validate the robustness against observational systematics while Tests 15-20
investigate internal consistency of the measurements, splitting the Y3 data
into subsets, as well as comparing to the DES Y1 data in Tests 21-23. We
distinguish variations on the baseline setup that are not necessarily required
to give consistent results (e.g., by neglecting observational systematics) by
an asterisk and an open symbol. We show the CMB Planck 2018 (Planck
Collaboration, 2018) (high $\ell$ TT+EE+TE + low $\ell$ TT+EE) constraint in
yellow, reanalysed using DES cosmological priors, with free neutrino mass
density. The numerical parameter values are listed in Table 3.

##  IX Internal consistency

In this section, we investigate the consistency of cosmic shear cosmological
results in both data space and parameter space. To do so, we analyse the data
when excluding particular subsets. We test the impact of each tomographic
redshift bin by excluding them one at a time from the data vector; we test the
consistency between the small and large angular scale measurements; and we
assess the $\xi_{-}$ measurements compared to the $\xi_{+}$ measurements.
Finally, we demonstrate the consistency between the Y3 results and previous
DES cosmic shear analyses.

Figure 8: Internal consistency: comparing Fiducial $\rm\Lambda CDM$
constraints in the $\Omega_{\rm m}-S_{8}$ plane from subsets of the data. The
filled green contours show the Fiducial analysis. On the left, redshift bins
from low- and high-redshift are excluded to test the sensitivity of the
results to redshift-dependent observational and astrophysical systematics. In
the centre, subsets of the data vector are excluded to demonstrate robustness
against angular scale-dependent systematics, with the red contour using
$\theta>\theta_{\rm lim}^{+}=30$ arcmin and $\theta>\theta_{\rm lim}^{-}=100$
arcmin for $\xi_{+}$ and $\xi_{-}$, respectively, and the yellow contour using
the small scales, divided at the same $\theta_{\rm lim}$. On the right, we
compare the posteriors from $\xi_{-}$-only (red) and $\xi_{+}$-only (yellow)
data.

###  IX.1 Internal consistency methodology

For tests in data space, the Posterior Predictive Distribution (PPD)
methodology detailed in Ref. (Doux  _et al._ , 2020) is used. We consider two
types of internal consistency tests: goodness of fit tests, and consistency
tests. Given two subsets $\mathbf{d}$ and $\mathbf{d}^{\prime}$ of the full
data vector, the distribution $P(\mathbf{d}^{\prime}_{\rm sim}|\mathbf{d}_{\rm
obs})$ of simulated realizations of the latter is considered, given
observations for the subset of the former, $\mathbf{d}_{\rm obs}$. This
distribution can be sampled and those samples can be compared to actual
observations of $\mathbf{d}^{\prime}$, denoted $\mathbf{d}^{\prime}_{\rm
obs}$. This comparison provides a meaningful way for evaluating the
consistency of $\mathbf{d}^{\prime}$ with $\mathbf{d}$, given a single model.
To sample the distribution, a sample from the posterior of model parameters is
first obtained given observations for $\mathbf{d}$,
$P(\textbf{p}|\mathbf{d}_{\rm obs})$. Then, for each such p, the distribution
$P(\mathbf{d}^{\prime}_{\rm sim}|\mathbf{d}_{\rm obs},\textbf{p})$ is sampled.
Two cases are considered. In goodness-of-fit tests, $\mathbf{d}$ and
$\mathbf{d}^{\prime}$ refer to independent realizations of the same subset,
such that $P(\mathbf{d}^{\prime}_{\rm sim}|\mathbf{d}_{\rm
obs},\textbf{p})=P(\mathbf{d}^{\prime}_{\rm sim}|\textbf{p})$ reduces to the
likelihood of $\mathbf{d}^{\prime}_{\rm sim}$ at parameters p. For consistency
tests, $\mathbf{d}$ and $\mathbf{d}^{\prime}$ refer to disjoint subsets of a
single realization of the full data vector. In this case,
$P(\mathbf{d}^{\prime}_{\rm sim}|\mathbf{d}_{\rm obs},\textbf{p})$ is the
conditional likelihood, that is, the distribution of $\mathbf{d}^{\prime}_{\rm
sim}$ at parameters p given observations for $\mathbf{d}_{\rm obs}$. Here a
multivariate Gaussian likelihood is assumed, so the conditional likelihood is
also multivariate Gaussian, with shifted mean and covariance accounting for
the correlation between the subsets $\mathbf{d}$ and $\mathbf{d}^{\prime}$
(see Doux  _et al._ , 2020, for details). To perform the comparison of the
results, a statistic is defined to compare PPD realizations and observations
of $\mathbf{d}^{\prime}$, integrated over parameter space. For this,
$\chi^{2}(\mathbf{d}^{\prime},\textbf{p})$ is used, defined with respect to
the model at parameters p. This statistic determines the statistical
significance of consistency through a $p$-value, which is given by the
fraction of parameter samples p for which $\chi^{2}(\mathbf{d}^{\prime}_{\rm
sim},\textbf{p})>\chi^{2}(\mathbf{d}^{\prime}_{\rm obs},\textbf{p})$.

As demonstrated in (Doux  _et al._ , 2020), this choice of test statistic may
yield overly conservative $p$-values, especially in the case where parameter
posteriors from $\mathbf{d}$ and $\mathbf{d}^{\prime}$ are very different. For
this reason, a calibration procedure is followed as outlined there, based on
repeated tests for simulated data vectors consistently drawn at the fiducial
cosmology. A threshold of $\tilde{p}>0.01$ is defined as consistency, where
$\tilde{p}$ is the calibrated $p$-value. Note that the calibration has little
impact for the particular set of tests presented here.

An alternative to data-space consistency metrics like the PPD is to consider
parameter-space consistency metrics. In parameter space, posteriors obtained
with the full data vector are compared to those obtained with subsets are
shown and parameter shifts are reported. It is important to note that these
shifts should be treated with caution: we do not expect identical constraints
as the subsets of the data, and therefore their posterior distributions, are,
on the one hand, correlated, and on the other, subject to noise.

###  IX.2 Tomographic redshift bins

In this section, we investigate the impact of removing individual redshift
bins from the analysis, as well as the two low and high redshift bins jointly.
For the analysis with ‘Bin 1 removed’, we remove all auto- and cross-
correlations involving redshift bin 1 (that is, 1-1, 1-2, 1-3, 1-4), and for
other cases correspondingly. This test captures potential inconsistencies
across redshift bins, which may arise both due to deviations from $\rm\Lambda
CDM$ as well as redshift-dependent systematics. Deviations from the model, in
particular from the non-linear power spectrum (e.g. the effect of baryons) and
intrinsic alignments, preferentially impact the low-redshift bins, which have
the lowest cosmological signal-to-noise. Thus, these consistency checks
complement the tests of model robustness in the companion paper (Secco  _et
al._ , 2021). On the other hand, the sources of uncertainty in the redshift
distributions described in Section III.1.1, such as the uncertainty in the
photometric calibration or in the choice of redshift sample, are largest for
the lowest and highest redshift bins, respectively. Furthermore, observational
shear systematics, such as blending or PSF uncertainty, may have a larger
impact on higher redshift bins with predominantly fainter and smaller
galaxies. The PPD test, in particular, addresses the question of the
consistency of observations in one redshift bin with respect to predictions
given observations in other redshift bins (and our modeling of $\rm\Lambda
CDM$).

Removing any one redshift bin at a time is found to produce consistent
constraints within $\sim$0.3$\sigma$ in the $S_{8}$ parameter. For each of
these configurations, we compare data for the redshift bin that was removed,
with PPD realizations for predictions of that bin by the others. We find each
subset to pass the chosen threshold, with $\tilde{p}$-values of
$\tilde{p}_{1-4}=[0.357,0.394,0.014,0.427]$. These are discussed in Appendix
C. Only bin 3 is found to yield a $\tilde{p}$-value close to our consistency
threshold. It is notable that firstly, this bin has the smallest reported
uncertainty on the mean redshift and secondly, when discarded from the
analysis, we observe the most significant change of the intrinsic alignment
$a_{1}-a_{2}$ parameter posterior, then centered on zero, as shown in Figure
16.

We inspect the consistency when both redshift bins 1 and 2 are removed from
the analysis (Test 16 ‘High-$z$’), or both bins 3 and 4 (Test 15 ‘Low-$z$’).
Figure 8 (left panel) overlays constraints derived from the Fiducial analysis
(green) with those obtained when using only the two low-$z$ (yellow) or two
high-$z$ (red) bins. We find consistency within $\sim 1\sigma$, with the high-
redshift variant preferring a lower value of $S_{8}$ by 0.8$\sigma$. When
repeating PPD tests removing either half, we find that predictions of the low-
redshift bins derived from high-redshift observations over-predict the data,
albeit with a large uncertainty pertaining to the loss of constraining power
on the intrinsic alignment parameters. We obtain $\tilde{p}$-values of 0.993
and 0.207, showing general good agreement in data space.

It is difficult to predict, either from first principles or empirically, how
the intrinsic alignment contamination in DES Y3 should evolve with redshift
(see e.g. Joachimi and Bridle, 2010). Intrinsic alignments are known to depend
significantly on luminosity and color, and therefore, redshift (see the
discussion in Troxel  _et al._ , 2018; Samuroff  _et al._ , 2019). In this
analysis, a power law redshift scaling is adopted to capture this effect. In
practice, therefore, the intrinsic alignment constraint requires accurate
knowledge of the redshift distributions and their errors. In order to ensure
that systematic errors in the data are not absorbed by the intrinsic alignment
model, we investigate the consistency of astrophysical parameters, discussed
in Appendix C (Efstathiou and Lemos, 2018).

###  IX.3 Angular ranges

In the Fiducial analysis, any scales from the $\xi_{\pm}$ data vector that
have a fractional contribution from baryonic effects, as predicted by the OWLS
‘AGN’ simulation, that exceeds $\Delta\chi^{2}<0.5$. This is summarised in
Section VII, and detailed in Ref. (Krause  _et al._ , 2021). The procedure
removes a large number of data points at small scales, particularly in
$\xi_{-}$ where the impact of baryonic physics is larger. On the other hand,
large scales might be sensitive to some unaccounted for additive systematics,
as described in Section VII.4.

Here, we inspect the consistency of the angular small scale and large scale
contributions to the data vector, by comparing their respective cosmological
constraints. As the analysis does not marginalise over baryonic feedback, and
as we subtract a bias due to residual mean shear in the signal, this test
ensures the robustness of the scale cuts applied. The split is chosen to be at
$\theta=$30\text{\,}\mathrm{\SIUnitSymbolArcminute}$$ for $\xi^{+}$ and
$\theta=$100\text{\,}\mathrm{\SIUnitSymbolArcminute}$$ for $\xi^{-}$, to
roughly balance the constraining power between the subsets. Results are shown
in the $\Omega_{\rm m}-S_{8}$ plane in Figure 8, central panel, with the
Fiducial analysis as the green outline, along with results from the smaller
(yellow) and larger (red) scale selection. We find consistent results with
respect to the Fiducial analysis, within $\sim 0.5\sigma$ in the $S_{8}$
parameter. The two PPD tests comparing large- and small-scale data pass, with
$\tilde{p}$-values of 0.66 when comparing data at large scales with
predictions from small-scale data, and $\tilde{p}$=0.083 for the opposite
test, as reported in Table 5.

An alternative approach to account for small-scale baryonic feedback effects
is to preserve the small-scale information and marginalise over any model
uncertainty with HMcode (Mead  _et al._ , 2015). To this end, an approach is
investigated in (Secco  _et al._ , 2021) that uses this modeling and maintains
the Fiducial scale cuts. This finds consistency within $1\sigma$, with a small
preference for higher values of $S_{8}$ (compared to the slightly lower
$S_{8}$ preferred by the large scales-only analysis) that is attributed to
prior volume effects with unconstrained baryon nuisance parameters. Overall,
given the conservative choice of scale selection and these complementary
robustness tests, we deem the Fiducial constraints robust to small-scale
systematics.

\-

Figure 9:  Comparison to the DES Y1 posteriors: The $\rm\Lambda CDM$
constraints in the $\Omega_{\rm m}-S_{8}$ plane for the Y3 $\rm\Lambda
CDM$-Optimized (green) are compared to DES Y1, as published in Ref. (Troxel
_et al._ , 2018) (blue). For a more robust comparison, Y1 reanalysed with the
Y3 modeling choices is shown in red (in particular, the intrinsic alignment
model switching from NLA to TATT). Within this homogeneous modeling framework,
we find the Y3 $\rm\Lambda CDM$-Optimized constraints to be consistent within
$\sim 0.5\sigma$ and substantially more precise than the Y1 case, with
$2\times$, $1.24\times$ and $1.22\times$ smaller uncertainty on $S_{8}$,
$\Omega_{\rm m}$ and $\sigma_{8}$, respectively. In yellow, the Y3 modeling
choices are again used, as well as the better-informed Y3 shear calibration
correction. The Y3 $\rm\Lambda CDM$-Optimized $S_{8}$ result, when compared to
this, is $2.2\times$ more constraining and consistent with those from Y1
within 0.25$\sigma$ in $S_{8}$.

###  IX.4 $\xi_{+}$ versus $\xi_{-}$

We test the consistency of the two components of the cosmic shear two-point
function, $\xi_{+}$ and $\xi_{-}$. The motivation for this split is that
$\xi_{-}$ is more sensitive to smaller scales than $\xi_{+}$, and is therefore
more likely to be impacted by any unaccounted for baryonic feedback and
intrinsic alignments. In addition, residual B-modes, sourced by either
observational or astrophysical systematics, or higher-order shear effects, are
more likely to be prominent on small-scales. Considering $\xi_{+}$
measurements conditioned on $\xi_{-}$ observations with a PPD test, we find a
$\tilde{p}$-value of 0.262, and 0.451 for the opposite test. When considering
the $S_{8}$ parameter, we find consistency within $\sim 0.5\sigma$, with
posteriors from these subsets of the data shown in Figure 8 (right panel).

Figure 10:  Robustness to redshift calibration for the $\rm\Lambda
CDM$-Optimized analysis: A comparison of $\rm\Lambda CDM$ constraints in the
$\Omega_{\rm m}-S_{8}$ plane for alternative redshift choices. In all panels,
the filled green contours show the optimised analysis. On the left, it is
shown in comparison to ‘pure’ redshift calibration samples, C (red), derived
purely from COSMOS and MB (yellow), the ‘maximally biased’ spectroscopic
sample, showing the (in)sensitivity of the analysis to spectroscopic vs.
photometric training samples. In the middle, the tiers of the methodology:
SOMPZ, WZ and SR, are stepped back, showing the consistency of the likelihoods
from each method. For this analysis, the SR information is effective, finding
25% tighter $S_{8}$ constraints, with the improvement primarily attributed to
the breaking of bimodal intrinsic alignment posteriors, which are degenerate
with $S_{8}$, as well as constraining the redshift uncertainty. On the right,
the fiducial modeling of the uncertainty in the redshift calibration that
accounts for only shifts in the mean of each redshift bin, $\Delta z$, is
compared to an analysis where the full-shape uncertainty is accounted for by
varying $n(z)$ realisations with Hyperrank (red). A constraint that neglects
any redshift calibration correction is shown in yellow.

###  IX.5 Previous DES results

In this section we investigate the consistency of the DES Y3 constraints with
DES Y1 cosmic shear results from (Troxel  _et al._ , 2018). In Figure 9, the
$\Omega_{\rm m}-S_{8}$ constraints are overlaid. Overall, we find that our
results are in agreement with the Y1 constraint to $0.5\sigma$ in $S_{8}$. We
caution that a comparison of the two results is not straightforward, as
analysis choices differ. More specifically, the Y1 analysis used the simpler
NLA intrinsic alignment model, rather than TATT, the baseline choice for the
Y3 analysis. Furthermore, as the two shape catalogs share a non-negligible
fraction of imaging data, the correlation between DES Y3 and Y1 is difficult
to quantify, which limits a real assessment of the statistical significance of
any deviation.

For a comparison that is more on equal footing, we re-analyze the DES Y1
measurements with the DES Y3 analysis pipeline, preserving the DES Y1
observational systematic choices and priors (red). This results in
significantly degraded constraints and an, albeit not significantly, lower
$S_{8}$ posterior compared to the published Y1 result (blue). This is
primarily attributed to the change in the intrinsic alignment model from NLA
to TATT, with a similar effect seen in Figure 15 of Ref. (Troxel  _et al._ ,
2018). Within this homogeneous modeling framework, we find the Y3 $\rm\Lambda
CDM$-Optimized constraint to be consistent within $\sim 0.5\sigma$ and
substantially more precise than the Y1 case, with $2\times$, $1.24\times$ and
$1.22\times$ smaller uncertainty on $S_{8}$, $\Omega_{\rm m}$ and
$\sigma_{8}$, respectively. Furthermore, the Y1 data vector is re-analyzed
with the Y3 shear calibration priors to give the shaded blue posteriors. The
significant improvements in the realism of the image simulations for Y3, as
well as more sophisticated understanding and modeling of the effects of
blending, suggest that the Y3 calibration is likely to be more accurate for
both the Y3 and the Y1 data set. These Y3 multiplicative corrections result in
a shift toward higher values of $S_{8}$. Compared to this Y1 re-analyzed
constraint, the $\rm\Lambda CDM$-Optimized Y3 cosmic shear result is
consistent within $\sim 0.25\sigma$ in $S_{8}$, with a factor of $2.2\times$
improvement in the precision on $S_{8}$.

##  X Robustness to redshift calibration

Determination of the true ensemble redshift distribution is critical for
cosmological weak lensing analyses. To lowest order, lensing is primarily
sensitive to the mean redshift and the width of the redshift distribution of
each bin (Amara and Réfrégier, 2007).

In this section we show how different choices for the redshift distribution
impact the results. In particular, we demonstrate the robustness to (A) the
choice of redshift sample, (B) the redshift methodology, and (C) the modeling
of redshift uncertainties. These tests are shown for the higher precision
$\rm\Lambda CDM$-Optimized analysis in Figure 10. Corresponding tests for the
Fiducial analysis are summarised in Table 3 and Figure 7 as Tests 1-8.

###  X.1 Redshift sample

In Section III.2, we summarise the choice of redshift samples that form the
basis of the ensemble $n(z)$, which we consider to be complete and to span any
photometric-spectrosopic differences. However, it can be argued that this
framework underestimates the uncertainty as it does not span the extremities
of available redshift information: neither an $n(z)$ derived solely from
COSMOS , nor one solely from spectroscopic information are included. In this
section, we test the sensitivity of our analysis to these extremities by
analysing the data with an $n(z)$ that is purely based on COSMOS , $C$. On the
other hand, since a solely spectroscopic-based $n(z)$ cannot be directly
calibrated for DES data without substantial selection biases, we use an
artificially-constructed training sample. To this end, we modify the redshift
sample that is least reliant on COSMOS redshifts, such that the 10% of
weighted information that does still derive from this sample is manipulated to
reflect the COSMOS -spectroscopic biases calibrated per magnitude bin. This MB
sample can be deemed as most different to $C$.

As a robustness test, we substitute $n(z)$ with the ‘pure’ redshift samples.
Figure 10 (left panel) shows results in the $\Omega_{\rm m}-S_{8}$ plane. The
two extreme choices of redshift calibration sample, used in the analysis of
the red and yellow contours, show that no significant difference in
cosmological parameters. Even the fractional shift compared to the baseline
case in this plot are due to minor differences in processing between the
samples. The figure here uses the most constraining, $\Lambda$CDM-Optimized
scales. A comparison for the case of the less-constraining Fiducial analysis
is shown in Tests 1 and 2 in Figure 7 and Table 3).

The test indicates that the DES Y3 redshift methodology of using the multi-
band deep information, building complete redshift samples, and accounting for
an uncertainty due to the choice of redshift sample, yield cosmological
constraints robust to this choice, even at the $\rm\Lambda CDM$-Optimized
precision. The differences are substantially smaller than previous analyses
had indicated for the COSMOS -spectroscopic calibration (Hildebrandt  _et al._
, 2020; Joudaki and Hildebrandt, 2020). We note that this is consistent with
tests done on the DES Y1 calibration of (Hoyle  _et al._ , 2018) post-
unblinding that avoid selection biases, including those in (Joudaki and
Hildebrandt, 2020, their appendix A).

###  X.2 Redshift method

The redshift estimation for the DES Y3 analysis uses a combination of
approaches. As described in Section III.1, the information is drawn jointly
from the flux/color self-organising map-based method, SOMPZ, the clustering
redshift method, WZ, and the shear-ratio method, SR. First, it is shown in
(Myles  _et al._ , 2020) that the posteriors on $\langle z\rangle$ from these
methods are consistent, before their information is combined at different
points in the analysis pipeline, with the SR incorporated at the point of
evaluation of cosmological likelihoods. For robustness, we test the impact of
including each of these methods on the cosmological parameter level
separately.

The central panel of Figure 10 shows the $\rm\Lambda CDM$-Optimized analysis
using the $n(z)$ created by SOMPZ (yellow) and that informed by SOMPZ+WZ
jointly (red), compared to the $\rm\Lambda CDM$-Optimized result, which
combines SOMPZ+WZ+SR. We find that the $S_{8}$ constraints do not shift when
each method is included and are consistent, and thus that our results are
robust to these variants in the methodology. The analogous test for the
Fiducial analysis is shown in Tests 3 and 4 in Figure 7 and Table 3).

In support of that, we show the posteriors on the $\Delta z$ systematic
parameters for these variant analyses in Appendix D. The WZ analysis requires
marginalizing over flexible models of the redshift evolution of clustering
bias of the weak lensing sources, which are largely degenerate with $\langle
z\rangle$ (Gatti  _et al._ , 2020; van den Busch  _et al._ , 2020). As a
result, the WZ primarily constrains the shape of the redshift distribution,
rather than the mean, and has relatively little impact on cosmological
posteriors. For this analysis, the SR information is effective, finding 25%
tighter constraints with the improvement primarily attributed to the breaking
of degeneracies in the TATT intrinsic alignment model posteriors, which are
bimodal (see Appendix B), rather than a substantially tightened posterior on
redshift distributions Sánchez  _et al._ (2021). Indeed, as discussed in that
appendix, when a less-conservative model for intrinsic alignments is used, our
cosmological posteriors with and without the inclusion of SR remain
consistent. In that test, while the $a_{1}$ posterior is significantly tighter
with the inclusion of SR, as that parameter is de-correlated with $S_{8}$, the
impact on the $S_{8}$ constraint is negligible. In addition, Appendix D
demonstrates the robustness of the posteriors to choices in the SR method.
More specifically, it compares to an analysis that uses the alternative lens
galaxy sample, as well as one that uses SR measured from large scales only.

###  X.3 Modeling redshift uncertainty

For the Fiducial and $\rm\Lambda CDM$-Optimized analyses, the uncertainty in
the redshift estimation is incorporated as an uncorrelated shift in the mean
redshift of each bin. The latter is computed as the spread spanned by the
ensemble of $n(z)$ realisations that result from the DES redshift calibration
and image simulation studies and is modeled as a Gaussian prior on $\Delta z$.
As discussed in Section VII.5, while the DES Y3 methodology is capable of
sampling the full realisations in the likelihood analysis with Hyperrank,
thereby capturing variations in the shape of the $n(z)$ as well as
correlations between redshift bins, we do not include that more accurate
framework in the baseline analysis. As this modeling choice was made based on
a weighing of the impact of this approximation on posteriors in a simulated
analysis, versus increased run-time, we test the robustness of the decision
here. We analyse the data using the full ensemble of realisations and
Hyperrank. To assess the overall impact of the redshift uncertainty on the
cosmological constraints, we perform an analysis ignoring these nuisance
parameters, using only the mean of the $n(z)$ realisations and setting $\Delta
z=0$ for all bins.

Figure 10 (right-hand panel) demonstrates consistency in the $\Omega_{\rm
m}-S_{8}$ plane for the $\rm\Lambda CDM$-Optimized case: modeling only the
mean of the redshift distribution (green) sufficiently captures the effect of
redshift bias uncertainty, as analysed with Hyperrank (yellow). The full-shape
constraint is only marginally degraded, and is unbiased, illustrating that
uncertainties in the shape of the redshift distribution are sub-dominant for
cosmic shear at the current statistical precision. Furthermore, it shows that
in this plane the $n(z)$ are calibrated to sufficient accuracy and precision,
as cosmological constraints are not significantly impacted ($\sim 0.3\sigma$)
by not marginalizing over redshift calibration (red). The analogous test for
the Fiducial analysis is shown in Tests 7,8 in Figure 7 and Table 3. In
addition, we demonstrate the validity of the claim of consistency when
considering the intrinsic alignment parameters and the effective redshift
parameter constraints in Appendix D.

##  XI Robustness to blending and shear calibration

The amplitude of the cosmic shear signal and thus the inferred $S_{8}$
parameter depends directly on the multiplicative shear calibration. Accurate
shear calibration relies on highly realistic image simulations that
sufficiently match the properties of the data. Ref. (MacCrann  _et al._ ,
2020) finds blending to be the dominant contribution to the mean
multiplicative bias of Y3 shape catalogs, at approximately $-2$%, and finds
that the magnitude of this correction increases with redshift. We infer that
in the presence of object blending, a systematic that is more prominent in
deeper data, it is important that multi-band simulations allow for the
redshift analysis applied to the data to be repeated on simulations in order
to capture the coupled effects on both shear and photometric redshift
calibration. This cosmic shear analysis is the first to account for the effect
of blending jointly on shear and redshift calibration. In this section we
illustrate how these advancements in the methodology impact the cosmology
constraints.

Figure 11:  Robustness to blending: A comparison of $\rm\Lambda CDM$
constraints in the $S_{8}-\Omega_{\rm m}$ plane for varying complexity in
shear calibration modeling. The shaded green contours show the $\rm\Lambda
CDM$-Optimized analysis. Also shown are variants that neglect the redshift-
dependent impact of blending (red), that ignores any shear calibration
systematics (yellow), as well as one that uses the full blending treatment,
including correlations between shear and full-shape redshift calibration,
analysed with Hyperrank (blue).

Figure 11 shows a variant of the $\rm\Lambda CDM$-Optimized analysis that
ignores the redshift-mixing blending impact, ‘m-only: no n(z)-blending
correction’ (yellow), by neglecting to account for the correction on the
ensemble and mean $n(z)$. This analysis uses the uncorrected $n(z)$ and the
priors derived for each redshift bin from the redshift-independent constant
shear simulation, $\Delta z_{\rm noz-blend}$ and $m_{\rm noz-blend}$, quoted
in Table 1. Comparing this variant to the $\rm\Lambda CDM$-Optimized analysis
(green) in the $\Omega_{\rm m}-S_{8}$ plane, we find consistency. The
redshift-mixing effect is subdominant for the DES Y3 cosmic shear cosmology.
On the other hand, neglecting to account for shear calibration uncertainty
entirely (red) incurs a significant bias ($0.5\sigma$) toward a low $S_{8}$.

Uncertainties on shear and redshift calibration are correlated due to the
limited volume of image simulations that impact both. The baseline analysis
ignores any such correlations. In order to test the robustness of this
assumption, an analysis is performed that uses Hyperrank to sample over
$n_{\gamma}(z)$ realisations, which naturally include both multiplicative
bias-type and $n(z)$-type biases, labelled ‘full blending treatment’. The
cosmological posteriors from this variant, Test 11, are compared to those in
Figure 11, and are shown to be consistent.

Finally, we test the impact on the cosmological analysis of an additional,
unaccounted for shear calibration uncertainty. While the image simulations
used to inform these choices are well-matched to the data, they do not account
for some effects such as clustering with undetected sources, which can
contribute additional blending effects. We assume this to take the form of a
1% effect that is fully correlated across redshift bins. This variant is found
to produce results that are totally consistent with the Fiducial analysis
within, indicated by Test 10 in Figure 7.

##  XII What limits lensing cosmological precision?

The cosmic-shear measurements are known to be altered from simple theoretical
predictions by various systematic effects. The importance of work to mitigate
systematics is two-fold: first, these must be calibrated accurately as to not
bias cosmological parameters, but second, they must be sufficiently controlled
such that they do not dominate the error budget, thereby limiting precision.
Of interest for this and future experiments are the questions: how much has
the cosmological accuracy been degraded by treatment of systematic effects?
Which effects dominate the uncertainty? How much could lower-noise
measurements improve cosmological accuracy in the presence of these systematic
effects?

Our baseline analysis marginalizes over 19 systematic parameters in addition
to the 6 cosmological parameters. Of course, this marginalization is necessary
— ignoring systematic effects, as is done in the Tests 13 and 14 of Figure 7,
will produce biased inferences. In this investigation, we are not interested
in the incurred biases. As such, we plot posterior distributions centered at
zero to understand how ‘shutting off’ the effects changes the size of
posterior uncertainties in the $S_{8}-\Omega_{\rm m}$ plane, as shown in
Figure 12.

Figure 12:  Systematics limiting cosmology: A comparison of the size of
$\rm\Lambda CDM$-Optimized constraints in filled green contours to those where
observational systematics, both shear and redshift priors, are fixed (yellow,
dashed) and theoretical systematics are ‘switched off’, by fixing intrinsic
alignment parameters and including using the full scale-set that is limited to
mitigate baryonic effects (red, solid). The analysis is not limited by
observational systematics, but we find an improvement in of $\times$1.8 in
$S_{8}$ uncertainty when fixing theoretical systematics. The constraint fixing
all systematics (and therefore also using the full scale-set) is shown in
blue. The mean is individually subtracted from every constraint for ease of
comparing the constraining power. Both 68% and 95% confidence levels are
shown.

We divide the systematic effects into two classes. First, we consider
observational systematics, namely the redshift-distribution and shear-
measurement calibrations investigated in detail in this paper. The
uncertainties in these calibrations are propagated by marginalizing over
relevant nuisance parameters, which expands the posterior distribution on
cosmological parameters of interest—we will focus here on $S_{8}$. The impact
of observational systematics on the fiducial $S_{8}$ constraint can be
illustrated by comparing to the case where the relevant nuisance parameters
are fixed. We will refer to this case as ‘Fixed observational systematics’.

The second class is astrophysical or theoretical systematics, which arise from
our incomplete knowledge of non-linear processes in the Universe. This
includes intrinsic alignments, baryonic physics, and growth of structure in
the nonlinear regime, which are investigated in the companion paper (Secco
_et al._ , 2021). The impact of marginalising over the intrinsic alignment
model on the posterior uncertainty can be tested by fixing its parameters. The
baryonic/non-linear effects are ameliorated in the analysis by imposing scale
cuts on the data vector to retain only elements for which the model is
believed to be secure. In other words, the intrinsic precision of the data is
degraded in order to maintain the accuracy of the model. To investigate the
loss of constraining power from this, we produce posteriors that use the full
$\xi_{\pm}$ data vector. We have already seen that moving from Fiducial to
$\rm\Lambda CDM$-Optimized scale selections reduces uncertainties on $S_{8}$
by $\approx 1.5\times.$ We will refer to the combination of fixing intrinsic
alignment parameters and expanding the $\xi_{\pm}$ data vector to all measured
scales as ‘Fixed theory systematics.’

From the starting point of the $\rm\Lambda CDM$-Optimized analysis, we find
that fixing observational systematics (yellow, dashed) produces no significant
improvement in the $S_{8}$ posterior uncertainty. Thus, the current
uncertainties in shear and redshift calibration are small enough that they do
not degrade the results and we are not limited by observational systematics.
On the other hand, fixing only the theoretical systematics (red) yields a
substantially narrower $S_{8}$ constraint, by roughly a factor $\sigma_{\rm
opt}/\sigma_{\rm fixth.}=1.8$. Therefore, the theoretical uncertainties are
costing about 2/3 of the Y3 power (in terms of variance) on the cosmological
parameters, with intrinsic alignment uncertainty contributing slightly more
than baryon-driven scale cuts in limiting the cosmological precision. This
suggests that future decreases in measurement noise in $\xi_{\pm}$ may not
lead to concomitant decreases in cosmological uncertainties. The precise
balance will depend upon how many more modes have to be sacrificed to keep
modelling accuracy below measurement noise, and on how well the intrinsic
alignment models can self-calibrate from data. We note that this is a
different scenario to the KiDS-1000 analysis, which is found to be statistics-
limited (Asgari  _et al._ , 2021). That analysis uses the simpler and more
constraining NLA intrinsic alignment model, COSEBIs measurements, including
small angular scales, and marginalises over non-linear evolution.

Fixing both observational and theoretical systematic errors leads to the blue
posteriors in Figure 12. A small improvement is seen over the fixed-theory
result ($\times 1.9$ versus $\times 1.8$ improvement, compared to the
$\rm\Lambda CDM$-Optimized case), i.e. the current state of knowledge on
observational errors would suffice for a DES Y3 cosmic-shear measurement with
negligible theory errors. The work on DES observational systematics, presented
in a series of papers (Everett  _et al._ , 2020; Hartley  _et al._ , 2020;
Myles  _et al._ , 2020; Gatti  _et al._ , 2020; Sánchez  _et al._ , 2021;
MacCrann  _et al._ , 2020), has been successful in avoiding a loss of
constraining power in cosmic shear. This does not imply that innovation is not
required for a similar statement to hold for any future cosmic shear analyses.
In particular, in DES Y3, uncertainties related to, for example, the redshift
calibration sample (Myles  _et al._ , 2020), or blending (MacCrann  _et al._ ,
2020) increased with redshift bin, highlighting that deeper data will require
innovation for observational systematics to stay sub-dominant. For the case of
lower measurement noise, forthcoming with DES Y6 and Rubin Observatory LSST,
redshift and shear calibration may become significant contributors and require
improvement.

More crucial to extracting maximal information out of weak-lensing data is an
improved modeling of astrophysical effects. Progress on this systematic is
substantially inhibited by the large variation demonstrated in the different
hydro-simulations (Chisari  _et al._ , 2018) for the degree in which baryon
feedback impacts the matter power spectrum. While there are several options
for attempting to model these small-scale effects (Mead  _et al._ , 2015;
Huang  _et al._ , 2020; Asgari  _et al._ , 2020), their current level of
uncertainty demands more refined hydrodynamic simulations, tailored by
complementary observations such as the Sunyaev-Zel’dovich effect. It is
noteworthy, however, that pushing to smaller scales will also require more
accurate modeling of the PSF on these scales and accurate modeling of higher-
order lensing corrections (MacCrann and DES Collaboration, 2017). Important
too, may be the need for a principled, yet not overly conservative, modeling
of intrinsic alignments. An intrinsic alignment model choice investigation,
similar to performed a posteriori in Ref. (Secco  _et al._ , 2021), which
finds that simpler, less-conservative models provide a sufficient fit to the
DES Y3 data, could be useful if done before the data has been unblinded. This
is important, as the more widely-used NLA model gives significantly reduced
uncertainty on $S_{8}$, ($\sigma_{\rm NLA}/\sigma_{\rm TATT}=0.85$) and the
amplitude of intrinsic alignment measured in Ref. (DES Collaboration, 2021),
is relatively smaller than forecasted (see discussion in Section VII B. of
Ref. (Secco  _et al._ , 2021)). Of course, these two theoretical systematics
are connected – advancements that allow for harnessing the small-scale
information in the analysis will in turn allow for self-calibration of
nuisance model parameters. The degradation in constraints when marginalizing
over a model for intrinsic alignments motivates combined 3$\times$2pt analyses
(DES Collaboration, 2021).

Table 3: Summary of constraints on the posterior mean value of and $68\%$ confidence bounds on $S_{8}$, $\Omega_{\rm m}$ and $\sigma_{8}$ in $\Lambda$CDM, as well as the Maximum Posterior $S_{8}$ value (denoted $\hat{S}_{8}$). The $\chi^{2}$ per degree of freedom (dof) and FoM (defined in equation 25) for the $S_{8}-\Omega_{\rm m}$ plane are also shown. We distinguish variations on the Fiducial model that are not required to give consistent results (for example, by neglecting observational systematics) by an asterisk and an open symbol. A visual summary of the $S_{8}$ constraints can be seen in Figure 7. Data | $S_{8}$ | $\hat{S}_{8}$ | $\Omega_{\mathrm{m}}$ | $\sigma_{8}$ | FoM${}_{S_{8},\Omega_{\mathrm{m}}}$ | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---|---|---
Fiducial | $0.759_{-0.023}^{+0.025}$ | 0.755 | $0.290^{+0.039}_{-0.063}$ | $0.783^{+0.073}_{-0.092}$ | 927 | $237.7/222=1.07$
LCDM-Optimized | $0.772^{+0.018}_{-0.017}$ | 0.774 | $0.289^{+0.036}_{-0.056}$ | $0.795^{+0.072}_{-0.076}$ | 1398 | $285.0/268=1.06$
C$:$ pure photometric sample * | $0.752^{+0.025}_{-0.024}$ | 0.766 | $0.303^{+0.043}_{-0.074}$ | $0.760^{+0.080}_{-0.102}$ | 831 | -
MB$:$pure spectroscopic sample * | $0.752^{+0.021}_{-0.025}$ | 0.729 | $0.298^{+0.042}_{-0.063}$ | $0.763^{+0.075}_{-0.091}$ | 968 | -
SOMPZ only | $0.760^{+0.035}_{-0.031}$ | 0.739 | $0.292^{+0.043}_{-0.062}$ | $0.780^{+0.079}_{-0.089}$ | 542 | -
SOMPZ$+$WZ only | $0.758^{+0.035}_{-0.028}$ | 0.730 | $0.298^{+0.044}_{-0.068}$ | $0.772^{+0.079}_{-0.092}$ | 516 | -
Alternative lens sample-SR | $0.768^{+0.021}_{-0.022}$ | 0.757 | $0.296^{+0.041}_{-0.060}$ | $0.783^{+0.079}_{-0.091}$ | 1055 | -
Large-scale-SR | $0.777^{+0.022}_{-0.023}$ | 0.780 | $0.331^{+0.043}_{-0.076}$ | $0.750^{+0.083}_{-0.088}$ | 860 | -
Hyperrank$:$ full redshift shape model | $0.755^{+0.023}_{-0.018}$ | 0.744 | $0.287^{+0.038}_{-0.055}$ | $0.780^{+0.067}_{-0.082}$ | 1032 | -
No redshift systematics * | $0.753^{+0.023}_{-0.023}$ | 0.752 | $0.271^{+0.032}_{-0.054}$ | $0.801^{+0.081}_{-0.078}$ | 1111 | -
No n$(z)$ blending correction | $0.761^{+0.024}_{-0.022}$ | 0.753 | $0.283^{+0.036}_{-0.052}$ | $0.791^{+0.067}_{-0.082}$ | 1091 | -
Additional shear uncertainty | $0.759^{+0.023}_{-0.023}$ | 0.758 | $0.300^{+0.044}_{-0.057}$ | $0.767^{+0.071}_{-0.088}$ | 971 | -
No shear systematics * | $0.743^{+0.026}_{-0.022}$ | 0.7566 | $0.292^{+0.035}_{-0.059}$ | $0.764^{+0.073}_{-0.079}$ | 864 | -
No observational systematics * | $0.742^{+0.022}_{-0.021}$ | 0.740 | $0.272^{+0.036}_{-0.052}$ | $0.788^{+0.071}_{-0.086}$ | 1104 | -
Only cosmological parameters * | $0.756^{+0.019}_{-0.017}$ | 0.753 | $0.282^{+0.038}_{-0.054}$ | $0.789^{+0.070}_{-0.082}$ | 1403 | -
High-$z$: Bins 1,2 removed | $0.733^{+0.023}_{-0.025}$ | 0.737 | $0.355^{+0.050}_{-0.080}$ | $0.683^{+0.069}_{-0.090}$ | 784 | -
Low-$z$: Bins 3,4 removed | $0.750^{+0.045}_{-0.039}$ | 0.806 | $0.289^{+0.034}_{-0.099}$ | $0.787^{+0.144}_{-0.122}$ | 355 | -
$\xi_{-}$ only | $0.753^{+0.028}_{-0.032}$ | 0.747 | $0.265^{+0.031}_{-0.061}$ | $0.812^{+0.090}_{-0.098}$ | 829 | -
$\xi_{+}$ only | $0.779^{+0.022}_{-0.020}$ | 0.777 | $0.288^{+0.039}_{-0.062}$ | $0.805^{+0.081}_{-0.093}$ | 1008 | -
Small angular scales | $0.770^{+0.021}_{-0.021}$ | 0.763 | $0.283^{+0.039}_{-0.061}$ | $0.803^{+0.082}_{-0.086}$ | 934 | -
Large angular scales | $0.737^{+0.027}_{-0.031}$ | 0.734 | $0.268^{+0.033}_{-0.054}$ | $0.790^{+0.079}_{-0.089}$ | 845 | -
DES Y1 | $0.780^{+0.027}_{-0.021}$ | - | $0.319^{+0.044}_{-0.062}$ | $0.764_{-0.072}^{+0.069}$ | 625 | $227/211=1.08$
KiDS-1000 | $0.759^{+0.024}_{-0.021}$ | - | $0.246^{+0.101}_{-0.060}$ | $0.838^{+0.140}_{-0.141}$ | 650 | $85.5/70.5=1.21$
HSC Y1 $C_{\ell}$ | $0.780^{+0.030}_{-0.033}$ | - | $0.162^{+0.086}_{-0.044}$ | - | 461 | $45.4/53=0.86$
HSC Y1 $\xi_{\pm}$ | $0.804^{+0.032}_{-0.029}$ | - | $0.346^{+0.052}_{-0.100}$ | $0.766^{+0.110}_{-0.093}$ | 402 | $162.3/167=0.97$
Planck 2018 TT + TE + EE + lowE | $0.827^{+0.019}_{-0.017}$ | - | $0.327^{+0.008}_{-0.017}$ | $0.793^{+0.024}_{-0.009}$ | 3938 | -

##  XIII Conclusions

This paper, and its companion (Secco  _et al._ , 2021), present the cosmic
shear analysis of the most powerful weak lensing catalog to date, from the
Dark Energy Survey Year 3 (DES Y3). The data the spans $\sim 5000$ deg2 of the
southern sky and contains over 100 million galaxy shapes. These constitute the
most significant cosmic shear measurements to date. We constrain cosmological
parameters in $\rm\Lambda CDM$, while also varying the neutrino mass, and find
a 3% fractional uncertainty on $S_{8}$, with $S_{8}=0.759_{-0.023}^{+0.025}$
(0.755 best-fit) at 68% confidence limits. A $\rm\Lambda CDM$-Optimized
analysis, which includes more small-scale information while still passing the
requirements for robustness against baryonic effects, finds a 2% precision
constraint of $S_{8}=0.772^{+0.018}_{-0.017}$. This is consistent with the
Fiducial result and roughly 1.5$\times$ more constraining. The goodness of fit
to the data is acceptable, with a $p$-value of 0.223 in $\rm\Lambda CDM$. In
$w$CDM, the cosmic shear constraint on the dark energy equation of state, $w$,
is found to be prior-dominated.

The DES Y3 cosmic shear more than triples the survey area and number of
galaxies with respect to previous lensing analyses (Troxel  _et al._ , 2018;
Asgari  _et al._ , 2021; Hamana  _et al._ , 2020). The posteriors in the
$S_{8}-\Omega_{\rm m}$ plane qualitatively agree with previous cosmic shear
results from KiDS-1000 (Asgari  _et al._ , 2021) and HSC (Hamana  _et al._ ,
2020; Hikage  _et al._ , 2019), which, favor a lower value of $S_{8}$ than the
most recent CMB measurements from Planck (Planck Collaboration, 2018).
Considering the full parameter space, the DES Y3 shear cosmological
constraints are statistically consistent with those from Planck, with a
2.3$\sigma$ difference (and a $p$-value of 0.02) for the Fiducial case and
2.1$\sigma$ for the $\rm\Lambda CDM$-Optimized analysis (and a $p$-value of
0.05). While this tension is not statistically significant, our results
continue a trend of weak lensing data that are in agreement with, but display
a lower value of $S_{8}$ than the CMB.

Beyond substantial gains in the signal to noise of the measurements, this
analysis incorporated major updates in weak-lensing methodology. Therefore,
these cosmological constraints are build upon innovations in data calibration,
specifically the shear and redshift calibration. Throughout this paper, we
examine the rigour of analysis choices related to aspects in the data, made
while blind. The robustness of theoretical modelling choices is investigated
in Ref. (Secco  _et al._ , 2021). The main conclusions of this work are:

  * •

We have demonstrated the results to be robust to photometric redshift
calibration, with variant analysis exhibiting consistency at the $\sim
0.5\sigma$-level. The cosmological constraints are insensitive to even extreme
variations of the redshift sample. This is enabled by the DES Wide-Deep
strategy (Hartley  _et al._ , 2020) and this is the first cosmic shear
analysis to model the selection function of the survey through injections of
fake galaxy images into the data (using a tool called Balrog). The
cosmological, astrophysical and observational cosmic shear posteriors are
mutually consistent when using either of three independent redshift methods,
that together, provide more precise constraints on the redshift distributions.
The three calibration methods are combined, rather than used as a cross-check:
a photometric re-weighting of redshifts in the DES Deep Fields via an
unsupervised machine learning algorithm (Myles  _et al._ , 2020); clustering
cross-correlations with photometric and spectroscopic samples of galaxies with
precise and accurate redshifts (Gatti  _et al._ , 2020); and ratios of small-
scale galaxy-galaxy lensing measurements (Sánchez  _et al._ , 2021). In
addition, for the DES Y3 cosmic shear, we find that the inclusion of the
shear-ratio data substantially improves the constraints on the intrinsic
alignment parameters. Finally, we demonstrate that the DES Y3 analysis is
robust to the impact of uncertainty on the shape of the redshift
distributions, such as a fluctuation localized in the high-redshift tail,
using Hyperrank (Cordero  _et al._ , 2021).

  * •

We detect, model, and calibrate the redshift-dependent impact of blending on
weak lensing cosmology, with state-of-the-art multi-band image simulations
described in (MacCrann  _et al._ , 2020). As the first cosmic shear analysis
to account for such an effect, we find it to be an important correction. For
our galaxy sample and shape measurement techniques, the image simulation-based
calibration results in a shift toward higher values of $S_{8}$ by $0.5\sigma$.
Furthermore, we demonstrate that the cosmology is insensitive to an
additional, unaccounted for shear uncertainty.

  * •

We evaluate the internal consistency of the cosmic shear measurements using a
PPD method (Doux  _et al._ , 2020). We find all $p$-values to pass our
threshold. At the analysis level, we find the $S_{8}$ posteriors to be
consistent across small and large scales, low and high redshift and between
two-point shear statistics to $\sim 1\sigma$, which serves as a useful
validation of some scale-dependent, or redshift-dependent unmodeled
uncertainty. In addition, given the high-dimensionality of the analysis, we
test the plausibility that inconsistencies in the intrinsic alignment
parameters may absorb unaccounted-for systematics and that these to be stable
removing subsets of the redshift bins.

  * •

We show that the level of PSF contamination in the analysis is subdominant,
and that the tomographic B-mode measurements are consistent with zero.

The companion paper contains analyses that yield the following conclusions
about our results (Secco  _et al._ , 2021):

  * •

We account for the intrinsic alignment of galaxies with a model that includes
the tidal alignment and tidal torquing (quadratic) alignment, motivated by
tests on analytically simulated data. We explore different choices of
intrinsic alignment parameterizations on (real) data, including the NLA model,
and find the cosmological parameters to be robust to these variations to
within 1$\sigma$.

  * •

The matter power spectrum is modeled as dark matter-only, but we a select
angular scales conservatively from hydrodynamical simulations to mitigate
baryonic effects. We test for the impact of residual, unmodeled effects by
considering alternative models that account for non-linear physics and find
the results are stable within $\sim 0.5\sigma$.

  * •

We demonstrate that our posteriors are stable when inference is carried out at
fixed neutrino mass, and demonstrate that higher-order shear contributions are
negligible.

  * •

We present details of the consistency assessment between DES and external
probes, finding statistical agreement between those data.

  * •

We perform a detailed Bayesian evidence-based model selection for intrinsic
alignments and find that our data shows a preference for simpler and more
constraining models that give substantially tighter cosmological constraints.

Finally, we investigate the limiting systematics for DES Y3 cosmic shear
analysis. We demonstrate that the observational systematics are calibrated to
sufficient precision, such that their uncertainties do not limit the analysis.
However, we find the $S_{8}$ constraint to be significantly limited by
systematics in the theoretical modeling, that is, due to the uncertainty in
modeling of small-scale baryonic effects, as well as intrinsic alignments,
which costs about two thirds of the Y3 cosmic shear power. The limitation in
the former is reduced in the joint analysis with galaxy clustering and galaxy-
galaxy lensing, which can be combined to give significantly more precise
cosmological constraints (DES Collaboration, 2021). On the other hand,
improvements on the latter demand advancements in hydro-dynamical simulations
and external complementary observations that probe astrophysical effects.

Looking ahead, we anticipate substantially improved cosmic shear power owing
to observations with roughly twice the effective integrated exposure time per
galaxy, and thus an abundance of fainter galaxies usable, in the Year 6 data.
New Deep Field observations in fields with key external multi-band photometry
and redshift information will enable robust redshift calibration at the
enhanced depth. Despite the fact that the data calibration methodology for DES
Y3 is substantially ahead of the requirements set by the ever-growing
statistical power, future lensing data sets will require advances in redshift
calibration and the accounting for blending, which are disproportionately more
difficult for the fainter and more blended galaxy populations. Collaborative
joint analyses with the complementary KiDS and HSC surveys would allow for an
exchange of methodological experience. This would provide a particularly
promising platform for further development in cosmic shear techniques and as a
training ground for the imminent and challenging data-rich era of the
groundbreaking surveys to be performed by Rubin Observatory, Euclid, and the
Roman Space Telescope.

## Acknowledgements

This research manuscript made use of Astropy Astropy Collaboration  _et al._
(2013); Astropy Collaboration and Astropy Contributors (2018), GetDist Lewis
(2019),ChainConsumer101010samreay.github.io/ChainConsumer (Hinton, 2016) and
Matplotlib Hunter (2007), and has been prepared using NASA’s Astrophysics Data
System Bibliographic Services. Funding for the DES Projects has been provided
by the U.S. Department of Energy, the U.S. National Science Foundation, the
Ministry of Science and Education of Spain, the Science and Technology
Facilities Council of the United Kingdom, the Higher Education Funding Council
for England, the National Center for Supercomputing Applications at the
University of Illinois at Urbana-Champaign, the Kavli Institute of
Cosmological Physics at the University of Chicago, the Center for Cosmology
and Astro-Particle Physics at the Ohio State University, the Mitchell
Institute for Fundamental Physics and Astronomy at Texas A&M University,
Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à
Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento
Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação,
the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the
Dark Energy Survey.

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

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

The DES data management system is supported by the National Science Foundation
under Grant Numbers AST-1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants
ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597,
and MDM-2015-0509, some of which include ERDF funds from the European Union.
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
Research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We
acknowledge support from the Brazilian Instituto Nacional de Ciência e
Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

## References

  * Secco  _et al._ (2021) L. F. Secco, S. Samuroff, _et al._ , To be submitted to PRD (2021).
  * Weinberg (1972) S. Weinberg,  _Gravitation and Cosmology_ (1972).
  * Peebles (1980) P. J. E. Peebles,  _The large-scale structure of the universe_ (1980).
  * Bertone and Hooper (2018) G. Bertone and D. Hooper, Rev. Mod. Phys. 90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO] .
  * Trimble (1987) V. Trimble, Ann. Rev. Astron. Astrophys. 25, 425 (1987).
  * Perlmutter  _et al._ (1999) S. Perlmutter _et al._ (Supernova Cosmology Project), Astrophys. J. 517, 565 (1999), arXiv:astro-ph/9812133 .
  * Riess  _et al._ (1998) A. G. Riess _et al._ (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201 .
  * Frieman  _et al._ (2008) J. A. Frieman, M. S. Turner, and D. Huterer, ARA&A 46, 385 (2008), arXiv:0803.0982 [astro-ph] .
  * Weinberg  _et al._ (2013) D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess, and E. Rozo, Phys. Rept. 530, 87 (2013), arXiv:1201.2434 [astro-ph.CO] .
  * Abazajian  _et al._ (2016) K. N. Abazajian _et al._ (CMB-S4), (2016), arXiv:1610.02743 [astro-ph.CO] .
  * Verde  _et al._ (2019) L. Verde, T. Treu, and A. G. Riess, Nature Astronomy 3, 891 (2019), arXiv:1907.10625 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, (2021), arXiv:2103.01183 [astro-ph.CO] .
  * Heymans and Grocutt (2013) C. Heymans and E. Grocutt, MNRAS 432, 2433 (2013), arXiv:1303.1808 .
  * Troxel  _et al._ (2018) M. A. Troxel, N. MacCrann, J. Zuntz, T. F. Eifler, E. Krause, S. Dodelson, D. Gruen, J. Blazek, O. Friedrich, S. Samuroff, and DES Collaboration, Phys. Rev. D 98, 043528 (2018), arXiv:1708.01538 [astro-ph.CO] .
  * Dark Energy Survey Collaboration (2018) Dark Energy Survey Collaboration, Phys. Rev. D 98, 043526 (2018), arXiv:1708.01530 [astro-ph.CO] .
  * Hikage  _et al._ (2019) C. Hikage, M. Oguri, T. Hamana, S. More, R. Mandelbaum, M. Takada, F. Köhlinger, H. Miyatake, and A. J. Nishizawa, PASJ 71, 43 (2019), arXiv:1809.09148 [astro-ph.CO] .
  * Asgari  _et al._ (2021) M. Asgari, C.-A. Lin, B. Joachimi, B. Giblin, C. Heymans, and H. Hildebrandt, A&A 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Heymans  _et al._ (2021) C. Heymans, T. Tröster, M. Asgari, C. Blake, H. Hildebrandt, B. Joachimi, and K. Kuijken, A&A 646, A140 (2021), arXiv:2007.15632 [astro-ph.CO] .
  * Krolewski  _et al._ (2021) A. Krolewski, S. Ferraro, and M. White, arXiv e-prints , arXiv:2105.03421 (2021), arXiv:2105.03421 [astro-ph.CO] .
  * Planck Collaboration (2018) Planck Collaboration, arXiv e-prints , arXiv:1807.06210 (2018), arXiv:1807.06210 [astro-ph.CO] .
  * Bartelmann and Schneider (2001) M. Bartelmann and P. Schneider, Phys. Rep. 340, 291 (2001), arXiv:astro-ph/9912508 [astro-ph] .
  * Hoekstra and Jain (2008) H. Hoekstra and B. Jain, Annual Review of Nuclear and Particle Science 58, 99 (2008), arXiv:0805.0139 [astro-ph] .
  * Kilbinger (2015) M. Kilbinger, Reports on Progress in Physics 78, 086901 (2015), arXiv:1411.0115 [astro-ph.CO] .
  * Kristian (1967) J. Kristian, ApJ 147, 864 (1967).
  * Bacon  _et al._ (2000) D. J. Bacon, A. R. Refregier, and R. S. Ellis, MNRAS 318, 625 (2000), astro-ph/0003008 .
  * Kaiser  _et al._ (2000) N. Kaiser, G. Wilson, and G. A. Luppino, arXiv:astro-ph/0003338 (2000), astro-ph/0003338 .
  * Van Waerbeke  _et al._ (2000) L. Van Waerbeke, Y. Mellier, T. Erben, J. C. Cuillandre, F. Bernardeau, R. Maoli, E. Bertin, H. J. McCracken, O. Le Fèvre, B. Fort, M. Dantel-Fort, B. Jain, and P. Schneider, A&A 358, 30 (2000), astro-ph/0002500 .
  * Wittman  _et al._ (2000) D. M. Wittman, J. A. Tyson, D. Kirkman, I. Dell’Antonio, and G. Bernstein, Nature 405, 143 (2000), arXiv:astro-ph/0003014 [astro-ph] .
  * Hoekstra  _et al._ (2002) H. Hoekstra, H. K. C. Yee, M. D. Gladders, L. F. Barrientos, P. B. Hall, and L. Infante, Astrophys. J. 572, 55 (2002), arXiv:astro-ph/0202285 .
  * Refregier  _et al._ (2002) A. Refregier, J. Rhodes, and E. J. Groth, Astrophys. J. Lett. 572, L131 (2002), arXiv:astro-ph/0203131 .
  * Jarvis  _et al._ (2003) M. Jarvis, G. M. Bernstein, P. Fischer, D. Smith, B. Jain, J. A. Tyson, and D. Wittman, Astron. J. 125, 1014 (2003), arXiv:astro-ph/0210604 .
  * Hamana  _et al._ (2003) T. Hamana _et al._ , Astrophys. J. 597, 98 (2003), arXiv:astro-ph/0210450 .
  * Brown  _et al._ (2003) M. L. Brown, A. N. Taylor, D. J. Bacon, M. E. Gray, S. Dye, K. Meisenheimer, and C. Wolf, Mon. Not. Roy. Astron. Soc. 341, 100 (2003), arXiv:astro-ph/0210213 .
  * Rhodes  _et al._ (2004) J. Rhodes, A. Refregier, N. R. Collins, J. P. Gardner, E. J. Groth, and R. S. Hill, Astrophys. J. 605, 29 (2004), arXiv:astro-ph/0312283 .
  * Heymans  _et al._ (2005) C. Heymans _et al._ , Mon. Not. Roy. Astron. Soc. 361, 160 (2005), arXiv:astro-ph/0411324 .
  * Jarvis  _et al._ (2006) M. Jarvis, B. Jain, G. Bernstein, and D. Dolney, Astrophys. J. 644, 71 (2006), arXiv:astro-ph/0502243 .
  * Hetterscheidt  _et al._ (2007) M. Hetterscheidt, P. Simon, M. Schirmer, H. Hildebrandt, T. Schrabback, T. Erben, and P. Schneider, Astron. Astrophys. 468, 859 (2007), arXiv:astro-ph/0606571 .
  * Massey  _et al._ (2007) R. Massey _et al._ , Astrophys. J. Suppl. 172, 239 (2007), arXiv:astro-ph/0701480 .
  * Leauthaud  _et al._ (2007) A. Leauthaud _et al._ , Astrophys. J. Suppl. 172, 219 (2007), arXiv:astro-ph/0702359 .
  * Benjamin  _et al._ (2007) J. Benjamin, C. Heymans, E. Semboloni, L. Van Waerbeke, H. Hoekstra, T. Erben, M. D. Gladders, M. Hetterscheidt, Y. Mellier, and H. K. C. Yee, Mon. Not. Roy. Astron. Soc. 381, 702 (2007), arXiv:astro-ph/0703570 .
  * Fu  _et al._ (2008) L. Fu _et al._ , Astron. Astrophys. 479, 9 (2008), arXiv:0712.0884 [astro-ph] .
  * Schrabback  _et al._ (2010) T. Schrabback _et al._ , Astron. Astrophys. 516, A63 (2010), arXiv:0911.0053 [astro-ph.CO] .
  * Huff  _et al._ (2014a) E. M. Huff, C. M. Hirata, R. Mandelbaum, D. Schlegel, U. Seljak, and R. H. Lupton, Mon. Not. Roy. Astron. Soc. 440, 1296 (2014a), arXiv:1111.6958 [astro-ph.CO] .
  * Huff  _et al._ (2014b) E. M. Huff, T. Eifler, C. M. Hirata, R. Mandelbaum, D. Schlegel, and U. Seljak, Mon. Not. Roy. Astron. Soc. 440, 1322 (2014b), arXiv:1112.3143 [astro-ph.CO] .
  * Lin  _et al._ (2012) H. Lin, S. Dodelson, H.-J. Seo, M. Soares-Santos, J. Annis, J. Hao, D. Johnston, J. M. Kubo, R. R. R. Reis, and M. Simet (SDSS), Astrophys. J. 761, 15 (2012), arXiv:1111.6622 [astro-ph.CO] .
  * Jee  _et al._ (2013a) M. J. Jee, J. A. Tyson, M. D. Schneider, D. Wittman, S. Schmidt, and S. Hilbert, Astrophys. J. 765, 74 (2013a), arXiv:1210.2732 [astro-ph.CO] .
  * Jee  _et al._ (2016) M. J. Jee, J. A. Tyson, S. Hilbert, M. D. Schneider, S. Schmidt, and D. Wittman, Astrophys. J. 824, 77 (2016), arXiv:1510.03962 [astro-ph.CO] .
  * Erben  _et al._ (2013) T. Erben _et al._ , Mon. Not. Roy. Astron. Soc. 433, 2545 (2013), arXiv:1210.8156 [astro-ph.CO] .
  * Miller  _et al._ (2013) L. Miller, C. Heymans, T. D. Kitching, L. van Waerbeke, and T. Erben, MNRAS 429, 2858 (2013), arXiv:1210.8201 [astro-ph.CO] .
  * Kilbinger  _et al._ (2013) M. Kilbinger _et al._ , Mon. Not. Roy. Astron. Soc. 430, 2200 (2013), arXiv:1212.3338 [astro-ph.CO] .
  * Kitching  _et al._ (2014) T. D. Kitching _et al._ (CFHTLenS), Mon. Not. Roy. Astron. Soc. 442, 1326 (2014), arXiv:1401.6842 [astro-ph.CO] .
  * Heymans and Van Waerbeke (2006) C. Heymans and L. Van Waerbeke, MNRAS 368, 1323 (2006), arXiv:astro-ph/0506112 [astro-ph] .
  * Aihara and HSC Collaboration (2018) H. Aihara and HSC Collaboration, PASJ 70, S4 (2018), arXiv:1704.05858 [astro-ph.IM] .
  * Kuijken  _et al._ (2015) K. Kuijken, C. Heymans, H. Hildebrandt, R. Nakajima, T. Erben, J. T. A. de Jong, M. Viola, A. Choi, H. Hoekstra, L. Miller, and E. van Uitert, MNRAS 454, 3500 (2015), arXiv:1507.00738 .
  * Jee  _et al._ (2013b) M. J. Jee, J. A. Tyson, M. D. Schneider, D. Wittman, S. Schmidt, and S. Hilbert, The Astrophysical Journal 765, 74 (2013b).
  * Dark Energy Survey Collaboration (2016a) Dark Energy Survey Collaboration, Phys. Rev. D 94, 022001 (2016a), arXiv:1507.05552 [astro-ph.CO] .
  * Hildebrandt  _et al._ (2017) H. Hildebrandt, M. Viola, C. Heymans, S. Joudaki, K. Kuijken, C. Blake, T. Erben, B. Joachimi, D. Klaes, L. t. Miller, _et al._ , Monthly Notices of the Royal Astronomical Society 465, 1454 (2017).
  * Troxel  _et al._ (2018a) M. A. Troxel _et al._ (DES), Mon. Not. Roy. Astron. Soc. 479, 4998 (2018a), arXiv:1804.10663 [astro-ph.CO] .
  * Hildebrandt  _et al._ (2020) H. Hildebrandt, F. Köhlinger, J. L. van den Busch, B. Joachimi, and C. Heymans, A&A 633, A69 (2020), arXiv:1812.06076 [astro-ph.CO] .
  * Hamana  _et al._ (2020) T. Hamana, M. Shirasaki, S. Miyazaki, C. Hikage, M. Oguri, and S. More, PASJ 72, 16 (2020), arXiv:1906.06041 [astro-ph.CO] .
  * Joudaki  _et al._ (2018) S. Joudaki, C. Blake, A. Johnson, A. Amon, M. Asgari, and A. Choi, MNRAS 474, 4894 (2018), arXiv:1707.06627 [astro-ph.CO] .
  * van Uitert  _et al._ (2018) E. van Uitert, B. Joachimi, S. Joudaki, A. Amon, and C. Heymans, MNRAS 476, 4662 (2018), arXiv:1706.05004 [astro-ph.CO] .
  * LSST Dark Energy Science Collaboration (2012) LSST Dark Energy Science Collaboration, arXiv e-prints , arXiv:1211.0310 (2012), arXiv:1211.0310 [astro-ph.CO] .
  * Laureijs  _et al._ (2011) R. Laureijs, J. Amiaux, S. Arduini, J. . Auguères, J. Brinchmann, R. Cole, M. Cropper, C. Dabin, L. Duvet, A. Ealet, and et al., ArXiv e-prints (2011), arXiv:1110.3193 [astro-ph.CO] .
  * Spergel (2015) D. Spergel, ArXiv e-prints (2015), arXiv:1503.03757 [astro-ph.IM] .
  * Mandelbaum (2018) R. Mandelbaum, ARA&A 56, 393 (2018), arXiv:1710.03235 [astro-ph.CO] .
  * Huff and Mandelbaum (2017) E. Huff and R. Mandelbaum, arXiv e-prints , arXiv:1702.02600 (2017), arXiv:1702.02600 [astro-ph.CO] .
  * Sheldon and Huff (2017) E. S. Sheldon and E. M. Huff, ApJ 841, 24 (2017), arXiv:1702.02601 .
  * Jarvis  _et al._ (2021) M. Jarvis _et al._ (DES), Mon. Not. Roy. Astron. Soc. 501, 1282 (2021), arXiv:2011.03409 [astro-ph.IM] .
  * Sheldon  _et al._ (2020) E. S. Sheldon, M. R. Becker, N. MacCrann, and M. Jarvis, ApJ 902, 138 (2020), arXiv:1911.02505 [astro-ph.CO] .
  * Bernstein and Armstrong (2014) G. M. Bernstein and R. Armstrong, Monthly Notices of the Royal Astronomical Society 438, 1880–1893 (2014).
  * Gatti  _et al._ (2021) M. Gatti, E. Sheldon, _et al._ (DES), Mon. Not. Roy. Astron. Soc. 504, 4312 (2021), arXiv:2011.03408 [astro-ph.CO] .
  * Giblin  _et al._ (2021) B. Giblin, C. Heymans, M. Asgari, and Hildebrandt, A&A 645, A105 (2021), arXiv:2007.01845 [astro-ph.CO] .
  * Mandelbaum  _et al._ (2018) R. Mandelbaum, H. Miyatake, T. Hamana, M. Oguri, and M. Simet, PASJ 70, S25 (2018), arXiv:1705.06745 [astro-ph.CO] .
  * Bernstein and Huterer (2010) G. Bernstein and D. Huterer, MNRAS 401, 1399 (2010), arXiv:0902.2782 [astro-ph.CO] .
  * Newman (2015) J. A. Newman, Astroparticle Physics 63, 81 (2015), arXiv:1309.5384 [astro-ph.CO] .
  * Gruen and Brimioulle (2017) D. Gruen and F. Brimioulle, MNRAS 468, 769 (2017), arXiv:1610.01160 [astro-ph.CO] .
  * Hoyle  _et al._ (2018) B. Hoyle, D. Gruen, G. M. Bernstein, M. M. Rau, and DES Collaboration, MNRAS 478, 592 (2018), arXiv:1708.01532 [astro-ph.CO] .
  * Wright  _et al._ (2020) A. H. Wright, H. Hildebrandt, J. L. van den Busch, C. Heymans, B. Joachimi, A. Kannawadi, and K. Kuijken, arXiv e-prints , arXiv:2005.04207 (2020), arXiv:2005.04207 [astro-ph.CO] .
  * Joudaki and Hildebrandt (2020) S. Joudaki and H. Hildebrandt, A&A 638, L1 (2020), arXiv:1906.09262 [astro-ph.CO] .
  * Hartley  _et al._ (2020) W. G. Hartley, C. Chang, and DES Collaboration, MNRAS 496, 4769 (2020), arXiv:2003.10454 [astro-ph.GA] .
  * Myles  _et al._ (2020) J. Myles, A. Alarcon, _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.08566 [astro-ph.CO] .
  * Gatti  _et al._ (2020) M. Gatti, G. Giannini, _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.08569 [astro-ph.CO] .
  * Sánchez  _et al._ (2021) C. Sánchez, J. Prat, _et al._ , To be submitted to PRD (2021).
  * Dawson  _et al._ (2016) W. A. Dawson, M. D. Schneider, J. A. Tyson, and M. J. Jee, ApJ 816, 11 (2016), arXiv:1406.1506 [astro-ph.CO] .
  * Mandelbaum and Lanusse (2018) R. Mandelbaum and F. Lanusse, MNRAS 481, 3170 (2018), arXiv:1710.00885 [astro-ph.CO] .
  * Jarvis  _et al._ (2016) M. Jarvis, E. Sheldon, and J. Zuntz, MNRAS 460, 2245 (2016), arXiv:1507.05603 [astro-ph.IM] .
  * Bosch (2018) J. Bosch, PASJ 70, S5 (2018), arXiv:1705.06766 [astro-ph.IM] .
  * Samuroff  _et al._ (2017) S. Samuroff, M. A. Troxel, S. L. Bridle, J. Zuntz, N. MacCrann, E. Krause, T. Eifler, and D. Kirk, MNRAS 465, L20 (2017), arXiv:1607.07910 [astro-ph.CO] .
  * Kannawadi  _et al._ (2019) A. Kannawadi, H. Hoekstra, L. Miller, M. Viola, I. Fenech Conti, R. Herbonnet, T. Erben, C. Heymans, H. Hildebrandt, K. Kuijken, M. Vakili, and A. H. Wright, A&A 624, A92 (2019), arXiv:1812.03983 [astro-ph.CO] .
  * MacCrann  _et al._ (2020) N. MacCrann _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.08567 [astro-ph.CO] .
  * Croft and Metzler (2000) R. A. C. Croft and C. A. Metzler, ApJ 545, 561 (2000), arXiv:astro-ph/0005384 [astro-ph] .
  * Heavens  _et al._ (2000) A. Heavens, A. Refregier, and C. Heymans, MNRAS 319, 649 (2000), arXiv:astro-ph/0005269 [astro-ph] .
  * Troxel and Ishak (2015) M. A. Troxel and M. Ishak, Phys. Rep. 558, 1 (2015), arXiv:1407.6990 [astro-ph.CO] .
  * Chisari  _et al._ (2018) N. E. Chisari, M. L. A. Richardson, J. Devriendt, Y. Dubois, A. Schneider, A. M. C. Le Brun, R. S. Beckmann, S. Peirani, A. Slyz, and C. Pichon, MNRAS 480, 3962 (2018), arXiv:1801.08559 [astro-ph.CO] .
  * Wechsler and Tinker (2018) R. H. Wechsler and J. L. Tinker, ARA&A 56, 435 (2018), arXiv:1804.03097 [astro-ph.GA] .
  * Krause and Eifler (2017) E. Krause and T. Eifler, MNRAS 470, 2100 (2017), arXiv:1601.05779 [astro-ph.CO] .
  * Efstathiou and Lemos (2018) G. Efstathiou and P. Lemos, MNRAS 476, 151 (2018), arXiv:1707.00483 [astro-ph.CO] .
  * Prat  _et al._ (2021) J. Prat _et al._ , To be submitted to PRD (2021).
  * Rodríguez-Monroy  _et al._ (2021) M. Rodríguez-Monroy _et al._ , To be submitted to MNRAS (2021).
  * Porredon  _et al._ (2021) A. Porredon _et al._ , To be submitted to PRD (2021).
  * Pandey  _et al._ (2021) S. Pandey _et al._ , To be submitted to MNRAS (2021).
  * Elvin-Poole  _et al._ (2021) J. Elvin-Poole, N. MacCrann, _et al._ , To be submitted to MNRAS (2021).
  * DES Collaboration (2021) DES Collaboration, To be submitted to PRD (2021).
  * Sevilla-Noarbe  _et al._ (2020) I. Sevilla-Noarbe _et al._ (DES), Submitted to ApJS (2020), arXiv:2011.03407 [astro-ph.CO] .
  * Hartley  _et al._ (2020) W. G. Hartley, A. Choi, _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.12824 [astro-ph.CO] .
  * Everett  _et al._ (2020) S. Everett _et al._ (DES), Submitted to ApJS (2020), arXiv:2012.12825 [astro-ph.CO] .
  * Buchs  _et al._ (2019) R. Buchs, C. Davis, D. Gruen, J. DeRose, and DES Collaboration, MNRAS 489, 820 (2019), arXiv:1901.05005 [astro-ph.CO] .
  * Porredon  _et al._ (2021) A. Porredon _et al._ (DES), Phys. Rev. D 103, 043503 (2021), arXiv:2011.03411 [astro-ph.CO] .
  * Cordero  _et al._ (2021) J. P. Cordero, I. Harrison, _et al._ , To be submitted to MNRAS (2021).
  * Krause  _et al._ (2021) E. Krause _et al._ , To be submitted to PRD (2021).
  * Friedrich  _et al._ (2020) O. Friedrich _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.08568 [astro-ph.CO] .
  * DeRose  _et al._ (2021a) J. DeRose _et al._ , To be submitted to MNRAS (2021a).
  * Doux  _et al._ (2020) C. Doux _et al._ (DES), Mon. Not. Roy. Astron. Soc. (2020), 10.1093/mnras/stab526, arXiv:2011.03410 [astro-ph.CO] .
  * Lemos  _et al._ (2020) M. Lemos, P. Raveri _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.09554 [astro-ph.CO] .
  * Flaugher  _et al._ (2015) B. Flaugher, H. T. Diehl, K. Honscheid, and DES Collaboration, AJ 150, 150 (2015), arXiv:1504.02900 [astro-ph.IM] .
  * Dark Energy Survey Collaboration (2016b) Dark Energy Survey Collaboration, MNRAS 460, 1270 (2016b), arXiv:1601.00329 [astro-ph.CO] .
  * Schneider  _et al._ (2010) P. Schneider, T. Eifler, and E. Krause, A&A 520, A116 (2010), arXiv:1002.2136 [astro-ph.CO] .
  * De Vicente  _et al._ (2016) J. De Vicente, E. Sánchez, and I. Sevilla-Noarbe, Monthly Notices of the Royal Astronomical Society 459, 3078–3088 (2016).
  * Rozo and Rykoff (2016) E. Rozo and E. S. Rykoff, Monthly Notices of the Royal Astronomical Society 461, 1431–1450 (2016).
  * Cawthon  _et al._ (2020) R. Cawthon _et al._ (DES), Submitted to MNRAS (2020), arXiv:2012.12826 [astro-ph.CO] .
  * Rykoff  _et al._ (2014) E. S. Rykoff, E. Rozo, M. T. Busha, C. E. Cunha, A. Finoguenov, A. Evrard, J. Hao, B. P. Koester, A. Leauthaud, B. Nord, and et al., The Astrophysical Journal 785, 104 (2014).
  * Huterer  _et al._ (2006) D. Huterer, M. Takada, G. Bernstein, and B. Jain, MNRAS 366, 101 (2006), arXiv:astro-ph/0506030 [astro-ph] .
  * Van Waerbeke  _et al._ (2006) L. Van Waerbeke, M. White, H. Hoekstra, and C. Heymans, Astroparticle Physics 26, 91 (2006), arXiv:astro-ph/0603696 [astro-ph] .
  * Bonnett  _et al._ (2016) C. Bonnett, M. A. Troxel, W. Hartley, and Dark Energy Survey Collaboration, Phys. Rev. D 94, 042005 (2016), arXiv:1507.05909 [astro-ph.CO] .
  * Masters  _et al._ (2015) D. Masters, P. Capak, D. Stern, and O. Ilbert, ApJ 813, 53 (2015), arXiv:1509.03318 [astro-ph.CO] .
  * Sánchez and Bernstein (2019) C. Sánchez and G. M. Bernstein, MNRAS 483, 2801 (2019), arXiv:1807.11873 [astro-ph.CO] .
  * Ross  _et al._ (2020) A. J. Ross, J. Bautista, and R. Tojeiro, MNRAS (2020), 10.1093/mnras/staa2416, arXiv:2007.09000 [astro-ph.CO] .
  * Jain and Taylor (2003) B. Jain and A. Taylor, Physical Review Letters 91 (2003), 10.1103/physrevlett.91.141302.
  * Schneider (2016) P. Schneider, A&A 592, L6 (2016), arXiv:1603.04226 [astro-ph.CO] .
  * Mandelbaum  _et al._ (2005) R. Mandelbaum, C. M. Hirata, U. Seljak, J. Guzik, N. Padmanabhan, C. Blake, M. R. Blanton, R. Lupton, and J. Brinkmann, Monthly Notices of the Royal Astronomical Society 361, 1287–1322 (2005).
  * Hoekstra  _et al._ (2006) H. Hoekstra, Y. Mellier, L. van Waerbeke, E. Semboloni, L. Fu, M. J. Hudson, L. C. Parker, I. Tereno, and K. Benabed, ApJ 647, 116 (2006), arXiv:astro-ph/0511089 [astro-ph] .
  * Heymans  _et al._ (2012) C. Heymans, L. Van Waerbeke, and Miller, MNRAS 427, 146 (2012), arXiv:1210.0032 .
  * Lilly  _et al._ (2009) S. J. Lilly, V. Le Brun, C. Maier, and V. Mainieri, ApJS 184, 218 (2009).
  * Masters  _et al._ (2017) D. C. Masters, D. K. Stern, J. G. Cohen, P. L. Capak, J. D. Rhodes, F. J. Castander, and S. Paltani, ApJ 841, 111 (2017), arXiv:1704.06665 [astro-ph.CO] .
  * Masters  _et al._ (2019) D. C. Masters, D. K. Stern, J. G. Cohen, P. L. Capak, S. A. Stanford, N. Hernitschek, A. Galametz, I. Davidzon, J. D. Rhodes, D. Sand ers, B. Mobasher, F. Castander, K. Pruett, and S. Fotopoulou, ApJ 877, 81 (2019), arXiv:1904.06394 [astro-ph.GA] .
  * Le Fèvre (2013) O. Le Fèvre, A&A 559, A14 (2013), arXiv:1307.0545 [astro-ph.CO] .
  * Guzzo and Scodeggio (2014) L. Guzzo and M. Scodeggio, A&A 566, A108 (2014), arXiv:1303.2623 [astro-ph.CO] .
  * Garilli and Guzzo (2014) B. Garilli and L. Guzzo, A&A 562, A23 (2014), arXiv:1310.1008 [astro-ph.CO] .
  * Scodeggio  _et al._ (2018) M. Scodeggio, L. Guzzo, and B. Garilli, A&A 609, A84 (2018), arXiv:1611.07048 [astro-ph.GA] .
  * Laigle  _et al._ (2016) C. Laigle, H. J. McCracken, and O. Ilbert, ApJS 224, 24 (2016), arXiv:1604.02350 [astro-ph.GA] .
  * Alarcon  _et al._ (2021) A. Alarcon, E. Gaztanaga, and P. Eriksen, MNRAS 501, 6103 (2021), arXiv:2007.11132 [astro-ph.GA] .
  * Troxel  _et al._ (2018b) M. A. Troxel, N. MacCrann, J. Zuntz, T. Eifler, E. Krause, S. Dodelson, D. Gruen, J. Blazek, O. Friedrich, S. Samuroff, _et al._ , Physical Review D 98, 043528 (2018b).
  * Leauthaud  _et al._ (2007) A. Leauthaud, R. Massey, J.-P. Kneib, and J. Rhodes, ApJS 172, 219 (2007), arXiv:astro-ph/0702359 [astro-ph] .
  * Wechsler  _et al._ (2021) R. H. Wechsler, J. DeRose, M. T. Busha, M. R. Becker, E. Rykoff, and A. Evrard, arXiv e-prints , arXiv:2105.12105 (2021), arXiv:2105.12105 [astro-ph.CO] .
  * DeRose  _et al._ (2021b) J. DeRose, M. R. Becker, and R. H. Wechsler, arXiv e-prints , arXiv:2105.12104 (2021b), arXiv:2105.12104 [astro-ph.CO] .
  * Springel (2005) V. Springel, MNRAS 364, 1105 (2005), arXiv:astro-ph/0505010 [astro-ph] .
  * Crocce  _et al._ (2012) M. Crocce, S. Pueblas, and R. Scoccimarro, “2LPTIC: 2nd-order Lagrangian Perturbation Theory Initial Conditions,” (2012), ascl:1201.005 .
  * Becker (2013) M. R. Becker,  _CALCLENS: Weak lensing simulations_ , Ph.D. thesis, The University of Chicago (2013).
  * Górski  _et al._ (2005a) K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann, ApJ 622, 759 (2005a), arXiv:astro-ph/0409513 [astro-ph] .
  * DeRose  _et al._ (2019) J. DeRose, R. H. Wechsler, and M. R. Becker, arXiv e-prints , arXiv:1901.02401 (2019), arXiv:1901.02401 [astro-ph.CO] .
  * Jarvis  _et al._ (2004) M. Jarvis, G. Bernstein, and B. Jain, MNRAS 352, 338 (2004), arXiv:astro-ph/0307393 [astro-ph] .
  * Fang  _et al._ (2020) X. Fang, T. Eifler, and E. Krause, MNRAS 497, 2699 (2020), arXiv:2004.04833 [astro-ph.CO] .
  * Hildebrandt  _et al._ (2016) H. Hildebrandt, A. Choi, C. Heymans, C. Blake, T. Erben, L. Miller, R. Nakajima, L. van Waerbeke, M. Viola, A. Buddendiek, and et al., Monthly Notices of the Royal Astronomical Society 463, 635–654 (2016).
  * Zuntz  _et al._ (2018) J. Zuntz, E. Sheldon, S. Samuroff, M. A. Troxel, M. Jarvis, N. MacCrann, D. Gruen, and DES Collaboration, MNRAS 481, 1149 (2018), arXiv:1708.01533 [astro-ph.CO] .
  * Muir  _et al._ (2020) J. Muir _et al._ (DES), Mon. Not. Roy. Astron. Soc. 494, 4454 (2020), arXiv:1911.05929 [astro-ph.CO] .
  * Olive (2016) K. Olive, Chinese Physics C 40, 100001 (2016).
  * Stebbins (1996) A. Stebbins, arXiv e-prints , astro-ph/9609149 (1996), arXiv:astro-ph/9609149 [astro-ph] .
  * Limber (1953) D. N. Limber, ApJ 117, 134 (1953).
  * LoVerde and Afshordi (2008) M. LoVerde and N. Afshordi, Phys. Rev. D 78, 123506 (2008).
  * Dodelson  _et al._ (2006) S. Dodelson, C. Shapiro, and M. White, Phys. Rev. D 73, 023009 (2006), arXiv:astro-ph/0508296 [astro-ph] .
  * Shapiro (2009) C. Shapiro, ApJ 696, 775 (2009), arXiv:0812.0769 [astro-ph] .
  * Schneider  _et al._ (2002) P. Schneider, L. van Waerbeke, M. Kilbinger, and Y. Mellier, A&A 396, 1 (2002), arXiv:astro-ph/0206182 [astro-ph] .
  * Schmidt  _et al._ (2009) F. Schmidt, E. Rozo, S. Dodelson, L. Hui, and E. Sheldon, ApJ 702, 593 (2009), arXiv:0904.4703 [astro-ph.CO] .
  * Seljak (1996) U. Seljak, ApJ 463, 1 (1996), arXiv:astro-ph/9505109 [astro-ph] .
  * Dodelson  _et al._ (2008) S. Dodelson, F. Schmidt, and A. Vallinotto, Phys. Rev. D 78, 043508 (2008), arXiv:0806.0331 [astro-ph] .
  * Hirata and Seljak (2004) C. M. Hirata and U. Seljak, Phys. Rev. D 70, 063526 (2004), arXiv:astro-ph/0406275 [astro-ph] .
  * Kiessling  _et al._ (2015) A. Kiessling, M. Cacciato, B. Joachimi, D. Kirk, T. D. Kitching, A. Leonard, R. Mandelbaum, B. M. Schäfer, C. Sifón, M. L. Brown, and A. Rassat, Space Sci. Rev. 193, 67 (2015), arXiv:1504.05546 [astro-ph.GA] .
  * Blazek  _et al._ (2019) J. A. Blazek, N. MacCrann, M. A. Troxel, and X. Fang, Phys. Rev. D 100, 103506 (2019), arXiv:1708.09247 [astro-ph.CO] .
  * Mackey  _et al._ (2002) J. Mackey, M. White, and M. Kamionkowski, MNRAS 332, 788 (2002), arXiv:astro-ph/0106364 [astro-ph] .
  * Codis  _et al._ (2015) S. Codis, C. Pichon, and D. Pogosyan, MNRAS 452, 3369 (2015), arXiv:1504.06073 [astro-ph.CO] .
  * Blazek  _et al._ (2015) J. Blazek, Z. Vlah, and U. Seljak, J. Cosmology Astropart. Phys 2015, 015 (2015), arXiv:1504.02510 [astro-ph.CO] .
  * Brown  _et al._ (2002) M. L. Brown, A. N. Taylor, N. C. Hambly, and S. Dye, Monthly Notices of the Royal Astronomical Society 333, 501 (2002), https://academic.oup.com/mnras/article-pdf/333/3/501/3215948/333-3-501.pdf .
  * van Daalen  _et al._ (2014) M. P. van Daalen, J. Schaye, I. G. McCarthy, C. M. Booth, and C. Dalla Vecchia, MNRAS 440, 2997 (2014), arXiv:1310.7571 [astro-ph.CO] .
  * Semboloni  _et al._ (2013) E. Semboloni, H. Hoekstra, and J. Schaye, Monthly Notices of the Royal Astronomical Society 434, 148 (2013).
  * Harnois-Déraps  _et al._ (2015) J. Harnois-Déraps, L. van Waerbeke, M. Viola, and C. Heymans, Monthly Notices of the Royal Astronomical Society 450, 1212 (2015).
  * Mead  _et al._ (2015) A. Mead, J. Peacock, C. Heymans, S. Joudaki, and A. Heavens, Monthly Notices of the Royal Astronomical Society 454, 1958 (2015).
  * Schneider  _et al._ (2019) A. Schneider, R. Teyssier, J. Stadel, N. E. Chisari, A. M. Le Brun, A. Amara, and A. Refregier, Journal of Cosmology and Astroparticle Physics 2019, 020 (2019).
  * Huang  _et al._ (2020) H.-J. Huang, T. Eifler, R. Mandelbaum, G. M. Bernstein, A. Chen, A. Choi, J. García-Bellido, D. Huterer, E. Krause, E. Rozo, _et al._ , arXiv preprint arXiv:2007.15026 (2020).
  * McAlpine  _et al._ (2016) S. McAlpine, J. C. Helly, M. Schaller, J. W. Trayford, Y. Qu, M. Furlong, R. G. Bower, R. A. Crain, J. Schaye, T. Theuns, _et al._ , Astronomy and Computing 15, 72 (2016).
  * van Daalen  _et al._ (2011) M. P. van Daalen, J. Schaye, C. M. Booth, and C. Dalla Vecchia, MNRAS 415, 3649 (2011), arXiv:1104.1174 [astro-ph.CO] .
  * Smith  _et al._ (2003) R. E. Smith, J. A. Peacock, A. Jenkins, S. D. M. White, C. S. Frenk, F. R. Pearce, P. A. Thomas, G. Efstathiou, and H. M. P. Couchman, MNRAS 341, 1311 (2003), arXiv:astro-ph/0207664 [astro-ph] .
  * Takahashi  _et al._ (2012) R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, and M. Oguri, ApJ 761, 152 (2012), arXiv:1208.2701 [astro-ph.CO] .
  * Choi  _et al._ (2016) A. Choi, C. Heymans, C. Blake, H. Hildebrandt, C. A. J. Duncan, T. Erben, R. Nakajima, L. Van Waerbeke, and M. Viola, MNRAS 463, 3737 (2016), arXiv:1512.03626 [astro-ph.CO] .
  * Joudaki  _et al._ (2017) S. Joudaki, C. Blake, C. Heymans, A. Choi, J. Harnois-Deraps, H. Hildebrandt, B. Joachimi, A. Johnson, A. Mead, D. Parkinson, M. Viola, and L. van Waerbeke, MNRAS 465, 2033 (2017), arXiv:1601.05786 [astro-ph.CO] .
  * Handley  _et al._ (2015) W. J. Handley, M. P. Hobson, and A. N. Lasenby, MNRAS 453, 4384 (2015), arXiv:1506.00171 [astro-ph.IM] .
  * Zuntz  _et al._ (2015) J. Zuntz, M. Paterno, E. Jennings, D. Rudd, A. Manzotti, S. Dodelson, S. Bridle, S. Sehrish, and J. Kowalkowski, Astronomy and Computing 12, 45 (2015), arXiv:1409.3409 [astro-ph.CO] .
  * Howlett  _et al._ (2012) C. Howlett, A. Lewis, A. Hall, and A. Challinor, Journal of Cosmology and Astroparticle Physics 2012, 027–027 (2012).
  * Lewis  _et al._ (2000) A. Lewis, A. Challinor, and A. Lasenby, ApJ 538, 473 (2000), arXiv:astro-ph/9911177 [astro-ph] .
  * Lemos  _et al._ (2020) P. Lemos, M. Raveri, A. Campos, Y. Park, C. Chang, N. Weaverdyck, D. Huterer, and A. R. Liddle, arXiv e-prints , arXiv:2012.09554 (2020), arXiv:2012.09554 [astro-ph.CO] .
  * Chang  _et al._ (2019) C. Chang, M. Wang, and LSST Dark Energy Science Collaboration, MNRAS 482, 3696 (2019), arXiv:1808.07335 [astro-ph.CO] .
  * Huterer and Turner (2001) D. Huterer and M. S. Turner, Phys. Rev. D 64, 123527 (2001), arXiv:astro-ph/0012510 [astro-ph] .
  * Wang (2008) Y. Wang, Physical Review D 77 (2008), 10.1103/physrevd.77.123525.
  * Albrecht  _et al._ (2006) A. Albrecht, G. Bernstein, R. Cahn, W. L. Freedman, J. Hewitt, W. Hu, J. Huth, M. Kamionkowski, E. W. Kolb, L. Knox, J. C. Mather, S. Staggs, and N. B. Suntzeff, arXiv e-prints , astro-ph/0609591 (2006), arXiv:astro-ph/0609591 [astro-ph] .
  * Raveri and Hu (2019) M. Raveri and W. Hu, Phys. Rev. D 99, 043506 (2019), arXiv:1806.04649 [astro-ph.CO] .
  * Handley and Lemos (2019) W. Handley and P. Lemos, arXiv e-prints , arXiv:1903.06682 (2019), arXiv:1903.06682 [astro-ph.CO] .
  * Joachimi and Bridle (2010) B. Joachimi and S. L. Bridle, A&A 523, A1 (2010), arXiv:0911.2454 [astro-ph.CO] .
  * Samuroff  _et al._ (2019) S. Samuroff, J. Blazek, M. A. Troxel, N. MacCrann, E. Krause, and DES Collaboration, MNRAS 489, 5453 (2019), arXiv:1811.06989 [astro-ph.CO] .
  * Amara and Réfrégier (2007) A. Amara and A. Réfrégier, MNRAS 381, 1018 (2007), arXiv:astro-ph/0610127 [astro-ph] .
  * van den Busch  _et al._ (2020) J. L. van den Busch, H. Hildebrandt, A. H. Wright, C. B. Morrison, C. Blake, B. Joachimi, T. Erben, C. Heymans, K. Kuijken, and E. N. Taylor, A&A 642, A200 (2020), arXiv:2007.01846 [astro-ph.CO] .
  * Asgari  _et al._ (2020) M. Asgari, T. Tröster, and C. Heymans, A&A 634, A127 (2020), arXiv:1910.05336 [astro-ph.CO] .
  * MacCrann and DES Collaboration (2017) N. MacCrann and DES Collaboration, MNRAS 465, 2567 (2017), arXiv:1608.01838 [astro-ph.CO] .
  * Astropy Collaboration  _et al._ (2013) Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, P. Greenfield, M. Droettboom, E. Bray, T. Aldcroft, M. Davis, A. Ginsburg, A. M. Price-Whelan, W. E. Kerzendorf, A. Conley, N. Crighton, K. Barbary, D. Muna, H. Ferguson, F. Grollier, M. M. Parikh, P. H. Nair, H. M. Unther, C. Deil, J. Woillez, S. Conseil, R. Kramer, J. E. H. Turner, L. Singer, R. Fox, B. A. Weaver, V. Zabalza, Z. I. Edwards, K. Azalee Bostroem, D. J. Burke, A. R. Casey, S. M. Crawford, N. Dencheva, J. Ely, T. Jenness, K. Labrie, P. L. Lim, F. Pierfederici, A. Pontzen, A. Ptak, B. Refsdal, M. Servillat, and O. Streicher, A&A 558, A33 (2013), arXiv:1307.6212 [astro-ph.IM] .
  * Astropy Collaboration and Astropy Contributors (2018) Astropy Collaboration and Astropy Contributors, AJ 156, 123 (2018), arXiv:1801.02634 [astro-ph.IM] .
  * Lewis (2019) A. Lewis,  (2019), arXiv:1910.13970 [astro-ph.IM] .
  * Hinton (2016) S. R. Hinton, The Journal of Open Source Software 1, 00045 (2016).
  * Hunter (2007) J. D. Hunter, Computing in Science & Engineering 9, 90 (2007).
  * Paulin-Henriksson  _et al._ (2008) S. Paulin-Henriksson, A. Amara, L. Voigt, A. Refregier, and S. L. Bridle, A&A 484, 67 (2008), arXiv:0711.4886 [astro-ph] .
  * Rowe (2010) B. Rowe, MNRAS 404, 350 (2010), arXiv:0904.3056 [astro-ph.CO] .
  * Hikage  _et al._ (2011) C. Hikage, M. Takada, T. Hamana, and D. Spergel, MNRAS 412, 65 (2011), arXiv:1004.3542 [astro-ph.CO] .
  * Asgari and Heymans (2019) M. Asgari and C. Heymans, A&A 624, A134 (2019), arXiv:1810.02353 [astro-ph.CO] .
  * Asgari  _et al._ (2019) M. Asgari, C. Heymans, H. Hildebrandt, L. Miller, P. Schneider, A. Amon, A. Choi, T. Erben, C. Georgiou, J. Harnois-Deraps, and K. Kuijken, A&A 624, A134 (2019), arXiv:1810.02353 [astro-ph.CO] .
  * Górski  _et al._ (2005b) K. M. Górski, E. Hivon, A. J. Banday, B. D. Wand elt, F. K. Hansen, M. Reinecke, and M. Bartelmann, ApJ 622, 759 (2005b), arXiv:astro-ph/0409513 [astro-ph] .
  * Alonso  _et al._ (2019) D. Alonso, J. Sanchez, A. Slosar, and LSST Dark Energy Science Collaboration, MNRAS 484, 4127 (2019), arXiv:1809.09603 [astro-ph.CO] .
  * Nicola  _et al._ (2020) A. Nicola, C. García-García, D. Alonso, J. Dunkley, P. G. Ferreira, A. Slosar, and D. N. Spergel, arXiv e-prints , arXiv:2010.09717 (2020), arXiv:2010.09717 [astro-ph.CO] .
  * Doux  _et al._ (2021) C. Doux, C. Chang, and (DES Collaboration), MNRAS 503, 3796 (2021), arXiv:2011.06469 [astro-ph.CO] .
  * Joachimi and Lin (2021) B. Joachimi and Lin, A&A 646, A129 (2021), arXiv:2007.01844 [astro-ph.CO] .

##  Appendix A Shear systematics

Figure 13: Impact of shear systematics on the $\xi_{\pm}(\theta)$ signal: The
fractional impact, $\delta\xi_{\pm}/{\xi}$, of the effects of (i) B-modes
(black) (ii) the approximation made in the response correction (red) (iii) the
PSF contamination (yellow) and (iv) the additive $c$-correction applied to the
shears (blue, dashed). To demonstrate how the fractional biases compare to the
uncertainties on a model fit to the data, the green shaded band represents the
uncertainty on the amplitude for each bin pair.

In addition to multiplicative biases, cosmic shear observables can also be
affected by additive systematic errors, potentially arising from issues with
the point-spread function modeling or the instrument. The DES Y3 catalog has
undergone extensive null testing, as summarised in Section II and presented in
(Gatti  _et al._ , 2021). Here, we demonstrate the robustness of the
tomographic cosmic shear analysis to PSF contamination (see Appendix A.1), a
spurious B-mode signal (detailed in Appendix A.2), and a residual additive
correction (detailed in Appendix A.3).

Figure 13 summarises and compares the predicted impact of these effects on the
tomographic cosmic shear measurements. The panels show the fractional impact,
$\delta\xi_{\pm}/{\xi}$, of each effect listed in the legend. For comparison,
the green shaded region shows the amplitude of the uncertainty or the inverse
signal to noise, defined for each bin pair as
$\sigma_{A}=\left(\xi^{T}C^{-1}\xi\right)^{-0.5}$.

###  A.1 PSF modeling

The observed shape of a galaxy inherits additional contributions due to
systematic errors, such as due to PSF misestimation,
$\delta\epsilon^{\textrm{sys}}_{\textrm{PSF}}$, and noise,
$\delta\epsilon^{\textrm{noise}}$, as

$\epsilon^{\rm
obs}=\gamma+\delta\epsilon^{\textrm{sys}}_{\textrm{PSF}}+\delta\epsilon^{\textrm{noise}}\,,$
(32)

neglecting any multiplicative biases (see equation 1).

It is common to parameterize the uncertainty in modeling the point spread
function in terms of $\alpha,\beta$ parameters Paulin-Henriksson  _et al._
(2008). This can be extended to include a parameter $\eta$ for the PSF size-
dependence of the additive error in shear measurement as

$\delta\epsilon^{\textrm{sys}}_{\textrm{PSF}}=\alpha\epsilon_{\rm
model}+\beta\left(\epsilon_{\rm*}-\epsilon_{\rm
model}\right)+\eta\left(\epsilon_{\rm*}\frac{T_{\textrm{\rm*}}-T_{\rm
model}}{T_{\rm*}}\right).$ (33)

$\epsilon_{\rm model},T_{\rm model}$ denote the shape and size of the PSF
model and $\epsilon_{*},T_{*}$ are measured directly from a field of reserved
stars that are not used in the fitting of the PSF model (Gatti  _et al._ ,
2021). The first term considers linear leakage, which is suppressed by the
parameter $\alpha\ll 1$. The second and third term are the shape and size
residual dependencies, where the parameters $\beta$ and $\eta$ are of order
unity in the unweighted moments approximation.

The uncertainty due to PSF systematics in DES Y3, quantified in terms of the
$\rho$-statistics (Rowe, 2010; Jarvis  _et al._ , 2016) are found to have a
substantially smaller amplitude compared to DES Y1, owing to improvements
detailed in Ref. (Jarvis  _et al._ , 2021). These statistics are used to fit
for $\alpha,\beta,\eta$ per redshift bin, with the best-fit values and their
corresponding $\chi^{2}$ reported in Table 4, with similar amplitudes to DES
Y1 (Gatti  _et al._ , 2021). As the inclusion of the $\eta$ parameter does not
significantly alter the reduced $\chi^{2}_{\nu}$ model fit, we deduce that the
effect of PSF size error is subdominant. However, we preserve this more
complete model.

Here, the impact of PSF modeling uncertainties on cosmological constraints is
tested. The best-fit $\alpha,\beta,\eta$ are propagated to determine the
expected additive contamination of the true cosmic shear signal as
$\xi_{+}^{\textrm{obs}}=\xi_{+}+\delta\xi^{\textrm{PSF}}$, where
$\delta\xi^{\textrm{PSF}}=\left\langle\delta\epsilon_{\textrm{PSF}}^{\textrm{sys}}\,\delta\epsilon_{\textrm{PSF}}^{\textrm{sys}}\right\rangle$.
The fractional PSF contamination with respect to the baseline simulated shear
correlation functions , $d\xi/\xi$, is shown in yellow in Figure 13. The
impact is most significant for the third redshift bin, but in the relevant
radial range is at most 30% of the physical cosmic shear signal and limited to
only small-scales of a few bin pairs. The 2$\sigma$ limits of the
$\delta\xi^{\textrm{PSF}}$ constraints are used to contaminate a simulated Y3
measurement in order to test the residual impact of the PSF bias on cosmic
shear cosmological parameters. The result, shown in Figure 14 in red and
yellow contours, compared to the baseline green, finds the $\Omega_{\rm
m}-\sigma_{8}$ parameters to be insensitive to PSF modeling errors in the Y3
analysis.

| Bin 1 | Bin 2 | Bin 3 | Bin 4
---|---|---|---|---
$\alpha$ | $0.010_{-0.005}^{+0.005}$ | $-0.001_{-0.005}^{+0.005}$ | $-0.004_{-0.005}^{+0.005}$ | $0.014_{-0.006}^{+0.006}$
$\beta$ | $0.6_{-0.2}^{+0.2}$ | $1.4_{-0.2}^{+0.2}$ | $2.5_{-0.2}^{+0.2}$ | $1.3_{-0.3}^{+0.3}$
$\eta$ | $-4.6_{-2.7}^{+2.6}$ | $-4.5_{-2.7}^{+2.7}$ | $3.0_{-2.7}^{+2.7}$ | $4.2_{-3.0}^{+3.1}$
$\chi^{2}_{\nu}$ | $1.02_{-0.01}^{+0.02}$ | $1.43_{-0.01}^{+0.02}$ | $1.20_{-0.01}^{+0.02}$ | $1.25_{-0.01}^{+0.02}$
Table 4: The values of the parameters $\alpha$,$\beta$ and $\eta$ for each
redshift bin, estimated from fits to the mean-substracted $\rho$-statistics,
according to equation 33, as well as the reduced goodness-of-fit,
$\chi^{2}_{\nu}$ for $\nu=117$ degrees of freedom.

###  A.2 B-mode contamination

Figure 14: Robustness to PSF modeling systematics: The $\pm 2\sigma$ limits of
the expected additive contamination of PSF residual uncertainty to the true
cosmic shear signal, $\delta\xi^{\textrm{PSF}}$ are propagated to cosmological
parameters and shown in red and yellow. For comparison, the uncontaminated
simulated data vector is shown in green shaded contours.

To first order, lensing does not produce B-modes in the shear field for
reasons of symmetry. Therefore, a detection of B-modes can indicate either a
contamination by observational systematic effects or higher-order lensing or
astrophysical effects, such as intrinsic alignments. For a fuller description
and discussion of E- and B-mode power spectra, we refer the reader to (Krause
_et al._ , 2021). Here, we test for the presence of B-modes in the shape
catalog and potential contamination of the two-point functions used in the
cosmological analysis. Section 6.4 of (Gatti  _et al._ , 2021) describes two
complementary approaches toward measuring B-modes for the full sample of
source galaxies without any redshift binning: the map-based pseudo-$C_{\ell}$
(Hikage  _et al._ , 2011) and Complete Orthogonal Sets of E/B-Integrals
(COSEBIs; Schneider  _et al._ , 2010; Asgari and Heymans, 2019). B-modes
computed using both pseudo-$C_{\ell}$ and COSEBIs methods were shown to be
consistent with zero, with $p$-values of 0.06 and 0.87, respectively. Here, we
additionally report tomographic measurements of B-modes using the
pseudo-$C_{\ell}$ approach. We note that a non-detection in harmonic space
does not fully rule out contamination, as was shown in (Asgari  _et al._ ,
2019).

For each redshift bin, we build two HEALPix (Górski  _et al._ , 2005b) maps
with resolution ${n_{\rm side}=1024}$ of the cosmic shear signal by computing
the weighted average of response-corrected ellipticities of galaxies within
each pixel. We then estimate the E- and B-mode power spectra of these maps by
the method of pseudo-$C_{\ell}$ using NaMaster (Alonso  _et al._ , 2019), an
open-source code that deconvolves the effects of masked regions from the
harmonic space coefficients. We use the inverse-variance weight masks given by
the weighted count maps and measure auto- and cross-spectra for multipoles in
the range $\ell\in[8-2048]$. We turn these into 32 bins evenly spaced on a
square-root scale, which spreads signal-to-noise more evenly than linear or
logarithmic binning. The measured auto power spectra receive an additive bias
from the shape-noise power spectrum $N_{\ell}$, which may diverge from the
approximation $N_{\ell}=\sigma_{e}^{2}/\bar{n}$ due to masking effects and
properties of the pseudo-$C_{\ell}$ estimator. We therefore employ the
analytic formula derived in (Nicola  _et al._ , 2020) to estimate the noise
power spectrum and subtract it from the initial measurements.

In order to evaluate the B-mode covariance matrix, we use $10\,000$ Gaussian
simulations of the shear fields at the baseline cosmology, following the
method outlined in Doux  _et al._ (2021). We draw shear maps at resolution
${n_{\rm side}=1024}$, which we then sample at the positions of galaxies in
the real data. The intrinsic ellipticities of galaxies are obtained by
randomly rotating measured ellipticities. This procedure preserves the shape
noise, in terms of galaxy density and ellipticity distribution, and produces a
null B-mode signal. For each simulation, we then measure the B-mode power
spectra using the data masks. Finally, we compute the sample covariance matrix
and use it to form a $\chi^{2}$ statistic to test the null hypothesis of no
B-modes. We have verified that the number of simulations is sufficient for
this statistic to converge. We report $\chi^{2}$ statistics for each bin
combination as well as the $\chi^{2}$ for the full data vector, accounting for
the full covariance. We obtain $\chi^{2}=344.0$ for 320 degrees of freedom,
consistent with the absence of B-modes.

Finally, we propagate the B-mode contamination to the shear two-point
correlation functions, shown in Figure 13. The two-point correlation functions
$\xi_{\pm}$ are related to angular spectra by expressions of the form (see
Eqn. 11)

$\xi_{\pm}^{ab}\quantity(\theta)=\sum_{\ell}F_{\ell}^{\pm}\quantity(\theta)\quantity(C_{\ell}^{ab,\rm
EE}\pm C_{\ell}^{ab,\rm BB}),$ (34)

where $a$ and $b$ denote redshift bin indices. This allows us to compute an
approximate covariance between two-point functions and the bin-averaged B-mode
power spectra $C_{L}^{ab,\rm BB}$, given as

$\begin{split}\mathbf{C}_{\rm B\pm}&\equiv\expectationvalue{\Delta
C_{L}^{ab,\rm BB}\Delta\xi_{\pm}^{cd}\quantity(\theta)}\\\
&\approx\sum_{\ell}F_{\ell}^{\pm}\quantity(\theta)\quantity[\expectationvalue{\Delta
C_{L}^{ab,\rm BB}\Delta C_{\ell}^{cd,\rm EE}}\pm\expectationvalue{\Delta
C_{L}^{ab,\rm BB}\Delta C_{\ell}^{cd,\rm BB}}].\end{split}$ (35)

We then approximate $F_{\ell}^{\pm}\quantity(\theta)$ and the signal to be
piece-wise constant and use the covariance matrices of E- and B-mode spectra
measured from the simulations described above to compute $\mathbf{C}_{\rm
B\pm}$. Assuming all data are Gaussian distributed, we finally compute the
conditional distribution of the additive bias to the two-point functions given
the B-mode measurements, which is Gaussian with mean

$\left.\Delta\xi_{\pm}\quantity(\theta)\,\middle|\,\hat{C}_{\ell,\rm
data}^{\rm BB}\right.=\mathbf{C}_{\rm B\pm}\dotproduct\mathbf{C}_{\rm
BB}^{-1}\dotproduct\hat{C}_{\ell,\rm data}^{\rm BB},$ (36)

where we omit redshift bin indices to imply vectorization.

Figure 15:  The impact of bimodality on constraining power in $S_{8}$: The
left-hand panel shows analyses of three synthetic Y3-like cosmic shear data
vectors with random realizations of noise, determined with the Y3 covariance,
one of which presents a bimodality in the $a_{1}-a_{2}$ constraints, using
with no SR and TATT modeling. The right-hand panel shows posteriors from the
data with permutations on the choice of intrinsic alignment model (TATT and
NLA-no-$z$) and the inclusion of SR. Compared to the Fiducial analysis
(green), the $\rm\Lambda CDM$-Optimized scales (black), and the redshift-
independent NLA model (red), the TATT-no-SR (yellow) case exhibits a degraded
$S_{8}$ constraint. The combination of the conservative Fiducial scale-cuts
and TATT model, results in intrinsic alignment parameters that are not well-
constrained that are degenerate with, and reduce the precision of the $S_{8}$
constraint. The inclusion of SR is effective in constraining the intrinsic
alignment parameters, and for the TATT case, alleviating the impact of the
bimodality. On the other hand, when using a less conservative intrinsic
alignment model, which is mildly preferred by our data (Secco  _et al._ ,
2021), while the $a_{1}$ constraint is substantially improved, the $S_{8}$
precision is negligibly impacted by including SR.

###  A.3 Additive shear

While we conclude that the propagation of PSF model uncertainty and the B-mode
signal is below the statistical uncertainty, the catalog gives an unidentified
source of mean shear or additive bias (see equation 1). This is too large in
some redshift bins to be consistent with shape noise and cosmic variance. The
low-level contribution from the constant additive ellipticity bias is not
explicitly marginalized over, but corrected as a global constant per redshift
bin and ellipticity component, $c=\langle e_{1,2}^{i}\rangle$ reported in
Table 1. The impact of this additive correction on the shear two-point
correlation functions is

$\xi_{\pm}^{ij}=\xi_{\pm,\rm{true}}^{ij}+(\langle e_{1}^{i}\rangle\langle
e_{1}^{j}\rangle\pm\langle e_{2}^{i}\rangle\langle e_{2}^{j}\rangle)\,,$ (37)

and the result of artificially introducing this bias due to mean shear to an
un-biased simulated data vector is shown in Figure 13 as a blue dashed line.
The residual mean shear, $c=\langle e_{1,2}^{i}\rangle$, is subtracted from
each galaxy’s ellipticity, which minimizes its impact on $\xi_{\pm}$. Note
that the residual mean shear may be a result of a systematic effect that
generates a scale-dependent additive bias in $\xi_{\pm}$, such that simply
subtracting the mean will not correct for this scale dependence.

The approximation assumes that the galaxies shapes are not correlated in the
absence of lensing, and the mean response is relatively homogeneous across the
footprint. The impact of this simplification in the computation of the shear
response is shown in red in Figure 13 to be negligible at the current level of
precision.

##  Appendix B Constraining power & bimodality in noisy data

In this section, the constraining power of the cosmic shear posteriors is
investigated in more detail. In particular, the occurance of bimodality in the
intrinsic alignment parameters is investigated for various analysis choices,
including the intrinsic alignment model and the inclusion/exclusion of SR
data.

Prior to un-blinding, the potential for bimodality in the $a_{1}-a_{2}$ space
was explored using a simulated data vector, created with the fiducial analysis
pipeline. We found that it is possible for particular noise realisations to
present such a feature, which, given the degeneracy with $S_{8}$,
substantially alters the precision of that parameter. Ten realisations of
noise were added, consistent with the Y3 covariance, and cosmological
inference was performed on each, with the baseline TATT model and no SR. The
resulting parameter constraints from three of these datavectors are shown in
the left-hand panel of Figure 15. One of the ten realisations, ‘noise 2’
(yellow contour), was found to exhibit bimodality in the $a_{1}-a_{2}$ plane.
The intrinsic alignment parameters are degenerate with $S_{8}$. This
particular realisation presents significantly degraded cosmological $S_{8}$
constraint that is scattered toward a lower value of $S_{8}$, with lower
signal-to-noise.

The Y3 data also exhibits a bimodal posterior in the $a_{1}-a_{2}$ plane that
is less pronounced. The right-hand panel of Figure 15 shows the Fiducial Y3
cosmic shear result in green filled contours and the $\rm\Lambda
CDM$-Optimized analysis, in black, with significantly improved constraints in
the $S_{8},a_{1},a_{2}$ parameters. This bimodality and the degraded $S_{8}$
constraint are consistent with findings of Ref. (Asgari  _et al._ , 2021), who
use the NLA-$a_{1}$ as their fiducial choice. Similarly, they report that when
opting for the more conservative NLA choice, they observe bimodality in these
astrophysical parameters, with degraded cosmological posteriors. Therefore, we
interpret this feature of a doubly-peaked posterior as an internal degeneracy
of the intrinsic alignment model that is reduced or eliminated as statistical
power increases.

Given the the degeneracy in $S_{8}-a_{1}/a_{2}$, we find reduced bimodality
and significantly improved $S_{8}$ constraints with either the inclusion of
small-scale measurements, such as the $\rm\Lambda CDM$-Optimized choice, the
inclusion of orthogonal SR instrinsic alignment information, or a more
aggressive intrinsic alignment model, such as NLA-$a_{1}$. Alongside the
Fiducial and Optimized constraints, Figure 15 shows a variant of the analysis
without SR (yellow). We see a substantial improvement in the precision of the
$a_{1}$, and therefore $S_{8}$, compared to the case with SR (green).
Moreover, a case that uses a simpler intrinsic alignment model results in
substantially tighter $S_{8}$ posteriors, with a $\times\sim 1.5$ improvement
from TATT (yellow) to NLA-$a_{1}$ (red). These improvements can be attributed
to the elimination of negative $a_{1}$ space and the upper and lower bounds of
the $a_{2}$ space.

We test the impact of SR in the case of the data-preferred NLA-$a_{1}$ model,
shown in Figure 15 (red to blue). Note that our companion paper (Secco  _et
al._ , 2021), performs a detailed model comparison that finds the data favours
a less conservative intrinsic alignment model than TATT (see their Table 3).
We find that the inclusion of SR has little effect on the $S_{8}$ constraint,
although it provides significantly tighter constraints on the intrinsic
alignment parameters, which are not degenerate with $S_{8}$ to begin with.

##  Appendix C Internal consistency

PPD test | Calibrated $\tilde{p}$-values | $\xi_{+}$ | $\xi_{-}$
---|---|---|---
Goodness-of-fit tests |  |  |
Cosmic shear | 0.268 | 0.252 | 0.422
$\xi_{+}$ | 0.234 | 0.234 | –
$\xi_{-}$ | 0.382 | – | 0.382
Data splits |  |  |
Bin 1 vs. no bin 1 | 0.357 | 0.196 | 0.759
Bin 2 vs. no bin 2 | 0.394 | 0.547 | 0.132
Bin 3 vs. no bin 3 | 0.014 | 0.041 | 0.070
Bin 4 vs. no bin 4 | 0.427 | 0.376 | 0.614
Low-z vs. High-z | 0.993 | 0.992 | 0.974
High-z vs. Low-z | 0.207 | 0.324 | 0.282
Large vs. small scales | 0.660 | 0.646 | 0.441
Small vs. large scales | 0.083 | 0.068 | 0.332
$\xi_{+}$ vs. $\xi_{-}$ | 0.601 | 0.601 | –
$\xi_{-}$ vs. $\xi_{+}$ | 0.422 | – | 0.422
Table 5: Summary of internal consistency test $\tilde{p}$-values. The ‘PPD
test’ column specifies the details of the comparison. For consistency tests,
‘A vs. B’ indicates a comparison of observations for data A with PPD
realizations for data A derived from data B. The second column shows the
calibrated $\tilde{p}$-value for each test, obtained by comparison of PPD
tests on simulated data. The third and fourth column show the calibrated
$\tilde{p}$-values when PPD metrics are restricted to the $\xi_{+}$ and
$\xi_{-}$ components respectively. These tests correspond to the parameter
constraints shown in Section IX. All internal consistency tests pass the pre-
defined (arbitrary) threshold of 0.01.

Table 5 summarizes the calibrated $\tilde{p}$-values for the full data vector
as well as each subset considered in Section IX (see Ref. Doux  _et al._
(2020) for details of the methodology). As stated previously, the threshold
for consistency is chosen to be 0.01. There are no obvious discrepancies
between the PPD realizations of cosmic shear and the actual data. We compute a
$\tilde{p}$-value for the goodness-of-fit of cosmic shear of $p=0.268$,
indicating no evidence for tension between the measurements and PPD
realizations.

When splitting the data into subsets, all overall $\tilde{p}$-values are above
0.01, indicating no sign of tensions between redshift bins, angular scales or
the two statistics that describe the DES Y3 cosmic shear data. These findings
are consistent with the agreement in cosmological parameters measured by the
subsets of the data shown in Section IX, validating an internally consistent
analysis.

The modeling of intrinsic alignments is degenerate with the $\Omega_{\rm
m}-S_{8}$ parameters, and systematic errors in the data can be absorbed by the
intrinsic alignment model, impacting the cosmological posteriors. Thus,
disentangling the true cosmic shear signal from intrinsic alignments requires
accurate knowledge of the redshift distributions and their errors. As such, we
demonstrate consistency of the intrinsic alignment constraints with redshift
(Efstathiou and Lemos, 2018). Figure 16 demonstrates the stability of the
$S_{8}$ and intrinsic alignment $a_{1}-a_{2}$ solution across redshift for the
DES data: the posteriors across the five parameters are consistent for each
subset of the data within $1\sigma$. The first point to note is that the
$S_{8}$ parameter is stable in the removal of any of these subsets of the
data, and unsurprisingly, Bin 4 holds the most cosmological constraining
power. Next, the intrinsic alignment amplitude, $a_{1}$, is stable to the
removal of photometric redshift bins, within 1$\sigma$. Moreover, all five
intrinsic alignment posteriors, $[a_{1},a_{2},\eta_{1},\eta_{2},b_{\rm ta}]$,
a subset of which are shown in Figure 16, are consistent with the intrinsic
alignment solution from the full dataset. Bins 1 and 4 carry a higher weight
in fixing the amplitudes, $a_{1}$ and $a_{2}$.

##  Appendix D Robust redshift calibration in the presence of intrinsic
alignment

The redshift uncertainty parameters, $\Delta z$, are shown in Figure 18 with
varying redshift methodologies: removing the SR method (red), removing the SR
and WZ methods (yellow) and using Hyperrank (blue). The corresponding
constraints on the $\Omega_{\rm{m}}-S_{8}$ parameters are shown in Figure 10.
We find that these nuisance parameters are most constrained by the addition of
the SR, but that each tier of the methodology gives consistent posteriors.
When accounting for the full-shape uncertainty in the $n(z)$ with Hyperrank,
we find that the need for shifts in $\Delta z$, such as the case for Bin 3,
are alleviated. This suggests that an uncertainty in the shape of the $n(z)$
is compensated for by a more substantial shift in the mean of the
distribution. While the approximation of the uncertainty as a shift in the
mean does not impact the cosmological parameters at the precision of this
analysis, this demonstrates that shifts in the mean have the potential to be
misleading.

Uncalibrated redshift error can result in an incorrectly inferred intrinsic
alignment signal or be absorbed by unconstrained model nuisance parameters in
the likelihood analysis, as suggested in (Wright  _et al._ , 2020; Joachimi
and Lin, 2021). To test that such an effect is not occurring in the DES Y3
analysis, we include a detailed account of the full uncertainty in the $n(z)$
estimates (Myles  _et al._ , 2020). Section X shows that approximating
uncertainty in the $n(z)$ to a shift in the mean leads to a consistent set of
cosmological parameters for the Y3 analysis. In this Appendix, we add to this
a demonstration that the intrinsic alignment parameter constraints are fully
consistent when marginalizing over the full shape uncertainty of $n(z)$ with
Hyperrank, instead of the approximation of the mean shift (green).
Interestingly, Figure 18 shows the $a_{1},a_{2}$ posteriors in the Hyperrank
analysis are significantly tighter, though consistent. This demonstrates that
shifts in the mean have the potential to be misleading and could bias the
measurement of intrinsic alignment parameters, a possibility we now explore.
We find no evidence that the nuisance model parameters absorb residual
observational systematics in the analysis.

Figure 16:  Intrinsic alignment consistency across redshift bins: The
posteriors for the $S_{8}$ and intrinsic alignment parameters, $a_{1}$ and
$a_{2}$, as correlations involving each redshift bin are removed. For
comparison, the Fiducial result using all redshift bins is shown in the grey
shaded contours. The 68 and 95% constraints are shown and a zero-line is
marked for reference.

Figure 17:  Stability of redshift uncertainty posteriors, $\Delta z$, from the
cosmological-inference chains, with the varying redshift methods and modeling.
The filled grey contours correspond to the priors, listed in Table 2. The
Fiducial analysis (green) is compared to decreasing complexity in methodology,
removing the SR, ‘SOMPZ+WZ’ (red), as well as the clustering redshift
likelihood, ‘SOMPZ only’ (yellow), as well as the translation of Hyperrank
parameters to $\Delta z$ (blue), when the uncertainty in the full shape of the
$n(z)$ is accounted for. While the approximation of the uncertainty as a shift
in the mean does not impact $S_{8}$ at the precision of this analysis, a
substantial shift in the mean of the $n(z)$ can compensate for a change in
shape, as seen in bin 3.

Figure 18: Stability of $S_{8}$, the intrinsic alignment model parameters,
$a_{1}$, $a_{2}$, $\eta_{1}$, when including additional information: The
filled green contours show the Fiducial analysis, which includes SR measured
with the MagLim sample, consistent with one that includes SR measured with the
redMaGiC lens sample (yellow). Using the large-scale-only MagLim SR (red),
tests the robustness of the modeling along with the result incorporating the
$\rm\Lambda CDM$-Optimized cosmic shear without any SR (black). In addition,
the Fiducial Hyperrank result (blue) modulates the shape of the $n(z)$ and
finds more constraining $a_{1}$-$a_{2}$ posteriors that are consistent.

In this Appendix, the robustness of the results of the analysis to choices in
the SR is also investigated. More specifically, Figure 18 shows the
constraints on the $S_{8}$ parameters, the intrinsic alignment amplitudes,
$a_{1}$ and $a_{2}$, and their redshift dependence, $\eta_{1}$, when the
alternative lens sample, redMaGiC, is used in the measurement (yellow) instead
of the Fiducial MagLim (green). In addition, a $\rm\Lambda CDM$-Optimized case
that includes smaller-scale measurements and neglects SR is shown in black.
While this method has been validated in a simulated framework (Sánchez  _et
al._ , 2021), here, the stability of the data constraints is demonstrated: The
$S_{8}$ and intrinsic alignment posteriors are consistent. In all variations
tested, the $a_{1}$ amplitude is consistent with zero, with the Fiducial
MagLim result finding the most negative value. The ‘$\rm\Lambda
CDM$-Optimized, no SR’ case, that does not depend on either lens sample, or
the SR method is centred on $a_{1}=0$.

Furthermore, Figure 18 shows the consistency of the constraints when large-
scale (LS) SR are used (red). The fiducial SR measurements use small angular
scales to compute the ratios, which can be subject to systematics in the
modeling of non-linear galaxy bias, baryonic effects, and intrinsic
alignments. The LS-SR measurements are assumed to be independent of those from
the small scales. For this consistency check, we assume any cross-covariance
between the LS-SR and the cosmic shear measurements is negligible. The results
are found to be consistent, demonstrating the robustness to these modeling
effects, with the LS-SR preferring a slightly higher value of $S_{8}$, and
$a_{1}$ and $a_{2}$ values closer to zero. Overall, the cosmic shear $S_{8}$
and intrinsic alignment posteriors are robust to the inclusion of SR, as well
as the lens sample and scales used in that method.

##  Appendix E The unblinding process

After unblinding the cosmological parameters, two updates to the analysis were
made that marginally impacted the results of this work: a standard, planned
update to the covariance matrix to the ‘3x2pt’ best-fit parameters, as well as
a change of the fiducial lens sample, from redMaGiC to Maglim, which impacts
the SR method (see Ref. (DES Collaboration, 2021) for a discussion). While the
use of either of the two SR give consistent cosmological constraints, it does
shift the $S_{8}$ parameter by 0.3$\sigma$. This is primarily attributed to a
shift in the intrinsic alignment $a_{1}-a_{2}$ space, as demonstrated in
Figure 18 (the difference between the green and yellow contours) and discussed
in more detail in Appendix D.

##  Appendix F Cosmological parameters

The 1D posteriors for the full $\rm\Lambda CDM$ cosmological parameter space
is shown in Figure 19. While the DES constraint on $A_{\rm s}$ is weak, it is
interesting that there is no evidence for a discrepancy with Planck in this
parameter. We find no significant constraints beyond the prior imposed on the
parameters $\Omega_{\rm b}$, $H_{0}$, $n_{\rm s}$, and $\Omega_{\nu}h^{2}$.
The priors for all six parameters are listed in Table 2.

Figure 19:  The posteriors for a subset of the $\rm\Lambda CDM$ cosmological
parameters for the Fiducial and $\rm\Lambda CDM$-Optimized analyses.