Hydrostatic Chandra X-ray analysis of SPT-selected galaxy clusters - I. Evolution of profiles and core properties

TL;DR

This study uses Chandra X-ray data and a new modeling code to analyze galaxy cluster profiles, finding no significant evolution in core properties or entropy distribution over a broad redshift range.

Contribution

It introduces MBProj2, a new forward-modeling projection code, and applies it to study the evolution of galaxy cluster core properties with no evidence of significant change.

Findings

No evidence for a central entropy floor in low-entropy systems.

The entropy distribution peaks near zero and around 130 keV cm^2.

No significant evolution of core properties with redshift.

Abstract

We analyse Chandra X-ray Observatory observations of a set of galaxy clusters selected by the South Pole Telescope using a new publicly-available forward-modelling projection code, MBProj2, assuming hydrostatic equilibrium. By fitting a powerlaw plus constant entropy model we find no evidence for a central entropy floor in the lowest-entropy systems. A model of the underlying central entropy distribution shows a narrow peak close to zero entropy which accounts for 60 per cent of the systems, and a second broader peak around 130 keV cm^2. We look for evolution over the 0.28 to 1.2 redshift range of the sample in density, pressure, entropy and cooling time at 0.015 R_500 and at 10 kpc radius. By modelling the evolution of the central quantities with a simple model, we find no evidence for a non-zero slope with redshift. In addition, a non-parametric sliding median shows no significant change. The fraction of cool-core clusters with central cooling times below 2 Gyr is consistent above and below z=0.6 (~30-40 per cent). Both by comparing the median thermodynamic profiles, centrally biased towards cool cores, in two redshift bins, and by modelling the evolution of the unbiased average profile as a function of redshift, we find no significant evolution beyond self-similar scaling in any of our examined quantities. Our average modelled radial density, entropy and cooling-time profiles appear as powerlaws with breaks around 0.2 R_500. The dispersion in these quantities rises inwards of this radius to around 0.4 dex, although some of this scatter can be fit by a bimodal model.

Tables

Table 1. Table 1: Models fitted to the surface-brightness profiles. Listed are the model names, the section in this paper which describes a model, whether wide- or finely-binned data are fitted, the mass models used and the parametrizations in the analysis. Other parameters included in the model are a background scaling parameter and the log outer pressure (if a mass model is used).

Name	Section	Binning	Mass model	Parametrization
BIN-NFW	2.5.1	Wide	NFW	Density in bins; mass model
BIN-GNFW	2.5.2	Wide	GNFW	Density in bins; mass model
BIN-NONHYDRO	2.5.3	Wide	None	Density and temperatures in bins
INT-NFW	2.5.4	Fine	NFW	Density at centres of wide bins with interpolation; mass model
MBETA-NFW	2.5.5	Fine	NFW	Modified-$\beta$ density profile; mass model
STEP-NFW	2.5.6	Fine	NFW	Constant densities between fixed radii; mass model
GRAD-NFW	2.5.7	Fine	NFW	Constant gradients between fixed radii; mass model
KPLAW-NFW	2.5.8	Fine	NFW	Powerlaw entropies inside and outside 300 kpc plus constant; mass model

Table 2. Table 2: Parameters from model fits to the inner quantities as a function of redshift (Fig.  11 ). The values given are the median and 1 ​ σ 1 𝜎 1\sigma uncertainties from the marginalised distributions.

Radius	Value	$c$	$m$	$w$
10 kpc	$n_e$	$-1.61\pm 0.04$	$0.14\pm 0.22$	$0.31\pm 0.03$
	$K_e$	$1.66\pm 0.04$	$-0.03\pm 0.23$	$0.29\pm 0.04$
	$t_{\mathrm{cool}}$	$9.39\pm 0.05$	$-0.07\pm 0.26$	$0.35\pm 0.04$
	$P_e$	$-1.01\pm 0.04$	$0.27\pm 0.22$	$0.31\pm 0.03$
$0.015R_{500}$	$n_e$	$1.36\pm 0.04$	$-0.31\pm 0.22$	$0.31\pm 0.03$
	$K_e$	$-1.14\pm 0.05$	$0.26\pm 0.25$	$0.33\pm 0.04$
	$t_{\mathrm{cool}}$	$-1.50\pm 0.05$	$0.34\pm 0.27$	$0.38\pm 0.04$
	$P_e$	$1.15\pm 0.03$	$-0.24\pm 0.18$	$0.25\pm 0.02$

Table 3. Table 3: Sample of objects analysed here. Listed is the cluster index, SPT identifier, main identifier (if any), redshift from Bleem et al. ( 2015 ) , Galactic equivalent hydrogen column density ( N H subscript 𝑁 H N_{\mathrm{H}} ; 10 20 ​ cm − 2 superscript 10 20 superscript cm 2 10^{20}\hbox{$\rm\thinspace cm^{-2}\,$} ), maximum radius examined ( r max subscript 𝑟 max r_{\mathrm{max}} ; arcmin), percentage uncertainty on the emissivities in each bin after binning (Rebin), total-cleaned exposure (ks), background-subtracted counts (and background counts) within the maximum radius (k) and list of Chandra observation identifiers.

Index	SPT ID	Identifier	$z$	$N_H$	$r_{\max }$	Rebin	Exp.	Counts	OBSIDs
1	SPT-CLJ0000-5748		0.702	1.4	3.0	20	29.7	$1.8$ ($2.6$)	9335
2	SPT-CLJ0013-4906		0.406	1.5	3.5	20	13.7	$2.3$ ($1.4$)	13462
3	SPT-CLJ0014-4952		0.752	1.5	3.0	20	54.5	$2.4$ ($3.3$)	13471
4	SPT-CLJ0033-6326		0.597	1.8	4.0	20	20.7	$1.1$ ($2.1$)	13483
5	SPT-CLJ0040-4407		0.35	3.5	5.0	20	7.9	$2.1$ ($1.6$)	13395
6	SPT-CLJ0058-6145		0.826	1.6	2.5	20	50.0	$1.1$ ($2.4$)	13479
7	SPT-CLJ0102-4603	El Gordo	0.722	1.7	2.5	20	59.0	$0.9$ ($3.0$)	13485
8	SPT-CLJ0102-4915		0.870	1.7	4.0	20	171.0	$36.0$ ($20.9$)	12258, 14023
9	SPT-CLJ0106-5943		0.348	1.8	5.0	20	17.3	$2.6$ ($2.8$)	13468
10	SPT-CLJ0123-4821		0.62	1.8	4.0	20	70.1	$2.3$ ($6.9$)	13491
11	SPT-CLJ0142-5032		0.73	2.2	3.0	20	28.6	$1.0$ ($1.8$)	13467
12	SPT-CLJ0151-5954		1.035	2.2	2.5	20	48.0	$0.6$ ($2.4$)	13480
13	SPT-CLJ0156-5541		1.221	3.1	2.0	20	76.6	$1.0$ ($2.3$)	13489
14	SPT-CLJ0200-4852		0.498	1.8	4.0	20	23.3	$1.3$ ($2.5$)	13487
15	SPT-CLJ0212-4657		0.655	1.7	3.0	20	27.8	$1.1$ ($1.9$)	13464
16	SPT-CLJ0217-5245	MCXC J0217.2-5244	0.343	2.7	5.5	20	19.4	$1.9$ ($5.0$)	12269
17	SPT-CLJ0232-4421	2MAXI J0231-440	0.284	1.7	6.0	20	11.3	$8.8$ ($3.0$)	4993
18	SPT-CLJ0232-5257		0.556	2.8	4.5	20	19.4	$1.1$ ($2.9$)	12263
19	SPT-CLJ0234-5831	1RXS J023443.1-583114	0.415	2.7	4.0	20	9.3	$2.0$ ($1.2$)	13403
20	SPT-CLJ0235-5121	PSZ1 G270.90-58.78	0.278	3.0	6.5	20	19.6	$5.4$ ($5.1$)	12262
21	SPT-CLJ0236-4938	ACT-CLJ0237-4939	0.334	2.5	4.5	20	38.6	$3.0$ ($6.2$)	12266
22	SPT-CLJ0243-5930		0.635	2.4	3.0	20	45.6	$2.5$ ($2.9$)	13484, 15573
23	SPT-CLJ0252-4824		0.421	2.5	4.0	20	29.6	$1.8$ ($3.3$)	13494
24	SPT-CLJ0256-5617		0.58	1.4	3.5	20	46.8	$2.4$ ($4.3$)	13481, 14448
25	SPT-CLJ0304-4401		0.458	1.3	4.5	20	14.7	$2.2$ ($2.1$)	13402
26	SPT-CLJ0304-4921		0.392	1.8	5.5	20	20.8	$4.1$ ($4.4$)	12265
27	SPT-CLJ0307-5042		0.55	1.8	3.5	20	38.2	$2.4$ ($3.4$)	13476
28	SPT-CLJ0307-6225		0.579	2.1	4.0	20	24.2	$1.2$ ($3.4$)	12191
29	SPT-CLJ0310-4647		0.709	1.8	3.0	20	36.1	$1.1$ ($2.4$)	13492
30	SPT-CLJ0324-6236		0.73	2.8	2.5	20	53.3	$1.7$ ($3.0$)	12181, 13137, 13213
31	SPT-CLJ0334-4659		0.485	1.2	4.0	20	25.3	$2.4$ ($2.6$)	13470
32	SPT-CLJ0346-5439	ACT-CLJ0346-5438	0.530	1.4	3.5	20	33.6	$2.4$ ($3.5$)	12270, 13155
33	SPT-CLJ0348-4515		0.358	0.9	4.5	20	12.4	$1.3$ ($1.7$)	13465
34	SPT-CLJ0352-5647		0.67	1.4	3.5	20	41.9	$1.5$ ($3.4$)	13490, 15571
35	SPT-CLJ0406-4805		0.737	1.3	3.0	20	25.8	$0.7$ ($1.8$)	13477
36	SPT-CLJ0411-4819	PLCKESZ G255.62-46.16	0.424	1.5	6.5	20	65.1	$11.9$ ($15.7$)	13396, 16355, 17536
37	SPT-CLJ0417-4748		0.581	1.3	3.0	20	21.2	$2.7$ ($1.5$)	13397
38	SPT-CLJ0426-5455		0.63	0.8	3.5	20	31.8	$1.0$ ($2.5$)	13472
39	SPT-CLJ0438-5419	PLCKESZ G262.71-40.91	0.421	1.0	6.0	20	19.6	$6.4$ ($5.0$)	12259
40	SPT-CLJ0441-4855		0.79	1.5	3.0	20	67.5	$2.3$ ($4.8$)	13475, 14371, 14372
41	SPT-CLJ0449-4901		0.792	1.2	2.5	20	49.8	$1.4$ ($2.4$)	13473
42	SPT-CLJ0456-5116		0.562	1.1	3.5	20	49.8	$2.5$ ($4.6$)	13474
43	SPT-CLJ0509-5342	ACT-CLJ0509-5341	0.461	1.5	3.5	20	28.2	$2.8$ ($3.2$)	9432
44	SPT-CLJ0516-5430	Abell S520	0.295	2.1	8.5	20	30.4	$13.5$ ($11.4$)	9331, 15099
45	SPT-CLJ0528-5300		0.768	3.2	2.5	20	122.1	$1.6$ ($6.9$)	11747, 11874, 12092, 13126, 9341, 10862, 11996
46	SPT-CLJ0533-5005		0.881	2.9	1.5	40	71.7	$0.4$ ($1.6$)	11748, 12001, 12002
47	SPT-CLJ0542-4100		0.642	3.2	3.5	20	48.7	$2.5$ ($3.7$)	914
48	SPT-CLJ0546-5345	1RXS J054638.7-534434	1.066	6.8	3.5	20	69.0	$2.1$ ($7.7$)	9332, 9336, 10851, 10864, 11739
49	SPT-CLJ0555-6406		0.345	4.0	6.5	20	10.9	$2.3$ ($2.9$)	13404
50	SPT-CLJ0559-5249	1RXS J055942.1-524950	0.609	5.1	3.5	20	106.9	$6.1$ ($11.5$)	12264, 13116, 13117
51	SPT-CLJ0655-5234		0.470	4.6	3.5	20	20.0	$0.9$ ($2.1$)	13486
52	SPT-CLJ0658-5556	Bullet cluster	0.296	4.9	-	5	531.2	$729.6$ ($282.5$)	5355, 5356, 5357, 5358, 3184, 5361, 4984, 4985, 4986
53	SPT-CLJ2031-4037	MCXC J2031.8-4037	0.342	3.4	8.0	20	9.9	$3.9$ ($3.2$)	13517
54	SPT-CLJ2034-5936		0.919	5.7	3.0	20	57.7	$1.4$ ($3.0$)	12182
55	SPT-CLJ2035-5251		0.528	2.9	3.5	20	18.0	$0.9$ ($1.8$)	13466
56	SPT-CLJ2043-5035		0.723	2.4	3.5	20	77.8	$5.7$ ($7.6$)	13478

Table 4. Table 5: Positions of the clusters. Shown are the J2000 coordinates of the peak in degrees used as the cluster centres for our analysis. Also shown are our positions of the centroid using a 250 to 500 kpc annulus. The offset between the two is shown in arcsec and kpc.

SPT ID	RA (peak)	Dec (peak)	RA (annulus)	Dec (annulus)	Offset (arcsec)	Offset (kpc)
SPT-CLJ0000-5748	$0.2499$	$-57.8093$	$0.2490$	$-57.8100$	$3$	$21$
SPT-CLJ0013-4906	$3.3306$	$-49.1103$	$3.3304$	$-49.1159$	$20$	$110$
SPT-CLJ0014-4952	$3.7044$	$-49.8837$	$3.6905$	$-49.8806$	$34$	$251$
SPT-CLJ0033-6326	$8.4691$	$-63.4444$	$8.4695$	$-63.4423$	$8$	$51$
SPT-CLJ0040-4407	$10.2102$	$-44.1317$	$10.2088$	$-44.1328$	$5$	$26$
SPT-CLJ0058-6145	$14.5886$	$-61.7678$	$14.5842$	$-61.7694$	$9$	$72$
SPT-CLJ0102-4603	$15.6774$	$-46.0716$	$15.6739$	$-46.0658$	$22$	$162$
SPT-CLJ0102-4915	$15.7423$	$-49.2743$	$15.7347$	$-49.2664$	$33$	$258$
SPT-CLJ0106-5943	$16.6166$	$-59.7210$	$16.6142$	$-59.7200$	$6$	$28$
SPT-CLJ0123-4821	$20.7980$	$-48.3559$	$20.7936$	$-48.3573$	$12$	$79$
SPT-CLJ0142-5032	$25.5422$	$-50.5400$	$25.5452$	$-50.5401$	$7$	$50$
SPT-CLJ0151-5954	$27.8457$	$-59.9079$	$27.8583$	$-59.9073$	$23$	$184$
SPT-CLJ0156-5541	$29.0437$	$-55.6984$	$29.0407$	$-55.6988$	$6$	$53$
SPT-CLJ0200-4852	$30.1455$	$-48.8711$	$30.1391$	$-48.8739$	$18$	$111$
SPT-CLJ0212-4657	$33.0998$	$-46.9540$	$33.1087$	$-46.9496$	$27$	$187$
SPT-CLJ0217-5245	$34.3047$	$-52.7632$	$34.2949$	$-52.7512$	$48$	$236$
SPT-CLJ0232-4421	$38.0778$	$-44.3466$	$38.0710$	$-44.3513$	$24$	$104$
SPT-CLJ0232-5257	$38.2039$	$-52.9529$	$38.1977$	$-52.9554$	$16$	$103$
SPT-CLJ0234-5831	$38.6745$	$-58.5235$	$38.6791$	$-58.5241$	$9$	$49$
SPT-CLJ0235-5121	$38.9356$	$-51.3519$	$38.9351$	$-51.3576$	$21$	$87$
SPT-CLJ0236-4938	$39.2595$	$-49.6365$	$39.2509$	$-49.6345$	$21$	$102$
SPT-CLJ0243-5930	$40.8620$	$-59.5172$	$40.8646$	$-59.5171$	$5$	$32$
SPT-CLJ0252-4824	$43.2074$	$-48.4163$	$43.1949$	$-48.4139$	$31$	$172$
SPT-CLJ0256-5617	$44.1054$	$-56.2983$	$44.1046$	$-56.2980$	$2$	$12$
SPT-CLJ0304-4401	$46.0700$	$-44.0253$	$46.0669$	$-44.0323$	$26$	$153$
SPT-CLJ0304-4921	$46.0675$	$-49.3569$	$46.0665$	$-49.3573$	$3$	$14$
SPT-CLJ0307-5042	$46.9604$	$-50.7021$	$46.9598$	$-50.7044$	$9$	$55$
SPT-CLJ0307-6225	$46.8160$	$-62.4474$	$46.8273$	$-62.4352$	$48$	$316$
SPT-CLJ0310-4647	$47.6352$	$-46.7855$	$47.6357$	$-46.7831$	$9$	$61$
SPT-CLJ0324-6236	$51.0529$	$-62.5982$	$51.0515$	$-62.5987$	$3$	$21$
SPT-CLJ0334-4659	$53.5460$	$-46.9958$	$53.5496$	$-46.9960$	$9$	$53$
SPT-CLJ0346-5439	$56.7328$	$-54.6484$	$56.7315$	$-54.6472$	$5$	$34$
SPT-CLJ0348-4515	$57.0739$	$-45.2477$	$57.0703$	$-45.2501$	$13$	$63$
SPT-CLJ0352-5647	$58.2406$	$-56.7959$	$58.2394$	$-56.7985$	$10$	$67$
SPT-CLJ0406-4805	$61.7311$	$-48.0819$	$61.7271$	$-48.0850$	$14$	$105$
SPT-CLJ0411-4819	$62.8183$	$-48.3153$	$62.8093$	$-48.3217$	$32$	$176$
SPT-CLJ0417-4748	$64.3463$	$-47.8135$	$64.3456$	$-47.8146$	$4$	$29$
SPT-CLJ0426-5455	$66.5226$	$-54.9217$	$66.5201$	$-54.9169$	$18$	$122$
SPT-CLJ0438-5419	$69.5725$	$-54.3226$	$69.5778$	$-54.3201$	$14$	$78$
SPT-CLJ0441-4855	$70.4489$	$-48.9236$	$70.4498$	$-48.9226$	$4$	$31$
SPT-CLJ0449-4901	$72.2765$	$-49.0267$	$72.2741$	$-49.0248$	$9$	$66$
SPT-CLJ0456-5116	$74.1147$	$-51.2789$	$74.1207$	$-51.2779$	$14$	$89$
SPT-CLJ0509-5342	$77.3388$	$-53.7037$	$77.3375$	$-53.7036$	$3$	$16$
SPT-CLJ0516-5430	$79.1572$	$-54.5134$	$79.1490$	$-54.5126$	$17$	$76$
SPT-CLJ0528-5300	$82.0217$	$-52.9969$	$82.0219$	$-52.9962$	$3$	$20$
SPT-CLJ0533-5005	$83.4068$	$-50.0971$	$83.4048$	$-50.0972$	$5$	$35$
SPT-CLJ0542-4100	$85.7090$	$-40.9987$	$85.7118$	$-41.0021$	$14$	$100$
SPT-CLJ0546-5345	$86.6552$	$-53.7593$	$86.6529$	$-53.7613$	$9$	$69$
SPT-CLJ0555-6406	$88.8578$	$-64.1070$	$88.8667$	$-64.1056$	$15$	$73$
SPT-CLJ0559-5249	$89.9282$	$-52.8317$	$89.9354$	$-52.8249$	$29$	$196$
SPT-CLJ0655-5234	$103.9714$	$-52.5695$	$103.9731$	$-52.5701$	$4$	$25$
SPT-CLJ0658-5556	$104.5829$	$-55.9418$	$104.6188$	$-55.9453$	$73$	$324$
SPT-CLJ2031-4037	$307.9696$	$-40.6227$	$307.9646$	$-40.6219$	$14$	$67$
SPT-CLJ2034-5936	$308.5378$	$-59.6052$	$308.5369$	$-59.6038$	$5$	$41$
SPT-CLJ2035-5251	$308.7974$	$-52.8556$	$308.7923$	$-52.8546$	$12$	$74$
SPT-CLJ2043-5035	$310.8238$	$-50.5923$	$310.8246$	$-50.5933$	$4$	$29$
SPT-CLJ2106-5844	$316.5224$	$-58.7422$	$316.5185$	$-58.7427$	$8$	$62$
SPT-CLJ2135-5726	$323.9093$	$-57.4411$	$323.9132$	$-57.4392$	$10$	$56$

Table 5. Table 7: Goodness of fits. Values show the difference in fit statistic from that expected on average for the model divided by the expected standard deviation.

Name	BIN-NFW	BIN-GNFW	BIN-NONHYDRO	INT-NFW	MBETA-NFW	KPLAW-NFW	GRAD-NFW
SPT-CLJ0000-5748	$-0.2$	$-0.2$	$-0.2$	$+0.4$	$+0.5$	$+0.6$	$+0.5$
SPT-CLJ0013-4906	$-0.9$	$-1.0$	$-1.2$	$+0.0$	$+0.5$	$+1.4$	$+0.1$
SPT-CLJ0014-4952	$+1.3$	$+1.2$	$+0.7$	$+1.1$	$+1.6$	$+1.6$	$+1.3$
SPT-CLJ0033-6326	$-1.9$	$-1.9$	$-1.9$	$-0.6$	$-0.5$	$-0.5$	$-0.5$
SPT-CLJ0040-4407	$-2.5$	$-2.5$	$-2.5$	$-1.8$	$-1.7$	$-1.7$	$-1.7$
SPT-CLJ0058-6145	$+1.9$	$+1.9$	$+1.5$	$+0.5$	$+0.5$	$+1.2$	$+0.5$
SPT-CLJ0102-4603	$-0.8$	$-0.9$	$-1.0$	$+1.2$	$+1.2$	$+1.2$	$+1.2$
SPT-CLJ0102-4915	$+0.5$	$+0.5$	$-0.1$	$+1.1$	$+12.9$	$+5.8$	$+5.9$
SPT-CLJ0106-5943	$-1.3$	$-1.3$	$-1.3$	$-0.4$	$+0.3$	$+0.4$	$-0.1$
SPT-CLJ0123-4821	$+0.8$	$+0.8$	$+1.0$	$+1.8$	$+1.9$	$+1.9$	$+1.8$
SPT-CLJ0142-5032	$-1.0$	$-1.0$	$-1.0$	$-0.4$	$-0.2$	$-0.3$	$-0.3$
SPT-CLJ0151-5954	$+1.5$	$+1.5$	$+1.5$	$+0.4$	$+0.4$	$+0.4$	$+0.3$
SPT-CLJ0156-5541	$+1.0$	$+1.0$	$+1.0$	$+2.5$	$+2.6$	$+2.5$	$+2.6$
SPT-CLJ0200-4852	$+1.3$	$+1.3$	$+1.3$	$+2.1$	$+2.3$	$+2.2$	$+2.2$
SPT-CLJ0212-4657	$-1.6$	$-1.6$	$-1.8$	$-1.6$	$-1.5$	$-1.2$	$-1.6$
SPT-CLJ0217-5245	$-0.4$	$-0.4$	$-0.5$	$-0.6$	$-0.5$	$-0.5$	$-0.5$
SPT-CLJ0232-4421	$-1.4$	$-1.6$	$-1.9$	$+0.3$	$+0.8$	$+1.3$	$+1.1$
SPT-CLJ0232-5257	$-0.4$	$-0.4$	$-0.4$	$-0.6$	$-0.4$	$-0.4$	$-0.6$
SPT-CLJ0234-5831	$-1.0$	$-1.0$	$-1.1$	$-0.7$	$-0.2$	$-0.2$	$-0.3$
SPT-CLJ0235-5121	$+1.3$	$+1.2$	$+1.4$	$+0.8$	$+1.0$	$+1.2$	$+1.0$
SPT-CLJ0236-4938	$-0.4$	$-0.5$	$-0.6$	$-1.6$	$-0.8$	$-0.8$	$-1.2$
SPT-CLJ0243-5930	$-1.1$	$-1.1$	$-1.1$	$+2.1$	$+2.3$	$+2.4$	$+2.4$
SPT-CLJ0252-4824	$-1.3$	$-1.3$	$-1.3$	$+0.7$	$+0.8$	$+0.9$	$+0.7$
SPT-CLJ0256-5617	$-2.1$	$-2.1$	$-2.0$	$-1.0$	$+0.3$	$+0.4$	$-0.7$
SPT-CLJ0304-4401	$+0.1$	$+0.1$	$+0.1$	$+1.6$	$+2.1$	$+2.8$	$+2.3$
SPT-CLJ0304-4921	$-0.2$	$-0.3$	$-0.4$	$+0.5$	$+1.0$	$+0.9$	$+0.8$
SPT-CLJ0307-5042	$-1.0$	$-1.1$	$-1.0$	$-0.1$	$+0.1$	$+0.1$	$+0.1$
SPT-CLJ0307-6225	$-0.5$	$-0.7$	$-0.5$	$-0.6$	$-0.0$	$-0.1$	$-0.6$
SPT-CLJ0310-4647	$-0.4$	$-0.4$	$-0.5$	$+0.6$	$+0.5$	$+0.6$	$+0.5$
SPT-CLJ0324-6236	$+1.2$	$+1.2$	$+1.2$	$+0.9$	$+1.0$	$+1.1$	$+0.9$
SPT-CLJ0334-4659	$-1.6$	$-1.7$	$-1.6$	$-0.2$	$+0.6$	$+0.6$	$+0.5$
SPT-CLJ0346-5439	$-0.8$	$-0.8$	$-0.8$	$-0.8$	$-0.7$	$-0.7$	$-0.7$
SPT-CLJ0348-4515	$-0.1$	$-0.2$	$-0.2$	$-0.4$	$-0.4$	$-0.3$	$-0.4$

Table 6. Table 9: Table detailing individual profiles, provided electronically. The profiles for each cluster are repeated for each model (BIN-NFW, BIN-GNFW, BIN-NONHYDRO, INT-NFW, MBETA-NFW and KPLAW-NFW). r in subscript 𝑟 in r_{\mathrm{in}} and r out subscript 𝑟 out r_{\mathrm{out}} are the inner and outer radii (kpc) of the annulus on the sky. The median, 84.1 percentile (suffix hi) and 15.9 percentile (suffix lo) values for each physical quantity are given in the table. The quantities and their units are, in order, temperature ( k ​ T 𝑘 𝑇 kT ; keV), electron density ( n e subscript 𝑛 e n_{\mathrm{e}} ; cm − 3 superscript cm 3 \rm\thinspace cm^{-3}\, ), electron entropy ( K e subscript 𝐾 e K_{\mathrm{e}} ; keV ​ cm 2 keV superscript cm 2 \rm\thinspace keV\rm\thinspace cm^{2} ), electron pressure ( P e subscript 𝑃 e P_{\mathrm{e}} ; keV ​ cm − 3 keV superscript cm 3 \rm\thinspace keV\hbox{$\rm\thinspace cm^{-3}\,$} ), gravitational acceleration ( g 𝑔 g ; cm ​ s − 2 cm superscript s 2 \rm\thinspace cm\rm\thinspace s^{-2} ; not valid for BIN-NONHYDRO), mean radiative cooling time ( t cool subscript 𝑡 cool t_{\mathrm{cool}} ; yr), mass deposition rate ( M ˙ ˙ 𝑀 \dot{M} ; M ⊙ ​ yr − 1 subscript M direct-product superscript yr 1 \hbox{$\rm\thinspace M_{\odot}$}\rm\thinspace yr^{-1}\, ), total cumulative mass ( M tot subscript 𝑀 tot M_{\mathrm{tot}} ; M ⊙ subscript M direct-product \rm\thinspace M_{\odot} ; not valid for BIN-NONHYDRO) and cumulative gas mass ( M gas subscript 𝑀 gas M_{\mathrm{gas}} ; M ⊙ subscript M direct-product \rm\thinspace M_{\odot} ; not valid for BIN-NONHYDRO).

Name	Model	$r_{\mathrm{in}}$	$r_{\mathrm{out}}$	$kT$	$k{T}_{\mathrm{hi}}$	$k{T}_{\mathrm{lo}}$	$n_e$	$n_{e,\mathrm{hi}}$	$n_{e,\mathrm{lo}}$	$K_e$	$K_{e,\mathrm{hi}}$	$K_{e,\mathrm{lo}}$	…
SPT-CLJ0000-5748	BIN-NFW	0.0	10.6	2.47	2.99	2.06	0.181	0.203	0.158	8	10	6	…
SPT-CLJ0000-5748	BIN-NFW	10.6	21.1	4.07	4.86	3.43	0.075	0.084	0.066	23	29	18	…
SPT-CLJ0000-5748	BIN-NFW	21.1	35.2	5.40	6.31	4.67	0.040	0.045	0.036	46	58	38	…
SPT-CLJ0000-5748	BIN-NFW	35.2	52.8	5.31	6.07	4.70	0.028	0.031	0.026	57	69	48	…
SPT-CLJ0000-5748	BIN-NFW	52.8	73.9	6.93	8.08	6.06	0.015	0.017	0.013	112	142	92	…
…	…	…	.. .	…	…	…	…	…	…	…	…	…	…

Equations

$$\rho (r)=\frac{{\delta }_{\mathrm{c}}{\rho }_{\mathrm{c}}}{(r/r_{\mathrm{s}})(1+r/r_{\mathrm{s}}{)}^2},$$

$${\delta }_{\mathrm{c}}=\frac{200}{3}\frac{c^3}{\log (1+c)-c/(1+c)}.$$

$$\rho (r)=\frac{{\rho }_0}{(1/r_{\mathrm{s}}{)}^{\alpha }(1+r/r_{\mathrm{s}}{)}^{3-\alpha }},$$

$$n_{\mathrm{e}}^2=n_0^2\frac{(r/r_{\mathrm{c}}{)}^{-\alpha }}{(1+r^2/r_{\mathrm{c}}^2{)}^{3\beta -\alpha /2}}\frac{1}{(1+r^{\gamma }/r_{\mathrm{s}}^{\gamma }{)}^{ϵ/\gamma }},$$

$$K(r)=K_0+K_{300}{\left(\frac{r}{300\mathrm{kpc}}\right)}^{{\alpha }_k(r)},$$

$$K(r)=K_0+K_{100}{\left(\frac{r}{100\mathrm{kpc}}\right)}^{\alpha },$$

$$r_{\mathrm{mass}}=\frac{3}{4}\frac{r_{\mathrm{out}}^4-r_{\mathrm{in}}^4}{r_{\mathrm{out}}^3-r_{\mathrm{in}}^3},$$

$$g_{\mathrm{gas}}=G\frac{3M_{\mathrm{gas}}(r<r_{\mathrm{in}})+\rho (r_{\mathrm{out}}-r_{\mathrm{in}})\left[(r_{\mathrm{out}}+r_{\mathrm{in}}{)}^2+2r_{\mathrm{in}}^2\right]}{r_{\mathrm{in}}^2+r_{\mathrm{in}}{r}_{\mathrm{out}}+r_{\mathrm{out}}^2},$$

$$g_i=\frac{G{M}_{DM}(r<r_{\mathrm{mass},i})}{r_{\mathrm{mass},i}}+g_{\mathrm{gas},i}$$

$$P_i=(r_{\mathrm{out},i}-r_{\mathrm{mass},i}){\rho }_i{g}_i+\sum _{j=i+1}^N(r_{\mathrm{out},j}-r_{\mathrm{in},j}){\rho }_j{g}_j+P_{\mathrm{out}}$$

Full text

1 Introduction

Within the dark-matter-dominated potential well of galaxy clusters lies the intracluster medium (ICM), a hot atmosphere primarily seen by its emission in the X-ray waveband. By examining the morphology and temperature of the ICM and assuming that it is in hydrostatic equilibrium, the mass, profile and shape of the underlying dark-matter halo can be inferred (e.g. Allen et al., 2001). The ICM is also sensitive to baryonic physics, such as the input of energy by AGN in galactic nuclei within clusters (e.g. Böhringer et al., 1993).

In many nearby clusters the central cooling times of the ICM are short. In the absence of heating a cooling flow would develop (Fabian, 1994), where material would rapidly cool out of the X-ray waveband at rates of 10-1000\hbox{\hbox{$\rm\thinspace M_{\odot}$}\rm\thinspace yr^{-1},}. Such high rates of cooling are not observed (e.g. Peterson & Fabian, 2006) and so there must be a mechanism by which the rapid cooling is prevented. AGN in cluster cores are observed to put energy into their surroundings by the inflation of bubbles of radio-emitting plasma, seen as cavities in X-ray images of the ICM (e.g. McNamara & Nulsen, 2007). The balance between the energy lost by X-ray emission and the cavity heating rates estimated from observations imply AGN feedback is the mechanism for how cooling flows are prevented (e.g. Fabian, 2012). An important question is how AGN maintain the close heating-cooling balance in nearby clusters and whether this balance is maintained in earlier epochs. This is not only of relevance to the cores of galaxy clusters and their central galaxies, but has widespread relevance to understanding galaxy formation.

One of the strongest indicators of non-gravitational heating in clusters is the entropy (Voit et al., 2002). The specific value can be written as $K_{\mathrm{e}}=kT\>n_{\mathrm{e}}^{-2/3}$, if $n_{\mathrm{e}}$ is the electron density and $T$ is the temperature. In the absence of conduction, convection ensures that low-entropy material moves to the centre, while high-entropy material goes to the outskirts. Non-gravitational processes show as deviations from the entropy distribution expected in pure-gravitational distributions.

Previous studies have found different behaviour of the entropy in cluster cores. Several groups have found evidence for entropy flattening in the cores of clusters and groups (e.g. David et al., 1996; Cavagnolo et al., 2009; McDonald et al., 2013). In a volume-limited sample of local clusters (Panagoulia et al., 2014) instead found that the entropy profiles were consistent with being powerlaws in the central regions. The measurement of central cluster properties is difficult because of projection effects, substructure, multiphase material and metallicity gradients. Hogan et al. (2017) also found no evidence for a floor in a small sample of clusters with deep X-ray observations. Resolving these disagreements over the central entropy is important for understanding the heating and cooling processes taking place in the centres of these objects.

The X-ray emission observed from galaxy clusters is projected along the line of sight. To extract the three-dimensional information some assumptions of the geometry have to be made, such as spherical symmetry. Various methods have been previously used to extract the three-dimensional thermodynamical properties, including deprojection of the X-ray surface-brightness profile (Fabian et al., 1981), projecting a spectral model in shells to fit projected spectra, as implemented as the PROJCT model in xspec (Arnaud, 1996), correcting projected quantities (Ettori et al., 2002), deprojection of the X-ray spectra (Sanders & Fabian, 2007; Russell et al., 2008), forward-fitting of a mass and temperature model to spectra extracted from shells (Mahdavi et al., 2008; Nulsen et al., 2010) and fitting a model to the X-ray event dataset (Olamaie et al., 2015).

In Sanders et al. (2014) we introduced a new forward-fitting code, mbproj, which fits surface-brightness profiles in multiple energy bands. It fits a model density profile and either a mass (assuming hydrostatic equilibrium) or a temperature model. An MCMC (Markov Chain Monte Carlo) analysis is used to generate profiles of physical quantities. The advantage a surface-brightness profile modelling code has over spectral fitting is that it is easier to visually inspect how the goodness of the fit changes as a function of radius and to check that the background modelling is correct. It is also easier to adapt the size of the radial bins as the modelling requires and to connect the obtained profiles to images of the cluster.

Selection of galaxy clusters using the Sunyaev-Zel’dovich (SZ; Sunyaev & Zeldovich 1972) effect has some advantages compared to other methods, including a uniform mass selection to higher redshifts and not being sensitive to cool cores (e.g. Birkinshaw, 1999). Nevertheless, it may be the case that there could be a large population of contaminating AGN at higher redshifts which could affect the SZ signal or selection (e.g. Bufanda et al., 2017).

In this paper we analyse Chandra observations of an SZ-selected sample of galaxy clusters obtained by the SPT telescope (Carlstrom et al., 2011) with the aid of a new version of the multi-band X-ray projection algorithm, mbproj2. In this paper we focus on the thermodynamic profiles from our hydrostatic analysis, to examine the evolution of the profiles and core properties, while we leave the comparison of our obtained hydrostatic masses with other mass measurements to a future work.

This analysis differs in several respects from the analyses of almost-same samples by McDonald et al. (2013) [hereafter referred to as MD13] and McDonald et al. (2014) [hereafter MD14]. Firstly, we did a self-consistent modelling of the X-ray profiles to obtain the profiles of physical quantities, using both parametric and binned density profiles. In MD13, the surface-brightness profiles were fit by a parametric model and the projected temperature profiles obtained by fitting spectra in wide spatial bins. A hydrostatic model was fit to both to obtain deprojected quantities. Our modelling is more sensitive to variations in temperature as we do not use these wide spectral bins. In addition, the published density profiles of MD13 can have central densities a factor of a few away from updated values due to an error in the fitting procedure (see Section 3.2).

In MD14 a joint analysis was done to all the clusters in different subsamples to obtain average physical profiles. In their analysis they assumed that the clusters in a subsample shared a common modelled temperature profile. The projected spectra in radial bins for the clusters were fit jointly with this temperature model, allowing the normalisation profiles to be different. To deproject the temperature profile a parametric model was fitted to the previous projected profile. The deprojected density profiles were taken from McDonald et al. (2013), but the mean density profile of a subsample was computed by weighting the individual density profiles by the number of counts in each radial bin. In comparison, our modelling assumed hydrostatic equilibrium but did not assume the same temperature profile for each system. We also examine some median cluster quantities, which are independent on the relative data quality of different clusters.

Our analysis also differs from MD13 and MD14 by the choice of cluster centres. We use the X-ray peak defined using a small (50 count) aperture as a cluster centre, whereas they use the centroid of a 250-500 kpc annulus for the main part of their analysis. When they compare to results using the X-ray peak, the aperture used to find the peak position is substantially larger than ours. This difference cluster centre is important for the differences in results we obtain in the cores of these systems from MD13 and MD14.

We assume a cosmology where H_{0}=70\hbox{\hbox{$\rm\thinspace km\rm\thinspace s^{-1}\,$}\rm\thinspace Mpc^{-1}}, $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.

2 Data analysis

Our sample of systems (listed in Appendix A.1) includes the sample of SPT clusters of Bleem et al. (2015) marked as having X-ray data, excluding SPT-CLJ0037-5047 which has low signal to noise. We also include two further systems in that paper which also have X-ray data and were examined in MD13, SPT-CLJ0236-4938 and SPT-CLJ0310-4647. We do not include SPT-CLJ0446-5849 which has low signal to noise. We also exclude SPT-CLJ0330-5228 and SPT-CLJ0551-5709 which are contaminated (MD14). As MD13 describes, their sample consists of strongly-detected clusters by SPT, with SPT detection significances between 5.7 and 43. The mass range of $M_{500}$ is between $2\times 10^{14}$ and 2\times 10^{15}\hbox{\rm\thinspace M_{\odot}}, while the redshift range is between 0.3 and 1.2. At the median redshift, the sample should be around 50 per cent complete at $M_{500}=4\times 10^{14}$, increasing to 100 per cent at 6\times 10^{14}\hbox{\rm\thinspace M_{\odot}}. We make use of any new public observations from the Chandra archive, where possible.

As detailed below, the cluster surface-brightness profiles are fit in multiple X-ray bands using the mbproj2 code (described in Appendix B), which can compute profiles assuming hydrostatic equilibrium or in its absence. A surface-brightness profile in a single band would only be sensitive to variations in gas density. However, by using profiles from multiple bands simultaneously, we are sensitive to temperature variations due to the change in the spectral shape. With low numbers of bands, this is similar to using X-ray colours to measure temperature (e.g. Allen & Fabian, 1997). As the number of bands increases this becomes equivalent to spectral fitting, which would also allow the metallicity to be fit given sufficient data quality.

When using fine radial bins it is difficult to obtain the gas temperature due to the lack of counts. However, by introducing the assumption of hydrostatic equilibrium with some underlying dark-matter potential, we can compute the pressure, and given the densities, the temperature. mbproj2 computes the projected surface-brightness profiles in multiple bands for a given gas-density profile and dark-matter profile. The uncertainties on the fits are explored using the observed profiles and MCMC.

2.1 Initial data preparation

We downloaded the data for each cluster from the Chandra archive. The datasets were reprocessed using ciao (Fruscione et al., 2006) acis_process_events, applying very-faint event grading where possible. We excluded bad time periods by iteratively $\sigma$-clipping the lightcurve to remove periods $2.5\sigma$ away from the median value, where $\sigma$ is the Poisson error on the median number of counts in a 200s time bin. For observations using ACIS-S the lightcurve was constructed in the 2.5 to 7 keV band, otherwise the 0.5 to 12 keV band was used. For each cluster, we reprojected observations to a common coordinate system.

Standard blank-sky background datasets were obtained for each CCD of each observation. We used the bad-pixel table from an observation to remove bad pixels from the respective background observations. For a particular observation, the exposure times of the background event files for each CCDs were adjusted so that each had the same 9 to 12 keV band count rate as the respective cluster data (this spectral range is dominated by the particle background). These background event files for an observation were then adjusted to have the same exposure time as the lowest exposure file by randomly discarding events. When multiple datasets were used for a cluster, we similarly adjusted the exposure times of the background datasets to have the same ratio of their exposure to the total background exposure, as their respective cluster observation to the total cluster exposure. The background event files for each cluster were reprojected to match the coordinate system of the respective cluster observation and then the common coordinate system for the cluster.

Total images were created using detector pixel binning in the 0.5 to 7 keV band. We detected point sources in these images using wavdetect, using scales of 2, 2.828, 4, 5.657, 8, 11.314 and 16 pixels, with a maximum of 5 iterations. The resulting point sources were manually verified, removing obvious false detections and adding missed sources, as appropriate. Some of the observations were contaminated by other extended sources and structures. We identified these in smoothed images and excluded them in our analysis. In a couple of cases the systems were too close to be separated and so they remain in the analysis.

2.2 Surface-brightness profiles

To identify the centre of each cluster, we initially found the brightest pixel in an adaptively-smoothed map, smoothed to have a minimum signal to noise of 10 in a top hat kernel. This position was then refined iteratively by repeatedly finding the centroid of a circle with a radius chosen to contain 50 counts (although its minimum radius was four 0.492 arcsec pixels). These peak positions are given in Appendix A.2. Also shown in this table for each object is a second position computed from the centroid of an annulus between radii of 250 and 500 kpc, which is the same technique as used by MD13 to define their cluster centres. As the centroid of this annulus does not always converge to a single point, we used the mean position of 100 iterations, after discarding an initial 100 iterations. The offset between the two positions in arcsec and kpc on the sky is also shown in the table. Although we used the same technique as MD13 for the annulus centroids, some of our positions show large differences from MD13. These include the positions for SPT-CLJ0217-5245 and SPT-CLJ0252-4824, which show differences by 47 and 40 arcsec, respectively. It is unclear why the positions differ given we use the same method.

We extracted total cluster and background images in ten energy bands between neighbouring energies of 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4, 5, 6 and 7 keV. These bands were chosen to capture most of the spectral information without overly increasing the processing time and storage used. We also created total exposure maps assuming monochromatic energies in the centre of each band. Radial profiles of cluster counts, background counts, average exposure and sky area were extracted around the cluster centre. These profiles were created with single-pixel (0.492 arcsec) radial binning, masking out excluded regions and not splitting pixels between bins. The profiles were truncated at radii where the cluster was not distinguishable from the background, found by manually examining the adaptively-smoothed maps and profiles (listed in in Appendix A.1). We did this truncation to improve the robustness of the radial binning procedure and to greatly reduce the time to model the cluster profiles. We manually chose a maximum radius for each cluster as automated procedures were insufficiently robust.

2.3 Radial binning

For the standard binned analysis (Section 2.5.1), we binned the projected profiles to have root-mean-square uncertainties on the inferred deprojected emissivities below a threshold (typically 20 per cent – see Appendix A.1), which we refer to as ‘wide binning’. This was done by optimizing the edges of the annuli to minimize the total-squared uncertainty on the emissivities. To compute the uncertainties on the deprojected emissivities we propagated the covariance matrix of the uncertainties in the surface brightness in the projected bins. This optimization procedure works well in most cases, although it can occasionally produce bins which do not meet the requirement.

We also created surface-brightness profiles used for the parametric fits, referred to as ‘fine binning’. In this case we split each of the wide-binned annuli into five annuli with approximately-equal radial size, rounding to integer pixel radii and splitting into fewer bins if it was not possible to split into five. For SPT-CLJ0658-5556 and SPT-CLJ0102-4915 we split by three instead to reduce the number of bins and increase the analysis speed. mbproj2 does not require the input profiles to be binned for parametric fits, but this substantially decreases the computing time required to analyse the profiles.

2.4 Background modelling

As our background model for each of the bands, we used background surface-brightness profiles extracted from the background event files using the same binning as the cluster observations. For each band these background profiles were rescaled to match the surface brightness of the cluster observation at large radius beyond the cluster emission, unless the cluster emission fills the entire field of view. The scaling was to account for cluster-to-cluster variation in the astrophysical background from the blank-sky backgrounds. At low energies, the scaling accounts for variation in Galactic and extragalactic emission, while at high energies the scaling accounts for particle background changes. In the softest bands the clusters are scaled by factors with a standard deviation of around 10 percent, which declines to 4 per cent at the highest energies. The softest 0.5 to 0.75 keV band is scaled up on average by around 6 per cent relative to the standard blank-sky data, while the other bands are consistent with no scaling on average.

The background was treated as an additional component added to the total cluster model. The exposure times of the background datasets are typically 10 times greater than the cluster observations (although in the case of SPT-CLJ0102-4915 this ratio decreases to 4). We can therefore ignore the Poisson uncertainty on the background as the statistical uncertainty on the surface brightness is dominated by the cluster emission, providing there are sufficient counts per radial bin. For the wide-binned profiles, the total cluster signal is on average 9 per cent above the background in the outermost bins. This decreases to around 2 per cent for the finely-binned profiles. The median number of total counts in the outermost bins of the wide-binned profiles is around 2400 in the data and 39000 in the background.

Although we match the backgrounds to the observed profiles for those clusters which do not fill the field of view, there may be additional unresolved substructures and point sources in the source. Larger scale fluctuations will be removed by the background scaling, but smaller scale features can remain. We quantified this by measuring the fluctuations in the cluster surface-brightness profiles in the radial range used for background matching, relative to the background model, assuming Gaussian fluctuations and taking account of the Poisson noise. We examined the variation on scales of $16$ arcsec between 0.5 and 7 keV. The typical variation is 3 per cent, with 90 per cent lying between 1 and 6 percent and a tail up to 10 per cent. We therefore conservatively add an additional free parameter allowing scaling of the background profiles using a Gaussian prior with $\sigma=10$ per cent.

2.5 Profile modelling

The profiles were fit using the mbproj2 multiband projection code (described in Appendix B). Given the predicted model (including background) and observed profiles for a cluster a total Poisson likelihood can be computed. An affine-invariant MCMC sampler (Goodman & Weare, 2010) as implemented in emcee (Foreman-Mackey et al., 2013) was used to sample the parameters for the model, starting from around the maximum-likelihood position. This sampler has the advantage of not requiring a proposal distribution and being affine-invariant, handles well covariance between the parameters. In our analyses we used a chain length of 2000 steps, a burn-in period of 1000 steps and 800 walkers. Profiles of various physical properties were then calculated by iterating through the resulting chain in jumps of 10 steps and computing the profile given the model parameters for each position. We then obtained the median profiles and uncertainties enclosing 68.3 percent of the produced profiles.

The metallicity was assumed to be 0.3\hbox{\thinspace\mathrm{Z}_{\odot}} using the solar relative abundances of Anders & Grevesse (1989). Equivalent hydrogen absorbing column densities were obtained from the LAB survey (Kalberla et al., 2005) and redshifts from Bleem et al. (2015), both given in Table 3.

Our main results come from fitting an NFW mass model to the binned profiles (model BIN-NFW). In some cases we use a different modelling to check sensitivity to the assumed model, to look at fixed radius or when modelling the entropy profiles. The models are detailed below and listed in Table 1. In Appendix A.3 we calculate the goodness of the fits for a subset of the models.

2.5.1 Binned NFW fits (model BIN-NFW)

Our most simple analysis is to use widely-binned surface-brightness profiles, chosen to have constant fractional uncertainties in derived emissivities (Section 2.2). One logarithmic parameter with a flat prior was used to parametrize the electron-density value in each radial bin. The dark matter was modelled with an NFW profile (Navarro et al., 1996), using parameters which are the logarithms of $R_{200,\mathrm{DM}}$ (the radius of an average overdensity of dark matter of 200 times the critical value) and $c$ (the concentration). The density of dark matter at a radius $r$ is given by

[TABLE]

where the scale radius $r_{\mathrm{s}}=R_{200,\mathrm{DM}}/c$, the critical density of the universe at redshift $z$ is $\rho_{\mathrm{c}}=(3H^{2}(z))/(8\pi G)$, $H(z)$ is the Hubble constant at the redshift and $G$ is the gravitational constant. The characteristic overdensity of the halo is

[TABLE]

Note that $R_{200,\mathrm{DM}}$ is not the usual $R_{200}$ for a galaxy cluster, as it is the radius where dark matter has an average overdensity of 200, rather than the total matter in the cluster having this overdensity.

As the data quality are often limited in the outskirts of the profiles, $R_{200,\mathrm{DM}}$ and $c$ can be strongly correlated parameters. We therefore used a flat prior of $0\leq\log_{10}c\leq 1$, to give a range similar to that found in high-mass systems in simulations (e.g. Duffy et al., 2008) and observed using gravitational lensing (e.g. Merten et al., 2015). A flat prior on the log cluster radius $-1\leq(\log_{10}R_{200,\mathrm{DM}}/\mathrm{Mpc})\leq 1$ was used. We used wide, flat logarithmic priors on the outer pressure of the cluster.

2.5.2 Binned GNFW fits (model BIN-GNFW)

To check the sensitivity of our results on the assumed form of the dark-matter profile, we used the generalized NFW (GNFW) profile (Zhao, 1996; Wyithe et al., 2001), applying it to the widely-binned surface-brightness profiles. The GNFW profile has a parametrized inner density slope, $\alpha$, which if $\alpha=1$ is the NFW profile. The mass density follows the form

[TABLE]

where $\rho_{0}$ and $r_{\mathrm{s}}$ are the central density and scale radius, respectively, calculated from $c$ and $r_{200}$. In the analysis $\alpha$ was allowed to vary between 0 and 2.5, assuming a flat prior. $c$ and $r_{200}$ were allowed to vary in the same ranges as for the NFW fits. We do not quantitatively examine the results of this model, but plot the resulting profiles.

2.5.3 Binned non-hydrostatic fits (model BIN-NONHYDRO)

To check the assumption of hydrostatic equilibrium we also fitted a non-hydrostatic model. Like BIN-NFW, the density was modelled by a logarithmic density parameter in each radial bin. The temperature was parametrized in every third bin, assuming a flat log prior between 0.1 and 50 keV. Values in intermediate bins were calculated using interpolation of log temperature in the radial bin index.

2.5.4 Interpolated NFW profiles (model INT-NFW)

To examine the effect of binning, we also fitted the finely-binned surface-brightness profiles (Section 2.2) with an interpolated density model. The model parametrizes the density at the particular radii with the density at intermediate radii calculated by log interpolation in log radius. We set the parametrized radii to be the centres of each of the wide bins, giving the same number of free parameters as BIN-NFW. In these fits we assume hydrostatic equilibrium using the NFW model, using the same flat priors as BIN-NFW.

2.5.5 Modified-$\beta$ model (model MBETA-NFW)

The MD13 paper assumes a parametric model for the gas density, given by

[TABLE]

which we refer to as a modified-$\beta$ profile. This form is based on the profile described by Vikhlinin et al. (2006), not including their second $\beta$ component. We set $\gamma$ to be 3 (following MD13) and apply the following flat priors: $0\leq\beta\leq 4$, $-1\leq\log_{10}(r_{\mathrm{c}}/\mathrm{kpc})\leq 3.7$, $1\leq\log_{10}(r_{\mathrm{s}}/\mathrm{kpc})\leq 3.7$, $0\leq\epsilon\leq 5$ and $0\leq\alpha\leq 4$. We note that $r_{\mathrm{s}}$ and $\alpha$ are not the same parameters as used in the GNFW or NFW models. The model was fit to the data assuming hydrostatic equilibrium (with the same NFW dark-matter profile and priors as BIN-NFW).

We note that this functional form is not capable of fitting all possible X-ray surface-brightness profiles. For example, we tested fitting the profiles of Abell 1795, a relatively nearby cluster with cool low-entropy gas in its core. The model was unable to fit the steep central X-ray peak inside 10 kpc radius. The parameters of parametric models are driven by the brightest radial regions and may not produce good results outside these regions.

Forcing the parameter $\alpha$ to be positive (following MD13) ensures the model central profiles are either flat or inwards-rising. We examine the effect of allowing $\alpha$ to be negative in Section 3.2.

2.5.6 Stepped-density models (model STEP-NFW)

To check how well we can measure the central density, we fitted a model where the density is constant within annuli with edges of fixed radii of 20, 40, 80, 160, 320, 640 and 1280 kpc. We assume hydrostatic equilibrium with an NFW dark-matter mass component and the usual priors.

2.5.7 Gradient model (model GRAD-NFW)

To investigate the density gradient (log density in log radius), we parametrized it at radii of $<50$, $50$–$100$, $100$–$200$, $200$–$400$, $400$–$800$ and $>800$ kpc. We used flat priors on the gradients, to better be able to examine their distributions. The model was normalised by the density at $200$ kpc. The surface brightness profiles were fitted with fine binning assuming hydrostatic equilibrium with an NFW dark-matter mass component and the priors used previously.

2.5.8 Powerlaw entropy (model KPLAW-NFW)

The previous models parametrize the density profile of the cluster. mbproj2 also allows a parametrization of the entropy profile of the cluster (Appendix B). To examine whether there is an entropy floor we assumed the form

[TABLE]

where $\alpha_{K}(r)$ is $\alpha_{K,\mathrm{inner}}$ at $r<300\rm\thinspace kpc$ and $\alpha_{K,\mathrm{outer}}$ otherwise. We assumed a flat prior on $K_{0}$ between $10^{-5}$ and $1000\rm\thinspace keV\rm\thinspace cm^{2}$. $K_{300}$ was given a flat prior in log space between 1 and $10^{4}\rm\thinspace keV\rm\thinspace cm^{2}$. We gave $\alpha_{K,\mathrm{inner}}$ and $\alpha_{K,\mathrm{outer}}$ Jeffreys priors between values of 0 and 4. A Jeffreys prior is an uninformative prior invariant under monotonic transformations. In the case of a gradient, gradient values which increase linearly give profiles which are ever more closely separated. A flat prior would therefore be weighted towards steep slopes, as every value is assumed equally likely. The Jeffreys prior removes this bias towards steep slopes.

We used the NFW gravitational potential with the same priors as the BIN-NFW model. The entropy model was fitted to the finely-binned surface-brightness profiles to better-resolve the core region. As described in Appendix B, under the entropy parametrization the model surface-brightness profiles were predicted from density and temperature profiles which were themselves calculated from the entropy and gravitational profiles assuming hydrostatic equilibrium. Densities and temperatures were calculated at the centres of each radial bin.

2.5.9 Central galaxy

The mass models we fitted in this paper do not include a special mass component for the central galaxy, which could have some effect on the central properties of the cluster. Whether this is important depends on how well resolved the centre is and whether the centre used actually lies on a central galaxy (or whether the cluster is relaxed). To check this, we fitted NFW models adding a point source to the potential to account for the central galaxy. The typical effect of this was to reduce temperatures, entropies and pressures by around 5 to 10 per cent. In most clusters there is no significant point source mass component, but in 10 clusters the inclusion of a central component led to largely unconstrained central temperatures and unphysical central masses. Most of these systems were disturbed and unlikely to have a real central galaxy and so this component appears unphysical.

If we compare the BIN-GNFW fits, where the central slope of the mass profile is a free parameter, with BIN-NFW, the inner temperature is increased by $5\pm 16$ per cent and the density by $0\pm 1$ per cent ($1\sigma$ percentile ranges). This also suggests that the bias caused by the exclusion of the central galaxy is small. However, forcing the addition of a King mass model with a velocity dispersion of 300\hbox{\rm\thinspace km\rm\thinspace s^{-1},} and a core radius of $10$ kpc into the BIN-NFW model increases inner temperatures by $12_{-5}^{+10}$ per cent and densities by $-1_{-2}^{+1}$ per cent ($1\sigma$ percentile ranges). Zappacosta et al. (2006) note pure-NFW profiles are good fits to relaxed cluster mass profiles, in particular Abell 2589 and Abell 2029 (Lewis et al., 2003); in Abell 2589 a central galaxy component degrades the fit (although this is based on one data point). This result suggests that a central galaxy component should not be forced into the mass model. In the remainder of this paper we do not include a central galaxy mass component.

2.5.10 Effect of priors

In the hydrostatic analyses, there are multiple priors assumed on the fitting parameters: the density/entropy profile priors, the mass model priors, the outer pressure prior and the background rescaling factor prior. These model parameters except density and entropy are not interesting for the purposes of this paper. The priors on these parameters are folded implicitly into the derived thermodynamic profiles. We found that the effects of the priors on the derived profiles are small.

For most of the parameters we have assumed wide, physically non-informative priors. In the binned and interpolated analysis, BIN-NFW and INT-NFW, we do not assume a strong parametric form for the density. We assume a flat logarithmic prior on the density parameters, which are very well constrained by the data and so any prior is unimportant. Likewise, we assume a flat logarithmic prior on the outer pressure, which does not influence the results significantly.

In the mass model, we use a flat logarithmic prior on the concentration between values of $c=1$ and $10$, which is a large range given existing simulations and datasets. The main effect of this prior is to constrain the dark matter mass at large radius, which is not examined in this paper. We tested increasing the range to $c=0.5-20$, finding there was no systematic change in average temperature, density, entropy or pressure values, while the uncertainties increased by around 6 per cent for temperature, entropy and pressure. The effect on the profiles was typically much smaller than the size of the error bars given. The prior on the background scaling factor affects whether the density is well constrained in around 14 systems in the outermost bins of the profiles. Inside the outermost bin the profiles are unaffected if the allowed range is increased or decreased by a factor of 2.

In our analysis we assume a fixed metallicity of 0.3\hbox{\thinspace\mathrm{Z}{\odot}}. As the clusters are relatively hot and massive, the effect of this assumption is weak. For example, if we vary the metallicity assumed to $0.2$ or 0.4\hbox{\thinspace\mathrm{Z}{\odot}}, in SPT-CLJ0000-5748, the temperatures are changed on average by around 2.5 per cent, densities by 1.5 per cent and entropies by 1.5 per cent.

2.6 Example cluster

As an example we consider SPT CLJ0000-5748, the first cluster in the sample, which is at a redshift of 0.70. The Chandra data have $\sim 1800$ counts in the 0.5 to 7 keV band after subtracting background, which is not atypical in our sample. The system appears relaxed with a bright central peak. Fig. 1 shows our profiles of the physical quantities using different models and binning. The quantities shown include the electron density ($n_{\mathrm{e}}$), temperature ($kT$), electron pressure ($P_{\mathrm{e}}$), electron entropy ($K_{\mathrm{e}}$), radiative cooling time ($t_{\mathrm{cool}}$), cumulative-gas mass ($M_{\mathrm{gas}}$) and cumulative-total mass ($M$). Cooling time here is defined as the ratio between the enthalpy per unit volume of the intracluster medium ($5nkT/2$, where $n$ is the total particle density) and its emissivity. The results plotted include those from the binned data assuming an NFW potential (BIN-NFW), a GNFW profile (BIN-GNFW) and without assuming hydrostatic equilibrium (BIN-NONHYDRO). We also show as a shaded region the interpolated density profile fit to the finely-binned data (INT-NFW). In the plot is marked the BIN-NFW $1\sigma$ range of $R_{500}$, obtained by scaling the dark-matter mass profile assuming a baryon fraction of 0.175, and the SPT value of $R_{500}$ (Bleem et al., 2015).

The results show good agreement between the different modelling techniques. The main differences occur in the outer bin, where the background is a large component of the observed surface brightness. The non-hydrostatic results agree well with the hydrostatic models, indicating that hydrostatic equilibrium is a reasonable assumption for this system. BIN-GNFW shows a flatter mass profile in the core, becoming steeper in the outskirts, with $\alpha=1.31^{+0.20}_{-0.31}$. In the central region of the cluster the density is high, there is a cool core, the cooling time drops to around $0.3\rm\thinspace Gyr$ and the entropy to $10\rm\thinspace keV\rm\thinspace cm^{2}$. The entropy profile shows no evidence for a central floor.

We show similar individual profiles for each of the clusters in Appendix C.

3 Central thermodynamic quantities

Here we examine the central thermodynamical properties of the clusters to look for the existence of an entropy or cooling time floor.

3.1 Central quantities as a function of radius

Fig. 2 shows the BIN-NFW model central values of the density, entropy and cooling time profiles plotted against the radial range of the inner bin. The results clearly show that the larger the size of the inner bin, the lower the density, the longer the cooling time and the higher the entropy. These bins were chosen to give the same 10 per cent uncertainty (for most objects) on the density. As found by Panagoulia et al. (2014), our ability to resolve these inner values is limited by the quality of the data. Therefore these central measured density values (the average in the annulus) are lower limits to the central density and the entropy and cooling times are upper limits. The values roughly scale (or inversely scale) with the size of the central bin to the power 1.5. Despite the density being correlated to bin radius (and therefore data quality), our ability to resolve spatial regions in a cluster also depends on the density of the ICM as more counts are emitted from denser regions. There are typically 100 to 200 counts in total in the central bin of the clusters with 20 per cent emissivity uncertainties. In order to measure a central entropy to a reasonable accuracy requires this number of counts, given the hydrostatic model.

As clusters with flat cores will have lower surface brightnesses than those with steep cores, it is not clear what fraction of the trend in Fig. 2 is due to data quality or cluster morphology. Using the results from the GRAD-NFW model we can look at the distribution of density gradients as a function of radius in the sample. Fig. 3 (top panel) plots the cumulative distribution of measured density gradients in the sample (taking median values from each chain) in radial regions and models fitted to the gradient posterior probability distributions for the sample in those regions. This modelling accounts for the uncertainties on the measurements, in particular in the central region, to obtain the intrinsic distribution. In this case we assume the gradient distributions can be modelled by a two-Gaussian-component model, but the results are very similar assuming a skewed-normal distribution instead. To see the radial distribution we plot the median model slope as a function of radius (bottom panel), showing the width of the distribution.

The results show that the average cluster is consistent with a central density gradient of around $-0.75$, with very few systems with completely flat cores. This average inner gradient is consistent with our later modelling of the thermodynamic profiles of the clusters (Section 4.3) and the results for a representative sample of nearby clusters (Croston et al., 2008). Therefore around half the $\sim-1.5$ power density trend seen in Fig. 2 is attributable to data quality and half due to the bias towards steeply-peaked surface brightness profiles.

3.2 Density comparison

It is important to check that our densities are accurate as the density profile is an important contribution to the other profiles, due to it being a parametrized profile in the majority of our analysis. We tested our mbproj2 binned density profiles by comparing with those produced by the PROJCT spectral model in xspec, using the same radial bins and assuming isothermality. We reproduced the density profiles well in these cases, subject to small factors due to the lack of temperature variation. In addition, the hydrostatic assumption does not appear to bias the densities, with excellent agreement between the non-hydrostatic BIN-NONHYDRO and the hydrostatic BIN-NFW central densities.

MD13 assumes the same functional form for the density as we use for the MBETA-NFW model. We compared our results at fixed 10 kpc radius against MD13, initially finding poor agreement. There were problems in the method used to obtain the published MD13 values,

but the updated values in McDonald et al. (2017)

matched our results much better (Fig. 4). The top panel compares the results for the profiles centred on the X-ray peak, while the lower panel shows the results using a 250–500 kpc annulus centroid (the main method used by MD13). Note that both these sets of cluster centres were independently obtained by us and MD13. For the annulus centroid we also show the originally-published MD13 results for comparison.

For the peak densities, there is a reasonable agreement between the two sets of results, with our densities being on average 30 per cent larger than the updated results of MD13. Using the 250–500 kpc annulus centroid, our densities are higher than the updated values of MD13 by 7 per cent. We have also checked the profiles for a few systems against those obtained by McDonald (private communication), finding reasonable agreement. The differences in the peak densities are due to the differences in how the centre is chosen. We optimize the position using a circle containing at least 50 counts, while MD13 considers a larger region.

However, when fitting profiles centred on the X-ray annulus the measured inner densities are strongly affected by the modified-$\beta$ model assumptions. Fig. 5 compares our standard densities at 10 kpc radius (assuming centrally flat or rising-inward profiles following MD13; $0\leq\alpha\leq 4$) against those obtained allowing declining central profiles ($-4\leq\alpha\leq 4$). For the profiles using the X-ray peak position, as might be expected there is little difference between the two results, with a trend allowing slightly lower central densities for systems with low central densities. However, if the annulus centre is used, the assumption of flat or inward-rising profiles has a large effect on the obtained densities. If the centre is not on the X-ray peak then it is physically possible to have a declining central profile. Assuming positive slopes will bias the central densities upwards in unrelaxed systems.

The density at 10 kpc radius is poorly constrained by the data (Fig. 2), so the assumed functional form can have a large impact on the resulting values. The STEP-NFW density model assumes a constant density inside a radius of 20 kpc. Even with this relatively large region, we can only constrain the density within this region to better than an order of magnitude in around 40 per cent of systems (Fig. 6 centre panel). Despite the difficulty in directly measuring densities at 10 kpc radius, if we compare the INT-NFW model, which interpolates in density between the wide bin centres, with MBETA-NFW, we obtain good agreement (Fig. 6 top panel). The INT-NFW densities are $10\pm 4$ per cent greater than the MBETA-NFW densities (examining median differences in log space), implying the central slopes are similar between the two models.

While the choice of density model has some effect on the central densities, the choice of centre also strongly influences the obtained values. Fig. 6 (bottom panel) compares the MBETA-NFW densities for the standard X-ray peak profile centre and for profiles using the 250–500 kpc radius centroid (following MD13). Some of the points are an order of magnitude lower than the peak densities. We note that these results assume a positive $\alpha$ parameter in the modified-$\beta$ model fits. If this is allowed to be negative (as in Fig. 5), the densities are moved to lower values, with some three orders of magnitude below the peak-centre profile values.

In conclusion, although our density values agree with MD13 given the same cluster centre and density model, the choice of cluster centre and model strongly influences the obtained central densities and other derived quantities such as entropy. Due to the variable quality of data (Section 3.1 and Fig. 6 centre panel) extrapolation has to be used to obtain the cluster properties at the 10 kpc radius used by MD13. The density profiles obtained by MD13 when using the annulus as cluster centre, are strongly biased upwards by the choice to force the inner density profiles to be flat or inwards-rising in the functional fit. However, this is partially compensated for by choosing a cluster centre based on the larger scale emission and missing the central peak. Our use of the X-ray peak as cluster centre is more robust against the choice of density model.

We note that using the X-ray peak as the centre of our NFW mass model may be inconsistent with simulations which use a mass centroid, possibly impacting the derived deprojected quantities. However, the NFW model is being used to fit a smooth pressure profile and not the cluster masses here, and so as long as the pressure profile is consistent with the data this should not be a problem. Indeed, if we compare the BIN-GNFW profiles, which have freedom in the inner slope of the pressure profile, with the BIN-NFW profiles (Appendix C) we see good consistency between the two, indicating that the NFW model and assumed priors is sufficient to fit the data.

3.3 Inner entropy values

Cavagnolo et al. (2009) fitted entropy profiles from a large sample of clusters with the functional form

[TABLE]

finding evidence for a bimodal distribution of values with peaks of $K_{0}$ at $\sim 15$ and $\sim 150\rm\thinspace keV\rm\thinspace cm^{2}$.

We examine our cluster entropy profiles using the KPLAW-NFW model, which uses a similar functional form for the entropy (equation 5), to see whether there is evidence for a floor. It parametrizes the slope of the entropy profile inside and outside a radius of 300 kpc separately to avoid the central profile fits being biased by the outskirts (300 kpc was a typical outer radius of a profile analysed by Cavagnolo et al. 2009). The entropy at 300 kpc, $K_{300}$, is parametrized instead of $K_{100}$, to avoid covariance with the $K_{0}$ parameter. We choose to fit the cluster X-ray profiles directly with the entropy model rather than fitting the posterior entropy profiles from the MCMC chain, to better take account of the covariance between the radial bins, which might otherwise bias the fit parameters (Lakhchaura et al., 2016), particularly if the posterior distributions are non-Gaussian. It should be noted, however, that a few clusters are fit poorly with this simple model (Appendix A.3), which could lead to biased results for those objects.

Fig. 7 (top panel) plots the $K_{0}$ and $K_{300}$ parameters for each system. The points plotted are the most-likely values in the chain, to avoid biasing the $K_{0}$ values upwards due to the lower bound on the parameter. In the centre panel is a probability density histogram of the median $K_{0}$ values. We also plot the histogram of the entropies obtained by Cavagnolo et al. (2009) and indicate the peaks of their bimodal distribution in the centre panel. The cumulative distribution of the two sets of $K_{0}$ values is shown in the bottom panel. Many of the clusters with entropies around zero are at lower entropies than the lower peak of values obtained by Cavagnolo et al. (2009). The strong zero peak is consistent with the findings of Panagoulia et al. (2014) and Hogan et al. (2017), who found no evidence for a common floor in entropy in their cluster samples.

We obtained a similar distribution of points using a break radius of 100 kpc instead of 300 kpc, or when we forced the parameter $\alpha_{K,inner}$ to be $0.64$, the average value obtained by Panagoulia et al. (2014). The peak around zero also remains if we fit the BIN-NFW posterior entropy profiles for each cluster, rather than reanalyse the surface-brightness profiles.

We modelled the underlying distribution of $K_{0}$ values before measurement errors with a probability-density function (PDF) made of two skew-normal components 111The Python code and input data used to model the distribution can be found at https://github.com/jeremysanders/K0dist/ and included with this paper as online-only material.. We used MCMC to sample this distribution given the marginalised $K_{0}$ PDFs for each cluster. The median model PDF and $1\sigma$ range is shown in the centre panel, while its cumulative distribution is shown in the bottom panel. In this analysis we excluded SPT-CLJ0658-5556 due to it having a very tightly constrained $K_{0}$ value given the high quality of data, but very poor fit quality.

In detail, to calculate the likelihood for a given a model PDF, for each cluster we multiplied the model PDF by the $K_{0}$ posterior PDF and integrated to compute the per-cluster likelihood. The log likelihoods for the individual clusters were summed with the prior to calculate a total log likelihood. We assumed the two components had centres between [math] and $1000\rm\thinspace keV\rm\thinspace cm^{2}$ and widths between $1$ and $500\rm\thinspace keV\rm\thinspace cm^{2}$, with flat priors. The model PDFs were forced to have no likelihood for $K_{0}<0$, adjusting to have a total integrated probability of 1. The skew parameters had a normal priors with a width of 20 in the analysis. The relative strength of the two components was modelled with a parameter $b$, where the fraction of probability give for the first component was the sigmoid function $1/(1+e^{-b})$, with a normal prior on $b$ of width 20.

The modelling appears to favour two components, although the second peak could be consistent with broad wings on the first peak to higher entropy values. Around 60 per cent of the integrated probability is in the lower entropy component. The second component is centred around $130\rm\thinspace keV\rm\thinspace cm^{2}$. Adding a third component does not produce an identifiable peak, but increases the width of the tails in the probability distribution. We note that we obtained similar model distributions if the $K_{0}$ model parameter was allowed to go negative, to avoid having a hard limit at 0, or if we did not bin the surface-brightness profiles within the central annulus. If we excluded those clusters with the worst fits for the KPLAW-NFW model (a goodness of fit greater than 2; Appendix A.3), we obtained a consistent model.

4 Evolution with redshift

4.1 Median profiles, with central cool-core bias

To look for evolution of the cluster properties in a model-independent fashion we examined the median profiles in two subsamples, $z<0.6$ and $z>0.6$, giving approximately equal numbers of systems (44 and 39, respectively). This approach has the advantage of being non-parametric, but we also examine the evolution using Gaussian and two-component modelling of the distribution of thermodynamic quantities in Sections 4.3 and 4.4, respectively.

Examining the unscaled BIN-NFW profiles in physical units, Fig. 8 shows the density, temperature, entropy, pressure, cooling time, mass deposition rate, cumulative gas and total mass profiles for the two subsamples. Plotted on each are the median and 68 per cent range of the profiles (calculated from percentiles) for the whole sample and redshift subset. To compute the medians and percentiles at a particular radius, we took all clusters where this radius was between the central radii of their inner and outer bins. For each cluster we constructed a sample of profiles using the MCMC chains, interpolating in radial log space between the bin centres. The median and range was computed from the combined set of samples from each cluster, weighting equally. Note that only including clusters which have valid profiles in the radial range examined, as we do here, is correct if data quality is the primary reason for poor spatial resolution (as indicated by Fig. 2), but could introduce biases if the cluster properties also affect the data quality (see Section 4.4). As cool-core systems have brighter cores and therefore smaller central annuli, this will bias the median profile towards cool-core systems (see Section 3.1). Assuming the density profiles are flat inwards when computing the median would produce the opposite bias. This bias is not present when we model the core (Section 4.2) or profile (Section 4.3) evolution.

Following Arnaud et al. (2010), we also scaled our physical profiles to the characteristic values of the isothermal self-similar model at a radius of $R_{500}$. When scaling we assumed $R_{500}$ and $M_{500}$ were the SPT-derived values (the hydrostatic values have larger and non-uniform uncertainties) and we did not account for the uncertainties in these quantities. We scaled the physical chains of quantities by the self-similar values and repeated the same analysis as for the median physical profiles. Fig. 9 show the profiles after scaling, plotted as a function of scaled radius. Similarly to the physical profiles we also plot the median and range of the scaled profiles.

To compare the median self-similar profiles of the subsamples, we computed the fractional difference between the two median profiles for each physical quantity as a function of scaled radius (Fig. 10). The uncertainties were obtained using bootstrap resampling, where for each quantity the uncertainty was taken as the standard deviation in the median profile from 1000 new random cluster subsamples created from the original cluster sample with replacement. The uncertainty on the fractional difference was calculated using the standard error propagation formulae.

Examining each of the physical quantities, there is no significant difference ($>2\sigma$) between the two subsamples at any scaled radius. The differences between the two samples are, if present, a small fraction of the dispersion within the sample (comparing the profiles in Fig. 9).

The median cluster increases in density inwards to around 0.02\hbox{\rm\thinspace cm^{-3},} at a radius of 20 kpc (Fig. 8). The low-redshift clusters have core densities which are 20 per cent greater than at higher redshift, although the difference is insignificant. The gas-mass profiles also show this similar trend. The median temperature profiles decline inwards to around 4 keV, or 1/2 of $T_{500}$ at this radius. There is little evidence for difference in the two samples, except at the largest radii where the background is more important.

The core entropies for the best-resolved clusters lie below the floor of Cavagnolo et al. (2009) (Fig. 8) and look similar to the central entropy powerlaw of Panagoulia et al. (2014) (note that their profile was fitted to radii of below 20 kpc which are not resolved in many of our systems). The central cooling times fall inwards to $\sim 1$ Gyr or around 1 per cent of $t_{\mathrm{cool,500}}$ (Fig. 9). Comparing the low- to the high-redshift samples shows mild indications for the entropy and cooling times being 20 per cent lower at low redshifts. If we compare our scaled entropy profiles to the baseline profile from the Voit et al. (2005) hydrodynamical simulations (Fig. 9), we see at $R_{500}$ that our results are consistent. At lower radii, we see entropy enhancement above this baseline, similar to that seen by Pratt et al. (2010), consistent with a picture of additional centrally-concentrated entropy increase not associated with the shocks in the outskirts of clusters. MD14 found that their entropy profiles flattened above $0.5R_{500}$ for their $z>0.6$ clusters, which we do not see in our median profile for this redshift bin.

In Fig. 9 we also plot the “universal” pressure profile of Arnaud et al. (2010) on our scaled pressure profiles, assuming median SPT masses for the two redshift subsamples. The profile was scaled following MD14, to account for the relatively cooler temperatures measured by XMM-Newton compared to Chandra (Schellenberger et al., 2015), by 10 per cent in pressure and 3 per cent in radius. However, it is not completely clear whether this scaling is valid in our case as our profiles were created assuming the SPT values of $R_{500}$, but the amount scaled is small. Our profiles match the Arnaud et al. (2010) values at $R_{500}$, as was also found by MD14 in their analysis of the same data, without assuming hydrostatic equilibrium. If there was some significant non-thermal pressure component, then it might be expected to appear as a difference at large radius between the hydrostatic and non-hydrostatic pressure profiles, which appears not to be the case. Scaling the pressure profiles by 20-30 per cent moves them significantly at large radius from the “universal” profile. At low radius, there is increasingly poor agreement between the Arnaud et al. (2010) profiles and ours, with ours around a factor of two lower at $0.03R_{500}$. MD14 also found that their pressure profiles were significantly lower than the “universal” profile at smaller radii. We note that the there is increasing scatter in the pressure profiles of Arnaud et al. (2010) below $0.2R_{500}$, with roughly an order of magnitude variation at $0.1R_{500}$, so the disagreement between our results and the universal profile is unlikely to be significant. If we compute the median pressure profile from the non-hydrostatic fits, it agrees well with the hydrostatic profiles at radii above $0.06R_{500}$, but increases at lower radii as the temperatures become more uncertain.

4.2 Modelling the core evolution

We looked for evolution in the ICM properties near the centre of the cluster. The binned profiles cannot be evaluated at a particular radius, so we used the interpolated density fits (the INT-NFW model). The results are very similar if the MBETA-NFW model is used instead, although we decided to use INT-NFW due to the MBETA-NFW fits being poor in some disturbed systems. Note that as we showed in Section 3.2, the values at this radius are model-dependent, although here the MBETA-NFW and INT-NFW models agree. We examine the gas properties at 10 kpc radius, although there is model uncertainty here. This choice is to compare against MD13 and MD14 who used this radius.

In Section 4.3 and Section 4.4 we examine the evolution of the whole profile, finding a consistent picture. Fig. 11 (left panel) shows the density, entropy, cooling time and pressure at 10 kpc radius for each cluster as a function of redshift. The right panel shows the quantities divided by the value at $R_{500}$ in the self-similar model at a radius of $0.015R_{500}$.

Shown in the plot is a sliding median, showing the median (with bootstrap uncertainties) of the 12 clusters nearest the redshift shown.