the miscentering of the X-ray center, the redshift distribution of sources, the bias from the photo-z, the choice of the radial bins in the fitting, and the inaccurate assumption about the halo mass distribution. In this way, we empirically establish the connection between the lensing observable $g_+$ and $M_{500}$ at the cluster redshift. Similarly, we calibrate the bias in the observed count rate $\eta$ with respect to the true count rate through a calibration by running the end-to-end eROSITA analysis pipeline on large simulations, enabling us to robustly connect $\eta$ to $M_{500}$. With these simulation-based calibrations, we calibrate the $\eta-M_{500}-z$ and $M_{WL}-M_{500}-z$ relations by simultaneously modeling the individual $\eta$ and $g_+$ in a forward-modeling framework. As a result, the count-rate-inferred and lensing-calibrated posterior of $M_{500}$ for each cluster is obtained.

The resulting mass of the eFEDS sample spans a range of $10^{13}h^{-1}M_{\odot} \lesssim M_{500} \lesssim 10^{15}h^{-1}M_{\odot}$ with a median value of $M_{500} \approx 10^{14}h^{-1}M_{\odot}$, of which the low-mass clusters ($M_{500} \lesssim 10^{14}h^{-1}M_{\odot}$) are mainly located at low redshift ($z \lesssim 0.4$). The characteristic uncertainty of the cluster mass, which is estimated as the dispersion of the mass posterior $P(M_{500})$, is at the level of $\approx 27\%$ for a cluster with $M_{500} \approx 10^{14}h^{-1}M_{\odot}$. The mass estimates are not sensitive to the weak-lensing modeling of the cluster core; excluding the radial range of $0.2h^{-1}\text{Mpc} \lesssim R \lesssim 0.5h^{-1}\text{Mpc}$ results in a negligible change ($\approx 0.01$ dex) in the cluster mass. Currently, the statistical uncertainty of $A_{\eta}$, the normalization of the X-ray-mass-proxy-to-mass-and-redshift relation, is constrained to be at the level of $\approx 20\%$. If the weak-lensing mass calibration can be controlled with a subpercent accuracy, the uncertainty on $A_{\eta}$ is expected to be constrained with a precision of $\approx 1.8\%$ to first order in the final eROSITA survey, assuming that half of the clusters ($\approx 100\,000/2 = 50\,000$) have weak-lensing follow-ups. The mass calibration shows that, as expected, eROSITA is able to reliably detect clusters down to much lower mass scales than any currently existing or planned SZ survey.

Leveraging the weak-lensing mass calibration, we further explore the relationships between cluster mass, redshift, and the follow-up X-ray observables, including the rest-frame soft-band (bolometric) luminosity $L_X$ ($L_b$), the core-included temperature $T_X$, the gas mass $M_g$, and the mass proxy $Y_X$. These X-ray observables can be well described by a power-law function of the cluster mass and redshift with log-normal intrinsic scatter at fixed mass. We make use of the full combination of the data from both the eROSITA and HSC surveys to significantly tighten the constraints on the X-ray observable relations, even for those observables, for example, $T_X$, which cannot be estimated for any of the detected clusters with high S/N. Except for $L_X$, the scaling relations of the follow-up X-ray observables show

- a mass trend that is statistically consistent with the self-similar prediction at a level of $\lesssim 1.7\sigma$,

- no significant deviation from self-similarity at the level of $\lesssim 1.7\sigma$ in terms of redshift scaling, and

- no clear sign of a redshift-dependent power-law index in the cluster mass.

Meanwhile, we find that the $L_X-M_{500}-z$ relation has a steeper mass trend than the self-similar prediction at the level of $\approx 3\sigma$ but a redshift scaling that is in good agreement with self-similarity. This suggests that the eFEDS clusters, which are dominated by low-mass clusters or galaxy groups, show X-ray scaling relations that are largely consistent with self-similar predictions in terms of mass and redshift scaling. A generally steeper mass dependence of the luminosity clearly indicates that nongravitational processes play an important role in low-mass systems at the scale of galaxy groups, which is the main population studied

in this work. The energy input from nongravitational processes lowers the gas content within $R_{500}$ and hence decreases the X-ray luminosity; moreover, this effect is mass-dependent and is more prominent at a low-mass scale, resulting in a steeper slope seen in the luminosity-to-mass relation. This picture is also in line with the $M_g-M_{500}-z$ relation of the eFEDS sample with a steeper mass trend than the self-similar prediction, despite the low significance due to large error bars.

We compare the mass and redshift trends of the X-ray scaling relations derived from the eFEDS sample with those from the previous work and find good consistency at a level of $\lesssim 2\sigma$. This work extends the study of the X-ray scaling relations to low-mass clusters over a wide range of redshift using the eROSITA-selected sample, which confers advantages over SZ-based selections. It is interesting to mention the comparison with Bulbul et al. (2019), the study of X-ray scaling relations based on the largest SZ-selected sample out to $z \approx 1.3$. Our work probes a similar range of redshift but a mass scale that is much lower than that of these latter authors at the scale of massive clusters ($M_{500} \approx 4.5 \times 10^{14}h^{-1}M_{\odot}$). A significantly steeper mass dependence than the self-similar prediction was found by Bulbul et al. (2019) for all the X-ray observables ($L_X$, $L_b$, $T_X$, $M_g$, and $Y_X$) at the level of $\ge 3\sigma$. Despite broad consistency ($\approx 2\sigma$) with their results, our eFEDS sample generally shows a mass scaling that is shallower than theirs and is statistically consistent with the self-similar predictions at $\lesssim 2\sigma$. This may suggest a broken power law (Le Brun et al. 2017) of the X-ray observables in the transition between groups and clusters, or a possible mass- and redshift-dependent systematic bias in the cluster mass and/or X-ray observables between Bulbul et al. (2019) and this work. Deep X-ray follow-ups of a large sample selected in the eROSITA All-Sky Survey will shed light on this subject.

We find that the gas mass $M_g$ shows the smallest intrinsic scatter compared to other follow-up observables, suggesting that the ICM mass-based mass proxy may be most useful for clusters with $\approx 100$ X-ray photons observed by large surveys.

This work clearly demonstrates the capability of the HSC weak-lensing data set in calibrating the mass of a sizable sample of galaxy clusters and groups detected by eROSITA out to high redshift ($z \approx 1.3$). This work not only presents the weak-lensing mass calibration of eROSITA-detected clusters in synergy with a wide and deep lensing survey, but also lays the foundation for the cluster cosmology in a combination of the revolutionary eROSITA sample and the state-of-the-art lensing data from the HSC.

*Acknowledgments.* The authors thank the referee for constructive comments that lead to improvements of this paper. I-Non Chiu thanks Hung-Hsu Chan and Chia-Ying Lin for useful discussions that lead to improvements of this paper. This work is supported in part by the national science foundation of China (Nos. 11833005, 11890691, 11890692, 11621303, 11890693, 11421303), 111 project No. B20019 and Shanghai Natural Science Foundation, grant No. 15ZR1446700, 19ZR1466800. This work made use of the Gravity Supercomputer at the Department of Astronomy, Shanghai Jiao Tong University. This work is supported in part by the Ministry of Science and Technology of Taiwan (grant MOST 106-2628-M-001-003-MY3 and MOST 109-2112-M-001-018-MY3) and by Academia Sinica (grant AS-IA-107-M01). This work made use of the computing resources in the National Center for High-Performance Computing (NCHC) in Taiwan. This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by
MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg and ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS/NRT software system developed by the German eROSITA consortium. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for the Legacy Survey of Space and Time carried out by the Vera C. Rubin Observatory. We thank the LSST Project for making our code available as free software at http://dm.lsst.org. This paper is based in part on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ. We are honored and grateful for the opportunity of observing the Universe from Maunakea, which has the cultural, historical and natural significance in Hawaii. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. This work was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI Grant Numbers JP19KK0076, JP20H00181, and JP20H05856. This work is possible because of the efforts in the LSST (Juric et al. 2017; Ivezic et al. 2019) and PS1 (Chambers et al. 2016; Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013), and in the HSC (Aihara et al. 2018a) developments including the deep imaging of the COSMOS field (Tanaka et al. 2017), the on-site quality-assurance system (Furusawa et al. 2018), the Hyper Suprime-Cam (Miyazaki 2015; Miyazaki et al. 2018; Komiyama et al. 2018), the design of the filters (Kawanomoto et al. 2018), the data pipeline (Bosch et al. 2018), the design of bright-star masks (Coupon et al. 2018), the characterization of the photometry by the code Synpipe (Huang et al. 2018), the photometric redshift estimation (Tanaka et al. 2018), the shear calibration (Mandelbaum et al. 2018a), and the public data releases (Aihara et al. 2018b, 2019). This work made use of the IPython package (Pérez & Granger 2007), SciPy (Virtanen et al. 2020), TOPCAT, an interactive graphical viewer and editor for tabular data (Taylor 2005, 2006), matplotlib, a Python library for publication quality graphics (Hunter 2007), Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration 2013), NumPy (Van Der Walt et al. 2011). This work made use of Pathos (McKerns et al. 2012) in parallel computing. This work made use of Bocquet & Carter (2016) and Hinton (2016) for producing the corner plots for the parameter constraints. The code Colossus (Diemer 2018) is heavily used to calculate cosmology-dependent quantities in this work. Data availability. The eROSITA X-ray detection parameters are publicly available in Liu et al. (2022a) and the redshifts are provided in Klein et al. (2022). The X-ray observables of the eFEDS sample will be made publicly available in Bahar et al. (2022). The HSC weak-lensing shape catalog will be made publicly available following the plan of the HSC Public Data Releases. The other data products underlying this article will be shared upon a reasonable request to the corresponding author.

## References

* Adami, C., Giles, P., Koulouridis, E., et al. 2018, A&A, 620, A5
* Aihara, H., Arimoto, N., Armstrong, R., et al. 2018a, PASJ, 70, S4

* Aihara, H., Armstrong, R., Bickerton, S., et al. 2018b, PASJ, 70, S8
* Aihara, H., AlSayyad, Y., Ando, M., et al. 2019, PASJ, 71, 114
* Alam, S., Albareti, F. D., Allende Prieto, C., et al. 2015, ApJS, 219, 12
* Andersson, K., Benson, B., Ade, P., et al. 2011, ApJ, 738, 48
* Applegate, D., von der Linden, A., Kelly, P., et al. 2014, MNRAS, 439, 48
* Arnaud, K. A. 1996, ASP Conf. Ser., 101, 17
* Arnaud, M., Pointecouteau, E., & Pratt, G. 2005, A&A, 441, 893
* Arnaud, M., Pointecouteau, E., & Pratt, G. 2007, A&A, 474, L37
* Arnaud, M., Pratt, G., Piffaretti, R., et al. 2010, A&A, 517, A92
* Astropy Collaboration (Robitaille, T. P., et al.) 2013, A&A, 558, A33
* Bahar, Y. E., Bulbul, E., Clerc, N., et al. 2022, A&A, 661, A7 (eROSITA EDR SI)
* Barnes, D. J., Kay, S. T., Bahé, Y. M., et al. 2017, MNRAS, 471, 1088
* Bartelmann, M., & Schneider, P. 2001, Phys. Rep., 340, 291
* Benson, B., de Haan, T., Dudley, J., et al. 2013, ApJ, 763, 147
* Biesiadzinski, T., McMahon, J., Miller, C. J., Nord, B., & Shaw, L. 2012, ApJ, 757, 1
* Bleem, L., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27
* Bleem, L. E., Bocquet, S., Stalder, B., et al. 2020, ApJS, 247, 25
* Bocquet, S., & Carter, F. W. 2016, J. Open Source Softw., 1, 46
* Bocquet, S., Saro, A., Mohr, J., et al. 2015, ApJ, 799, 214
* Bocquet, S., Saro, A., Dolag, K., & Mohr, J. 2016, MNRAS, 456, 2361
* Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
* Böhringer, H., Schuecker, P., Guzzo, L., et al. 2001, A&A, 369, 826
* Böhringer, H., Schuecker, P., Guzzo, L., et al. 2004, A&A, 425, 367
* Böhringer, H., Dolag, K., & Chon, G. 2012, A&A, 539, A120
* Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103
* Bonamente, M., Joy, M., LaRoque, S., et al. 2008, ApJ, 675, 106
* Bonnett, C., Troxel, M. A., Hartley, W., et al. 2016, Phys. Rev. D, 94, 042005
* Borm, K., Reiprich, T. H., Mohammed, I., & Lovisari, L. 2014, A&A, 567, A65
* Bosch, J., Armstrong, R., Bickerton, S., et al. 2018, PASJ, 70, S5
* Brunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1 (eROSITA EDR SI)
* Bulbul, E., Chiu, I. N., Mohr, J. J., et al. 2019, ApJ, 871, 50
* Castro, T., Borgani, S., Dolag, K., et al. 2021, MNRAS, 500, 236
* Cavaliere, A., & Fusco-Femiano, R. 1976, A&A, 500, 95
* Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, ArXiv e-prints [arXiv:1612.05560]
* Chen, K.-F., Oguri, M., Lin, Y.-T., & Miyazaki, S. 2020, ApJ, 891, 139
* Chiu, I., Mohr, J., McDonald, M., et al. 2016a, MNRAS, 455, 258
* Chiu, I., Saro, A., Mohr, J., et al. 2016b, MNRAS, 458, 379
* Chiu, I., Mohr, J. J., McDonald, M., et al. 2018a, MNRAS, 478, 3072
* Chiu, I.-N., Umetsu, K., Sereno, M., et al. 2018b, ApJ, 860, 126
* Costanzi, M., Okumura, T., Oguri, M., et al. 2020a, MNRAS, 498, 2030
* Chiu, I. N., Umetsu, K., Murata, R., Medezinski, E., & Oguri, M. 2020b, MNRAS, 495, 428
* Clowe, D., De Lucia, G., & King, L. 2004, MNRAS, 350, 1038
* Coil, A. L., Blanton, M. R., Burles, S. M., et al. 2011, ApJ, 741, 8
* Comparat, J., Merloni, A., Salvato, M., et al. 2019, MNRAS, 487, 2005
* Comparat, J., Eckert, D., Finoguenov, A., et al. 2020, Open J. Astrophys., 3, 13
* Cool, R. J., Moustakas, J., Blanton, M. R., et al. 2013, ApJ, 767, 118
* Corless, V. L., & King, L. J. 2007, MNRAS, 380, 149
* Costanzi, M., Rozo, E., Rykoff, E. S., et al. 2019a, MNRAS, 482, 490
* Costanzi, M., Rozo, E., Simet, M., et al. 2019b, MNRAS, 488, 4779
* Costanzi, M., Saro, A., Bocquet, S., et al. 2021, Phys. Rev. D, 103, 043522
* Coupon, J., Czakon, N., Bosch, J., et al. 2018, PASJ, 70, S7
* de Haan, T., Benson, B., Bleem, L., et al. 2016, ApJ, 832, 95
* DES Collaboration (Abott, T., et al.) 2020, Phys. Rev. D, 102, 023509
* Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168
* Diemer, B. 2018, ApJS, 239, 35
* Diemer, B., & Kravtsov, A. V. 2015, ApJ, 799, 108
* Dietrich, J. P., Böhnert, A., Lombardi, M., Hilbert, S., & Hartlap, J. 2012, MNRAS, 419, 3547
* Dietrich, J. P., Bocquet, S., Schrabback, T., et al. 2019, MNRAS, 483, 2871
* Eckert, D., Ettori, S., Coupon, J., et al. 2016, A&A, 592, A12
* Eddington, A. 1913, MNRAS, 73, 359
* Evrard, A. E., Arnault, P., Huterer, D., & Farahi, A. 2014, MNRAS, 441, 3562
* Finoguenov, A., Reiprich, T. H., & Böhringer, H. 2001, A&A, 368, 749
* Foreman-Mackey, D., Hogg, D., Lang, D., & Goodman, J. 2013, PASP, 125, 306
* Foreman-Mackey, D., Farr, W., Sinha, M., et al. 2019, J. Open Source Softw., 4, 1864
* Furusawa, H., Koike, M., Takata, T., et al. 2018, PASJ, 70, S3
* Gatti, M., Giannini, G., Bernstein, G. M., et al. 2022, MNRAS, 510, 1223
* Ghirardini, V., Bulbul, E., Hoang, D. N., et al. 2021, A&A, 647, A4
* Giles, P. A., Maughan, B. J., Pacaud, F., et al. 2016, A&A, 592, A3
* Giodini, S., Lovisari, L., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 247
Grandis, S., Mohr, J. J., Dietrich, J. P., et al. 2019, MNRAS, 488, 2041

Grandis, S., Klein, M., Mohr, J. J., et al. 2020, MNRAS, 498, 771

Grandis, S., Bocquet, S., Mohr, J. J., Klein, M., & Dolag, K. 2021, MNRAS, 507, 5671

Gruen, D., & Brimioulle, F. 2017, MNRAS, 468, 769

Gruen, D., Seitz, S., Brimioulle, F., et al. 2014, MNRAS, 442, 1507

Hamana, T., Shirasaki, M., Miyazaki, S., et al. 2020, PASJ, 72, 16

Hasinger, G., Capak, P., Salvato, M., et al. 2018, ApJ, 858, 77

Henden, N. A., Puchwein, E., & Sijacki, D. 2019, MNRAS, 489, 2439

Hikage, C., Oguri, M., Hamana, T., et al. 2019, PASJ, 71, 43

Hildebrandt, H., Köhlinger, F., van den Busch, J. L., et al. 2020, A&A, 633, A69

Hilton, M., Sifón, C., Naess, S., et al. 2021, ApJS, 253, 3

Hinton, S. R. 2016, J. Open Source Softw., 1, 00045

Hoekstra, H. 2003, MNRAS, 339, 1155

Hoekstra, H., Bartelmann, M., Dahle, H., et al. 2013, Space Sci. Rev., 177, 75

Hoekstra, H., Herbonnet, R., Muzzin, A., et al. 2015, MNRAS, 449, 685

Hoshino, H., Leauthaud, A., Lackner, C., et al. 2015, MNRAS, 452, 998

Hsieh, B. C., & Yee, H. K. C. 2014, ApJ, 792, 102

Huang, S., Leauthaud, A., Murata, R., et al. 2018, PASJ, 70, S6

Hunter, J. D. 2007, Comput. Sci. Eng., 9, 90

Ikebe, Y., Reiprich, T. H., Böhringer, H., Tanaka, Y., & Kitayama, T. 2002, A&A, 383, 773

Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236

Ivezie, Z., Axelrod, T., Brandt, W. N., et al. 2008, Serbian Astron. J., 176, 1

Ivezie, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111

Jee, M. J., Dawson, K. S., Hoekstra, H., et al. 2011, ApJ, 737, 59

Johnston, D. E., Sheldon, E. S., Tasitsiomi, A., et al. 2007a, ApJ, 656, 27

Johnston, D. E., Sheldon, E. S., Wechsler, R. H., et al. 2007b, ArXiv e-prints [arXiv:0709.1159]

Juric, M., Kantor, J., Lim, K. T., et al. 2017, ASP Conf. Ser., 512, 279

Kaiser, N. 1986, MNRAS, 222, 323

Kawanomoto, S., Uraguchi, F., Komiyama, Y., et al. 2018, PASJ, 70, 66

Kelly, P. L., von der Linden, A., Applegate, D. E., et al. 2014, MNRAS, 439, 28

Kettula, K., Finoguenov, A., Massey, R., et al. 2013, ApJ, 778, 74

King, L., & Corless, V. 2007, MNRAS, 374, L37

Klein, M., Mohr, J. J., Desai, S., et al. 2018, MNRAS, 474, 3324

Klein, M., Grandis, S., Mohr, J. J., et al. 2019, MNRAS, 488, 739

Klein, M., Oguri, M., Mohr, J. J., et al. 2022, A&A, 661, A4 (eROSITA EDR SI)

Kodwani, D., Alonso, D., & Ferreira, P. 2019, Open J. Astrophys., 2, 3

Komiyama, Y., Obuchi, Y., Nakaya, H., et al. 2018, PASJ, 70, S2

Kravtsov, A., Vikhlinin, A., & Nagai, D. 2006, ApJ, 650, 128

Kravtsov, A. V., Vikhlinin, A. A., & Meshcheryakov, A. V. 2018, Astron. Lett., 44, 8

Laigle, C., McCracken, H. J., Ilbert, O., et al. 2016, ApJS, 224, 24

Lauer, T. R., Postman, M., Strauss, M. A., Graves, G. J., & Chisari, N. E. 2014, ApJ, 797, 82

Laureijs, R., Duvet, L., Escudero Sanz, I., et al. 2010, SPIE Conf. Ser., 7731, 1

Le Brun, A. M. C., McCarthy, I. G., Schaye, J., & Ponman, T. J. 2017, MNRAS, 466, 4442

Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473

Li, X., Miyatake, H., Luo, W., et al. 2022, PASJ, 74, 421

Lieu, M., Smith, G. P., Giles, P. A., et al. 2016, A&A, 592, A4

Lilly, S. J., Le Brun, V., Maier, C., et al. 2009, ApJS, 184, 218

Lin, Y.-T., & Mohr, J. 2004, ApJ, 617, 879

Lin, Y., Mohr, J., & Stanford, S. 2003, ApJ, 591, 749

Lin, Y.-T., Hsieh, B.-C., Lin, S.-C., et al. 2017, ApJ, 851, 139

Liu, F. S., Mao, S., & Meng, X. M. 2012, MNRAS, 423, 422

Liu, J., Mohr, J., Saro, A., et al. 2015, MNRAS, 448, 2085

Liu, A., Bulbul, E., Ghirardini, V., et al. 2022a, A&A, 661, A2 (eROSITA EDR SI)

Liu, T., Merloni, A., Comparat, J., et al. 2022b, A&A, 661, A27 (eROSITA EDR SI)

Lovisari, L., Reiprich, T. H., & Schellenberger, G. 2015, A&A, 573, A118

Magnier, E. A., Schlafly, E., Finkbeiner, D., et al. 2013, ApJS, 205, 20

Mahdavi, A., Hoekstra, H., Babul, A., et al. 2013, ApJ, 767, 116

Mandelbaum, R., Hirata, C. M., Seljak, U., et al. 2005, MNRAS, 361, 1287

Mandelbaum, R., Miyatake, H., Hamana, T., et al. 2018a, PASJ, 70, S25

Mandelbaum, R., Lanusse, F., Leauthaud, A., et al. 2018b, MNRAS, 481, 3170

Mantz, A. B., von der Linden, A., Allen, S. W., et al. 2015, MNRAS, 446, 2205

Mantz, A., Allen, S., Morris, R., et al. 2016, MNRAS, 463, 3582

Masters, D., Capak, P., Stern, D., et al. 2015, ApJ, 813, 53

Masters, D. C., Stern, D. K., Cohen, J. G., et al. 2019, ApJ, 877, 81

Maughan, B. 2007, ApJ, 668, 772

McClintock, T., Varga, T. N., Gruen, D., et al. 2019, MNRAS, 482, 1352

McKerns, M. M., Strand, L., Sullivan, T., Fang, A., & Aivazis, M. A. G. 2012, ArXiv e-prints [arXiv:1202.1056]

Medezinski, E., Battaglia, N., Umetsu, K., et al. 2018a, PASJ, 70, S28

Medezinski, E., Oguri, M., Nishizawa, A. J., et al. 2018b, PASJ, 70, 30

Melchior, P., Gruen, D., McClintock, T., et al. 2017, MNRAS, 469, 4899

Merloni, A., Predehl, P., Becker, W., et al. 2012, ArXiv e-prints [arXiv:1209.3114]

Miyatake, H., Battaglia, N., Hilton, M., et al. 2019, ApJ, 875, 63

Miyazaki, S. 2015, IAU General Assembly, 22, 2255916

Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018, PASJ, 70, S1

Mohr, J. J., & Evrard, A. E. 1997, ApJ, 491, 38

Mohr, J. J., Mathiesen, B., & Evrard, A. E. 1999, ApJ, 517, 627

Momcheva, I. G., Brammer, G. B., van Dokkum, P. G., et al. 2016, ApJS, 225, 27

Murata, R., Oguri, M., Nishimichi, T., et al. 2019, PASJ, 71, 107

Nagai, D., Kravtsov, A., & Vikhlinin, A. 2007, ApJ, 668, 1

Navarro, J., Frenk, C., & White, S. 1997, ApJ, 490, 493

Oguri, M., Takada, M., Okabe, N., & Smith, G. P. 2010, MNRAS, 405, 2215

Oguri, M., Lin, Y.-T., Lin, S.-C., et al. 2018a, PASJ, 70, S20

Oguri, M., Miyazaki, S., Hikage, C., et al. 2018b, PASJ, 70, S26

O'Hara, T., Mohr, J., Bialek, J., & Evrard, A. 2006, ApJ, 639, 64

Okabe, N., & Smith, G. P. 2016, MNRAS, 461, 3794

Okabe, N., Oguri, M., Akamatsu, H., et al. 2019, PASJ, 71, 79

Pacaud, F., Pierre, M., Melin, J. B., et al. 2018, A&A, 620, A10

Páris, I., Petitjean, P., Aubourg, É., et al. 2018, A&A, 613, A51

Pérez, F., & Granger, B. E. 2007, Comput. Sci. Eng., 9, 21

Pierre, M., Pacaud, F., Adami, C., et al. 2016, A&A, 592, A1

Piffaretti, R., Arnaud, M., Pratt, G. W., Pointecouteau, E., & Melin, J. B. 2011, A&A, 534, A109

Pillepich, A., Porciani, C., & Reiprich, T. H. 2012, MNRAS, 422, 44

Pillepich, A., Reiprich, T. H., Porciani, C., Borm, K., & Merloni, A. 2018, MNRAS, 481, 613

Planck Collaboration XXIV. 2016, A&A, 594, A24

Poole, G. B., Babul, A., McCarthy, I. G., Sanderson, A. J. R., & Fardal, M. A. 2008, MNRAS, 391, 1163

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

Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25

Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1

Raihan, S. F., Schrabback, T., Hildebrandt, H., Applegate, D., & Mahler, G. 2020, MNRAS, 497, 1404

Reichert, A., Böhringer, F., Fassbender, R., & Mühlegger, M. 2011, A&A, 535, A4

Reiprich, T., & Böhringer, H. 2002, ApJ, 567, 716

Ritchie, B. W., & Thomas, P. A. 2002, MNRAS, 329, 675

Rozo, E., & Rykoff, E. S. 2014, ApJ, 783, 80

Rozo, E., Rykoff, E., Koester, B., et al. 2011, ApJ, 740, 53

Saro, A., Bocquet, S., Rozo, E., et al. 2015, MNRAS, 454, 2305

Schellenberger, G., & Reiprich, T. H. 2017, MNRAS, 471, 1370

Schlafly, E. F., Finkbeiner, D. P., Jurić, M., et al. 2012, ApJ, 756, 158

Schrabback, T., Hartlap, J., Joachimi, B., et al. 2010, A&A, 516, A63

Schrabback, T., Applegate, D., Dietrich, J. P., et al. 2018a, MNRAS, 474, 2635

Schrabback, T., Schirmer, M., van der Burg, R. F. J., et al. 2018b, A&A, 610, A85

Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, MNRAS, 505, 3923

Sehgal, N., Addison, G., Battaglia, N., et al. 2013, ApJ, 767, 38

Seitz, C., & Schneider, P. 1997, A&A, 318, 687

Sereno, M., Umetsu, K., Ettori, S., et al. 2020, MNRAS, 492, 4528

Silverman, J. D., Kashino, D., Sanders, D., et al. 2015, ApJS, 220, 12

Skelton, R. E., Whitaker, K. E., Momcheva, I. G., et al. 2014, ApJS, 214, 24

Smith, B. W., Mushotzky, R. F., & Serlemitsos, P. J. 1979, ApJ, 227, 37

Sommer, M. W., Schrabback, T., Applegate, D. E., et al. 2022, MNRAS, 509, 1127

Song, J., Zenteno, A., Stalder, B., et al. 2012, ApJ, 761, 22

Straatman, C. M. S., van der Wel, A., Bezanson, R., et al. 2018, ApJS, 239, 27

Sunayama, T., Park, Y., Takada, M., et al. 2020, MNRAS, 496, 4468

Sunyaev, R., & Zel'dovich, Y. 1972, Comments on Astrophysics and Space Physics (Berlin: Springer), 4, 173

Sunyaev, R., Arefiev, V., Babyshkin, V., et al. 2021, A&A, 656, A132

Tanaka, M., Hasinger, G., Silverman, J. D., et al. 2017, ArXiv e-prints [arXiv:1706.00566]

Tanaka, M., Coupon, J., Hsieh, B.-C., et al. 2018, PASJ, 70, S9

Taylor, M. B. 2005, ASP Conf. Ser., 347, 29

Taylor, M. B. 2006, ASP Conf. Ser., 351, 666

The Dark Energy Survey Collaboration 2005, ArXiv e-prints [arXiv: astro-ph/0510346]

To, C., Krause, E., Rozo, E., et al. 2021, Phys. Rev. Lett., 126, 141301

Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, ApJ, 750, 99

Truemper, J. 1982, Adv. Space Res., 2, 241

Umetsu, K. 2020, A&ARV, 28, 7

Umetsu, K., Medezinski, E., Nonino, M., et al. 2014, ApJ, 795, 163
Umetsu, K., Sereno, M., Lieu, M., et al. 2020, ApJ, 890, 148
van der Burg, R., Muzzin, A., Hoekstra, H., et al. 2014, A&A, 561, A79
Van Der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Comput. Sci. Eng., 13, 22

Varga, T. N., DeRose, J., Gruen, D., et al. 2019, MNRAS, 489, 2511

Vikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, ApJ, 640, 691

Vikhlinin, A., Burenin, R., Ebeling, H., et al. 2009, ApJ, 692, 1033

Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nat. Methods, 17, 261

von der Linden, A., Mantz, A., Allen, S., et al. 2014a, MNRAS, 443, 1973

von der Linden, A., Allen, M. T., Applegate, D. E., et al. 2014b, MNRAS, 439, 2

Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2006, MNRAS, 373, 1159

Zhang, Y.-Y., Laganá, T., Pierini, D., et al. 2012, A&A, 544, C3

Zhang, Y., Jeltema, T., Hollowood, D. L., et al. 2019, MNRAS, 487, 2578

Zitrin, A., Bartelmann, M., Umetsu, K., Oguri, M., & Broadhurst, T. 2012, MNRAS, 426, 2944

Zu, Y., Mandelbaum, R., Simet, M., Rozo, E., & Rykoff, E. S. 2017, MNRAS, 470, 551

¹ Tsung-Dao Lee Institute, and Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, PR China
e-mail: inchiu@sjtu.edu.cn

² Department of Astronomy, School of Physics and Astronomy, and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, PR China

³ Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), 11F of AS/NTU Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan

⁴ Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany

⁵ Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany

⁶ IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France

⁷ Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan

⁸ Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwashi, Chiba 277-8583, Japan

⁹ Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University, Nagoya 464-8602, Japan

¹⁰ Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan

¹¹ Division of Physics and Astrophysical Science, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan

¹² The Inter University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune 411007, India

¹³ Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan

¹⁴ Core Research for Energetic Universe, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

¹⁵ Physics Program, Graduate School of Advanced Science and Engineering, Hiroshima University, 1-3-1 Kagamiyama, HigashiHiroshima, Hiroshima 739-8526, Japan

¹⁶ Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

¹⁷ Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
## Appendix A: The richness-to-mass-and-redshift relation of the eFEDS sample

In this section, we show the richness-to-mass-and-redshift ($\lambda - M_{500-z}$) relation of the eFEDS sample, where the richness $\lambda$ is estimated by the MCMF algorithm (see Sect. 2.3). We use the same fitting framework as described in Sect. 5.3 with $\chi = \lambda$ in Eq. (56). The resulting relation is

$$ \langle \ln \lambda | M_{500} \rangle = \ln (36.2^{+3.7}_{-3.5}) + \left[ (0.881^{+0.077}_{-0.088}) + (-0.56^{+0.91}_{-0.40}) \ln \left( \frac{1+z}{1+z_{\text{piv}}} \right) \right] \times \ln \left( \frac{M_{500}}{M_{\text{piv}}} \right) + (-0.46^{+0.54}_{-0.51}) \times \ln \left( \frac{1+z}{1+z_{\text{piv}}} \right), \quad (A.1) $$

with log-normal intrinsic scatter of $0.274^{+0.078}_{-0.055}$ at fixed mass. At the pivotal mass ($M_{\text{piv}} = 1.4 \times 10^{14} h^{-1} M_{\odot}$) and redshift ($z_{\text{piv}} = 0.35$), the richness is constrained to be $36.2^{+3.7}_{-3.5}$ scaled as $M_{500}^{0.881^{+0.077}_{-0.088}}$ without a significant redshift trend ($\lambda \propto (1+z)^{-0.46^{+0.54}_{-0.51}}$). We show the richness of the eFEDS sample as a function of mass (redshift) in the left (right) panel of Fig. A.1, where we re-normalize the richness to the pivotal redshift (mass) following the prescription in producing Fig. 14. The red line in Fig. A.1 represents a self-similar prediction in terms of richness, which is $\lambda \propto M_{500}(1+z)^0$. Our result shows a shallower mass trend deviating from one at a level of $\approx 1.5\sigma$ and no significant redshift-dependent mass trend ($\langle \ln \lambda | M_{500} \rangle \propto -0.56^{+0.91}_{-0.40} \times \ln (M_{500}) \ln (1+z)$).

We note one caveat as follows. The result of the $\lambda - M_{500-z}$ relation is obtained using the secure sample with a cut on $f_{\text{cont}}$, for which this is effectively equivalent to a redshift-dependent cut on the observed richness (Klein et al. 2018, 2019). The fitting does not account for this richness-dependent selection, for which a more detailed modeling of the selection function is needed (as in Grandis et al. 2020). This effect is likely small compared to the current uncertainty, given the sample size used here. We therefore defer a more complete modeling to a future work.

## Appendix B: Post-unblinding analysis

After the submission of the paper and during the process of the referee review, we discovered two errors in the X-ray analysis that need to be corrected after the unblinding. The first error is that incorrect redshifts were assigned to a small fraction of clusters (11 systems), such that the X-ray follow-up observables of those clusters were incorrect. The second error is due to a bug in the code used to derive the bolometric luminosity, resulting in a systematic bias in $L_b$ at the level of $\approx -13\%$. To correct these errors, we reran the modeling of $\chi - M_{500-z}$, where $\chi \in \{L_X, L_b, T_X, M_g, Y_X\}$, without any further changes except the uniform prior applied on the normalization of the $L_b - M_{500-z}$ relation$^{12}$, $A_{L_b}$. We find the results are not significantly affected by these errors, except for the normalization of the $L_b - M_{500-z}$ relation, which becomes higher due to the systematic bias. We note that only the modeling of the follow-up X-ray scaling relations were affected by these errors, while the mass calibration results are intact in the post-unblinding analysis.

## Appendix C: The cluster mass

We present the estimates of the cluster true mass (see Sect. 6.1) for the eFEDS clusters in Table C.1. In addition to the secure sample of 434 clusters with $f_{\text{cont}} < 0.2$, we also show the mass of clusters with $0.2 \le f_{\text{cont}} < 0.3$. This leads to a total number of 457 clusters in Table C.1.

## Appendix D: Corrections for the cluster mass in the literature

Different methods were used to estimate the cluster mass in the literature, therefore causing systematic discrepancy in $M_{500}$. For example, the mass estimated from X-ray assuming the hydrostatic equilibrium in ICM is suggested to be biased low by $\approx 20\%$ compared to that inferred from weak lensing (see e.g., Bocquet et al. 2015; Dietrich et al. 2019). Therefore, we need to correct the different mass estimates from the literature to make a fair comparison in Fig. 16.

We follow the scheme introduced in Chiu et al. (2018a) to correct for the systematic difference in $M_{500}$ from the literature. For Vikhlinin et al. (2009), Pratt et al. (2009), Mahdavi et al. (2013), and Lovisari et al. (2015), the cluster mass $M_{500}$ is estimated in X-rays assuming the hydrostatic equilibrium. We therefore multiply their mass by a constant factor of 1.12, which is quantified by comparing the masses inferred from X-ray and the weak-lensing technique (see more in Dietrich et al. 2019). For Mahdavi et al. (2013), Mantz et al. (2016), and Bulbul et al. (2019), the mass estimates are obtained from either a direct weak-lensing technique or a mass proxy with the weak-lensing calibration; we therefore do not apply any correction to them, assuming that their weak-lensing inferred mass is the true underlying mass. For the simulated clusters presented in Barnes et al. (2017), the true cluster mass is used, and hence no correction is required.

<footnote n="12">12 We took this opportunity of the rerun to modify the uniform prior applied on $A_{L_b}$ from $\mathcal{U}(10^{41} \text{erg/s}, 10^{44} \text{erg/s})$ to $\mathcal{U}(10^{41} \text{erg/s}, 10^{46} \text{erg/s})$, since we found that the former, which was used, is only slightly above the upper limit of the 68% confidence level of the resulting $A_{L_b}$.</footnote>
<figure><img src="image_24.png" /><figcaption>**Fig. A.1.** Observed richness $\lambda$ as a function of the cluster mass $M_{500}$ (left panel) and redshift $z$ (right panel). This plot is produced and shown in the same way as in Fig. 14.</figcaption></figure>