Mpc^{-1}$. The fact that our constraints are stable when moving to larger
scales, or using the fully perturbative model, suggests that our analysis is
under excellent perturbative control: any differences in the modeling between
CLEFT and HEFT, which must exist beyond one-loop order, are not important at
the accuracy level of our data, or indeed for the more stringent requirements
of our mock tests.

We can also consider the $S_{8}$ constraints from individual lens bins to
check the internal consistency of the data. As shown in Figure 15, all of our
lens bins yield consistent results, although BGS1 shows a $\sim 1\sigma$
preference for higher $S_{8}$ than the other two bins. The preference for
lower values of $S_{8}$ in the first and third lens bin is largely driven by
projection effects that become more substantial when fitting to individual
bins. As expected, the BGS1 bin yields the tightest $S_{8}$ constraints, as it
has comparably low IA contamination compared with BGS0 but is at higher
redshift and thus we are able to model to higher $\ell$ given our scale cuts
of $k_{\rm max}=0.4\,h\,\rm Mpc^{-1}$, while the LRG1 bin drives our
constraints on $\Omega_{m}$.

Figure 14: Comparison of constraints varying the maximum scale included in the
analysis, as well as the galaxy bias model employed. Our fiducial constraints
using HEFT with $k_{\rm max}=0.4\,h\,\rm Mpc^{-1}$ are shown in black. A
purely perturbative analysis, i.e., using CLEFT rather than HEFT, but with all
other settings kept the same is shown in blue. When using CLEFT we adopt
broader counterterm priors, as described in the text. The orange and green
contours show similar analyses using $k_{\rm max}=0.2\,h\,\rm Mpc^{-1}$ with
HEFT and CLEFT respectively. Our fiducial analysis is $20\%$ more constraining
than fitting to $k_{\rm max}=0.2\,h\,\rm Mpc^{-1}$, but otherwise the
differences that we observe are consistent with tests done with noiseless
simulations. Figure 15: Comparison of constraints from individual lens bins
(colored contours) and the fiducial combined constraints (black contours). It
is apparent that the second BGS bin (orange, labeled $l1$) is the most
constraining bin on $S_{8}$, while the first LRG bin (green, labeled $l2$)
drives our $\Omega_{m}$ constraining power. Both of these finding are
consistent with tests we have performed on noiseless simulated data.

###  VI.1 Including BAO

The line-of-sight projection in two-dimensional data like those used in this
paper washes out much of the shape information in the three-dimensional power
spectrum that can be used to constrain cosmological parameters beyond the
lensing amplitude $S_{8}$ such as $\Omega_{m}$ and $H_{0}$.666While we note
that there is still some information, particularly in $C^{gg}_{\ell}$ (see
e.g. Figure 16), in order to be conservative in our modeling choices our
fiducial analysis setup is not optimized to extract this information since (a)
we only use the auto-spectrum of the lens galaxies within the DES footprint to
ensure sample homogeneity, thus reducing the area over which $C^{gg}_{\ell}$
is measured and (b) we do not include $C^{gg}_{\ell}$ from lenses where the
cross-spectrum is IA-dominated to avoid sampling over an excessive number of
free parameters. We have made these choices considering that their effect on
$S_{8}$ is minimal and that the constraints on other cosmological parameters
from these data alone are not competitive. While this does not significantly
affect our ability to constrain $S_{8}$, we can slightly tighten our
constraints and avoid sampling ruled-out cosmological parameter space by
including additional large-scale structure data. To avoid including additional
amplitude information we include constraints on the expansion history from
baryon acoustic oscillations (BAO) as measured in the Sloan Digital Sky
Survey—specifically, we use the BAO data measured in the LRGs from BOSS (DR12)
[152] and LRGs, QSOs and Ly$\alpha$ in eBOSS (DR16) [153]. These data were
deemed sufficiently statistically independent to be combined in ref. [153] and
together (when combined with a BBN prior on $\omega_{b}$) significantly
constrain the available parameter space in $\Omega_{m}$ and $H_{0}$.

The orange contour in Figure 12 shows the cosmological constraints from our
data when combined with these BAO data, which yield

$\displaystyle S_{8}$ $\displaystyle=0.840^{+0.038}_{-0.045}\,(0.843)$
$\displaystyle\Omega_{m}$
$\displaystyle=0.300^{+0.012}_{-0.015}\,(0.295)\quad\text{(w. BAO)}$ (62)
$\displaystyle\sigma_{8}$ $\displaystyle=0.840\pm 0.044\,(0.850)\,.$

The BAO allow us to break the $\sigma_{8}$-$\Omega_{m}$ degeneracy to measure
$\sigma_{8}$ and, in addition, since the degeneracy is not perfect, slightly
tighten the $S_{8}$ constraint by $9\%$. The resulting constraints are also in
excellent agreement with the cosmological constraints from the Planck CMB
data, which is not surprising since the BAO and lensing constraints are
independently consistent with them.

###  VI.2 Dependence on IA Parametrization and Priors

Figure 16: Comparison of constraints varying the IA model parameterization
employed in our analysis. The black contours show our fiducial analysis,
allowing one set of IA parameters per cross-spectrum, and restricting the
source–lens bin combinations used to only those with relatively small
potential IA contamination, as described in §V.1.1. The blue, orange and green
contours show analyses analyzing all source and lens bin combinations assuming
that the IA parameters evolve linearly in redshift. The blue contours assume
our fiducial second order EFT IA model and one linear function
$c_{\mathcal{O}}(z)$ for each source bin. The orange contours assume the same
linear function $c_{\mathcal{O}}(z)$ for all source bins, while the green
contours assume the same but use the nonlinear alignment (NLA) model. These
more restrictive IA models result in $35\%$, $46\%$ and $59\%$ more
constraining power respectively, but make strong assumptions about the
redshift and scale dependence of the IA signal beyond the evidence provided by
current IA measurements. Until better IA measurements from the data or IA
simulations are available, we favor the constraints provided by our fiducial
model and quote these for the duration of this work. Figure 17: The width of
our constraint on $S_{8}$ as a function of the width of the $c_{s}$ prior in
each cross-correlation pair $\Delta c_{s}$. The trend is roughly fit assuming
that the theoretical IA error adds to the width of the constraint in
quadrature, with the theoretical error overtaking the sample variance around
$\Delta c_{s}=8$.

Given the novelty of the fiducial IA model that we apply in this analysis, we
now present how our constraints change when making IA assumptions more in line
with those used in previous galaxy lensing analyses. Figure 16 shows these
variations, where the black contours again represent our fiducial constraints
that use our EFT model for IA, keeping operators up to second order, and
allowing for one set of IA parameters per source–lens bin pair (i.e., our
MAIAR parameterization). The blue, orange and green contours fit to all bin
combinations with varying levels of IA model complexity. The blue contours
make the same assumptions as our fiducial analysis, except there we use a
linear spline model for the redshift evolution of the IA parameters, allowing
for one linear function $c_{\mathcal{O}}^{i}(z)$ per source bin. The orange
contours make the additional assumption that the linear IA functions
$c_{\mathcal{O}}^{i}(z)$ are shared between source bins, i.e.,
$c_{\mathcal{O}}^{i}(z)=c_{\mathcal{O}}^{0}(z)$ for all $i$. Finally, the
green contours show an analysis assuming the nonlinear alignment (NLA) model,
although we use the one-loop prediction for $P_{\delta,s_{ij}}(k)$ rather than
the fully nonlinear prediction sometimes used when applying the NLA model.
These models all yield $S_{8}$ posterior means that are consistent with each
other, but result in significantly different levels of constraining power.
Indeed, adopting the two-parameter NLA model artificially improves our
constraining power by $63\%$ over our fiducial results. The majority of the
improvement comes from constraining the redshift evolution of the IA
parameters to be linear, even when allowing separate linear functions per
source bin. Doing so improves our $S_{8}$ constraining power by $36\%$.
Further restricting to one linear function, $c_{\mathcal{O}}(z)$, for all
source bins shrinks our $S_{8}$ errors by $51\%$ from our fiducial model.

As a further comparison to IA models used in the literature, we conducted an
analysis using a Lagrangian version of the common tidal alignment and tidal
torquing model (TATT), where we set $\alpha_{s}$ and $c_{t}$ to zero while
assuming a linear redshift evolution of the other IA parameters. Given the
additional terms generated by the advection of shapes in LPT, this is not
entirely equivalent to the standard TATT model, but it is reasonably close
with the same number of degrees of freedom. The resulting constraint shown in
Figure 13 is very similar to using our full IA model while assuming linear
redshift evolution.

We can also explore the dependence of our constraints on the width of our IA
priors, i.e., the extent to which the $S_{8}$ constraint is weakened by
conservative assumptions about the ranges that IA parameters can assume for
weak-lensing source samples. Figure 17 shows our $S_{8}$ constraint as a
function of the prior width $\Delta c_{s}$, where we also scale the nonlinear
IA parameter priors by an equal factor for consistency. The theoretical error
induced on our measurement by unknown IA bias parameters adds to the $S_{8}$
posterior width roughly in quadrature

$\sigma^{2}=\sigma_{0}^{2}\Big{(}1+\Big{(}\frac{\sigma_{c_{s}}}{\sigma_{c_{s},0}}\Big{)}^{2}\Big{)}\
,$ (63)

becoming dominant at $\sigma_{c_{s,0}}=8$. This number is large, even relative
to (conservative) expectations from simulations, because we have purposefully
picked only those bins where the IA contamination is small in our fiducial
setup. Had we chosen to utilize all the cross-correlation pairs in our
measurements, the IA-free error $\sigma_{0}$ would be reduced, but
$\sigma_{c_{s},0}$ would decrease, making the total constraint relatively
unchanged for reasonable choices of prior width as we saw in Figure 8.

###  VI.3 Fixed-Cosmology IA Measurement

As we have seen, allowing wide IA-parameter priors substantially degrade our
ability to constrain the matter clustering amplitude $S_{8}$. The flip side of
this observation is that our measurements allow us to tomographically
constrain the intrinsic alignments of each source sample $c_{s}(z)$ at the
lens redshifts. We show these constraints, where the cosmology is fixed to the
mean Planck PR4 cosmology, and we analyze all source and lens bin
combinations, in Figure 18. Since our aim is now to measure $c_{s}(z)$ we
widen the priors on $c_{s}^{ij}$ to $\mathcal{N}(0,10)$ so as to not bias the
resulting constraints. Intrinsic alignments, or deviations from the Planck
prediction for lensing amplitudes, are detected to be nonzero at more than
$2\sigma$ in several bins and, importantly, also deviate by similar amounts
for a given lens sample redshift. The tightest constraints come from lens-
source pairs with substantial redshift overlap, such that the signal becomes
IA dominated and, conversely, the pairs we have opted to include in our
fiducial analysis show rather weak constraints, with $\sigma(c_{s})$ wider
than our fiducial priors in many cases. This indicates that the statistical
error in the lensing cross-correlation amplitude
$C^{\delta_{g}\gamma_{E}}_{\ell}$ is not subdominant to the theoretical error
in these bins. Our IA constraining power also degrades slightly when going to
higher redshift, as uncertainties in the magnification contributions to these
spectra become a confounding factor.

In order to illustrate the differences of these tomographic measurements of IA
with the two-parameter source-independent IA parameterizations common in the
literature we also perform a fixed-cosmology fit to a model that assumes a
linear redshift evolution for the IA parameters (Eqn. 52). The results are
shown in the black shaded regions—since all IA amplitudes are forced to be the
same at a given redshift in this model the constraints are driven by the most
tightly-constrained bins, which as noted above do not always have the same
implied clustering amplitudes as the lensing-dominated bins constraining
$S_{8}$. For example, for the cross-correlations between LRG0 and the first
two source bins, whose shapes we omit in our fiducial analysis, prefer IA
amplitudes closer to zero than the last two. Even the more tightly constrained
bins, such as BGS0xS0 and BGS0xS1, differ by several $\sigma$ in their
preferred value of $c_{s}$. Taking into account the covariance between the
individual bins, the pairwise IA posteriors in Figure 18 deviate from the
source-independent 2-parameter model in black by $\chi^{2}=29.0$, with nonzero
linear IA amplitudes being detected at $\chi^{2}=34.8$. It is worth noting
that since these differences exist at the same redshift they cannot be due to
a mismatch with the true and assumed cosmologies. However, if we assume that
IAs can only be anti-aligned with the tidal field, the fact that several bins
prefer a positive IA signal, rather than the negative radial alignment signal
predicted by simulations, may be taken as evidence that either $S_{8}$ is
higher than Planck, as our cosmology fits indicate, or the presence of a
systematic or statistical fluctuation.

Figure 18: Tomographic constraints on the intrinsic alignment amplitude of
each source sample. The black line and shaded bands show the mean and 1- and 2
sigma regions when fitting $c_{s}(z)$ independently of source sample in the
linear model.

##  VII Conclusions

Figure 19: A comparison of our $S_{8}$ constraints with a number of results in
the literature. Blue points denote constraints from the primary CMB, while
orange show CMB lensing auto-spectrum constraints. Green, red, purple and
brown points show cosmic shear, $3\times 2$-point, CMB-galaxy lensing and
galaxy–galaxy lensing constraints respectively. Our results with and without
BAO are shown at the bottom in black. See §VII for more discussion. Figure 20:
Comparison of the constraints derived in this work to those of [67], using the
Maglim lens sample and the Metacalibration source sample. The black contours
here show our fiducial results, while the blue and orange contours show the
linear and nonlinear bias constraints from [67], although note that these
constraints also use different minimum scales as described in §VII. Of
particular note is that our fiducial constraints agree quite well with the
nonlinear bias constraints derived using the Maglim sample, although the
latter use a very different nonlinear bias model. These constraints should be
very correlated, given that they share the same source catalog and sky area,
so this finding is reassuring. Furthermore, when we restrict to a similar IA
treatment, assuming a single linear function of redshift for each IA parameter
(green), we obtain similar uncertainties on $S_{8}$, although the analysis
presented here is slightly more constraining likely due to our inclusion of
higher redshift lens bins.

The cross correlations of the projected densities and shapes of galaxies, as
measured in imaging surveys, directly probe the clustering of matter on
cosmological scales, providing a precision test of the growth of structure in
the standard model of cosmology. In this work we present measurements of these
cross correlations using the target imaging samples for the DESI BGS and LRG
samples and the Metacalibration galaxy shape catalog derived from the first
three years of the Dark Energy Survey. In addition to having comparable
signal-to-noise to other state-of-the-art galaxy–galaxy lensing measurements,
the spectroscopic calibration of the DESI samples means that our measurements
do not suffer from the photometric uncertainties typical of imaging surveys.
Firstly, these samples have well measured and extremely small contamination
from stars and systematic variations in galaxy photometry. Additionally, due
to the accurate photometric redshift estimates available for these samples
derived from the abundant spectroscopy that has been gathered for them, each
lens sample can be well-localized along the line of sight. This second
property makes our measurements particularly amenable to perturbative models,
which use the separation of spatial scales to construct effective theories of
structure formation that are non-local in time: the fixed line-of-sight
distance translates angular scales to fixed physical ones, enabling us to
robustly distinguish between large and small scales, and the unknown time
evolution of the effective-theory parameters is restricted by fitting in well-
localized redshift bins.

We present the “$2\times 2$-point” cosmological analysis of these data using
perturbation theory techniques, yielding a constraint on the matter clustering
amplitude $S_{8}=0.850^{+0.040}_{-0.051}$. For the first time, we include a
full accounting of nonlinear contributions to the intrinsic alignments of
galaxies to one-loop order in the effective theory using the Lagrangian
formalism recently developed in ref. [36]. This enables us to consistently
model the matter density, galaxy densities, and galaxy shapes at each lens
redshift on equal footing from first principles within the formalism of
Lagrangian perturbation theory. In addition, we augment the LPT predictions
for matter and galaxy clustering using hybrid effective field theory (HEFT) by
combining the predictions of Lagrangian bias with non-linear dark matter
displacements from N-body simulations using the Aemulus $\nu$ emulators [27]
for the first time, along with [9]. We test our modeling pipeline on mock data
constructed from the Buzzard simulations [74, 75], finding that both allow us
to place unbiased constraints on cosmological parameters including $S_{8}$ out
to angular scales corresponding to $k_{\rm max}=0.4\ h\mathrm{Mpc}^{-1}$,
substantially beyond the range of validity for linear theory, and improving
upon the $S_{8}$ constraint of an analysis using $k_{\rm max}=0.2\
h\mathrm{Mpc}^{-1}$ by $20\%$. This is comparable to the improvement found in
the first application of a linear Taylor series HEFT model to data in [154],
although a thorough comparison with that analysis is beyond the scope of this
work. Since the two-dimensional data mostly constrain the lensing amplitude,
we also obtain constraints combining our measurements with baryon-acoustic
oscillation data, finding a small improvement in
$S_{8}=0.838^{+0.038}_{-0.045}$.

Our $S_{8}$ constraints in both cases are in excellent agreement with those
inferred from the primary cosmic microwave background ([7, 151, 155] and auto
power spectrum of CMB lensing ([11]), within the $\Lambda$CDM model. This is
in contrast to many works in the literature which have found lower lensing
amplitudes than predicted by the CMB at $1-2\sigma$ level, as shown in Figure
19 [156, 157, 158, 4, 9, 159], though our error bars are somewhat wider, and
assumptions somewhat more conservative, than many of these constraints.

One particularly relevant exception is the $2\times 2$-point analysis of ref.
[67], who conduct a similar analysis as we do but using the DES Maglim sample
for their lenses. Their fiducial analysis, using only linear bias and modeling
to a minimum scale of 8 and 6 $h^{-1}\,\rm Mpc$ for clustering and
galaxy–galaxy lensing respectively, infers a lower value of $S_{8}$ than ours
by approximately $1.5\sigma$, but an analysis of the same data using a
nonlinear bias model [160, 68], infers a value of $S_{8}$ that is $\sim
0.5\sigma$ higher than the DES linear bias analysis. Furthermore, an analysis
using the DES nonlinear bias model and fitting to 4 $h^{-1}\,\rm Mpc$ infers a
very similar value of $S_{8}$ to this work, with differences in constraining
power due to our use of the more conservative intrinsic-alignments (IA) scheme
(Fig. 20).

The large shift observed between the DES linear and nonlinear bias results
underscores the importance of a proper treatment of nonlinear galaxy bias. The
source of the additional shift to higher $S_{8}$ when analyzing smaller scales
in the DES analysis is not readily apparent, and a full investigation of this
is beyond the scope of this work. We note that there is also an observed $\sim
0.5\sigma$ shift to lower $S_{8}$ when analyzing even larger scales than the
fiducial linear bias analysis (see Figure 12 of [67]), lending some evidence
to an explanation involving residual angular systematics on large scales in
the Maglim sample, however we have not investigated the statistical
significance of this shift. In this scenario, analyzing smaller scales with
the nonlinear bias model may downweight these residual systematics, leading to
the observed shift. It is also possible that the nonlinear bias model used in
the DES analysis and that used in our analysis, which were validated on the
same set of simulations as used here [75], are insufficiently flexible to fit
the data leading to shifts to higher $S_{8}$ when analyzing smaller scales,
although this possibility is less likely given the stability of our
constraints when varying our scale cuts as shown in Figure 14.

We also note that our $S_{8}$ constraints derive fundamentally from large
scales, as marginalizing over nonlinear bias and counterterms effectively
dilutes the clustering and lensing information at high $k$. Many references
have suggested that the low $S_{8}$ measured by weak lensing surveys can be
alleviated by invoking baryonic or beyond-$\Lambda$CDM physics at small scales
[161, 15, 162, 163, 164]—these scenarios cannot be easily ruled out by
perturbative analyses such as ours, where these effects are absorbed into the
counterterms ($b_{\nabla^{2}x},b_{\nabla^{2}m}$), although the fact that our
weak constraints on these effective-theory parameters, particularly in HEFT,
are not in any tension with the input priors suggests that any small scale
suppression of power in the data should be “small” at
$k<0.4h\mathrm{Mpc}^{-1}$. On the other hand, our $S_{8}$ constraints should
be quite degenerate with large-scale suppressions of power, for example due to
neutrinos or ultralight axions [165], and a bound on these effects can be
inferred from our consistency with Planck $\Lambda$CDM. While our constraints
are slightly weaker than many other weak-lensing results reported in the
literature, their reliance on first-principles calculations and fundamental
symmetries makes possible inferences about exotic physical scenarios rather
robust, making it a priority to understand and hopefully reduce the sources of
theoretical error in a rigorous way.

Indeed, the leading source of error, both statistical and theoretical, in our
analysis setup is the uncertain contribution of galaxy intrinsic alignments to
the GGL signal. At leading order, a change in the linear shape bias, $c_{s}$,
and a change in the matter clustering amplitude, $S_{8}$, contribute equally
to any given density-shape cross correlation, with the only difference being
that $S_{8}$ changes cross-spectra coherently across redshifts, while the IA
amplitude of each source sample at each lens redshift is an independent degree
of freedom (Eqn 58). In our fiducial analysis we attempt to mitigate this
degeneracy with a combined strategy of (a) putting priors on the size of the
IA amplitude based conservatively on measurements of halos in simulations,
which are known to be enhanced relative to galaxies and (b) deriving our
constraints from pairs of lens and source samples with maximal redshift
separation—specifically cross correlating the three lowest redshift lens
samples with the two highest redshift source samples—such that the IA
contamination is further suppressed by the overlap integral between their
redshift distributions. Simultaneously, we put the shapes of galaxies on the
same, consistent footing with their densities by allowing the equivalent
degrees of freedom for each independent sample and redshift required of a—in
this case spin 2—biased tracer at one-loop in perturbation theory. We show
that, when all these degrees of freedom are properly accounted for in our
maximally agnostic IA redshift dependence (MAIAR) prescription, the subset of
data used in our fiducial setup returns essentially identical constraints to
the full set, because IA contaminated measurements now have their proper
theory error accounted for (§V.1.1).

Our modeling, as described above, is robust against the redshift evolution of
intrinsic alignments, as well as variations between source bins, and we
demonstrate this by injecting a number-density dependent IA signal into mock
data. On the other hand, conventional modeling choices which describe IAs
through a source-sample independent two-parameter power law in redshift yield
biased constraints with tightened error bars by deriving their IA constraints
from the most IA-dominated data points. If we were to adopt these conventional
choices in our analysis, our constraints would artificially improve to
$S_{8}=0.848^{+0.028}_{-0.033}$, a $51\%$ improvement. Further restricting to
the NLA model in addition to using the more conventional redshift evolution
model results in $S_{8}=0.836^{+0.026}_{-0.031}$, a $63\%$ improvement over
our fiducial analysis. Conversely, by fixing the cosmological parameters in
our fits to their Planck best fits, we can place tomographic constraints on
the IA parameters of the DES source galaxies, particularly in the IA dominated
bins we drop in our fiducial analysis. Such an analysis shows that the source-
independent two-parameter model is strongly disfavored by the data, which show
variations in the cross-spectrum amplitudes across sources at the same lens
redshift that cannot be due to deviations from the Planck predictions for
growth alone.

The methods introduced in this paper have immediate implications for future
surveys and GGL analyses. On the data side, the careful selection of galaxy
lens and source samples that can be well-localized and separated in redshift,
e.g. through spectroscopic calibration of redshift distributions as we have
done here, will be critical. Alternatively, the impact of the IA signal could
be mitigated by placing priors based on direct measurements. Importantly,
these measurements will have to probe the same effective redshifts and samples
from which the lensing constraints are derived, rather than the IA dominated
cross correlations from which past lensing surveys have tended to constrain
their more restrictive IA models. One promising avenue is to measure the
three-dimensional clustering of galaxy shapes through spectroscopic
surveys—these measurements have recently attracted renewed interested as
probes of fundamental physics in their own right [125, 41, 126, 166,
167].Without the significant expenditure of resources, particularly in the
form of wide field spectroscopy, IAs will continue to be a dominant source of
uncertainty in galaxy lensing analyses.

On the modeling side, our work can be directly extended to include the
clustering of the lens galaxies in redshift space. Including these three-
dimensional data will allow us to break degeneracies in our constraints by
accessing cosmological information in the shape of the linear power spectrum,
and improve clustering constraints through redshift-space distortions. Indeed,
the Lagrangian effective theory models used in this work have been extensively
used to analyze spectroscopic clustering [20, 168, 169] and can be
straightforwardly and consistently combined with the analysis in this work
with minimal new free parameters. In addition, comparing measurements of the
matter clustering amplitude in lensing and peculiar velocities will allow us
to test the predictions of general relativity and study the effect of
anisotropic selection bias, which are a major “known unknown” in spectroscopic
galaxy samples [170, 171, 82, 172].

The modeling in this paper can also be extended to the auto-correlation of
galaxy shapes, utilizing the effective theory of galaxy shapes in order to
extract the matter clustering information in cosmic shear, although modeling
connected trispectrum terms may be important for this extension if considering
smaller surveys than DES Y3. Unlike the GGL signal modeled in this paper, the
auto-correlation is not well-localized in redshift, such that the MAIAR
prescription used in this work will not be sufficient. However, in this case
the redshift dependence can be treated using the spline formalism described in
Section III.4, where the number free parameters are determined by the
correlation length of bias parameters across redshift (i.e., the separation
scale of the spline basis functions), and the values of the spline
coefficients can be tomographically constrained by the GGL signal as we have
done here. This analysis can also be extended to include cross correlations of
galaxy shapes with CMB lensing, whose kernels likewise span a broad range of
redshift. Since the constraining power of these setups will likely be
dependent on the redshift correlations assumed, it will be critical to study
the robustness of the resulting constraints varying the spline spacing.
Conversely, such an analysis will be an important stress test of current
cosmic shear constraints. We intend to return to this topic in the near
future.

## Data Availability

Data from the plots in this paper are available on at
https://zenodo.org/records/12642934 as part of DESI’s Data Management Plan. We
intend to make our likelihood code, measurement pipeline and fiducial data
vectors publicly available at https://github.com/j-dr/DESIxDES upon
publication.

## Acknowledgements

The authors thank Nickolas Kokron for discussions related to galaxy bias and
intrinsic alignment modeling, Elisabeth Krause and Anna Porredon for useful
discussions regarding comparisons with DES Y3 results and Alejandro Aviles and
Cristhian Garcia-Quintero for their helpful comments as DESI internal
reviewers of this work. SC acknowledges the support of the National Science
Foundation at the Institute for Advanced Study. JD is supported by the
Chamberlain Fellowship at Lawrence Berkeley National Lab.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of High-Energy Physics, under Contract No.
DE–AC02–05CH11231, and by the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility under the same contract. This
work used Stampede2 at the Texas Advanced Computing Center and Bridges2 at the
Pittsburgh Supercomputing Center through allocation PHY200083 from the Extreme
Science and Engineering Discovery Environment (XSEDE) [173], which was
supported by National Science Foundation grant number 1548562. Additional
support for DESI was provided by the U.S. National Science Foundation (NSF),
Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s
National Optical-Infrared Astronomy Research Laboratory; the Science and
Technology Facilities Council of the United Kingdom; the Gordon and Betty
Moore Foundation; the Heising-Simons Foundation; the French Alternative
Energies and Atomic Energy Commission (CEA); the National Council of
Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of
Science and Innovation of Spain (MICINN), and by the DESI Member Institutions:
https://www.desi.lbl.gov/collaborating-institutions. Any opinions, findings,
and conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the U. S. National
Science Foundation, the U. S. Department of Energy, or any of the listed
funding agencies.

The DESI Legacy Imaging Surveys consist of three individual and complementary
projects: the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona
Sky Survey (BASS), and the Mayall $z$-band Legacy Survey (MzLS). DECaLS, BASS
and MzLS together include data obtained, respectively, at the Blanco
telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok
telescope, Steward Observatory, University of Arizona; and the Mayall
telescope, Kitt Peak National Observatory, NOIRLab. NOIRLab is operated by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation. Pipeline
processing and analyses of the data were supported by NOIRLab and the Lawrence
Berkeley National Laboratory. Legacy Surveys also uses data products from the
Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of
the Jet Propulsion Laboratory/California Institute of Technology, funded by
the National Aeronautics and Space Administration. Legacy Surveys was
supported by: the Director, Office of Science, Office of High Energy Physics
of the U.S. Department of Energy; the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility; the U.S. National
Science Foundation, Division of Astronomical Sciences; the National
Astronomical Observatories of China, the Chinese Academy of Sciences and the
Chinese National Natural Science Foundation. LBNL is managed by the Regents of
the University of California under contract to the U.S. Department of Energy.
The complete acknowledgments can be found at https://www.legacysurvey.org/.

The authors are honored to be permitted to conduct scientific research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation.

The author list in this work was compiled using
mkauthlist777https://github.com/DarkEnergySurvey/mkauthlist. We made use of
bibmanager888https://bibmanager.readthedocs.io/en/latest/ to edit our
bibliography. This research has made use of NASA’s Astrophysics Data System
and adstex (https://github.com/yymao/adstex).

##  Appendix A BGS and LRG Sample Calibration

|  $z_{\rm eff}$ (Full) |  $z_{\rm eff}$ (DES) |  $\sigma(z)$ (Full) |  $\sigma(z)$ (DES) |  $f_{\rm star}$ (Full) |  $f_{\rm star}$ (DES)
---|---|---|---|---|---|---
BGS0 | 0.228 | 0.229 (0.33%) | 0.0591 | 0.0597 (0.952%) | 0.00495 | 0.00278 (-43.8%)
BGS1 | 0.362 | 0.363 (0.345%) | 0.0603 | 0.0621 (2.88%) | 0.0025 | 0.00216 (-13.8%)
LRG0 | 0.47 | 0.469 (-0.222%) | 0.0632 | 0.0636 (0.677%) | 0.00141 | 0.000634 (-55.1%)
LRG1 | 0.628 | 0.626 (-0.311%) | 0.0734 | 0.0715 (-2.48%) | 0.00156 | 0.000602 (-61.3%)
LRG2 | 0.791 | 0.794 (0.371%) | 0.0781 | 0.0766 (-1.91%) | 0.00194 | 0.00146 (-24.5%)
LRG3 | 0.924 | 0.932 (0.82%) | 0.0956 | 0.0913 (-4.48%) | 0.00376 | 0.00218 (-41.9%)
Table 4: Redshift distribution summary statistics and stellar contamination
fractions for our lens samples measured over the full DESI Y1 footprint, and
over the overlap area between DESI Y1 and DES Y3 footprints. Figure 21:
Overdensity trends with relevant potential systematics for the two BGS
photometric bins used in this work. The solid lines use maps without weights,
while the dashed lines include weights to correct for the observed trends,
assuming a linear trend between the systematic and observed galaxy density.
Figure 22: Difference between BGS auto-power spectra with and without applying
linear systematic weights. Figure 23: Comparison of the redshift distributions
measured using the full Iron data set (solid) and only the overlap region with
the DES Y3 footprint (dashed) for our all of our lens bins. The differences in
the mean and standard deviation of the redshift distributions for these two
footprints as well as differences in stellar contamination are listed in Table
4.

In this appendix we describe the systematics tests that we performed on the
BGS and LRG redshift bins used in this work. Many of these systematics tests
for the LRG samples are detailed in ref. [53], and we repeat them for the BGS
samples here. Firstly, we consider the trends in angular galaxy number density
as a function of various survey properties, and potential contaminants. Figure
21 shows this for $E(B-V)$, the depth and psf size in each band used in the
BGS sample selection, as well as stellar density as measured using GAIA. The
blue line shows these trends for the first BGS bin, while the orange line
shows the same for the second. In general, we see that these correlations are
quite small in the regime where there are appreciable numbers of galaxies, as
shown by the blue histograms. We fit a linear function to these trends, and
use this to construct weights to remove them. The correlations between
weighted galaxy number densities and these fields are shown by dashed lines.
We note that there is very little change when applying these weights.

In order to investigate the impact of the systematic weights that we have
constructed, we examine the difference in $C^{\delta_{g},\delta_{g}}_{\ell}$
measured with and without these systematic weights. Figure 22 shows this
difference, compared to the statistical error on
$C^{\delta_{g},\delta_{g}}_{\ell}$ given by our fiducial covariance matrix. We
see that any shift that is derived from applying weights for the BGS sample is
well within these errors, and as such we have chosen to not apply systematics
weights for this sample, since it is well known that such weights can
potentially remove cosmological power, and there is no evidence that they are
correcting for any detectable systematic trends in our data for this sample.

An additional concern that one might have about our analysis is that the
redshift distributions that we use in our analysis are computed over the
overlap region between DES Y3 and DESI Y1, which comprises just of the 4143
sq. degrees used in our analysis. While the area used to measure our redshift
distributions is large enough to have negligible statistical error, it is
possible that the average $n(z)$ over the full footprint differs from the
$n(z)$ measured over the overlap region due to varying survey conditions. In
order to set a reasonable upper bound on this variation, we compare the $n(z)$
estimated for each of the samples used in this work measured over the overlap
region between DESI Y1 and DES Y3 to that measured over the full DESI Y1 area.
The BGS and LRG selections in the NGC of the DESI footprint use photometry
from the Bok and Mayall telescopes, rather than DeCAM, and as such the
$n(z)$’s could potentially differ significantly between the NGC and the
overlap region if the selections were particularly sensitive to variations in
photometry.

These two sets of $n(z)$s are shown in Figure 23, and differences in these
samples are further summarized in Table 4. In general we see measure
statistically significant, but small shifts in the mean and widths of the
redshift distributions of our lenses. The largest shifts are observed in the
last LRG bin, which is unsurprising given that it is comprised of the faintest
galaxies, whose selection will be most influenced by changes in photometry. We
propagate these differences to cosmology as shown in Figure 13 for our
fiducial lens bins, i.e., the BGS bins and first LRG bin and find these
differences to be entirely negligible in terms of constraining power. As such,
we conclude that this systematic is negligible at the constraining power of
this analysis, although this and related concerns will need to be revisited
should we combine these measurements with anisotropic power spectra of these
samples measured with DESI spectroscopy.

##  Appendix B Fitting All Source and Lens Bin Combinations

For the results that fit all source and lens bin combinations, we use a
covariance that takes as input the best fit of our fiducial model analyzing
all of these bin combinations. This best-fit model is shown in Figure 24.

Figure 24: Same as Figure 2, but using our fiducial model and the full set of
source and lens bins. We find $\chi^{2}=201.2$ for 280 data points using 178
free parameters, 132 of which are analytically marginalized over. Assuming
again that IA, magnification and source uncertainities are prior-dominated and
thus do not enter into the reduced $\chi^{2}$ calculation we obtain
$\chi^{2}_{\rm red}=0.86$, though we note that this is likely an undercounting
of the free parameters in the model.

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