′ < r < 5 ′ from the central AGN, where the ICM emission is negligible. We used the data at observed energies E = 0.7 − 7.0 keV to minimize the effects of the ACIS contamination layer and the instrumental background (e.g., Bartalucci et al. 2014;Plucinsky et al. 2018). Throughout the analysis, we assumed that the ICM is in collisional ionization equilibrium, the abundance ratio of elements heavier than helium is that of Anders & Grevesse (1989), and the Galactic hydrogen column density toward the Phoenix cluster is 1.52 × 10 20 cm −2 (Kalberla et al. 2005). We fixed the helium mass fraction at Y = 0.25, which is nearly unchanged between the primordial gas and the solar photosphere (e.g., Asplund et al. 2009;Planck Collaboration 2018). Spectral fitting was done with XSPEC version 12.10.0e (Arnaud 1996).
There is an X-ray bright type II quasar near the center of the Phoenix cluster (Ueda et al. 2013;McDonald et al. 2015;McDonald et al. 2019). The 2.0 − 7.0 keV brightness has a prominent point-like peak at (23 h 44 m 43. s 962, −42 • 43 ′ 12. ′′ 412) and we refer to this position as "the Xray center" of this cluster; it agrees with the position of the central AGN (source C1) in the ALMA 92 GHz map within 0.6 ′′ . For definiteness, we denote the projected distance . Chandra X-ray spectrum of the Phoenix cluster at θX < 1.5 ′′ (top) and the residuals with respect to the best-fit models (bottom). Left: Lines show the best-fit models of the ICM without intrinsic absorption (red solid), the central AGN (red dot-dashed), and their sum (blue solid). Right: Same as the left panel except that intrinsic absorption within the cluster is included in the fit. For reference, a green dashed line shows the ICM spectrum at E < 2 keV free from intrinsic absorption.
from the X-ray center by θX. The emission around the Xray center is dominated by the quasar and the ICM at E > 2 keV and E < 2 keV, respectively (figure 5). We modeled the spectrum of the former by an obscured (by a torus) powerlaw plus an iron fluorescent line at the rest-frame energy of 6.4 keV as in Ueda et al. (2013). Following McDonald et al. (2019), we also took into account additional photoelectric absorption by cool gas within the Phoenix cluster (see the next paragraph for details). Unless otherwise stated, there were in total five position-dependent free parameters in our spectral model; projected temperature of the ICM, projected metallicity of the ICM, column density of an intrinsic absorber within the cluster, spectral normalization factors for the ICM and for the central AGN. In addition, three parameters of the central AGN (the spectral index, column density of an obscuring torus, and the flux ratio between the 6.4 keV line and the power-law continuum) were varied when fitting the spectrum at θX < 1.5 ′′ and fixed at their best-fit values elsewhere.
Figure 5 explicitly shows the impact of intrinsic absorption within the Phoenix cluster mentioned above. If intrinsic absorption is not taken into account, the projected ICM temperature at θX < 1.5 ′′ is kT = 9.8 ± 1.5 keV with χ 2 /dof = 472/401 (left panel), where dof denotes a degree of freedom of the fit. Inclusion of intrinsic absorption significantly decreases the temperature to kT = 2.7 +0.4 −0.3 keV and improves the fit to χ 2 /dof = 443/400 (right panel); the best-fit hydrogen column density of the absorber NH,int = (6.7 ± 1.1) × 10 21 cm −2 is also consistent with figure 6 of McDonald et al. (2019). In other words, intrinsic absorption largely modifies the spectral shape at 0.7-2.0 keV and leads to a reduction of the best-fit tem-perature by more than a factor of 3. This reflects the fact that the ICM continuum is overwhelmed by the AGN at E > 2 keV and the ICM metal lines are sensitive to intrinsic absorption at E < 2 keV. It is hence meaningful to perform an independent measurement of the ICM temperature without relying on the X-ray spectrum. We will discuss such a complementary probe using the SZE data in section 4.4.
As in Kitayama et al. (2016), we performed X-ray thermodynamic mapping using the contour binning algorithm (Sanders 2006). In addition to the central region at θX < 1.5 ′′ mentioned above, we defined subregions with nearly equal photon counts by adopting the signal-to-noise ratio (S/N) threshold of 100 (i.e., ∼ 10000 counts) in the 0.7 − 7.0 keV band. The 0.7 − 7.0 keV spectrum in each subregion was fitted by the model mentioned above. Typical statistical errors are 25 % and 5 % for the temperature and the density of the ICM, respectively; the errors are mainly driven by the limited energy range, particularly near the central AGN, available for determining the ICM properties. We also checked that the AGN emission is consistent with the point spread function (PSF) of ACIS-I and becomes negligible at θX > 5 ′′ ; we excluded the AGN emission in the spectral fitting at such distances. The absorption-corrected intrinsic AGN luminosity is LX(0.7 − 7.0 keV) = (1.81 ± 0.09) × 10 45 h −2 erg s −1 , which amounts to ∼ 40% of the X-ray luminosity from the entire ICM. the central AGN. On the other hand, projected temperature (panel b) drops to ∼ 3 keV near the center and is lower in the north-south direction. Pseudo electron density assuming a uniform line-of-sight depth of L (panel c) tends to be higher in the north-south direction across the center. Consequently, pseudo electron pressure (panel d), a product of the quantities plotted in panels b and c, is less concentrated than the density; it decreases only by a factor of ∼ 2 from the center to θ ∼ 15 ′′ . Note that absolute values of the pseudo pressure are arbitrary (∝ L −1/2 ) and the plotted values are also subject to statistical errors of ∼ 25%. Overall trend of the pseudo pressure map still shows qualitative agreement with the ALMA SZE image.
To further examine the departure from symmetry in the observed X-ray and SZE data, we plot in figure 7 residual images after subtracting the mean signal in a modelindependent manner. For this purpose, we first searched for an ellipse that minimizes the variance of the 0.7 − 2.0 keV X-ray brightness relative to its mean atθX < 5 ′′ , wherē θX is the geometrical mean of semi-major and semi-minor axis lengths around the X-ray center. We found that such an ellipse has the axis ratio of 0.97, the position angle of −112 • , and the center at (∆R.A., ∆Dec)= (0.1 ′′ , −0.2 ′′ ) from the X-ray center. We then computed the mean X-ray brightness over this ellipse as a function ofθX and subtracted it from the X-ray brightness at each sky position. We also subtracted from the SZE image the mean SZE brightness over the same ellipse as used for the X-ray image. Figure 7 confirms the presence of X-ray cavities and overdense regions around the central AGN reported previously (McDonald et al. 2015;McDonald et al. 2019). It further shows the absence of significant disturbance in the residual SZE image above the noise level (1σ = 0.025 mJy/beam at 5 ′′ FWHM). This suggests that the ICM in this region is consistent with being isobaric.

Imaging simulations
To understand the degree of the missing flux of the ALMA SZE data, we performed imaging simulations using the Xray pseudo pressure map (figure 6d) of the Phoenix cluster. Given that absolute values of the pseudo pressure were arbitrary, they were normalized so that the peak signal corresponds to the Compton y-parameter of y peak = 8 × 10 −4 , which is a typical value for massive cool core clusters.
To take into account uncertainties of this normalization, we also examined the cases of y peak = 4 × 10 −4 and 12 × 10 −4 . A relativistic correction to the SZE intensity by Itoh & Nozawa (2004) was applied adopting the projected temperature shown in figure 6b at each sky position.
We created model images separately at four spectral windows centered at 85, 87, 97, and 99 GHz with an effective bandwidth of 1.875 GHz each. The pointing directions, the array configuration, the hour angle, the total effective integration time, and the average precipitable water vapor were set to match those of each executing block of real observations. Visibility data were then produced using the CASA task simobserve including both instrumental and atmospheric thermal noise in each spectral window. The rms levels of dirty images are consistent with the values given in table 1. The mock visibility was deconvolved in the same way as the real data as described in section 3.2. For each value of y peak , we repeated the above procedure 10 times adopting different noise realizations. To correct for any potential bias in producing an image at a single frequency from the data taken over finite bandwidths, the simulation results are compared with an input model evaluated at the central frequency 92 GHz.  figure 9. The simulated images for y peak = 8×10 −4 show similar amplitude and spatial extension to the real data plotted in figures 3 and 4. Figures 8 and 9 also illustrate that the simulated images are in good agreement with the input model once the correction by equation (1) described below is applied. The rms values at θ < 45 ′′ in figures 8d and 8f are both 0.022 mJy/beam and fully consistent with noise.
As in Kitayama et al. (2016), we find that the simulation results on average follow the linear relation where I out ν and I in ν are respectively the intensities of output and input images at the same sky position. Figure  10 shows that the coefficient c0 is sensitive to the adopted value of y peak whereas c1 remains nearly unchanged and is close to unity. We hence determined c0 and c1 by fitting the results of 10 simulation realizations for each value of y peak , where c1 is assumed to be common among different values of y peak . This yielded c1 = 0.94 ± 0.02 and c0 = (1.19 ± 0.16, 2.07 ± 0.20, 3.02 ± 0.28) µJy/arcsec 2 for y peak = (4 × 10 −4 , 8 × 10 −4 , 12 × 10 −4 ), respectively. The error bars of I out ν plotted in figure 10 show the standard deviation in each bin and are dominated by statistical errors; each bin contains on average 8000 pixel data (i.e., 800 per realization). The rms deviation of the mean values of (I out ν , I in ν ) from the best-fitting relation is ∆I in ν = 0.17µJy/arcsec 2 and gives an estimate of intrinsic deviation from equation (1) apart from the statistical errors. We regard √ 2 times this value (i.e., ∆I in ν = 0.24µJy/arcsec 2 ) as the 1σ systematic uncertainty in the true intensity when a constant offset (corresponding to c0) is subtracted. In real observations, the true intensity is unknown and we will subtract the mean signal at the edge of the emission profile instead of assuming any specific value of c0 in sections 4.3 and 4.4.
The fact that the simulated images of the Phoenix cluster are well reproduced by equation (1) with the value of c1 close to unity (figures 8-10) confirms that the observed ALMA SZE image gives a reasonable representation of differential values of the true intensity (Kitayama et al. 2016). Figure 9 further implies that the extended signal is retained out to θ ∼ 40 ′′ irrespectively of the adopted value of y peak , i.e., normalization of the intrinsic signal.

Compton y-parameter map and the inner pressure profile
A high angular resolution SZE image provides a direct probe of the projected electron pressure, or equivalently, the Compton y-parameter, of the ICM. Figure 11 shows the Compton y-parameter map reconstructed from the observed ALMA image in figure 3 using the results of section 4.2. Specifically, the ALMA image was divided by the correction factor for the missing flux of 0.94 (c1 in equation [1]) and the mean signal (consistent with zero) at θ = 42 ′′ is subtracted. At each sky position, the relativistic correction by Itoh & Nozawa (2004) was applied adopting the projected X-ray spectroscopic temperature shown in figure 6b. The plotted values corre-  spond to the incremental y-parameter (∆y) relative to the sky positions at θ = 42 ′′ . The peak value is ∆y = (5.3 ± 0.4 [statistical] ± 0.3 [systematic]) × 10 −4 , where the systematic error is from the absolute calibration of ALMA (6%: Kitayama et al. 2016) and the missing flux correction (0.24 µJy/arcsec 2 : section 4.2). Within the inferred range of statistical and systematic uncertainties, overall morphology of the Compton y-parameter map does not exhibit deviation from that of X-rays. Given the regularity of the Compton y-parameter map, we compare its azimuthal average to the mean radial pressure profiles of various galaxy cluster samples in figure 12. Vertical error bars include both statistical and systematic errors, whereas horizontal ones indicate the bin size; the bins are geometrically spaced with the inner-most bin at 0 < θ < 4 ′′ and the size increasing by a factor of 1.1 so that a statistical error is smaller than 15% in each bin. The mean radial pressure profiles are computed by integrating along the line-of-sight the generalized NFW profile (Nagai et al. 2007) with the model parameters given in the literature (mentioned below) assuming M500 = 8.8 × 10 14 h −1 M⊙ and r500 = 0.92h −1 Mpc at z = 0.597 for the Phoenix cluster 2 (McDonald et al. 2012), being convolved with the synthesized beam size of ALMA (2.25 ′′ × 1.92 ′′ FWHMs), and 2 M500 is the total mass enclosed within the radius r500 at which the enclosed overdensity is 500 times the critical density of the Universe at that redshift. Fig. 10. Top: Relation between the output intensity and the input intensity at 92 GHz from the imaging simulations with y peak = 4×10 −4 (blue), 8×10 −4 (red), and 12 × 10 −4 (green). Error bars denote the standard deviation in each bin. Bottom: Same as the left panel, except that the fitted value of c0/c1 in equation (1) has been added to I in ν to give zero intercept in each case. The thick dashed line shows the best fitting relation with c1 = 0.94. taking a difference with respect to the positions at 42 ′′ from the center. We find that the inner pressure profile of the Phoenix cluster is in good agreement with the expectation from the local cool core clusters by Arnaud et al. (2010) and is clearly steeper than that from more distant counterparts by Planck Collaboration (2013) and McDonald et al. (2014b). McDonald et al. (2015) showed that the Chandra X-ray data of this cluster also agree with the pressure profile of the local cool core clusters of Arnaud et al. (2010). These results further imply that azimuthally averaged SZE and X-ray data of the Phoenix cluster are consistent with each other.

Deprojected electron temperature and density
A combination of SZE and X-ray images provides a measure of temperature and density of the ICM independently of the X-ray spectroscopy (e.g., Kitayama et al. 2004;Nord et al. 2009;Basu et al. 2010). Assuming spherical symmetry and h = 0.7 for this particular purpose, we performed non-parametric deprojection of electron density ne and temperature T as described in detail below. A major advantage of this method is that the impact of X-ray Overlaid are the expectations from the generalized NFW pressure profile of X-ray selected cool core clusters at z < 0.2 (long dashed line) by Arnaud et al. (2010), Planck selected cool core clusters at z < 0.5 (short dashed line) by Planck Collaboration (2013), and SPT selected cool core clusters at z > 0.6 (dotted line) by McDonald et al. (2014b). Both the data points and the expectations are relative to the positions at θ = 42 ′′ .
absorbing gas described in section 4.1 is minimized. If the absorbed and unabsorbed X-ray brightness is related by I abs X = αI unabs X (0 ≤ α ≤ 1), deprojected density and temperature vary approximately as n abs e = n unabs e / √ α and T abs = T unabs √ α, respectively. The intrinsic absorption inferred in section 4.1 gives α ∼ 0.75 at θ < 1.5 ′′ in the 0.7-2.0 keV band, implying that the central density and temperature are expected to change only by ∼ 13%.
We first took geometrically-spaced bins on the sky with the inner-most bin at 0 < θ < 4 ′′ (corresponding to the deprojected spherical shell of 0 < r < 26.7 kpc) and the size increasing by a factor of 1.3 so that a statistical error is smaller than 10% in each bin. Systematic errors from the flux calibration of ALMA (6%), the missing flux correction of the SZE (0.24 µJy/arcsec 2 ), and the effective area of Chandra ACIS-I (4%) were added in quadrature to the statistical error in each bin. We then fitted the volume averaged brightness in each bin of the SZE at θ < 40 ′′ and of the 0.7-2.0 keV X-rays at θ < 100 ′′ together varying the temperature and the density in each spherical shell. As the temperature at r > 250 kpc (θ > 37 ′′ ) cannot be constrained by the ALMA data, it was fixed at the projected mean value of 15.8 keV by the X-ray spectral analysis described in section 4.1; we checked that the deprojected quantities at r < 100 kpc are essentially unchanged by this assump- Fig. 13. Deprojected temperature profiles of the Phoenix cluster from SZE and X-ray images (left panel) and from X-ray spectroscopy (right panel) assuming h = 0.7. The error bars show the results with (solid) or without (dashed) intrinsic absorption inside the cluster. For clarity, dashed error bars are slightly shifted horizontally. Lines indicate the mean profiles of cool core clusters at z > 0.6 (long-dashed) by McDonald et al. (2014b) and at z < 0.23 (short-dashed) by Vikhlinin et al. (2006). tion. For the SZE, we modeled and fitted the incremental brightness relative to the bin centered at θ = 43.6 ′′ ; the missing flux correction factor of 0.94 and the temperaturedependent relativistic correction were applied to the model brightness in each spherical shell. The X-ray emissivity was computed by SPEX v3.0.5.00 (Kaastra et al. 1996), fixing the metal abundance at 0.35 times the Solar value. To examine the impact of intrinsic absorption within the Phoenix cluster, the hydrogen column density in the innermost bin was assumed to be either NH,int = 6.7×10 21 cm −2 or NH,int = 0; the former value was chosen as a limiting case of strong absorption throughout the inner 27 kpc. Note that the X-ray emission from the central AGN is negligible in the 0.7-2.0 keV band used in the analysis.
For comparison, we separately performed a deprojection analysis using solely the X-ray data. The 0.7-7.0 keV spectra in six circular annuli at θ = 0 ′′ − 3 ′′ , 3 ′′ − 6 ′′ , 6 ′′ − 12 ′′ , 12 ′′ − 24 ′′ , 24 ′′ − 48 ′′ , and 48 ′′ − 96 ′′ were fitted together using the model projct implemented in XSPEC version 12.10.0e (Arnaud 1996). For simplicity, only statistical errors were taken into account in this particular analysis. The position-dependent free parameters were the same as those described in section 4.1 except that they were assigned to each of six spherical shells corresponding to the above mentioned annuli. The spectral index, column density of an obscuring torus, and the flux ratio between the 6.4 keV line and the power-law continuum of the central AGN were fixed at their best-fit values obtained in section 4.1. Figure 13 shows deprojected temperatures obtained by two different methods. As expected, the results from the SZE and X-ray images are insensitive to intrinsic absorption; the temperature at r < 27 kpc is 2.70 ± 1.71 keV and 3.01 ± 1.86 keV with and without intrinsic absorption, respectively. Those solely from the X-ray data are much more sensitive, albeit with smaller statistical errors; the temperature at r < 20 kpc changes by more than a factor of two from 2.73 +0.24 −0.22 keV to 6.37 +0.73 −0.58 keV once intrinsic absorption is neglected. If intrinsic absorption is taken into account, the temperature profile from the latter deprojection method comes to a good agreement with that from the former.
Also plotted for reference in figure 13 are the mean profiles of local and distant cool core clusters (Vikhlinin et al. 2006;McDonald et al. 2014b), using the average temperature within r500 of this cluster, kT500 = 12.6 keV, expected from the X-ray scaling relation by Reichert et al. (2011). We find that the deprojected temperatures at r < 100 kpc are systematically lower than the average profiles of both local and distant clusters. The inferred central temperature is ∼ 5 times lower than the mean temperature (∼ 16 keV) at r > 300 kpc. This is a much larger factor of reduction than observed in other clusters and supports that radiative cooling is efficient in the Phoenix cluster. Abell 1795 (short-dashed) by Vikhlinin et al. (2006); both profiles are scaled to match the radius and redshift of the Phoenix cluster as described in the text. In the left panel, the data points at r > 250 kpc are obtained solely from the X-ray image assuming kT = 15.8 keV.
Given that the pressure profile of the Phoenix cluster well matches that of local cool core clusters (section 4.3), gas density should have been enhanced to compensate for the reduced temperature. Figure 14 shows that this is indeed the case; overlaid are the best-fit density profiles by Vikhlinin et al. (2006) of typical local cool core clusters, Abell 2029 and Abell 1795. To correct for different size and redshift of individual clusters, these profiles have been scaled so that r gives the same fraction of r500 as the Phoenix cluster and that the normalization of ne evolves to z = 0.597 by the same fraction as the critical density of the Universe. Gas density of the Phoenix cluster is in good agreement with those of Abell 2029 and Abell 1795 at r > 0.2r500 ∼ 300 kpc, whereas it is systematically higher at smaller radii. Figure 14 also confirms that density profiles obtained from two deprojection methods are consistent with each other. Figure 15 further illustrates that the entropy K ≡ kT n −2/3 e decreases to < 10 keV cm 2 at r < 27 kpc. Also plotted are a model K ∝ r 1.2 which tends to give a lower limit to non-radiative clusters at r < ∼ 0.5r500 (Voit et al. 2005) and a prediction K ∝ r 1.4 for the steady-state cooling flow (Voit 2011); both profiles are normalized to match the data point in the outer-most bin in the left panel. The observed entropy profile shows a better agreement with K ∝ r 1.4 than K ∝ r 1.2 and strongly supports that radiative cooling is efficient in the Phoenix cluster.
Finally, we plot in Figure 16 the radiative cooling time, i.e., total thermal energy of the gas divided by the bolometric luminosity in each radial bin. As the bolometric luminosity of the gas at kT > ∼ 1 keV is dominated by Xrays, it is computed by integrating the rest-frame X-ray emissivity provided by SPEX over photon energies (e.g., Schure et al. 2009). The radial profile of the cooling time is well represented by a single power-law of t cool ∝ r 1.7 and drops to ∼ 0.1 Gyr at r < 27 kpc.

How much gas is cooling in the Phoenix cluster?
The results in section 4.4 immediately yield the amount of ∼ 3 keV gas within r = 27 kpc to be M cool = (6.4 ± 0.9) × 10 11 M⊙. This corresponds to ∼ 20% of the stellar mass ∼ 3 × 10 12 M⊙ in the central galaxy of the Phoenix cluster (McDonald et al. 2012;McDonald et al. 2013). If the mass deposition rate at kT > ∼ 3 keV isṀ cool ∼ 2000 M⊙ yr −1 as indicated by the X-ray data (e.g., McDonald et al. 2013;Ueda et al. 2013), the cool gas should have accumulated over the period  is also much smaller than the adiabatic sound speed cs ≃ 900(kT /3 keV) 1/2 km s −1 over the range of radii considered in this paper.
Our results consistently imply that radiative cooling is hardly suppressed down to kT ∼ 3 keV. At even lower temperatures, previous works infer that cooling may weaken moderately with the mass deposition rate ofṀ cool = 130 − 480 M⊙ yr −1 for the gas below 2 keV (Pinto et al. 2018) and the star formation rate ofṀSF = 400 − 900 M⊙ yr −1 in the central galaxy (McDonald et al. 2012;McDonald et al. 2013;Mittal et al. 2017). If this is the case, the time required for the above mentioned cool gas to turn into stars will be i.e., the stellar mass of the central galaxy is expected to increase by ∼ 20% over this period.

Conclusions
We have presented the SZE image of the Phoenix galaxy cluster at z = 0.597 taken by ALMA in Band 3. The SZE is imaged at 5 ′′ resolution (or 23 h −1 kpc) within 200 h −1 kpc from the central AGN with the peak S/N exceeding 11. Combined with the Chandra X-ray image, the ALMA SZE data further allow for non-parametric deprojection of electron temperature, density, and entropy. Our method can minimize contamination by the central AGN and the Xray absorbing gas within the cluster, both of which largely affect the X-ray spectrum.
We find no significant asymmetry or disturbance in the SZE image within the current measurement errors. The detected signal shows much higher central concentration than other distant clusters and agrees well with the average pressure profile of local cool-core clusters. Unlike typical clusters at any redshift, gas temperature drops by at least a factor of 5 toward the center. In the inner 20 h −1 kpc, we identify the presence of ∼ 6 × 10 11 M⊙ cool gas with kT ∼ 3 keV, the amount of which corresponds to ∼ 20% of the stellar mass in the central galaxy. The low entropy (∼ 10 keV cm 2 ) and the short cooling time (∼ 0.1 Gyr) of this gas further corroborates that radiative cooling is hardly suppressed between kT ∼ 16 keV and kT ∼ 3 keV. Taken together, our results imply that the gas is cooling efficiently and nearly isobarically down to the inner 20h −1 kpc in the Phoenix cluster.