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Bourdin, H., Sauvageot, J. L., Slezak, E., Bijaoui, A., & Teyssier, R. 2004, A&A, 414, 429

Bourdin, H., Mazzotta, P., & Rasia, E. 2015, ApJ, 815, 92

Bourdin, H., De Luca, F., Mazzotta, P., et al. 2023, A&A, 678, A181

Bryan, G. L., Norman, M. L., O'Shea, B. W., et al. 2014, ApJS, 211, 19

Campitiello, M. G., Ettori, S., Lovisari, L., et al. 2022, A&A, 665, A117

Cappellari, M., & Copin, Y. 2003, MNRAS, 342, 345

Cash, W. 1979, ApJ, 228, 939

CHEX-MATE Collaboration (Arnaud, M., et al.) 2021, A&A, 650, A104

Churazov, E., Vikhlinin, A., Zhuravleva, I., et al. 2012, MNRAS, 421, 1123

Cui, W., Knebe, A., Yepes, G., et al. 2018, MNRAS, 480, 2898

Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977

de Vries, M., Mantz, A. B., Allen, S. W., et al. 2023, MNRAS, 518, 2954

Diehl, S., & Statler, T. S. 2006, MNRAS, 368, 497

Dolag, K., Jubelgas, M., Springel, V., Borgani, S., & Rasia, E. 2004, ApJ, 606, L97

Dupourqué, S., Clerc, N., Pointecouteau, E., et al. 2023, A&A, 673, A91

Eckert, D., Ghirardini, V., Ettori, S., et al. 2019, A&A, 621, A40

Ettori, S., & Eckert, D. 2022, A&A, 657, L1

Ettori, S., Ghirardini, V., Eckert, D., et al. 2019, A&A, 621, A39

Ferrari, C., Arnaud, M., Ettori, S., Maurogordato, S., & Rho, J. 2006, A&A, 446, 417

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Gaspari, M., & Churazov, E. 2013, A&A, 559, A78

Gaspari, M., Ruszkowski, M., & Sharma, P. 2012, ApJ, 746, 94

Gaspari, M., Churazov, E., Nagai, D., Lau, E. T., & Zhuravleva, I. 2014, A&A, 569, A67

Gaspari, M., Tombesi, F., & Cappi, M. 2020, Nat. Astron., 4, 10

Ghirardini, V., Eckert, D., Ettori, S., et al. 2019, A&A, 621, A41

Ghizzardi, S., Rossetti, M., & Molendi, S. 2010, A&A, 516, A32

Gianfagna, G., Rasia, E., Cui, W., et al. 2023, MNRAS, 518, 4238

Herbonnet, R., Sifón, C., Hoekstra, H., et al. 2020, MNRAS, 497, 4684

Hitomi Collaboration (Aharonian, F., et al.) 2016, Nature, 535, 117

Hofmann, F., Sanders, J. S., Nandra, K., Clerc, N., & Gaspari, M. 2016, A&A, 585, A130

Jones, C., & Forman, W. 1999, ApJ, 511, 65

Joye, W. A., & Mandel, E. 2003, in Astronomical Data Analysis Software and Systems XII, eds. H. E. Payne, R. I. Jedrzejewski, & R. N. Hook, ASP Conf. Ser., 295, 489

Kawahara, H., Suto, Y., Kitayama, T., et al. 2007, ApJ, 659, 257

Khatri, R., & Gaspari, M. 2016, MNRAS, 463, 655

Kuntz, K. D., & Snowden, S. L. 2008, A&A, 478, 575

Laganá, T. I. Durret, F., & Lopes, P. A. A. 2019, MNRAS, 484, 2807

Lau, E. T., Kravtsov, A. V., & Nagai, D. 2009, ApJ, 705, 1129

Leccardi, A., & Molendi, S. 2008, A&A, 486, 359

Liu, A., Yu, H., Tozzi, P., & Zhu, Z.-H. 2015, ApJ, 809, 27

Liu, A., Yu, H., Tozzi, P., & Zhu, Z.-H. 2016, ApJ, 821, 29

Lovisari, L., & Reiprich, T. H. 2019, MNRAS, 483, 540

Lovisari, L., Kapferer, W., Schindler, S., & Ferrari, C. 2009, A&A, 508, 191

Lovisari, L., Schindler, S., & Kapferer, W. 2011, A&A, 528, A60

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Lovisari, L., Ettori, S., Sereno, M., et al. 2020a, A&A, 644, A78

Lovisari, L., Schellenberger, G., Sereno, M., et al. 2020b, ApJ, 892, 102

Marelli, M., Molendi, S., Rossetti, M., et al. 2021, ApJ, 908, 37

Markevitch, M., & Vikhlinin, A. 2007, Phys. Rep., 443, 1

Mathiesen, B., Evrard, A. E., & Mohr, J. J. 1999, ApJ, 520, L21

Mazzotta, P., Rasia, E., Moscardini, L., & Tormen, G. 2004, MNRAS, 354, 10

Miniati, F. 2014, ApJ, 782, 21

Mohapatra, R., Federrath, C., & Sharma, P. 2020, MNRAS, 493, 5838

Mohapatra, R., Federrath, C., & Sharma, P. 2021, MNRAS, 500, 5072

Mohapatra, R., Jetti, M., Sharma, P., & Federrath, C. 2022, MNRAS, 510, 2327

Molendi, S., De Grandi, S., Rossetti, M., et al. 2023, A&A, 670, A104

Nagai, D., & Lau, E. T. 2011, ApJ, 731, L10

O'Sullivan, E., Giacintucci, S., David, L. P., Vrtilek, J. M., & Raychaudhury, S. 2011, MNRAS, 411, 1833

Planck Collaboration XXVII. 2016, A&A, 594, A27

Pratt, G. W., Croston, J. H., Arnaud, M., & Böhringer, H. 2009, A&A, 498, 361

Rasia, E., Ettori, S., Moscardini, L., et al. 2006, MNRAS, 369, 2013

Rasia, E., Meneghetti, M., Martino, R., et al. 2012, New J. Phys., 14, 055018P

Rasia, E., Lau, E. T., Borgani, S., et al. 2014, ApJ, 791, 96

Roncarelli, M., Ettori, S., Borgani, S., et al. 2013, MNRAS, 432, 3030

Roncarelli, M., Gaspari, M., Ettori, S., et al. 2018, A&A, 618, A39

Ryu, D., Kang, H., Cho, J., & Das, S. 2008, Science, 320, 909

Sabol, E. J., & Snowden, S. L. 2019, Astrophysics Source Code Library [record ascl:1904.001]

Sanders, J. S. 2006, MNRAS, 371, 829

Sanders, J. S., & Fabian, A. C. 2001, MNRAS, 325, 178

Sanders, J. S., & Fabian, A. C. 2012, MNRAS, 421, 726

Sanders, J. S., Dennerl, K., Russell, H. R., et al. 2020, A&A, 633, A42

Schuecker, P., Finoguenov, A., Miniati, F., Böhringer, H., & Briel, U. G. 2004, A&A, 426, 387

Sereno, M., & Ettori, S. 2015, MNRAS, 450, 3633

Sijacki, D., & Springel, V. 2006, MNRAS, 371, 1025

Simionescu, A., ZuHone, J., Zhuravleva, I., et al. 2019, Space Sci. Rev., 215, 24

Simonte, M., Vazza, F., Brighenti, F., et al. 2022, A&A, 658, A149

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Snowden, S. L., Mushotzky, R. F., Kuntz, K. D., & Davis, D. S. 2008, A&A, 478, 615

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Valdarnini, R. 2019, ApJ, 874, 42

Vazza, F., Brunetti, G., Kritsuk, A., et al. 2009, A&A, 504, 33

Vazza, F., Roediger, E., & Brüggen, M. 2012, A&A, 544, A103

Vazza, F., Eckert, D., Simionescu, A., Brüggen, M., & Ettori, S. 2013, MNRAS, 429, 799

Vazza, F., Jones, T. W., Brüggen, M., et al. 2017, MNRAS, 464, 210

Vikhlinin, A. 2006, ApJ, 640, 710

Willingale, R., Starling, R. L. C., Beardmore, A. P., Tanvir, N. R., & O'Brien, P. T. 2013, MNRAS, 431, 394

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Zhuravleva, I., Churazov, E., Kravtsov, A., et al. 2013, MNRAS, 428, 3274

Zhuravleva, I., Churazov, E. M., Schekochihin, A. A., et al. 2014, ApJ, 788, L13

Zhuravleva, I., Churazov, E., Arévalo, P., et al. 2016, MNRAS, 458, 2902

Zhuravleva, I., Allen, S. W., Mantz, A., & Werner, N. 2018, ApJ, 865, 53

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ZuHone, J. A., Kunz, M. W., Markevitch, M., Stone, J. M., & Biffi, V. 2015, ApJ, 798, 90

ZuHone, J. A., Markevitch, M., & Zhuravleva, I. 2016, ApJ, 817, 110
Bourdin, H., & Mazzotta, P. 2008, A&A, 479, 307

Bourdin, H., Sauvageot, J. L., Slezak, E., Bijaoui, A., & Teyssier, R. 2004, A&A, 414, 429

Bourdin, H., Mazzotta, P., & Rasia, E. 2015, ApJ, 815, 92

Bourdin, H., De Luca, F., Mazzotta, P., et al. 2023, A&A, 678, A181

Bryan, G. L., Norman, M. L., O'Shea, B. W., et al. 2014, ApJS, 211, 19

Campitiello, M. G., Ettori, S., Lovisari, L., et al. 2022, A&A, 665, A117

Cappellari, M., & Copin, Y. 2003, MNRAS, 342, 345

Cash, W. 1979, ApJ, 228, 939

CHEX-MATE Collaboration (Arnaud, M., et al.) 2021, A&A, 650, A104

Churazov, E., Vikhlinin, A., Zhuravleva, I., et al. 2012, MNRAS, 421, 1123

Cui, W., Knebe, A., Yepes, G., et al. 2018, MNRAS, 480, 2898

Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977

de Vries, M., Mantz, A. B., Allen, S. W., et al. 2023, MNRAS, 518, 2954

Diehl, S., & Statler, T. S. 2006, MNRAS, 368, 497

Dolag, K., Jubelgas, M., Springel, V., Borgani, S., & Rasia, E. 2004, ApJ, 606, L97

Dupourqué, S., Clerc, N., Pointecouteau, E., et al. 2023, A&A, 673, A91

Eckert, D., Ghirardini, V., Ettori, S., et al. 2019, A&A, 621, A40

Ettori, S., & Eckert, D. 2022, A&A, 657, L1

Ettori, S., Ghirardini, V., Eckert, D., et al. 2019, A&A, 621, A39

Ferrari, C., Arnaud, M., Ettori, S., Maurogordato, S., & Rho, J. 2006, A&A, 446, 417

Frank, K. A., Peterson, J. R., Andersson, K., Fabian, A. C., & Sanders, J. S. 2013, ApJ, 764, 46

Gaspari, M., & Churazov, E. 2013, A&A, 559, A78

Gaspari, M., Ruszkowski, M., & Sharma, P. 2012, ApJ, 746, 94

Gaspari, M., Churazov, E., Nagai, D., Lau, E. T., & Zhuravleva, I. 2014, A&A, 569, A67

Gaspari, M., Tombesi, F., & Cappi, M. 2020, Nat. Astron., 4, 10

Ghirardini, V., Eckert, D., Ettori, S., et al. 2019, A&A, 621, A41

Ghizzardi, S., Rossetti, M., & Molendi, S. 2010, A&A, 516, A32

Gianfagna, G., Rasia, E., Cui, W., et al. 2023, MNRAS, 518, 4238

Herbonnet, R., Sifón, C., Hoekstra, H., et al. 2020, MNRAS, 497, 4684

Hitomi Collaboration (Aharonian, F., et al.) 2016, Nature, 535, 117

Hofmann, F., Sanders, J. S., Nandra, K., Clerc, N., & Gaspari, M. 2016, A&A, 585, A130

Jones, C., & Forman, W. 1999, ApJ, 511, 65

Joye, W. A., & Mandel, E. 2003, in Astronomical Data Analysis Software and Systems XII, eds. H. E. Payne, R. I. Jedrzejewski, & R. N. Hook, ASP Conf. Ser., 295, 489

Kawahara, H., Suto, Y., Kitayama, T., et al. 2007, ApJ, 659, 257

Khatri, R., & Gaspari, M. 2016, MNRAS, 463, 655

Kuntz, K. D., & Snowden, S. L. 2008, A&A, 478, 575

Laganá, T. I. Durret, F., & Lopes, P. A. A. 2019, MNRAS, 484, 2807

Lau, E. T., Kravtsov, A. V., & Nagai, D. 2009, ApJ, 705, 1129

Leccardi, A., & Molendi, S. 2008, A&A, 486, 359

Liu, A., Yu, H., Tozzi, P., & Zhu, Z.-H. 2015, ApJ, 809, 27

Liu, A., Yu, H., Tozzi, P., & Zhu, Z.-H. 2016, ApJ, 821, 29

Lovisari, L., & Reiprich, T. H. 2019, MNRAS, 483, 540

Lovisari, L., Kapferer, W., Schindler, S., & Ferrari, C. 2009, A&A, 508, 191

Lovisari, L., Schindler, S., & Kapferer, W. 2011, A&A, 528, A60

Lovisari, L., Forman, W. R., Jones, C., et al. 2017, ApJ, 846, 51

Lovisari, L., Ettori, S., Sereno, M., et al. 2020a, A&A, 644, A78

Lovisari, L., Schellenberger, G., Sereno, M., et al. 2020b, ApJ, 892, 102

Marelli, M., Molendi, S., Rossetti, M., et al. 2021, ApJ, 908, 37

Markevitch, M., & Vikhlinin, A. 2007, Phys. Rep., 443, 1

Mathiesen, B., Evrard, A. E., & Mohr, J. J. 1999, ApJ, 520, L21

Mazzotta, P., Rasia, E., Moscardini, L., & Tormen, G. 2004, MNRAS, 354, 10

Miniati, F. 2014, ApJ, 782, 21

Mohapatra, R., Federrath, C., & Sharma, P. 2020, MNRAS, 493, 5838

Mohapatra, R., Federrath, C., & Sharma, P. 2021, MNRAS, 500, 5072

Mohapatra, R., Jetti, M., Sharma, P., & Federrath, C. 2022, MNRAS, 510, 2327

Molendi, S., De Grandi, S., Rossetti, M., et al. 2023, A&A, 670, A104

Nagai, D., & Lau, E. T. 2011, ApJ, 731, L10

O'Sullivan, E., Giacintucci, S., David, L. P., Vrtilek, J. M., & Raychaudhury, S. 2011, MNRAS, 411, 1833

Planck Collaboration XXVII. 2016, A&A, 594, A27

Pratt, G. W., Croston, J. H., Arnaud, M., & Böhringer, H. 2009, A&A, 498, 361

Rasia, E., Ettori, S., Moscardini, L., et al. 2006, MNRAS, 369, 2013

Rasia, E., Meneghetti, M., Martino, R., et al. 2012, New J. Phys., 14, 055018P

Rasia, E., Lau, E. T., Borgani, S., et al. 2014, ApJ, 791, 96

Roncarelli, M., Ettori, S., Borgani, S., et al. 2013, MNRAS, 432, 3030

Roncarelli, M., Gaspari, M., Ettori, S., et al. 2018, A&A, 618, A39

Ryu, D., Kang, H., Cho, J., & Das, S. 2008, Science, 320, 909

Sabol, E. J., & Snowden, S. L. 2019, Astrophysics Source Code Library [record ascl:1904.001]

Sanders, J. S. 2006, MNRAS, 371, 829

Sanders, J. S., & Fabian, A. C. 2001, MNRAS, 325, 178

Sanders, J. S., & Fabian, A. C. 2012, MNRAS, 421, 726

Sanders, J. S., Dennerl, K., Russell, H. R., et al. 2020, A&A, 633, A42

Schuecker, P., Finoguenov, A., Miniati, F., Böhringer, H., & Briel, U. G. 2004, A&A, 426, 387

Sereno, M., & Ettori, S. 2015, MNRAS, 450, 3633

Sijacki, D., & Springel, V. 2006, MNRAS, 371, 1025

Simionescu, A., ZuHone, J., Zhuravleva, I., et al. 2019, Space Sci. Rev., 215, 24

Simonte, M., Vazza, F., Brighenti, F., et al. 2022, A&A, 658, A149

Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91

Snowden, S. L., Mushotzky, R. F., Kuntz, K. D., & Davis, D. S. 2008, A&A, 478, 615

Towler, I., Kay, S. T., & Altamura, E. 2023, MNRAS, 520, 5845

Valdarnini, R. 2019, ApJ, 874, 42

Vazza, F., Brunetti, G., Kritsuk, A., et al. 2009, A&A, 504, 33

Vazza, F., Roediger, E., & Brüggen, M. 2012, A&A, 544, A103

Vazza, F., Eckert, D., Simionescu, A., Brüggen, M., & Ettori, S. 2013, MNRAS, 429, 799

Vazza, F., Jones, T. W., Brüggen, M., et al. 2017, MNRAS, 464, 210

Vikhlinin, A. 2006, ApJ, 640, 710

Willingale, R., Starling, R. L. C., Beardmore, A. P., Tanvir, N. R., & O'Brien, P. T. 2013, MNRAS, 431, 394

Zhang, Y.-Y., Reiprich, T. H., Finoguenov, A., Hudson, D. S., & Sarazin, C. L. 2009, ApJ, 699, 1178

Zhuravleva, I., Churazov, E., Kravtsov, A., et al. 2013, MNRAS, 428, 3274

Zhuravleva, I., Churazov, E. M., Schekochihin, A. A., et al. 2014, ApJ, 788, L13

Zhuravleva, I., Churazov, E., Arévalo, P., et al. 2016, MNRAS, 458, 2902

Zhuravleva, I., Allen, S. W., Mantz, A., & Werner, N. 2018, ApJ, 865, 53

Zhuravleva, I., Chen, M. C., Churazov, E., et al. 2023, MNRAS, 520, 5157

ZuHone, J., & Su, Y. 2022, Hand-book of X-ray and Gamma-ray Astrophysics (Singapore: Springer), 93

ZuHone, J. A., Markevitch, M., Ruszkowski, M., & Lee, D. 2013, ApJ, 762, 69

ZuHone, J. A., Kunz, M. W., Markevitch, M., Stone, J. M., & Biffi, V. 2015, ApJ, 798, 90

ZuHone, J. A., Markevitch, M., & Zhuravleva, I. 2016, ApJ, 817, 110
**Appendix A: Gallery**

In Fig. A.1 we show the recovered surface brightness, electron density, and temperature maps within $R_{500}$ for all the clusters analyzed in this work.

<figure><img src="image_15.png" /><figcaption>**Fig. A.1.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to $R_{500}$; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to $0.15R_{500}$, $0.5R_{500}$, and $R_{500}$.</figcaption></figure>
**Appendix A: Gallery**

In Fig. A.1 we show the recovered surface brightness, electron density, and temperature maps within $R_{500}$ for all the clusters analyzed in this work.

<figure><img src="image_15.png" /><figcaption>**Fig. A.1.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to $R_{500}$; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to $0.15R_{500}$, $0.5R_{500}$, and $R_{500}$.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. A.2.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. A.2.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. A.3.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. A.3.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. A.4.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. A.4.** From left to right: (a) bkg-subtracted and exposure-corrected images in the 0.3-7 keV band; The size of the boxes corresponds to *R*<sub>500</sub>; (b) binned surface brightness maps; (c) projected electron density maps (d) projected temperature maps; (e) relative temperature error maps. The white circles in the voronoi maps corresponds to 0.15*R<sub>500</sub>*, 0.5*R<sub>500</sub>*, and *R<sub>500</sub>*.</figcaption></figure>
## Appendix B: Comparison with the curvelet maps

<figure><img src="image_19.png" /></figure>

**Fig. B.1.** Comparison of the temperatures measured in the maps reconstructed in this work and the ones obtained from the curvelet technique. The histogram represents the distribution of the difference in spectroscopic measurements normalized to the statistical error $\epsilon = \sqrt{\epsilon_{\text{this work}}^2 + \epsilon_{\text{cur}}^2}$ for 1,916 spatial cells obtained in 10 objects. The vertical dashed lines indicate 0 (red) and -3, 3 (green).

Thermodynamical 2D maps have been extensively used in the study of galaxy clusters, thanks to their great potential to characterize the dynamical state of a system. For this reason, a number of spectral-mapping methods have been developed. Based on the approach, these methods can be divided into three main categories: (i) hardness ratio method which combines X-ray imaging in several energy bands (e.g., Ferrari et al. 2006); (ii) wavelet (see Bourdin et al. 2004; Bourdin & Mazzotta 2008) or curvelet (see Bourdin et al. 2015) analysis using the wavelet or curvelet coefficients to couple a multi-scale spectroscopic analysis with a structure detection scheme; (iii) spatially resolved spectroscopy of independent cells sampled by a required S/N or a minimum number of counts for which at least three different binning strategies have been proposed in this context: adaptive binning (e.g., Sanders & Fabian 2001, Lovisari et al. 2009, 2011, O'Sullivan et al. 2011); contour binning (Sanders 2006, Hofmann et al. 2016); Weighted Voronoi Tessellation (WVT) method (e.g., Cappellari & Copin 2003). In general, the Voronoi algorithms define the smallest possible regions for the spectral analysis, given the statistics of the observation. It is based on the distribution of the signal and noise in a given band and it puts together pixels to reach a required S/N, regardless of the “temperature” that would be measured in that cell. The wavelet or curvelet analysis has the advantage of being not driven by the distribution of the brightness of the emission. Conversely, the advantage of Voronoi algorithms is that (i) regions can be treated as independent from each other (when neglecting the effect of the PSF) and (ii) it does not try to follow the gradients of the ICM emission.

For ten objects in our sample, we compared the temperature map obtained through the spectral analyses of the Voronoi tessellation regions with the one obtained through the curvelet analysis. For each Voronoi region, we computed the weighted tem-

perature from the curvelet map and we then compared the two T values. In Fig. B.1, we show a summary plot to demonstrate that the temperatures recovered between the two methods are in good agreement, with an average value of the ratio $(T_{\text{this work}} - T_{\text{cur}})/\epsilon$ of 0.005 (median: 0.06; 1st-3rd quartile: -0.42, 0.53).

We also compared the radial temperature fluctuations estimated from the Voronoi and curvelet temperature maps. We found a remarkably good agreement between the $\sigma_T/T$ recovered with the two methods in the regions between 0.2-0.8$R_{500}$. At large radii (i.e., $> 0.8R_{500}$) the comparison is not straightforward because of the coarse mapping with the Voronoi method and of the masking of low signal-to-noise regions in the curvelet analysis. Within $0.2R_{500}$, the $\sigma_T/T$ values in the Voronoi maps are slightly larger than what was found by the curvelet analysis. A detailed investigation is required to understand the cause of this slight difference but it is beyond the scope of the paper. However, this does not affect any of the main conclusions.

## Appendix C: Temperature cell weights

<figure><img src="image_20.png" /></figure>

**Fig. C.1.** Comparison between the temperature derived from the maps (i.e., $T_{1D,500}$) and the global temperature (i.e., $T_{spec}$) obtained by fitting a single spectrum extracted within $R_{500}$. In each subplot the text in magenta indicates the weighting used to recover $T_{1D,500}$ from the maps, while the values of $\mu$ and $\sigma$ indicate the median of the distribution and its dispersion. In blue we show the results using the maps obtained with S/N=30 while in red we show the ones obtained with S/N=50.

In Fig. C.1, we show how well the overall cluster temperatures recovered from the maps (i.e., $T_{1D,500}$) match the fitted values using one single extraction region (i.e., $T_{spec}$). We used six different weightings, each of them including a proxy of the X-ray emissivity (i.e., either the value of the surface brightness of the cells or the electron density obtained using Eq. 1) multiplied by: (i) the area of the cell, (ii) the area of the cell multiplied by $T^\alpha$ (with $\alpha=-0.75$ from Mazzotta et al. 2004), and (iii) the area multiplied by the relative errors of the cells. Since the relative errors
## Appendix B: Comparison with the curvelet maps

<figure><img src="image_19.png" /></figure>

**Fig. B.1.** Comparison of the temperatures measured in the maps reconstructed in this work and the ones obtained from the curvelet technique. The histogram represents the distribution of the difference in spectroscopic measurements normalized to the statistical error $\epsilon = \sqrt{\epsilon_{\text{this work}}^2 + \epsilon_{\text{cur}}^2}$ for 1,916 spatial cells obtained in 10 objects. The vertical dashed lines indicate 0 (red) and -3, 3 (green).

Thermodynamical 2D maps have been extensively used in the study of galaxy clusters, thanks to their great potential to characterize the dynamical state of a system. For this reason, a number of spectral-mapping methods have been developed. Based on the approach, these methods can be divided into three main categories: (i) hardness ratio method which combines X-ray imaging in several energy bands (e.g., Ferrari et al. 2006); (ii) wavelet (see Bourdin et al. 2004; Bourdin & Mazzotta 2008) or curvelet (see Bourdin et al. 2015) analysis using the wavelet or curvelet coefficients to couple a multi-scale spectroscopic analysis with a structure detection scheme; (iii) spatially resolved spectroscopy of independent cells sampled by a required S/N or a minimum number of counts for which at least three different binning strategies have been proposed in this context: adaptive binning (e.g., Sanders & Fabian 2001, Lovisari et al. 2009, 2011, O'Sullivan et al. 2011); contour binning (Sanders 2006, Hofmann et al. 2016); Weighted Voronoi Tessellation (WVT) method (e.g., Cappellari & Copin 2003). In general, the Voronoi algorithms define the smallest possible regions for the spectral analysis, given the statistics of the observation. It is based on the distribution of the signal and noise in a given band and it puts together pixels to reach a required S/N, regardless of the “temperature” that would be measured in that cell. The wavelet or curvelet analysis has the advantage of being not driven by the distribution of the brightness of the emission. Conversely, the advantage of Voronoi algorithms is that (i) regions can be treated as independent from each other (when neglecting the effect of the PSF) and (ii) it does not try to follow the gradients of the ICM emission.

For ten objects in our sample, we compared the temperature map obtained through the spectral analyses of the Voronoi tessellation regions with the one obtained through the curvelet analysis. For each Voronoi region, we computed the weighted tem-

perature from the curvelet map and we then compared the two T values. In Fig. B.1, we show a summary plot to demonstrate that the temperatures recovered between the two methods are in good agreement, with an average value of the ratio $(T_{\text{this work}} - T_{\text{cur}})/\epsilon$ of 0.005 (median: 0.06; 1st-3rd quartile: -0.42, 0.53).

We also compared the radial temperature fluctuations estimated from the Voronoi and curvelet temperature maps. We found a remarkably good agreement between the $\sigma_T/T$ recovered with the two methods in the regions between 0.2-0.8$R_{500}$. At large radii (i.e., $> 0.8R_{500}$) the comparison is not straightforward because of the coarse mapping with the Voronoi method and of the masking of low signal-to-noise regions in the curvelet analysis. Within $0.2R_{500}$, the $\sigma_T/T$ values in the Voronoi maps are slightly larger than what was found by the curvelet analysis. A detailed investigation is required to understand the cause of this slight difference but it is beyond the scope of the paper. However, this does not affect any of the main conclusions.

## Appendix C: Temperature cell weights

<figure><img src="image_20.png" /></figure>

**Fig. C.1.** Comparison between the temperature derived from the maps (i.e., $T_{1D,500}$) and the global temperature (i.e., $T_{spec}$) obtained by fitting a single spectrum extracted within $R_{500}$. In each subplot the text in magenta indicates the weighting used to recover $T_{1D,500}$ from the maps, while the values of $\mu$ and $\sigma$ indicate the median of the distribution and its dispersion. In blue we show the results using the maps obtained with S/N=30 while in red we show the ones obtained with S/N=50.

In Fig. C.1, we show how well the overall cluster temperatures recovered from the maps (i.e., $T_{1D,500}$) match the fitted values using one single extraction region (i.e., $T_{spec}$). We used six different weightings, each of them including a proxy of the X-ray emissivity (i.e., either the value of the surface brightness of the cells or the electron density obtained using Eq. 1) multiplied by: (i) the area of the cell, (ii) the area of the cell multiplied by $T^\alpha$ (with $\alpha=-0.75$ from Mazzotta et al. 2004), and (iii) the area multiplied by the relative errors of the cells. Since the relative errors
are smaller for lower temperatures, both the latter approaches weight more the cells with lower temperatures.

In general, using SB instead of $n_e$ as a proxy for the X-ray emissivity returns $T_{1D,500}$ in better agreement with $T_{spec}$. This is possible because there is no assumption on the scale of the line of sight (i.e., the volume to be used to recover the electron density). If so, in principle this result could be used to provide a constraint on the elongation along the line of sight (helpful for a measure of the triaxiality), but this is beyond the scope of this paper.

Using only the area of the cells as weights is clearly not sufficient (see top panels of Fig. C.1) because it does not account for the energy dependence of the effective area. Using the relative errors of each cell in the weighting (bottom panels) returns $T_{1D,500}$ in much better agreement with $T_{spec}$ (e.g., compare the $\mu$ and $\sigma$ values of the different subplots). Weighting each cell by $T^\alpha$ works also relatively well (middle panels), although there is an increasing deviation at lower temperatures. However, since for comparison with the simulations using the factor $T^\alpha$ is the best option (e.g., Mazzotta et al. 2004), and after verifying that the impact on the results of the paper is not significant, we used this latter weighting as our default method.

## Appendix D: Voronoi tessellation and map resolution

Given a certain image, the binning mask obtained with WVT is deterministic (i.e., by running the algorithm multiple times, one obtains always the same result). Depending on the resolution of the mask, it is possible that some inhomogeneities are missed because they are spanning multiple cells, with the net result that the feature is washed out. In order to investigate how the choice of the map center impacts the obtained map and our results, we derived maps centered offset from the X-ray peak. The center of the new maps was computed from a grid with the X values obtained as $X_{center}=X_{peak}+C \times R_{500}$ and the Y values as $Y_{center}=Y_{peak}+C \times R_{500}$, where C can assume 3 values: -0.1, 0, +0.1. As an example, in the top panel of Fig. D.1, we show the results for G041.45+29.10. Despite the different binning, the same features are observed in all the maps. In the bottom panel we show that the recovered profiles from the individual maps differ only by a few percent (and are always consistent within the uncertainties) as far as the map resolution is good enough (typically within ~0.6$R_{500}$). At large radii, when the binning becomes quite coarse there is an increase of the scatter, although the data points are still consistent within the $1\sigma$ errors. Similar results are obtained with the other clusters in the sample, so that the results of the paper are not affected by the deterministic nature of the Voronoi binning.

The results presented in the current analysis are based on the binning of temperature maps obtained with S/N=30. Such a choice is set by the requirement to have typical temperature uncertainties of the order of 10-20%. However, the resolution of the maps can hide some relevant features and reduce or increase the observed scatter, in particular in the electron density maps used to infer the type of fluctuations. This is probably due to the increasing size of the binning in the outer regions, amplified by the presence of steep surface brightness gradients. To verify that, we produced SB maps with S/N=15, 30, and 50 and we derived the scatter as a function of radius. In Fig. D.2 we show the results with the electron densities calculated under the assumption of $n_e = \sqrt{SB}$. Indeed, the ratio $\sigma_{n_e}/n_e$ increases from low to high

<figure><img src="image_21.png" /><figcaption>**Fig. D.1.** Impact of the Voronoi tessellation on the 2D temperature distribution. Top: temperature maps obtained for G041.45+29.10 by offsetting the center of the map wrt to the X-ray peak. The green circles identify $R_{500}$ centered on the peak. Bottom: ratio between the temperature recovered from the nine maps (using Eq. 2) and the temperature estimated using the spectra extracted from azimuthal annuli. Each color (from blue to red) represents one of the maps in the top panel (from top-left to bottom-right).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. D.2.** Impact of the map resolution on the density profile in the case of G041.45+29.10. We truncate the plot at $0.6R_{500}$ because beyond that radius we have a poor resolution (in particular with S/N=50). Each point represents the value of $\sigma_{n_e}/n_e$ computed at the radius of the cells in the map.</figcaption></figure>
are smaller for lower temperatures, both the latter approaches weight more the cells with lower temperatures.

In general, using SB instead of $n_e$ as a proxy for the X-ray emissivity returns $T_{1D,500}$ in better agreement with $T_{spec}$. This is possible because there is no assumption on the scale of the line of sight (i.e., the volume to be used to recover the electron density). If so, in principle this result could be used to provide a constraint on the elongation along the line of sight (helpful for a measure of the triaxiality), but this is beyond the scope of this paper.

Using only the area of the cells as weights is clearly not sufficient (see top panels of Fig. C.1) because it does not account for the energy dependence of the effective area. Using the relative errors of each cell in the weighting (bottom panels) returns $T_{1D,500}$ in much better agreement with $T_{spec}$ (e.g., compare the $\mu$ and $\sigma$ values of the different subplots). Weighting each cell by $T^\alpha$ works also relatively well (middle panels), although there is an increasing deviation at lower temperatures. However, since for comparison with the simulations using the factor $T^\alpha$ is the best option (e.g., Mazzotta et al. 2004), and after verifying that the impact on the results of the paper is not significant, we used this latter weighting as our default method.

## Appendix D: Voronoi tessellation and map resolution

Given a certain image, the binning mask obtained with WVT is deterministic (i.e., by running the algorithm multiple times, one obtains always the same result). Depending on the resolution of the mask, it is possible that some inhomogeneities are missed because they are spanning multiple cells, with the net result that the feature is washed out. In order to investigate how the choice of the map center impacts the obtained map and our results, we derived maps centered offset from the X-ray peak. The center of the new maps was computed from a grid with the X values obtained as $X_{center}=X_{peak}+C \times R_{500}$ and the Y values as $Y_{center}=Y_{peak}+C \times R_{500}$, where C can assume 3 values: -0.1, 0, +0.1. As an example, in the top panel of Fig. D.1, we show the results for G041.45+29.10. Despite the different binning, the same features are observed in all the maps. In the bottom panel we show that the recovered profiles from the individual maps differ only by a few percent (and are always consistent within the uncertainties) as far as the map resolution is good enough (typically within ~0.6$R_{500}$). At large radii, when the binning becomes quite coarse there is an increase of the scatter, although the data points are still consistent within the $1\sigma$ errors. Similar results are obtained with the other clusters in the sample, so that the results of the paper are not affected by the deterministic nature of the Voronoi binning.

The results presented in the current analysis are based on the binning of temperature maps obtained with S/N=30. Such a choice is set by the requirement to have typical temperature uncertainties of the order of 10-20%. However, the resolution of the maps can hide some relevant features and reduce or increase the observed scatter, in particular in the electron density maps used to infer the type of fluctuations. This is probably due to the increasing size of the binning in the outer regions, amplified by the presence of steep surface brightness gradients. To verify that, we produced SB maps with S/N=15, 30, and 50 and we derived the scatter as a function of radius. In Fig. D.2 we show the results with the electron densities calculated under the assumption of $n_e = \sqrt{SB}$. Indeed, the ratio $\sigma_{n_e}/n_e$ increases from low to high

<figure><img src="image_21.png" /><figcaption>**Fig. D.1.** Impact of the Voronoi tessellation on the 2D temperature distribution. Top: temperature maps obtained for G041.45+29.10 by offsetting the center of the map wrt to the X-ray peak. The green circles identify $R_{500}$ centered on the peak. Bottom: ratio between the temperature recovered from the nine maps (using Eq. 2) and the temperature estimated using the spectra extracted from azimuthal annuli. Each color (from blue to red) represents one of the maps in the top panel (from top-left to bottom-right).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. D.2.** Impact of the map resolution on the density profile in the case of G041.45+29.10. We truncate the plot at $0.6R_{500}$ because beyond that radius we have a poor resolution (in particular with S/N=50). Each point represents the value of $\sigma_{n_e}/n_e$ computed at the radius of the cells in the map.</figcaption></figure>
S/N. However, it is worth noticing that the trend remains the same with an increase as a function of the radius (as discussed in Sect. 3.4 and shown in Fig. 9), although there is a flattening beyond 0.3-0.4*R<sub>500</sub>.

**Appendix E: Projection effects to the density and temperature perturbations**

By analyzing a set of 3D cosmological simulations (Vazza et al. 2017; Simonte et al. 2022) we tested how well Eq. 11 (derived following Schuecker et al. 2004) provides a good representation of the correlation between the projected density and temperature fluctuations (i.e., the relation of the 2D perturbations being a factor of ~2 flatter than the relation between 3D density and temperature perturbations). The simulations are performed with the code ENZO (Bryan et al. 2014) and accurately capture the dynamical evolution of the ICM as galaxy clusters undergo matter accretion throughout their evolutionary process. Although the employed simulations are classified as non-radiative, previous studies have indicated that the influence of non-gravitational effects on the larger scales of interest (> few tenths of kpc) is comparatively limited, in contrast to the significant impact of mergers and accretion phenomena in shaping the properties of ICM (e.g., Vazza et al. 2012; Valdarnini 2019). We refer the reader to Simonte et al. (2022, and references therein) for further details. The 3D ICM perturbations are estimated in spherical shells with a size of 0.1*R<sub>500</sub>. In Fig. E.1 we show as red points the fitted slope (i.e., γ-1) between the 3D density and temper-

<figure><img src="image_23.png" /><figcaption>**Fig. E.1.** For each of the 8 simulated clusters, we show the mean ratio between the radial values of $\delta T_{3D}/T_{3D}$ and $\delta n/n$ (red points), $\delta T_{2D}/T_{2D}$ and $\delta n_{2D}^2/n_{2D}^2$ in three different projections (blue diamonds), and the latter values corrected by the factor of 2 from Schuecker et al. (2004; the green squares represent the average values while the error bars represent the minimum and maximum values from the different projections).</figcaption></figure>

ature perturbations and as blue diamonds the 2D slopes from three different projections. It is clear that projection effects tend to flatten the relation as expected. The flattening is roughly a factor of 2 as derived following Schuecker et al. (2004; see the green squares).
S/N. However, it is worth noticing that the trend remains the same with an increase as a function of the radius (as discussed in Sect. 3.4 and shown in Fig. 9), although there is a flattening beyond 0.3-0.4*R<sub>500</sub>.

## Appendix E: Projection effects to the density and temperature perturbations

By analyzing a set of 3D cosmological simulations (Vazza et al. 2017; Simonte et al. 2022) we tested how well Eq. 11 (derived following Schuecker et al. 2004) provides a good representation of the correlation between the projected density and temperature fluctuations (i.e., the relation of the 2D perturbations being a factor of ~2 flatter than the relation between 3D density and temperature perturbations). The simulations are performed with the code ENZO (Bryan et al. 2014) and accurately capture the dynamical evolution of the ICM as galaxy clusters undergo matter accretion throughout their evolutionary process. Although the employed simulations are classified as non-radiative, previous studies have indicated that the influence of non-gravitational effects on the larger scales of interest (> few tenths of kpc) is comparatively limited, in contrast to the significant impact of mergers and accretion phenomena in shaping the properties of ICM (e.g., Vazza et al. 2012; Valdarnini 2019). We refer the reader to Simonte et al. (2022, and references therein) for further details. The 3D ICM perturbations are estimated in spherical shells with a size of 0.1*R<sub>500</sub>. In Fig. E.1 we show as red points the fitted slope (i.e., γ-1) between the 3D density and temper-

<figure><img src="image_23.png" /><figcaption>**Fig. E.1.** For each of the 8 simulated clusters, we show the mean ratio between the radial values of $\delta T_{3D}/T_{3D}$ and $\delta n/n$ (red points), $\delta T_{2D}/T_{2D}$ and $\delta n_{2D}^2/n_{2D}^2$ in three different projections (blue diamonds), and the latter values corrected by the factor of 2 from Schuecker et al. (2004; the green squares represent the average values while the error bars represent the minimum and maximum values from the different projections).</figcaption></figure>

ature perturbations and as blue diamonds the 2D slopes from three different projections. It is clear that projection effects tend to flatten the relation as expected. The flattening is roughly a factor of 2 as derived following Schuecker et al. (2004; see the green squares).