SPT clusters with DES and HST weak lensing. II.
Cosmological constraints from the abundance of massive
halos

S. Bocquet, S. Grandis, L. E. Bleem, M. Klein, J. J. Mohr, T. Schrabback, T.

M. C. Abbott, P. A. R. Ade, M. Aguena, A. Alarcon, et al.

► To cite this version:

S. Bocquet, S. Grandis, L. E. Bleem, M. Klein, J. J. Mohr, et al., SPT clusters with DES and HST weak lensing. II. Cosmological constraints from the abundance of massive halos. Physical Review D, 2024, 110 (8), pp.083510. 10.1103/PhysRevD.110.083510. hal-04800745

HAL Id: hal-04800745

https://hal.science/hal-04800745v1

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Distributed under a Creative Commons Attribution 4.0 International License
SPT Clusters with DES and HST Weak Lensing. II. Cosmological Constraints from
the Abundance of Massive Halos

S. Bocquet,<fnref n="1" />,* S. Grandis,<fnref n="2" />,<fnref n="1" /> L. E. Bleem,<fnref n="3" />,<fnref n="4" /> M. Klein,<fnref n="1" /> J. J. Mohr,<fnref n="1" />,<fnref n="5" /> T. Schrabback,<fnref n="2" />,<fnref n="6" /> T. M. C. Abbott,<fnref n="7" />
P. A. R. Ade,<fnref n="8" /> M. Aguena,<fnref n="9" /> A. Alarcon,<fnref n="3" /> S. Allam,<fnref n="10" /> S. W. Allen,<fnref n="11" />,<fnref n="12" />,<fnref n="13" /> O. Alves,<fnref n="14" /> A. Amon,<fnref n="15" />,<fnref n="16" />

A. J. Anderson,<fnref n="10" /> J. Annis,<fnref n="10" /> B. Ansarinejad,<fnref n="17" /> J. E. Austermann,<fnref n="18" />,<fnref n="19" /> S. Avila,²⁰ D. Bacon,²¹ M. Bayliss,²²
J. A. Beall,<fnref n="18" /> K. Bechtol,²³ M. R. Becker,<fnref n="3" /> A. N. Bender,<fnref n="3" />,<fnref n="4" />,²⁴ B. A. Benson,²⁴,<fnref n="4" />,<fnref n="10" /> G. M. Bernstein,²⁵

S. Bhargava,²⁶ F. Bianchini,<fnref n="11" />,<fnref n="12" />,<fnref n="13" /> M. Brodwin,²⁷ D. Brooks,²⁸ L. Bryant,²⁹ A. Campos,³⁰ R. E. A. Canning,³¹

J. E. Carlstrom,²⁴,<fnref n="4" />,³²,<fnref n="3" />,²⁹ A. Carnero Rosell,³³,<fnref n="9" />,³⁴ M. Carrasco Kind,³⁵,³⁶ J. Carretero,²⁰ F. J. Castander,³⁷,³⁸

R. Cawthon,³⁹ C. L. Chang,<fnref n="4" />,<fnref n="3" />,²⁴ C. Chang,²⁴,<fnref n="4" /> P. Chaubal,<fnref n="17" /> R. Chen,⁴⁰ H. C. Chiang,⁴¹,⁴² A. Choi,⁴³

T-L. Chou,<fnref n="4" />,³² R. Citron,⁴⁴ C. Corbett Moran,⁴⁵ J. Cordero,⁴⁶ M. Costanzi,⁴⁷,⁴⁸,⁴⁹ T. M. Crawford,<fnref n="4" />,²⁴

A. T. Crites,⁵⁰ L. N. da Costa,<fnref n="9" /> M. E. S. Pereira,⁵¹ C. Davis,<fnref n="11" /> T. M. Davis,⁵² J. DeRose,⁵³ S. Desai,⁵⁴

T. de Haan,⁵⁵,⁵⁶ H. T. Diehl,<fnref n="10" /> M. A. Dobbs,⁴¹,⁵⁷ S. Dodelson,³⁰,⁵⁸ C. Doux,²⁵,⁵⁹ A. Drlica-Wagner,²⁴,<fnref n="10" />,<fnref n="4" />

K. Eckert,²⁵ J. Elvin-Poole,⁶⁰ S. Everett,⁶¹ W. Everett,⁶² I. Ferrero,⁶³ A. Ferté,<fnref n="13" /> A. M. Flores,<fnref n="12" />,<fnref n="11" /> J. Frieman,<fnref n="10" />,<fnref n="4" />

J. Gallicchio,<fnref n="4" />,⁶⁴ J. García-Bellido,⁶⁵ M. Gatti,²⁵ E. M. George,⁶⁶ G. Giannini,²⁰,<fnref n="4" /> M. D. Gladders,²⁴,<fnref n="4" /> D. Gruen,<fnref n="1" />

R. A. Gruendl,³⁵,³⁶ N. Gupta,⁶⁷ G. Gutierrez,<fnref n="10" /> N. W. Halverson,⁶²,<fnref n="19" /> I. Harrison,⁶⁸ W. G. Hartley,⁶⁹ K. Herner,<fnref n="10" />

S. R. Hinton,⁵² G. P. Holder,³⁶,⁷⁰,⁵⁷ D. L. Hollowood,⁷¹ W. L. Holzapfel,⁷² K. Honscheid,⁷³,⁷⁴ J. D. Hrubes,⁴⁴
N. Huang,⁷² J. Hubmayr,<fnref n="18" /> E. M. Huff,⁶¹ D. Huterer,<fnref n="14" /> K. D. Irwin,<fnref n="13" />,<fnref n="12" /> D. J. James,⁷⁵ M. Jarvis,²⁵

G. Khullar,<fnref n="4" />,²⁴ K. Kim,²² L. Knox,⁷⁶ R. Kraft,⁷⁵ E. Krause,⁷⁷ K. Kuehn,⁷⁸,⁷⁹ N. Kuropatkin,<fnref n="10" /> F. Kéruzoré,<fnref n="3" />

O. Lahav,²⁸ A. T. Lee,⁷²,⁸⁰ P.-F. Leget,⁸¹ D. Li,<fnref n="18" />,<fnref n="13" /> H. Lin,<fnref n="10" /> A. Lowitz,²⁴ N. MacCrann,⁸² G. Mahler,⁸³,⁸⁴

A. Mantz,<fnref n="11" />,<fnref n="12" /> J. L. Marshall,⁸⁵ J. McCullough,⁸¹ M. McDonald,⁸⁶ J. J. McMahon,<fnref n="4" />,³²,²⁴ J. Mena-Fernández,⁸⁷
F. Menanteau,³⁵,³⁶ S. S. Meyer,<fnref n="4" />,³²,²⁴,²⁹ R. Miquel,⁸⁸,²⁰ J. Montgomery,⁴¹ J. Myles,⁸⁹ T. Natoli,²⁴,<fnref n="4" /> A.

Navarro-Alsina,⁹⁰ J. P. Nibarger,<fnref n="18" /> G. I. Noble,⁹¹ V. Novosad,⁹² R. L. C. Ogando,⁹³ Y. Omori,<fnref n="4" /> S. Padin,⁹⁴

S. Pandey,²⁵ P. Paschos,²⁹ S. Patil,<fnref n="17" /> A. Pieres,<fnref n="9" />,⁹³ A. A. Plazas Malagón,⁸¹,<fnref n="13" /> A. Porredon,⁹⁵ J. Prat,²⁴,<fnref n="4" />

C. Pryke,⁹⁶ M. Raveri,⁹⁷ C. L. Reichardt,<fnref n="17" /> J. Roberson,²² R. P. Rollins,⁴⁶ C. Romero,⁷⁵ A. Roodman,⁸¹,<fnref n="13" />

J. E. Ruhl,⁹⁸ E. S. Rykoff,⁸¹,<fnref n="13" /> B. R. Saliwanchik,⁹⁹ L. Salvati,¹⁰⁰,¹⁰¹,¹⁰² C. Sánchez,²⁵ E. Sanchez,¹⁰³ D. Sanchez
Cid,¹⁰³ A. Saro,¹⁰⁴,¹⁰²,¹⁰¹,¹⁰⁵,¹⁰⁶ K. K. Schaffer,<fnref n="4" />,²⁹,¹⁰⁷ L. F. Secco,<fnref n="4" /> I. Sevilla-Noarbe,¹⁰³ K. Sharon,¹⁰⁸

E. Sheldon,¹⁰⁹ T. Shin,¹¹⁰ C. Sievers,⁴⁴ G. Smecher,⁴¹,¹¹¹ M. Smith,¹¹² T. Somboonpanyakul,¹¹³ M. Sommer,<fnref n="6" />
B. Stalder,⁷⁵ A. A. Stark,⁷⁵ J. Stephen,²⁹ V. Strazzullo,¹⁰¹,¹⁰² E. Suchyta,¹¹⁴ G. Tarle,<fnref n="14" /> C. To,⁷³

M. A. Troxel,⁴⁰ C. Tucker,<fnref n="8" /> I. Tutusaus,¹¹⁵ T. N. Varga,¹¹⁶,<fnref n="5" />,¹¹⁷ T. Veach,¹¹⁸ J. D. Vieira,³⁶,⁷⁰ A. Vikhlinin,⁷⁵

A. von der Linden,¹¹⁰ G. Wang,<fnref n="3" /> N. Weaverdyck,<fnref n="14" />,⁵³ J. Weller,<fnref n="5" />,¹¹⁷ N. Whitehorn,¹¹⁹ W. L. K. Wu,<fnref n="13" /> B. Yanny,<fnref n="10" />
V. Yefremenko,<fnref n="3" /> B. Yin,³⁰ M. Young,⁹¹ J. A. Zebrowski,<fnref n="4" />,²⁴,<fnref n="10" /> Y. Zhang,<fnref n="7" /> H. Zohren,<fnref n="6" /> and J. Zuntz¹²⁰

(the SPT and DES Collaborations)

<footnote n="1">¹University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany</footnote>

<footnote n="2">²Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria</footnote>

<footnote n="3">³High-Energy Physics Division, Argonne National Laboratory,
9700 South Cass Avenue, Lemont, IL 60439, USA</footnote>

<footnote n="4">⁴Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA</footnote>

<footnote n="5">⁵Max Planck Institute for Extraterrestrial Physics, Gießenbachstr. 1, 85748 Garching, Germany</footnote>

<footnote n="6">⁶Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany</footnote>

<footnote n="7">⁷Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile</footnote>

<footnote n="8">⁸School of Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United Kingdom</footnote>

<footnote n="9">⁹Laboratório Interinstitucional de e-Astronomia - LINEA,
Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil</footnote>

<footnote n="10">¹⁰Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="11">¹¹Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA</footnote>

<footnote n="12">¹²Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA</footnote>

<footnote n="13">¹³SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA</footnote>

<footnote n="14">¹⁴Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="15">¹⁵Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="16">¹⁶Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="17">¹⁷School of Physics, University of Melbourne, Parkville, VIC 3010, Australia</footnote>

<footnote n="18">¹⁸NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO 80305, USA</footnote>

<footnote n="19">¹⁹Department of Physics, University of Colorado, Boulder, CO 80309, USA</footnote>
²⁰ Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology,
Campus UAB, 08193 Bellaterra (Barcelona), Spain

²¹ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK

²² Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA

²³ Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390, USA

²⁴ Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA

²⁵ Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA

²⁶ Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK

²⁷ Department of Physics and Astronomy, University of Missouri,
5110 Rockhill Road, Kansas City, MO 64110, USA

²⁸ Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK

²⁹ Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

³⁰ Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA

³¹ Institute of Cosmology & Gravitation, University of Portsmouth,
Dennis Sciama Building, Portsmouth, PO1 3FX, UK

³² Department of Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA

³³ Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain

³⁴ Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain

³⁵ Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA

³⁶ Department of Astronomy, University of Illinois Urbana-Champaign,
1002 West Green Street, Urbana, IL 61801, USA

³⁷ Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

³⁸ Institute of Space Sciences (ICE, CSIC), Campus UAB,
Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

³⁹ Physics Department, William Jewell College, Liberty, MO, 64068

⁴⁰ Department of Physics, Duke University Durham, NC 27708, USA

⁴¹ Department of Physics and McGill Space Institute, McGill University,
3600 Rue University, Montreal, Quebec H3A 2T8, Canada

⁴² School of Mathematics, Statistics & Computer Science,
University of KwaZulu-Natal, Durban, South Africa

⁴³ NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA

⁴⁴ University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

⁴⁵ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91011, USA

⁴⁶ Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK

⁴⁷ Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy

⁴⁸ INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy

⁴⁹ Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

⁵⁰ Cornell University, Ithaca, NY 14853, USA

⁵¹ Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany

⁵² School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia

⁵³ Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

⁵⁴ Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India

⁵⁵ Institute of Particle and Nuclear Studies (IPNS),
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

⁵⁶ International Center for Quantum-field Measurement Systems for Studies of the Universe and Particles (QUP),
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

⁵⁷ Canadian Institute for Advanced Research, CIFAR Program in
Gravity and the Extreme Universe, Toronto, ON, M5G 1Z8, Canada

⁵⁸ NSF AI Planning Institute for Physics of the Future,
Carnegie Mellon University, Pittsburgh, PA 15213, USA

⁵⁹ Université Grenoble Alpes, CNRS, LPSC-IN2P3, 38000 Grenoble, France

⁶⁰ Department of Physics and Astronomy, University of Waterloo,
200 University Ave W, Waterloo, ON N2L 3G1, Canada

⁶¹ Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Dr., Pasadena, CA 91109, USA

⁶² Department of Astrophysical and Planetary Sciences,
University of Colorado, Boulder, CO 80309, USA

⁶³ Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway

⁶⁴ Harvey Mudd College, 301 Platt Boulevard, Claremont, CA 91711, USA
⁶⁵ Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, 28049 Madrid, Spain

⁶⁶European Southern Observatory, Karl-Schwarzschild-Str.,DE-85748 Garching b. Munchen, Germany

⁶⁷CSIRO Space & Astronomy, PO Box 1130, Bentley WA 6102, Australia

⁶⁸School of Physics and Astronomy, Cardiff University, CF24 3AA, UK

⁶⁹Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290 Versoix, Switzerland

⁷⁰Department of Physics, University of Illinois Urbana-Champaign,
1110 West Green Street, Urbana, IL 61801, USA

⁷¹Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA

⁷²Department of Physics, University of California, Berkeley, CA 94720, USA

⁷³Center for Cosmology and Astro-Particle Physics,
The Ohio State University, Columbus, OH 43210, USA

⁷⁴Department of Physics, The Ohio State University, Columbus, OH 43210, USA

⁷⁵Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

⁷⁶Department of Physics, University of California, One Shields Avenue, Davis, CA 95616, USA

⁷⁷Department of Astronomy/Steward Observatory, University of Arizona,
933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

⁷⁸Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia

⁷⁹Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA

⁸⁰Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

⁸¹Kavli Institute for Particle Astrophysics & Cosmology,
P. O. Box 2450, Stanford University, Stanford, CA 94305, USA

⁸²Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK

⁸³Centre for Extragalactic Astronomy, Durham University, South Road, Durham DH1 3LE, UK

⁸⁴Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK

⁸⁵George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA

⁸⁶Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA

⁸⁷LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France

⁸⁸Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain

⁸⁹Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA

⁹⁰Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil

⁹¹David A. Dunlap Department of Astronomy & Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada

⁹²Materials Sciences Division, Argonne National Laboratory,
9700 South Cass Avenue, Lemont, IL 60439, USA

⁹³Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil

⁹⁴California Institute of Technology, 1200 East California Boulevard., Pasadena, CA 91125, USA

⁹⁵Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute,
German Centre for Cosmological Lensing, 44780 Bochum, Germany

⁹⁶School of Physics and Astronomy, University of Minnesota,
116 Church Street SE Minneapolis, MN 55455, USA

⁹⁷Department of Physics, University of Genova and INFN, Via Dodecaneso 33, 16146, Genova, Italy

⁹⁸Department of Physics, Case Western Reserve University, Cleveland, OH 44106, USA

⁹⁹Brookhaven National Laboratory, Upton, NY 11973, USA

¹⁰⁰Université Paris-Saclay, CNRS, Institut d'Astrophysique Spatiale, 91405, Orsay, France

¹⁰¹INAF - Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, 34143 Trieste, Italy

¹⁰²IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

¹⁰³Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain

¹⁰⁴Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, 34131 Trieste, Italy

¹⁰⁵INFN - National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy

¹⁰⁶ICSC - Italian Research Center on High Performance Computing, Big Data and Quantum Computing, Italy

¹⁰⁷Liberal Arts Department, School of the Art Institute of Chicago,
112 South Michigan Avenue, Chicago, IL 60603, USA

¹⁰⁸Department of Astronomy, University of Michigan,
1085 S. University Ave, Ann Arbor, MI 48109, USA

¹⁰⁹Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA

¹¹⁰Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA

¹¹¹Three-Speed Logic, Inc., Victoria, B.C., V8S 3Z5, Canada

¹¹²School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK

¹¹³Department of Physics, Faculty of Science, Chulalongkorn University,
254 Phayathai Road, Pathumwan, Bangkok 10330, Thailand

¹¹⁴Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
We present cosmological constraints from the abundance of galaxy clusters selected via the thermal Sunyaev-Zel'dovich (SZ) effect in South Pole Telescope (SPT) data with a simultaneous mass calibration using weak gravitational lensing data from the Dark Energy Survey (DES) and the *Hubble Space Telescope* (HST). The cluster sample is constructed from the combined SPT-SZ, SPTpol ECS, and SPTpol 500d surveys, and comprises 1,005 confirmed clusters in the redshift range 0.25 - 1.78 over a total sky area of 5,200 deg². We use DES Year 3 weak-lensing data for 688 clusters with redshifts $z < 0.95$ and HST weak-lensing data for 39 clusters with $0.6 < z < 1.7$. The weak-lensing measurements enable robust mass measurements of sample clusters and allow us to empirically constrain the SZ observable-mass relation without having to make strong assumptions about, e.g., the hydrodynamical state of the clusters. For a flat $\Lambda$CDM cosmology, and marginalizing over the sum of massive neutrinos, we measure $\Omega_m = 0.286 \pm 0.032$, $\sigma_8 = 0.817 \pm 0.026$, and the parameter combination $\sigma_8 (\Omega_m/0.3)^{0.25} = 0.805 \pm 0.016$. Our measurement of $S_8 \equiv \sigma_8 \sqrt{\Omega_m/0.3} = 0.795 \pm 0.029$ and the constraint from *Planck* CMB anisotropies (2018 TT,TE,EE+lowE) differ by $1.1\sigma$. In combination with that *Planck* dataset, we place a 95% upper limit on the sum of neutrino masses $\sum m_\nu < 0.18$ eV. When additionally allowing the dark energy equation of state parameter $w$ to vary, we obtain $w = -1.45 \pm 0.31$ from our cluster-based analysis. In combination with *Planck* data, we measure $w = -1.34_{-0.15}^{+0.22}$, or a $2.2\sigma$ difference with a cosmological constant. We use the cluster abundance to measure $\sigma_8$ in five redshift bins between 0.25 and 1.8, and we find the results to be consistent with structure growth as predicted by the $\Lambda$CDM model fit to *Planck* primary CMB data.

# I. INTRODUCTION

Understanding the cause for the accelerated expansion of our universe is currently one of the biggest challenges in physics. To address this robustly, a multitude of different probes and techniques have been proposed and are being used to observationally constrain the cosmological parameters. Interestingly, the comparison of results obtained from these probes have recently uncovered two anomalies: First, the present-day value of the Hubble parameter determined from primary anisotropies of the cosmic microwave background (CMB) at high redshift is in at least $4\sigma$ tension with measurements in the local universe using the distance ladder e.g., [1-4]. Second, measurements of the $S_8$-parameter<fnref n="1" /> by low-redshift probes have tended to be lower than predicted by the $\Lambda$CDM model with parameters set by high-redshift CMB data e.g., [5-13].

Over recent years, new large-scale structure probes such as joint analyses of galaxy clustering and weak-lensing 2-point correlation functions (so-called 3×2 pt analyses) [8, 9, 14], the CMB lensing power spectrum [15-17], or the combination of all of these [18] have emerged. Measurements of the abundance of massive dark mat-

ter halos (and the galaxy clusters they host) probe the matter density field through a different mechanism and are subject to different systematics, thus offering an ideal and powerful complement [19, 20]. However, for a study of halo abundance to be successful, the relationship between the underlying, un-observable halo mass and the observable properties of the clusters needs to be known (see e.g., reviews [21, 22]).

In this work, we present cosmological constraints derived from a sample of 1,005 galaxy clusters detected using the South Pole Telescope (SPT) [23] in combination with galaxy and weak-lensing data from the Dark Energy Survey (DES) [24-26], photometric data from the Wide-field Infrared Survey Explorer (WISE) [27], and targeted weak-lensing measurements from the *Hubble Space Telescope* (HST). We select the galaxy clusters via the thermal Sunyaev-Zel'dovich (hereafter SZ) [28] effect in data from the first two SPT cameras, SPT-SZ and SPTpol (SPT-3G being the third, currently operating camera). The SZ effect is sourced by the electron pressure in the hot gas component of the cluster (the intra-cluster medium, or ICM) and arises when CMB photons scatter with these high-energy electrons, leading to a distinct spectral signature. The SZ effect signal is thus a clean tracer of the hot ICM and enables the detection of galaxy clusters out to the highest redshifts at which massive halos exist, thereby creating an essentially mass-limited sample whose limiting mass does not evolve strongly with redshift. Optical and near-infrared data from DES, WISE, and other targeted observation pro-

<footnote n="115"><sup>115</sup>Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse,
CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France</footnote>

<footnote n="116"><sup>116</sup>Excellence Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany</footnote>

<footnote n="117"><sup>117</sup>Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany</footnote>

<footnote n="118"><sup>118</sup>Space Science and Engineering Division, Southwest Research Institute, San Antonio, TX 78238, USA</footnote>

<footnote n="119"><sup>119</sup>Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA</footnote>

<footnote n="120"><sup>120</sup>Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK</footnote>

<footnote>(Phys. Rev. D accepted 24 May 2024)</footnote>

<footnote>* sebastian.bocquet@physik.lmu.de</footnote>

<footnote n="1">¹ $S_8 \equiv \sigma_8 \sqrt{\Omega_m/0.3}$ is a combination of the amplitude of fluctuations in the linear matter density field on scales of $8h^{-1}\text{Mpc}$ ($\sigma_8$) and the matter density $\Omega_m$.</footnote>
grams are then used to confirm clusters (and to exclude false detections) and to assign redshifts to all confirmed systems. With these follow-up data taken from wide-field surveys, we can robustly calibrate the probability of chance association of random optical structures with an SZ noise fluctuations through measurements along random lines of sight [29, 30].

To relate the strength of the SZ effect signal to the underlying halo mass, we cannot rely on first principles because modeling the dynamical state of the ICM is highly complex. In particular, the ICM cannot be assumed to be in hydrostatic equilibrium. Instead, we use measurements of the weak gravitational shear, which depends on the total halo mass irrespective of its dynamical state, and which can thus be mapped to halo mass with excellent control over systematic uncertainties (for a review on cluster lensing, see [31]). We now leverage the strategic overlap of the SPT survey and DES footprints and—for the first time—use DES lensing data for mass calibration in a cosmological analysis of SPT-selected clusters. Within our modeling framework, we empirically constrain the SZ-mass relation in a robust, weak-lensing-based way, thereby addressing and overcoming the main challenge in cluster cosmology: the accurate calibration of cluster masses.

Using 5,200 deg<fnref n="2" /> of SPT cluster surveys, we select 1,005 clusters at redshift $z > 0.25$. In the 3,567 deg<fnref n="2" /> overlap region between the DES and SPT surveys, we obtain measurements of weak-lensing shear from DES Year 3 data (hereafter DES Y3) for 688 clusters. Since DES lensing runs out of constraining power for high-redshift ($z \gtrsim 0.9$) clusters, we supplement the DES lensing dataset with targeted lensing observations of 39 clusters in the redshift range $0.6 < z < 1.7$ using the HST. This joint cluster and lensing dataset represents a significant improvement over the previous analysis of SPT clusters [7], which was based on 344 clusters in the SPT-SZ survey [32]. In that analysis, we used cluster lensing data from targeted programs at Magellan (19 clusters, [33]) and HST (13 clusters, [34]). The reduced statistical and systematic uncertainties of the dataset we consider here enables significantly improved cosmological constraints.

In this paper, we present cosmological constraints from the joint analysis of the SPT cluster abundance with a weak-lensing mass calibration using DES and HST data. The data products, the cluster lensing measurements using DES Y3 data, the modeling choices, the analysis pipeline, and its validation are presented in detail in a companion paper ([35], hereafter Paper I). We briefly review the dataset in Sec. II. We summarize the analysis method in Sec. III. We discuss our blinding strategy and the robustness tests that were performed prior to unblinding of the results in Sec. IV. We present our cosmological constraints in Sec. V, and close with a summary in Sec. VI.

Throughout this analysis, we assume spatial flatness. The parameter combination $S_8 \equiv \sigma_8 (\Omega_m/0.3)^{0.5}$ is defined to be tightly constrained by cosmic shear. We

empirically determine that our cluster dataset optimally constrains a parameter combination with an exponent of 0.225. However, for comparability, we adopt $S_8^{\text{opt}} \equiv \sigma_8 (\Omega_m/0.3)^{0.25}$, which is the optimal combination for CMB lensing [17].<fnref n="2" /> Halo masses $M_{200c}$ refer to the mass enclosed within a sphere of radius $r_{200c}$, within which the mean density is 200 times larger than the critical density $\rho_c(z)$ at the cluster redshift $z$. We express the (multivariate) normal distribution with mean $\mu$ and (co)variance $K$ as $N(\mu, K)$.

## II. DATA

We summarize the data products used in this analysis. For a detailed description, we refer the reader to Paper I.

### A. SPT Cluster Catalog

We use the cluster catalogs from the SPT-SZ, SPT-pol ECS, and SPTpol 500d surveys [30, 32, 36, 37]. Combined, they cover over 5,200 deg<fnref n="2" /> of the southern sky. Note that the deep SPTpol 500d survey lies within the footprint of the SPT-SZ survey; in the overlap area, we use only data from SPTpol 500d. The cluster candidate list is selected on the basis of SPT detection significance $\xi$ and redshift $z$.

Over the part of the SPT survey that is not covered by DES (27% of the total SPT-SZ and SPTpol cluster survey area), we confirm clusters and assign redshifts using targeted optical observations (using among others, the PISCO imager [38]) as described in [30, 32]. We select all candidates with

$$ \begin{cases} \xi > 5, \\ z > 0.25, \end{cases} \qquad (1) $$

which, after follow up, results in a highly pure cluster sample ($\gtrsim 95\%$) [30, 32].

Over the 3,567 deg<fnref n="2" /> of the SPT survey that is covered by DES (73% of the total survey area), we obtain measurements of optical richness $\lambda$ and of redshift $z$ using the multi-component matched filter cluster confirmation tool (MCMF) [29, 36]. At high cluster redshifts $z > 1.1$, which are beyond the reach of DES, we use data from WISE to compute richness and redshift. To characterize the probability of chance associations, we also compute richness and redshift for random lines of sight in the DES footprint. With this information, we can define a minimum richness $\lambda_{\text{min}}$ above which we consider an SZ cluster candidate as confirmed. In practice, $\lambda_{\text{min}}$ depends on redshift, and we define $\lambda_{\text{min}}(z)$ such that the

<footnote n="2">² The uncertainties in our constraints on $\sigma_8 (\Omega_m/0.3)^{0.225}$ and $\sigma_8 (\Omega_m/0.3)^{0.25}$ differ by less than a percent.</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. Projection of the 1,005 clusters in the SPT sample along the dimensions of redshift, SPT detection significance, and optical richness. Orange histograms show the data. Thick lines show the mean recovered model in an analysis of the cluster abundance and cluster lensing, thin solid lines show the 1σ uncertainty on the mean, and dotted lines show the 1σ Poisson uncertainty (statistical shot noise). There is qualitative agreement between the model and the data.</figcaption></figure>

final sample purity is > 98% (see Fig. 2 in Paper I).<fnref n="3" /> At fixed SPT detection significance, the different SPT surveys have different levels of purity because of the varying depths. To keep the overall sample purity approximately constant, over the joint SPT-DES area, we select clusters according to

$$ 
\begin{align} 
\xi &> 4.25 / 4.5 / 5 \ (500d / SZ / ECS), \\ 
\lambda &> \lambda_{\min}(z), \\ 
z &> 0.25. 
\end{align} 
\tag{2} $$

The final sample comprises 1,005 confirmed clusters with redshift measurements.<fnref n="4" /> All clusters in the DES region also have richness measurements and optically determined center positions (from DES or WISE). Figure 1 shows the distribution of clusters as a function of redshift, SPT detection significance, and optical richness.

## B. DES Y3 Weak-Lensing Data

The DES covers 5,000 deg² of the southern sky in the g, r, i, z, and Y bands. The DES Y3 cosmology dataset covers 4,143 deg² after masking, of which 3,567 deg² overlap with the SPT cluster surveys. The weak-lensing shape catalog [39] is created by applying the METACALIBRATION pipeline [40, 41] to data from the r, i, and z bands. Detailed information about the photometric dataset [42], the modeling of the point-spread function [43], and image and survey simulations [44, 45] can be found in dedicated

DES Y3 publications. Following the 3×2 pt analysis [9], we select lensing source galaxies in four tomographic bins. The redshift distributions of these source bins are calibrated using self-organizing maps [46].

For every SPT cluster in the DES footprint, we extract a weak-lensing shear profile within the radial range $0.5 < r/(h^{-1}Mpc) < 3.2(1+z_{cluster})^{-1}$, centered on the optically determined cluster center.<fnref n="5" /> This radial cut avoids the problematic central region of the cluster (affected by feedback from active galactic nuclei, miscentering, blending, cluster member contamination, non-linear shear) and ensures that only the 1-halo term regime is considered [47]. We restrict the use of DES weak-lensing data to clusters below redshift $z = 0.95$, which corresponds to the median redshift of the highest-redshift source bin. We extract 688 cluster shear profiles from a total of 555,912 source galaxies. For illustrative purposes, we show stacked shear profiles in bins of cluster redshift and SPT detection significance in Fig. 2.

For the purpose of robustness checks (see Sec. IV), we also measure shear profiles in a more conservative radial range $0.8 < r/(h^{-1}Mpc) < 3.2(1+z_{cluster})^{-1}$. As another alternative, we measure shear around the cluster centers as determined in the SZ analysis. We estimate the level of cluster member contamination using two different redshift estimators (DNF [48] as the default, and BPZ [49] as an alternative).

We quantify and discuss the relevant sources of systematic and statistical uncertainties related to the weak-lensing measurements (cluster member contamination, miscentering of the shear profile, shear and photo-z calibrated

<footnote n="3">³ Any remaining level of sample contamination is thus within the shot noise of the sample.</footnote>

<footnote n="4">⁴ The statistical uncertainties in the redshift measurements do not limit the cosmological constraining power of our cluster dataset.</footnote>

<footnote n="5">⁵ We convert the measured angular separations to physical separations assuming $\Omega_m = 0.3$.</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 2. Stacked DES Y3 lensing shear profiles. Upper panel: Tangential shear, along with the mean model prediction from an analysis of the cluster abundance and cluster lensing. The likelihood analysis is performed on a cluster-by-cluster basis, the stacks are only shown for validation purposes. The dark shading shows the radial ranges used in the analysis, light shaded data points outside that range are only shown for reference. For the 26 data points included in the analysis, we measure a signal-to-noise ratio of 32.2 and we obtain a reduced $\chi^2_{red} = 1.48$ between the data and the model. Lower panel: Within the radial ranges used in the analysis, the cross shear is consistent with null with $\chi^2_{red} = 0.90$.</figcaption></figure>

bration, halo mass modeling, impact of large-scale structure) in detail in Paper I, Sec. V.

### C. High-Redshift HST Weak-Lensing Data

The DES lensing data does not have substantial constraining power for lenses with $z \gtrsim 0.9$, and we use HST data to enable weak-lensing mass calibration at higher redshifts. Using the HST-39 dataset presented and analyzed in [34, 50–53], we have a sample of 39 clusters in the redshift range 0.6–1.7 with space-based weak-lensing measurements.

## III. ANALYSIS METHOD

The analysis method is introduced and described in detail in Paper I. Here, we briefly review the key features.

### A. Summary of Analysis Strategy

We set up a Bayesian (hierarchical) population model to describe our dataset (see also [54–56]). We explicitly model the selection criteria that we adopted to construct the cluster catalog. Specifically, these are cuts in the SPT detection significance $\xi$, cuts in the optical richness $\lambda$ (over the footprint that is common with DES), and a cut in redshift [see Eqs. (1) and (2)]. Clusters with weak-lensing data additionally have measurements of tangential shear profiles.

We follow analytical and simulation-based models for observable-mass relations e.g., [57, 58], and describe the scaling relation between the unbiased SPT detection significance $\zeta$ and mass as

$$ \langle \ln \zeta \rangle = \ln A_{SZ} + B_{SZ} \ln \left( \frac{M_{200c}}{3 \times 10^{14} h^{-1} M_{\odot}} \right) + C_{SZ} \ln \left( \frac{E(z)}{E(0.6)} \right) \quad (3) $$

with lognormal intrinsic scatter of width $\sigma_{\ln \zeta}$, and relate the detection significance $\xi$ to $\zeta$ as

$$ P(\xi|\zeta) = \mathcal{N}\left(\sqrt{\zeta^2 + 3}, 1\right) \quad (4) $$

to account for the maximization bias when measuring $\xi$ in noisy data [59]. We account for the varying depth within the SPT survey by rescaling $A_{SZ}$ and $C_{SZ}$ for each individual field [30, 32, 37]. For the SPTpol ECS survey, the overall normalization $\gamma_{ECS}$ of $A_{SZ}$ is difficult to calibrate (owing to detector linearity changes under high atmospheric loading at the elevation of these fields, see Sec. 2.2 [30]), and we thus consider $\gamma_{ECS}$ as an additional fit parameter.

We choose the same ansatz for the relation between the intrinsic richness $\tilde{\lambda}$ and mass

$$ \langle \ln \tilde{\lambda} \rangle = \ln A_{\lambda} + B_{\lambda} \ln \left( \frac{M_{200c}}{3 \times 10^{14} h^{-1} M_{\odot}} \right) + C_{\lambda} \ln \left( \frac{1+z}{1.6} \right), \quad (5) $$

with lognormal intrinsic scatter of width $\sigma_{\ln \tilde{\lambda}}$, and relate the observed richness $\lambda$ to $\tilde{\lambda}$ as

$$ P(\ln \lambda | \ln \tilde{\lambda}) = \mathcal{N}(\ln \tilde{\lambda}, 1/\tilde{\lambda}). \quad (6) $$

We consider two different types of measured richness; those based on DES data, which we use for clusters with redshift $z < 1.1$, and those based on WISE, which we use for clusters with redshifts beyond 1.1. Rather than attempting to match the two types of richness within an overlap range in redshift, we fit a separate observable-mass relation for each of the two types of richness, with a clean transition from DES-based to WISE-based richness at redshift $z = 1.1$.

Using the relations defined in Eqs. (3)–(6), we can model the $\xi-\lambda-z$ sample by convolving the halo mass
function with the observable-mass relations (accounting for the covariance in intrinsic scatter $\rho_{SZ,\tilde{\lambda}}\sigma_{\ln\zeta}\sigma_{\ln\tilde{\lambda}}$) and applying the sample selection cuts. There is no external information on the parameters A, B, C of the scaling relations or on the intrinsic scatter $\sigma_{\ln\zeta}$, $\sigma_{\ln\tilde{\lambda}}$, and we use weak-lensing data of sample clusters to calibrate these observable-mass relation parameters empirically. Because we can infer the halo mass from lensing data with excellent control over systematic uncertainties, this is a robust analysis strategy.

We introduce the modeling framework with which we analyze the DES Y3 lensing data in Paper I. Here, we briefly review the method. We adopt a simple model for the projected halo mass distribution $\Sigma$ based on a modified Navarro-Frenk-White profile (NFW) [60].<fnref n="6" /> We introduce a latent variable that we call the “weak-lensing mass” $M_{WL}$, which we define such that the reduced tangential shear is

$$g_t(r, M_{WL}) = \frac{\Delta\Sigma(r, M_{WL}) \Sigma_{crit}^{-1}}{1 - \Sigma(r, M_{WL}) \Sigma_{crit}^{-1}} (1 - f_{cl}(r)). \quad (7)$$

The density contrast $\Delta\Sigma(r) \equiv \langle\Sigma(<r)\rangle - \Sigma(r)$ is computed from $\Sigma$, and the lensing efficiency $\Sigma_{crit}^{-1}$ is computed from the distribution of source redshifts and the lens redshift. Note that $\Sigma_{crit}^{-1}$ explicitly depends on cosmology. The cluster member contamination $f_{cl}(r)$ accounts for contaminants in the lensing source sample that are not sheared and thus bias the measurement low [therefore, $(1 - f_{cl})^{-1}$ is often referred to as the boost factor].

The cluster lensing model we just introduced is not perfect (e.g., halos do not individually match the NFW profile) and $M_{WL}$ inferred from the shear profile $g_t$ is thus a biased and noisy estimator of the true halo mass $M_{halo}$ [61, 62]. To account for this, we establish a mean relationship,

$$\left\langle \ln \left( \frac{M_{WL}}{M_0} \right) \right\rangle = b_{WL}(z) + b_{WL,M} \ln \left( \frac{M_{200c}}{M_0} \right), \quad (8)$$

and describe the width of the lognormal scatter around the mean as

$$\ln \sigma_{\ln M_{WL}} = \frac{1}{2} \left[ s_{WL}(z) + s_{WL,M} \ln \left( \frac{M_{200c}}{M_0} \right) \right], \quad (9)$$

with a pivot mass $M_0 = 2 \times 10^{14} h^{-1} M_{\odot}$.

We create synthetic cluster shear maps by applying the source redshift distribution, cluster miscentering, and cluster member contamination to halo mass maps from numerical simulations, and we then use these maps to calibrate the free parameters of the $M_{WL}-M_{halo}$ relation

[47].<fnref n="7" /> Note that we use the mass maps from full-physics hydrodynamical simulations and establish the relationship between $M_{WL}$ and the “gravity-only mass” $M_{halo}$ as measured in paired gravity-only simulations with identical initial conditions. This has the benefit that our model includes the effect of baryons, but we are still able to use the arguably more robust halo mass function predictions from gravity-only simulations [such as 63] (including emulators e.g., [64–66]). In practice, we apply this method to the Magneticum [67–70] and the Illustris TNG simulations [71–76]. The recovered constraints on the $M_{WL}-M_{halo}$ relation differ somewhat, which we interpret as an uncertainty in the modeling of baryonic effects. We inflate the uncertainty on all parameters of the $M_{WL}-M_{halo}$ relation by this additional uncertainty given in Table 2 [47] that, e.g., for the amplitude $b_{WL}$, amounts to 2%. As we will show below, this level of systematic uncertainty is smaller than the current level of statistical uncertainty in the lensing dataset, and our analysis therefore does not strongly depend on the assumption that the Magneticum or Illustris TNG simulation correctly reproduces the real universe.

For the HST-39 dataset, we use a similar modeling approach, and also adopt $M_{WL}$ as a latent variable. Note, however, that several analysis choices differ. For example, the selection of HST lensing sources via color cuts leads to a much smaller level of cluster member contamination than in the DES Y3 analysis. Therefore, we do not explicitly model this contamination, and we do not explicitly model the effect of miscentering, either. We do, of course, take all sources of uncertainty into account in the $M_{WL}-M_{halo}$. The full HST cluster lensing model is presented in detail in the original analyses [34, 52, 53].

## B. Likelihood, Pipeline, Priors, and Sampling

We infer parameters $p$ from our data assuming a cluster population model. Bayes’ theorem states

$$P(p|\text{data, model}) \propto P(\text{data}|p, \text{model}) P(p|\text{model}), \quad (10)$$

where $P(p|\text{model})$ is the prior probability distribution of the parameters. The factor $P(\text{data}|p, \text{model})$ is the likelihood $\mathcal{L}$ of the data given the (fixed) model and the parameters. A detailed discussion of our likelihood function and its numerical implementation can be found in Paper I, Sec. VII. Briefly, we adopt a log-likelihood

$$\ln \mathcal{L} = \ln \text{Poisson} \left( \text{sample } \{\xi_i, \lambda_i, z_i\}_{i=1}^{N_{\text{clusters}}} \left| \frac{d^3 N(\mathbf{p})}{d\xi \, d\lambda \, d\mathbf{z}} \right. \right) \\ + \sum_i \ln P(\lambda_i, g_{t,i} | \lambda_i > \lambda_{\min}(z), \xi_i, z_i, \mathbf{p}), \quad (11)$$

<footnote n="6">⁶ To approximately account for the effect of miscentering, we modify the NFW profile such that it is constant within the typical cluster miscentering radius $R_{mis}$ (see Paper I, Sec. V B).</footnote>

<footnote n="7">⁷ All systematic and stochastic uncertainties in the lensing model and data, in halo morphology, and in the projected large-scale structure along the cluster line of sight are thus accounted for in our $M_{WL}-M_{halo}$ relation.</footnote>
where the first term is the unbinned Poisson likelihood of the cluster sample {$\xi, \lambda, z$} and its distribution $\frac{d^3N(p)}{d\xi d\lambda dz}$ in $\xi-\lambda-z$ space, and the second term is the “mass calibration likelihood”: the conditional probability to observe the tangential shear profile $g_t$ and richness $\lambda$ for a given cluster in the sample with measured $\xi$ and $z$, and whose richness is above the minimum richness $\lambda_{min}$.

We implement the likelihood function as a Python module in the COSMOSIS framework [77].<fnref n="8" /> We validate the implementation of the analysis pipeline by analyzing several statistically independent mock catalogs that we create from the model. Because creating those mocks is less challenging than analyzing them, this is a meaningful test of the analysis pipeline (rather than a test of the mock creation). For further details, we refer the reader to Paper I, Sec. VIII.

The parameters that we vary in the baseline analysis are defined in Table I, along with the informative priors where applied. The model for the DES cluster lensing mass bias and scatter are functions of cluster redshift, and the parameters $\sigma_{\ln b_{WL,1}}$ and $s_{WL}$ modulate the uncertainty of the model (therefore, the priors are centered on 0 with unit width) [see Paper I, Sec. V B]. We follow the same approach for the HST lensing data, where each cluster has its own bias and scatter, and the shared uncertainties are modulated via $b_{HST}$ and $s_{HST}$. The priors on the scatter ($> 0.05$) improve the numerical stability of the code. The priors on the correlation coefficients $U(-0.5, 0.5)$ are chosen such that the resulting correlation matrix is nonsingular for all parameter combinations. The parameters $h, \Omega_b h^2$, and $n_s$ are not constrained by the cluster dataset, and are not significantly correlated with the parameters of interest. We apply Gaussian priors on $\Omega_b h^2$ and $n_s$ from Planck [2], and a prior on the Hubble parameter $h \sim N(0.70, 0.05^2)$ that is wide enough to encompass all currently favored constraints. When analyzing the cluster dataset jointly with Planck, we do not apply these three Gaussian priors, and we additionally consider the optical depth to reionization $\tau$ as a free parameter.

We explore the parameter space using either the MULTINEST or the NAUTILUS sampler [78, 79]. The potential amount by which these samplers may or may not underestimate the parameter uncertainties is small and does not change our findings in any qualitatively relevant way [80].

## IV. BLINDING, ROBUSTNESS TESTS, AND UNBLINDING

In this section, we describe our blinding strategy and the robustness tests that were performed prior to unblinding of the results. We check for internal consistency

<table><caption>TABLE I. Fit parameters in the baseline analysis of the abundance of SPT clusters with DES Y3 and HST-39 lensing. A missing entry (-) for the prior indicates that a uniform prior is applied that is wide enough to not be informative. Priors with a lower limit (> ) are uniform above that bound with an un-informative upper bound. The parameters of the lensing models are prior-dominated and inform the empirical calibration of the other observable-mass relations. The prior on $\Omega_{\nu}h^2$ corresponds to a prior on the sum of neutrino masses $\sum m_{\nu} \sim U(0, 0.6)$ eV.</caption><thead><tr><th>Parameter</th><th>Description</th><th>Informative Prior</th></tr></thead><tbody><tr><td colspan="3"><strong>DES Y3 cluster lensing</strong></td></tr><tr><td>σ<sub>ln b<sub>WL,1</sub></sub></td><td>scaling of bias</td><td>N(0, 1)</td></tr><tr><td>σ<sub>ln b<sub>WL,2</sub></sub></td><td>scaling of bias</td><td>N(0, 1)</td></tr><tr><td>b<sub>WL,M</sub></td><td>mass slope of bias</td><td>N(1.029, 0.006<sup>2</sup>)</td></tr><tr><td>s<sub>WL</sub></td><td>scaling of scatter</td><td>N(0, 1)</td></tr><tr><td>s<sub>WL,M</sub></td><td>mass slope of scatter</td><td>N(-0.226, 0.040<sup>2</sup>)</td></tr><tr><td colspan="3"><strong>HST cluster lensing</strong></td></tr><tr><td>b<sub>HST</sub></td><td>overall scaling of bias</td><td>N(0, 1)</td></tr><tr><td>s<sub>HST</sub></td><td>overall scaling of scatter</td><td>N(0, 1)</td></tr><tr><td colspan="3"><strong>SZ-mass parameters</strong></td></tr><tr><td>ln A<sub>SZ</sub></td><td>amplitude</td><td>...</td></tr><tr><td>B<sub>SZ</sub></td><td>mass slope</td><td>...</td></tr><tr><td>C<sub>SZ</sub></td><td>redshift evolution</td><td>...</td></tr><tr><td>σ<sub>ln ζ</sub></td><td>intrinsic scatter</td><td>> 0.05</td></tr><tr><td>γ<sub>ECS</sub></td><td>depth of SPTpol ECS</td><td>...</td></tr><tr><td colspan="3"><strong>DES richness-mass parameters (used for z &lt; 1.1)</strong></td></tr><tr><td>ln A<sub>λ</sub></td><td>amplitude</td><td>...</td></tr><tr><td>B<sub>λ</sub></td><td>mass slope</td><td>...</td></tr><tr><td>C<sub>λ</sub></td><td>redshift evolution</td><td>...</td></tr><tr><td>σ<sub>ln λ̄</sub></td><td>intrinsic scatter</td><td>> 0.05</td></tr><tr><td colspan="3"><strong>WISE richness-mass parameters (used for z &gt; 1.1)</strong></td></tr><tr><td>ln A<sub>λ</sub></td><td>amplitude</td><td>...</td></tr><tr><td>B<sub>λ</sub></td><td>mass slope</td><td>...</td></tr><tr><td>C<sub>λ</sub></td><td>redshift evolution</td><td>...</td></tr><tr><td>σ<sub>ln λ̄</sub></td><td>intrinsic scatter</td><td>> 0.05</td></tr><tr><td colspan="3"><strong>Correlation coefficients</strong></td></tr><tr><td>ρ<sub>SZ,WL</sub></td><td>SZ-weak-lensing</td><td>U(-0.5, 0.5)</td></tr><tr><td>ρ<sub>SZ,λ̄</sub></td><td>SZ-richness</td><td>U(-0.5, 0.5)</td></tr><tr><td>ρ<sub>WL,λ̄</sub></td><td>weak-lensing-richness</td><td>U(-0.5, 0.5)</td></tr><tr><td colspan="3"><strong>Cosmology</strong></td></tr><tr><td>Ω<sub>m</sub></td><td>matter density</td><td>> 0.1</td></tr><tr><td>Ω<sub>ν</sub>h<sup>2</sup></td><td>neutrino density</td><td>U(0, 0.00644)</td></tr><tr><td>Ω<sub>b</sub>h<sup>2</sup></td><td>baryon density</td><td>N(0.02236, 0.00015<sup>2</sup>)</td></tr><tr><td>h</td><td>Hubble parameter</td><td>N(0.7, 0.05<sup>2</sup>)</td></tr><tr><td>ln 10<sup>10</sup> A<sub>s</sub></td><td>amplitude of P(k)</td><td>...</td></tr><tr><td>n<sub>s</sub></td><td>scalar spectral index</td><td>N(0.9649, 0.0044<sup>2</sup>)</td></tr><tr><td>w</td><td>dark energy equation<br>of state parameter</td><td>-1 (ΛCDM)<br>U(-2, -0.4) (wCDM)</td></tr></tbody></table>

<footnote n="8">⁸ https://cosmosis.readthedocs.io/</footnote>
of the dataset and verify the robustness to several analy-
sis choices in the processing of the DES Y3 lensing data.

A. Blinding Strategy

To avoid compromising our analysis with any expec-
tations we may have regarding what the results should
or should not look like, we proceed as follows. First, the
SPT cluster catalogs and the DES Y3 lensing data were
finalized before we started this analysis. Second, we did
not compare any weak-lensing mass estimates or prelimi-
nary SZ-mass relations with literature values. Third, we
blinded the astrophysical and cosmological constraints
obtained in this work until the robustness of the results
against specific analysis choices was demonstrated. To do
so, we created unknown, random, but constant blinding
offset parameters with which we shifted the parameters
when reading in any MCMC chain. It was thus possible
to compare different blinded analyses with each other to
check for relative shifts in the recovered parameters (or
changes in the parameter uncertainties) without knowing
the actual parameter values.

The HST-39 cluster lensing dataset was finalized, and
its analysis published before we started this analysis.
Therefore, the parameter constraints favored by this
dataset were known, and we preferred to not use these
data in the blinded phase of the analysis.

The blinded runs were set up assuming a ΛCDM model
with wide flat priors on the fit parameters. Failure to
choose sufficiently wide priors might lead to the blinded
results showing hard cuts, thereby effectively unblinding
the respective parameter constraint. Note that arbitrar-
ily wide priors are not always possible, because, e.g., the
scatter parameters or Ω_m need to be strictly positive. All
but the first test are summarized in Fig. 3, which shows
the constraints on key parameters. The results in the
first row (blue) are obtained from the full dataset (with-
out HST-39), marginalized over all parameters in Table I.
The second row (black) shows the constraints from the
same analysis, except that we fix a selection of model pa-
rameters to their fiducial values: the parameters of the
M_WL–M_halo relations, the correlation coefficients on in-
trinsic scatter, Ω_bh², Ω_νh², and n_s. The parameter shifts
are small, which also suggests that our constraints are
not limited by the systematic uncertainty in the DES Y3
lensing model. To save computational resources, and to
make the robustness tests somewhat more stringent, all
cross-check analyses discussed in the next section are run
in this restricted parameter space, with a selection of pa-
rameters fixed as just discussed.

Before running the blinded robustness tests, we had
agreed that we would unblind the results if no test re-
veals a statistically significant difference with the base-
line analysis.

B. Tests Performed prior to Unblinding

We now summarize the tests that were conducted with
the blinding setup discussed above.

1. Combination of Mass Calibration and Abundance

We derive constraints from just the cluster abundance
[first term in the likelihood Eq. (11)] and just mass cali-
bration (second term in the equation). We do not expect
interesting constraints on the cosmological parameters
from these individual analyses, and Fig. 4 shows broad
uncertainties in the cosmological parameters and elon-
gated parameter degeneracies (gray and red contours).<fnref n="9" />
The two posterior distributions overlap, and we combine
the two analyses to obtain constraints from the cluster
abundance with weak-lensing-informed mass calibration
(blue contours in Fig. 4). The constraints on select pa-
rameters are shown labeled as “full sample” in Fig. 3.
We note that the situation is similar for the parameters
of the richness-mass relations, for which the individual
constraints overlap, and tighten when we combine the
abundance with the lensing mass calibration.

2. Goodness of Fit

We validate that the model is an adequate descrip-
tion of the data by extracting posterior predictive dis-
tributions from the blinded MCMC chains (of the clus-
ter abundance and lensing analysis) and comparing them
with the data. In Fig. 1, blue lines show the mean pre-
dicted model for the cluster abundance. Similarly, in
Fig. 2, we show the mean predicted stacked shear profiles
and the error on the mean. We compute the statistical
difference between each stack and the mean model, and
obtain values for $\chi^2 \in \{11.3, 8.1, 7.3, 10.6\}$ for $\{7, 7, 6, 6\}$
data points in the $\{(z < 0.5, \xi \ge 5.6), (z < 0.5, \xi < 5.6), (z \ge 0.5, \xi \ge 5.6), (z \ge 0.5, \xi < 5.6)\}$ bins. For the
full dataset, we measure $\chi^2 = 37.4$ for 26 data points,
and conclude that the model adequately describes the
data.

3. Uncertainty in the Halo Mass Function

We marginalize over a 3.5% uncertainty in the am-
plitude and a 2% uncertainty in the slope of the mass

<footnote n="9">⁹ The mass calibration analysis constrains the parameters of the observable-mass relations but does place interesting constraints on the cosmological parameters. The analysis of the abundance measurement cannot tightly constrain the cosmological parameters, either, because these are degenerate with the scaling relation parameters (this is simply a manifestation of the fact that cluster abundance cosmology requires mass information).</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 3. Impact of different analysis choices on the constraints on key cosmological and SZ-mass relation parameters (mean and 68% credible interval), assuming a ΛCDM model. During the blinded phase of the analysis, the parameter values were artificially offset by an unknown amount. The vertical gray bands show the 68% credible interval of the “full sample” analysis (second row) for reference. All results shown here are based on mass calibration using only the DES Y3 lensing data, the HST-39 lensing data are not used. Only the results in the first row shows constraints that are marginalized over all nuisance parameters; for simplicity, for all other tests, we fix the parameters of the $M_{WL}-M_{halo}$ relations, the intrinsic scatter correlation coefficients, and $\Omega_{\nu}h^2$, $\Omega_b h^2$, and $n_s$ to their nominal values. DNF and BPZ are two redshift estimators that we use to determine the amount of cluster member contamination in the lensing signal. MCMF and SPT refer to cluster centers as measured in optical data by MCMF or as determined in the SZ analysis of SPT data.</figcaption></figure>

function (following [81]). The lack of parameter shifts in Fig. 3 suggests that we can neglect uncertainties in the halo mass function, which we will do from here on out.<fnref n="10" />

4. *Restricting the Analysis to Low Redshift $z < 1$*

At redshifts beyond $z \gtrsim 1$, it remains unclear how strong the correlated astrophysical emission, and thus the contamination of the cluster SZ signal is, although observational studies favor an effect that is only at the few-percent level e.g., [82–85]. Also recall that we use DES data for cluster confirmation up to redshift $z = 1.1$, beyond which we use data from WISE. When limiting the analysis to $z < 1$, we thus test the robustness of using WISE data for cluster confirmation and redshift assignment jointly with the robustness against astrophysical contamination of the SZ signal. We caution that disagreement between the fiducial sample and the low-redshift subset could also indicate interesting new physics

(e.g., different growth of structure beyond $z \gtrsim 1$) rather than pointing to the effects we might worry about.

The constraints from the $0.25 < z_{\text{cluster}} < 1$ dataset (869 clusters compared to 1,005 in the full sample) show negligible shifts compared to the analysis of the full sample (see Fig. 3). Since the remaining robustness checks are not expected to be sensitive to the redshift range of the sample, we perform the following tests on the $0.25 < z_{\text{cluster}} < 1$ sample.

5. *The $\xi - \zeta$ Relation*

The $\xi - \zeta$ relation [Eq. (4)] cannot hold down to arbitrarily low significance as the formalism breaks down for $\xi < \sqrt{3}$. As in previous work, we define a limit $\zeta_{\min}$, below which the maximization bias in the $\xi - \zeta$ relation does not apply. Using simulations of the SZ sky, we estimate $\zeta_{\min} = 1$. Here, we confirm that raising the threshold to $\zeta_{\min} = 2$ (which was applied in, e.g., [86]) only has a small impact on the recovered parameters ("$\zeta_{\min}: 1 \rightarrow 2$" in Fig. 3).

<footnote n="10">¹⁰ We note that this test did not necessarily need to be run before unblinding.</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 4. Constraints (68% and 95% credible regions) on select scaling relation and cosmology parameters, as obtained from just the DES Y3 weak-lensing-based mass calibration (gray) or just the cluster abundance (red). The abundance and mass calibration results do not show signs of significant disagreement, and we combine them to obtain cosmological constraints from the joint analysis of cluster abundance and lensing (blue).</figcaption></figure>

6. *Selection Cut in SPT Detection Significance ξ*

Enabled by the DES data and the MCMF analysis,
in this study, we select clusters down to lower detection
significances than previously done [see Eq. (2)]. We com-
pare the constraints obtained using this selection scheme

(869 clusters with $z < 1$) to the constraints obtained us-
ing the more conservative $\xi > 5$ selection (654 clusters
with $z < 1$). As shown in Fig. 3 as “$\xi > 5$” in purple, the
constraint on $S_8^{\text{opt}}$ is robust to this change, but the SZ
mass slope $B_{\text{SZ}}$ and the redshift evolution $C_{\text{SZ}}$ both shift
to higher values, and, due to parameter degeneracies, $\Omega_m$
also shifts to larger values.
We quantify these shifts more carefully. Since $S_8^{\text{opt}}$ is effectively unchanged, and because of the parameter degeneracies, it is sufficient to focus on the constraints on $\Omega_m$. First, we consider the analysis of the “low-redshift” $0.25 < z_{\text{cluster}} < 1$ dataset (green in Fig. 3), and examine the impact of additionally applying $\xi > 5$. Naively, one would compare the difference in the mean recovered values for $\Omega_m$ with the expected variance, computed as the sum of the variances from each analysis. Doing so, we observe a shift in $\Omega_m$ that is at the $0.5\sigma$ level. However, this simple estimate neglects the fact that the $\xi > 5$ sample is a subset of the larger catalog. In this case, “the covariance of the parameter differences between the partial and full analyses is simply the difference of the covariances” [87]. With this estimator, we report that $\Omega_m$ shifts at the $0.9\sigma$ level.

To empirically validate that this level of parameter shift is expected, we analyze pairs of mock catalogs (restricted to $z < 1$) for which one catalog is additionally restricted to $\xi > 5$.<fnref n="11" /> After running two such mock analyses, we do indeed observe parameter shifts that are very similar to the one obtained from the real data.

Finally, we compute the consistency of the $\Omega_m$ measurement between the full cluster sample ($z > 0.25$) and the $\xi > 5$, $0.25 < z < 1$ subsample (purple vs. black in Fig. 3). Naively adding the variances gives a shift at the $0.9\sigma$ level, and correctly subtracting the variances yields $1.2\sigma$. We caution the reader that in this comparison, we consider the combined effects of restricting ourselves to low redshifts (which shifts $\Omega_m$ toward higher values) and to high significances (which further shifts $\Omega_m$ high). To summarize, we find that, while the observed shifts are interesting and might reveal new properties of SZ-selected clusters in upcoming analyses of larger samples, there is no statistically significant evidence of internal tension in the dataset.

## 7. Cluster Member Contamination

In the DES Y3 lensing analysis, we account for cluster member contamination of the lensing source catalog as determined by applying the $P(z)$ decomposition method to DNF source redshifts (see Paper I, Sec. IV D). However, the level of contamination derived using BPZ redshifts is significantly different. We test the robustness of our cosmological results to adopting the cluster member contamination as derived using BPZ instead of the fiducial one derived using DNF. The resulting parameter shifts are $\lesssim 1\sigma$ (see “DES WL boost: DNF $\rightarrow$ BPZ” in Fig. 3). While this level of shifts is acceptable given the overall level of uncertainty, this test highlights the relative importance of the cluster member contamination

in our analysis. Future lensing surveys are expected to enable a cleaner source selection, e.g., [88], which would reduce that relative importance.

## 8. Centers Adopted for the Lensing Analysis

Our default analysis uses lensing data extracted around cluster positions as determined using the DES imaging data. We constrain the optical miscentering distribution and account for it in the lensing analysis (see Paper I, Secs. IV C and V B). As an alternative, we use the cluster centers as determined in the SZ analysis, and account for the corresponding miscentering distribution in the lensing model. The recovered constraints on $A_{\text{SZ}}$ and $S_8^{\text{opt}}$ shift by $\lesssim 1\sigma$, while the other parameters remain essentially unchanged (“DES WL center: MCMF $\rightarrow$ SPT” in Fig. 3). We conclude that our miscentering models and corrections are adequate.

## 9. Radial Range in the Lensing Analysis

Our fiducial setup ($r > 500 h^{-1}\text{kpc}$) is a compromise between avoiding the cluster central regions (where miscentering, cluster member contamination, baryonic feedback, and possible non-linear shear responses are particularly challenging to model), and including scales with high weak-lensing signal-to-noise ratios [47]. We validate this choice by comparing the fiducial results to an analysis with an even more conservative radial cut, $r > 800 h^{-1}\text{kpc}$. The outer limit remains unchanged: $r < 3.2 (1 + z_{\text{cluster}})^{-1} h^{-1}\text{Mpc}$. As expected, the recovered parameter constraints slightly broaden compared to the fiducial setup, but the central values remain almost identical (last row in Fig. 3, “DES WL $r_{\text{min}}$: $500 \rightarrow 800 h^{-1}\text{Mpc}$). This confirms that our model can adequately describe cluster shear profiles down to $r = 500 h^{-1}\text{kpc}$.

## C. Unblinding

The validation tests discussed in the previous section were designed to probe what we expected to be the most probable points of failure of our model. We performed these tests while being blinded to the actual values of the mean recovered parameter constraints. Since no alternative analysis led to significant parameter shifts, we proceeded and removed the so-far unknown blinding parameter offsets. Throughout this work, we present the results as obtained after unblinding, with no further modification of the data or the analysis.

<footnote n="11">¹¹ We use the mock catalogs created for validating the analysis pipeline in Paper I.</footnote>
<figure><img src="image_6.png" /></figure>

FIG. 5. Constraints on $\Omega_m$ and $\sigma_8$ from the abundance of SPT-selected clusters (68% and 95% credible regions). In comparison with the precursor analysis of 343 SPT-SZ clusters (gray contours; [7]), the larger cluster sample and the improved weak-lensing mass calibration enable tighter constraints (red contours). We also show the constraints obtained from the SPT cluster abundance with DES Y3 lensing, without using the HST-39 high-redshift cluster lensing dataset (blue contours). Green contours show the constraints obtained when restricting the cluster sample to $\xi > 5$, as was done in precursor analyses (this result also corresponds to item 6 in Sec. IV).

## V. RESULTS

We present and discuss constraints on the $\Lambda$CDM and $w$CDM models, and on the scaling relation parameters. Our baseline setup includes the sum of neutrino masses $\sum m_\nu$ as a free parameter; however, we can only constrain it in combination with measurements of primary CMB anisotropies from Planck. We also present constraints on an empirical extension of the $\Lambda$CDM model, in which we allow the rate of structure growth to deviate from the standard prediction. We implement this test as a direct measurements of $\sigma_8$ at different redshifts between 0.25 and 1.8.

### A. $\Lambda$CDM with Massive Neutrinos

In Fig. 5, we show the constraints on $\Omega_m$ and $\sigma_8$ as derived from the SPT cluster abundance. We note that supplementing the DES lensing data with high-redshift cluster lensing data from HST leads to a slight shift of the constraints along the degeneracy axis. The results obtained from the fiducial, combined analysis setup—

<figure><img src="image_7.png" /></figure>

FIG. 6. Constraints on $\Omega_m$ and $\sigma_8$ from various analyses of the cluster abundance (68% and 95% credible regions). Along with our work, we also show results obtained from X-ray selected clusters (ROSAT [89], eFEDS [90]) and optically selected clusters (DES Y1 [81, 91], KiDS [92]). Our analysis places the tightest constraints on $\sigma_8$ and $S_8^{\text{opt}}$.

SPT(SZ+pol) cluster abundance with (DES Y3 + HST-39) lensing—are summarized in Table II. As demonstrated in Fig. 5, our new analysis significantly improves over the precursor study, which used 343 clusters in the SPT-SZ survey and lensing information from 32 targeted observations [7]. Our new results are shifted toward higher values of $\sigma_8$, which is caused by using DES Y3 lensing data for mass calibration. Remember that our analysis also uses a larger cluster sample than the predecessor analysis, but the SPT cluster samples are consistent with one another, ruling out the possibility that the shift in $\sigma_8$ can be explained with changes in the cluster sample [30, 37].$^{12}$ Finally, we note that neglecting the systematic uncertainties in the lensing model only has a small impact on the recovered constraints (first vs. second row in Fig. 3). This implies that our constraints are still limited by the precision of the lensing measurements, rather than the uncertainties in the modeling. The statistical uncertainty will be improved in future analyses by increasing the number of lenses (larger cluster sample), and/or by using lensing datasets with higher source densities and higher lensing efficiencies.

<footnote n="12">12 We checked that analyzing the SPT cluster sample used in this work with the lensing data used in [7] leads to results that are centered on the gray contours in Fig. 5, whereas analyzing the 343 SPT-SZ clusters with the DES Y3 + HST-39 lensing data leads to constraints centered on the results presented in this work.</footnote>
<table><caption>TABLE II. Parameter constraints for the ΛCDM and wCDM models (mean and 68% credible intervals, or 95% limit). The SPTclusters+lensing dataset corresponds to SPT(SZ+pol) clusters + (DES Y3 + HST-39) WL. The parameters $n_s$ and $\Omega_b h^2$ are not meaningfully constrained by the cluster dataset and the results are thus omitted here. Similarly, the cluster dataset cannot constrain $h$ and $\sum m_\nu$ on its own, and we only quote the results recovered in combination with Planck 2018 TT,TE,EE+lowE [2].</caption><thead><tr><th>Dataset</th><th>Ω<sub>m</sub></th><th>σ<sub>8</sub></th><th>S<sub>8</sub></th><th>S<sub>8</sub><sup>opt</sup></th><th>h</th><th>Σ m<sub>ν</sub> [eV]</th><th>w</th></tr></thead><tbody><tr><td colspan="8"><strong>ΛCDM</strong></td></tr><tr><td>SPTclusters+lensing</td><td>0.286 ± 0.032</td><td>0.817 ± 0.026</td><td>0.795 ± 0.029</td><td>0.805 ± 0.016</td><td>...</td><td>...</td><td>-1</td></tr><tr><td>SPTclusters+lensing+Planck</td><td>0.315 ± 0.011</td><td>0.807 ± 0.013</td><td>0.827 ± 0.013</td><td>0.817 ± 0.013</td><td>0.674 ± 0.008</td><td>&lt; 0.18</td><td>-1</td></tr><tr><td colspan="8"><strong>wCDM</strong></td></tr><tr><td>SPTclusters+lensing</td><td>0.268 ± 0.037</td><td>0.820 ± 0.026</td><td>0.772 ± 0.040</td><td>0.796 ± 0.020</td><td>...</td><td>...</td><td>-1.45 ± 0.31</td></tr><tr><td>SPTclusters+lensing+Planck</td><td>0.257 ± 0.026</td><td>0.848 ± 0.027</td><td>0.783 ± 0.026</td><td>0.814 ± 0.016</td><td>0.75 ± 0.04</td><td>&lt; 0.6</td><td>-1.34<sup>+0.22</sup><sub>-0.15</sub></td></tr></tbody></table>

We compare our results with other analyses of the cluster abundance and lensing mass calibration (see Fig. 6). The results from the DES Y1 cluster abundance and lensing are shifted toward lower values of $\Omega_m$ than expected from most cosmological probes [81], and are wider than the constraints we report here. The same cluster sample, cross-correlated with clusters, galaxies, and shear on large scales (“DES Y1 clusters + 4×2 pt”), yields constraints that largely overlap with ours [91]. The analyses of optically selected clusters in the KiDS survey with KiDS lensing [92], and the analysis of clusters selected in the eROSITA Science Verification data (eFEDS) with HSC lensing [90, 93] both exhibit constraints that are less tightly constrained along the direction of the $\Omega_m-\sigma_8$ degeneracy than our analysis, but they overlap with our results. The result obtained from ROSAT clusters with Subaru/HSC lensing “Weighing the Giants” (WtG) is almost centered on ours, but is somewhat less constraining along the width of the degeneracy [89]. We note that their constraint also includes a constraint on $\Omega_m$ through measurements of gas mass fractions $f_{gas}$, whereas our constraints are solely derived from the cluster abundance.

In Fig. 7, we compare our constraints in the $\Omega_m-\sigma_8$ plane with those obtained from other cosmological probes. Current measurements on CMB lensing from ACT and SPT cannot break the $\Omega_m-\sigma_8$ parameter degeneracy, but their degeneracy goes through our constraints [16, 94, 95]. Our measurement $S_8^{opt} = 0.805 \pm 0.016$ and $S_8^{opt} = 0.818 \pm 0.022$ from ACT DR6 CMB lensing alone [17] agree to within $0.5\sigma$. As an example of joint analyses of weak lensing and galaxy clustering, we show the DES Y3 3×2 pt results [9]. They constrain $S_8^{opt} = 0.754 \pm 0.031$, which is $1.5\sigma$ lower than our measurement. In the $\Omega_m-\sigma_8$ plane, the difference between their results and ours has a probability to exceed of 0.25 ($1.1\sigma$).<fnref n="13" /> Finally, Planck 2018 TT,TE,EE+lowE data (without lensing) constrain $S_8^{opt} = 0.820 \pm 0.016$ [2],

<figure><img src="image_8.png" /><figcaption>FIG. 7. Constraints on $\Omega_m$ and $\sigma_8$ (68% and 95% credible regions) in a ΛCDM universe with massive neutrinos from the abundance of SPT clusters, a selection of large-scale structure lensing probes [9, 17], Planck primary CMB measurements (TT,TE,EE+lowE) [2], and BAO [96–99]. Our measurement of the cluster abundance places competitive constraints especially on $S_8^{opt}$.</figcaption></figure>

which differs from our measurement by $0.7\sigma$, and with an uncertainty that is equal to ours. In the $\Omega_m-\sigma_8$ plane, the difference is essentially identical (probability to exceed of 0.47, $0.7\sigma$). For the $S_8$ parameter, the difference is $1.1\sigma$, which does not strengthen the claim of an “$S_8$ tension” between large-scale structure measurements at late times and the primary CMB. We also show constraints from baryonic acoustic oscillations (BAO), which constrain a combination of $\Omega_m$ and $H_0$, but are unable to constrain

<footnote n="13">¹³ We compute the probability to exceed using https://github.com/SebastianBocquet/PosteriorAgreement (see Sec. 5.2 in [56]).</footnote>
<figure><img src="image_9.png" /><figcaption>FIG. 8. Constraints (68% and 95% credible regions) on $\Omega_m$, $\sigma_8$, and on the sum of neutrino masses $\sum m_\nu$ from the abundance of SPT clusters and its combination with Planck primary CMB anisotropies (TT,TE,EE+lowE) [2]. The joint analysis places an upper limit $\sum m_\nu < 0.18$ eV (95% limit).</figcaption></figure>

$\sigma_8$.<fnref n="14" />

We proceed and combine our cluster abundance measurements with primary CMB anisotropies (Planck 2018 TT,TE,EE+lowE). As shown in Fig. 8, the Planck analysis exhibits a well-known degeneracy between the sum of neutrino masses $\sum m_\nu$, $\Omega_m$, and $\sigma_8$. Conversely, the cluster abundance—like other probes of the large-scale structure—constrains $\sigma_8$ essentially independently of $\sum m_\nu$. The combined analysis therefore leads to improved constraints by breaking that degeneracy. Our measurement of $\sigma_8$ fully encompasses the Planck measurement, and the cluster measurements only help to exclude the high-$\sum m_\nu$ tail. We report an improved constraint on the upper limit,

$$ \sum m_\nu < 0.18 \text{ eV (95\% limit)}, \quad (12) $$

compared to $\sum m_\nu < 0.26$ eV from Planck data alone. The constraints on the other cosmological parameters only improve mildly (see Table II).

## B. Constraints on the Observable–Mass Relations

In our analysis, we simultaneously calibrate the observable–mass relations while fitting for cosmology. In Table III, we present the recovered constraints on the parameters of the relations defined in Eqs. (3) and (5). In the Appendix, we show marginalized parameter constraints for all scaling relation and cosmological parameters that are not prior-dominated.

### 1. The SZ–Mass Relation

The SZ parameter constraints are similar to, but tighter than the ones presented in the precursor SPT cluster analysis based on targeted lensing observations [7], with the exception of the amplitude $A_{SZ}$.<fnref n="15" /> Our new amplitude is somewhat lower than the old result, which also explains the shift toward larger $\sigma_8$ seen in Fig. 5 (a lower $A_{SZ}$ implies that the observed clusters are more massive). As discussed in Sec. V A, this shift is caused by the updated weak-lensing mass calibration using DES Y3 data. Finally, our constraint on $\gamma_{ECS}$, the relative depth of the SPTpol ECS survey, is centered on the value obtained from an SZ–richness scaling relation analysis (see Sec. 6.1.3 in [30]), but we recover a tighter constraint, most likely due to simultaneously analysing the cluster abundance and multi-observable scaling relations.

### 2. The Richness–Mass Relation

Throughout this work, we use richness measurements as computed by MCMF (see Sec. II). At low cluster redshift ($z_{cluster} < 1.1$), where our richness measurements are based on DES data, the recovered richness–mass relation implies that a halo of mass $M_{200c} = 3 \times 10^{14} h^{-1} M_{\odot}$ has a typical richness of 40. We find a power-law dependence with mass that is steeper than unity at $> 3\sigma$, and a dependence with redshift that is consistent with no evolution at $< 1\sigma$. The intrinsic scatter in $\ln \lambda$ is small ($0.18 \pm 0.03$), but note that the relation between observed richness $\lambda$ and mass contains an additional log-normal scatter of width $\lambda^{-1/2}$ [see Eq. (6)].<fnref n="16" />

The parameters of the mean richness–mass relation at high redshift $z > 1.1$, where we use WISE data, do not quite match those of the low-redshift richness–mass relation. This is not entirely surprising, given how challenging it is to measure richness at these high redshifts and with the modest angular resolution of WISE. Interestingly though, the intrinsic scatter is smaller than for

<footnote n="14">¹⁴ Specifically, we analyze BAOs as measured using the 6dFGS [96], SDSS DR7 Main Galaxy Sample [MGS, 97], BOSS DR12 luminous red galaxies [LRGs, 98], and eBOSS DR16 LRGs [99] datasets.</footnote>

<footnote n="15">¹⁵ Note that we are using $M_{200c}$ in this work, as opposed to $M_{500c}$ in [7]. Therefore, the recovered constraints on $A_{SZ}$ can only be compared after converting between the two mass definitions.</footnote>

<footnote n="16">¹⁶ For intrinsic richness $\lambda < 31$, the “Poisson” part of the uncertainty dominates over the intrinsic scatter.</footnote>
<table><caption>TABLE III. Constraints on the parameters of the observable–mass relations (mean and 68% credible interval or 95% limit). We do not show the prior-dominated parameters of the $M_{WL}$–$M_{halo}$ relation. Note that the constraints on the correlation coefficients are heavily affected by the prior $\mathcal{U}(-0.5, 0.5)$.</caption><thead><tr><th>Parameter</th><th>SPTclusters+lensing</th><th>SPTclusters+lensing<br>+Planck</th></tr></thead><tbody><tr><td colspan="3"><strong>SZ parameters</strong></td></tr><tr><td>ln A<sub>SZ</sub></td><td>0.72 ± 0.09</td><td>0.69 ± 0.06</td></tr><tr><td>B<sub>SZ</sub></td><td>1.69 ± 0.06</td><td>1.73 ± 0.04</td></tr><tr><td>C<sub>SZ</sub></td><td>0.50 ± 0.27</td><td>0.74 ± 0.11</td></tr><tr><td>σ<sub>ln ζ</sub></td><td>0.20 ± 0.05</td><td>0.21 ± 0.05</td></tr><tr><td>γ<sub>ECS</sub></td><td>1.05 ± 0.03</td><td>1.05 ± 0.03</td></tr><tr><td colspan="3"><strong>DES richness parameters (used for z &lt; 1.1)</strong></td></tr><tr><td>ln A<sub>λ</sub></td><td>3.74 ± 0.07</td><td>3.73 ± 0.05</td></tr><tr><td>B<sub>λ</sub></td><td>1.22 ± 0.06</td><td>1.25 ± 0.04</td></tr><tr><td>C<sub>λ</sub></td><td>-0.03 ± 0.22</td><td>0.15 ± 0.12</td></tr><tr><td>σ<sub>ln λ̄</sub></td><td>0.18 ± 0.03</td><td>0.18 ± 0.04</td></tr><tr><td colspan="3"><strong>WISE richness parameters (used for z &gt; 1.1)</strong></td></tr><tr><td>ln A<sub>λ</sub></td><td>4.33 ± 0.20</td><td>4.33 ± 0.21</td></tr><tr><td>B<sub>λ</sub></td><td>0.93 ± 0.09</td><td>0.96 ± 0.09</td></tr><tr><td>C<sub>λ</sub></td><td>-2.1 ± 0.6</td><td>-2.0 ± 0.6</td></tr><tr><td>σ<sub>ln λ̄</sub></td><td>0.12 ± 0.04</td><td>0.12 ± 0.05</td></tr><tr><td colspan="3"><strong>Correlation coefficients</strong></td></tr><tr><td>ρ<sub>SZ,WL</sub></td><td>&lt; 0.17</td><td>&lt; 0.17</td></tr><tr><td>ρ<sub>SZ,λ̄</sub></td><td>&lt; 0.03</td><td>&lt; 0.08</td></tr><tr><td>ρ<sub>WL,λ̄</sub></td><td>-0.09 ± 0.24</td><td>-0.10 ± 0.24</td></tr></tbody></table>

DES-based richness at $1.5\sigma$. The mass slope is consistent with unity within $1\sigma$, whereas the preferred redshift evolution is negative at the ~3$\sigma$ level. The significant difference between the richness measurements from the two observatories motivates the separate empirical calibration of each relation, as performed in this work.

We discuss our low-redshift richness–mass relation in the light of existing DES results. These results are based on richness measurements using redMaPPer [100], whereas our richness measurements are based on MCMF, with median ($\lambda_{MCMF}/\lambda_{redMaPPer}$) = 1.09 and 24% scatter [101]. The redMaPPer richness–mass relation was determined by cross-matching with SPT-confirmed clusters, and calibrating the SZ–mass relation assuming a fixed fiducial cosmology [30, 102], and through an analysis of the DES Y1 cluster abundance with mass calibration enabled by cross-matching with SPT clusters, and using lensing data for SPT clusters [103]. Qualitatively, our richness–mass relation behaves as in these analyses, with residual differences being explained by the different richness measurements and by the fact that in this work, we calibrate the observable–mass relations using different lensing data (that is, DES Y3). The DES Y1 weaklensing based calibration of the redMaPPer richness–mass relation [104] was shown to yield a significantly

shallower mass slope ($B_\lambda \approx 1/1.356 = 0.74$) than the SPT-based analyses (e.g., $B_\lambda = 1.020 \pm 0.080$ in [30]). Within the limits of comparability, it appears that our constraint on the mass slope is in significant tension with the DES Y1 lensing result [104].

### 3. The Correlation Coefficients of Intrinsic Scatter

The constraints on the correlation coefficients (bottom rows in Table III) are heavily affected by the uniform priors $\rho \sim \mathcal{U}(-0.5, 0.5)$ that we apply to make the covariance matrix nonsingular. Note that as part of the blinding tests, we investigated the impact of fixing a set of fit parameters. This included setting the correlation coefficients to 0. The recovered scaling relation and cosmology constraints did not exhibit any significant shifts compared to the full analysis (see Fig. 3). We therefore conclude that the exact range of the priors that we apply to the correlation coefficients does not have a significant impact on the recovered constraints on other parameters.

The posterior distributions of the two correlation coefficients that involve the SZ observable ($\rho_{SZ,\bar{\lambda}}$ and $\rho_{SZ,WL}$) both peak at the lower bound $\rho = -0.5$ and we present the 95% upper limits in Table III. A negative correlation coefficient $\rho_{SZ,\bar{\lambda}} < 0$ is also found in the analysis of the LoCuSS cluster sample [105], whereas simulation works tend to favor positive correlation coefficients between the SZ signal and other observables, e.g., [58, 106, 107]. Negative correlation coefficients would imply that cluster triaxiality—which causes positive correlation—is not driving the joint scatter. Our constraint on the correlation coefficient between lensing and richness ($\rho_{WL,\bar{\lambda}}$) peaks within the prior range, but the quoted constraints are also strongly affected by the prior.

In our hierarchical modeling of the cluster population, the correlation coefficients are degenerate with other parameters of the observable–mass relations. Within our approach of empirical mass calibration, we do not expect strong constraints on the correlated scatter from current datasets (see also Sec. 4.1 in [108]).

### 4. Observable–Mass Relations from Joint Analysis with Planck primary CMB

In the last column in Table III, we show the constraints obtained from the joint analysis of the cluster dataset and Planck primary CMB anisotropies. Some parameters ($B_{SZ}$, $C_{SZ}$, $C_{\lambda,DES}$) shift within $1\sigma$ while most remain essentially unchanged. This is explained by the fact that our cluster-based cosmology constraints overlap with the Planck results (see Fig. 7). Therefore, their combination does not pull the parameters away from their mean in the clusters-only analysis. The uncertainties on most scaling relation parameters only tighten mildly when adding Planck CMB data. However, there is a noticeable improvement on the constraints on the redshift evolutions
$C_{SZ}$ and $C_{\lambda,DES}$. These parameters are degenerate with $\Omega_m$ (see Fig. 11), and the uncertainty on that parameter significantly tightens when adding Planck CMB data, as shown in Table II and Fig. 8.

### C. wCDM with Massive Neutrinos

We consider an extension of the ΛCDM model by opening up the dark energy equation of state parameter $w$ as an additional degree of freedom. As before, we also marginalize over the sum of neutrino masses. The constraints are summarized in Table II and shown in Fig. 9. Considering only the cluster dataset, we note that the constraints on $\Omega_m$ and $\sigma_8$ are very similar to those obtained assuming the ΛCDM model. We obtain $w = -1.45 \pm 0.31$, a constraint that is more negative than that corresponding to a cosmological constant, with a probability to exceed of 0.10 (1.7$\sigma$) (note that the posterior distribution is affected by the hard cut $w > -2$).$^{17}$ We point out that the uncertainty is comparable to the DES Y3 3×2 pt result ($w = -0.98_{-0.20}^{+0.32}$), underscoring the constraining power of cluster abundance measurements. Figure 9 also shows wide degeneracies between $\Omega_m$, $\sigma_8$, and $w$ in the analysis of Planck primary CMB data. In combination with the cluster dataset, some of these degeneracies can be broken, and the constraints are tightened.$^{18}$ The joint constraints are qualitatively similar to those obtained from the cluster dataset alone, except for $w$, for which the uncertainty reduces significantly. The mean recovered value for $w$ is less negative for the joint analysis, but the reduced uncertainty leads to a larger statistical difference with a cosmological constant (probability to exceed of 0.03, 2.2$\sigma$) than for the clusters-only analysis.

Overall, the recovered constraints on $w$ are still quite weak. As expected, $w$ exhibits a rather narrow degeneracy with the parameter of the SZ redshift evolution $C_{SZ}$ (see Fig. 11), which is only poorly constrained with the current weak-lensing data. In $w$CDM, we observe a shift to larger values ($C_{SZ} = 1.00 \pm 0.32$), and away from its preferred value in ΛCDM ($C_{SZ} = 0.50 \pm 0.27$). More powerful lensing data, especially at intermediate and high cluster redshifts, or a tight informative prior on the evolution of the ICM with redshift (as applied in the WtG analysis [89]) would enable tighter constraints on $C_{SZ}$, and thus on $w$. Alternatively, a measurement of the gas fraction $f_{gas}$ (as also applied in the WtG analysis) would constrain $\Omega_m$ independently, and through the $\Omega_m-w$ parameter degeneracy, reduce the uncertainty on $w$.

<figure><img src="image_10.png" /><figcaption>FIG. 9. Constraints on $\Omega_m$, $\sigma_8$, and the dark energy equation of state parameter $w$ (68% and 95% credible regions) in a universe with massive neutrinos. We show results obtained from the analysis of the SPT cluster abundance, Planck primary CMB anisotropies, and their combination. The cluster dataset prefers a value that is more negative than a cosmological constant $w = -1$ at the 1.7$\sigma$ level. Note that the constraints on $\Omega_m$ and $\sigma_8$ are only mildly affected by opening up $w$ as an additional degree of freedom.</figcaption></figure>

### D. Modified Growth of Structure: $\sigma_8(z)$

We consider a second, and final, extension of the ΛCDM model, in which we allow the rate of structure growth to depart from the standard prediction. We do so by directly measuring $\sigma_8$ at different redshifts. Because the SPT cluster sample spans a wide range of redshifts, this exercise allows us to map the growth of cosmic structures since the universe was about 3 billion years old.

We define five bins in redshift such that each bin contains an equal number of clusters. The six bin edges are $z \in \{0.25, 0.42, 0.57, 0.71, 0.93, 1.81\}$. Then, we modify the computation of the matter power spectrum such that $\sigma_8$ at each bin center is normalized to one of five new fit parameters $\sigma_8(z_i)$ (see Table IV). The observable-mass relations and the lensing data are treated the same as in the rest of this work, and we also marginalize over the sum of neutrino masses. When analyzing the cluster dataset in this way, the recovered constraint on $\Omega_m$ degrades significantly compared to the ΛCDM or $w$CDM analyses.$^{19}$ Therefore, we consider an additional

<footnote n="17">17 Because the posterior distributions we recover for $w$ do not appear to be well-approximated by Gaussian distributions we compute the probability to exceed instead of applying Gaussian statistics.</footnote>

<footnote n="18">18 Note that the sum of neutrino masses $\sum m_\nu$ remains poorly constrained in $w$CDM even in the joint analysis.</footnote>

<footnote n="19">19 The cluster abundance constrains a combination of $\Omega_m$ and $\sigma_8$.</footnote>
<table><caption>TABLE IV. Constraints on $\Omega_m$ and $\sigma_8(z)$ for a $\Lambda$CDM cosmology with massive neutrinos (mean and 68% credible interval). Using the abundance of SPT clusters with DES Y3 + HST weak-lensing mass calibration, and a prior on the sound horizon at recombination $\theta_*$ from Planck, we constrain $\sigma_8$ at discrete redshifts. We also quote $\sigma_8(z)$ for $\Lambda$CDM and parameters from Planck 2018 TT,TE,EE+lowE.</caption><thead><tr><th>Parameter</th><th>SPT cluster abundance+θ<sub>*</sub></th><th>Planck</th></tr></thead><tbody><tr><td>Ω<sub>m</sub></td><td>0.320 ± 0.029</td><td>0.321 ± 0.015</td></tr><tr><td>σ<sub>8</sub>(z = 0.34)</td><td>0.670 ± 0.026</td><td>0.676 ± 0.015</td></tr><tr><td>σ<sub>8</sub>(z = 0.49)</td><td>0.599 ± 0.022</td><td>0.625 ± 0.014</td></tr><tr><td>σ<sub>8</sub>(z = 0.64)</td><td>0.603 ± 0.023</td><td>0.580 ± 0.013</td></tr><tr><td>σ<sub>8</sub>(z = 0.82)</td><td>0.524 ± 0.020</td><td>0.535 ± 0.013</td></tr><tr><td>σ<sub>8</sub>(z = 1.35)</td><td>0.406 ± 0.017</td><td>0.425 ± 0.010</td></tr></tbody></table>

constraint on the matter density through the measurement of the sound horizon at recombination $100\theta_* = 1.04109 \pm 0.00030$ by Planck [2].$^{20}$

We summarize our results in Fig. 10. The gray band shows the 68% credible interval of $\sigma_8$ as a function of redshift, assuming the $\Lambda$CDM model with massive neutrinos, and parameters as measured by Planck primary CMB anisotropies. Red data points show direct measurements of $\sigma_8$ at five redshifts, as probed by the cluster abundance in combination with the $\theta_*$ prior. Note that the $\theta_*$ prior represents a subset of the information contained in the Planck CMB data, and we are thus comparing growth measurements for similar background cosmologies (thereby avoiding regimes where differences in $\sigma_8(z)$ could be caused by differences in $\Omega_m$ instead of differences in structure growth). Blue data points are obtained from a cross-correlation analysis of DESI LRGs and Planck CMB lensing, with $\theta_*$ fixed [109].$^{21}$ Brown data points show the constraint from DES Y3 galaxy clustering and lensing (3×2 pt) and shear ratios [110].

Our measurements of $\sigma_8(z)$ presented in Table IV result in a difference with the $\Lambda$CDM prediction with Planck parameters of $\chi^2 = 2.8$ for five parameters, which indicates good agreement. However, the recovered parameter constraints are degenerate [the correlation coefficients between the individual $\sigma_8(z_i)$ are all $> 0.76$], and taking these parameter correlations into account increases the $\chi^2$ value. We anticipate that a more precise determination of the observable-mass relations, enabled by future lensing dataset, will reduce the level of correlation among the measured $\sigma_8(z_i)$ and will therefore enable

<figure><img src="image_11.png" /><figcaption>FIG. 10. Evolution of $\sigma_8$ with time. Red diamonds show constraints from the SPT cluster abundance (mean and 68% credible interval). Blue dots are obtained from the cross-correlation of DESI LRGs and Planck CMB lensing [109]. Both analyses include a Planck-based prior on the sound horizon at recombination $\theta_*$, with the value of $\theta_*$ fixed to the mean Planck value for DESI $\times$ Planck, and marginalized over the Planck posterior on $\theta_*$ for SPT clusters. Brown squares show the constraints from DES Y3 3×2 pt + shear ratios [110]. The gray band shows the 68% credible interval of the $\Lambda$CDM prediction, assuming parameters determined by Planck. Our measurement of the growth of structure is consistent with that prediction.</figcaption></figure>

more robust tests of the evolution of growth of structure.

## VI. SUMMARY

In this work, we present cosmological constraints derived from the abundance of galaxy clusters selected in the SPT-SZ and SPTpol surveys [30, 32, 36, 37], and a simultaneous weak-lensing based mass calibration using DES Y3 and HST data. The cluster sample (1,005 confirmed clusters above $z > 0.25$), the analysis of DES weak-lensing data for 688 sample clusters, the Bayesian analysis framework, and the validation of the analysis pipeline are presented in companion Paper I. The HST lensing measurements for 39 SPT clusters and their analysis were presented in earlier works [34, 50–53].

In a first phase of this work, we performed a blinded analysis in which we artificially offset the parameter constraints from their true output values by unknown *blinding offset parameters* (see Sec. IV). This allowed us to perform a series of robustness tests with which we investigated the relative impact of alternative analysis setups on the results without knowing the actual parameter values. These tests included various cuts in the cluster sample (excluding high-redshift and/or low-mass objects), details of the modeling of the SPT observable, and three alternative analyses of the DES Y3 weak-lensing data.

<footnote>Because the two parameters are degenerate, allowing more freedom for $\sigma_8$ weakens the constraint on $\Omega_m$.</footnote>

<footnote n="20">20 In combination with our priors on the Hubble parameter and the baryon density $\Omega_b h^2$, the measurement of $\theta_*$ enables a relatively loose constraint $\Omega_m = 0.295 \pm 0.065$.</footnote>

<footnote n="21">21 Note that these measurements appear to be biased low due to degeneracies and parameter volume effects, as described in the original publication.</footnote>
Since none of these alternative analyses resulted in con-
straints that were significantly offset from the fiducial
ones, we unblinded the results. We did not further mod-
ify the data or the analysis, and we present the results
as obtained after unblinding.

Assuming a flat ΛCDM cosmology with massive neu-
trinos, we find that our constraints are consistent with,
but tighter than, results obtained from previous weak-
lensing calibrated cluster abundance measurements (see
Fig. 6 and Table II). Our constraints are competitive with
other leading probes of the large-scale structure such as
cosmic shear, joint 3×2 pt analyses, and the CMB lens-
ing power spectrum. Our dataset best constrains the
parameter combination σ<sub>8</sub>(Ω<sub>m</sub>/0.3)<sup>0.25</sup> = 0.805 ± 0.016,
with an uncertainty that is equal to the one recovered
from Planck 2018 TT,TE,EE+lowE data. The difference
between our constraint on S<sub>8</sub> ≡ σ<sub>8</sub>√Ω<sub>m</sub>/0.3 and the one
from Planck is 1.1σ, which does not strengthen the claim
of an “S<sub>8</sub> tension”. The difference in the Ω<sub>m</sub>–σ<sub>8</sub> plane
has a probability to exceed of 0.47 (0.7σ). The combina-
tion with the Planck dataset further tightens parameter
constraints, and we place a 95% upper limit on the sum
of neutrino masses ∑m<sub>ν</sub> < 0.18 eV.

Additionally allowing the dark energy equation of state
parameter *w* to vary does not change the recovered con-
straints on Ω_m and σ_8 in any significant way. We recover
*w* = -1.45 ± 0.31. The analysis of *Planck* primary CMB
data exhibits a degeneracy between *w* and Ω_m, σ_8, and
∑ m_ν, which is broken in a joint analysis with our cluster
dataset. The resulting constraint on *w* is still low, and we
report *w* = -1.34<sup>+0.22</sup><sub>-0.15</sub>, which differs from a cosmological
constant by 2.2σ.

Finally, we consider a phenomenological extension of
the ΛCDM model, in which we directly constrain the
growth of structure. Using the cluster dataset and a prior
on the sound horizon at recombination θ*, we constrain
σ₈ at five discrete redshifts between 0.34 and 1.35. Our
measurements of σ₈(z) are consistent with the ΛCDM
prediction with Planck 2018 TT,TE,EE+lowE parame-
ters (see Fig. 10 and Table IV), but we note that our
individual measurements are highly correlated.

A distinctive feature of this analysis is that, for the first
time, we leverage the exquisite quality of weak-lensing
data as provided by wide-field galaxy lensing surveys to
perform a robust and precise weak-lensing informed mea-
surement of the abundance of SZ-selected clusters. In
future work, we will use our cluster and lensing dataset
to calibrate other cluster observables (X-rays, dynamics
of cluster member galaxies) and to investigate further
extensions of the ΛCDM model, such as dark matter–
dark radiation interactions and modified gravity mod-
els. We also note that our analysis is the first cluster
cosmology work that uses DES Y3 lensing data. This
lensing dataset is also being used for mass calibration of
redMaPPer-selected clusters in the DES Y3 data and for
mass calibration of further ICM-selected cluster samples.
Therefore, the DES lensing data play an important role
in enabling cosmological constraints from the abundance

of galaxy clusters.

Our measurements are still limited by size of the clus-
ter sample, and by shape noise in the lensing measure-
ments. Therefore, increasing the cluster sample (thereby
also increasing the number of lenses), increasing the size
(and depth) of the lensing source sample, and the addi-
tional analysis of cluster lensing of the CMB e.g., [111–
113], will lead to improved cosmological constraints [114].
However, our analysis also highlights areas that deserve
further attention (e.g., low-significance clusters, cluster
member contamination, miscentering). While the rela-
tive importance of these effects is currently at the ≲ 1σ
level (see Fig. 3), more work is needed to ensure that
these do not become limiting factors in future analyses.
Opportunities for near-term improvements are numerous,
with large ICM-selected cluster samples from the Ata-
cama Cosmology Telescope [115], eROSITA [116], the
SPT-3G survey [117], and the upcoming Simons Observa-
tory [118]. These cluster samples will be analyzed jointly
with a weak-lensing mass calibration using DES Year 6
data, soon to be followed by galaxy and lensing data
from Euclid [119] and the Legacy Survey of Space and
Time (LSST) [120] at the Vera C. Rubin Observatory,
whose combination is expected to be particularly pow-
erful for extending sensitive cluster mass calibrations to
higher redshifts [121]. Another promising avenue is to
additionally consider the cluster 2-pt correlation func-
tion (and cluster–galaxy 2 pt), which further boosts the
constraining power [91, 122]. Finally, the CMB-S4 survey
will improve upon the aforementioned mm-wave surveys,
and will extend analyses of SZ-selected cluster samples
to well into the 2030s [123].

ACKNOWLEDGMENTS

This research was supported by the Excellence Cluster
ORIGINS, which is funded by the Deutsche Forschungs-
gemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy - EXC-2094-
390783311, the MPG Faculty Fellowship program and the
Ludwig-Maximilians-Universität München. Parts of the
MCMC computations were carried out on the computing
facilities of the Computational Center for Particle and
Astrophysics (C2PAP). The Bonn and Innsbruck authors
acknowledge support from the German Federal Min-
istry for Economic Affairs and Energy (BMWi) provided
through DLR under projects 50OR2002 and 50OR2302,
from the German Research Foundation (DFG) under
grant 415537506, and the Austrian Research Promotion
Agency (FFG) and the Federal Ministry of the Repub-
lic of Austria for Climate Action, Environment, Mobil-
ity, Innovation and Technology (BMK) via grants 899537
and 900565. The Melbourne authors acknowledge sup-
port from the Australian Research Council’s Discovery
Projects scheme (No. DP200101068).

The South Pole Telescope program is supported by the
National Science Foundation (NSF) through the Grant
No. OPP-1852617. Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago. PISCO observations were supported by US NSF grant AST-0126090. Work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of High Energy Physics, under Contract No. DE-AC02-06CH11357.

Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians-Universität München and the associated Excellence Cluster Origins, the University of Michigan, NSF's NOIRLab, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at NSF's NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MICINN under grants ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded

by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

This work is based on observations made with the NASA/ESA Hubble Space Telescope, using imaging data from the SPT follow-up GO programs 12246 (PI: C. Stubbs), 12477 (PI: F. W. High), 13412 (PI: T. Schrabback), 14252 (PI: V. Strazzullo), 14352 (PI: J. Hlavacek-Larrondo), and 14677 (PI: T. Schrabback). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. It is also based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programs 086.A-0741 (PI: Bazin), 088.A-0796 (PI: Bazin), 088.A-0889 (PI: Mohr), 089.A-0824 (PI: Mohr), 0100.A-0204 (PI: Schrabback), 0100.A-0217 (PI: Hernández-Martín), 0101.A-0694 (PI: Zohren), and 0102.A-0189 (PI: Zohren). It is also based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), Ministério da Ciência, Tecnologia e Inovação (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea), under programs 2014B-0338 and 2016B-0176 (PI: B. Benson).

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. This research has made use of the SAO/NASA Astrophysics Data System and of adstex.<fnref n="22" /> We thank Ludwig, the SPT support cat for this analysis. Inquiries about SPT support cats shall be directed to T.C.

## Appendix A: Full Parameter Posterior Distributions

In Fig. 11, we show the marginalized one- and two-dimensional parameter constraints obtained in the $\Lambda$CDM and wCDM analyses. We show all parameters that are not strongly prior-dominated. We highlight the joint constraint on $\Omega_m$ and $\sigma_8$, which is quite robust to the extension with w as a free parameter. We also highlight the degeneracy between $C_{SZ}$ and w, which implies that a tighter constraint on the redshift evolution

<footnote n="22">²² https://github.com/yymao/adstex</footnote>
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<figure><img src="image_12.png" /></figure>

FIG. 11. Parameter constraints on all parameters that are not strongly prior-dominated in the ΛCDM and wCDM analyses (68% and 95% credible regions). Between the two analyses, only the constraints on the redshift evolution parameters of the observable–mass relations $C_{SZ}$ and $C_{\lambda}$ shift by an appreciable amount. Conversely, the joint constraints on $\Omega_m$ and $\sigma_8$ are very similar (highlighted toward the bottom-right of the plot). The constraint on $w$ shows some significant degeneracies with other parameters, and in particular with the redshift evolution parameters $C_{SZ}$ (highlighted at the bottom-left of the plot) of the SZ–mass relation, and $C_{\lambda}$ of the richness–mass relation.

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