621.016.402 (JTAdJ) and 639.043.512 (HHo). AA acknowledges the support of a
Kavli Fellowship. AC acknowledges support from the SPHEREx project under a
contract from NASA/Goddard Space Flight Center to the California Institute of
Technology. MB is supported by the Polish National Science Center through
grants 2020/38/E/ST9/00395, 2018/30/E/ST9/00698, 2018/31/G/ST9/03388 and
2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher
Education through grant DIR/WK/2018/12. RD acknowledges support from the NSF
Graduate Research Fellowship Program under grant number DGE-2039656. BG
acknowledges the support of the Royal Society through an Enhancement Award
(RGF/EA/181006) and the Royal Society of Edinburgh through the Saltire Early
Career Fellowship (ref. number 1914). HHi is supported by a Heisenberg grant
of the Deutsche Forschungsgemeinschaft (Hi 1495/5-1). HHi and AHW acknowledge
funding from the German Science Foundation DFG, via the Collaborative Research
Center SFB1491 “Cosmic Interacting Matters - From Source to Signal”. KK
acknowledges support from the Royal Society and Imperial College. SL is
supported by NOVA, the Netherlands Research School for Astronomy. XL is
supported in part by the Department of Energy grant DE-SC0010118 and in part
by a grant from the Simons Foundation (Simons Investigator in Astrophysics,
Award ID 620789). NRN acknowledges financial support from the National Science
Foundation of China, Research Fund for Excellent International Scholars
(12150710511), and from the China Manned Space Project research grant CMS-
CSST-2021-A01. SS and JB are supported in part by NSF grant AST-2206563. HYS
acknowledges the support from CMS-CSST-2021-A01 and CMS-CSST-2021-A04, NSFC of
China under grant 11973070, and Key Research Program of Frontier Sciences,
CAS, Grant No. ZDBS-LY-7013 and Program of Shanghai Academic/Technology
Research Leader. CS acknowledges support from the Agencia Nacional de
Investigación y Desarrollo (ANID) through FONDECYT grant no. 11191125 and
BASAL project FB210003. TT acknowledges funding from the Swiss National
Science Foundation under the Ambizione project PZ00P2_193352. AHW is supported
by the Deutsches Zentrum für Luft- und Raumfahrt (DLR), made possible by the
Bundesministerium für Wirtschaft und Klimaschutz. For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC BY)
licence to any Author Accepted Manuscript version arising from this
submission.

Dark Energy Survey: This project used public archival data from the Dark
Energy Survey (DES). 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 Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University. The DES results in this paper
are based on observations at Cerro Tololo Inter-American Observatory, National
Optical Astronomy Observatory, which is operated by the Association of
Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation.

Hyper Suprime-Cam Survey: The HSC collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University. The HSC results in this paper make use of
software developed for Vera C. Rubin Observatory. We thank the Rubin
Obsertevatory for making their code available as free software at
http://pipelines.lsst.io/. The HSC results in this paper are based on data
collected at the Subaru Telescope and retrieved from the HSC data archive
system, which is operated by the Subaru Telescope and Astronomy Data Center
(ADC) at NAOJ. Data analysis was in part carried out with the cooperation of
Center for Computational Astrophysics (CfCA), NAOJ. We are honoured and
grateful for the opportunity of observing the Universe from Maunakea, which
has the cultural, historical and natural significance in Hawaii.

Kilo-Degree Survey: The KiDS results in this paper are based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under programme
IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products
produced by the KiDS consortium. The KiDS production team acknowledges support
from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the
University of Padova, and the University Federico II (Naples). Data processing
for VIKING has been contributed by the VISTA Data Flow System at CASU,
Cambridge and WFAU, Edinburgh.

## Author contributions

All authors contributed to this paper, to the data products, the data
calibration and/or the scientific analysis and interpretation. A core team of
DES and KiDS members brought together the wide body of expertise from the two
survey teams along with two extensive banks of software to create the Hybrid
pipeline and conduct the cosmological analysis: Alexandra Amon, Marika Asgari,
Ami Choi, Catherine Heymans, Anna Porredon, Simon Samuroff and Joe Zuntz.
Additional support was provided by Felipe Andrade-Oliveira, Gary Bernstein,
Jonathan Blazek, Hugo Camacho, Roohi Dalal, Konrad Kuijken, Xiangchong Li,
Chieh-An Lin and Jessie Muir and from the internal reviewers Chihway Chang,
Scott Dodelson, Hendrik Hildebrandt and Michael Troxel.

## Data and Software Availability

The DES Y3 and KiDS-1000 catalogues and data products are freely available for
download. The fiducial chains from the DES Y3, KiDS-1000 and Joint analysis
can be accessed at https://des.ncsa.illinois.edu/releases/y3a2/Y3key-joint-
des-kids. In CosmoSIS v3.3 we include our joint survey likelihood with the
example file des-y3_and_kids-1000.ini to allow for the easy reproduction of
this Hybrid analysis. This version of CosmoSIS allows for $S_{8}$ sampling,
the use of correlated priors for nuisance parameters, and includes a subset of
the software from the Asgari COSEBIs library.

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## Affiliations

1 Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile 2 Laboratório
Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de
Janeiro, RJ - 20921-400, Brazil 3 Argonne National Laboratory, 9700 South Cass
Avenue, Lemont, IL 60439, USA 4 Department of Physics, University of Michigan,
Ann Arbor, MI 48109, USA 5 Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK 6 Kavli Institute for Cosmology,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 7 E.A Milne
Centre, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom and
Centre of Excellence for Data Science, AI, and Modelling (DAIM), University of
Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX 8 Institut de Física
d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,
Campus UAB, 08193 Bellaterra (Barcelona) Spain 9 Institute of Cosmology and
Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 10 Physics
Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150
University Avenue Madison, WI 53706-1390 11 Department of Physics and
Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 12 CNRS,
UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 13
Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique
de Paris, F-75014, Paris, France 14 Center for Theoretical Physics, Polish
Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland 15 Department
of Physics, Northeastern University, Boston, MA 02115, USA 16 University
Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr.
1, 81679 Munich, Germany 17 Department of Physics & Astronomy, University
College London, Gower Street, London, WC1E 6BT, UK 18 Argelander-Institut für
Astronomie, Auf dem Hügel 71, D-53121 Bonn, Germany 19 Kavli Institute for
Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University,
Stanford, CA 94305, USA 20 SLAC National Accelerator Laboratory, Menlo Park,
CA 94025, USA 21 Instituto de Física Teórica, Universidade Estadual Paulista,
São Paulo, Brazil 22 Department of Physics, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15312, USA 23 Instituto de Astrofisica de Canarias,
E-38205 La Laguna, Tenerife, Spain 24 Universidad de La Laguna, Dpto.
Astrofísica, E-38206 La Laguna, Tenerife, Spain 25 Center for Astrophysical
Surveys, National Center for Supercomputing Applications, 1205 West Clark St.,
Urbana, IL 61801, USA 26 Department of Astronomy, University of Illinois at
Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA 27 Institut
d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain 28 Institute
of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193
Barcelona, Spain 29 Physics Department, William Jewell College, Liberty, MO,
64068 30 Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL 60637, USA 31 Kavli Institute for Cosmological Physics, University
of Chicago, Chicago, IL 60637, USA 32 Department of Physics, Duke University
Durham, NC 27708, USA 33 NASA Goddard Space Flight Center, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA 34 Jodrell Bank Center for Astrophysics, School of
Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13
9PL, UK 35 University of Nottingham, School of Physics and Astronomy,
Nottingham NG7 2RD, UK 36 Hamburger Sternwarte, Universität Hamburg,
Gojenbergsweg 112, 21029 Hamburg, Germany 37 Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08544, USA 38 School of
Mathematics and Physics, University of Queensland, Brisbane, QLD 4072,
Australia 39 Leiden Observatory, Leiden University, P.O.Box 9513, 2300RA
Leiden, The Netherlands 40 Kapteyn Astronomical Institute, University of
Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 41 Lawrence Berkeley
National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 42 Department
of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 43 Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 44 NSF AI
Planning Institute for Physics of the Future, Carnegie Mellon University,
Pittsburgh, PA 15213, USA 45 Université Grenoble Alpes, CNRS, LPSC-IN2P3,
38000 Grenoble, France 46 Ruhr University Bochum, Faculty of Physics and
Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological
Lensing, 44780 Bochum, Germany 47 Department of Astronomy/Steward Observatory,
University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA 48
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Dr., Pasadena, CA 91109, USA 49 Department of Physics and Astronomy,
University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada 50
Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA 51 Institute of Theoretical Astrophysics,
University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway 52 Instituto
de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid,
Spain 53 Instituto de Ciencias del Cosmos (ICC), Universidad de Barcelona,
Martí i Franquès, 1, 08028 Barcelona, Spain 54 Institute for Astronomy,
University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9
3HJ, UK 55 School of Physics and Astronomy, Cardiff University, CF24 3AA, UK
56 Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290
Versoix, Switzerland 57 Santa Cruz Institute for Particle Physics, Santa Cruz,
CA 95064, USA 58 Center for Cosmology and Astro-Particle Physics, The Ohio
State University, Columbus, OH 43210, USA 59 Department of Physics, The Ohio
State University, Columbus, OH 43210, USA 60 Department of Physics, University
of Arizona, Tucson, AZ 85721, USA 61 Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 62 Department of
Physics and Astronomy, University College London, Gower Street, London WC1E
6BT, UK 63 Department of Astrophysical Sciences, Princeton University, 4 Ivy
Lane, Princeton, NJ 08544, USA 64 Australian Astronomical Optics, Macquarie
University, North Ryde, NSW 2113, Australia 65 Lowell Observatory, 1400 Mars
Hill Rd, Flagstaff, AZ 86001, USA 66 Department of Physics, Université de
Montréal, Montréal, Canada 67 Mila - Quebec Artificial Intelligence Institute,
Montréal, Canada 68 Ciela - Montreal Institute for Astrophysical Data Analysis
and Machine Learning, Montréal, Canada 69 Center for Computational
Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA 70
McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon
University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA 71 Kavli Institute for
the Physics and Mathematics of the Universe (WPI),The University of Tokyo
Institutes for Advanced Study (UTIAS), The University of Tokyo, Chiba
277-8583, Japan 72 Instituto de Astrofísica e Ciências do Espaço, Faculdade de
Ciências, Universidade de Lisboa, 1769-016 Lisboa, Portugal 73 Departamento de
Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318,
São Paulo, SP, 05314-970, Brazil 74 WPC Systems Ltd., 55 Qingyun Road,
Tucheng, New Taipei, Taiwan 75 Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK 76 George
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 77 Centro de Investigaciones Energéticas,
Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 78 Institució Catalana
de Recerca i Estudis Avançats, E-08010 Barcelona, Spain 79 Max Planck
Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching,
Germany 80 Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada 81 Department of Physics, Stanford University,
382 Via Pueblo Mall, Stanford, CA 94305, USA 82 School of Physics and
Astronomy, Sun Yat-sen University, Guangzhou 519082, Zhuhai Campus, P.R. China
83 Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,
13083-859, Campinas, SP, Brazil 84 Observatório Nacional, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 85 Kavli Institute for the
Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan 86 INAF - Osservatorio Astronomico di Padova,
via dell’Osservatorio 5, 35122 Padova, Italy 87 Department of Physics,
University of Genova and INFN, Via Dodecaneso 33, 16146, Genova, Italy 88
Department of Physics and Astronomy, Pevensey Building, University of Sussex,
Brighton, BN1 9QH, UK 89 Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA 90 Shanghai Astronomical Observatory (SHAO),
Nandan Road 80, Shanghai 200030, China 91 Key Laboratory of Radio Astronomy
and Technology, Chinese Academy of Sciences, A20 Datun Road, Chaoyang
District, Beijing, 100101, P. R. China 92 University of Chinese Academy of
Sciences, Beijing 100049, China 93 Brookhaven National Laboratory, Bldg 510,
Upton, NY 11973, USA 94 Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794, USA 95 Instituto de Física, Pontificia
Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile 96 School
of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
97 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831 98 Institute for Particle Physics and Astrophysics, ETH
Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland 99 Institut de
Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse,
CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France 100 Excellence
Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany 101 Universitäts-
Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München,
Scheinerstr. 1, 81679 München, Germany

##  Appendix A Survey overlap: excising the KiDS area from the DES footprint

Figure 7: Survey footprints from DES Y3 (green) and KiDS-1000 (orange). The
HSC-Y1 footprint (purple) overlaps KiDS in the North and DES in the South
complicating the modelling of cross-survey covariance. For this reason, we
limit our joint-survey analysis to DES and KiDS.

Summary: In this appendix we use simulations to estimate the cross-covariance
between the DES and KiDS surveys when removing the KiDS-overlap region from
the DES footprint. We conclude that the cross-covariance is sufficiently small
and can therefore be neglected in the joint survey cosmic shear analysis. We
confirm that the DES footprint modification, which reduces the total DES area
by 8%, does not significantly impact the DES Y3 cosmological parameter
constraints.

In Figure 7 we compare the DES and KiDS on-sky footprints, highlighting the
Southern overlap region covering 471 square degrees, 337/246 of which are
unmasked in the DES/KiDS area respectively. Data in the overlap region
represents $8\%$ of DES Y3, and $32\%$ of KiDS-1000, introducing a cross-
covariance between the two surveys. Given the different redshift distributions
of the two surveys, it is non-trivial to analytically calculate the cross-
covariance. We therefore choose to reduce the cross-covariance by excising the
KiDS-overlap region from the DES survey444444To excise the KiDS-overlap, we
remove DES galaxies with $-0.63^{\circ}<{\rm RA}<53.94^{\circ}$ and
$-35.62^{\circ}<{\rm Dec}<-26.98^{\circ}$., given the relatively smaller
impact on the total DES area. We assume that DES is sufficiently homogenous
that the redshift distributions are unchanged by the reduction in area. We re-
measure the effective number density, the ellipticity shape noise and the mean
shear and selection response, for every tomographic source bin454545The
effective number density, response and shape noise are weighted quantities.
The DES Y3 per-galaxy weights are defined from empirical measurements of the
observed ellipticity variance across the survey, binned by galaxy-to-PSF size
ratio and signal-to-noise (see section 4.3 of Gatti et al., 2021). It is not
necessary to recalculate these galaxy weights for the reduced survey area as
the binned measurements automatically allow us to account for the effects of
varying PSF size and survey depth in our re-defined footprint.. We find
$<0.1$% changes for $\sigma_{e}$ in all bins, $<1$% changes for the mean shear
response and $<1$% changes for the mean selection response in all but the
second bin where the change is 2.6%. There are small changes in $n_{\rm eff}$,
which is systematically higher in all bins ranging from 2% in the highest
redshift source bin to 4% in the lowest redshift source bin. We then re-
calculate the analytical covariance for the revised DES footprint following
Friedrich et al. (2021).

In Figure 8 we compare the DES data vector from Amon et al. (2022); Secco,
Samuroff et al. (2022) to the excised DES data vector and covariance that we
analyse in Section 3. We find that excising the KiDS area has little impact on
the DES cosmological parameter constraints with $S_{8}^{\rm
original}-S_{8}^{\rm excised}=-0.0004$, using a DES-like analysis and
$S_{8}^{\rm original}-S_{8}^{\rm excised}=-0.004$, using the Hybrid pipeline.
The constraining power of the reduced survey area decreases when using a DES-
like analysis with the 68% credible interval $\sigma_{S_{8}}^{\rm
original}/\sigma_{S_{8}}^{\rm excised}=0.91$. For the Hybrid pipeline
analysis, however, we find little impact on the constraining power with
$\sigma_{S_{8}}^{\rm original}/\sigma_{S_{8}}^{\rm excised}=1.02$. We note
these numbers will be subject to chain-to-chain variance which we expect to be
at the level of $\sim 0.1\sigma$ (Joachimi et al., 2021).

Figure 8: The DES Y3 cosmic shear two-point correlation function,
$\xi_{+}(\theta)$ (left) and $\xi_{-}(\theta)$ (right), for 10 tomographic bin
combinations (see label). The fiducial DES Y3 scale cuts are indicated in pale
blue, with the $\Lambda$CDM-optimised scale cuts that are adopted in this
cosmic shear analysis indicated in dark blue. In the upper section of each
panel we compare measurements from the full DES Y3 footprint (green) and the
footprint with the KiDS-overlap region excised (black) which covers 8% less
area. The lower section of each panel shows the signal difference as a
fraction of the best-fit theory. The analytical covariance is calculated
following Friedrich et al. (2021), accounting for the reduction in area.

To assess the residual cross-survey covariance that remains after excising the
overlap region, we estimate the cross-covariance from 1250 lognormal
simulations created with FLASK (Xavier et al., 2016; Friedrich et al., 2021).
For simplicity, we only simulate the properties of DES Y3, measuring the
$\xi_{\pm}(\theta)$ statistic464646For this analysis we measure
$\xi_{\pm}(\theta)$ using TreeCorr with the bin slop parameter $b=0.1$ (Jarvis
et al., 2004) and adopt the fiducial DES Y3 scale cuts. These cuts are more
conservative than the $\Lambda$CDM-optimised scale cuts adopted in our cosmic
shear analysis (see Figure 8). This difference is unlikely to impact our
conclusions, however, as any cross-survey covariance will predominantly impact
the larger angular scales that feature in both analyses. in 4 tomographic bins
for the primary DES area where the KiDS-overlap region is excised, and the
remaining DES data within the KiDS-overlap region. The additional KiDS depth
and different sampling of $k$-space with the COSEBIs statistic should serve to
reduce the cross-survey covariance estimated here. This analysis therefore
provides an upper limit of the cross-covariance between the DES and KiDS data
vectors analysed in Section 3.

Figure 9 presents the FLASK-simulation estimated joint-survey correlation
matrix. We find the expected level of intra-survey covariance between the
different $\xi_{+}(\theta)$ and $\xi_{-}(\theta)$ tomographic bins. For the
cross-survey covariance, the majority of correlation coefficients are within a
threshold of $\pm 8\%$. These levels are similar to the study in Joachimi et
al. (2021) which concludes the cross-survey covariance can be neglected in a
$3\times 2$pt analysis when the majority of correlation coefficients are
within a threshold of $\pm 5\%$.

We follow Fang et al. (2020); Andrade-Oliveira et al. (2021) to assess the
impact of neglecting cross-survey covariance through a cosmic shear $\chi^{2}$
analysis of $i=[1..500,000]$ mock joint-survey data vectors with correlated
noise randomly sampled from the FLASK-estimated covariance $\bf{C}_{\rm
full}$. We measure

$\chi^{2}_{\rm A,i}=[\bf{d}_{i}-\bf{d}_{\rm th}]^{\rm T}\bf{C}_{\rm
A}^{-1}[\bf{d}_{i}-\bf{d}_{\rm th}]\,,$ (52)

where $\bf{d}_{i}$ is the data vector for realisation $i$ and $\bf{d}_{\rm
th}$ is the theoretical data vector. The label A indicates the covariance
matrix with A=‘full’ or A=‘no-cross’ where we zero the cross-survey blocks
shown in Figure 9. We calculate $\Delta\chi^{2}_{i}=\chi^{2}_{{\rm no-
cross},i}-\chi^{2}_{{\rm full},i}$ finding $\langle\Delta\chi^{2}\rangle=0\pm
11$ for $N_{\rm d}=454$ data points474747There are $N_{\rm d}=227$ data points
in the fiducial DES Y3 cosmic shear analysis, which doubles to $N_{\rm d}=454$
with the inclusion of the KiDS-overlap region as an additional data vector,
analysed in the same way as the primary DES-patch.. These values are
consistent with analytical estimates of $\Delta\chi^{2}$ from the different
covariance matrices using equations 20 and 21 from Andrade-Oliveira et al.
(2021). We therefore conclude that the use of an inaccurate covariance matrix
that neglects cross-survey covariance will not introduce any significant
effects on the inferred goodness of fit or the best-fit parameter values. Our
joint-survey analysis therefore adopts a covariance matrix for the KiDS and
excised-DES data vectors with zero cross-survey correlation.

Our approach differs from that of Longley et al. (2023) who account for the
cross-survey covariance between KiDS and HSC in their joint cosmic shear
analysis by conservatively assuming 100% correlation between the Northern
stripe of KiDS and half of HSC. They then enlarge either survey’s covariance
by the ratio of the full survey footprint area and the Northern footprint
area. In lieu of an accurate analytical model, we find our data excision
approach to be better suited for KiDS and DES, given the relatively small
fractional area of DES which overlaps with KiDS. In principle, we could have
adopted the same approach to also include the public HSC-Y1 catalogues into
this joint-survey analysis, excising the smaller HSC-Y1 footprint from the DES
equatorial stripe and KiDS-North (see Figure 7). Given that this would not
increase our overall joint-survey area, however, and that the uncertainty in
the $S_{8}$ constraints from this first HSC data release (Hikage et al., 2019)
are $\sim 40\%$ larger than the KiDS-1000 or DES Y3 results, we choose not to
extend our analysis with the inclusion of HSC-Y1. As half of the complete HSC
footprint has full overlap with half of the complete KiDS footprint, any
future combination of the final data releases from KiDS and HSC will require
an analytical solution to account for cross-survey covariance.

Figure 9: Correlation coefficient matrix for the DES Y3 tomographic cosmic
shear data vector measured in two areas: the DES Y3 footprint with the KiDS
overlap region excised, denoted $\xi_{\pm}(\theta)^{\rm DES}$, and the KiDS-
excised region, denoted $\xi_{\pm}(\theta)^{\rm KiDS}$. The upper triangle
shows the correlation matrix as calculated from FLASK mocks; the lower
triangle shows the correlation coefficients above $\pm 8\%$. The low-levels of
correlation between the main DES area and the KiDS-excised area, in the lower-
left quadrants, supports our decision to neglect the cross-survey covariance
between DES and KiDS in our joint-survey analysis.

##  Appendix B Mitigating the uncertain impact of baryon feedback with scale
cuts

Summary: In this appendix we determine scale cuts for the KiDS COSEBIs data
vector, following the baryon feedback mitigation strategy of Krause et al.
(2021). When used in a joint-survey analysis alongside the $\Lambda$CDM-
optimised scale cuts for the DES $\xi_{\pm}(\theta)$ data vector, the Hybrid
pipeline is shown to be robust to the presence of baryon feedback at the level
of $<0.22\sigma_{\rm 2D}$. Here $\sigma_{\rm 2D}$ is the 68% credible interval
in the 2D $(S_{8},\Omega_{\rm m})$ marginalised posterior for a joint survey
cosmic shear analysis.

We compare constraints in the $(S_{8},\Omega_{\rm m})$ plane between a joint-
survey analysis of a mock dark-matter only and a mock van Daalen et al. (2011)
OWLS-AGN baryon-feedback contaminated data vector484848We note that the OWLS
AGN model introduces less baryon feedback suppression to the matter power
spectrum than the Le Brun et al. (2014) Cosmo-OWLS AGN simulations that were
included in the Huang et al. (2021) baryon feedback analysis of DES Y1. We
choose to exclude the Cosmo-OWLS models in this analysis as, in contrast to
BAHAMAS which guides our chosen $T_{\rm AGN}$ prior range in the Hybrid
pipeline analysis, they are unable to recover both the observed global stellar
mass function (McCarthy et al., 2017) and the observed baryon fraction on
group scales (van Daalen et al., 2020).. We then progressively remove small-
scale information until the two analyses agree to within $0.3\sigma_{\rm 2D}$.
We conduct this study following Krause et al. (2021) by using the DES
methodology, summarised in Table 2, and adopting the best fit
flat-$\Lambda$CDM cosmological parameters from Planck Collaboration (2020).

In contrast to the DES-adopted shear correlation function,
$\xi_{\pm}(\theta)$, the KiDS-adopted COSEBIs statistic does not
straightforwardly allow for redshift-dependent scale cuts. The theoretical
prediction for each of the $E_{\rm n}$ and $B_{\rm n}$ tomographic modes is
dependent on a fixed angular range $\theta_{\rm min}<\theta<\theta_{\rm max}$
(see Equation 5). Whilst predictions for varied angular ranges for each
tomographic bin combination, and the corresponding analytical covariance
calculation, can in principle be calculated, this approach would require an
extensive update to the existing COSEBIs software. We therefore take a
pragmatic approach in this study, fixing the redshift-dependent DES scale cuts
to the Amon et al. (2022); Secco, Samuroff et al. (2022) defined $\Lambda$CDM-
optimised scales, which predict a baryon feedback bias in a DES Y3
$\Lambda$CDM cosmic shear analysis of $<0.14\sigma_{\rm 2D}$. We then
progressively increase $\theta_{\rm min}$ for the KiDS-COSEBIs analysis, in
all tomographic bins, until the joint-survey analysis meets the Krause et al.
(2021) robustness criteria.

For the KiDS-adopted $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5, we do not meet the
$<0.3\sigma_{\rm 2D}$ offset criteria for a KiDS-only analysis using the DES
framework. We find that raising $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 is sufficient to mitigate the
bias from OWLS-AGN baryon feedback, increasing the error on $S_{8}$ by only
$\sim 10\%$. In Figure 10 we show that with this limit for KiDS, in addition
to the redshift-dependent DES scale cuts from Amon et al. (2022); Secco,
Samuroff et al. (2022), the Krause et al. (2021) methodology predicts a baryon
feedback bias of $<0.18\sigma_{\rm 2D}$ in a DES-like joint-survey cosmic
shear analysis. We note that raising $\theta_{\rm min}>$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 was found to significantly
reduce the joint survey constraining power, without any significant reduction
in the bias. For example with $\theta_{\rm
min}=$5$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0, the fractional error on
$S_{8}$ increased, relative to the $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 case, by $\sim 30\%$, with the
bias remaining at $\sim 0.2\sigma_{\rm 2D}$.

We find that this scale cut combination is also sufficient for our Hybrid set-
up, using the mock analysis in Appendix E. Here we measure a baryon feedback
bias for the joint cosmic shear analysis of $<0.14\sigma_{\rm 2D}$ with scale
cuts alone, and $<0.22\sigma_{\rm 2D}$ for the combination of scale cuts and
marginalisation over the $T_{\rm AGN}$ parameter in the HMCode2020 non-linear
model. We remind the reader that these results will be subject to chain-to-
chain variance. For Multinest, marginal constraints are found to differ
between successive runs at the level of $\sim 0.1\sigma$ (Joachimi et al.,
2021). We have not carried out a systematic analysis to quantify the level of
chain-to-chain variance for Polychord, but find $\sim 0.1\sigma$ variations in
the handful of Polychord chains that have been repeated in this analysis.

Figure 10: Cosmological constraints in the $\Omega_{\rm m}-S_{8}$ plane from a
mock DES-like joint-survey analysis of data vectors from a dark-matter only
Universe (‘Baseline’, purple), and an OWLS-AGN baryon-feedback contaminated
Universe (‘Contaminated’, pink), for an input cosmology $\Omega_{\rm m}=0.3$
and $S_{8}=0.826$ (grey-dashed). The contours are set at $0.3\sigma_{\rm 2D}$,
where $\sigma_{\rm 2D}$ is the 68% credible interval in the 2D marginalised
posterior. In this analysis the KiDS COSEBIs data vector has been created with
an increased minimum angular scale of $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0, leading to an offset from the
Baseline dark-matter only case of $0.18\sigma_{\rm 2D}$. We discuss the
projection effects that impact both the Baseline and Contaminated analysis in
Appendix C.3.

##  Appendix C Mock data analysis

Summary: In this appendix we quantify the impact of the different analysis
choices in the DES and KiDS fiducial pipelines using a series of mock data
vectors. More specifically:

  * •

In Appendix C.1 we find that a DES-like analysis increases the $1\sigma$
uncertainty on $S_{8}$ by up to $\sim 60\%$ compared to the corresponding
KiDS-like analysis of the same data set. This distinction primarily arises
from the TATT or NLA choice of IA model, with little difference seen in the
$S_{8}$ constraints when switching between each team’s baryon feedback
mitigation strategy.

  * •

In Appendix C.2 we review the intrinsic alignment literature, selecting
survey-independent IA parameters for our fiducial analysis. We show this does
not degrade the expected constraining power of the analysis.

  * •

In Appendix C.3 we quantify projection effects in the multi-dimensional
parameter space. We measure the offset of the mean and maximum value of the
marginal $S_{8}$ posterior from the true input $S_{8}$ value. When using KiDS
priors for a KiDS-like analysis, and DES priors for a DES-like analysis the
offset introduced by projection effects on the $S_{8}$ marginal is negligible.
Adopting the KiDS cosmological parameter priors for a DES-like analysis or the
DES cosmological parameter priors for KiDS-like analysis, however, results in
a $\sim\pm 0.5\sigma$ offset.

  * •

In Appendix C.4 we analyse a mock data vector created with a TATT IA model and
an HMCode2020 non-linear matter power spectrum with baryon feedback. In this
scenario the different astrophysical modelling choices of the two teams bias
the $S_{8}$ constraints in opposite directions leading to a $\sim 2\sigma$
level tension between a DES-like and a KiDS-like and analysis of the same
joint-survey data vector.

Parameter | Mock Input | DES Prior | KiDS Prior
---|---|---|---
Cosmological parameters: |
$10^{-9}A_{\mathrm{s}}$ | 2.41 | $\left[0.5,\,5.0\right]$ | -
$S_{8}$ | $0.759$ | - | $\left[0.1,1.3\right]$
$h$ | 0.767 | $\left[0.55,0.91\right]$ | $\left[0.64,0.82\right]$
$\Omega_{\mathrm{m}}$ | 0.246 | $\left[0.1,0.9\right]$ | -
$\omega_{\rm c}$ | 0.118 | - | $\left[0.051,0.255\right]$
$\Omega_{\mathrm{b}}$ | 0.044 | $\left[0.03,0.07\right]$ | -
$\omega_{\mathrm{b}}$ | 0.026 | - | $\left[0.019,0.026\right]$
$n_{\mathrm{s}}$ | 0.90 | $\left[0.87,\,1.07\right]$ | $\left[0.84,\,1.1\right]$
$1000\,\Omega_{\nu}h^{2}$ | 0.64 | $\left[0.6,6.44\right]$ | -
Astrophysical systematic model parameters: |
TATT Intrinsic Alignments: ‘Strong’ |
$a_{1}$ | -0.44 | $\left[-5,5\right]$ | -
$\eta_{1}$ | 4.33 | $\left[-5,5\right]$ | -
$a_{2}$ | 1.25 | $\left[-5,5\right]$ | -
$\eta_{2}$ | 2.12 | $\left[-5,5\right]$ | -
$b_{\rm TA}$ | 1.81 | $\left[0,2\right]$ | -
TATT Intrinsic Alignments: ‘Weak’ |
$a_{1}$ | 0.15 | $\left[-5,5\right]$ | -
$\eta_{1}$ | -4.09 | $\left[-5,5\right]$ | -
$a_{2}$ | -0.02 | $\left[-5,5\right]$ | -
$\eta_{2}$ | 2.31 | $\left[-5,5\right]$ | -
$b_{\rm TA}$ | 0.04 | $\left[0,2\right]$ | -
NLA Intrinsic Alignments: |
$A_{\rm IA}$ | 0.26 | $\left[-6,6\right]$ | $\left[-6,6\right]$
Baryon Feedback: |
$\log_{10}(T_{\rm AGN}/{\rm K})$ | 7.8 | - | $\left[7.6,8.0\right]$
Table 6: Cosmological and astrophysical systematic parameters adopted for the
mock survey analysis. The input parameters can be compared to the top-hat
prior ranges of the DES-like and KiDS-like pipelines. The TATT ‘Strong’
parameters are the mean marginal values from the DES Y3 cosmic shear with
shear ratio analysis (Amon et al. 2022; Secco, Samuroff et al. 2022). The TATT
‘Weak’ parameters are the mean marginal values from the DES Y3 $3\times 2$pt
without shear ratio analysis (DES Collaboration et al., 2022). The TATT
‘Strong’ model is used to create the mocks analysed with the DES-like pipeline
in Appendices C.1, C.2 and C.3. The impact of using the TATT ‘Weak’ or
‘Strong’ parameters as an input is quantified for both a DES-like and KiDS-
like mock survey analysis in Appendix C.4.

This study was motivated as an alternative safety measure to a blinded data
analysis, as the standard DES and KiDS blinding procedures became redundant
when each survey’s result entered the public domain. Given the wide range of
conclusions drawn from this mock survey analysis, we recommend that in
addition to existing blinding tools, future studies use mocks to validate
pipelines in both the fiducial setup (as in Joachimi et al., 2021; Krause et
al., 2021), and any proposed alternative combinations of models and priors.

The suite of mock data was created using the cosmological parameters listed in
Table 6 which are in agreement with both the DES Y3 and KiDS-1000 cosmic shear
constraints. Specifically, we choose $S_{8}=0.759$ to match the identical
reported best-fit $S_{8}$ values from the fiducial cosmic shear analysis from
Amon et al. (2022); Secco, Samuroff et al. (2022), and the fiducial COSEBIs
cosmic shear analysis from Asgari et al. (2021). The choice is well within
$1\sigma$ of the best-fit $S_{8}$ value from the DES Y3 optimised $\Lambda$CDM
cosmic shear analysis where $S_{8}=0.772^{+0.018}_{-0.017}$, and the best-fit
$S_{8}$ from the KiDS-1000 band power spectra cosmic shear analysis where
$S_{8}=0.760^{+0.016}_{-0.038}$.

Throughout Appendix C, when we refer to the ‘KiDS-like’ pipeline, we use the
KiDS-1000 Asgari et al. (2021) pipeline with one update: the HMCode2016 non-
linear model has been updated to the improved HMCode2020 model adopted in the
Tröster et al. (2021) KiDS-1000 cosmic shear analysis. At the outset of this
project it was our intention to only report constraints from a DES-like and
KiDS-like analysis and the KiDS team wished to use the version of their
pipeline that they considered to be the most accurate. They chose a
$\log_{10}(T_{\rm AGN}/{\rm K})$ prior of $\left[7.6,8.0\right]$, to match the
allowed baryon feedback range where BAHAMAS reproduces the observed baryon
fraction at group scale. In the process of using this updated pipeline to
conduct the mock survey study documented in this appendix, we appreciated the
necessity to extend the scope of project and define a new Hybrid setup for our
headline joint-survey results. For consistency with the most well known
KiDS-1000 cosmic shear results, we chose to revert to the original Asgari et
al. (2021) pipeline for the KiDS-like data analysis reported in Section 3.6.
In Appendix F we find the KiDS-1000 constraints using HMCode2016 and
HMCode2020 differ by $0.2\sigma$. The conclusions that we draw in this
appendix are therefore also applicable to the original Asgari et al. (2021)
pipeline.

In Appendices C.1, C.2 and C.3, when reporting DES-like constraints, the mock
surveys have been created and analysed using Halofit, with no baryon feedback
and a TATT IA model with 5 free parameters. For the same appendices, when
reporting KiDS-like constraints, the mock surveys have been created and
analysed using HMCode2020, including baryon feedback and an NLA IA model
without redshift dependence. We choose to use a ‘noise-free’ mock data vector,
such that for these mocks which match the expectation of each pipeline, the
maximum a posteriori (MAP) is found at the set of input parameters. This data
vector is then analysed, adopting the DES Y3, KiDS-1000 and DES Y3+KiDS-1000
cosmic shear covariances, assuming zero correlation between KiDS-1000 and DES
Y3 in the joint-survey analysis.

The parameters encoded for each astrophysical systematic are listed in Table
6, as informed by the best-fit values for these parameters in Amon et al.
(2022); Secco, Samuroff et al. (2022); DES Collaboration et al. (2022); Asgari
et al. (2021); Tröster et al. (2021).

Figure 11: $S_{8}$ constraints reported from a DES-like (blue circle) and KiDS-like (pink star) analysis of mock survey data representing DES, KiDS and a joint survey. In this figure the noise-free mock survey data differs for each analysis, as they are created to match the astrophysical systematic framework within which they are analysed. For the DES-like analysis (blue), the mock surveys are created using Halofit and a TATT ‘Strong’ IA model (see Table 6). For the KiDS-like analysis (pink), the mock surveys are created using HMCode and an NLA IA model. The input cosmological model and MAP at $S_{8}=0.759$ is marked in black. Any offset in $S_{8}$ is caused by projection effects (see Appendix C.3). For the joint-survey analysis we compare an analysis where the IA nuisance parameters are shared between the surveys (Joint shared IA), to an analysis where the IA signal for each survey is considered to be independent, doubling the number of IA parameters (Joint Mock and colour bars). For the KiDS-like analysis we display the survey-preferred maximum-marginal $S_{8}$ and 68% credible intervals. For the DES-like analysis we display the survey-preferred mean-marginal $S_{8}$ and 68% credible intervals. These data are tabulated in Table 7. Analysis | KiDS | DES
---|---|---
Non-linear | HMCode2020 | Halofit
IA | NLA | TATT
Baryons | $T_{\rm AGN}=7.8$ | Scale Cuts
Quoted $S_{8}$ | Max | Mean | Max | Mean
$S_{8}^{\rm KiDS}$ | $0.752^{+0.022}_{-0.023}$ | $0.747^{+0.026}_{-0.017}$ | $0.758^{+0.028}_{-0.033}$ | $0.748^{+0.036}_{-0.018}$
$S_{8}^{\rm DES}$ | $0.758^{+0.018}_{-0.017}$ | $0.758^{+0.016}_{-0.017}$ | $0.758^{+0.026}_{-0.028}$ | $0.753^{+0.030}_{-0.020}$
$S_{8}^{\rm J1IA}$ | $0.758^{+0.012}_{-0.014}$ | $0.757^{+0.013}_{-0.012}$ | $0.761^{+0.020}_{-0.021}$ | $0.757^{+0.024}_{-0.015}$
$S_{8}^{\rm Joint}$ | $0.757^{+0.013}_{-0.013}$ | $0.757^{+0.011}_{-0.011}$ | $0.764^{+0.016}_{-0.019}$ | $0.759^{+0.019}_{-0.014}$
$\Delta S_{8}^{\rm KiDS}$ | $-0.31\sigma$ | $-0.55\sigma$ | $-0.02\sigma$ | $-0.42\sigma$
$\Delta S_{8}^{\rm DES}$ | $-0.08\sigma$ | $-0.04\sigma$ | $-0.04\sigma$ | $-0.26\sigma$
$\Delta S_{8}^{\rm J1IA}$ | $-0.05\sigma$ | $-0.15\sigma$ | $0.10\sigma$ | $-0.10\sigma$
$\Delta S_{8}^{\rm Joint}$ | $-0.12\sigma$ | $-0.14\sigma$ | $0.27\sigma$ | $0.01\sigma$
$\sigma^{\rm KiDS}/\sigma^{\rm KiDS}_{\rm min}$ | $1.06$ | $1.00$ | $1.42$ | $1.24$
$\sigma^{\rm DES}/\sigma^{\rm DES}_{\rm min}$ | $1.05$ | $1.00$ | $1.64$ | $1.53$
$\sigma^{\rm J1IA}/\sigma^{\rm J1IA}_{\rm min}$ | $1.06$ | $1.00$ | $1.67$ | $1.58$
$\sigma^{\rm Joint}/\sigma^{\rm Joint}_{\rm min}$ | $1.13$ | $1.00$ | $1.60$ | $1.48$
$\sigma^{\rm KiDS}/\sigma^{\rm Joint}$ | $1.80$ | $1.92$ | $1.71$ | $1.61$
$\sigma^{\rm DES}/\sigma^{\rm Joint}$ | $1.37$ | $1.47$ | $1.51$ | $1.52$
$\sigma^{\rm J1IA}/\sigma^{\rm Joint}$ | $1.03$ | $1.10$ | $1.15$ | $1.18$
Table 7: $S_{8}$ constraints for a DES-like and KiDS-like analysis of noise-
free mock survey DES Y3 and KiDS-1000 data. In all cases the mock is created
to match the non-linear, IA and baryon feedback astrophysical systematic
framework within which they are analysed (see upper panel). The different rows
present the results for the analysis of the KiDS, DES and Joint mocks. The
columns present the results from each pipeline, reporting both the maximum-
marginal $S_{8}$ (Max) and the mean-marginal $S_{8}$ (Mean) both with 68%
credible intervals. $\Delta S_{8}$ quantifies the bias of the recovered
$S_{8}$ relative to the true $S_{8}=0.759$, as a fraction of the $1\sigma$
error. This offset is induced by a projection effect when reporting one-
dimensional marginal constraints from prior-dominated and multi-dimensional
degenerate posteriors (see Appendix C.3). As the signal-to-noise of each mock
data vector is identical across trials, an inspection of the relative $S_{8}$
constraining power, $\sigma^{\rm survey}/\sigma^{\rm survey}_{\rm min}$,
reflects the impact of different analysis choices. Here $\sigma^{\rm
survey}_{\rm min}$ is the minimum reported 68% credible interval for each mock
survey across the four analysis trials (KiDS-like, DES-like, and Max and Mean
defined constraints). For the joint-survey analysis we compare an analysis
where the IA nuisance parameters are shared between the surveys (J1IA), to an
analysis setup where the IA parameters for each survey are independent
(Joint). For reference the lower section compares the constraining power for
the KiDS-only, DES-only and joint shared IA analyses (J1IA), to the joint-
survey analysis in each scenario. Some of this data is displayed in Figure 11.

###  C.1 The constraining power of a KiDS-like and DES-like cosmic shear
analysis

In Table 7 and Figure 11 we present $S_{8}$ constraints from a KiDS-like and
DES-like analysis of mock cosmic shear data from DES Y3, KiDS-1000 and a
joint-survey analysis. We find that the DES-like constraints increase the
$1\sigma$ uncertainty on $S_{8}$ by up to $\sim 60\%$ compared to the
corresponding KiDS-like analysis. From the NLA-modified DES-like analysis in
Table 8 and Figure 12, where the difference with the KiDS-like fiducial
$S_{8}$ uncertainties is reduced to $\sim 10\%$, we can conclude that the
contrast in constraining power between the KiDS and DES pipelines is mainly
driven by the choice of IA model494949We remind the reader that Amon et al.
(2022); Secco, Samuroff et al. (2022) show that the decrease in constraining
power when adopting TATT can be mitigated with the inclusion of additional
data from the shear ratio test.. The remaining 10% difference is likely driven
by the chosen samplers, where the KiDS-chosen Multinest sampler is known to
underestimate uncertainty at this level, in comparison to the DES-chosen
Polychord sampler (see Appendix D and Lemos, Weaverdyck et al. 2023).

The different mitigation strategies for baryon feedback were not found to
significantly impact the constraining power in the KiDS-like analysis. Table 8
and Figure 12 present constraints from a KiDS-like analysis that also adopted
DES-like scale cuts to mitigate baryon feedback (see Appendix B). Given the
similar results from these two analyses we can conclude that the small-scale
information removed by the DES scale cuts is already being effectively removed
by the baryon feedback, $T_{\rm AGN}$, nuisance parameter marginalisation in
the KiDS pipeline.

Comparing the constraints from the individual mock surveys and the joint mock
constraints in Table 7, we draw similar conclusions on the expected fractional
gain for constraining power, irrespective of the framework used in the
analysis. In the mocks we find the $1\sigma$ uncertainty on $S_{8}$ from a
joint-survey analysis decreases by a factor of $1.4-1.7$ relative to KiDS-
only, and a factor of $\sim 1.3$ relative to DES-shear-only. These results are
replicated in our Hybrid pipeline analysis of the DES Y3, KiDS-1000 and joint-
survey data in Section 3.

Joint Analysis | Modification | Maximum Marginal | Mean Marginal
---|---|---|---
|  | $S_{8}$ | $\Delta S_{8}$ | $\sigma/\sigma_{\rm min}$ | $S_{8}$ | $\Delta S_{8}$ | $\sigma/\sigma_{\rm min}$
KiDS-like: | Fiducial | $0.758^{+0.012}_{-0.014}$ | $-0.05\sigma$ | $1.1$ | $0.757^{+0.013}_{-0.012}$ | $-0.15\sigma$ | $1.0$
| with DES Priors | $0.754^{+0.013}_{-0.013}$ | $-0.41\sigma$ | $1.1$ | $0.754^{+0.013}_{-0.012}$ | $-0.43\sigma$ | $1.0$
| with DES Scale cuts | $0.758^{+0.014}_{-0.013}$ | $-0.11\sigma$ | $1.1$ | $0.758^{+0.013}_{-0.012}$ | $-0.11\sigma$ | $1.0$
DES-like: | Fiducial | $0.761^{+0.020}_{-0.021}$ | $0.10\sigma$ | $1.7$ | $0.757^{+0.024}_{-0.015}$ | $-0.10\sigma$ | $1.6$
| with KiDS Priors | $0.770^{+0.018}_{-0.020}$ | $0.58\sigma$ | $1.6$ | $0.766^{+0.020}_{-0.014}$ | $0.39\sigma$ | $1.4$
| with NLA | $0.752^{+0.013}_{-0.014}$ | $-0.49\sigma$ | $1.1$ | $0.752^{+0.014}_{-0.013}$ | $-0.49\sigma$ | $1.1$
| with $\Sigma m_{\nu}=0.06{\rm eV}$ | $0.772^{+0.017}_{-0.021}$ | $0.68\sigma$ | $1.6$ | $0.766^{+0.022}_{-0.013}$ | $0.42\sigma$ | $1.4$
Table 8: Quantifying the impact of analysis choices in a joint mock survey
analysis where the noise-free mock survey data matches the non-linear, IA and
baryon feedback astrophysical systematic framework within which they are
analysed. We report both the maximum-marginal $S_{8}$ and 68% credible
intervals, and the mean-marginal $S_{8}$ and 68% credible intervals. $\Delta
S_{8}$ quantifies the offset of the recovered $S_{8}$ from the true
$S_{8}=0.759$, as a fraction of the $1\sigma$ error (see appendix C.3). The
fiducial constraints from the KiDS-like and DES-like joint mock analysis can
be compared to a modified analysis where the alternative set of cosmological
parameter priors are adopted (as listed in Table 2). We also explore the
impact of using an alternative astrophysical systematic mitigation strategy,
with the KiDS-like analysis adopting scale cuts to mitigate baryon feedback
(see Appendix B), and the DES-like analysis adopting the NLA model to mitigate
IA. As the signal-to-noise of each mock data vector is identical for all
tests, an inspection of the $S_{8}$ constraining power between the trials,
$\sigma/\sigma_{\rm min}$, reflects the impact of each survey’s analysis
choices. These data are displayed in Figure 12.  Figure 12: Mock joint-survey
$S_{8}$ constraints, comparing the KiDS-like (upper section) and DES-like
(lower section) fiducial analyses with modifications: adopting an alternative
set of cosmological parameter priors (as listed in the second panel of Table
1); adopting alternative astrophysical systematic mitigation strategies with
the KiDS-like analysis adopting scale cuts to mitigate baryon feedback (see
Appendix B), and the DES-like analysis adopting the NLA model to mitigate IA.
We present both the maximum-marginal $S_{8}$ and 68% credible intervals (blue
circle), and the mean-marginal $S_{8}$ and 68% credible intervals (orange
star). In this figure the mock data matches each survey’s chosen astrophysical
systematic framework, with an identical input cosmology and signal-to-noise.
For reference the colour bars show $0.1\sigma,0.3\sigma$ and $1\sigma$ bands
from the fiducial analysis, centred on the input $S_{8}$, marked in black.
These data are tabulated in Table 8.

###  C.2 Adopting shared or independent intrinsic alignment parameters

In a DES-like analysis, five nuisance parameters are used to marginalise over
uncertainty in the intrinsic alignment of galaxies. In a KiDS-like analysis a
single parameter is adopted (see Section 2.2 for details). In direct IA
measurements significant variation is seen as a function of galaxy type and
luminosity (see for example Mandelbaum et al. 2006; Johnston, Georgiou et al.
2019, and references therein). Furthermore Singh & Mandelbaum (2016);
Georgiou, Johnston et al. (2019) measure a dependence of the IA amplitude on
the isophote used to define galaxy ellipticity. This means that the IA
nuisance parameters in our cosmic shear analyses constrain a shape-measurement
dependent model of the average intrinsic alignment contamination for the
source galaxy population of each survey505050Note that some cosmic shear
analyses also include a red/blue galaxy split in their intrinsic alignment
analysis (Heymans et al., 2013; Samuroff et al., 2019; Li et al., 2021)..

Both Amon et al. (2022); Secco, Samuroff et al. (2022) and Asgari et al.
(2021) measure an effective intrinsic alignment amplitude that is consistent
with zero. Given the broad similarities between the DES and KiDS survey
depths, redshift distributions and shape measurement approaches, we do not
expect to detect any statistically significant differences between the
effective intrinsic alignment contamination of the two surveys. As such, it
could be preferable to adopt a single set of intrinsic alignment parameters to
coherently model IA contamination in a joint-survey analysis, and we denote
this approach, ‘Joint shared IA’.

It has been noted that the IA parameter constraints are somewhat sensitive to
systematic errors in the photometric redshift distributions (see for example
the $\lesssim 0.5\sigma$ parameter variations in Figure D.1 of Wright et al.
2020b, Figure 18 of Amon et al. 2022 and Figure 15 of Secco, Samuroff et al.
2022, when different redshift distributions are adopted in the cosmic shear
analysis). Fischbacher et al. (2023) quantify the interplay between the free
parameters of the IA model and photometric redshift calibration corrections
which can lead to biases that are sensitive to prior volume effects. Given the
differences between the DES and KiDS approach to redshift calibration, it
could therefore be preferable to adopt independent sets of IA parameters,
accounting for both the extra freedom that IA-marginalisation affords to
absorb redshift errors and the uncertainty over the impact of variations in
galaxy population515151See also figure 16 of Amon et al. (2022) which finds
significant variation in the recovered TATT amplitude parameters when
different tomographic bins are excluded from the analysis. and shape
measurement methods. In our fiducial analyses we adopt this ‘Joint’ approach,
doubling the number of IA parameters.

In Table 7 and Figure 11 we compare the two scenarios, finding the $1\sigma$
constraints on $S_{8}$ from the joint analysis to be $\sim 10\%$ smaller than
the constraints from the joint shared IA analysis. This result is counter-
intuitive, given that a larger number of free nuisance parameters would
typically lead to a less constrained cosmological model. This behaviour is,
however, not replicated in our Hybrid analysis of the data in Section 3.3
where the $S_{8}$ constraints from the shared IA analysis and fiducial joint
analysis are indistinguishable. We therefore consider this unexpected result
to be an artefact related to the noise-free nature of the mock data vector.

###  C.3 Quantifying projection effects and the offset introduced on $S_{8}$
marginals

For the mocks analysed so far in this appendix, the MAP is the input parameter
set by definition, as we do not apply noise and we use priors that are either
flat, or peaked at the input values. We therefore use these mocks to quantify
the bias525252Throughout Appendix C we use the term ‘bias’, defined as the
measured offset between the reported $S_{8}$ value and the input truth. As
this word commonly implies inaccuracy, it is worth highlighting that an offset
induced by a projection effect is not a reflection of any error with the
measured posterior. Instead these offsets demonstrate the non-Gaussian nature
of the cosmic shear multi-dimensional posterior: the conventional expectation
that the mean and maximum marginal distribution will recover the MAP no longer
applies. This analysis is intended to draw attention to the problematic
practice of directly comparing 1D marginal values from a range of different
surveys and probes where the projection effects will differ. It also allows
for quantification of the marginal offsets inherent to each parameter and
prior set, before determining the inaccuracy that arises when the models
adopted for the analysis differ from the underlying truth (see Appendices C.4
and E.2). that is introduced, relative to the MAP, when reporting projected 1D
marginal constraints (see also Chintalapati et al., 2022). In Table 7 and
Figure 11 we compare $S_{8}$ values and their 68% credible intervals defined
from the maximum-marginal535353The maximum-marginal posterior is the KiDS-
preferred alternative to their fiducial, but computationally expensive,
MAP+PJ-HPD estimate. Here we use chainconsumer with the settings
statistics=‘max’ and kde=1.0 (Hinton, 2016). The credible intervals are
defined using a 1D kernel density estimator (KDE), reporting the asymmetric
iso-likelihood levels above and below the KDE maximum., and the mean-
marginal545454The mean-marginal posterior is the DES-preferred estimator. Here
we use postprocess within CosmoSIS using the default settings (Zuntz et al.,
2015). The credible intervals are defined using a 1D KDE with the equal-
posterior asymmetric limits defined to contain the correct number of samples.
This is in contrast to the intervals defining the credible region relative to
the smoothed probability volume (see appendix F of Zuntz et al., 2015, for
details).. We find that the reported maximum-marginal credible intervals are
$\sim 10\%$ larger than the reported mean-marginal credible intervals for all
chains. This difference is likely a result of the different approaches taken
by chainconsumer and CosmoSIS-postprocess to define credible intervals (Zuntz
et al., 2015; Hinton, 2016). We find a $\sim 0.2\sigma$ offset between the
mean and maximum marginal for the more skewed Polychord chains that include a
TATT IA model, which we discuss further in Appendix D.

In Table 8 and Figure 12 we find that the KiDS-like and DES-like fiducial
joint-survey analyses recover the input cosmology within $\sim 0.1\sigma$,
which is the expected run-to-run variance between chains (Joachimi et al.,
2021). Interestingly, however, we find a significant offset when the
alternative survey’s cosmological priors are adopted (see Table 1). In a
modified KiDS-like analysis which adopts the DES cosmological priors, the
reported $S_{8}$ is underestimated by $0.4\sigma$. In a modified DES-like
analysis which adopts the KiDS cosmological priors, the reported $S_{8}$ is
overestimated by $0.4-0.6\sigma$. Note in both these modified analyses the
astrophysical systematics mitigation strategy for each survey was retained.

In order to understand the different projection effects with each survey’s
choice of cosmological priors, we conducted a modified DES-like analysis using
the NLA IA model, as both input to the simulated mock and as part of the
analysis pipeline, and the DES cosmological priors. The resulting marginal
$S_{8}$ value is underestimated by $0.5\sigma$, consistent with the bias found
with a KiDS-like NLA analysis adopting DES priors. We also conducted a
modified DES-like analysis adopting a fixed neutrino mass prior in addition to
the other DES cosmological priors. We find the resulting marginal $S_{8}$ to
be overestimated by $0.4-0.7\sigma$, similar to the bias found for a DES-like
analysis which adopts the KiDS cosmological priors with a fixed neutrino mass.
From these modified analyses we conclude that there is a complex interplay
between the cosmological parameter priors and the astrophysical systematic
prior space, especially the neutrino and IA priors. As an example of where
complexity could arise we note that the non-NLA component of the TATT model,
the tidal torque term modulated by the $A_{2}$ and $\eta_{2}$ parameters,
depends on the linear matter power spectrum. This means that the calculated
contribution to the observed cosmic shear signal from the TATT tidal torque
term is insensitive to variations in the neutrino mass or the baryon feedback
parameter in an inference analysis.

Chintalapati et al. (2022) show that the picture is further complicated when
priors for unconstrained cosmological parameters are asymmetric about the true
parameter value, as in the case of the neutrino mass prior in the DES-like
analysis of these mocks. Projection effects therefore need to be carefully
considered when comparing marginal statistics of surveys that adopt different
primary parameter sets and priors (see for example Longley et al., 2023, who
find significant shifts in poorly constrained parameters, such as $\Omega_{\rm
m}$, with different prior choices).

###  C.4 Cosmological constraints when the underlying astrophysical model
differs from the survey-specific framework

In this section we analyse mock joint-survey data where the underlying
astrophysical systematic differs from the theoretical models adopted by each
survey, in one way. In contrast to the mocks previously analysed, the MAP will
no longer be found at the set input parameters. This study therefore
quantifies potential systematic biases in the recovered cosmological
parameters from a joint-survey analysis which adopts a different astrophysical
systematic model from the truth. This analysis extends and complements
existing studies in this area, which we briefly review before presenting the
mock analysis.

In comparing IA models, Blazek et al. (2019) demonstrate that a future Rubin
cosmic shear analysis that adopts the NLA IA model, when the underlying IA
truth is TATT, may be subject to a significant bias in the recovered
cosmological parameters. They find an offset $\Delta S_{8}\sim 0.08$ for their
preferred set of TATT parameters, which would translate into a $\sim
0.7\sigma$ offset for a joint DES-KiDS analysis. Fortuna et al. (2021) reach a
similar conclusion for a future Euclid-like survey cosmic shear analysis
adopting the NLA IA model, when the underlying IA truth was an aligned
analytical halo model. The offset they find is less significant, however, with
$\Delta S_{8}\sim 0.005$, which would be negligible for this joint-survey
study. Secco, Samuroff et al. (2022), analysing mock DES Y3 cosmic shear data
created with the DES Y1 best-fit TATT parameters from Samuroff et al. (2019),
find a significant $\sim 3\sigma$ offset in $\Omega_{\rm m}$ with a good
recovery of the input $S_{8}$ when adopting an NLA model. Campos et al. (2023)
show that the different implications of these studies can be understood by
recognising the strong dependence of model tests on the assumed IA parameters.
Defining a range of IA models from 21 samples of the DES Y1 TATT parameter
posteriors, they find bias from an NLA analysis of DES Y3-like mocks could
range from $0.04\sigma_{\rm 2D}$ to $5\sigma_{\rm 2D}$, where $\sigma_{\rm
2D}$ is the 68% credible interval in the 2D $(S_{8},\Omega_{\rm m})$
marginalised posterior. They propose an empirical model selection approach to
better inform the choice of IA model in future cosmic shear analyses.

Turning to the non-linear power spectrum model, Krause et al. (2021) use
N-body simulations from Mira Titan, Cosmic Emu (Lawrence et al., 2017) and the
EuclidEmulator1 (Euclid Collaboration: Knabenhans et al., 2019), to
demonstrate that the Halofit non-linear model for the matter power spectrum is
sufficiently accurate for the DES Y3 baseline $3\times 2$pt analysis with
biases at $<0.15\sigma_{\rm 2D}$. They also conclude that HMCode2016 fails
their requirements for a $2\times 2$pt joint galaxy-galaxy lensing and galaxy
clustering analysis (see section B1 of Krause et al., 2021). In contrast,
Joachimi et al. (2021) use the Cosmic Emu simulations (Heitmann et al., 2014)
to demonstrate that the HMCode non-linear model for the matter power spectrum
is sufficiently accurate for the KiDS-1000 baseline analysis with biases at
$<0.1\sigma$ in $S_{8}$. They find, however, an unacceptable $0.3\sigma$
offset in the recovered value of $S_{8}$ when adopting Halofit555555The
Joachimi et al. (2021) result is consistent with the $0.4\sigma$ offset in
$S_{8}$ between the Halofit and BaccoEmu DES Y3 cosmic shear analysis
presented in Aricò et al. (2023) which they conclude arises from the lower
accuracy of Halofit over the wide parameter space that is sampled by the
posterior. It is also consistent with the $0.5\sigma$ offset in $S_{8}$
between the Halofit and HMCode DES Y3 cosmic shear analysis presented in
Secco, Samuroff et al. (2022) which they conclude originates from projection
effects from the adopted baryon feedback priors which are significantly wider
than the HMCode2020 $T_{\rm AGN}$ priors studied in this analysis.. We find
that these apparently inconsistent conclusions can be resolved by recognising
the differing goals of the two surveys, with DES focussed on optimising the
accuracy of the $3\times 2$pt analysis and KiDS more focussed on the accuracy
of the cosmic shear measurement. Furthermore, a fixed tolerance level in terms
of $\sigma_{\rm 2D}$ in the $\Omega_{\rm m}-S_{8}$ plane, allows for a less
restricted tolerance level in the 1D $S_{8}$ marginals (see the discussion in
Appendix E).

|  | Maximum Marginals: | Mean Marginals:
---|---|---|---
|  | KiDS-like: | DES-like: | KiDS-like: | DES-like:
Non-Linear | Intrinsic Alignment | $S_{8}$ | $\Delta S_{8}$ | $S_{8}$ | $\Delta S_{8}$ | $S_{8}$ | $\Delta S_{8}$ | $S_{8}$ | $\Delta S_{8}$
Fiducial | Fiducial | $0.758^{+0.012}_{-0.014}$ | $-0.05\sigma$ | $0.761^{+0.020}_{-0.021}$ | $0.10\sigma$ | $0.757^{+0.013}_{-0.012}$ | $-0.15\sigma$ | $0.757^{+0.024}_{-0.015}$ | $-0.10\sigma$
HMCode2020 | TATT strong | $0.775^{+0.012}_{-0.012}$ | $1.30\sigma$ | $0.741^{+0.019}_{-0.022}$ | $-0.88\sigma$ | $0.775^{+0.012}_{-0.012}$ | $1.28\sigma$ | $0.735^{+0.024}_{-0.015}$ | $-1.28\sigma$
HMCode2020 | TATT weak | $0.763^{+0.012}_{-0.013}$ | $0.32\sigma$ | $0.736^{+0.013}_{-0.017}$ | $-1.55\sigma$ | $0.762^{+0.013}_{-0.013}$ | $0.24\sigma$ | $0.733^{+0.014}_{-0.014}$ | $-1.88\sigma$
Table 9: Joint survey $S_{8}$ constraints from mock data where the underlying
astrophysical systematic differs from the adopted survey-specific model. The
fiducial analyses can be compared to a scenario where the non-linear matter
power spectrum, including baryon feedback, is defined using HMCode2020,
deviating from the Halofit model adopted by DES. The IA model is then defined
using TATT, deviating from the NLA model adopted by KiDS, exploring two
strengths for the TATT IA contamination (see Table 6). We report the $S_{8}$
constraints and 68% credible interval using both the maximum-marginal and
mean-marginal approach, with $\Delta S_{8}$ quantifying the offset from the
true $S_{8}=0.759$, as a fraction of the $1\sigma$ error. These data are
displayed in Figure 13. . Figure 13: Joint survey $S_{8}$ constraints reported
from a DES-like (blue) and KiDS-like (pink) analysis of mock survey data where
the underlying astrophysical systematics differ in one way from the
theoretical models adopted by each survey: the non-linear matter power
spectrum, including baryon feedback, is defined using HMCode2020 as expected
by the KiDS-like pipeline, and the IA contamination is defined using two
different strengths of TATT as expected by the DES-like pipeline (see Table
6). For reference the colour bars show the KiDS-like (pink) and DES-like
(blue) constraints from the fiducial analyses where the mocks are constructed
using the astrophysical systematics expected by each pipeline. In addition, we
present DES-like and KiDS-like constraints for an NLA (no-z) mock where the
input power spectrum changes from an HMCode2020 model including baryon
feedback, to a dark matter only EuclidEmulatorv2 model. Constraints from our
Hybrid analysis are shown in green, analysing the same EuclidEmulatorv2 mock
with the addition of OWLS-AGN baryon feedback. For the KiDS-like analysis we
display the survey-preferred maximum-marginal $S_{8}$ and 68% credible
intervals. For the DES-like and Hybrid analysis we display the survey-
preferred mean-marginal $S_{8}$ and 68% credible intervals. These data are
tabulated in Tables 9 and 12.

Table 9 and Figure 13 present the $S_{8}$ constraints from a DES-like and
KiDS-like analysis of a joint survey cosmic shear data vector that has been
created with one underlying astrophysical systematic that differs from the
model adopted in each framework. The non-linear matter power spectrum is
defined using HMCode2020. The baryon feedback is set with an AGN heating
temperature, $\log_{10}(T_{\rm AGN}/{\rm K})=7.8$, to match the BAHAMAS
simulation which best recovers the observed hot gas mass fraction of groups
and clusters in addition to the galaxy stellar mass function (McCarthy et al.,
2017). We include intrinsic alignments using the TATT model comparing two sets
of TATT parameters. We take the best-fit TATT parameters565656The $\eta_{1}$,
$\eta_{2}$, and $b_{\rm TA}$ TATT parameters are formally unconstrained in all
DES Y3 analysis combinations. We denote the DES Y3 cosmic shear with SR
parameters TATT ‘Strong’, because the best-fit $\eta_{1}$ and $b_{\rm TA}$
parameters are close to the allowed upper prior limit for these values. We
denote the TATT parameters from the DES Y3 $3\times 2$pt without SR MagLim
analysis, TATT ‘Weak’ because the $a_{2}$ and $b_{\rm TA}$ values are close to
zero, such that the TATT model is tending towards an NLA model. from the
optimised $\Lambda$CDM cosmic shear analysis of Amon et al. (2022); Secco,
Samuroff et al. (2022), denoted TATT ‘Strong’ in Table 6. We also take the
best-fit TATT parameters from the $3\times 2$pt analysis in DES Collaboration
et al. (2022), denoted TATT ‘Weak’ in Table 6. Both of these scenarios are
within the range allowed by the most constraining Y3 analysis ($3\times 2$pt
with shear ratio data), and so are considered plausible given the current
data.

Assessing the KiDS-like pipeline, a comparison of the fiducial analysis with
the analyses from both HMCode2020+TATT mocks reveals the systematic bias in
the recovered $S_{8}$ that is introduced by adopting the NLA model when the
underlying truth is a DES Y3-like TATT model. The mismatch between intrinsic
alignment models leads to an overestimate of the true $S_{8}$ at the level of
$1.3\sigma$, in the TATT-strong case, and $0.3\sigma$ in the TATT-weak case.
This result is consistent with the analysis in figure 15 of Amon et al.
(2022), where the $S_{8}$ NLA constraint is $\sim 0.9\sigma$ higher than the
corresponding TATT analysis. In contrast to the mock analysis of Secco,
Samuroff et al. (2022), we find that the $\Omega_{\rm m}$ constraints for the
HMCode2020+TATT mocks are unchanged compared to the fiducial analysis. This is
in agreement with the NLA-TATT analysis comparison in Amon et al. (2022),
demonstrating how sensitive the conclusions of mock studies are to the default
TATT parameters adopted for the study.

Assessing the DES-like pipeline, we can compare the fiducial analysis with
either of the HMCode2020+TATT mock analyses. This reveals the systematic bias
in the recovered $S_{8}$ that results from adopting a Halofit non-linear
correction with scale cuts to mitigate baryon feedback, when the underlying
truth is a non-linear matter power spectrum from HMCode2020 including AGN
baryon feedback. The mismatch between non-linear power spectrum models leads
to an underestimate of the true $S_{8}$ at the level of $1.3\sigma$, in the
TATT-strong case, and $1.9\sigma$ in the TATT-weak case when reporting mean
marginals. As can be seen in Figure 13 the difference between these two
results is primarily driven by the reduced $\sigma$ error on $S_{8}$ in the
TATT-weak case. These offsets reduce to $0.9\sigma$, and $1.6\sigma$, when
considering the maximum marginal constraints. Given the scale-cut analysis in
Appendix B, we expect any bias from neglecting baryon feedback to be within
$\sim 0.3\sigma_{\rm 2D}$. We therefore expect that this offset is primarily
caused by the mismatch in the non-linear modelling of the power spectrum. We
explore this further with a dark matter-only mock survey analysis in Appendix
E.2.

From this mock joint survey cosmic shear analysis we conclude that neither the
KiDS-like nor DES-like analysis choices pass the accuracy requirements set by
each survey, when considering an $S_{8}=0.759$ cosmology with a TATT IA model
and an HMCode2020 non-linear matter power spectrum. In this specific scenario,
the combination of different systematic modelling choices bias the constraints
in opposing directions leading to a $\sim 2\sigma$ level tension between a
KiDS-like and DES-like analysis of identical data vectors. To address this
potential issue we developed the concept of the Hybrid pipeline, described in
Section 2, which is validated using the EuclidEmulatorv2 in Appendix E. The
Hybrid setup was finalised before the joint survey data analysis commenced to
minimise any confirmation bias in our decision making process.

##  Appendix D Sampler comparison

Summary: In this appendix we find that cosmic shear posteriors sampled with
Multinest underestimate the width of the 68%/95% marginal credible interval
for $S_{8}$ by 12%/15%. Polychord is shown to provide an accurate recovery,
within a few percent, of both the 68% and 95% $S_{8}$ marginal credible
interval when compared to a Metropolis-Hastings sampler and Emcee.

In this appendix we compare the posteriors recovered from a DES Y3 cosmic
shear analysis575757We note that this cosmic shear analysis does not include
additional information from the shear ratio test. using two nested sampling
algorithms, Multinest (Feroz et al., 2009) and Polychord (Handley et al.,
2015), and two Markov Chain Monte Carlo (MCMC) algorithms, Metropolis-Hastings
and emcee (Metropolis et al., 1953; Goodman & Weare, 2010; Foreman-Mackey et
al., 2013). Polychord and Multinest vary in their implementation of nested
sampling. Multinest uses a series of multivariate ellipses to define
likelihood thresholds which Lemos, Weaverdyck et al. (2023) show can lead to
an undersampling of the posterior tails for many-parameter unimodal non-
Gaussian posteriors. Polychord uses one-dimensional slice sampling, starting
at a given set of parameter values to determine the posterior. Samples are
then taken across one parameter only to find a new parameter set with a higher
posterior, continuing the iteration one dimension at a time. This approach
makes Polychord significantly slower585858The Polychord DES Y3 cosmic shear
analysis adopting a TATT (NLA) IA model, consumed a factor of $\sim 3(\sim 7)$
times as many CPU hours as the equivalent Multinest analysis. than Multinest,
but it was found to be more accurate in the DES Y1 lensing and clustering
sampling study of Lemos, Weaverdyck et al. (2023). This appendix extends this
study by analysing DES Y3 cosmic shear data from Amon et al. (2022); Secco,
Samuroff et al. (2022) within the flat $\Lambda$CDM model. Our analysis
duplicates and agrees with the recent HSC Y3 cosmic shear sampler comparison
(Li et al., 2023a).

We adopt the DES Y3 sampler settings595959Polychord settings: $n_{\rm
live}=500$, $n_{\rm repeats}=60$, tolerance=0.01, fast_fraction = 0.1,
fast_slow = True. for Polychord, and the KiDS-1000 sampler
settings606060Multinest settings: $n_{\rm live}=1000$, efficiency = 0.3,
tolerance = 0.01, constant efficiency = False, max_iterations = 1,000,000. for
Multinest (see section 4.2 and 4.3 of Lemos, Weaverdyck et al. 2023 for
details of the nested sampling hyper parameters and convergence diagnostics
for each software package). To benchmark the results from these two nested
samplers we use both a classical Metropolis-Hastings sampler616161We use the
CosmoSIS Metropolis sampler, running eight independent chains. We follow
Lemos, Weaverdyck et al. (2023) in burning the first 20% of each chain and
determine convergence using the Gelman-Rubin test with a tolerance of 0.01.
and the emcee affine invariant MCMC sampler626262emcee simultaneously evolves
an ensemble of walkers, where the position in parameter space of each walker
depends on the positions of the remaining walkers. We combine four independent
emcee chains run with the parameter settings walkers = 160, samples = 12000.
The resulting chains have a total of $\sim 6$M samples with $\sim 2$M burn-in
samples removed using Chainconsumer diagnostic statistics to indicate
convergence according to both the Gelman-Rubin and Geweke metrics (Hinton,
2016). We note that these metrics provide only approximate estimates of
convergence for emcee given the correlations between the walkers. A more
appropriate metric is the integrated autocorrelation time (Goodman & Weare,
2010). Unfortunately this statistic was found to be highly sensitive to
variations in the methodology chosen to estimate it, suggesting that our
chains are relatively short compared to the quantity. Our best estimate
suggests that after burn-in removal, there are $\sim 5$ autocorrelation times
in each of the four independent chains being combined to produce emcee
posteriors.. Both algorithms start from a proposal distribution which we take
from the output of the Multinest or Polychord analysis. We do not use these
two samplers for our cosmological analysis with the Hybrid pipeline due to the
lack of Bayesian evidence estimation. Furthermore, when analysing large-
dimensional parameter spaces and non-Gaussian posterior distributions, the run
times to reach convergence can become prohibitive636363Even when started with
accurate proposal distributions, the emcee (Metropolis-Hastings) DES Y3 cosmic
shear analysis consumed a factor of $\sim 10$ ($\sim 1.5$) times as many CPU
hours as the equivalent Polychord analysis..

Figure 14 shows the 68% and 95% credible intervals of the sampled posteriors,
marginalised in the $\Omega_{\rm m}-S_{8}$ plane, where we have adopted the
DES methodology framework summarised in Table 2. We find significant
differences between the credible intervals, with a narrower Multinest
posterior compared to the Polychord, emcee and Metropolis-Hastings posterior.
Specifically we find the width of the 68% credible interval,
$\Delta\sigma_{68}=\sigma_{68}^{\rm upper}-\sigma_{68}^{\rm lower}$, for the
$S_{8}$ marginal, to be $\sim 12\%$ ($\sim 9\%$) narrower compared to the
Metropolis-Hastings (emcee or Polychord) posterior (see Figure D and Table
10).

Inspecting the width of the 95% marginal credible interval, the $\sim 15\%$
narrowing of the Multinest posterior relative to the ‘truth’ is even more
pronounced. The $S_{8}$ credible interval from Polychord is, however, accurate
to a few percent. We note the emcee and Metropolis-Hastings analyses require a
proposal distribution of starting guesses, finding a few percent variation
between two analyses using initial starting points drawn from the Polychord or
Multinest posteriors (see Table D for details, where Figures 14 and D display
the Polychord-starting option). As such, even though we use the emcee and
Metropolis-Hastings result as our ‘truth’, there is still some degree of
uncertainty in it.

In Figure D we compare the 1D marginalised posteriors for $S_{8}$ for both the
fiducial DES analysis which adopts the TATT IA model (upper panel), and an
alternative analysis adopting the NLA (no-z) IA model (lower panel). We note
that the narrowing of the posterior is asymmetrical about the maximum, which
is likely a side-effect of the Multinest ellipsoidal sampling (see Figure 1 of
Lemos, Weaverdyck et al. 2023 for an illustration). The skew of the Polychord
$S_{8}$ posterior, relative to the Multinest posterior leads to significant
differences between the lower $S_{8}$ credible interval, and larger offsets
between mean $S_{8}$ marginal estimates, compared to estimates of the maximum
of the marginalised $S_{8}$ posterior.

Figure 14: Comparing the Polychord (black), Multinest (red), emcee (blue,
dashed) and Metropolis-Hastings (MH, light blue, filled) samplers for a DES Y3
cosmic shear analysis: the 68% and 95% credible intervals are shown for the
two-dimensional posterior distribution in the $\Omega_{\rm m}-S_{8}$ plane.

Figure 15: Comparing DES Y3 cosmic shear one-dimensional marginalized
posteriors for $S_{8}$ obtained using Polychord (black), Multinest (red),
emcee (blue, dashed) and Metropolis-Hastings (MH, light blue) samplers.
Results from an analysis using the TATT IA model, matching the posteriors
shown in Figure 14, are shown in the upper row, while the lower row adopts an
NLA IA model.

| $S_{8}(\rm mean)$ | $\Delta\sigma_{68}$ | $\Delta\sigma_{95}$
---|---|---|---
TATT: |  |  |
Polychord | 0.759 | 0.063 | 0.134
Multinest | 0.763 | 0.058 | 0.115
emcee (PC) | 0.760 | 0.064 | 0.134
emcee (MN) | 0.761 | 0.064 | 0.132
Metropolis-Hastings (PC) | 0.760 | 0.065 | 0.135
Metropolis-Hastings (MN) | 0.760 | 0.067 | 0.137
NLA: |  |  |
Polychord | 0.782 | 0.043 | 0.091
Multinest | 0.786 | 0.039 | 0.081
Table 10: The impact of sampler choice on one-dimensional marginalized mean
$S_{8}$ constraints from the DES Y3 cosmic shear analyses. For the emcee and
Metropolis-Hastings analyses, ‘PC’ and ‘MN’ indicate whether the initial
starting points were drawn from the Polychord or Multinest posteriors,
respectively. Columns show the CosmoSIS-postprocess-defined posterior mean,
the full width of the 68% credible interval
($\Delta\sigma_{68}=\sigma_{68}^{\rm upper}-\sigma_{68}^{\rm lower}$) and the
full width of the 95% credible interval ($\Delta\sigma_{95}=\sigma_{95}^{\rm
upper}-\sigma_{95}^{\rm lower}$).

##  Appendix E Verifying the Hybrid analysis pipeline

Summary: In this appendix we quantify the accuracy of the Hybrid pipeline
analysis through a series of mock analyses using the EuclidEmulatorv2 (Euclid
Collaboration: Knabenhans et al., 2021).

  * •

In Appendix E.1 we show HMCode2020 predictions of the DES $\xi_{\pm}(\theta)$
and KiDS $E_{n}$ cosmic shear observables agree with the EuclidEmulatorv2
model at the percent level across a wide range of cosmological parameters.

  * •

In Appendix E.2 we find that using the Appendix B-defined scale cuts alone
results in a $0.5\sigma$ bias on $S_{8}$ in the presence of OWLS AGN baryon
feedback. Combining scale cuts with marginalisation over a free AGN feedback
parameter $T_{\rm AGN}$, however, reduces the bias to $0.0\sigma$. We find the
inclusion of an free neutrino mass parameter introduces a projection offset in
the marginal value of $S_{8}$ at the level of $0.3\sigma$.

###  E.1 Quantifying the accuracy of non-linear models for DES and KiDS cosmic
shear

In Figure 16 we compare the DES and KiDS cosmic shear statistics predicted
from the EuclidEmulatorv2 to predictions adopting different non-linear
models646464We refer readers interested in a direct comparison between the
different $P_{\delta}(k,z)$ models to figures 2 and D.1 of Mead et al. (2021),
figures 13 and 14 of Euclid Collaboration: Knabenhans et al. (2021) and figure
7 of Adamek et al. (2023).. The EuclidEmulatorv2 has been shown to be accurate
to $<1$% for $k\lesssim 10h{\rm Mpc}^{-1}$ and $z\lesssim 3$ (Euclid
Collaboration: Knabenhans et al., 2021; Adamek et al., 2023). For the mock
survey analysis in this appendix we have modified our input set of
cosmological parameters to lie within the emulator’s parameter range (see
Table 11), retaining the input $S_{8}=0.759$, to match the fiducial $S_{8}$
cosmic shear result from both KiDS-1000 and DES Y3. With this set of
cosmological parameters we find significant offsets in the small-scale cosmic
shear signal when adopting Halofit.

Figure 16: The ratio for a range of KiDS $E_{n}$ (upper) and DES $\xi_{\pm}(\theta)$ (lower) and cosmic shear models to the prediction from EuclidEmulatorv2. Each auto-correlation tomographic bin is shown, with similar trends found for the cross-correlations (not shown). The dark matter-only predictions for Halofit (blue) and HMCode2020 (cyan) can be compared to models that include baryon feedback using OWLS-AGN (pink) and the HMCode2020 $T_{\rm AGN}$ parameter with $\log_{10}(T_{\rm AGN}/{\rm K})=7.3$ (green), $\log_{10}(T_{\rm AGN}/{\rm K})=7.6$ (yellow) and $\log_{10}(T_{\rm AGN}/{\rm K})=8.0$ (purple). We have verified that cosmic shear signals predicted by the HMCode2020 $T_{\rm AGN}$ model (green, yellow, purple), agree at the percent level to the signals predicted when using the equivalent BAHAMAS power spectrum to directly suppress the EuclidEmulatorv2 dark matter power spectrum. The grey shaded regions indicate the adopted DES scale cuts. The upper two rows compare the KiDS fiducial COSEBIs $E_{n}$ with $\theta_{\rm min}=$ 0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5, to the scale cut version with $\theta_{\rm min}=$ 2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0. Figure 17: Comparison of the DES $\xi_{\pm}(\theta)$ (upper) and KiDS $E_{n}(\theta_{\rm min}=$0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5$)$ (lower) cosmic shear predictions from HMCode2020 (green), HMCode2016 (blue) and Halofit (pink), as a ratio with the prediction from the EuclidEmulatorv2 (see Equation 136) for a range of cosmological parameters. Each data point represents one sample from the DES Y3 $3\times 2$pt posterior that lies within the allowed parameter range of the EuclidEmulatorv2. The solid lines show an unweighted linear best fit to the data points as a function of input $S_{8}$ (left), $\Omega_{\rm m}$ (middle) and $\Sigma m_{\nu}$ (right). Only weak trends are seen with all the parameters including those not shown here. The shaded region brackets $\pm 1\%$ accuracy. Parameter | Mock Input |  EuclidEmulator2 range
---|---|---
$\Omega_{\rm b}$ | $0.041$ | $\left[0.04,0.06\right]$
$\Omega_{\rm m}$ | $0.246$ | $\left[0.24,0.40\right]$
$n_{\mathrm{s}}$ | $0.99$ | $\left[0.92,\,1.00\right]$
$h$ | $0.7251$ | $\left[0.61,0.73\right]$
$w_{0}$ | $-1.0$ | $\left[-1.30,-0.70\right]$
$w_{\rm a}$ | $0.0$ | $\left[-1.7,0.5\right]$
$\Sigma m_{\nu}$ |  $0.06$ eV | $\left[0.0,0.15\right]{\rm eV}$
$A_{\rm s}$ | $2.445\times 10^{-9}$ | $\left[1.7,2.5\right]\times 10^{-9}$
Table 11: Cosmological parameters adopted for the EuclidEmulator2-based mock
joint-survey analysis which correspond to an input $S_{8}=0.759$. The chosen
parameters lie within the range where the emulator has been shown to be
accurate to $<1\%$ for $k\lesssim 10h{\rm Mpc}^{-1}$ and $z\lesssim 3$ (Euclid
Collaboration: Knabenhans et al., 2021).

In order to determine whether the accuracy of each non-linear model changes
across a wide range of cosmological parameters, in Figure 17 we compress the
theoretical $\xi_{\pm}(\theta)$ or $E_{n}$ tomographic data vector, $\bf{d}$,
into a single signal-to-noise estimate, $\nu$, as

$\nu_{\rm H}=\sqrt{\bf{d}_{\rm H}^{\rm T}\,\bf{C}^{-1}\,\bf{d}_{\rm H}}\,.$
(136)

Here $\bf{C}$ is the corresponding DES or KiDS covariance matrix and ${\rm H}$
labels the adopted non-linear model Halofit, HMCode2016 or HMCode2020. Figure
17 shows the ratio $\nu_{\rm{EuclidEmulator}}/\nu_{\rm H}$ for a range of
different cosmologies that fall within the region of parameter space covered
by the EuclidEmulatorv2 (see Table 11). In order to test a realistic
combination of cosmological parameters, each data point represents a sample
from the DES Y3 $3\times 2$pt Polychord chain. We find only weak trends with
each cosmological parameter, with the HMCode2016 and HMCode2020 predictions
agreeing with the EuclidEmulatorv2 model at the percent level for both
statistics. Halofit provides a poorer fit, but is nevertheless accurate at the
$\sim 2\%$ level overall.

Table 12 and Figure 13 present the $S_{8}$ constraints from the
EuclidEmulatorv2-based mock. With the impact of the IA model choice assessed
in Appendix C.4, we adopt the NLA (no-z) model as an input for this analysis.
For the DES-like analysis we find an underestimate of the true $S_{8}$ at the
level of $\sim 1.2\sigma$. This offset is likely caused from the mismatch
between the Halofit non-linear power spectrum and EuclidEmulatorv2 shown in
Figures 16 and 17, combined with the use of the TATT model which can introduce
a bias in the projected marginal $S_{8}$ posterior656565When analysing an NLA-
input mock with TATT and HMCode2020 we find a $\sim 0.5\sigma(0.8\sigma)$
underestimate of the input $S_{8}$ when using the maximum (mean) marginalised
posterior estimate (see Test G in Table 12).. For the KiDS-like analysis, we
find an overestimate of the true $S_{8}$ at the level of $\sim 0.9\sigma$,
which can be understood when considering the $T_{\rm AGN}$ baryon feedback
marginalisation within the KiDS pipeline. As shown in Figure 16, the KiDS
BAHAMAS-range prior on $\log_{10}(T_{\rm AGN}/{\rm K})$, with
$\left[7.6,8.0\right]$, does not allow for a dark-matter-only cosmology. It
only includes feedback models which reduce power on small physical scales,
such that baryon feedback marginalisation, in a dark-matter-only Universe,
leads to an overestimate of the marginal $S_{8}$. In a KiDS-like analysis
modified to use a dark-matter-only matter correction for the non-linear matter
power spectrum, we recover an unbiased value for $S_{8}$ with
$S_{8}=0.760^{+0.013}_{-0.016}$. These results lead to the same conclusion as
Appendix C.4: the KiDS-like and DES-like analysis choices can lead to $\sim
2\sigma$ $S_{8}$ offsets using identical data vectors, this time when
considering the scenario of an $S_{8}=0.759$ EuclidEmulatorv2-based dark-
matter-only Universe.

###  E.2 A Hybrid pipeline analysis of EuclidEmulatorv2 mock data

EuclidEmulatorv2 | Maximum Marginal | Mean Marginal
---|---|---
Analysis | IA | $\Sigma m_{\nu}$ | +OWLS? | COSEBIs Cut? | $\log_{10}(T_{\rm AGN}/{\rm K})$ | $S_{8}$ | $\Delta S_{8}$ | $\sigma/\sigma_{\rm min}$ | $S_{8}$ | $\Delta S_{8}$ | $\sigma/\sigma_{\rm min}$
DES-like | TATT | free | $\times$ | ✓ | $\times$ | $0.738^{+0.017}_{-0.020}$ | $-1.15\sigma$ | $1.42$ | $0.734^{+0.020}_{-0.015}$ | $-1.44\sigma$ | $1.35$
KiDS-like | NLA | 0.06eV | $\times$ | $\times$ | $\left[7.6-8.0\right]$ | $0.771^{+0.013}_{-0.013}$ | $0.89\sigma$ | $1.03$ | $0.771^{+0.013}_{-0.013}$ | $0.90\sigma$ | $1.00$
A | NLA-z | 0.06eV | $\times$ | ✓ | $\times$ | $0.759^{+0.017}_{-0.018}$ | $0.01\sigma$ | $1.36$ | $0.758^{+0.016}_{-0.015}$ | $-0.09\sigma$ | $1.22$
B | " | " | ✓ | $\times$ | $\times$ | $0.745^{+0.017}_{-0.018}$ | $-0.83\sigma$ | $1.35$ | $0.742^{+0.018}_{-0.013}$ | $-1.10\sigma$ | $1.21$
C | " | " | ✓ | ✓ | $\times$ | $0.750^{+0.018}_{-0.018}$ | $-0.52\sigma$ | $1.42$ | $0.747^{+0.019}_{-0.013}$ | $-0.75\sigma$ | $1.27$
D | " | " | ✓ | $\times$ | $\left[7.6-8.0\right]$ | $0.761^{+0.017}_{-0.022}$ | $0.10\sigma$ | $1.53$ | $0.757^{+0.019}_{-0.015}$ | $-0.14\sigma$ | $1.33$
E | " | " | ✓ | ✓ | $\left[7.6-8.0\right]$ | $0.759^{+0.019}_{-0.020}$ | $-0.00\sigma$ | $1.51$ | $0.756^{+0.019}_{-0.015}$ | $-0.16\sigma$ | $1.35$
F | " | " | ✓ | ✓ | $\left[7.3-8.0\right]$ | $0.759^{+0.018}_{-0.020}$ | $-0.03\sigma$ | $1.48$ | $0.755^{+0.020}_{-0.015}$ | $-0.22\sigma$ | $1.39$
Hybrid | NLA-z | free | ✓ | ✓ | $\left[7.3-8.0\right]$ | $0.754^{+0.020}_{-0.021}$ | $-0.25\sigma$ | $1.62$ | $0.751^{+0.021}_{-0.015}$ | $-0.46\sigma$ | $1.43$
G | TATT | free | ✓ | ✓ | $\left[7.3-8.0\right]$ | $0.748^{+0.022}_{-0.023}$ | $-0.49\sigma$ | $1.78$ | $0.743^{+0.025}_{-0.015}$ | $-0.81\sigma$ | $1.58$
Table 12: Joint survey $S_{8}$ constraints from mock data created using the
EuclidEmulatorv2 including an NLA IA model. The fiducial KiDS-like and DES-
like analyses can be compared to our Hybrid pipeline. A series of tests
compare the baryon feedback mitigation strategies of adopting scale cuts
and/or marginalising over a free nuisance parameter $T_{\rm AGN}$ using KiDS-
like priors with the addition of the NLA-z $\eta$ parameter. Where relevant,
the baryon feedback is modelled using OWLS-AGN. We report the $S_{8}$
constraints and 68% credible interval using both the maximum-marginal and
mean-marginal approach, with $\Delta S_{8}$ quantifying the offset from the
true $S_{8}=0.759$, as a fraction of the $1\sigma$ error. Some of these data
are displayed in Figure 13.

In defining a unified Hybrid pipeline it was clear that we would adopt the
PolyChord sampler (Appendix D) and HMCode2020 for the non-linear power
spectrum model as this outperforms both Halofit and HMCode2016 for the DES
data vector in Figure 17. For the IA model choice we lack high signal-to-noise
IA observations with which to make an informed decision between TATT and NLA.
After discussion, the NLA-z model was selected as sufficiently complex for the
signal-to-noise of our current surveys (see for example Fortuna et al., 2021).
The KiDS cosmological parameter priors were then selected as these were found
to minimise projection effects with the NLA model in Appendix C.3, with the
addition of the DES-like prior on the NLA-z $\eta_{\rm IA}$ parameter. With
these choices finalised, Tests A-F in Table 12 were designed to compare the
baryon feedback mitigation strategies of adopting scale cuts and/or
marginalising over a free nuisance parameter $T_{\rm AGN}$.

Starting with a mock survey that includes no baryon feedback, in Test A we
recover the input $S_{8}$ within $0.1\sigma$ using only scale cuts. Test A
therefore confirms the analysis in Appendix E.1 that HMCode2020 provides a
sufficiently accurate description of the EuclidEmulatorv2 non-linear matter
power spectrum, and the analysis in Appendix C.3 that the prior choice has
minimised projection effects. In Test B and C, we quantify the impact of
including scale cuts for DES and KiDS by including baryon feedback modelled
through the OWLS-AGN hydrodynamical simulation (van Daalen et al., 2011) which
introduces a significant reduction of power on small physical scales (see
Figure 16). Adopting scale cuts for only the DES part of the joint-survey data
vector (Test B) we underestimate the value of $S_{8}$ by $\sim 1.1\sigma$.
Including the COSEBIs scale cuts for KiDS (Test C) reduces this bias, as
designed, but we are still left with an underestimate of the value of $S_{8}$
by $\sim 0.5\sigma$ with the maximum marginal, and $\sim 0.75\sigma$ with the
mean marginal. These tests may seem to contradict the analysis in Appendix B
where we reported an offset of $\sim 0.14\sigma_{\rm 2D}$ in the
$S_{8}-\Omega_{\rm m}$ plane for this OWLS AGN-contaminated data vector. These
results can however be understood by considering the weak constraints from
cosmic shear alone on $\Omega_{\rm m}$. We find that the relative error on
$\Omega_{\rm m}$ is an order of magnitude larger than the relative error on
$S_{8}$, such that a small bias within the $S_{8}-\Omega_{\rm m}$ plane can
translate into a large relative bias in $S_{8}$. Note that the Krause et al.
(2021) scale cut methodology, that we have adopted, was designed for the DES
$3\times 2$pt analysis, where $\Omega_{\rm m}$ is more tightly constrained.

Figure 18: Comparing the degeneracy in the $S_{8}-\log_{10}(T_{\rm AGN}/{\rm
K})$ plane for an analysis of noise-free EuclidEmulatorv2+OWLS-AGN mocks using
different scale cuts and baryon feedback priors (see Table 12). Test D
(orange, dot-dashed) applies no additional scale cuts to the KiDS mock data
and shows the strongest degeneracy. Test E (green, dotted) shows that
including scale cuts reduces the degeneracy. Test F (blue, solid) extends the
lower bound of the $\log_{10}(T_{\rm AGN}/{\rm K})$ prior to include dark-
matter only scenarios. The inner 68% and outer 95% credible intervals are
shown contoured, accurately recovering the input cosmology where
$S_{8}=0.759$.

In Tests D, E, and F we analyse an OWLS AGN-contaminated data vector,
marginalising over the $T_{\rm AGN}$ baryon feedback parameter in HMCode2020.
In all tests we recover the input $S_{8}$ within $0.2\sigma$ using the mean
marginal value and within $\sim 0.1\sigma$ using the maximum marginal value.
These results are expected from Figure 16 where the $T_{\rm AGN}$ prior range
encompasses the OWLS-AGN model. Comparing Test D and Test E we quantify the
impact of including the Appendix B-defined scale cuts to the KiDS data vector.
In common with the analysis in Table 8, we find that the addition of scale
cuts to KiDS does not significantly impact the $S_{8}$ constraining power,
suggesting that the $T_{\rm AGN}$ marginalisation to mitigate baryon feedback
is already removing the majority of information from these scales666666Longley
et al. (2023) find that the addition of scale cuts to a joint cosmic shear
analysis of DES, HSC and KiDS increases the error on $S_{8}$ by $15-30\%$,
relative to an all-scale analysis with baryon feedback marginalisation. We can
understand the difference with our findings by noting that in their analysis,
scale cuts are applied to DES, HSC and KiDS, in contrast to this analysis
where we only investigate changes to scale cuts in KiDS. In addition, the
Longley et al. (2023) $\Delta\chi^{2}$ scale cut methodology leads to more
stringent limits on the OWLS-AGN induced parameter bias with $\Delta
S_{8}<0.2\sigma$. This can be compared to the Krause et al. (2021)
methodology, implemented in Appendix B, with a bias threshold of
$<0.3\sigma_{\rm 2D}$.. In Figure 18 we show that the degeneracy between the
baryon feedback nuisance parameter $T_{\rm AGN}$ and $S_{8}$ is, however,
reduced when scale cuts are included. This is important as it implies that the
scale cut analysis is less sensitive to the $T_{\rm AGN}$ prior which in this
case is informative, based on our current best knowledge from hydrodynamical
simulations. When expanding the informative baryon feedback prior to
$\log_{10}(T_{\rm AGN}/{\rm K}):\left[7.0,10.0\right]$, Amon & Efstathiou
(2022) find a $0.55\sigma$ increase in the KiDS-1000 mean marginal
value676767The projection effects for the Amon & Efstathiou (2022) prior set
have not been quantified. for $S_{8}$ and a 40% increase in the uncertainty on
that value. This study analysed the two-point shear correlation function with
angular limits 0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5$<\theta<$
300$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 for $\xi_{+}(\theta)$ and
4$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0$<\theta<$
300$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 for $\xi_{-}(\theta)$.
Adopting an uninformative prior on $T_{\rm AGN}$ would further increase the
uncertainty on $S_{8}$, with a stronger impact expected for statistics where
the high-$k$-scale information has been removed686868Amon & Efstathiou (2022)
find the constraints from a COSEBIs $E_{n}(\theta_{\rm min})=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 analysis with wide, but still
informative, priors on a phenomenological power spectrum suppression parameter
almost doubles the uncertainty on $S_{8}$ compared to the $\xi_{\pm}(\theta)$
analysis adopting the same priors., reducing the ability of the data to self-
calibrate the baryon feedback model.

Amongst the authors there is a broad range of views on the preferred approach
to mitigate baryon feedback. Those with a high degree of confidence in the
scale and redshift dependence of the BAHAMAS model for baryon feedback and the
HMCode2020 model for the non-linear power spectrum recommend a self-
calibrating all-scale cosmic shear analysis with uninformative $T_{\rm AGN}$
priors. Those with a high degree of confidence in the BAHAMAS-defined maximum
range of baryon feedback suppression, but less confidence in the finer
high-$k$ details of the non-linear modelling recommend a limited scale cosmic
shear analysis with informative $T_{\rm AGN}$ priors. Those with a low degree
of confidence in the results from hydrodynamical simulations recommend the use
of more conservative scale cuts, similar to the approach taken in Appendix B
but adopting a more extreme worst case scenario for baryon feedback such as
Cosmo-OWLS-AGN:8.7. Other opinions include the use of highly flexible baryon
feedback models, with or without the combination of scale cuts. As we have
seen in Appendix C.3, any change in prior and/or astrophysical model leads to
different projection effects, some of which may be unexpected (see for example
the strong degeneracy between the HMCode2016 baryon feedback parameter $A_{\rm
bary}$ and $n_{\rm s}$ when using uninformative priors in Yoon & Jee, 2021). A
comparison of the results from the authorship’s range of preferred baryon
feedback mitigation strategies would therefore require both the analysis of
mocks and data which we defer to a future study. We also note that data
comparisons using different angular ranges will also be subject to random
noise fluctuations (see Appendix F).

We conclude our baryon feedback study with Test F, where we expand the lower
limit of the $T_{\rm AGN}$ prior to include low feedback models that are
equivalent to a dark matter only model (see Figure 16). The decision to allow
the non-linear model of the matter power spectrum to include no baryon
feedback followed the preferred approach of Amon et al. (2022); Secco,
Samuroff et al. (2022) and also Asgari et al. (2021) who allow for a dark
matter only model within their baryon feedback prior. We find that extending
the informative $T_{\rm AGN}$ prior to include no feedback, in combination
with scale cuts, does not change the recovered $S_{8}$ value or constraining
power. This can be understood from Figure 18 where we see little degeneracy
between $T_{\rm AGN}$ and $S_{8}$ for $\log_{10}(T_{\rm AGN}/{\rm K})\lesssim
7.5$. For our Hybrid analysis we therefore choose to combine the mitigation
strategies from both surveys using scale cuts and a $T_{\rm AGN}$ prior that
spans a dark matter-only Universe through to the strongest BAHAMAS AGN
feedback model.

The ‘Hybrid’ row in Table 12 repeats Test F with the Hybrid prior set,
extending the KiDS prior set with the DES prior on neutrino mass. This
introduces an underestimate of $S_{8}$ by $0.46\sigma$ in our recovered mean
marginal $S_{8}$ value, arising from projection effects. Here our mock input
neutrino mass, $\Sigma m_{\nu}=0.06$ eV, is at the lower edge of the allowed
tophat prior range which extends to $\Sigma m_{\nu}<0.6$ eV. The weak
correlation between $\Sigma m_{\nu}$ and $S_{8}$ then leads to a bias in the
mean or maximum of the projected marginal $S_{8}$ distribution. This issue
should be largely circumvented, however, by reporting the MAP in contrast to
the marginal values.

In one final Test G, we replace the NLA-z model with a TATT model in our
analysis to quantify the $\sim 0.8\sigma$ level of bias from the truth when
adopting this model to analyse an NLA-z mock Universe. As shown in Appendix
C.3, however, this bias decreases to zero when adopting the DES priors and
analysing a mock Universe that includes a TATT IA signal.

We review the different constraining power of each setup in Table 12, focusing
on the mean-marginal constraints. Comparing the KiDS-like analysis with Test
D, we find a $\sim 33\%$ increase in the error on $S_{8}$ resulting from the
change in both IA model and sampler, which is consistent with expectations
(see Appendix D; Secco, Samuroff et al. 2022; Asgari et al., 2021). We find a
modest improvement of $\sim 10\%$ in constraints between the DES-like
analysis, and Tests A, B and C, demonstrating that the combination of changing
from TATT to NLA-z and from DES to KiDS+ priors does not make a significant
difference when the input IA model is NLA, consistent with the results in
Appendix C.4. Comparing Test C and E, we find a $\sim 6\%$ increase in the
error on $S_{8}$ when including baryon feedback marginalisation. The inclusion
of a free neutrino mass parameter raises the error on $S_{8}$ by a further
$\sim 6\%$. For this EuclidEmulatorv2 mock we find that our Hybrid analysis
choices lead to an increase in the error on $S_{8}$ by $\sim 43\%$ relative to
the KiDS-like fiducial analysis, with similar constraining power relative to
the DES-like fiducial analysis. The difference we see between the constraints
from this suite of mock analyses is largely replicated in our data analysis in
Section 3.

##  Appendix F Comparing KiDS-1000 COSEBIs with and without scale cuts

Summary: In this appendix we measure a $0.7-0.8\sigma$ increase in the
KiDS-1000 $S_{8}$ constraint when including scale cuts in a KiDS-like COSEBIs
analysis. We find this result to be consistent with random noise fluctuations,
where a $\geq 0.7\sigma(\geq 0.8\sigma)$ offset between a $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 and $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 COSEBIs analysis occurs
$23\%(14$%) of the time in noisy mock KiDS-1000 simulations. To address
whether this offset could alternatively/additionally indicate an issue with
unmodeled baryon feedback, we test a mock using the most extreme hydrodynamic
simulation Cosmo-OWLS:8.7. In this case, we estimate a $0.4\sigma$ offset in
$S_{8}$ between a $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 and a $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 noise-free COSEBIs analysis that
adopts the HMCode2020 model, fixing $\log_{10}(T_{\rm AGN}/{\rm K})=7.8$.

2pt | $\theta_{\rm min}$ | HM | $S_{8}$ | $\Delta S_{8}$ | $\sigma/\sigma_{\rm fid}$ | $A_{\rm IA}$ | $\Delta A_{\rm IA}$
---|---|---|---|---|---|---|---
$E_{n}$ | 0.5´ | 2016 | $0.758^{+0.017}_{-0.026}$ | $0.00\sigma$ | 1.0 | $0.39^{+0.35}_{-0.41}$ | $0.00\sigma$
$\xi_{\pm}$ | 0.5´ | 2016 | $0.765^{+0.019}_{-0.017}$ | $0.33\sigma$ | 0.83 | $0.37^{+0.36}_{-0.34}$ | $0.12\sigma$
$E_{n}$ | 2.0´ | 2016 | $0.774^{+0.020}_{-0.024}$ | $0.73\sigma$ | $1.01$ | $0.62^{+0.35}_{-0.43}$ | $0.61\sigma$
$\xi_{\rm\pm}$ | 2.0´ | 2016 | $0.769^{+0.018}_{-0.020}$ | $0.52\sigma$ | $0.87$ | $0.56^{+0.34}_{-0.35}$ | $0.45\sigma$
$E_{n}$ | 0.5´ | 2020 | $0.762^{+0.023}_{-0.028}$ | $0.19\sigma$ | $1.19$ | $0.41^{+0.34}_{-0.46}$ | $0.04\sigma$
$E_{n}$ | 2.0´ | 2020 | $0.780^{+0.020}_{-0.024}$ | $1.01\sigma$ | $1.03$ | $0.60^{+0.34}_{-0.44}$ | $0.56\sigma$
Table 13: KiDS-1000 maximum marginal constraints on $S_{8}$, and the intrinsic
alignment parameter, $A_{\rm IA}$, using a KiDS-like pipeline for a range of
different two-point statistics (2pt). We compare COSEBIs, $E_{n}$, with the
two-point shear correlation function, $\xi_{\pm}$, for two different scale
cuts $\theta_{\rm min}=$ 0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 (the
fiducial KiDS setting) and $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 (the Appendix B DES-like scale
cut setting). We also compare the COSEBIs results for two versions of HMCode
(HM). With HMCode2016 we use KiDS-like priors on the baryon feedback parameter
$A_{\rm bary}=\left[2,3.13\right]$. With HMCode2020 we use the BAHAMAS-defined
baryon feedback range $\log_{10}(T_{\rm AGN}/{\rm K}):\left[7.6,8.0\right]$.
$\Delta S_{8}$ and $\Delta A_{\rm IA}$ compares the fiducial Asgari et al.
(2021) $E_{n}$ result with the different variations, in units of $\sigma$, the
68% credible interval for each parameter. We also compare the $S_{8}$
constraining power for each variation through the ratio $\sigma/\sigma_{\rm
fid}$.

In Table 13 we report the maximum marginal constraints on $S_{8}$ and $A_{\rm
IA}$ for a series of KiDS-1000 cosmic shear analyses using: COSEBIs, $E_{n}$,
and the two-point correlation function696969We refer the reader to section 2.4
and figure 1 of Asgari et al. (2021) for a discussion on the advantages and
disadvantages for each cosmic shear statistic. Of note is the finding that the
$\xi_{\pm}(\theta)$-optimised definition $S_{8}=\sigma_{8}(\Omega_{\rm
m}/0.3)^{\alpha}$, where $\alpha=0.5$, does not provide an optimal description
of the COSEBIs $\sigma_{8}-\Omega_{\rm m}$ degeneracy where the best fit
$\alpha=0.54$., $\xi_{\pm}(\theta)$; two different minimum angular scales,
$\theta_{\rm min}=$ 0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 and
$\theta_{\rm min}=$ 2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0; and two
versions of HMCode. The COSEBIs analysis with $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 and HMCode2016 is the fiducial
result from Asgari et al. (2021). The COSEBIs analysis with $\theta_{\rm
min}=$ 2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 and HMCode2020 adopts the
scale cuts and updated non-linear power spectrum model used in the Hybrid
analysis presented in Section 3.1. We find a $1.01\sigma$ increase in $S_{8}$
between these two analyses. An increase of $\sim 0.2-0.3\sigma$ is found when
changing from HMCode2016 to HMCode2020, which can be understood from the
percent-level change in the amplitude of the cosmic shear predictions from
these two models (see Figure 17), and the use of different baryon feedback
priors707070With HMCode2016 the most extreme baryon feedback simulation
considered at the time was OWLS-AGN which corresponds to $\log_{10}(T_{\rm
AGN}/{\rm K})\sim 7.8$. With HMCode2020, more extreme models were permitted by
the BAHAMAS simulation, raising the prior limit to $\log_{10}(T_{\rm AGN}/{\rm
K})=8.0$. Owing to the degeneracy between $S_{8}$ and $T_{\rm AGN}$, stronger
feedback allows for higher $S_{8}$ values.. An increase of $\sim
0.7-0.8\sigma$ is found when changing from $\theta_{\rm min}=$
0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5 to $\theta_{\rm min}=$
2$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$0 for the COSEBIs $E_{n}$
statistic717171Defining $\sigma=(\sigma_{68}^{\rm upper}+\sigma_{68}^{\rm
lower})/2$, using constraints from the
$E_{n}(\theta=$0$\stackrel{{\>{}^{\prime}}}{{\textstyle.}}$5$)$ analysis, we