DES Collaboration

# Dark Energy Survey Year 3 Results:
Cosmology from Cosmic Shear and Robustness to Data Calibration

A. Amon<EMAIL_ADDRESS>Kavli Institute for Particle Astrophysics &
Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA  D.
Gruen  Department of Physics, Stanford University, 382 Via Pueblo Mall,
Stanford, CA 94305, USA  Kavli Institute for Particle Astrophysics &
Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA  SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  M. A. Troxel
Department of Physics, Duke University Durham, NC 27708, USA  N. MacCrann
Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, UK  S. Dodelson  Department of Physics, Carnegie
Mellon University, Pittsburgh, Pennsylvania 15312, USA  A. Choi  Center for
Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH
43210, USA  C. Doux  Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104, USA  L. F. Secco  Department of Physics
and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA  Kavli
Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637,
USA  S. Samuroff  Department of Physics, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15312, USA  E. Krause  Department of
Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue,
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Ohio State University, Columbus, OH 43210, USA  Institute of Physics,
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Lima  Departamento de Física Matemática, Instituto de Física, Universidade de
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and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA
Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA
Radcliffe Institute for Advanced Study, Harvard University, Cambridge, MA
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Miquel  Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona,
Spain  Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
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de Janeiro, RJ - 20921-400, Brazil  A. K. Romer  Department of Physics and
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Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
08193 Barcelona, Spain  M. Smith  School of Physics and Astronomy, University
of Southampton, Southampton, SO17 1BJ, UK  M. Soares-Santos  Department of
Physics, University of Michigan, Ann Arbor, MI 48109, USA  G. Tarle
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA  D.
Thomas  Institute of Cosmology and Gravitation, University of Portsmouth,
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Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA
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J. Weller  Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse, 85748 Garching, Germany  Universitäts-Sternwarte, Fakultät
für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679
München, Germany

###### Abstract

This work, together with its companion paper, Secco, Samuroff _et al._ (2021),
presents the Dark Energy Survey Year 3 cosmic shear measurements and
cosmological constraints based on an analysis of over 100 million source
galaxies. With the data spanning 4143 deg2 on the sky, divided into four
redshift bins, we produce the highest significance measurement of cosmic shear
to date, with a signal-to-noise of 40. We conduct a blind analysis in the
context of the $\Lambda$CDM model and find a 3% constraint of the clustering
amplitude, $S_{8}\equiv\sigma_{8}(\Omega_{\rm
m}/0.3)^{0.5}=0.759^{+0.025}_{-0.023}$. A $\rm\Lambda CDM$-Optimized analysis,
which safely includes smaller scale information, yields a 2% precision
measurement of $S_{8}=0.772^{+0.018}_{-0.017}$ that is consistent with the
fiducial case. The two low-redshift measurements are statistically consistent
with the Planck Cosmic Microwave Background result, however, both recovered
$S_{8}$ values are lower than the high-redshift prediction by $2.3\sigma$ and
$2.1\sigma$ ($p$-values of 0.02 and 0.05), respectively. The measurements are
shown to be internally consistent across redshift bins, angular scales and
correlation functions. The analysis is demonstrated to be robust to
calibration systematics, with the $S_{8}$ posterior consistent when varying
the choice of redshift calibration sample, the modeling of redshift
uncertainty and methodology. Similarly, we find that the corrections included
to account for the blending of galaxies shifts our best-fit $S_{8}$ by
$0.5\sigma$ without incurring a substantial increase in uncertainty. We
examine the limiting factors for the precision of the cosmological constraints
and find observational systematics to be subdominant to the modeling of
astrophysics. Specifically, we identify the uncertainties in modeling baryonic
effects and intrinsic alignments as the limiting systematics.

gravitational lensing: weak; dark matter; dark energy; methods: data analysis;
cosmology: observations; cosmological parameters

††preprint: DES-2019-0479††preprint: FERMILAB-PUB-21-250-AE

##  I Introduction

The current era of precision cosmology has delivered measurements of
cosmological parameters at percent-level accuracy, and a standard cosmological
model that is an excellent fit to most data. This era was enabled by decades
of progress at the nexus of instrumentation, observations, cosmology theory
and analysis (Weinberg, 1972; Peebles, 1980). On the one hand, increasingly
strong evidence for dark matter, made over many decades (Bertone and Hooper,
2018; Trimble, 1987), along with the discovery of dark energy and the
accelerating universe in the late 1990s (Perlmutter  _et al._ , 1999; Riess
_et al._ , 1998), paved the way for the current standard model. On the other
hand, advances in our ability to collect, process, and analyze data from such
diverse observations such as type Ia supernovae (SNIa), the cosmic microwave
background (CMB) anisotropies, and the distribution of galaxies and other
tracers of large-scale structure (LSS), improve our ability to test theories
for the accelerating universe (Frieman  _et al._ , 2008; Weinberg  _et al._ ,
2013), as well as physics at moments after the Big Bang (Abazajian  _et al._ ,
2016).

While the model’s success has been reinforced by agreement across a broad
range of observations, the advancing precision has revealed some tensions
among cosmological parameters measured by different observational probes. Most
notably, there is a 3-5$\sigma$ tension in the Hubble constant, $H_{0}$,
between low-redshift measurements made by the distance-ladder technique and
those from the CMB at $z\approx 1100$ (see Refs. Verde  _et al._ (2019); Di
Valentino  _et al._ (2021) for a summary). Another widely discussed
discrepancy, though less statistically significant, is that between
constraints on the parameter $S_{8}\equiv\sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$
— the amplitude of matter density fluctuations, $\sigma_{8}$, scaled by the
square root of the matter density, $\Omega_{\rm m}$. This quantity is
consistently found to be 2-3$\sigma$ lower when measured in LSS data,
including from the Dark Energy Survey111 https://www.darkenergysurvey.org/
(DES) Heymans and Grocutt (2013); Troxel  _et al._ (2018); Dark Energy Survey
Collaboration (2018); Hikage  _et al._ (2019); Asgari  _et al._ (2021);
Heymans  _et al._ (2021); Krolewski  _et al._ (2021), than the constraint by
the CMB Planck Collaboration (2018). New and improved data and analysis
methods are key to bring these tensions into sharp focus, in order to test
whether they are attributed to new physics, or are caused by unforeseen
systematic errors.

Weak gravitational lensing of large-scale structure, cosmic shear, is a
powerful method that is sensitive to both the geometry and the growth of
cosmic structure in the Universe. This is the measurement of small but
coherent distortions of the observed shapes of galaxies as their light passes
through the intervening structure on its way to Earth. Measurements of these
distortions carry information about the projected mass density to the source
galaxy, and hence the amplitude, shape, and time evolution of the matter power
spectrum. They are also sensitive to geometrical factors such as distances to
both the lens structures and the source galaxy (for reviews, see (Bartelmann
and Schneider, 2001; Hoekstra and Jain, 2008; Kilbinger, 2015)).

Although proposed half a century ago (Kristian, 1967), weak lensing by large-
scale structure was not detected until 2000 (Bacon  _et al._ , 2000; Kaiser
_et al._ , 2000; Van Waerbeke  _et al._ , 2000; Wittman  _et al._ , 2000).
Cosmic shear has made strides since its first detection two decades ago with
multiple surveys from the ground and space, owing to rapid advancements in
technology, galaxy surveys and methodology Hoekstra  _et al._ (2002);
Refregier  _et al._ (2002); Jarvis  _et al._ (2003); Hamana  _et al._ (2003);
Brown  _et al._ (2003); Rhodes  _et al._ (2004); Heymans  _et al._ (2005);
Jarvis  _et al._ (2006); Hetterscheidt  _et al._ (2007); Massey  _et al._
(2007); Leauthaud  _et al._ (2007); Benjamin  _et al._ (2007); Fu  _et al._
(2008); Schrabback  _et al._ (2010); Huff  _et al._ (2014a, b); Lin  _et al._
(2012); Jee  _et al._ (2013a, 2016). A significant step forward, both in
quantity and quality of the data, was made with the Canada-France-Hawaii
Lensing Survey (CFHTLens) (Erben  _et al._ , 2013; Miller  _et al._ , 2013;
Kilbinger  _et al._ , 2013; Kitching  _et al._ , 2014; Heymans and Van
Waerbeke, 2006) while the current state of the art is being pursued by DES,
the Hyper Suprime-Cam Subaru Strategic Program222
https://hsc.mtk.nao.ac.jp/ssp/ (HSC; Aihara and HSC Collaboration (2018)) and
the ESO Kilo-Degree Survey333http://kids.strw.leidenuniv.nl/ (KiDS; Kuijken
_et al._ (2015)). These ‘Stage III’ surveys use an increasingly sophisticated
set of tools to make cosmic shear measurements over large areas on the sky and
provide competitive constraints on cosmological parameters (Jee  _et al._ ,
2013b; Dark Energy Survey Collaboration, 2016a; Hildebrandt  _et al._ , 2017;
Hikage  _et al._ , 2019; Troxel  _et al._ , 2018, 2018a; Hildebrandt  _et al._
, 2020; Hamana  _et al._ , 2020; Asgari  _et al._ , 2021). Cosmic shear is
highly complementary to probes utilizing the positions of galaxies. Of
particular note is the joint analysis of cosmic shear with galaxy-galaxy
lensing and galaxy clustering, commonly referred to as 3$\times$2pt, performed
either within a single survey (Dark Energy Survey Collaboration, 2018), or in
tandem with spectroscopic surveys (Heymans  _et al._ , 2021), and varying
statistics (Joudaki  _et al._ , 2018; van Uitert  _et al._ , 2018). Such an
analysis utilizes multiple sources of information to break degeneracies in
cosmological parameters, and as importantly, nuisance parameters that describe
observational and astrophysical systematic effects. The field is at the brink
of a new epoch, both in terms of the advances in analysis and with upcoming
surveys spanning ‘full-sky’ to unprecedented depth. These ‘Stage IV’ surveys
will revolutionize the field in the decades to come. They include the ground-
based Vera C. Rubin Observatory’s Legacy Survey of Space and Time
(LSST444http://www.lsst.org/lsst; (LSST Dark Energy Science Collaboration,
2012)), and the space missions Euclid555sci.esa.int/euclid/ (Laureijs  _et
al._ , 2011) and the Nancy Grace Roman Space
Telescope666https://roman.gsfc.nasa.gov/ (Spergel, 2015).

While cosmic shear is a cornerstone to the future precise measurements of dark
matter, dark energy, neutrino mass, and other fundamental quantities, its
accurate measurement and modeling are challenging. Firstly, cosmic shear is a
percent-level effect that is statistically extracted from millions of
galaxies. Second, the signal encoded in the shapes can be contaminated with a
number of systematic effects (for a review, see (Mandelbaum, 2018)), such as
leakage of the Point-Spread Function (PSF) that must be modeled and accounted
for through robust shape measurement techniques (Miller  _et al._ , 2013; Huff
and Mandelbaum, 2017; Sheldon and Huff, 2017; Jarvis  _et al._ , 2021; Sheldon
_et al._ , 2020; Bernstein and Armstrong, 2014) and validated through
rigourous testing (Gatti  _et al._ , 2021; Giblin  _et al._ , 2021; Mandelbaum
_et al._ , 2018).

Furthermore, interpreting the cosmic shear signal demands accurate estimation
of the distribution of the galaxies’ redshifts, in order to not incur a bias
in cosmological constraints (see, for example Bernstein and Huterer, 2010).
Approaches to this calibration challenge may employ template-fitting or
machine-learning to empirically learn the relationship between photometry and
redshift based on a training sample. Neither method is a priori designed to
meet the needs of, and quantify the resulting uncertainty for, weak lensing
analyses. Methods are limited by incomplete information: in the case of
template fitting, a complete description of the distribution of galaxy SEDs
across luminosity and redshift; in the case of machine learning, a very large,
perfectly representative spectroscopic training sample (Newman, 2015). The
resulting selection biases have spurred debate on the use of complete
photometric redshift or incomplete spectroscopic redshift training samples
(Newman, 2015; Gruen and Brimioulle, 2017; Hoyle  _et al._ , 2018; Wright  _et
al._ , 2020; Hildebrandt  _et al._ , 2020; Joudaki and Hildebrandt, 2020;
Hartley  _et al._ , 2020). Regardless of the choice taken, consistency of
different techniques, and the use of independent information increase
confidence in the calibration (see Myles  _et al._ , 2020; Gatti  _et al._ ,
2020; Sánchez  _et al._ , 2021, for DES Y3).

Notably, the robust calibration of both the shear signals and redshift
distributions necessitates understanding and mitigating the impact of
blending, or crowded images, through realistic image simulations (Dawson  _et
al._ , 2016; Mandelbaum and Lanusse, 2018). The rejection of galaxies with
nearby neighbors (Miller  _et al._ , 2013; Jarvis  _et al._ , 2016) is limited
to handling only recognized blends, does not account for the increased
occurrence of blending in high-density regions, and is unfeasible for deeper
surveys (Bosch, 2018). Detailed image simulation analyses are required to shed
light on the resulting systematic errors (Mandelbaum  _et al._ , 2018;
Samuroff  _et al._ , 2017; Kannawadi  _et al._ , 2019). The corresponding DES
Y3 analysis has shown that the selection of galaxies based on their multi-band
photometry has to be performed consistently between the data and simulations
in order to understand the impact of blended sources at different redshift on
both the shape measurement and on the effective redshift distribution for
lensing (MacCrann  _et al._ , 2020).

Finally astrophysical effects, such as intrinsic alignments (e.g. Croft and
Metzler, 2000; Heavens  _et al._ , 2000; Troxel and Ishak, 2015) and the
impact of the physics of galaxy formation on weak lensing observables (e.g.
Chisari  _et al._ , 2018; Wechsler and Tinker, 2018), must be sufficiently
modeled, or nulled, such that no residual impact is detected (Krause and
Eifler, 2017). This is particularly challenging as the nuisance parameters
incurred in intrinsic alignment modeling can absorb systematic errors in the
calibrations of photometric redshift distributions (e.g. Wright  _et al._ ,
2020). Tests of internal consistency across redshift, angular scales and
statistics can provide essential checks of sufficient mitigation against both
theoretical and observational systematics (Efstathiou and Lemos, 2018).

This work presents the cosmological constraints from cosmic shear measurements
with the DES wide-field survey, using data taken during its first three years
of observations and presents its robustness to data calibration. A companion
paper to this work, Secco, Samuroff _et al._ (2021), demonstrates the
robustness of these cosmic shear cosmological constraints to modeling choices,
in particular, to intrinsic alignments, baryonic effects, higher-order lensing
effects and neutrinos. Cosmic shear is analysed using a common framework with
those from galaxy-galaxy lensing (Prat  _et al._ (2021)) and galaxy clustering
(Rodríguez-Monroy  _et al._ (2021)), combined as 2$\times$2pt in Porredon _et
al._ (2021); Pandey _et al._ (2021); Elvin-Poole, MacCrann _et al._ (2021) in
a joint DES Y3 3$\times$2pt analysis presented in DES Collaboration (2021).
The measurements presented in this work are supported by a number of
accompanying infrastructure papers:

  * •

The construction and validation of the photometric ‘Gold’ catalog of high-
quality objects in DES Y3 is described in Sevilla-Noarbe  _et al._ (2020). New
PSF modeling (Jarvis  _et al._ (2021)) combined with weak lensing shape
measurement based upon Ref. (Sheldon and Huff, 2017) gives a catalog of 100
million selected galaxies that are validated in Gatti, Sheldon _et al._
(2021).

  * •

Redshift calibration methodology is summarised in Myles, Alarcon _et al._
(2020). This framework utilises external training data from narrow-band
photometric and spectroscopic sources and DES deep observations with
overlapping near-infrared data, presented in Hartley, Choi _et al._ (2020),
mapped to the wide-field with an improved image injection measurements
(Everett  _et al._ (2020)). The full scheme incorporates new, independent
methods, a two-step reweighting with self-organising maps (Buchs  _et al._
(2019)), clustering redshifts (Gatti, Giannini _et al._ (2020)) derived from
correlations with the DES foreground redMaGiC lens samples and small-scale
shear ratios (Sánchez, Prat _et al._ (2021)), to accurately constrain the
redshift distributions. The latter two methods use the two DES foreground lens
samples, redMaGiC and Maglim (Porredon  _et al._ (2021)), in their
measurements and are included at different stages in the pipeline, such that
the ratios are able to constrain other nuisance parameter in the analysis. We
test alternative techniques for modeling and marginalising over the
uncertainty on the tomographic distributions, implementing a new tool,
Hyperrank, (Cordero  _et al._ (2021)).

  * •

State-of-the-art shear calibration with realistic image simulations and new
methodology to account for the impact of blending on the effective redshift
distribution for lensing measurements in MacCrann  _et al._ (2020).

  * •

The general methodology, likelihood analysis and covariance used in the
cosmological analyses shown in this work and DES Collaboration (2021) is
presented in Krause  _et al._ (2021) and Friedrich  _et al._ (2020) and this
methodology is independently validated using realistic simulations in DeRose
_et al._ (2021a).

  * •

The statistical framework to assess the internal consistency of the DES data
and measurements is presented in Doux  _et al._ (2020) and the consistency
with independent, external data in Lemos  _et al._ (2020).

This paper is structured as follows: in Sections II, III, and IV the DES Y3
data and the calibration of the shear and redshift distributions are
described. Section V provides an overview of the cosmological simulations used
in validating the model and methods. Section VI presents the cosmic shear
measurements, covariance matrix validation and the blinding methodology and
Section VII, the model. We detail the cosmological constraints in Section
VIII, and their internal consistency in Section IX. Finally, in Sections X and
Section XI, the robustness of the cosmological constraints to systematics in
the data calibration is assessed through a range of validation tests. More
technical details of the analysis are provided in the appendices.

##  II Dark Energy Survey Year 3 Data

Figure 1: DES Y3 footprint showing the variation in the number density across
the sky, as determined with the Metacalibration catalog (left) and the
variation in mean redshift of that catalog (right). Overlaid on the left is
the red outline of the Y1 footprint and on the right, the locations of the
four DES Deep Fields (Hartley  _et al._ , 2020) (the fourth field, COSMOS, is
positioned at $\sim 150$deg, outside of the DES footprint, but has been
rotated here to be shown on the map). The catalog spans a final effective area
of 4143 deg2 with an average number density of 5.59 arcmin-2 and a mean
redshift of 0.63.

The Dark Energy Survey has completed a six year observing program at the Cerro
Tololo Inter-American Observatory (CTIO), Chile, using the Blanco telescope
and the 570-megapixel Dark Energy Camera (DECam) (Flaugher  _et al._ , 2015).
Ultimately, the complete survey spans 5000 square degrees of the Southern
hemisphere in the $grizY$ bands with exposure times of 90 seconds in $griz$
and 45/90 seconds in $Y$ (Dark Energy Survey Collaboration, 2016b). With 10
overlapping exposures in each of the $griz$ bands over the full wide-field
area, the survey reaches a limiting magnitude of $i\sim 24$.

This DES Year 3 (Y3) analysis exploits the data acquired over the first three
years of observations, from August 15, 2013 to February 12, 2016,
approximately 345 nights. The survey has a mean depth of four exposures that
reaches an $i$-band signal-to-noise of $\sim 10$ for extended objects up to
$i_{AB}\sim 23.0$ (Sevilla-Noarbe  _et al._ , 2020). The median recorded
seeing (FWHM) in the $riz$ bands is 0.98, 0.89, 0.85 arcsec, respectively.

The DES Y3 dataset exhibits a number of improvements compared to DES Y1,
including updates in brighter-fatter correction, sky-background modeling,
better morphological star-galaxy classification and astrometric solutions,
reduced photometric residuals compared to Gaia with photometric calibration
uniformity $<3$ mmag, the introduction of per-object chromatic corrections,
improvements to PSF modeling (Jarvis  _et al._ , 2021) and better flagging of
imaging artifacts. These improvements, as well as the production and
validation of a ‘Gold’ catalog of 390 million objects are described in detail
in Ref. Sevilla-Noarbe  _et al._ (2020).

###  II.1 Shape measurement

Table 1: DES Y3 data properties per redshift bin: the unweighted number of objects passing the weak-lensing selection, the effective number density $n_{\rm eff}$ (gal/arcmin2), calculated with $A_{\rm eff}=4143$ deg2, the per-component shape noise, $\sigma_{e}$, the mean redshift, $z_{\rm mean}$, the spread in the mean redshift of the redshift distribution realisations, $\Delta z$, the mean shear per component, $c_{1,2}=\langle e_{1,2}\rangle$, the mean shear response, $\left<\bm{R}_{\gamma}\right>$, selection response, $\left<\bm{R}_{\rm s}\right>$, and the shear calibration parameter, $m$. Also shown are the variations of observational systematic uncertainties, $\Delta z_{\rm noz-blend}$ and $m_{\rm noz-blend}$, which do not account for the redshift-mixing effect due to blending, discussed in Section IV.  Bin | no. objects | $n_{\rm eff}$ | $\sigma_{e}$ | $z_{\rm mean}$ | $\langle e_{1}\rangle\times 10^{4}$ | $\langle e_{2}\rangle\times 10^{4}$ | $\Delta z$ | $m$ | $\Delta z_{\rm noz-blend}$ | $m_{\rm noz-blend}$ | $\left<\bm{R}_{\gamma}\right>$ | $\left<\bm{R}_{\rm s}\right>$
---|---|---|---|---|---|---|---|---|---|---|---|---
Full | 100 204 026 | 5.590 | 0.268 | 0.633 | - | - | - | - | - | - |  |
0 | 24 940 465 | 1.476 | 0.243 | 0.336 | 3.22 | 1.60 | 0.018 | -0.006 $\pm$ 0.009 | 0.016 | -0.013 $\pm$ 0.003 | 0.7636 | 0.0046
1 | 25 280 405 | 1.479 | 0.262 | 0.521 | 3.36 | 0.38 | 0.015 | -0.020 $\pm$ 0.008 | 0.013 | -0.018 $\pm$ 0.004 | 0.7182 | 0.0083
2 | 24 891 859 | 1.484 | 0.259 | 0.742 | 3.77 | 0.07 | 0.011 | -0.024 $\pm$ 0.008 | 0.006 | -0.023 $\pm$ 0.004 | 0.6887 | 0.0126
3 | 25 091 297 | 1.461 | 0.301 | 0.964 | 4.06 | -0.27 | 0.017 | -0.037 $\pm$ 0.008 | 0.015 | -0.036 $\pm$ 0.006 | 0.6154 | 0.0145

The relation of measured galaxy shape to gravitational shear can expressed for
each of their two components as

$\epsilon^{\rm obs}=(1+m)(\epsilon^{\rm int}+\gamma)+c\,,$ (1)

i.e., a combination of the intrinsic ellipticity of the galaxy, $\epsilon^{\rm
int}$, and an additional lensing-induced shear, $\gamma$, which is the
cosmologically interesting weak lensing signal. Contamination of the
measurement, or systematics, come in two variants. Additive bias $c$ can
result, for example, from insufficient modeling of the PSF that causes a
leakage into the measured galaxy ellipticity (see Section A). Multiplicative
bias, $m\neq 0$, can arise, for example, from crowded images with blended
detections (see Section IV), or, for methods unlike Metacalibration, from
noise bias or model bias.

For the DES Y3 analysis, Metacalibration is used to produce the shear catalog
(Sheldon and Huff, 2017; Huff and Mandelbaum, 2017). This method calibrates
shear statistics from the imaging data itself, without reliance on prior
information about galaxy properties. In its Y3 implementation, Metacalibration
measures the properties of a galaxy, including its ellipticity, using a single
Gaussian as a pre-seeing model that is fit to each detected object, for all
the available epochs, using the $riz$-band images. The $g$-band data are
excluded in the fit due to insufficient PSF modeling, as detailed in (Jarvis
_et al._ , 2021). During the fit, light from neighbouring objects is masked in
order to reduce blending effects. The galaxy image is then deconvolved with
the PSF, artificially sheared, reconvolved by a symmetrized version of the PSF
and the ellipticity remeasured. Done repeatedly, this results in one unsheared
and four artificially-sheared versions of the shape catalog (each component of
ellipticity is sheared both positively and negatively). These are used to
construct the shear response matrix, $R_{\gamma}$, via numerical derivatives
of the ellipticity. The response matrix is a noisy quantity on a single galaxy
basis, but ensemble averages, $\left<\bm{R}_{\gamma}\right>$, are precisely
measured for large samples, and known to a sufficient degree of accuracy given
the size of the DES Y3 sample. The pipeline is largely based upon that
employed in the DES Y1 analysis, now with the inclusion of an inverse variance
weighting for the galaxies, employed to boost the signal-to-noise of the data.
The details of this implementation can be found in (Gatti  _et al._ , 2021).

The Metacalibration pipeline is designed to self-calibrate biases in the shear
estimation by correcting for not only the response of the shear estimator, but
also the selection biases (Huff and Mandelbaum, 2017). To this end,
measurements of flux, size, signal-to-noise ratio and other selection-relevant
properties are made on the unsheared and sheared images. Selection response
can be estimated by selecting objects based on sheared measurements and
computing $R_{\rm s}$ as

$\displaystyle\left<\mbox{\boldmath$R_{\rm
s}$}\right>_{i,j}=\frac{\left<e_{i}\right>^{\rm{s}_{+}}-\left<e_{i}\right>^{\rm{s}_{-}}}{\Delta\gamma_{j}}\,.$
(2)

Here $\left<e_{i}\right>^{\rm{S}_{+}}$ represents the mean of the $i$th
ellipticity component measured on images without applied shear,
$\Delta\gamma_{j}$, in component $j$, over the group of galaxies selected
using the parameters extracted from positively sheared images.
$\left<e_{i}\right>^{\rm{S}_{-}}$ is the analogous quantity for negatively
sheared images. To calibrate the mean shear of the catalog, it is sufficient
to consider the total response matrix per redshift bin as the sum of shear and
selection response:

$\displaystyle\left<\bm{R}\right>=\left<\bm{R}_{\gamma}\right>+\left<\bm{R_{\rm
s}}\right>\,,$ (3)

which are quoted in Table 1. The response matrix is diagonal within a very
good approximation, and as a consequence, the mean shear response correction
reduces to element-wise division. The Metacalibration response accounts for
shear biases at a level of a few parts in a thousand in the absence of blends
and detection biases (Sheldon and Huff, 2017). The current Metacalibration
implementation, however, does not correct for a shear-dependent detection bias
(Sheldon  _et al._ , 2020) and the redshift-mixing effect of blending, which
leave a multiplicative factor at the level of $m\sim 2-3$% and are calibrated
using image simulations in (MacCrann  _et al._ , 2020) and discussed in
Section IV.

###  II.2 The shape catalog

Of the 390 million detected and measured objects, only those that pass a
number of criteria are included in the weak lensing catalog (Gatti  _et al._ ,
2021). This is encoded in an index column in the catalog and encompasses
selections based on a combination of galaxy and PSF properties, which are
designed to reduce potential systematic biases due to blending, PSF mis-
estimation and stellar contamination. In addition, magnitude selections are
imposed ($i=18-23.5$ and $rz=15-26$) to exclude from the analysis galaxies for
which robust redshift estimation is difficult. Finally, we mask the catalog to
limit to the area that enables coherent combination with the DES lens sample
(Section II.3), bringing the footprint’s final effective area to span $A_{\rm
eff}=4143$ deg2. The resulting Metacalibration catalog yields a total of
$100,204,026$ galaxies, with a weighted number density of $n_{\rm eff}=5.59$
galaxies per square arcmin, as defined by Heymans and Grocutt (2013). Table 1
lists the effective number density for the full catalog and the corresponding
weighted ellipticity variance. The statistical power of the Metacalibration
catalog without galaxy weights is tripled compared to Y1, with weighting
increasing this further by $\approx 25\%$ Gatti  _et al._ (2021).

The spatial variation of number density over the survey footprint is shown in
the left-hand panel of Figure 1 in orthographic projection and equatorial
coordinates, with a map that utilises
skymap777https://github.com/kadrlica/skymap. Overlaid are the bounds of the
previous DES Y1 survey.

The Metacalibration shape catalog has passed a library of tests aimed at
identifying residual biases, detailed in (Gatti  _et al._ , 2021). These have
validated the measurements against systematic errors connected to PSF mis-
modeling, which are negligible for the full catalog with a smaller amplitude
compared to the DES Y1 analysis. In addition, the catalog was tested for
robustness against the presence of spurious B-modes using two different
estimators, COSEBIs (Schneider  _et al._ , 2010) and Pseudo-$C_{\ell}$, which
consistently resulted in a null detection (see also Appendix A.2). In Section
XI, these tests are extended to the tomographic case. Other tests included
checking the dependence of the two components of the shear on a number of
galaxy or survey properties, finding no significant correlations, except for a
linear dependence between $\langle e_{1}\rangle$ and the ratio between the
galaxy size and PSF size. While the origin of this trend is unknown, its
amplitude is three orders of magnitude smaller than the cosmic shear signal at
all scales and thus can be safely neglected in the cosmological analysis.

###  II.3 The lens sample

The DES fiducial foreground galaxy sample, MagLim, is divided into six
redshift bins and used in the measurement of the shear ratios (see Section
III.1.3). These galaxies are defined by a magnitude cut that evolves linearly
with their photometric redshift estimate, $z_{\rm ML}$ (De Vicente  _et al._ ,
2016), as $i<az_{\rm ML}+b$, with $a=4$ and $b=18$, as well as a lower
magnitude bound $i>17.5$. The sample selection is optimized to prioritize
brighter galaxies at low redshift and balance number density and photometric
redshift accuracy in terms of cosmological constraints obtained from galaxy
clustering and galaxy-galaxy lensing (Porredon  _et al._ , 2021).

An alternative lens sample is selected by redMaGiC (Rozo and Rykoff, 2016),
designed to find luminous red galaxies with precise photometric redshift
estimates (Cawthon  _et al._ , 2020). It does so by selecting galaxies above
some luminosity threshold based on how well they fit a red sequence template.
The template is calibrated using redMaPPer (Rykoff  _et al._ , 2014) and a
subset of galaxies with spectroscopic-verified redshifts. The goodness of fit
threshold is chosen to maintain a desired comoving number density of galaxies.
Clustering weights are assigned to redMaGiC galaxies to eliminate spurious
correlations with observational systematics.

##  III Redshift calibration

Any cosmological interpretation of weak lensing signals requires accurate
knowledge of the distribution of distances to the source galaxies used in the
measurement (Huterer  _et al._ , 2006; Van Waerbeke  _et al._ , 2006). A
tomographic cosmic shear measurement requires the distribution of source
galaxies into several redshift bins, each of which is characterized by a
redshift distribution $n(z)$. Along any line of sight, $\bm{\theta}$, the
expected observed shear, $\bm{\gamma}^{\rm obs}(\bm{\theta})$, is related to
an integral of the shear experienced by sources at redshift $z$ along that
line of sight, $\bm{\gamma}(\bm{\theta},z)$, as

$\bm{\gamma}^{\rm
obs}(\bm{\theta})=\int\mathrm{d}z\,n(z)\,\bm{\gamma}(\bm{\theta},z)\;.$ (4)

The cosmic shear signal, as the angular correlation of two such
$\bm{\gamma}^{\rm obs}(\bm{\theta})$ fields, is highly sensitive to biases in
the estimates of $n(z)$. Such biases can result in significant shifts in the
inferred cosmological parameters (Bonnett  _et al._ , 2016; Hildebrandt  _et
al._ , 2020; Joudaki and Hildebrandt, 2020). The increased statistical
precision of DES Y3 requires errors in the redshift distributions to be
unprecedentedly small so as not to dominate the uncertainty budget.

In practice, photometric redshift calibration for weak lensing source galaxies
relies on DES galaxies for which there exists accurate redshift information,
defining the redshift sample. To this end, the DES Y3 methodology follows the
idea that galaxies can be categorised as phenotypes that are distinguishable
and tightly constrained in redshift by their color in many-band photometric
information (Masters  _et al._ , 2015; Sánchez and Bernstein, 2019; Buchs  _et
al._ , 2019). The DES Deep Fields (Hartley  _et al._ , 2020), a combination of
optical and near-infrared multi-band, deep photometry over a smaller area of
5.88 deg2 (after masking), are used to define these phenotypes. These
observations act as an intermediary between redshift sample galaxies and the
wide field photometry over the DES footprint. This framework successfully
reduces both the statistical and systematic uncertainty in redshift
calibration (Myles  _et al._ , 2020).

This section summarises the strategy and choices made for the calibration of
redshift distributions, described fully in (Myles  _et al._ , 2020). First, it
summarises the primary method employed that relates redshift samples to sets
of galaxies distinguishable by their wide-field photometry (Section III.1.1),
and how clustering cross-correlation measurements and small-scale shear ratios
are folded in to validate and further constrain the proposed redshift
distributions (Sections III.1.2 and III.1.3). Next, the DES Deep Fields and
redshift samples used are described in (Section III.2). The estimated mean
redshift of the source sample is shown in the right panel of Figure 1 as a
function of position in the sky, showing the survey to be homogeneous across
the footprint. In Figure 2, the overall redshift distributions of the four
bins are shown.

###  III.1 Methodology

Figure 2: Estimated redshift distributions for the weak lensing catalog,
divided into four redshift bins (upper panel). Fainter lines indicate the
realisations, whose spread represents the uncertainty in their estimation.
Darker, solid lines denote the mean of the ensemble. These are derived using
the joint SOMPZ and WZ methodology, summarised in Section III.1 and detailed
in (Myles  _et al._ , 2020). The stair-step appearance is an artifact of using
a binned representation for the $n(z)$ and is immaterial to the cosmological
results. In the lower panel, the lensing kernel (defined in equation 14) for
each of the source redshift bins demonstrates that the DES Y3 sample is most
sensitive between $z=0.1-0.7$.

The DES Y3 weak lensing redshift methodology combines three likelihoods with
complementary information (Myles  _et al._ , 2020). The cornerstone to the
method for estimating the photometric redshift distributions is a scheme based
on two self-organizing maps (hereafter SOMPZ). The estimate of the redshift
distributions is uncertain and this framework provides a means to generate
samples of the $n(z)$ for each redshift bin that encompasses various sources
of error. The clustering redshift method (Gatti  _et al._ , 2020, WZ) uses the
angular cross-correlation of the weak lensing sources with galaxies of known
redshift to generate realisations of $n(z)$ from the combined SOMPZ+WZ
likelihoods. The shear ratio method, SR, (Sánchez  _et al._ , 2021)
incorporates the ratios of small-scale galaxy-galaxy lensing measurements
between different source bins with the same lens bin to give an independent
likelihood for each proposed redshift distribution. Ideally, weak lensing
analyses approach redshift estimation with redundancy, comparing alternative
methods as a means of validation. The DES Y3 cosmological analysis first
assesses the consistency of these methods, demonstrated in (Myles  _et al._ ,
2020, Figure 12) and then combines their information: we sample over an $n(z)$
generated by SOMPZ+WZ, with a joint likelihood from cosmic shear and SR.

####  III.1.1 SOMPZ

The estimates of the redshift distribution of $\sim$100 million DES Y3 wide
field galaxies rely on a redshift sample, or external samples of $\sim$60
thousand galaxies that have spectroscopic or many-band photometric redshifts,
as well as deep DECam and near-infrared photometry (Hartley  _et al._ , 2020).
To relate redshift information to the wide field galaxies, a self-organizing
map formalism, SOMPZ, is used to compress the multi-dimensional information of
both the deep and the wide field photometry (Masters  _et al._ , 2015; Buchs
_et al._ , 2019). This is done as a two-step process that connects the three
samples: the DES wide field source galaxies with $riz$ information, the DES
Deep Fields that span a smaller area with $ugrizJHK_{s}$, and the redshift
sample. The DES Deep Fields are key to the redshift process as they contain
deep eight-band photometry for 1.6 million galaxies. This information limits
degeneracies of galaxy spectral energy distributions present in lower-
dimensional color spaces when calibrating the weak-lensing galaxies.

A deep-field color defines a galaxy’s phenotype, $c,$ and the wide field $riz$
color-magnitude is used to determine $\hat{c}$, both of which are cell
identifiers for two separate self-organizing maps (Buchs  _et al._ , 2019;
Myles  _et al._ , 2020). Thus, the redshift distributions of wide field
galaxies whose photometry best matches cell $\hat{c}$ can be written as

$p(z|\hat{c})=\sum_{c}p(z|c,\hat{c})p(c|\hat{c})\;.$ (5)

The choice of sample that informs the redshift of a given phenotype, $p(z|c)$,
is discussed in Section III.2. To estimate the statistical connection between
the wide-field and deep-field photometry $p(c|\hat{c})$, the fitted light
profile of each Deep Field galaxy is drawn into real DES Y3 science images
multiple times in random positions with Balrog (Everett  _et al._ , 2020).
Processing these renderings with the DES photometry and shape measurement
pipeline delivers the mapping of galaxies with noisy wide field $riz$ and
successful shape measurement to the deep $ugrizJHK_{s}$ color space.

Four redshift bins, $\hat{b}_{i}$, are constructed as sets of wide field
photometry SOM cells, $\hat{c}\in\hat{b}_{i}$, by defining redshift bin edges
such that each contains an approximately equal number of source galaxies. Once
the four redshift bins have been established, an estimate of the redshift
distribution of each bin, $n_{i}(z)$, is given as

$n_{i}(z)=p(z|\hat{b}_{i})\approx\sum_{\hat{c}\in\hat{b}_{i}}\sum_{c}\underbrace{p(z|c)}_{\text{Redshift}}\underbrace{p(c)}_{\text{Deep}}\underbrace{\frac{p(c,\hat{c})}{p(c)p(\hat{c})}}_{\text{Balrog}}\underbrace{p(\hat{c})}_{\text{Wide}}\,.$
(6)

In each probability density term, contributing galaxies need to be weighted
appropriately by their shear response and inverse variance weight. See (Myles
_et al._ , 2020) for more details of this procedure.

An ensemble of $n(z)$ realisations is constructed to encompass four main
sources of uncertainty. Specifically, as outlined in Section III.2, these
include i) the sample variance in the color-redshift inference due to the
limited Deep Field area, ii) the Deep Field zero-point error in the color
information, particularly at redshifts where prominent features in the galaxy
spectra transition between filters, iii) an uncertainty on the method due to
the finite number of galaxies and Balrog realizations, and iv) the uncertainty
due to limitations in the redshift sample, incorporated with an ensemble of
realisations built from multiple redshift samples.

####  III.1.2 Clustering redshifts

The tomographic $n(z)$ are further constrained by the measured small-scale
angular cross-correlations, $w_{br}(z)$, between the source galaxies in bin
$b$ and a reference population $r$ of galaxies that is known to be within a
small range of redshift centered at $z$. The WZ method correlates the Y3
sources with both the redMaGiC galaxy sample (which has photo-$z$s determined
with $\sigma_{z}\approx 0.02$) and eBOSS galaxies (Ross  _et al._ , 2020), as
detailed in (Gatti  _et al._ , 2020). Using a linear bias model, one can
predict the observed $w_{br}$ data, given assumed values for $n(z),$ the
biases and lensing magnification coefficients of the source and reference
populations, and several nuisance parameters, $N$, that describe potential
errors in the linear-bias model. Comparison of this model to the observed
$w_{br}$ measures allows the construction of a likelihood.

With the SOMPZ photometric and WZ clustering constraints on the redshift
distribution in hand, a Hamiltonian Monte Carlo (HMC) method is used to draw
samples of the $n(z)$ functions from the joint likelihood. This yields
SOMPZ+WZ samples of possible sets of $n(z)$ for the redshift bins. By
marginalizing over these, one simultaneously accounts for uncertainty in the
nuisance parameters. The final set of redshift distributions are shown in
Figure 2. The mean, median and spread of the realisations for each source bin
are quoted in Table 1.

####  III.1.3 Shear ratios

The ratios of small-scale galaxy-galaxy lensing measurements from two source
bins and a shared lens sample provide geometric information (Jain and Taylor,
2003). The primary dependence of shear ratios, SR, on distances has
established them as a means of constraining and testing redshift distributions
(Schneider, 2016; Mandelbaum  _et al._ , 2005; Hoekstra  _et al._ , 2006;
Heymans  _et al._ , 2012; Hildebrandt  _et al._ , 2017, 2020; Giblin  _et al._
, 2021). When source and lens redshift bins overlap, the SR data also respond
to intrinsic alignments, in an orthogonal direction to the cosmic-shear
2-point functions. In this analysis, SR are incorporated at the likelihood
level to constrain the $n(z)$ and other nuisance parameters of our model,
particularly those of intrinsic alignments (Sánchez  _et al._ , 2021).

The SR data vector consists of nine scale-averaged lensing ratios, each
constructed from the combination of tangential shear signals with a MagLim
lens bin and two weak-lensing source bins. As validation, the SR using
RedMaGiC are analysed and the consistency of the results are tested in
Appendix D. The lens data is limited to the three low-redshift MagLim bins,
defined in (Porredon  _et al._ , 2021), to mitigate the impact of lens
magnification. It also utilises only small angular scales, $\sim 2-6$ Mpc/$h$,
which correspond to a maximum angular scale of $\sim 2.5-9$ arcmin depending
on lens redshift, that are not used in, and therefore mostly independent of,
the joint 3$\times$2pt analysis (Sánchez  _et al._ , 2021, Table 2). As a
robustness test of the use of these scales and their impact on the analysis,
the large-scale SR using RedMaGiC are incorporated in a variant analysis in
Appendix D.

In the inference, a Gaussian likelihood for the SR is assumed, using an
analytical covariance matrix (Sánchez  _et al._ , 2021). The addition of the
shear-ratio data to the analysis necessitates additional modeling and lens
observational parameters, summarised in Table 2. These describe the
uncertainties in the redshift distribution of lens galaxies, as well as the
relation between galaxies and dark matter, parametrized using a per-bin linear
galaxy bias. The impact of the SR information, as well as the WZ method, on
cosmological constraints is tested in Section X.

###  III.2 Deep Fields and redshift samples

The subset of these DES Deep Field galaxies with Balrog (Everett  _et al._ ,
2020) wide-field realizations that pass the weak-lensing selection and have
external high-quality redshift information forms the redshift sample. It is
constructed from both spectroscopic and multi-band photometric redshifts as
detailed in (Myles  _et al._ , 2020, Section 3.3). The former consists of
spectra from the following surveys: zCOSMOS-bright and deep (Lilly  _et al._ ,
2009), C3R2 (Masters  _et al._ , 2015, 2017, 2019), VVDS (Le Fèvre, 2013), and
VIPERS (Guzzo and Scodeggio, 2014; Garilli and Guzzo, 2014; Scodeggio  _et
al._ , 2018). The two many-band photo-$z$ catalogs used are those based on the
COSMOS 30-band (Laigle  _et al._ , 2016) and the PAUS+COSMOS 66-band catalog
(Alarcon  _et al._ , 2021), which adds narrow band filters from the PAUS to
the COSMOS photometry. These are the underlying data for three equally-
weighted redshift samples that are used in conjunction to span the uncertainty
of the redshift sample selection. These prioritize either spectroscopic or
high quality photometric information, where available, and are designed to be
complete by using COSMOS 30-band photometric redshifts elsewhere.

Spectroscopic calibration samples on their own can suffer from selection
effects (Newman, 2015; Bonnett  _et al._ , 2016; Gruen and Brimioulle, 2017;
Hoyle  _et al._ , 2018; Wright  _et al._ , 2020; Hildebrandt  _et al._ , 2020;
Joudaki and Hildebrandt, 2020; Hartley  _et al._ , 2020): at a given color or
color-magnitude, the subset of galaxies with successful spectroscopic redshift
measurements may not have redshifts that are representative of the full
sample. This motivates our choice to never discard those galaxies in the
COSMOS field from our redshift sample that do not have spectroscopic
information, but rather to use their photometrically estimated redshifts where
necessary. Conversely, even high-quality photometric catalogs suffer from
biases (Laigle  _et al._ , 2016; Hildebrandt  _et al._ , 2020; Joudaki and
Hildebrandt, 2020), or missing templates or photometric outliers in photo-$z$
surveys. The maximal impact of redshift sample uncertainty on cosmic shear
cosmological parameters, and therefore the robustness of this analysis to
these effects, is tested by analysing DES cosmic shear with ‘pure’ redshift
samples in Section X:

  * •

C: This sample includes only information from the COSMOS catalog and would
therefore suffer from the systematic calibration biases, claimed by
(Hildebrandt  _et al._ , 2020; Joudaki and Hildebrandt, 2020).

  * •

MB: Even the redshift sample that is least reliant on COSMOS has ten percent
of the redshift information derived from that sample at the faintest
magnitudes. In order to test the impact of any residual calibration biases due
to this subset, this ‘maximally-biased’ (MB) sample is complete, by design,
and artificially constructed to realistically alter the COSMOS galaxies that
are not matched to spectroscopic information, by altering those redshifts with
a magnitude-dependent prescription (Myles  _et al._ , 2020).

This approach differs to that used in the cosmic shear analysis for both DES
Y1 Troxel  _et al._ (2018b), which focuses on COSMOS as the redshift sample,
adjusted with clustering redshift information for the former and HSC (Hikage
_et al._ , 2019; Hamana  _et al._ , 2020), which uses the same redshift sample
with a reweighting method. It is also unlike the KiDS-1000 approach (Asgari
_et al._ , 2021), which uses only spectroscopic information, but mitigates
selection biases with 9-band information over the footprint and by removing
the subset of their data that is not well-represented by the redshift sample.

##  IV Shear Calibration

Accurate galaxy shape estimates are essential for cosmic shear studies. They
are hindered by the fact that the majority of galaxies used in any weak
lensing measurement are faint, noisy, pixellated, barely resolved due to
convolution with an anisotropic PSF, or blended with neighboring galaxies.
Their detection, deblending, and inclusion in the source sample is subject to
biases. The Metacalibration algorithm corrects several, but not all these
sources of bias, particularly not ones related to detection and (de-)blending.
Our ability to calibrate biases in the shear estimation therefore relies on
producing sufficiently realistic image simulations to calibrate the shear
estimates with.

The suite of image simulations used for DES Y3 shear calibration are presented
in (MacCrann  _et al._ , 2020). They are based on drawing model fits with
morphological information of galaxies from HST imaging (Leauthaud  _et al._ ,
2007) and DES Deep Field observations (Hartley  _et al._ , 2020) in the COSMOS
field to generate mock DES Y3 observations in $riz$ bands. Objects are
simulated with realistic DES observing conditions, i.e., convolved with the
DES PSF models of sets of $riz$ exposures overlapping a tile of the sky, with
noise and masking matching the data. Image coaddition, object detection, shape
measurement, assignment to redshift bins and redshift estimation are performed
as in the fiducial DES pipelines. As such, the multi-band suite affords a
state-of-the-art testing ground that is well-matched to the DES observations
in each redshift bin.

Shape measurement biases are taken into account through a multiplicative bias
correction, as defined in equation 1. The corresponding correction is applied
as an average $m$ over all the galaxies in a redshift bin. From a second
rendering of simulation tiles with a shear $\Delta\gamma(z)$ applied to all
galaxy images, we can estimate biases in our shear measurement. Multiplicative
bias can be accounted for by replacing, in equation 4, $n(z)$ by
$n(z)\times(1+m)$. In this way, we account not only for the fraction of
galaxies in an ensemble at a given redshift, but also for the sensitivity of
that subset of the galaxies to the applied shear. Thus substituting, equation
4 reads

$\Delta\gamma^{\rm obs}=\int\mathrm{d}z\,(1+m)n(z)\,\Delta\gamma(z)\,.$ (7)

Blending is expected to introduce a distortion of the mean and shape of
$n(z)$, in addition to its normalization. For one thing, the $m$ in equation 7
could well be redshift dependent, due to the changing morphology of galaxies.
But a distortion of the $n(z)$ is also caused by the response of the measured
shape of one galaxy to the shearing of light at a blended galaxy’s different
redshift, which leads to a perturbation on the $n(z)$ that should be used for
lensing analyses. Constraints on this effect are possible by using multiple
different redshift-dependent $\Delta\gamma(z)$ in an image simulation. We find
it to be significantly non-zero for the DES Y3 analysis choices and level of
precision (MacCrann  _et al._ , 2020). A joint understanding of multiplicative
shear bias and blending-related redshift calibration errors is a necessary
development for future weak lensing analyses. This is especially important for
deeper imaging, where blending becomes even more ubiquitous than the case of
DES Y3. A comparison to simpler shear calibration simulations, in which galaxy
images are sheared uniformly, or constrained to be equally separated over a 2D
grid, determines that the dominant contributor to shear calibration in DES Y3
is indeed this redshift-dependent blending effect, in some regimes by factors
of several.

These effects are disentangled in Ref. (MacCrann  _et al._ , 2020): first,
uncertainties on $m$ are derived for each redshift bin using a redshift
independent shear simulation $m_{\rm noz-blend}$, noted in Table 1. In
addition, image simulations with an applied redshift-dependent shear allow for
constraints on the distortion of the mean and shape of the $n(z)$. In Section
VII.4, details of how these are used to model the calibration bias and
uncertainty due to the impact of blending are given.

##  V Cosmological simulations

Aspects of this analysis have been validated using the Buzzard v2.0 suite of
18 cosmological simulations, the process of which we describe in brief here.
We refer the reader to the comprehensive discussion in (DeRose  _et al._ ,
2021a).

The Buzzard v2.0 simulations are $N$-body lightcone simulations that have been
populated with galaxies using the Addgals algorithm (Wechsler  _et al._ ,
2021; DeRose  _et al._ , 2021b), endowing each galaxy with a position,
velocity, spectral energy distribution, broad band photometry, half-light
radius and ellipticity. Each pair of Y3 simulations is produced from a set of
3 independent $N$-body lightcones with box sizes of
$[1.05,\,2.6,\,4.0]\,(h^{-3}\,\rm Gpc^{3})$, mass resolutions of
$[0.33,\,1.6,\,5.9]\,\times 10^{11}\,h^{-1}M_{\odot}$, spanning redshift
ranges in the intervals $[0.0,\,0.32,\,0.84,\,2.35]$ respectively. The
lightcones are run with the L-Gadget2 $N$-body code, a memory optimized
version of Gadget2 (Springel, 2005), with initial conditions generated using
2LPTIC at $z=50$. Together these produce $10,313$ deg2 of unique lightcone
area (Crocce  _et al._ , 2012).

The simulations are ray-traced using the spherical-harmonic transform
configuration of Calclens, performed on an $N_{\rm side}=8192$ HEALPix grid
(Becker, 2013; Górski  _et al._ , 2005a). The lensing distortion tensor is
computed at each galaxy position and is used to deflect the galaxy angular
positions, shear galaxy intrinsic ellipticities (including effects of reduced
shear), and magnify galaxy shapes and photometry. Convergence tests conducted
on this algorithm find that resolution effects on $\xi_{\pm}$ are negligible
on the scales used for this analysis DeRose  _et al._ (2019).

The DES Y3 footprint mask was applied to the ray-traced simulations and each
set of three $N$-body simulations yields two Y3 footprints, with 520 deg2 of
overlap. We apply a photometric error model to the mock wide field photometry
in our simulations based on a relation measured from Balrog (Everett  _et al._
, 2020). A weak lensing source selection is applied to the simulations using
the PSF-convolved sizes and $i$-band S/N in order to match the non-tomographic
source number density, $n_{\rm eff}=5.9$ arcmin-2, from an earlier iteration
of the Metacalibration source catalog. The SOMPZ framework is applied to
divide source galaxies into redshift bins, each with a number density of
$n_{\rm eff}=1.48$ arcmin-2, and to obtain estimates of the redshift
distribution of galaxies which resemble those of the data well (DeRose  _et
al._ , 2021a; Myles  _et al._ , 2020). The shape noise of the simulations is
then matched to that measured in the Metacalibration catalog per bin.
Following the methodology in (Myles  _et al._ , 2020), mock $n(z)$
realisations are produced, limited to uncertainties from sample variance only.
These samples are found to be consistent with the true redshift distributions,
and the inclusion of WZ and SR likelihoods finds consistency and reduces their
variance.

Cosmic shear two-point functions are measured in the Buzzard v2.0 simulations
without shape noise using the same pipeline as that used for the data, with
Metacalibration responses and inverse variance weights set to 1 for all
galaxies. The model for these measurements at the true Buzzard v2.0 cosmology,
assuming the true source redshift distributions and no intrinsic alignments,
reproduces the mean of our measurements on 18 simulations with a chi-squared
value of $1.37$ per degree of freedom, for $207$ data points using the
Fiducial scale selection (see Section VII.3) and assuming the covariance of a
single simulation. Simulated likelihood analyses assuming either the true
simulated source redshift distributions or calibrated photometric redshift
distributions results in cosmological constraints that are unbiased.

##  VI Cosmic shear measurement

This section presents the real-space two-point shear correlation functions,
$\xi_{\pm}$, from the weak-lensing science catalog. Derived from the
contiguous 4143 deg2 sky footprint and over 100 million galaxies, divided into
redshift bins according to Table 1, these constitute the highest signal-to-
noise cosmic shear measurements to date, with a total $S/N=40$. After
eliminating the small scales, in order to mitigate model uncertainties in the
cosmology analysis, as motivated in Section VII.3 (Krause  _et al._ , 2021;
Secco  _et al._ , 2021), the measurement is found to have a $S/N=27$. Using
the $\rm\Lambda CDM$-Optimized scale selection, described in Section VII.7, a
$S/N=31$ is obtained, where the signal-to-noise is computed as

$S/N=\frac{\xi_{\pm}^{\rm data}\mathbf{C}^{-1}\xi_{\pm}^{\rm
model}}{\sqrt{\xi_{\pm}^{\rm model}\mathbf{C}^{-1}\xi_{\pm}^{\rm model}}}\,.$
(8)

It is notable that the S/N for the Y1 measurement without scale cuts (26.8)
(Troxel  _et al._ , 2018) is matched by the Y3 $\rm\Lambda CDM$-Optimized
measurement after limiting the scales.

###  VI.1 Correlation function measurements

Figure 3:  Measured tomographic DES Y3 cosmic shear two-point correlation
functions: $\xi_{+}(\theta)$ (left) and $\xi_{-}(\theta)$ (right), scaled by
the angular separation, $\theta$, to emphasize differences relative to the
best-fit model (upper panels). The correlation functions are measured for each
redshift bin pair, indicated by the label and the error bar represents the
square root of the diagonal of the analytic covariance matrix. The best-fit
$\rm\Lambda CDM$ theoretical prediction from the cosmic shear-only tomographic
analysis is denoted by a green line. Scales excluded from the analysis, due to
their sensitivity to small-scale systematics, are shaded in light blue for the
Fiducial analysis and darker blue for the $\rm\Lambda CDM$-Optimized analysis.
The signal-to-noise of the measurement is 40 using all angular scales and 27
(31) using the Fiducial ($\rm\Lambda CDM$-Optimized) scale-selection. For
comparison, the yellow shaded region shows the Y1 uncertainty, with a factor
of $\sim\sqrt{2}$ lower signal-to-noise. The lower panels plot the fractional
difference between the measurements and best-fit,
$\delta\xi_{\pm}/\xi_{\pm}=\big{(}\xi_{\pm}-\xi_{\pm}^{\rm
theory}\big{)}/\xi_{\pm}^{\rm theory}$. We find that the $\chi^{2}$ per
effective d.o.f of the $\rm\Lambda CDM$ model is $237.7/222.2=1.07$, and the
$p$-value is 0.223.

The estimator of the two-point correlation function or shear-shear correlation
function can be written in terms of the measured radial, $\epsilon_{\rm x}$,
and tangential ellipticities, $\epsilon_{\rm t}$, of a galaxy as,

$\xi^{ij}_{\pm}(\theta)=\langle\epsilon_{\rm t}\epsilon_{\rm
t}\pm\epsilon_{\times}\epsilon_{\times}\rangle(\theta)\,.$ (9)

It is determined by averaging over all galaxy pairs $(a,b)$ separated by an
angle $\theta$, for two redshift bins, $i$, and $j$, as,

$\centering\hat{\xi}^{ij}_{\pm}(\theta)=\frac{\sum_{ab}w_{a}w_{b}[\epsilon_{\rm
t}^{i}\epsilon_{\rm
t}^{j}\pm\epsilon^{i}_{\times}\epsilon^{j}_{\times}]}{\sum_{ab}w_{a}w_{b}R_{a}R_{b}}\,,\@add@centering$
(10)

where $R$ represents a response correction from Metacalibration, given in
equation 3. The weighted sum utilises the per-galaxy inverse-variance weights,
$w$, and is taken over galaxy pairs whose angular separation is within an
interval $\Delta\theta$ around $\theta$.

The tomographic DES Y3 cosmic shear data vector is shown in Figure 3. It is
computed via equation 10 using the public code
TreeCorr888https://rmjarvis.github.io/TreeCorr (Jarvis  _et al._ , 2004).
Twenty angular logarithmic bins are chosen to span 2.5 to 250.0 arcmin. Of
these, small-scale measurements are discarded from the cosmological inference,
represented as a light blue shaded region for the ‘Fiducial’ analysis, defined
in Section VII.3, and a darker blue for a ‘$\rm\Lambda CDM$-Optimized’
analysis (see Section VII.7). In the upper panel, the green line denotes the
best-fit cosmological and astrophysical parameters from the tomographic
analysis. The error bound indicates the square root of the diagonal of the
analytic covariance matrix. For comparison, the level of uncertainty from the
DES Y1 analysis is shown as the yellow shaded bands. In the lower panels, the
fractional residuals, $(\xi_{\pm}-\xi_{\pm}^{\rm theory})/\xi^{\rm theory}$
demonstrate the fit of the model to the measurements.

The data vector, $D$, comprises four auto-correlations, and six unique cross-
correlations between redshift bins for each $\xi_{+}$ and $\xi_{-}$. The small
angular scales of the measurements are eliminated from the analysis primarily
to mitigate the impact of baryonic effects, indicated by the shaded region,
leaving 167 (60) data points for $\xi_{+}$ ($\xi_{-}$).

We have verified that an independent pipeline produces the same $\xi_{\pm}$
measurements to numerical precision. The data points shown represent the
weighted mean of pair separation, but the theoretical prediction is averaged
over the bin using the geometric approximation, following equation 10 in Ref.
(Friedrich  _et al._ , 2020).

###  VI.2 Covariance Matrix

To model the statistical uncertainties of our measurements of $\xi_{\pm}$ we
assume a multi-variate Gaussian distribution for our combined data vector. The
modeling of the disconnected four-point function part of the covariance matrix
of that data vector (also known as the Gaussian part of the covariance) is
described in (Friedrich  _et al._ , 2020) and includes analytic treatment of
bin averaging and sky curvature. The connected four-point function part and
the contribution from super-sample covariance use the public code
CosmoCov999https://github.com/CosmoLike/CosmoCov (Fang  _et al._ , 2020) based
on the CosmoLike framework (Krause and Eifler, 2017).

The covariance matrix, $\mathbf{C}$, is a function of both the cosmological
and nuisance parameters that are required to describe the data vector, as well
as the redshift distributions. In this work, following previous cosmic shear
analyses, an iteratively fixed covariance matrix is used. That is, we assume a
fiducial set of input parameters for the computation of the covariance matrix
used in the initial unblind analysis. Then, the covariance is recomputed at
the best-fit from the first iteration, and the final analysis is performed.
This step incurs a change in cosmological parameters of much less then
1$\sigma$ (DES Collaboration, 2021).

The robustness of our analysis with respect to the details of our covariance
modeling is demonstrated in (Friedrich  _et al._ , 2020). FLASK simulations
are used to test for the impact of effective number density and shape-noise
dispersions in the presence of complex survey footprints (Troxel  _et al._ ,
2018b). There, it is also demonstrated that deviations from the Gaussian
likelihood assumption are negligible for this analysis. If the covariance
model was perfect and there were no tight priors on any of the parameters of
the model for cosmic shear measurements, then the $\chi^{2}$ between any
measurement of our combined data vector of cosmic shear correlation functions
and our best-fit model to that measurement should on average be $\approx
N_{\mathrm{data}}-N_{\mathrm{param}}$ = 227-28 = 199. Furthermore, in that
situation the typical scatter of $\chi^{2}$ ($1\sigma$) is expected to be
$\approx\sqrt{2(N_{\mathrm{data}}-N_{\mathrm{param}})}$. Any errors in our
covariance model may cause deviations from that behaviour. We estimate the
impact of numerous potential covariance errors in (Friedrich  _et al._ , 2020)
and find that none of them significantly impact the cosmic shear part of the
DES Y3 2-point function analysis — neither with respect to $\chi^{2}$ nor with
respect to parameter constraints. The aforementioned behaviour of $\chi^{2}$
will also be altered in the presence of priors on any of the model parameters.
Taking into account our Gaussian priors on multiplicative shear bias and on
shifts of the redshift distributions we find in (Friedrich  _et al._ , 2020)
that within a 68% confidence interval our best-fit $\chi^{2}$ should be
$\approx 220.2\pm 20.7$.

###  VI.3 Mitigating observer bias

In order to ensure that analysis decisions are not influenced, even
unconsciously, by the comparison of results to experimenters’ expectations, we
apply transformations to the data that are designed to obscure the
cosmological results. Although the approaches differ, a philosophy of blind
analyses is necessary and accepted for all recent cosmological weak lensing
analyses (Hildebrandt  _et al._ , 2016; Dark Energy Survey Collaboration,
2016a; Hildebrandt  _et al._ , 2017; Troxel  _et al._ , 2018; Dark Energy
Survey Collaboration, 2018; Hikage  _et al._ , 2019; Hildebrandt  _et al._ ,
2020; Hamana  _et al._ , 2020; Asgari  _et al._ , 2021; Heymans  _et al._ ,
2021). For this DES Y3 analysis, these transformations, and the procedures for
deciding when to remove them, make up a three-stage blinding strategy, at the
levels of the shape catalog, data vector, and parameter inference. Stages,
described below, are removed sequentially as a pre-determined set of criteria
is fulfilled.

During stage one, galaxy shapes are transformed by altering their
ellipticities, e, via |$\eta$| $\equiv 2{\rm arctanh}|\textbf{e}|\rightarrow
f$|$\eta$$|$, with an unknown and random value 0.9 < f < 1.1. This
transformation is sufficient to change the cosmology results for shear-only
analyses and is similar to previous work (Zuntz  _et al._ , 2018; Hikage  _et
al._ , 2019; Hildebrandt  _et al._ , 2020). The transformed shear catalog is
used until the non-tomographic shear validation tests (Gatti  _et al._ ,
2021), and the tomographic versions (see Appendix A.3) are passed, the three
redshift methods were proved to be consistent in the $\Delta z$ parameter
space (Myles  _et al._ , 2020) and similarly, Section III.1, the analysis
choices and model pipeline fixed (Krause  _et al._ , 2021) and all relevant
papers had completed a first round of the DES internal review process.

As the shear catalog transformation breaks the internal consistency between
shear and galaxy clustering observables, it cannot be used for a blinded
combined 3$\times$2pt analysis. The second stage of blinding is implemented at
the data vector level using the method detailed in Ref. (Muir  _et al._ ,
2020). Here, a transformation adds to each two-point shear correlation
function data point a quantity equal to the difference between model
predictions for that observable computed at two sets of cosmological parameter
values, $\Theta_{\rm ref}$ and $\Theta_{\rm ref}+\Delta\Theta$. This has been
shown to change the best-fit cosmology associated with the data vector by
approximately $\Delta\Theta$ while preserving the internal consistency between
components of the data vector. This blinding step is performed automatically
in the measurement pipeline. $\Theta_{\rm ref}$ is chosen to be equal to the
same fiducial cosmology used for our modeling tests, and the parameter shift
$\Delta\Theta$ is drawn pseudo-randomly in $w$CDM parameter space, using a
fixed seed to ensure that different measurements are transformed consistently.
The final stage of our blinding procedure is a simple obscuring of axes of the
cosmological parameters in 1 or 2D contours, implemented as a final safeguard.

The measurement and analysis pipeline was tested on mocks, confirmed with a
secondary pipeline, and shown to recover unbiased parameters. When the
analysis framework, scale selection and systematic priors were fixed, and the
correlation functions measured consistently with two pipelines (see Section
VI), the criteria for removing the third stage of blinding is met. At this
point, internal consistency is assessed in terms of goodness of fit and
consistency between the measurements from cosmic shear, galaxy-galaxy lensing
(Prat  _et al._ , 2021) and galaxy clustering (Rodríguez-Monroy  _et al._ ,
2021), required to meet an arbitrary $\chi^{2}$ criterion of $p$-value $>0.01$
with the posterior predictive distribution process (see Section IX). Finally,
parameter constraints could be revealed. Any changes to the analysis after
this point are documented in Appendix E.

##  VII Modeling and Analysis Choices

This section outlines the baseline theoretical model for the cosmic shear
correlation functions used in this analysis and discusses the evidence for its
robustness. Systematic errors in the model are demonstrated to be subdominant
to the precision of the data, in order to derive unbiased cosmological
parameters (Krause  _et al._ , 2021), as validated specifically for cosmic
shear in Ref. Secco  _et al._ (2021). Cosmological, astrophysical, and
systematic parameters are constrained for the $\Lambda$CDM model. For the case
of cosmological parameters that are not well-constrained by cosmic shear,
informative priors with boundaries that widely encompass allowed values from
external experiments are used, reported in Table 2. For the massive neutrino
density parameter, $\Omega_{\nu}$, we vary $\Omega_{\nu}h^{2}$, where $h$ is
the Hubble parameter. We note that this is often fixed in other analyses
(Planck Collaboration, 2018) at zero, or to the minimum mass allowed by
oscillation experiments, $m_{\nu}=0.06$eV (Olive, 2016).

###  VII.1 Cosmic shear signal

The observed angular two-point correlation for two redshift bins $i$ and $j$
is expressed in terms of the convergence power spectrum $C_{\kappa}(\ell)$ at
an angular wavenumber $\ell$ as

$\displaystyle\xi_{+}(\theta)=\sum_{\ell}\frac{2\ell+1}{4\pi}G^{\pm}_{\ell}(\cos\theta)\quantity[C_{\kappa,\textrm{EE}}^{ij}(\ell)+C_{\kappa,\textrm{BB}}^{ij}(\ell)]\,,$
(11)
$\displaystyle\xi_{-}(\theta)=\sum_{\ell}\frac{2\ell+1}{4\pi}G^{\pm}_{\ell}(\cos\theta)\quantity[C_{\kappa,\textrm{EE}}^{ij}(\ell)-C_{\kappa,\textrm{BB}}^{ij}(\ell)],$
(12)

where the functions $G^{\pm}_{\ell}(x)$ are computed from Legendre polynomials
$P_{\ell}(x)$ (see e.g. (Stebbins, 1996)). Although cosmological lensing does
not produce B-modes except due to multiple-deflection effects, our baseline
model does allow for a B-mode contribution from intrinsic alignments, and so
we show the more general expression here.

The 2D convergence power spectrum $C_{\kappa}$ can be related to the non-
linear 3D matter power spectrum $P_{\delta}$ via the flat-sky and Limber
approximations (Limber, 1953; LoVerde and Afshordi, 2008) as,

$\centering C_{\kappa}^{ij}(\ell)=\int_{0}^{\chi_{\rm
H}}d\chi\,\frac{W_{i}(\chi)W_{j}(\chi)}{\chi^{2}}P_{\delta}\left(\chi,k=\frac{\ell+0.5}{z(\chi)}\right)\,,\@add@centering$
(13)

where $\chi_{\rm H}$ is the horizon distance and a flat spatial geometry is
assumed, so that $f_{K}(\chi)$, the comoving angular diameter distance, is
simplified to $\chi$. The lensing efficiency kernel $W_{i}(\chi)$ for the
redshift bin $i$ is defined as

$\centering W_{i}(\chi)=\frac{3H_{0}^{2}\Omega_{\rm
m}}{2c^{2}}\frac{\chi}{a(\chi)}\int_{\chi}^{\chi_{\rm
H}}d\chi^{\prime}\,n_{i}(\chi^{\prime})\frac{\chi^{\prime}-\chi}{\chi^{\prime}}\,,\@add@centering$
(14)

where $n_{i}(\chi)d\chi$ is the effective number of galaxies in $d\chi$ in the
$i$th redshift bin, normalised such that $\int_{0}^{\chi_{\rm
H}}n_{i}(\chi)d\chi=1$. The convergence power spectrum can be described by the
amplitude of matter density fluctuations on an 8$h^{-1}$Mpc scale in linear
theory, $\sigma_{8}$, which is related to the amplitude of the primordial
scalar density perturbations, $A_{\rm s}$, and is degenerate with the matter
density parameter $\Omega_{\rm m}$. That is, the power spectrum at small $k$
increases with an increase in either $\Omega_{\rm m}$ or the $\sigma_{8}$
normalisation.

Higher-order contributions to the observed two-point statistics are caused by
reduced shear (Dodelson  _et al._ , 2006; Shapiro, 2009), source clustering
and magnification (Schneider  _et al._ , 2002; Schmidt  _et al._ , 2009) and
the deflection of source positions (Seljak, 1996; Dodelson  _et al._ , 2008).
The impact of these higher order effects is verified to be negligible at the
precision of this analysis, with the derivation and computation of the
contributions accounted in Ref. (Krause  _et al._ , 2021). The predicted
impact on cosmic shear measurements is illustrated in Figure 5 of Ref. (Secco
_et al._ , 2021).

Table 2: Summary of cosmological, observational and astrophysical parameters and priors used in the analysis. In the case of flat priors, the prior bound to the range indicated in the ‘value’ column while Gaussian priors are described by their mean and 1$\sigma$ width. Parameter | Type | Value
---|---|---
Cosmological |  |
$\Omega_{\rm m}$, Total matter density | Flat | [0.1, 0.9]
$\Omega_{\rm b}$, Baryon density | Flat | [0.03, 0.07]
$10^{-9}A_{\rm s}$, Scalar spectrum amplitude | Flat | [0.5, 5.0]
$h$, Hubble parameter | Flat | [0.55, 0.91]
$n_{\rm s}$, Spectral index | Flat | [0.87,1.07]
$\Omega_{\nu}h^{2}$, Neutrino mass density | Flat | [0.00060,0.00644]
$w$, Dark energy parameter | Fixed | [-2,-1/3]
Observational |  |
$\Delta z^{1}$, Source redshift 1 | Gaussian | ( 0.0, 0.018 )
$\Delta z^{2}$, Source redshift 2 | Gaussian | ( 0.0, 0.015 )
$\Delta z^{3}$, Source redshift 3 | Gaussian | ( 0.0, 0.011 )
$\Delta z^{4}$, Source redshift 4 | Gaussian | ( 0.0, 0.017 )
$m^{1}$, Shear calibration 1 | Gaussian | ( -0.006, 0.009 )
$m^{2}$, Shear calibration 2 | Gaussian | ( -0.020, 0.008 )
$m^{3}$, Shear calibration 3 | Gaussian | ( -0.024, 0.008 )
$m^{4}$, Shear calibration 4 | Gaussian | ( -0.037, 0.008 )
Intrinsic alignment |  |
$a_{1}$, Tidal alignment amplitude | Flat | $[-5,5]$
$a_{2}$, Tidal torque amplitude | Flat | $[-5,5]$
$\eta_{1}$, Tidal alignment redshift index | Flat | $[-5,5]$
$\eta_{2}$, Tidal torque redshift index | Flat | $[-5,5]$
$b_{\rm ta}$, Tidal alignment bias | Flat | $[0,2]$
Shear-ratio |  |
Galaxy bias, $b^{1-3}_{\rm g}$ | Flat | $[0.8,3]$
$\Delta z_{l}^{1}$, Lens redshift 1 | Gaussian | ( -0.009, 0.007 )
$\Delta z_{l}^{2}$, Lens redshift 2 | Gaussian | ( -0.035, 0.011 )
$\Delta z_{l}^{3}$, Lens redshift 3 | Gaussian | ( -0.005, 0.006 )
$\sigma z_{\rm l}^{1}$, Lens redshift width 1 | Gaussian | ( 0.98,0.06 )
$\sigma z_{\rm l}^{2}$, Lens redshift width 2 | Gaussian | (1.31,0.09 )
$\sigma z_{\rm l}^{3}$, Lens redshift width 3 | Gaussian | ( 0.87,0.05 )

###  VII.2 Intrinsic Alignments

If galaxy orientation was truly random, any measured $\xi_{\pm}$ signal would
be attributed to gravitational lensing. In reality, we expect intrinsic
alignment, whereby galaxy shapes are correlated with the local environments in
which they formed and evolved (Hirata and Seljak, 2004; Troxel and Ishak,
2015; Kiessling  _et al._ , 2015). One can approximate the apparent shape of a
galaxy as the superposition of an intrinsic shape alignment, $\gamma_{\rm I}$,
and a true shear due to lensing, $\gamma_{\rm G}$, such that the observed
shape is given by $\epsilon^{\mathrm{obs}}=\epsilon+\gamma_{\rm G}+\gamma_{\rm
I}$, where the galaxy shape in the absence of lensing and intrinsic
alignments, $\epsilon$, is uncorrelated across the sky. The total harmonic-
space power spectrum can be written as a sum over corresponding terms,

$C_{l}^{\rm total}=C^{ij}_{\rm GG}(\ell)+C^{ij}_{\rm GI}(\ell)+C^{ji}_{\rm
IG}(\ell)+C^{ij}_{\rm II}(\ell).$ (15)

The ‘gravitational-gravitational’ term, $C^{ij}_{\rm GG}(\ell)$, corresponds
to the convergence power spectrum given by equation 13. The ‘intrinsic-
intrinsic’ contribution, $C^{ij}_{\rm II}(\ell)$, arises due to correlations
between the intrinsic shapes of two physically nearby galaxies, while the
‘gravitational-intrinsic’ terms, $C^{ij}_{\rm GI/IG}(\ell)$, arise in pairs of
galaxies for which some common structure affects the intrinsic shape of one of
the galaxies, and the shear on the other.

The baseline intrinsic alignment model used in this work, a nonlinear
perturbative prescription, the Tidal Alignment and Tidal Torquing model (TATT;
Blazek  _et al._ (2019)), is motivated by tests on simulated data (Krause  _et
al._ , 2021; Secco  _et al._ , 2021). This model choice represents a departure
from previous weak lensing analyses where the more simple nonlinear alignment
(NLA) model was opted for (e.g. Troxel  _et al._ , 2018; Hildebrandt  _et al._
, 2020; Hamana  _et al._ , 2020; Asgari  _et al._ , 2021). In brief, the TATT
model allows three contributions in the gravitational tidal field which
capture the ‘tidal alignment’, linear in the tidal field, the ‘tidal torquing’
(Mackey  _et al._ , 2002; Codis  _et al._ , 2015), quadratic in the tidal
field, and the impact of source density weighting (Blazek  _et al._ , 2015).
At fixed redshift, the TATT power spectra depends on a tidal alignment
amplitude, $A_{1}$, a tidal torquing amplitude, $A_{2}$, and an effective
source linear bias of the galaxies, $b_{\rm ta}$. The redshift evolution of
$A_{1}$ and $A_{2}$ is parametrized as a power law, governed by $\eta_{1}$ and
$\eta_{2}$ given by,

$\displaystyle A_{1}(z)=-a_{1}\bar{C}_{1}\frac{\rho_{\rm crit}\Omega_{\rm
m}}{D(z)}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{1}}\,,$ (16) $\displaystyle
A_{2}(z)=5a_{2}\bar{C}_{1}\frac{\rho_{\rm crit}\Omega_{\rm
m}}{D(z)^{2}}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{2}}\,,$ (17)
$\displaystyle A_{1\delta}(z)=b_{\rm ta}A_{1}(z)\,,$ (18)

where a normalization constant, by convention is fixed to $\bar{C}_{1}=5\times
10^{-14}{\rm M}h^{-1}{\rm Mpc}^{2}$ (Brown  _et al._ , 2002), the pivot
redshift is fixed at $z_{0}=0.62$ (Troxel  _et al._ , 2018) and $b_{\rm ta}$
is assumed to be constant in redshift. Given the absence of informative
priors, the analysis marginalises over the five intrinsic alignment parameters
that govern the amplitude and redshift dependence of the signal,
$(a_{1},a_{2},\eta_{1},\eta_{2},b_{\rm ta})$, with wide flat priors summarised
in Table 2. In the limit $A_{2},b_{\rm ta}\rightarrow 0$, TATT reduces to the
more commonly used NLA.

###  VII.3 The matter power spectrum and baryonic effects

Modeling the impact of baryonic feedback effects on the small-scale matter
power spectrum is a leading systematic uncertainty in cosmic shear surveys
(van Daalen  _et al._ , 2014; Semboloni  _et al._ , 2013; Harnois-Déraps  _et
al._ , 2015; Chisari  _et al._ , 2018). Power is surpressed by Active Galactic
Nuclei (AGN) feedback processes at $k\sim 10h$/Mpc, as well as enhanced at
smaller scales due to the more efficient cooling and star formation. These
effects on the matter power spectrum can be modeled using, for example,
empirical halo models fitted to hydrodynamic simulations (Mead  _et al._ ,
2015; Hildebrandt  _et al._ , 2020; Asgari  _et al._ , 2021),
‘baryonification’ models (Schneider  _et al._ , 2019) and Principle Component
Analysis (PCA) (Huang  _et al._ , 2020), though these are limited by the large
range of behaviours exhibited.

Instead, to mitigate any bias, this analysis adopts a gravity-only power
spectrum and limits the measurements to larger angular scales (Krause  _et
al._ , 2021; Secco  _et al._ , 2021). To define the maximum scale up to which
plausible models may cause significant impact, synthetic cosmic shear data
vectors are contaminated with baryonic effects measured from EAGLE (McAlpine
_et al._ , 2016) and OWLS-AGN (van Daalen  _et al._ , 2011) hydrodynamic
simulations, according to

$P_{\delta,b}(k,z)=\frac{P_{\rm hydro}(k,z)}{P_{\rm DMO}(k,z)}P_{\delta}(k,z)$
(19)

where $P_{\rm hydro}(k,z)$ and $P_{\rm DMO}(k,z)$ are the matter power spectra
measured from hydrodynamic and dark matter only simulations of the same suite.
$P_{\delta}(k,z)$ is the non-linear matter power spectrum analytically
calculated with the Halofit model (Smith  _et al._ , 2003; Takahashi  _et al._
, 2012). A threshold for the minimum set of angular scales is determined to
ensure that the bias in the 3$\times$2pt analysis is below $0.3\sigma_{2D}$ in
$\Omega_{\rm m}-S_{8}$ for $\rm\Lambda CDM$ (found to be $0.09\sigma$, in the
baseline analysis) and in $\Omega_{\rm m}-w$ for $w$CDM ($0.23\sigma_{2D}$),
by balancing the cuts made for cosmic shear and 2$\times$2pt. With that
criteria, the cosmic shear scale selection is identified where the bias
incurred in the contaminated analysis compared to the baseline is
$\Delta\chi^{2}=0.5$, which corresponds to a residual bias of
$0.02\sigma_{2D}$ (see Figure 5 in Ref. (Krause  _et al._ , 2021)). The
angular bounds that satisfy this requirement are indicated by the shaded
region in Figure 3. Taking these into account, the data vector has 166 and 61
angular bins in $\xi_{+}$ and $\xi_{-}$ and a total of 227 data points.

###  VII.4 Modeling blending

Figure 4:  Impact of choices in redshift calibration on predicted cosmic shear
observables. The fractional difference between the fiducial simulated signal
and one with an alternative analysis choice is shown,
$\delta\xi_{\pm}/\xi_{\pm}$. Plotted are predicted data vectors (i) with the
purely COSMOS C-redshift sample (blue solid line) (ii) with the artificially
biased spectroscopic MB-redshift sample (yellow dash-dotted line) (iii)
without accounting for the redshift-mixing effects of blending (red dotted
line). Fiducial ($\rm\Lambda CDM$-Optimized) scale cuts are shown as (dark)
blue shaded regions. The shaded green regions represent the simulated signals
corresponding to the full range of hyperrank $n(z)$ realisations described in
Section VII.5.1, and the dashed grey lines show the 5th and 95th percentiles
of these simulated signals.

The DES Y3 catalog uses the Metacalibration method to remove the largest
calibration biases. However, blending is not fully accounted for in this
approach, resulting in percent-level residual multiplicative bias. This is
usually modeled in the shear correlation functions as (Heymans and Van
Waerbeke, 2006; Huterer  _et al._ , 2006)

$\xi^{ij}_{\rm obs}=(1+m^{i})(1+m^{j})\xi^{ij}\;,$ (20)

where $m^{i}$ is the multiplicative bias for photometric redshift bin $i$, and
is marginalized over with some prior usually inferred from image simulations,
as in the case of this analysis. As detailed in Ref. (MacCrann  _et al._ ,
2020), shear calibration biases for a given redshift bin cannot be fully
described by a single $m^{i}$ factor, firstly, because the $m^{i}$ may evolve
over the redshift bin, i.e. $m^{i}=m^{i}(z)$. Moreover, in the presence of
blending, the shape estimation for galaxies at one redshift may be influenced
by the shear that blended galaxies at a different redshift are subject to. A
general approach allowing for both of these effects is to quantify biases in
the effective redshift distribution $n_{\gamma}(z)$. A mean multiplicative
bias is related to the normalization of this effective redshift distribution,
but biases to its shape are also expected due to redshift evolution of $m$ and
blending.

Ref. MacCrann  _et al._ (2020) calibrates this, assuming a model for the
correct effective $n_{\gamma}^{i}(z)$ for a redshift bin $i$ as

$n_{\gamma}^{i}(z)=[1+F_{i}(z)]n^{i}_{\rm obs}(z)+G_{i}(z).$ (21)

where $F_{i}(z)$ and $G_{i}(z)$ are functions of $z$, constrained using image
simulations in (MacCrann  _et al._ , 2020). They allow for multiplicative and
additive deviations from the Metacalibration response-weighted redshift
distribution, $n^{i}_{\rm obs}(z)$, presented in Section III. The term
$F_{i}(z)$ models the impact on the effective weighting of the discrete
sources used to construct $n_{i}(z)$, such as (possibly redshift-dependent)
multiplicative bias effects. The term $G_{i}(z)$ captures responses to shear
of light at redshifts other than that of the primary galaxy of a detection,
due to blending of that primary galaxy with galaxies at different redshifts.
Uncertainty in both effects are incorporated into the redshift calibration by
producing samples of possible $n_{\gamma}^{i}(z)$. The impact of this
modification to the $n^{i}_{\rm obs}(z)$ on the shear correlation functions is
demonstrated in Figure 4 as ‘no blending(z)’. The use of the normalized $n(z)$
samples in the analysis pipeline gives shear calibration priors per bin,
$m^{i}$, inferred from the $n_{\gamma}(z)$ samples, that are sampled. The
blending-based perturbations are also reflected in widened priors on the mean
redshift of each redshift bin. Both priors are listed in Table 1.

To demonstrate the importance of this previously uncalibrated effect, a
variant analysis is performed in Section XI that neglects the impact of the
redshift-mixing blending. In this test, the redshift distributions without
shear-calibration correction are used with the $m_{\rm noz-blend}$ priors,
listed in Table 1 of 3-6% width, that represent the impact from masking,
detection biases and blending, drawn from (MacCrann  _et al._ , 2020, their
Table 3) and quoted. The impact of the assumption that ignores the correlation
between an individual $n(z)$ realisation and the residual multiplicative
factor, $m$, is tested at the likelihood level. To do so the analysis is
modified to use Hyperrank to sample the ensemble $n(z)$, with an $m$
associated with each realisation, labelled ‘Full blending treatment’.

###  VII.5 Redshift uncertainty

An approach previously used for propagating photometric redshift calibration
uncertainties into cosmic shear cosmological parameter constraints (e.g., Dark
Energy Survey Collaboration, 2016a; Choi  _et al._ , 2016; Joudaki  _et al._ ,
2017; Troxel  _et al._ , 2018; Hikage  _et al._ , 2019; Asgari  _et al._ ,
2021; Hamana  _et al._ , 2020) relies on introducing a nuisance parameter,
$\Delta z^{i}$, for the offset of the mean redshift of each source bin $i$
from its estimate,

$n_{i}(z)\rightarrow n_{i}(z-\Delta z^{i})\,.$ (22)

The prior on the $\Delta z^{i}$ encapsulates the statistical and systematic
calibration uncertainty. However, uncertainties in the estimated redshift
distributions are not limited to the mean redshift, but rather include, e.g.,
the extendedness of a redshift distribution’s tail. The assumption of a prior
purely on $\Delta z^{i}$ could potentially lead to a mis-estimation of the
confidence intervals in cosmological parameters, or possibly biases in
inferred cosmological, or particularly, intrinsic alignment parameters,
especially as the statistical power of the measurement improves. Some studies
in the literature have thus chosen to account for uncertainty in the redshift
distributions differently than with such a prior on their mean. For instance,
several recent cosmic shear studies (Hildebrandt  _et al._ , 2017; Troxel  _et
al._ , 2018; Hildebrandt  _et al._ , 2020; Joudaki and Hildebrandt, 2020)
compare cosmological constraints achieved with different redshift calibration
strategies, e.g., with different redshift samples, to gauge the uncertainty.

####  VII.5.1 Full shape uncertainty

While the ensemble variance of mean redshift can be expressed as a set of
$\Delta z^{i}$ priors, it is more accurate to account for the full uncertainty
in the shape of the $n(z)$. The DES Y3 effort has developed techniques to
marginalize over the full shape uncertainty of the redshift distribution in
the likelihood analysis. The set of candidate $n(z)$ samples described in
Section III.1.1 encapsulates the full uncertainty on redshift calibration.
These samples preserve correlations between redshift bins, uncertainties on
higher-order moments of the $n(z)$, and non-Gaussianities in those systematic
errors. The methodology for an alternative to the $\Delta z^{i}$
marginalization approach, Hyperrank, is presented in Ref. (Cordero  _et al._ ,
2021). Here, a single realization of the $n(z)$ of the four bins is selected
in each likelihood evaluation. The sampling across a set of such realizations
describes the full uncertainty and preserves the correlation of $n(z)$
variations across redshift and between bins.

To avoid low Markov chain sampling efficiency by selecting a random
realization on each evaluation, Hyperrank constructs a mapping between the
$n(z)$ ensemble, characterized by a set of descriptive parameters (in this
case, the mean redshift of the three bins with the largest variance across the
realizations of the ensemble) and a multidimensional grid. Coordinates of this
grid are sampled in the likelihood analysis and the $n(z)$ realization mapped
to the closest point is used in each step.

Prior to unblinding, in a simulated analysis, the impact of accounting for the
full shape uncertainty was contrasted against marginalising over an
approximate uncertainty on the mean, as well as neglecting any redshift
uncertainty. This Y3 analysis was found to be insensitive to any redshift
uncertainty, given the high precision of the redshift calibration priors, in
tandem with the current level of statistical power of the DES Y3 data. While
this is an important step forward in the methodology for future analyses,
given the significant additional computing time required for Hyperrank, the
primary analysis uses the mean of the ensemble and the uncertainty
approximated as a shift in the mean, with the priors given in Table 2. As
validation, in Section X, the robustness of this approximation is tested.

####  VII.5.2 Impact on cosmic shear

Figure 4 illustrates the impact of the effects of sources of uncertainty in
redshift calibration on the expected cosmic shear data vector. In particular,
the predicted signals that use redshift distributions calibrated from the pure
COSMOS photometric sample, C, (blue) and the (artificially) pure spectroscopic
sample, MB, (yellow), defined in Section III.2 are compared to the fiducial
simulated data vector, computed with each $n(z)$ realisation. Their spread,
and the uncertainty on the redshift distribution is indicated by the shaded
green region with the 5th and 95th percentile denoted by the grey dashed
lines. The red line indicates the impact of the redshift-mixing effect of
blending on the redshift distribution (see Section VII.4). Differences are
small — well within the range of uncertainty allowed by the $n(z)$ ensemble.
The highest significance of differences is seen in the lowest redshift bin,
which is of relatively small importance for cosmological constraints. The size
of the effects of redshift sample choice and of accounting for the impact of
blending on the $n(z)$ are comparable. Due to the coherence of the impact
across scales and bins, investigating the impact on cosmology is still
warranted, as explored in Section X.

###  VII.6 Bayesian Inference

For parameter inference, the likelihood of the data vector, $D$, given the
model parameters, p, can be expressed as $\mathcal{L}(D|\textbf{p})$. The
latter is assumed to be a multivariate Gaussian,

$\ln{\mathcal{L}(D|\textbf{p})}=-\frac{1}{2}\sum_{ij}\Big{(}D_{i}-T_{i}(\textbf{p})\Big{)}\,\,[\mathbf{C}^{-1}]_{ij}\,\,\Big{(}D_{j}-T_{j}(\textbf{p})\Big{)}\,.$
(23)

$D_{i}$ represents the $i$th element of the data vector $\xi_{\pm}$, presented
in Section VI. It initially contains 20 angular data points each over the
combinations of 4 redshift bins and 2 correlation functions, which amounts to
227 data points after limiting the angular scales. The corresponding
theoretical prediction for the statistics, $T_{i}(\textbf{p})$, are detailed
in this section. The Bayesian posteriors of the cosmological parameters,
$\mathcal{P}(\textbf{p}|D)$, are computed as the product of the likelihood
with the priors, $P(\textbf{p})$, listed in Table 2, following Bayes’ theorem:

$\mathcal{P}(\textbf{p}|D)=\frac{P(\textbf{p})\mathcal{L}(D|\textbf{p})}{P(D)},$
(24)

where $P(D)$ is the evidence of the data.

The posterior is sampled with the Polychord sampler Handley  _et al._ (2015).
The analysis pipeline is built upon the framework CosmoSIS (Zuntz  _et al._ ,
2015) and validated through an independent implementation of the analysis
pipeline in cosmolike (Krause and Eifler, 2017; Krause  _et al._ , 2021).
CosmoSIS is a modular cosmological parameter estimation code and in order to
calculate the linear matter power spectrum $P_{\rm\delta}(k,z)$, it uses CAMB
(Howlett  _et al._ , 2012; Lewis  _et al._ , 2000). Although the fiducial
sampler settings (500 live points, tolerance 0.01) have been tested to
demonstrate the accuracy of the posteriors and Bayesian evidence estimates
(see Lemos  _et al._ , 2020), the position of the peak inferred from the
posterior samples in 28-dimensional space is noisy. Hence we use the MaxLike
minimizer after the chain has run to have a reliable estimate of the maximum a
posteriori point (MAP).

Even in the case of the baseline framework applied to a synthetic, noiseless
data vector generated from the same model, the marginalized parameter
posteriors can appear biased from the input parameter values due to parameter
volume or projection effects, which occur when parameters of interest are not
well-constrained by the data or are degenerate with other parameters that are
prior informed (see Figure 2 in Ref. (Krause  _et al._ , 2021)). Related to
projection effects, prior to unblinding, the scope of noise realisations was
studied. In particular, the possibility of bimodal astrophysical nuisance
parameter posteriors and their impact was investigated, discussed in Appendix
B. The MAP is equivalent to the best-fit (b.f.) and recovers the input
parameter values. In presence of noise, the mean provides a stable single
point estimate of any single parameter value so, following Section 4 of Ref.
(Krause  _et al._ , 2021), we report the 1D marginalized mean and its
asymmetric $\pm 34\%$ confidence intervals, together with the MAP.

###  VII.7 An ‘Optimized’ analysis

The DES Y3 approach emphasizes conservative modeling choices for robust
cosmological posteriors. In addition to the Fiducial analysis, an approach
that is optimized for $\rm\Lambda CDM$ is investigated. In particular, the
scale cuts implemented to account for small-scale baryonic physics, are
revisited to optimise the analysis post-unblinding. The Fiducial scale
selection was chosen to be robust for the joint $w$CDM 3$\times$2pt analysis
using the procedures described in Section VII.3. As a result, for cosmic
shear-only in $\rm\Lambda CDM$, this resulted in a conservative choice of
scale cuts, with potential biases in cosmological parameters inferred to be
significantly lower than the required threshold.

We consider an ‘$\rm\Lambda CDM$-Optimized’ analysis, where the scale cuts are
relaxed maintaining that potential biases for 3$\times$2pt satisfy the limit
of $\leq 0.3\sigma_{2D}$. For cosmic shear in $\rm\Lambda CDM$, this scale
selection is still conservative, with potential biases at most
$0.14\sigma_{2D}$. This scale selection is illustrated in Figure 3 by the
darker shaded region, giving 184 and 89 angular bins in $\xi_{+}$ and
$\xi_{-}$, and a total of 273 data points. Throughout this work, the
$\rm\Lambda CDM$-Optimized analysis is presented as a robust $\rm\Lambda CDM$
result alongside the Fiducial, which was used for the un-blinding and internal
consistency requirements.

##  VIII Cosmological constraints

Figure 5: Cosmological constraints on the clustering amplitude, $\sigma_{8}$,
(left) and $S_{8}$ (right) with the matter density, $\Omega_{\rm m}$ in
$\rm\Lambda CDM$. The marginalised posterior contours (inner 68% and outer 95%
confidence levels) are shown for the Fiducial DES Y3 analysis in green and
Planck 2018 CMB in yellow (Planck Collaboration, 2018). The black dashed
contours represent the $\rm\Lambda CDM$-Optimized analysis, that preserves
more small-scale information compared to the Fiducial analysis, as described
in Section VII.7.

In this section, we present constraints on cosmological parameters: the
Fiducial analysis is described in Section VIII.1 and the results of the
$\rm\Lambda CDM$-Optimized analysis (see Section VII.7) in Section VIII.2.
This analysis represents the most precise cosmic shear analysis to date, with
the highest significance measurements and correspondingly tight constraints in
the $\Omega_{\rm m}-S_{8}$ parameter space. These are compared to constraints
from the CMB measurements by Planck (Planck Collaboration, 2018) for a test of
the $\Lambda$CDM model over the history of cosmic evolution. In addition, the
posteriors are compared to those from other weak-lensing analyses, with
caution that, for this comparison, these analyses differ in many aspects and
cannot be quantitatively assessed without a homogeneous framework (Chang  _et
al._ , 2019). In (DES Collaboration, 2021), these cosmic shear results are
combined with measurements of galaxy-galaxy lensing and galaxy clustering.

Although we sample over the normalization of the matter power spectrum $A_{\rm
s}$, results are presented in terms of the commonly used $S_{8}$ parameter,
defined as $S_{8}=\sigma_{8}(\Omega_{\rm m}/0.3)^{\alpha}$, in terms of the
matter density parameter and the amplitude of fluctuations, with $\alpha=0.5$.
The constraints are quantified in terms of the 68% confidence limit, which
defines the area around the peak of the posterior within which 68% of the
probability lies and the figure of merit (FoM), to compare the relative
constraining power of results in 2D. The FoM is defined for two parameters, in
this case $\Omega_{\rm m}$ and $S_{8}$, and their covariance,
$\mathbf{C}_{\Omega_{\rm m},S_{8}}$, as (Huterer and Turner, 2001; Wang, 2008)

$\textrm{FoM}_{\Omega_{\rm m},S_{8}}=(\textrm{det}\,\mathbf{C}_{\Omega_{\rm
m},S_{8}})^{-0.5}\,,$ (25)

which is in analogy to the Dark Energy Task Force recommendation for the dark
energy FoM (Albrecht  _et al._ , 2006).

###  VIII.1 Fiducial results

The constraints are obtained by marginalising over 6 cosmological parameters
in the $\Lambda$CDM model, including a free neutrino mass density (assuming a
normal mass hierarchy), and 22 systematic and astrophysical parameters, as
summarised in Table 2. In Figure 5 we show the Fiducial DES Y3 cosmic shear
posteriors for the $\rm\Lambda CDM$ model, projected into 2D parameters
$\Omega_{\rm m},\sigma_{8}$, and $S_{8}$. These are represented by green
filled contours, denoting the 68% and 95% confidence levels.

The marginalized mean (and MAP) values of $S_{8}$, $\Omega_{\rm m}$ and
$\sigma_{8}$ and are found with 68% confidence intervals to be

$\displaystyle S_{8}$ $\displaystyle{}=0.759_{-0.025}^{+0.023}\quad(0.755)$
(26) $\displaystyle\Omega_{\rm m}$
$\displaystyle{}=0.290_{-0.063}^{+0.039}\quad(0.293)$ (27)
$\displaystyle\sigma_{8}$
$\displaystyle{}=0.783^{+0.073}_{-0.092}\quad(0.763)\,,$ (28)

constituting a 3% fractional uncertainty on $S_{8}$.

The best-fit $\rm\Lambda CDM$ prediction is over-plotted on the cosmic shear
two-point measurements in Figure 3 as a black line. A total $\chi^{2}$ of
237.7 is found for the Fiducial measurement with the $\rm\Lambda CDM$ best-fit
model. The analysis has 202 degrees of freedom (227 data points and 28 free
parameters), but when accounting for the informative priors used (Raveri and
Hu, 2019), the effective dimensionality of parameter space is reduced to
$\sim$5 and from that, 222 effective degrees of freedom, giving
$\chi^{2}/$d.o.f = 237.7 / 222 = 1.07. The probability of getting a higher
$\chi^{2}$ value can be derived assuming our data vector is drawn from a
multivariate Gaussian likelihood with our assumed covariance matrix is
precisely and fully characterized. This leads to $p$-value = 0.223. The FoM
for the analysis is found to be 927. These constraints are summarised in
Figure 7, alongside a raft of robustness tests. The parameter constraints,
goodness of fit and FoM are tabulated in Table 3. In both, we distinguish
robustness tests that are not expected to give consistent results, such as by
neglecting to account for systematics, by an asterisk and an open symbol.

In comparison, Figure 5 shows constraints from the Planck satellite CMB
temperature and polarization measurements (Planck Collaboration, 2018) in
yellow. These include the Planck measurements of the auto power spectra of
temperature $C_{\ell}^{\rm TT}$, of $E$-modes $C_{\ell}^{\rm EE}$, and their
cross-power spectra $C_{\ell}^{\rm TE}$, using the ‘Plik’ version for $\ell$
>30 in addition to the temperature and $E$-mode power spectra, $C_{\ell}^{\rm
TT}$, $C_{\ell}^{\rm EE}$ measurements in the range 2< $\ell$ <29 and using no
lensing, for a distinct high-redshift result. These are reanalysed using the
DES Fiducial cosmological priors, primarily to allow variations in
$\Omega_{\nu}h^{2}$, in order to assess consistency between the two
measurements. For the DES Y3 analysis, we compute the Bayesian Suspiciousness
(Handley and Lemos, 2019; Lemos  _et al._ , 2020), an evidence-based method
that corrects for the prior dependence of the constraints and in the full
parameter-space. As in Ref. (DES Collaboration, 2021), we conclude that two
data sets are statistically consistent if the $p$-value implied by our tension
metrics is greater than 0.01.

There has been significant discussion in the literature regarding the
consistency of low- and high-redshift cosmological probes and specifically of
cosmic shear constraints with those from measurements by the Planck satellite
CMB. We, nevertheless, find no significant evidence for disagreement with CMB,
compatible at the level of $2.30\pm 0.34\sigma$, yielding a $p$-value of $\sim
0.02$. It is notable that the constraint on the value of the $S_{8}$ parameter
is determined to be lower than that from Planck by 2.3$\sigma$. In Appendix F,
the $\Omega_{\rm m}-A_{\rm s}$ posterior is also shown, along with the other
cosmological parameters. While the DES constraint in this parameter space is
weak, it is interesting that there is no evidence for a discrepancy in this
parameter direction. A more detailed interpretation of consistency of DES
cosmic shear with external probes is given in the companion paper (Secco  _et
al._ , 2021), and in greater detail and with higher significance, Ref. (DES
Collaboration, 2021), where the cosmic shear measurements are jointly analysed
with galaxy clustering and galaxy-galaxy lensing to give tighter cosmological
constraints.

(a)  (b)

(c)

Figure 6:  Recent cosmic shear results: The constraints on $\Omega_{\rm m}$,
$\sigma_{8}$ (upper left) and $S_{8}$ (upper right) are shown for the DES Y3
$\rm\Lambda CDM$-Optimized analysis (green) and Planck 2018 CMB (Planck
Collaboration, 2018) (with free neutrino mass density, yellow). In addition,
shown are KiDS-1000 COSEBIs analysis (Asgari  _et al._ , 2021) (blue) and the
HSC $C_{\ell}$ (Hikage  _et al._ , 2019) (red, dashed) and HSC $\xi_{\pm}$
(Hamana  _et al._ , 2020) (red, solid) results. Lower: Summary of 1D
constraints, including the previous DES and KiDS results (Troxel  _et al._ ,
2018; Hildebrandt  _et al._ , 2020). The mean 1D values are indicated with
filled symbols and 68% confidence limits as horizontal bars. The $\rm\Lambda
CDM$-Optimized DES Y3 result is also indicated by the green shaded region.
Note that external weak-lensing experiments employ different analysis choices,
including the less general NLA intrinsic alignment model, and measure
alternative statistics using different scales than DES Y3, which limits the
ability to compare these results on equal footing.

The Fiducial posteriors exhibit a bimodality intrinsic alignment parameters,
that are degenerate with the $S_{8}$ parameter. In Appendix B, we discuss the
extent of the impact of noise on cosmological analyses with an investigation
of multiple noisy simulated runs.

###  VIII.2 $\rm\Lambda CDM$-Optimized results

The ‘$\rm\Lambda CDM$-Optimized’ analysis, introduced in Section VII.7, allows
for additional small-scale information to be used safely. Figures 5 and 6 show
the cosmological constraints using the $\rm\Lambda CDM$-Optimized scales as a
dotted black contour, finding substantially improved precision. Compared to
the fiducial result, it is found to be consistent within 0.5$\sigma$ in the
$S_{8}$ parameter. The 1D mean (and best-fit) values of $S_{8}$, $\Omega_{\rm
m}$ and $\sigma_{8}$ and are found with 68% confidence intervals to be

$\displaystyle S_{8}$ $\displaystyle{}=0.772^{+0.017}_{-0.017}\quad(0.774)$
(29) $\displaystyle\Omega_{\rm m}$
$\displaystyle{}=0.289^{+0.036}_{-0.056}\quad(0.279)$ (30)
$\displaystyle\sigma_{8}$
$\displaystyle{}=0.80^{+0.072}_{-0.076}\quad(0.802)\,,$ (31)

with a $\sigma_{\rm Fid}/\sigma_{\rm Opt}=1.47\times$ smaller uncertainty on
$S_{8}$, now at a level of a relative error of 2%, surpassing Planck’s
constraint on this parameter.

As is the case for the Fiducial analysis, in the full parameter-space, the
$\rm\Lambda CDM$-Optimized results find no significant evidence for
disagreement with Planck CMB data. Using the Suspiciousness statistic, we
enhanced compatibility at the level of $2.0\pm 0.4\sigma$, corresponding to a
tension probability of $p$value$\sim 0.05$. We note that while the inclusion
of smaller scales does result in a smaller uncertainty on $S_{8}$, the mean
does shift slightly towards higher values, such that a $2.1\sigma$-level
difference with Planck is preserved. Therefore, we observe that the moderate
$\sim 2.3\sigma$ tension observed in the main analysis reduces to $\sim
2.0\sigma$ when the additional small-scale data is used. While the evidence is
inconclusive, it is interesting that this behaviour of the tension weakening
with the inclusion of more data is consistent with the hypothesis of a
statistical fluctuation, and the consistency with Planck. The $\chi^{2}$/d.o.f
= 285/268.2 = 1.06 which gives a $p$-value = 0.22. The FoM for the analysis is
found to be 1362, as quoted in Table 3.

The gain in $S_{8}$ constraining power and small shift can be, in part,
attributed to eliminating the most extreme lobes of the Fiducial intrinsic
alignment $a_{1}-a_{2}$ posterior that is degenerate with $S_{8}$ (see
Appendix B for further discussion). In terms of $\Omega_{\rm m}$ and
$\sigma_{8}$, the small scales now included contribute a 7% and 3% improvement
in precision, respectively. In Section XII, we assess the limiting systematics
for this $\rm\Lambda CDM$-Optimized analysis.

In Figure 6, the $\rm\Lambda CDM$-Optimized $S_{8}$, $\Omega_{\rm m}$ and
$\sigma_{8}$ results are shown alongside the most recent COSEBIs cosmic shear
constraints from the KiDS-1000 survey (Asgari  _et al._ , 2021) shown in blue
and the HSC-Y1 constraints from their Fourier-space analysis in red (Hikage
_et al._ , 2019). While plotted for illustrative purposes, we caution the
reader against direct comparisons of cosmological parameters as the priors,
measurement statistics, scales allowed and other analysis choices such as the
intrinsic alignment modeling adopted by other surveys vary and can lead to
different conclusions (Chang  _et al._ , 2019). Nevertheless, it is evident
that in recent years, all cosmic shear analyses find a lower value of $S_{8}$
than the inferred Planck 2018 constraint under a flat $\rm\Lambda CDM$ model,
although with varying levels of significance. A summary of the most recent
$S_{8}$ result from each ‘Stage 3’ weak-lensing survey is shown in Figure 6. A
more quantitative comparison using a unified set of analysis choices is left
to future, collaborative work.

Figure 7:  Summary of marginalised 1D constraints on $S_{8}$ in $\rm\Lambda
CDM$, testing various choices. The mean of the $S_{8}$ marginalized posterior
is indicated by the symbol and 68% confidence intervals are shown as
horizontal bars. Empty symbols represent those analyses which are not
necessarily expected to agree with the Fiducial result. For the Fiducial
analysis, this is also represented as the green shaded region. Tests 1-14