<EMAIL_ADDRESS>
# Joint constraints from cosmic shear, galaxy-galaxy lensing and galaxy
clustering:
internal tension as an indicator of intrinsic alignment modelling error

S. Samuroff1∗ A. Campos2,3 A. Porredon4,5 and J. Blazek1 1 Department of
Physics, Northeastern University, Boston, MA, 02115, USA  2 McWilliams Center
for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh,
PA 15213, USA  3 NSF AI Planning Institute for Physics of the Future, Carnegie
Mellon University, Pittsburgh, PA 15213, USA  4 Ruhr University Bochum,
Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German
Centre for Cosmological Lensing, 44780 Bochum, Germany  5 Institute for
Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill,
Edinburgh, EH9 3HJ, UK  [

###### Abstract

In cosmological analyses it is common to combine different types of
measurement from the same survey. In this paper we use simulated Dark Energy
Survey Year 3 (DES Y3) and Legacy Survey of Space and Time Year 1 (LSST Y1)
data to explore differences in sensitivity to intrinsic alignments (IA)
between cosmic shear and galaxy-galaxy lensing. We generate mock shear,
galaxy-galaxy lensing and galaxy clustering data, contaminated with a range of
IA scenarios. Using a simple 2-parameter IA model (NLA) in a DES Y3 like
analysis, we show that the galaxy-galaxy lensing + galaxy clustering
combination ($2\times 2$pt) is significantly more robust to IA mismodelling
than cosmic shear ($1\times 2$pt). IA scenarios that produce up to $5\sigma$
biases for shear are seen to be unbiased at the level of $\sim 1\sigma$ for
$2\times 2$pt. We demonstrate that this robustness can be largely attributed
to the redshift separation in galaxy-galaxy lensing, which provides a cleaner
separation of lensing and IA contributions. We identify secondary factors
which may also contribute, including the possibility of cancellation of
higher-order IA terms in $2\times 2$pt and differences in sensitivity to
physical scales. Unfortunately this does not typically correspond to equally
effective self-calibration in a $3\times 2$pt analysis of the same data, which
can show significant biases driven by the cosmic shear part of the data
vector. If we increase the precision of our mock analyses to a level roughly
equivalent to LSST Y1, we find a similar pattern, with considerably more bias
in a cosmic shear analysis than a $2\times 2$pt one, and significant bias in a
joint analysis of the two. Our findings suggest that IA model error can
manifest itself as internal tension between $\xi_{\pm}$ and $\gamma_{t}+w$
data vectors. We thus propose that such tension (or the lack thereof) can be
employed as a test of model sufficiency or insufficiency when choosing a
fiducial IA model, alongside other data-driven methods.

###### keywords:

cosmological parameters – gravitational lensing: weak – methods: statistical

##  1 Introduction

The study of weak lensing as a cosmological probe has evolved considerably in
the last few years and decades. Although we talk about “weak lensing" fairly
loosely, the term actually encompasses several distinct measurements. Cosmic
shear is the correlation of galaxy shapes with each other. That is, light from
galaxies on adjacent lines of sight must pass through a similar cross-section
of the Universe, and so the lensing distortions are correlated. It has been
known for some time that shear-shear two-point correlations are a useful way
to learn about cosmology (Bartelmann & Schneider, 2001; Huterer, 2002; Hu &
Jain, 2004; Frieman et al., 2008). Indeed, to date, cosmic shear analyses
(referred to as $1\times 2$pt analyses) have been key results of almost all
galaxy imaging cosmology surveys (Benjamin et al. 2007; Kilbinger et al. 2013;
Heymans et al. 2013; Jee et al. 2016; Hildebrandt et al. 2017; Troxel et al.
2018; Chang et al. 2019; Hikage et al. 2019; Hamana et al. 2020; Asgari et al.
2021; Loureiro et al. 2022; Secco, Samuroff et al. 2022; Amon et al. 2022;
Doux et al. 2022; Longley et al. 2023; Li et al. 2023; Dalal et al. 2023; DES
& KiDS Collaboration 2023).

Alternatively, instead of trying to detect lensing caused by background large
scale structure, we can measure weak lensing around specific foreground
lenses. One can do this with massive clusters, and so study their density
profiles and total mass (Schneider et al., 2000; King & Schneider, 2001).
Alternatively, one can use foreground galaxies as lenses, a measurement known
as galaxy-galaxy lensing. Since the clustering pattern of galaxies traces out
the large scale structure of the Universe, these shear-position correlations
measure a similar physical effect to shear-shear ones. If we have a good
estimate for the galaxy-to-dark matter mapping (i.e. galaxy bias) and the
distribution of redshifts (or better yet, precise redshifts for individual
galaxies), we can use galaxy-galaxy lensing to probe the properties of the
Universe. There is a relatively long history of this, using both spectroscopic
and photometric surveys (Sheldon et al. 2004; Baldauf et al. 2010; Mandelbaum
et al. 2013; Kwan et al. 2017; Prat, Sánchez et al. 2018; Alam et al. 2017;
Leauthaud et al. 2017; Yoon et al. 2019; Blake et al. 2020; Singh et al. 2020;
Miyatake et al. 2022; Lee et al. 2022; Prat et al. 2022; Porredon et al. 2022;
Pandey et al. 2022). Often studies of this sort (including many of those cited
above) will also incorporate galaxy clustering auto-correlations (Cole et al.
2005; Blake et al. 2012; Aubourg et al. 2015; Elvin-Poole et al. 2018; Alam et
al. 2021; Zhou et al. 2021; Rodríguez-Monroy et al. 2022; Sánchez, Alarcon et
al. 2023), in order to break the degeneracy between galaxy bias and the
clustering amplitude $\sigma_{8}$. This combination is often referred to as
$2\times 2$pt.

It is also worth noting briefly that there is a subfield of weak lensing
studies looking at the cross-correlation between galaxy surveys (lensing and
positions) and Cosmic Microwave Background (CMB) lensing. These are
complementary to other types of weak lensing (Namikawa et al. 2019; Marques et
al. 2020; Robertson et al. 2021; Krolewski et al. 2021; Omori et al. 2023;
Chang et al. 2023; DES Collaboration 2023). The nomenclature of the
combinations beyond $3\times 2$pt is slightly less well-defined, but CMB
lensing is not the focus of this work, so we will not go into the details
here.

Although cosmic shear and galaxy-galaxy lensing are powerful probes in their
own right, combining the two with galaxy clustering in a joint analysis
($3\times 2$pt) has become an increasingly mainstream part of modern
cosmology. This is for a number of reasons, not least that the two have
different sensitivities to cosmological parameters, and so together can break
internal degeneracies (Hu & Jain, 2004; Frieman et al., 2008; Cacciato et al.,
2009; Yoo & Seljak, 2012). Another factor is the ability of joint data vectors
to self-calibrate photometric redshift error and other systematic
uncertainties (Huterer et al., 2006; Bridle & King, 2007; Bernstein, 2009;
Joachimi & Bridle, 2010; Samuroff et al., 2017). In the past five or so years,
joint-probe analyses using cosmic shear, galaxy-galaxy lensing and galaxy
clustering together have provided some of the most powerful late-universe
cosmology constraints to date (DES Collaboration, 2018; van Uitert et al.,
2018; Joudaki et al., 2018; Heymans et al., 2021; DES Collaboration, 2022;
Miyatake et al., 2023; Sugiyama et al., 2023).

Before combining any given set of measurements, however, it is usual to
demonstrate their consistency. If two data sets are found to be discrepant, it
points either to systematics in one or the other, or the need for a new model.
This can be an interesting new result or a nuisance, but either way, one
should not combine the discrepant data sets. Judging consistency using
projected confidence contours can be misleading, but fortunately a range of
metrics using the full parameter space have been proposed. For uncorrelated
data (e.g. from non-overlapping surveys or very different observables),
various statistics have been developed, both Bayesian evidence- and
likelihood-based (Lemos, Raveri, Campos et al. 2021; Raveri & Hu 2019). For
probes that are correlated, such as cosmic shear and galaxy-galaxy lensing
measurements from the same survey, alternatives are commonly used. For
example, the DES Y3 analysis used a method based on the Posterior Predictive
Distribution (PPD), as described in Doux, Baxter et al. (2021).

It has long been known that an effect known as intrinsic alignment (IA) has
the potential to bias weak lensing based measurements. IA is the name given to
correlations arising from the fact that the _intrinsic_ (i.e. pre
gravitational shear) shapes of galaxies are not entirely random. This induces
correlations both between physically close pairs (known as intrinsic-intrinsic
or II correlations) and between the shear of one galaxy and the intrinsic
shape of another (known as shear-intrinsic or GI correlations). A related but
slightly different correlation also arises between the clustering density and
intrinsic shapes of close-by objects ($g$I).

The reason IAs can cause bias is simple: they introduce new features into two-
point lensing data, which appear similar to lensing (although not identical,
allowing self-calibration of the sort discussed below). If the IA model used
to analyse that data cannot match them perfectly, then other parameters may
need to adjust in order to maintain a good overall fit. Sometimes this can
happen inside the IA model space, in which case the mapping between IA
parameters and underlying physical processes becomes more complicated.
However, unmodelled IAs can quite can easily (if imperfectly; see Campos et
al. 2023) be absorbed by cosmological parameters too (Krause et al. 2016;
Blazek et al. 2019; Fortuna et al. 2021a; Secco, Samuroff et al. 2022; Campos
et al. 2023). This subject, and the question of IA model sufficiency, has been
discussed relatively widely in the context of cosmic shear. One aspect that
has received somewhat less attention is how parameter biases play out in the
context of a multi-probe analysis. That is, how different the sensitivities of
galaxy-galaxy lensing and cosmic shear are to IA model error, and how
effectively the combination can mitigate such error. In this work we consider
exactly this question, using simulated Dark Energy Survey Year 3 (DES Y3) and
Legacy Survey of Space and Time Year 1 (LSST Y1) like data.

This paper is structured as follows. In Section 2 we discuss how our joint
data vector is modelled. We pay specific attention to the modelling of the
various IA terms, since these are important for our conclusions. Section 3
then considers a number of simulated setups with different lens and source
sample configurations. We also set out the details of the mock analyses,
including the nuisance parameters associated with each sample. In Section 4 we
discuss our main results and test their robustness to variations in the
analysis and mock data. We conclude in Section 5.

##  2 Modelling

This section describes the modelling framework used in this paper. In short,
we use a DES Y3 like $3\times 2$pt analysis, combining cosmic shear, galaxy-
galaxy lensing and galaxy clustering. Although we highlight the key features
relevant for this work, much of the infrastructure was developed for the DES
Y3 analyses and is documented more extensively in Krause et al. (2021) and the
accompanying DES papers. For part of our results, we also consider an LSST Y1
like analysis. We use many of the same tools for this, but with key
modifications that we will highlight. We consider below the details of how
$\xi_{\pm}$ and $\gamma_{t}$ are calculated, since these are both sensitive to
IAs, but in slightly different ways – a point that will be important later.
For details about how $w(\theta)$ is computed, see the papers cited above.

###  2.1 Cosmic shear

Cosmic shear is the name given to small distortions in the observed shapes of
galaxies due to weak gravitational lensing by foreground large scale
structure. At a particular point on the sky, the shear $\gamma^{\rm G}$ can be
written as a projection of the matter overdensity field $\delta$ (or, rather,
the spatial derivatives of $\delta$) along the line-of-sight. Since light from
galaxies that appear close together on the sky trace similar paths through the
Universe, $\gamma^{\rm G}$ is spatially correlated. Indeed, two-point
functions, which measure the shear-shear correlation $\left\langle\gamma^{\rm
G}\gamma^{\rm G}\right\rangle$ as a function of angular separation $\theta$,
are the most common way to observe cosmic shear. Since shear is a spin-2
quantity, there are two different cosmic shear correlations that can be
constructed, $\xi_{+}$ and $\xi_{-}$. These are sensitive to different
physical scales, and so are complimentary to each other (see e.g. Secco,
Samuroff et al. 2022 Figure 2). Most commonly, these correlations are
constructed between galaxies in redshift bins $i,j$, which is useful in order
to break degeneracies and constrain the evolution of $\delta$ with redshift.

Starting with the 3D nonlinear matter power spectrum $P_{\delta}(k,z)$, and
assuming the Limber approximation (Limber, 1953; LoVerde & Afshordi, 2008), we
can write down an expression for the 2D angular power spectrum of convergence
$\kappa$ as:

$C^{ij}_{\kappa}(\ell)=\int_{0}^{\chi_{\rm hor}}\mathrm{d}\chi\frac{W_{\rm
s}^{i}(\chi)W_{\rm s}^{j}(\chi)}{\chi^{2}}\\\ \times
P_{\delta}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right).$ (1)

We have defined a few things here. First $\chi$ is a comoving distance from
the observer and $z(\chi)$ is the corresponding redshift (assuming some
cosmology). The integral limit $\chi_{\rm hor}$ is the comoving horizon
distance. The lensing kernel $W_{\rm s}^{i}$ is given by:

$W_{\rm
s}^{i}(\chi)=\frac{3}{2c^{2}}H^{2}_{0}\Omega_{\mathrm{m}}\frac{\chi}{a(\chi)}\int^{\chi_{\rm
hor}}_{\chi}\mathrm{d}\chi^{\prime}n_{\rm
s}^{i}\left(z(\chi)\right)\frac{\chi^{\prime}-\chi}{\chi^{\prime}}\frac{\mathrm{d}z}{\mathrm{d}\chi^{\prime}}.$
(2)

The source redshift distribution $n_{\rm s}(z)$ is normalised to integrate to
1. For an illustration of what $n_{\rm s}$ and $W_{\rm s}$ look like as a
function of redshift for the fiducial analyses presented in this paper, see
the upper two panels of Figure 1.

The conversion from angular multipoles $\ell$ to separation $\theta$ can be
written as:

$\displaystyle\xi^{ij}_{+}(\theta)=\sum_{\ell}\frac{2\ell+1}{2\pi\ell^{2}(\ell+1)^{2}}\left(G^{+}_{\ell,2}(\mathrm{cos}\theta)+G^{-}_{\ell,2}(\mathrm{cos}\theta)\right)$
$\displaystyle\times\left(C^{ij}_{\gamma,\mathrm{EE}}(\ell)+C^{ij}_{\gamma,\mathrm{BB}}(\ell)\right),$
(3)

$\displaystyle\xi^{ij}_{-}(\theta)=\sum_{\ell}\frac{2\ell+1}{2\pi\ell^{2}(\ell+1)^{2}}\left(G^{+}_{\ell,2}(\mathrm{cos}\theta)-G^{-}_{\ell,2}(\mathrm{cos}\theta)\right)$
$\displaystyle\times\left(C^{ij}_{\gamma,\mathrm{EE}}(\ell)-C^{ij}_{\gamma,\mathrm{BB}}(\ell)\right).$
(4)

In the above $G^{+}_{\ell,2}$ and $G^{-}_{\ell,2}$ are bin-averaged functions
defined in Krause et al. (2021). In the simplest case, the convergence and
E-mode shear spectra are equivalent here, $C_{\gamma,\mathrm{EE}}\rightarrow
C_{\kappa}$ and $C_{\gamma,\mathrm{BB}}=0$. In practice, however, intrinsic
alignments can contribute to both the E- and B-mode $C_{\gamma}$ spectra (see
Section 2.3).

For a given set of cosmological parameters, we evaluate $P_{\delta}(k,z)$
using the Boltzmann code CAMB (for the linear part; Lewis et al. 2000) and
HaloFit (for the nonlinear corrections; Takahashi et al. 2012). Note that
unlike Secco, Samuroff et al. (2022) and Amon et al. (2022), our fiducial
results do not make use of shear ratios (ratios of galaxy-galaxy lensing
measurements on small scales, which were included as an additional likelihood
in the fiducial Y3 analyses; see Sánchez, Prat et al. 2022).

###  2.2 Galaxy-galaxy lensing

Galaxy-galaxy lensing is a similar measurement to cosmic shear, described
above. Instead of the auto-correlation of shear, however, we are now measuring
the cross-correlation with galaxy density $\left\langle\delta_{g}\gamma^{\rm
G}\right\rangle$. One can write down the equivalent transformation to Eq. (1),
again assuming the Limber approximation, as:

$C^{ij}_{\delta_{g}\kappa}(\ell)=\int_{0}^{\chi_{\rm
hor}}\mathrm{d}\chi\frac{n_{\mathrm{l}}^{i}(\chi)W_{\rm
s}^{j}(\chi)}{\chi^{2}}\\\ \times
P_{\delta_{g}\delta}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right).$ (5)

As before, $n_{\rm l}(z)$ is a redshift distribution that integrates to 1, but
this time of the lens galaxies. We model the galaxy-matter 3D power spectrum
$P_{\delta_{g}\delta}$ by assuming linear bias, such that:

$P_{\delta_{g}\delta}(k,z)=b_{1}P_{\delta}(k,z).$ (6)

The linear bias coefficient $b_{1}$ is expected to depend on the details of
the sample and to have some evolution with redshift. In our fiducial setup, we
thus allow one independent parameter $b_{1}^{i}$ for each lens redshift bin
$i$, which are marginalised with uncorrelated uninformative priors (see
Section 3.3 for discussion).

To test our results in Section 4, a subset of our analyses use a nonlinear
bias model, which is based on a perturbative expansion to third order (see
Pandey et al. 2020). In short, this includes three extra parameters: $b_{2}$,
$b_{s^{2}}$ and $b_{\rm 3nl}$. For the sake of simplicity (in order to allow
the nonlinear bias model to reproduce mock data containing linear bias), we
fix the latter two parameters to zero, rather than their non-zero coevolution
values (see Pandey et al. 2020 Sec. 2D). We also follow Porredon et al. (2022)
and Pandey et al. (2022) in choosing to vary the product $\sigma_{8}b_{1}$ and
$\sigma_{8}^{2}b_{2}$ rather than $b_{1}$ and $b_{2}$ directly. This is a
practical decision to limit projection effects111Projection effects are shifts
in marginalised posteriors that can arise from the projection of a
multidimensional volume down to one or two axes. These are not biases in the
usual sense, since they do not indicate any sort of model/data mismatch, and
there are statistics (e.g. the global _Maximum a Posteriori_) that are
unaffected. See e.g. Krause et al. 2021 Sec IV-A for discussion., but is not
expected to otherwise affect the results. This amounts to one extra free
parameter per lens bin. We vary the bias parameters with wide flat priors
$\sigma_{8}b^{i}_{1}=[0.67,3.0]$ and $\sigma_{8}^{2}b^{i}_{2}=[-4.2,4.2]$.
Note that use of this model is limited to the tests in Section 4.2.3 – for all
other chains we use the linear bias model.

It is also worth mentioning that, in principle, cross-terms arise between
nonlinear galaxy bias and higher-order IA contributions. These are expected to
be small, and so they were neglected in the modelling pipeline for DES Y3.
Also note that no part of our analysis includes _both_ TATT and nonlinear
galaxy bias; chains (and data vectors) run with the former assume linear bias
and those run with the latter assume NLA.

With the angular power spectrum in hand, we can evaluate the real space
galaxy-galaxy lensing correlation as:

$\gamma^{ij}_{t}(\theta)=\sum_{\ell}\frac{2\ell+1}{4\pi\ell(\ell+1)}\bar{P}^{2}_{\ell}(\mathrm{cos}\theta)C^{ij}_{g\gamma}(\ell),$
(7)

where $\bar{P}^{2}_{\ell}$ are associated Legendre polynomials, averaged
within each $\theta$ bin (see e.g. Prat et al. 2022 Eq. 25). Similarly to with
cosmic shear above, in the absence of intrinsic alignments and magnification,
$C_{g\gamma}\rightarrow C_{\delta_{g}\kappa}$ in Eq. (7).

###  2.3 Intrinsic Alignments & Magnification

In practice, both cosmic shear and galaxy-galaxy lensing have contributions
from effects other than pure lensing and galaxy clustering. First, IAs add a
spatially correlated shape component, such that the observed shear is
$\gamma=\gamma^{\rm G}+\gamma^{\rm I}$, giving rise to
$\left\langle\gamma^{\rm G}\gamma^{\rm I}\right\rangle$ and
$\left\langle\gamma^{\rm I}\gamma^{\rm I}\right\rangle$ correlations at the
two-point level (often referred to simply as GI and II). Analogously, the
observed density of galaxy counts on a patch of sky is altered by
magnification, $\delta_{g,\mathrm{obs}}=\delta_{g}+\delta_{\mu}$. When writing
out the angular correlation functions, we have several additional
correlations.

$C^{ij}_{\gamma,\mathrm{EE}}=C^{ij}_{\kappa}+C^{ij}_{\rm GI}+C^{ij}_{\rm
IG}+C^{ij}_{\rm II,EE}$ (8)

$C^{ij}_{\gamma,\mathrm{BB}}=C^{ij}_{\rm II,BB}$ (9)

$C^{ij}_{g\gamma}=C^{ij}_{\delta_{g}\kappa}+C^{ij}_{\delta_{g}\mathrm{I}}+C^{ij}_{\mu\kappa}+C^{ij}_{\mu\mathrm{I}}$
(10)

Since, to first order, there is no B-mode contribution from lensing,
$C^{ij}_{\gamma,\mathrm{BB}}$ is non-zero only due to intrinsic
alignments222We are ignoring other theoretical sources of B-modes such as
source clustering, which are commonly assumed to be small (at least for two-
point statistics).. The Limber integrals for each of the terms above can be
expressed in a similar way to those in Sections 2.1 and 2.2:

$C^{ij}_{\rm GI}(\ell)=\int_{0}^{\chi_{\rm hor}}\mathrm{d}\chi\frac{W_{\rm
s}^{i}(\chi)n_{\mathrm{s}}^{j}(\chi)}{\chi^{2}}\\\ \times P_{\rm
GI}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right),$ (11)

$C^{ij}_{\rm II,EE/BB}(\ell)=\int_{0}^{\chi_{\rm
hor}}\mathrm{d}\chi\frac{n_{\rm
s}^{i}(\chi)n_{\mathrm{s}}^{j}(\chi)}{\chi^{2}}\\\ \times P_{\rm
II,EE/BB}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right),$ (12)

$C^{ij}_{\delta_{g}\mathrm{I}}(\ell)=\int_{0}^{\chi_{\rm
hor}}\mathrm{d}\chi\frac{n_{\rm
l}^{i}(\chi)n_{\mathrm{s}}^{j}(\chi)}{\chi^{2}}\\\ \times
P_{\delta_{g}\mathrm{I}}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right),$ (13)

$C^{ij}_{\mu\mathrm{I}}(\ell)=\int_{0}^{\chi_{\rm
hor}}\mathrm{d}\chi\frac{W_{\rm
l}^{i}(\chi)n_{\mathrm{s}}^{j}(\chi)}{\chi^{2}}\\\ \times
P_{\mu\mathrm{I}}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right),$ (14)

$C^{ij}_{\mu\kappa(\ell)}=\int_{0}^{\chi_{\rm hor}}\mathrm{d}\chi\frac{W_{\rm
l}^{i}(\chi)W_{\mathrm{s}}^{j}(\chi)}{\chi^{2}}\\\ \times
P_{\mu\delta}\left(k=\frac{\ell+0.5}{\chi},z(\chi)\right).$ (15)

The power spectra in the latter three can be related to the GI IA spectrum and
the nonlinear matter power spectrum by assuming linear bias:

$P_{\delta_{g}\mathrm{I}}(k,z)=b_{1}P_{\rm GI}(k,z),$ (16)

$P_{\mu\mathrm{I}}(k,z)=CP_{\rm GI}(k,z),$ (17)

$P_{\mu\delta}(k,z)=CP_{\delta}(k,z),$ (18)

with the factor $C$ being the magnification coefficient (see Elvin-Poole,
MacCrann et al. 2023 for a definition and further discussion), which describes
the overall impact of magnification333We follow Elvin-Poole, MacCrann et al.
2023 here in parameterising the overall impact of magnification with a single
amplitude per redshift bin. In terms of physics, one has two competing effects
due to the fact that magnification both boosts the observed fluxes of galaxies
and also expands the apparent area of a given patch on the sky. In the
notation of Joachimi & Bridle (2010), $C_{\rm mag}=2(\alpha-1)$, where
$\alpha$ is the logarithmic slope of the faint end of the luminosity
function.. Note that $C$ is not included as a free parameter in our analyses,
but rather fixed to the fiducial values measured by Elvin-Poole, MacCrann et
al. 2023 in each bin (see also Table 1). As mentioned in the previous section,
a subset of our chains include nonlinear galaxy bias. Note that the Y3
nonlinear bias model does not include an expansion of Eq. 16 to include all
relevant terms – the model always assumes a linear relation between
$P_{\delta_{g}\mathrm{I}}$ and $P_{\rm GI}$ (as in Krause et al. 2021).

Although we include it in all our modelling, the magnification-intrinsic term
$\mu$I is expected to be much smaller in magnitude than $\delta_{g}$I, and the
impact is thus expected to be minimal (see the turquoise dashed line in Prat
et al. 2022’s Figure 7). By considering the equations above, we can see that
(with certain assumptions), all of the 2D angular spectra entering our cosmic
shear and galaxy-galaxy lensing measurements are derived from four 3D power
spectra: the matter power spectrum $P_{\delta}$, and three intrinsic alignment
spectra $P_{\rm GI}$, $P_{\rm II,EE}$ and $P_{\rm II,BB}$.

We have discussed how we estimate $P_{\delta}$ in Section 2.1. Using the
formalism of Blazek et al. (2019), one can write the three IA power spectra
as:

$P_{\rm GI}=C_{1}P_{\delta}+b_{\rm TA}C_{1}P_{0|0E}+C_{2}P_{0|E2},$ (19)

$P_{\rm II,EE}=C^{2}_{1}P_{\delta}+2b_{\rm TA}C^{2}_{1}P_{0|0E}+b^{2}_{\rm
TA}C^{2}_{1}P_{0E|0E}\\\ +C^{2}_{2}P_{E2|E2}+2C_{1}C_{2}P_{0|E2}+2b_{\rm
TA}C_{1}C_{2}P_{0E|E2},$ (20)

$P_{\rm II,BB}=b^{2}_{\rm TA}C^{2}_{1}P_{0B|0B}+C^{2}_{2}P_{B2|B2}\\\ +2b_{\rm
TA}C_{1}C_{2}P_{0B|B2}.$ (21)

We should note here that $C_{1,2}$ in Eq. 19-21 are IA amplitudes, and are
_not_ related to the magnification coefficients discussed earlier (despite the
similar notation). The various scale dependent terms, $P_{X}$, can all be
calculated to one-loop order as integrals of the linear matter power spectrum
over $k$ (Blazek et al., 2019). We perform these integrals within CosmoSIS
using FastPT
444https://cosmosis.readthedocs.io/en/latest/reference/standard_library/fast_pt.html
(McEwen et al., 2016; Fang et al., 2017). The amplitudes $C_{1}$ and $C_{2}$
are given by:

$C_{1}(z)=-A_{1}\frac{\bar{C}_{1}\rho_{\rm
c}\Omega_{\mathrm{m}}}{D(z)}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{1}},$ (22)

$C_{2}(z)=5A_{2}\frac{\bar{C}_{1}\rho_{\rm
c}\Omega_{\mathrm{m}}}{D^{2}(z)}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{2}}.$
(23)

The pivot redshift $z_{0}$ is set to $z_{0}=0.62$ and the constant
$\bar{C}_{1}$ is fixed at a value of $\bar{C}_{1}=5\times
10^{-14}M_{\odot}h^{-2}\mathrm{Mpc}^{2}$ (Brown et al., 2002; Hirata & Seljak,
2004). The implementation of all of the above has been validated in Krause et
al. (2021) for DES Y3. Note that the sign convention in Eq. 22 and 23 ensures
consistency at the level of the power spectrum contributions. That is, if
$A_{1}$ and $A_{2}$ have the same sign, then the power spectrum contributions
in Eq. 19-21 will do too (see Blazek et al. 2019 for discussion).

We consider a few different IA setups in this work. The most complex is the
TATT model with five free parameters $(A_{1},A_{2},\eta_{1},\eta_{2},b_{\rm
TA})$, which are varied with the priors shown in Table 1. Alternatively the
NLA model is a subspace of TATT with $A_{2},\eta_{2},b_{\rm TA}=0$ (i.e. 2
free parameters). In this case, both GI and II spectra have the same shape as
the matter power spectrum $P_{\delta}$, modulated by the amplitude in Eq.
(22). Note that this is a specific variant of the NLA model; the version first
proposed in Hirata & Seljak (2004); Bridle & King (2007); Hirata et al. (2007)
did not have the extra freedom in redshift, and had only one free parameter,
$A_{1}$ (equivalent to fixing $\eta_{1}=0$ in Eq. (22) above). Where relevant,
we will refer to the simpler version with only $A_{1}$ free, as “1-parameter
NLA" or NLA-1.

###  2.4 Free parameters, sampling and priors

All likelihood analyses used in this work are carried out within the framework
of CosmoSIS 555https://cosmosis.readthedocs.io/en/latest/ (Zuntz et al.,
2015). Our main results make use of the PolyChord (Handley et al., 2015)
nested sampling algorithm666500 live points, $\mathrm{num\\_repeats}=30$,
$\mathrm{tolerance}=0.01$ (see Campos et al. 2023 Appendix D for a comparison
with the MultiNest sampling algorithm for our Y3 setup). When discussing best
fit values in the following sections we use $10\times$ oversampled chains
generated by PolyChord, in the same way as Secco, Samuroff et al. (2022) and
DES & KiDS Collaboration (2023). This reduces sampling noise, and is
effectively equivalent to running a likelihood maximiser. When calculating
degrees of freedom (e.g. for $p-$values), we use an estimate for the effective
number of parameters $N_{\rm par,eff}$ rather than the naive value. This is
given by the expression $N_{\rm par,eff}=N_{\rm
par,nai}-\mathrm{Tr}[C_{\Pi}^{-1}C_{\rm p}]$, where $C_{\Pi}^{-1}$ and $C_{\rm
p}$ are the covariance matrices of the prior and posterior distributions
respectively (see footnote 14 of Secco, Samuroff et al. 2022 and Raveri & Hu
2019).

For a subset of chains (the LSST Y1-like ones), instead of PolyChord we opt to
use a sampler called Nautilus 777https://nautilus-
sampler.readthedocs.io/en/stable/ (Lange, 2023). This choice was primarily
driven by speed – Nautilus uses neural networks to find an efficient way of
choosing the boundary around each set of live points, and as such is
relatively fast. It is seen to produce comparably accurate posteriors to
PolyChord. See Appendix C for further discussion.

In addition to either six or seven free cosmological parameters
$(A_{\mathrm{s}},n_{\mathrm{s}},\Omega_{\mathrm{m}},\Omega_{\mathrm{b}},h,\Omega_{\nu}h^{2}$,
plus $w$ in a subset of chains run in $w$CDM) and up to five free parameters
for IAs (see Section 2.3 above), we have a selection of nuisance parameters.
These are designed to account for uncertainties related to the data and
measurements, and so differ between $2\times 2$pt and $1\times 2$pt analyses.
For the cosmic shear part of the data vector, we allow one free multiplicative
shear calibration factor $m$ per redshift bin. These parameters enter simply
as $\xi^{ij}_{\pm}\rightarrow(1+m^{i})(1+m^{j})\xi^{ij}_{\pm}$. Since galaxy-
galaxy lensing measurements rely on the same (potentially imperfectly
calibrated) shape catalogue, the same parameters also enter the $2\times 2$pt
data as $\gamma^{ij}_{t}\rightarrow(1+m^{j})\gamma^{ij}_{t}$. The shear
catalogues also have associated redshift uncertainties. We thus allow one free
shift parameter $\Delta z_{\rm s}$ per bin, which translates the redshift
distribution entering the equations above as $n^{i}_{\rm s}(z)\rightarrow
n^{i}_{\rm s}(z-\Delta z_{\rm s}^{i})$. This parameterisation, though simple,
is thought to be sufficient for cosmic shear in the current generation of
surveys (DES Collaboration, 2022).

For the lens sample, we similarly have various nuisance parameters. First of
all, each lens bin has an independent bias factor $b_{1}^{i}$, which is
marginalised with wide flat priors $b_{1}\in[0.8,3]$. Note that for most of
this work we will only consider linear bias both for generating and analysing
data vectors. The exception is Section 4.2.3, where we consider the impact of
using a more complex model on our findings. For this alternative setup we have
two parameters per bin, which are varied with flat priors
$b_{1}^{i}\sigma_{8}\in[0.67,3.0]$ and $b_{2}^{i}\sigma_{8}^{2}\in[-4.2,4.2]$
respectively. Each lens bin also has a shift $\Delta z_{\rm l}$, applied in
the same way as with the source $n(z)$, and a stretch factor $\sigma z_{\rm
l}$, which enters as:

$n^{i}_{\rm l}(z)\rightarrow\frac{1}{\sigma z^{i}_{\rm l}}n^{i}_{\rm
l}\left(\frac{z-\left\langle z\right\rangle_{i}}{\sigma z^{i}_{\rm
l}}+\left\langle z\right\rangle_{i}\right)$ (24)

(see e.g. Porredon et al. 2022 Eq. 19). These two parameters, $\Delta z$ and
$\sigma z$ allow changes in the mean and width of each lens $n(z)$ (though
they cannot affect the higher order details of the shape).

It is also worth noting that all $2\times 2$pt and $3\times 2$pt analyses
discussed in this paper include point mass marginalisation (see MacCrann et
al. 2020, Prat, Zacharegkas et al. 2023b and Krause et al. 2021 for details
and discussion). In brief, the procedure analytically marginalises the impact
of non-local lensing contributions to $\gamma_{t}$. This does not require
extra parameters, but effectively removes information coming from very small
scales.

We consider a few different analysis setups, which we will describe in the
next section. The choice of lens and source sample determines the priors on
these extra nuisance parameters. We refer the reader to Section 3.2 and Table
1 for a summary.

##  3 Synthetic Data & IA Model Error

Figure 1: The fiducial redshift distributions used in this work. From top, we
have (a) the DES Y3 source sample in 4 bins; (b) the associated lensing
kernels, (c) the Y3 redMaGiC lens $n(z)$ in five bins and (d) the equivalent
distributions, but for the MagLim lens sample. Note that the fiducial DES Y3
setup (DES Collaboration, 2022) made use of the lower four MagLim bins only;
in this work we consider both 4- and 6-bin setups. Each of these
distributions, and the overlap between them, enters the mock data as described
in Section 2. For reference, the mock LSST Y1 like $n(z)$s, used in Section
4.5, are shown in Appendix C.

We make use of much the same infrastructure as in Campos et al. (2023). Our
mock data are created with the same flat $\Lambda$CDM cosmology,
$\Omega_{\mathrm{m}}=0.29$, $A_{s}=2.38\times 10^{-9}$,
$\Omega_{\mathrm{b}}=0.052$, $h=0.75$, $n_{s}=0.99$,
$\Omega_{\nu}h^{2}=0.00083$, $w=-1$. This corresponds to $\sigma_{8}=0.79$,
$S_{8}=0.77$ and $\sum m_{\nu}c^{2}=0.077$ eV. When considering DES Y3 like
data, we also use the same analytic non-Gaussian estimate for the joint
$\xi_{\pm}+\gamma_{t}+w$ covariance matrix, as described in Friedrich et al.
(2021) and used in DES Collaboration (2022). This was estimated using CosmoCov
888https://github.com/CosmoLike/CosmoCov (Fang et al., 2020), which uses a
halo model to estimate covariances, including connected non-Gaussian and
super-sample variance terms, and also includes the impact of the Y3 survey
mask.

###  3.1 Mock cosmic shear data

We create noiseless mock data simply by running the theory pipeline with a
particular set of input parameter values. As in Campos et al. (2023), for our
fiducial setup we use the DES Y3 source redshift distributions, shown in the
upper panel of Figure 1. We create an ensemble of 21 data vectors, each one
with the same input cosmology, but different IA scenarios. A given IA scenario
is defined by a set of TATT parameters (5 values per scenario), chosen using
the process set out in Campos et al. (2023) Section 3.2, which uses a Latin
hypercube to generate an initial set of samples, before rotating them to
capture correlations seen in real DES Y1 data (see also their Figure 1). Note
that this process is not restricted by any priors, and so we include a small
number of cases outside the limits shown in Table 1.

###  3.2 Mock galaxy-galaxy lensing & galaxy clustering data

We also generate mock $\gamma_{t}$ and $w(\theta)$ data in the same way as
described above. One complication here is the choice of lens sample; whereas
in DES Y3 we had only one source catalogue, we had two separate lens
catalogues. The fiducial setup made use of a magnitude limited sample
(MagLim), as set out in Porredon et al. (2021). The idea here was to optimise
the constraining power of the joint lensing and clustering data by trading
redshift quality for density. Analyses using MagLim lenses have slightly more
nuisance parameters related to redshift error, but smaller statistical
uncertainties. Due to residual systematics, the fiducial Y3 cosmology analyses
made use of the first four MagLim bins only (Porredon et al., 2022). As
discussed below, however, we will consider both 4- and 6-bin MagLim
configurations in this work. In addition to MagLim, we consider a DES Y3 like
redMaGiC sample, as described in Pandey et al. (2022). Our redMaGiC sample has
five bins up to $z\sim 0.9$, with some slight overlap between neighbouring
bins.

The redshift distributions $n_{\mathrm{l}}(z)$ for both our MagLim and
redMaGiC lens samples are shown in the lower panels of Figure 1. Although it
would be simpler to consider one lens sample only, the reason we include two
is because sensitivity to IAs is a function both of the shape of the lens
distributions and the associated redshift uncertainties. It is, then, useful
to consider a range of plausible lens samples. Note that, as in DES
Collaboration (2022) but unlike some analyses in the literature, we include
all possible lens-source pairs in $\gamma_{t}$. That is, we do not exclude bin
pairs for which the mean lens redshift is higher than that of the sources.
This can help to constrain IAs as well as photometric redshift error, since
the lensing signal in these bin pairs is expected to be small.

###  3.3 Combining the data: analysis setups

All analyses in this work assume a flat cosmology with massive neutrinos. In
total, our fiducial $\Lambda$CDM cosmological model has six free parameters:
$\Omega_{\mathrm{m}}$, $A_{\mathrm{s}}$, $n_{\mathrm{s}}$, $h$,
$\Omega_{\mathrm{b}}$, $\Omega_{\nu}h^{2}$. In Section 4 we also run $w$CDM
analyses, which have an extra free parameter, $w$, corresponding to the dark
energy equation of state. One can find a summary of our priors in Table 1. We
constrain these parameters alongside other probe-specific ones using
combinations of data, as described below.

  * •

Cosmic shear ($1\times 2$pt): Our fiducial shear setup does not make use of
any information from galaxy-galaxy lensing. We apply the fiducial scale cuts
of Secco, Samuroff et al. (2022) and Amon et al. (2022) to the $\xi_{+}$ \+
$\xi_{-}$ data vector, which are driven by uncertainty due to the effect of
baryonic feedback on small scales. In total we have up to 19 free parameters
in our fiducial setup: 6 for cosmology in $\Lambda$CDM, 4 for shear
calibration $m^{i}$, 4 for redshift error $\Delta z^{i}$, and either 2 (NLA)
or 5 (TATT) for intrinsic alignments.

  * •

$\gamma_{t}+w$ ($2\times 2$pt), MagLim lenses, 4 bins : In this setup, we
combine galaxy-galaxy lensing and galaxy clustering using the lowest four
MagLim bins as lenses. This was the fiducial choice of DES Collaboration
(2022), with cuts removing the upper two bins, a choice motivated by
unacceptably poor fits in $\Lambda$CDM when those bins were included (see
Porredon et al. 2022 Section VII A and Appendix B for discussion). Again, the
scale cuts are not changed from those used in the fiducial Y3 analysis. The
cuts on the $2\times 2$pt data correspond to lower limits on comoving
separation at 6 $\mathrm{Mpc}/h$ and 8 $\mathrm{Mpc}/h$ for $\gamma_{t}$ and
$w$ respectively (see Porredon et al. 2022 Section VI-A). In this setup, we
have the same parameters for shear-related systematics and cosmology as above,
plus another 4 galaxy bias parameters (one per lens bin), and 8 lens redshift
parameters (1 stretch and 1 shift per bin). This gives a total of 31 free
parameters for an analysis using TATT, and 28 for NLA.

  * •

$\gamma_{t}+w$ ($2\times 2$pt), MagLim lenses, 6 bins : This setup is the same
as the one described above, but we now include the upper two lens bins. This
is our most constraining and our most optimistic setup, since it uses the
MagLim lens sample in a regime where it was not possible in the fiducial Y3
$3\times 2$pt analysis of DES Collaboration 2022 (although the upper bins have
been successfully included in some variants of analyses based on CMB cross-
correlations; see e.g. Chang et al. 2023 Figure 8). Incorporating the extra
lens bins adds another 6 parameters (2 bias + 4 redshift), making a total of
37/34 (TATT/NLA).

  * •

$\gamma_{t}+w$ ($2\times 2$pt), redMaGiC lenses: This analysis setup uses a
mock lens sample generated with the properties of the DES Y3 redMaGiC
catalogue in five bins. The key difference compared with MagLim is that
redMaGiC is designed to optimise redshift quality alone, and so the priors are
slightly different (see Table 1 and Pandey et al. 2022). In particular, the
first four width parameters are fixed. In total, then, we have 11 lens-related
parameters, and so 30/27 parameters overall. Note that this setup matches the
main Y3 redMaGiC selection discussed in Pandey et al. (2022). This is slightly
different from the “broad-$\chi^{2}$" redMaGiC sample, which is also discussed
in that paper. We discuss the distinction briefly in Section 4.2.1.

  * •

$\xi_{\pm}+\gamma_{t}+w$ ($3\times 2$pt): In addition to the analyses detailed
above, we also consider $3\times 2$pt analyses using each lens sample. This
does not increase the number of parameters relative to the corresponding
$2\times 2$pt analysis, or change any of the other analysis choices, but it
does significantly increase the constraining power, as we will see in Section
4.

Finally, in Section 4.2.3 and Appendix D only we consider $1\times 2$pt and
$2\times 2$pt analysis setups that include shear ratios. The mock shear ratio
data are constructed using our simulated $\gamma_{t}$ measurements on small
scales. Essentially shear ratios add nine data points to the data vector (the
lower 3 lens bins, each with 3 different combinations of source bin pairs; see
Sánchez, Prat et al. 2022 for details). When modelling these data, the
nuisance parameters affecting the relevant lens and source bins are propagated
through consistently.

Parameter | Fiducial | Prior
---|---|---
Cosmology
$\Omega_{\mathrm{m}}$ | 0.3 | [0.1, 0.9]
$A_{\mathrm{s}}\times 10^{9}$ | 2.19 | [$0.5$, $5.0$]
$n_{\rm s}$ | 0.97 | [0.87, 1.07]
$\Omega_{\mathrm{b}}$ | 0.048 | [0.03, 0.07]
$h$ | 0.69 | [0.55, 0.91]
$\Omega_{\nu}h^{2}\times 10^{3}$ | 0.83 | [0.6, 6.44]
$w$ | -1.0 | Fixed or $[-2,-0.333]$
Intrinsic alignment∗
$A_{1},\eta_{1}$ | 0.85, 0.0 | [$-5,5$ ]
$A_{2},\eta_{2}$ | -3.08, 2.79 | [$-5,5$ ] or $\delta(0)$
$b_{\mathrm{TA}}$ | 0.12 | [$0,2$] or $\delta(0)$
Source photo$-z$ uncertainty
$\Delta z^{1}_{\rm s}$ | 0.0 |  $\mathcal{N}(0.0,0.018$)
$\Delta z^{2}_{\rm s}$ | 0.0 |  $\mathcal{N}(0.0,0.013$)
$\Delta z^{3}_{\rm s}$ | 0.0 |  $\mathcal{N}(0.0,0.006$)
$\Delta z^{4}_{\rm s}$ | 0.0 |  $\mathcal{N}(0.0,0.013$)
Shear calibration uncertainty
$m^{1}$ | 0.0 |  $\mathcal{N}(0.0,0.008$)
$m^{2}$ | 0.0 |  $\mathcal{N}(0.0,0.013$)
$m^{3}$ | 0.0 |  $\mathcal{N}(0.0,0.009$)
$m^{4}$ | 0.0 |  $\mathcal{N}(0.0,0.012$)
Galaxy bias
$b_{\rm MagLim}^{i}$ | $1.5,1.8,1.8,1.9,2.3,2.3$ | [0.8,3.0]
$b_{\rm redMaGiC}^{i}$ | $1.7,1.7,1.7,2.0,2.0$ | [0.8,3.0]
Lens magnification
$C_{\rm MagLim}^{i}$ | 0.43, 0.30, 1.75, 1.94, 1.56, 2.96 | Fixed
$C_{\rm redMaGiC}^{i}$ | $1.31,-0.52,0.34,2.25,1.97$ | Fixed
Lens photo$-z$ uncertainty (MagLim)
$\Delta z^{1}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.007$)
$\Delta z^{2}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.011$)
$\Delta z^{3}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.006$)
$\Delta z^{4}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.006$)
$\Delta z^{5}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.007$)
$\Delta z^{6}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.008$)
$\sigma z^{1}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.062$)
$\sigma z^{2}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.093$)
$\sigma z^{3}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.054$)
$\sigma z^{4}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.051$)
$\sigma z^{5}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.067$)
$\sigma z^{6}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.073$)
Lens photo$-z$ uncertainty (redMaGiC)
$\Delta z^{1}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.004$)
$\Delta z^{2}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.003$)
$\Delta z^{3}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.003$)
$\Delta z^{4}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.005$)
$\Delta z^{5}_{\rm l}$ | 0.0 |  $\mathcal{N}(0.0,0.010$)
$\sigma z^{1}_{\rm l}$ | 1.0 | Fixed
$\sigma z^{2}_{\rm l}$ | 1.0 | Fixed
$\sigma z^{3}_{\rm l}$ | 1.0 | Fixed
$\sigma z^{4}_{\rm l}$ | 1.0 | Fixed
$\sigma z^{5}_{\rm l}$ | 1.0 |  $\mathcal{N}(1.0,0.054$)
Table 1: A summary of the parameters used in the various analyses considered
in this work. Here we show both fiducial (input) values used to generated data
vectors and also the priors used when analysing the mock data. Square brackets
$[a,b]$ indicate a flat prior within the bounds $a$ and $b$, while
$\mathcal{N}(\mu,\sigma)$ is a Gaussian prior with mean $\mu$ and width
$\sigma$. The upper four sections (cosmology, IAs, source redshift error and
shear calibration, relate to the lensing source sample, and so enter both
$1\times 2$pt and all $2\times 2$pt in exactly the same way. The intrinsic
alignments section is marked with a star because various different input
values are used in this work. We show two sets of priors in this section, for
NLA and TATT setups. For the other sections, we show both MagLim and redMaGiC
values.

###  3.4 Rubin LSST Year 1 Setup

Although our primary results are based on mock DES Y3 like data, we also run a
subset of our analyses using a Rubin Legacy Survey of Space and Time Year 1
(LSST Y1; Ivezić et al. 2019) like setup. A major difference here is the
covariance matrix, which we recompute using CosmoCov (including non-Gaussian
contributions). We assume a total area of 12,300 square degrees (compared with
$\sim 4143$ for DES Y3) and source and lens samples with effective number
densities of $n_{\rm eff}=11.2$ and $n_{\rm eff}=18$ galaxies per square
arcmin respectively. These are each divided between five redshift bins (all of
these numbers are taken from the projections in DESC Collaboration 2018 and
Fang et al. 2020). We adopt the analytic $n(z)$s suggested by the DESC Science
Requirements Document (SRD; DESC Collaboration 2018), but note that there is a
fair degree of uncertainty in the future sample selection and redshift
methods. The binning and redshift distributions are described in more detail
in Appendix C. We base the input galaxy bias and lens magnification values on
the MagLim values shown in Table 1.

When analysing the mock Rubin data, we maintain much of our previous DES Y3
pipeline including priors and other modelling choices. We update the scale
cuts using the Krause et al. (2021) method of comparing baryon-contaminated
and uncontaminated data vectors (although we rely on a $\chi^{2}$ threshold
alone – since we are only attempting a simple forecast, we do not go as far as
running chains on the baryonic mock data; see Appendix C for discussion and
further details). This setup is clearly an approximation. In reality, the
priors in any future Rubin analysis will (hopefully) be more informative and
the approach to scale cuts will likely differ from DES Y3. For the sake of
simplicity, however, we do not seek to anticipate these changes. There is a
strong element of unpredictability here; we cannot, for example, say how
advances in either modelling of, or observational constraints on, baryonic
feedback might change the eventual LSST Y1 scale cuts. Even at the precision
of current data sets there are different approaches to this question.
Likewise, improved understanding and control of systematics will hopefully
allow tighter priors on the redshift and shear calibration (as well as
potentially IA) parameters. Exactly how much tighter, however, is not
something we can predict with any degree of confidence, and so we will not
attempt to. We reiterate, however, that we do not need to predict all the
elements perfectly in order to test the effects we are interested in here.

##  4 Results

In this section we set out our results from analysing the 21 IA scenarios
discussed in Section 3. We begin in Section 4.1 using a single IA scenario as
an example, to illustrate a core finding of this paper. In Section 4.2 we
consider the robustness of our results to plausible changes in the analysis
setup and data. We explore the generality of our results using the collection
of 21 IA samples in Section 4.3. Section 4.4 then draws together some of the
evidence to try to build a coherent understanding of why the data behave as
they do. Finally we demonstrate the robustness of our findings in an LSST-like
analysis setup in Section 4.5.

###  4.1 Internal tension as an indicator of bias – a high bias example

Figure 2: An example of a joint probes analysis in which the IA model (TATT)
is sufficient to describe the data. Inner and outer contours represent $68\%$
and $95\%$ confidence levels. We show cosmic shear only ($1\times 2$pt; dotted
line), galaxy-galaxy lensing + galaxy clustering ($2\times 2$pt; dashed line)
and the combination ($3\times 2$pt; shaded) separately. All three analyses are
run on noiseless simulated data. Although there are projection effects, all
recover the input (black star) to within $1\sigma$, with the differing
orientations of the contours producing a relatively tight correctly-centred
$3\times 2$pt constraint. The lens sample used in this example is redMaGiC,
but a similar picture can be seen when using either MagLim setup.

Figure 3: The same as Figure 2, but now using the 2-parameter NLA model
instead of TATT. All other aspects of the analysis are the same. The top panel
shows the projected $S_{8}-\Omega_{\mathrm{m}}$ plane. While the cosmic shear
only ($1\times 2$pt; dotted) results are offset from the input by several
$\sigma$, $2\times 2$pt (dashed) is relatively unbiased. The two probes are
discordant and the joint $3\times 2$pt contours (shaded black) sit in between
the two. The lower panel shows the NLA amplitude $A_{1}$ and $S_{8}$. Whereas
shear predominantly absorbs the unmodelled IA signal into $S_{8}$, in the
$2\times 2$pt case it is $A_{1}$ that shifts.

For illustrative purposes, we choose a particular IA scenario to consider in
more detail. This scenario has TATT parameters
$A_{1}=0.848,A_{2}=-3.08,\eta_{1}=0.0,\eta_{2}=2.79,b_{\rm TA}=0.123$ (also
listed in Table 1). We chose this data vector from the various ones considered
in Campos et al. (2023) because it was found to be relatively extreme, as
assessed in the context of cosmic shear, resulting in a bias when using the
2-parameter NLA model of $>3\sigma$. Note that this particular parameter
combination is disfavoured at $\sim 3\sigma$ but has not been completely ruled
out by the DES Y3 analyses DES Collaboration (2022). Although it was not
selected for this reason, this scenario ended up being an fairly pronounced
example of the differences between contamination in $1\times 2$pt and $2\times
2$pt cosmological constraints. As we will see in the following sections,
however, the general trends hold over the range of IA scenarios.

We will start with the simplest case, where the data and the model match
exactly, shown in Figure 2. For all three of the posteriors shown here, the IA
model is TATT (with 5 free parameters). As we can see, both cosmic shear
(dotted) and galaxy-galaxy lensing plus clustering (dashed) recover the input
cosmology to $<<1\sigma$. For this example we’ve chosen to show the redMaGiC
rather than MagLim 2- and $3\times 2$pt results, but this choice makes no
qualitative difference (we will discuss this more in Section 4.2). As
discussed in Krause et al. (2021) and Pandey et al. (2022), the $2\times 2$pt
case is subject to slightly stronger projection effects, offsetting the dashed
contour downwards by a fraction of a $\sigma$. The constraints are entirely
consistent, however, and the joint $3\times 2$pt analysis recovers the input
almost perfectly. The difference in contour shape nicely illustrates the point
of the joint analysis – the intersection of the two allows a relatively tight
constraint on both $S_{8}$ and $\Omega_{\mathrm{m}}$. This also demonstrates
that while $S_{8}$ is, by design, the combination best constrained by cosmic
shear, it is not quite optimal for $2\times 2$pt – there is some slope in the
dashed contours in Figure 2.

We next consider the same IA scenario, but analysed with an overly simple
model (2-parameter NLA). There is now a relatively strong bias in the cosmic
shear fit, shown by the dotted contour in Figure 3. The bias here is similar
in nature to that seen in the simulated tests of Secco, Samuroff et al. (2022)
(see their Figure 6), with the best fitting constraints being shifted to high
$S_{8}$ and low $\Omega_{\mathrm{m}}$ by several $\sigma$. This shift is
accompanied by a worsening goodness-of-fit, with $\Delta\chi^{2}\sim 18$ for
222 degrees of freedom (227 data points; see Section 4.2.1 of Campos et al.
2023). Note that, as described in Section 2.4, the degrees of freedom quoted
here are calculated using an effective number of free parameters $N_{\rm
par,eff}$, which accounts for the fact that some parameters are prior
dominated, rather than the naive value from counting variables. The dashed
lines in Figure 3 show what happens when we analyse the same TATT scenario
using the same over-simple IA model, but now with a $2\times 2$pt data vector.
Interestingly, we can see that the bias is significantly reduced compared with
the cosmic shear only case. The $2\times 2$pt contour is centred on the input
to well within $1\sigma$. The goodness-of-fit is slightly worse, with
$\Delta\chi^{2}\sim 50$ for $\sim 297$ degrees of freedom999We have assumed
the effective number of parameters in the Y3 $2\times 2$pt analysis to be
roughly 5, in line with cosmic shear. This is partially guided by Dacunha et
al. (2022), who estimated $N_{\rm p,eff}=2.51$ for a Y1 like DES $3\times 2$pt
analysis. Our guess accounts for the fact that Y3 is more constraining than
Y1, and also has extra nuisance parameters. Although not precise, it is
expected to be correct to a factor of 2 or so, and so gives us a rough idea of
the quality of the fit. (302 data points: 248 in $\gamma_{t}$ and 54 in $w$).
In both cases, however, we can see that the change in the goodness-of-fit due
to the IA contamination is relatively small compared with the width of the
$\chi^{2}$ distribution, given the degrees of freedom. That is, under the null
hypothesis that the model fits the data perfectly, we expect a reduced
$\chi^{2}$ of $\sim 1$, on average. If we treat the $\Delta\chi^{2}$ values
above as coherent shifts in the goodness-of-fit, we obtain $\chi^{2}/N_{\rm
dof}\sim(222+18)/222=1.08$ and $(297+50)/297\sim 1.17$. In other words,
despite the biases seen in Figure 3, it is unlikely that the goodness-of-fit
would flag either the cosmic shear or galaxy-galaxy lensing + clustering
analyses as problematic.

We illustrate this by rerunning both with a noisy data vector (we use the
fiducial noise realisation from Campos et al. 2023 here). We obtain best fits
of $\chi^{2}_{2\times 2\mathrm{pt}}=313.3$ ($\chi^{2}/N_{\rm dof}\sim 1.05$,
again with 222 dof) and $\chi^{2}_{1\times 2\mathrm{pt}}=238.2$
($\chi^{2}/N_{\rm dof}\sim 1.07$, 297 dof), respectively. This gives a
$p-$value $p(>\chi^{2})=0.22$ for cosmic shear and $p(>\chi^{2})=0.24$ for
$2\times 2$pt101010Even using the naive calculation for the degrees of
freedom, ignoring the impact of priors, we have $302-27=275$ degrees of
freedom in the redMaGiC $2\times 2$pt NLA analysis and $227-16=211$ for
$1\times 2$pt. This still gives a $p>0.05$ in both cases.. To summarise, there
are data vector level residuals, which are apparent in the non-zero $\chi^{2}$
values from noiseless data vectors. Once we are in a scenario with (DES Y3
like) noise, however, it is difficult to tell anything is wrong from a
theoretical interpretation of the goodness-of-fit statistics. Both $p-$values
are well above commonly used thresholds such as $p=0.01$ or $p=0.05$, despite
the fact that there are considerable parameter biases. This is consistent with
the findings of Campos et al. (2023), who reported that for cosmic shear
alone, theoretically motivated thresholds based on $p-$values or information
criteria are not a reliable way of identifying parameter bias (see the
discussion in their Section 5.3).

It is also interesting to note that while $S_{8}$ (and other cosmological
parameters) are largely unbiased in the $2\times 2$pt analysis, there is a
significant shift in $A_{1}$ (lower panel of Figure 3). While clearly it
cannot do so perfectly (hence the non-zero $\Delta\chi^{2}$), the IA model is
absorbing some of the error, resulting in a slightly negative best fit
$A_{1}$. We will explore this trend further, and what it tells us, in Section
4.2. Interestingly, we do not see significant shifts in other parameters
(illustrated in Appendix A). Although there are minor offsets in the lowest
source bin, $\Delta z^{1}_{\rm s}$, these are well within $1\sigma$; all of
the other shift and stretch redshift parameters are remarkably stable in all
cases. Although it is often stated that redshift errors can easily be absorbed
by IA parameters (and the other way round), and this is certainly true in the
case where we have little/no prior knowledge, it is worth remembering that the
redshift parameters in our case are relatively tightly controlled (see the
priors in Table 1). This restricts the amount of possible interaction between
the two, and likely explains the relative stability that we see. A similar
picture is seen with shear biases. Although galaxy bias is varied with wide
priors, it is tightly constrained by $w(\theta)$, and we see no signs of
significant offsets in $b^{i}$. Nor is there any evidence that the parameter
bias in the $2\times 2$pt case is simply being absorbed into other parts of
cosmology parameter space; $h$, $n_{\mathrm{s}}$, $A_{\mathrm{s}}$,
$\Omega_{\mathrm{b}}$ are all well centred compared with their input values.

The joint $3\times 2$pt analysis (shaded black contour in Figure 3) is clearly
biased, but this is driven by the cosmic shear data favouring high $S_{8}$.
That is, what we are seeing is a differential sensitivity to IA mis-modeling
between galaxy-galaxy lensing and cosmic shear, but that does not translate
into self-calibration in the joint analysis. Accurate modelling is still
clearly needed to produce unbiased joint probe results. These results do,
however, suggest that IA model error can manifest itself as a form of internal
tension between different parts of the $3\times 2$pt data vector. This has a
number of practical implications, which we will return to later on. Note that
this observation – that cosmic shear is considerably more susceptible to
parameter bias than $2\times 2$pt when the IA model is wrong – is a core
finding of this paper. In the following sections we will explore what is going
on in more detail.

When considering our results, it is also worth keeping in mind that, in
practice, these sorts of analyses are typically significantly correlated. That
is, $\gamma_{t}$ and $\xi_{\pm}$ are measured using a common shear catalogue
with the same realisation of both shape noise and cosmic variance. While one
might expect to see random shifts of $1-2\sigma$ between independent data sets
relatively often, the chance of this happening in a case like ours is
considerably lower (see e.g. Doux, Baxter et al. 2021 for discussion). In
other words, the “tension" as assessed using metrics such as PPD may be more
significant than implied by a naive interpretation of the offset between the
contours in terms of $\sigma$.

###  4.2 Dependence on analysis choices

Given the observations above it is reasonable to ask how far our results
extend to other analysis configurations. In this section we consider a range
of variations on the fiducial setup. These tests help to confirm the
generality of our main result, and feed into the discussion of the
mechanism(s) producing it in Section 4.4.

####  4.2.1 Dependence on the choice of lens sample

Figure 4: Simulated parameter constraints from galaxy-galaxy lensing plus
galaxy clustering. Three different lens configurations are shown here (see
Section 3.2 for details). In each case the input IA scenario is the same, as
specified in Section 4. All three lens setups recover the input cosmology
(black star) to within $1\sigma$. For reference, a cosmic shear analysis with
the same input IA signal is biased by $\sim 4\sigma$.

One plausible variation is the choice of lens sample. In particular,
throughout the previous section we used the Y3 redMaGiC lens sample, which has
high quality redshift estimates. In a sample like MagLim, the redshift
uncertainties are slightly larger, requiring extra free parameters for the
shapes and positions of the lens bins.

In Figure 4, we show the same IA scenario as discussed in Section 4.1, but
with a few different lens samples. The black contours are the same as in
Figure 3, but overlain in purple and pink are the equivalent $2\times 2$pt
MagLim results (see Section 3 for an explanation of the difference between the
4- and 6-bin MagLim setups). As expected, these contours are tighter, for the
reasons discussed in Porredon et al. (2021). Although the centring here is
less accurate, we can see a similar qualitative picture as before. The
$2\times 2$pt results are less biased than $1\times 2$pt, and there is a
considerable offset between $1\times 2$pt and $2\times 2$pt analyses on the
same joint data. We consider the difference in more detail in Appendix B, but
note that the basic results are robust to reasonable changes in the lens
sample.

The offsets we do see in Figure 4 are thought to arise from the additional
redshift uncertainty in the cases with MagLim lenses. The stretch and shift
parameters have some freedom to increase the lens-source bin overlap in
certain parts of the data vector (and, so, change the way in which IAs enter
the data). We will return to this in Section 4.4, but it effectively creates
regions of parameter space where $S_{8}$ is more degenerate with the
unmodelled IA signal, and so reduces the ability of the data to distinguish
the two.

In addition to the main Y3 redMaGiC lens sample, Pandey et al. (2022) also
considered what became known as “broad-$\chi^{2}$ redMaGiC ". It was shown in
that paper that relaxing the redMaGiC goodness-of-fit criterion reduces
colour-dependent photometric selection biases. Since not all of the
calibration steps were run on the new broad-$\chi^{2}$ redMaGiC sample, nor
has it been validated to the same extent as the main redMaGiC and MagLim lens
samples, we do not run mock analyses in this setup. We note, however, that any
broad-$\chi^{2}$ redMaGiC $2\times 2$pt analysis would most likely sit between
the black and pink contours in Figure 4, both in size and position. In terms
of number density, the broad-$\chi^{2}$ sample is between MagLim and fiducial
redMaGiC, and so the $2\times 2$pt constraining power is expected to also be
somewhere between the two (Porredon et al., 2021). The predicted
broad-$\chi^{2}$ redMaGiC redshift priors are also wider than for fiducial
redMaGiC, but still somewhat tighter than those for MagLim (see Pandey et al.
2022 Table II and Appendix C). For this reason, we would not expect to see
shifts more extreme than those in Figure 4.

####  4.2.2 Dependence on IA model

Figure 5: The impact of the choice of IA model. The shaded contours here are
the same as those shown in the upper panel of Figure 3 (NLA with 2 free
parameters). Overlain (open contours) are analyses of the same three
contaminated data vectors, but using an even simpler IA model (NLA with only
one free parameter in the upper panel, no IA model at all in the lower).
Though there are shifts, the qualitative picture is the same: galaxy-galaxy
lensing plus clustering is significantly less biased than either shear alone
or the full $3\times 2$pt combination.

Another question one could ask is: how specific are these observations to our
choice of IA model. After all, the unmodelled TATT signal in $\gamma_{t}$
appears to be preferentially absorbed by the NLA parameters rather than
cosmology (see Figure 3 and the discussion in Section 4.1). It is reasonable
to ask how far this is a function of the flexibility of the IA model. For our
main results, we chose to analyse TATT-generated data with a 2-parameter
($A_{1}$, $\eta_{1}$) variant of the NLA model (which was the fiducial choice
of DES & KiDS Collaboration 2023 and was also run as an analysis variant in
Secco, Samuroff et al. 2022 and Asgari et al. 2021). Now we try running a
simpler sub-model, with the redshift index fixed $\eta_{1}=0$. This is closer
to the original NLA model of Hirata & Seljak (2004), and was used by KiDS-1000
(Asgari et al., 2021; Heymans et al., 2021; Tröster et al., 2022). For
comparison we also run a version assuming no IAs at all (i.e.
$A_{1},A_{2}=0$). This is not a realistic modelling choice, but we include it
here for illustrative purposes.

The results are summarised in Figure 5. Switching to NLA-1 (upper panel)
causes small shifts in both the shear and $2\times 2$pt contours. In the
former case, the contours are already at the edge of the prior space in almost
all of the cosmology parameters – although not quite hitting the edge in the
projected $\Omega_{\mathrm{m}}$ direction, it _is_ in the 2D
$\sigma_{8}-\Omega_{\mathrm{m}}$ plane, as well as in $h$, $n_{\mathrm{s}}$
and $\Omega_{\mathrm{b}}$ (this can be seen in Appendix A, which shows the
wider cosmological parameter space with the prior bounds). The purple contour
is thus restricted in how much further it can go in the
high-$S_{8}$/low-$\Omega_{\mathrm{m}}$ direction. The impact of switching
models on $2\times 2$pt is also small, but for different reasons. Here we see
a very small upwards shift, with the $1\sigma$ bound still comfortably
enclosing the input. If we consider the extreme model setup in the bottom
panel, we see interestingly little difference. Here we are not modelling IAs
at all. The $1\times 2$pt contour is seen to shrink and shift downwards
slightly. This, again, is likely due to the restriction of the prior
preventing it moving further upwards111111Note that the fan-shaped prior in
the $S_{8}-\Omega_{\mathrm{m}}$ plane (see Figure 14) is a result of our
decision to sample $A_{\mathrm{s}}$ rather than $S_{8}$. One could conceivably
widen it by expanding the $A_{\mathrm{s}}$ prior, but we note that the bounds
are already considerably wider than the range allowed by Planck 2018 (at the
level of 10s of $\sigma$)..

The $2\times 2$pt case is interesting here – unlike with cosmic shear, the
constraints are well contained by the prior, and so edge effects are not a
factor. In both (1- and $2\times 2$pt) cases, the best $\chi^{2}$ gets
steadily worse as one simplifies the model from 2-parameter NLA to 1-parameter
NLA to zero IA. The change is still relatively modest, however, with
$\Delta\chi_{2\times 2\mathrm{pt}}^{2}\sim 50\rightarrow 56\rightarrow 58$. We
can conclude a few things here. First, given the small change in $\chi^{2}$,
it may be that we are seeing some level of cancellation between IA
contributions, such that the preferred $A_{1}$ happens to be close to zero.
This can happen in galaxy-galaxy lensing in a way that is not, in general,
possible with cosmic shear (although, as we will come to in Section 4.4, there
are reasons to think this is not the full story). Another conclusion we can
draw is that while there _is_ residual model error, which gets worse as the
model is simplified, it simply is not strongly degenerate with $S_{8}$ in the
way it is for cosmic shear.

The net result is that, regardless of the details of the wrong IA model,
internal tension can arise due to differences in how IAs enter the different
probes. This is interesting as it points to the effect being relatively
general, rather than specific to our Y3 setup.

####  4.2.3 Dependence on scale cuts and galaxy bias model

As discussed in Section 2 the main DES Y3 analyses opted to model galaxy bias
using a linear approximation, with scale cuts to facilitate this. An
alternative approach would be to extend to smaller scales, at the cost of
needing marginalise over a more complex bias model. To test the impact, we re-
run a subset of our $2\times 2$pt chains in such a small scales + nonlinear
bias setup. More specifically, we use the parameterisation of Pandey et al.
(2020), which has been shown to be accurate at the level of a few percent down
to scales of $4$ Mpc$/h$ (see also Section 2.2). This setup was used on the
real data, and so has been fairly extensively validated for both the redMaGiC
and MagLim lens samples at Y3 precision (Porredon et al., 2022; Pandey et al.,
2022). As explained in Section 2, our modelling follows DES Y3, and so both
our fiducial and our small scale/nonlinear bias $2\times 2$pt setups include
point mass marginalisation (see DES Collaboration 2022 and Krause et al.
2021).

Figure 6: The impact of the galaxy bias model and scale cuts on our fiducial
$2\times 2$pt analysis. We show three different combinations of bias model and
scale cuts: linear bias only with large scale cuts ($>8$ Mpc$/h$ for $w$, $>6$
Mpc$/h$ for $\gamma_{t}$; upper left, black); nonlinear bias + large scale
cuts (upper right, pink); nonlinear bias + small scale cuts ($>4$ Mpc$/h$ for
both $w$ and $\gamma_{t}$; bottom centre, purple). In all cases shown, the
simulated data contain linear bias. In each case we show chains run on an
“uncontaminated" NLA data vector, as well as our fiducial TATT scenario. The
difference between the two represents the bias due to IA mismodelling,
excluding projection effects. As discussed in Section 4.2.3, the lower panel
also shows a case using linear bias combined with $4,4$Mpc$/h$ cuts to
illustrate the effects of bias model flexibility.

Our findings are summarised in Figure 6. To begin we repeat our baseline
$2\times 2$pt analysis on the TATT data vector introduced in Section 4.1, but
using the third-order bias model mentioned above. As before, these data
contain an unmodelled higher order IA signal, but the input galaxy bias is
linear, and there are no other sources of error. The impact can be seen by
comparing the upper two panels in Figure 6. For reference, we also show the
same model run on an NLA version of the data vector. Although there is an
overall shift due to projection effects, in both cases the bias due to the IA
contamination is negligible (i.e. chains run on TATT and NLA data vectors
agree very well).

Next we repeat the same NLA + nonlinear bias analysis, but incorporating
scales down to 4 Mpc$/h$ in both parts of the $2\times 2$pt data vector. In
total this increases the number of data points from 302 (248 in $\gamma_{t}$
\+ 54 in $w$) to 350 (280 + 70). This results in the black posterior in the
lower panel of Figure 6. Although still considerably less biased than the
cosmic shear analysis of the same IA scenario (see Section 4.1), there is an
upwards shift of around $0.7\sigma$. Unlike in the upper panels, we cannot
ascribe this to projection effects, since the chains run on NLA and TATT data
vectors are also offset from each other.

It is worth briefly considering what this tells us about the mechanisms behind
the robustness of the $2\times 2$pt data. The shift between the purple and
black contours in the lower panel of Figure 6 could be caused by two things.
One explanation is that shifting to $4$ Mpc$/h$ simply includes more data
points that are strongly affected by the higher order TATT contamination; this
can worsen the overall $\chi^{2}$, but also change the balance of bin pairs
affected/unaffected by IAs. This can potentially affect the ability of the
data to self-calibrate, as we will come to in Section 4.4. An alternative
explanation is that the extra model freedom allowed by the (now relatively
well constrained) $b_{2}$ parameters is allowing IA error to cross over into
cosmological error. The idea of degeneracy-breaking, discussed further in
Section 4.4, relies on the lensing dominated parts of $\gamma_{t}$ being able
to exclude shifts in cosmology that could otherwise absorb error in the IA
dominated parts. Certain kinds of extra model freedom can break (or weaken)
this self-calibration by allowing the lensing dominated parts of the data to
adjust. We can test the second hypothesis by rerunning the $4$ Mpc$/h$
analysis with linear bias. This is not a realistic setup, but is designed to
shed light on the effects at work here. As we can see from these results (the
dashed contours in the lower panel of Figure 6), the bias model freedom alone
accounts for roughly half of the bias. The rest, we conclude, is coming from
the additional contaminated scales.

In addition to the above tests, we also consider the impact of including
information from small scale shear ratios, in the way described in Sánchez,
Prat et al. (2022). More detail can be found in Appendix D, but in brief there
is very little change in how $2\times 2$pt responds to IA error even including
shear ratio measurements down to $2$ Mpc$/h$.

When thinking about the impact of scale cuts, we should note some basic
differences between our cosmic shear and $2\times 2$pt data, which affect the
physical scales they respond to. Our cuts for $2\times 2$pt, are defined in
terms of a fixed physical separation (see Section 3.2), and translated into an
angular cut in each lens bin. This puts a fairly simple lower limit on the
scales on which we are sensitive to IAs. The same approach is not possible for
cosmic shear, both because of the much broader redshift distributions, but
also because the lensing kernel tends to mix physical scales. Our $1\times
2$pt cuts are, then, defined in angular space. The result is that the physical
cutoff varies between bin pairs, and a range of scales are included in the
final analysis (see Secco, Samuroff et al. 2022 Section IV-B and Figure 4).
Shear analyses tend to be sensitive to physics on smaller scales, and also to
have slightly less control over exactly which are used, compared with $2\times
2$pt. Additionally, point mass marginalisation further removes information
coming from small scales. Given all this, it makes some intuitive sense that
scale cuts may be more effective in mitigating higher-order IA terms in
$\gamma_{t}$ than $\xi_{\pm}$. For reasons we will come to in Section 4.4, we
do not think this alone can explain the robustness of $2\times 2$pt analyses,
but it is a factor that likely plays a role.

In summary, we do see some sensitivity to the choice of $2\times 2$pt cuts,
with our small-scale galaxy-galaxy lensing + clustering analysis being more
susceptible to IA-related biases than our fiducial large scale one. As we have
seen, however, for realistic setups the impact is still relatively small. Our
result from the previous sections does not change qualitatively, with the bias
in a $2\times 2$pt analysis being considerably smaller than in a cosmic shear
one based on the same data. We have presented tests using models that are
currently thought to be robust. In coming years it is likely that there will
be advances in modelling of $\gamma_{t}$ on small scales (see e.g. Zacharegkas
et al. 2022 and the discussion in that paper). In such cases we should give
careful consideration to sensitivity to IA error, and how the mechanisms we
have described in this paper may change.

###  4.3 Alternative input IA scenarios

Figure 7:  Bias in the 2D $S_{8}-\Omega_{\mathrm{m}}$ plane in mock cosmic
shear (horizontal axis) and galaxy-galaxy lensing plus galaxy clustering
(vertical axis) analyses. Each point represents a particular input IA
scenario, and the bias arises from an insufficient IA model (in this case
2-parameter NLA). The fiducial IA scenario, discussed in Section 4.1 is
represented by the open stars. The diagonal line marks bias equality (i.e. the
bias in the $2\times 2$pt and $1\times 2$pt analyses are exactly equal). The
dashed lines represent $1\sigma$ bias, and the shaded bands show the
$<0.3\sigma$ region for each probe. We show results using two alternative lens
samples here, redMaGiC and MagLim (pink and purple, as labelled; see Section
3.2 for details). Figure 8: Data vector contamination and the resulting
cosmological bias for cosmic shear (purple circles) and galaxy-galaxy lensing
plus clustering (black diamonds). “Contamination" (the x-axis quantity) is
defined as a $\chi^{2}$, evaluated between noiseless TATT and NLA data vectors
at the same fiducial input cosmology. This is effectively a summary statistic
for the impact of higher-order IA contributions. In general, we can see that
for a given level of contamination, the bias in the cosmic shear analyses
tends to be larger than for $2\times 2$pt. For clarity, we only show the
redMaGiC lens sample here. As in Figure 7, the grey band and dashed line show
$0.3\sigma$ and $1\sigma$ thresholds in bias. We again highlight our fiducial
IA scenario as open stars.

We next move from our single example to look at whether our conclusions bear
out when considering a range of IA scenarios. Although we do not think
cancellation is the driving factor in the differential $1\times 2/2\times 2$pt
sensitivity, it is still worth testing that our fiducial IA scenario is not
unrepresentative in some way. As discussed in Section 3, we have a collection
of 21 data vectors with different input IA scenarios, sampled from the DES
Year 1 TATT posteriors. These scenarios were chosen to cover a wide range of
plausible IA parameter space, including cases that are unlikely but not
completely ruled out by existing data. The selection process is set out in
more detail in Campos et al. (2023) Section 3 (see also their Figure 1).

In Figure 7 we show the relation between cosmological bias in $1\times 2$pt
and $2\times 2$pt NLA analyses of each of these samples. Our metric for
cosmological bias here is defined as an offset in the 2D
$S_{8}-\Omega_{\mathrm{m}}$ plane relative to the input, and is estimated as
described in Campos et al. (2023) Section 4.1. We show 21 different input IA
scenarios, with each point representing a particular scenario. For reference
we indicate our fiducial case as a star. As we can see, the fiducial case is
not particularly unusual. In general, $2\times 2$pt is less biased than
$1\times 2$pt for a particular input (the points tend to lie below the
diagonal dotted line). Although it is not always the case that the residual
bias is completely insignificant (particularly in the MagLim setup; purple
diamonds), we see that $3-5\sigma$ in cosmic shear tend to translate into
$1\sigma$ or less in $2\times 2$pt. We should note that there is a relative
rotation of the posteriors between $1\times 2$pt and $2\times 2$pt (i.e.
$S_{8}$ is not the optimal combination of $\sigma_{8}$ and
$\Omega_{\mathrm{m}}$ for $2\times 2$pt). This can reduce the size of a shift
in the $S_{8}$ direction, as measured in terms of $\sigma$. Note however, that
the IA induced shifts do not act purely in the $S_{8}$ direction – considering
the 21 scenarios, we find a fairly isotropic distribution of biases relative
to the input, and there is no reason to think the contour shape will make the
biases as measured in the 2D $S_{8}-\Omega_{\mathrm{m}}$ plane larger or
smaller. Even measured solely in the $S_{8}$ direction, the difference tends
to be significant in absolute terms. That is to say, the trend seen in Figure
7 is not thought to arise from the geometry of the contours

We should note that in the most extreme cases the cosmic shear posteriors are
hitting the prior bounds in at least some parts of the parameter volume. This
limits the size of the bias in absolute terms, but it also distorts the
posterior, leading to an artificial reduction in the $1\sigma$ confidence
region. We do not consider this an issue, since the bias in these cases is
already large (i.e. we are not concerned particularly in distinguishing
between $5\sigma$ and $10\sigma$ biases – both are in the regime of
unacceptably large).

Of the two lens configurations, MagLim $2\times 2$pt is consistently more
sensitive to IA mismodelling than redMaGiC $2\times 2$pt (the pink points tend
to be lower than the purple for a given TATT input). This, again, is
consistent with what we saw previously (Section 4.2.1 and Figure 4), and is
likely due to the better redshift quality in redMaGiC, as we discuss in
Appendix B. In both cases, however, the general conclusion that $2\times 2$pt
tends to be more robust than cosmic shear holds.

In Figure 8, we show the data vector level contamination for the same
collection of IA samples. That is, for each sample we evaluate the $\chi^{2}$
between the contaminated TATT data vector, and an equivalent NLA data vector
(identical input parameters, but with $A_{2},\eta_{2},b_{\rm TA}=0$). This
serves as a summary statistic for the level of higher-order IA contamination
in each simulated data vector. Note that this is not quite the same as the
size of the overall IAs signal, but is the impact of the specific terms that
are not captured by NLA. As we can see in Figure 8, in the case of cosmic
shear, there is a clear positive correlation between the amplitude of the TATT
contamination and the cosmological bias. This makes intuitive sense – as the
signal NLA cannot reproduce gets larger, the bias due to mismodelling gets
larger. The trend in the black ($2\times 2$pt) points is interesting in
comparison. Here we see much lower levels of bias for a given $\chi^{2}$; the
black points tend to lie below the purple, on average. We should also note
that there is much less (if any) correlation between bias and contamination
for $2\times 2$pt. Indeed, the cases with the largest bias are not the ones
with the largest contamination level. This is, we should note, consistent with
the hypothesis we will come to in Section 4.4 that the reason for the
robustness of $2\times 2$pt analyses is due to a lack of degeneracy with
cosmology rather than inherently low levels of IA contamination in
$\gamma_{t}$. While the distribution of black points in Figure 8 is noticeably
lower for a given $\chi^{2}$, it is also true that the data-level
contamination in the $2\times 2$pt cases does not reach the very extreme
values as $1\times 2$pt (i.e. there are no black points on the far right hand
side of Figure 8). This is a sign that the higher order terms are having less
impact, either because of internal cancellation of TATT contributions or
because $\gamma_{t}$ (after scale cuts) is less sensitive to small scale
information. Even if these are not the primary factors behind the lack of bias
in $2\times 2$pt, they may be acting to limit the maximum contamination.

Considering the set of IA samples shown in Figures 7 and 8 together, we can
identify some basic trends with input parameters. First of all, the bulk of
the scenarios have $A_{1}$ and $A_{2}$ with opposite signs due to the way in
which they were generated. It does not appear, however, that the bias in
either probe is notably smaller (or larger) in the few samples where this is
not true. This implies, again, that although internal cancellation in the
$2\times 2$pt case may occur, it is not the primary mechanism at work. Overall
we find a fairly strong correlation between the cosmic shear bias and the low-
redshift $A_{2}$ amplitude in the data (that is,
$A_{2}(z)=A_{2}\times[(1+z)/(1+z_{0})]^{\eta_{2}}$, evaluated at $z=0.3$). If
we choose to evaluate $A_{2}(z)$ above the pivot redshift (i.e. $z>0.62$), the
correlation weakens considerably. This suggests that, at least with current
data sets, we are primarily sensitive to the behaviour of IAs at low redshift.
Put another way, the IA scenarios with large positive $\eta_{2}$ (implying an
$A_{2}$ signal that goes to 0 at $z\sim 0$ and increases at high $z$) are
often cases in which NLA is close to unbiased. There is no obvious correlation
between the same low-$z$ $A_{2}$ amplitude and the $2\times 2$pt bias.

If we similarly consider the density weighting term $A_{1\delta}(z)=b_{\rm
TA}A_{1}\times[(1+z)/(1+z_{0})]^{\eta_{1}}$, evaluated at low $z$, we see no
clear correlation with the cosmic shear bias. For the bulk of samples we also
see no clear trend with $2\times 2$pt bias either. It is noticeable, however,
the two cases with the largest $2\times 2$pt bias are the two with the largest
$|A_{1\delta}|$. In particular the one case with bias $>1\sigma$ (the pair of
points at the top of Figure 7) has fairly small input amplitudes
($A_{1}=0.48$, $A_{2}=-0.13$), combined with very strong redshift dependence
($\eta_{1}=-6.6$, $\eta_{2}=-6.8$) and $b_{\rm TA}=1.0$. This combination
gives the largest value of $A_{1\delta}(z=0)$ of all of the samples by at
least a factor of $2$. It is possible that this is a particular mode of IA
error that is able to break the self-calibration mechanism in our $2\times
2$pt analysis. We should also note, however, that this is an unusually extreme
set of input values, with $\eta_{1}$ outside of our prior $\eta_{1}\in[-5,5]$.
This means that even ignoring the higher order TATT terms, the NLA model as we
have defined it cannot reproduce the data perfectly. It is somewhat difficult
to draw general conclusions from this without further experimentation with
more IA samples. We should note, however, that is an unusual case and it is
not critical for the conclusions of this paper.

It is worth finally noting that these results rely on the assumption that any
significant unmodelled IA contributions that are missed by NLA are encompassed
by the 5 parameter TATT model. This is reasonable given that TATT is
physically motivated, and is expected to capture the relevant processes on
scales $>\sim 2$ Mpc$/h$ (see Blazek et al. 2019 for discussion). We have also
been careful to select a wide range of parameter values in order to explore
the scope of possible impacts (see Figure 1 from Campos et al. 2023). Finally,
we should note that, as we will come to in Section 4.4, there is some evidence
to suggest that our findings are the result of degeneracy breaking by the
combination of contaminated and uncontaminated bin pairs in the $\gamma_{t}$
data. This is a generic effect, determined by the differential sensitivity of
the data rather than the details of the unmodelled IA power spectra. Given
this, it should be relatively independent of the “complete" IA model we use
for contamination.

###  4.4 Possible mechanisms for the difference in sensitivity to IA error

Figure 9: The fractional contributions of the GI and II IA terms to our mock
cosmic shear data. Each IA component is shown as a fraction of the pure
lensing signal GG, as evaluated at the input cosmology. Solid lines show the
input TATT signal used to create the mock data vector discussed in Section
4.1. The dashed lines is the IA signal as predicted at the best fit from an
NLA cosmic shear analysis of the same mock data vector. Shaded grey regions
indicate scales removed from our analysis (the fiducial Y3 scale cuts). We see
a relatively strong IA signal, at $10-20\%$ or more on intermediate scales in
most bins. Note that although we show our fiducial IA scenario here for
illustrative purposes, a similar pattern can be seen when plotting the other
cases discussed in Section 4.3. Figure 10: The fractional contributions of the
$\delta_{g}$I and $\mu$I (galaxy-intrinsic and magnification-intrinsic) IA
terms to our galaxy-galaxy lensing data. Both are shown relative to the
cosmological signal (the sum of galaxy-shear and magnification-shear
contributions $\delta_{g}$G and $\mu$G). As in Figure 9, shaded grey bands
represent scale cuts. The numbers shown in each panel specify a particular bin
pair $(l,s)$. The solid lines show the input IA signal for our mock
$\gamma_{t}$ data, and dashed lines show the best fit prediction from an NLA
$2\times 2$pt analysis of that data. Note we are showing the redMaGiC lens
configuration here. See Figure 16 for the equivalent of this plot using the
MagLim lens sample. As with cosmic shear, the overall patterns of where in the
data vector IAs are significant hold when plotting the other IA scenarios
discussed in Section 4.3.

In the above subsections we have discussed the robustness of our central
result. During the course of the discussion, we touched on some of the
possible mechanisms that may contribute to this finding. Now we will draw
these strands together to address the question more directly.

There are a few possible explanations for the observation that a cosmological
analysis based on cosmic shear is substantially more sensitive to IA
mismodelling than an equivalent $2\times 2$pt one. First, it could be that the
IA contribution to $\gamma_{t}$, as a fraction of the total signal, is just
generically smaller than in $\xi_{\pm}$. On the face of it, this is plausible,
since the power spectra entering $\gamma_{t}$ and $\xi_{\pm}$ are filtered
through different kernels in $z$ and $k$. Scale cuts and point mass
marginalisation also act to remove sensitivity to smaller scales in
$\gamma_{t}$, reducing the significance of higher order IA terms. We can rule
this out fairly quickly, however using some basic observations. Comparing
Figures 9 and 10, which show the fractional contribution of IAs to our
fiducial shear and $\gamma_{t}$ data vectors, we can see that there is a
substantial contamination in both data vectors; the maximal fractional
contamination (relative to the lensing signal) is in fact larger in
$\gamma_{t}$ than in cosmic shear. This can also be seen in Figure 8, which
shows significant levels of non-NLA contamination in our $2\times 2$pt data
vectors. Note also that, as seen in the literature, $2-$ and $3\times 2$pt
data typically constrain IA parameters more tightly than $1\times 2$pt (this
holds for TATT model constraints as well as NLA; see e.g. our Figure 3; Figure
8 of Samuroff et al. 2019; Figure 8 of DES Collaboration 2022). This is true
in general, unless the lens and source samples are specifically constructed to
avoid IA contamination (see e.g. Miyatake et al. 2023). Given this, the
generic insensitivity theory seems unlikely.

A related idea is that there is internal cancellation of the terms
contributing to the GI power spectrum, leading to the TATT contributions being
smaller in galaxy-galaxy lensing than shear. In principle, if the signal is
entirely controlled by $P_{\rm GI}$, cancellation can occur in a way that is
not possible when $P_{\rm II}$ (which is sensitive to the squares of IA
amplitudes) is relevant121212Note that cancellation can in principle occur
between II and GI power spectra. This is, however, more difficult to achieve
in practice since the two IA components tend to affect different parts of the
data vector (i.e. it is relatively rare to have bin pairs that are equally
affected by GI and II contributions on a given scale).. Referring back to Eq.
(19), we can see that if $A_{1}$ and $A_{2}$ have opposite signs (as is the
case in our fiducial example), the first two and final terms act against each
other. Such cancellation can only ever be partial, and will affect some
scales/redshifts more than others, since $P_{\delta}$, $P_{0|0E}$ and
$P_{0|E2}$ have different shapes (e.g. Figure 1 Blazek et al. 2019). It can,
however, potentially work to reduce the amplitude of the IA signal. This is
slightly different from the previous explanation – cancellation would imply
the _actual_ TATT signal in the data is more often than not small, but does
not imply a lack of constraining power (i.e. higher-order IAs can have an
impact, in theory, but in practice they tend not to). There are some reasons
to think such cancellation is a plausible factor. Considering the constraints
in the literature, the degeneracies between $A_{1}$ and $A_{2}$ are such that
it is much more common to find a combination of amplitudes with opposite signs
than the same (see e.g. Figure 8 from Secco, Samuroff et al. 2022 and Figure 7
from Sánchez, Prat et al. 2022). Note also that in Figure 3, the posterior on
$A_{1}$ from $2\times 2$pt is closer to zero than the input value. That said,
it does not appear that this is the whole picture. As mentioned above, there
is relatively large data vector level contamination seen in Figure 10, which
is inconsistent with the idea that cancellation is _the_ main driver of the
relative robustness of the $2\times 2$pt data. There is also a significant
impact coming from the non-NLA terms, as illustrated in Figure 8. Considering
Figure 10 we should note that although the NLA fit has a low amplitude, this
is likely at least partly because of limited flexibility of the model. That
is, because NLA is approximating an input TATT signal that oscillates between
positive and negative on different scales, rather than because the TATT signal
is necessarily small. We also note that although the bulk of the IA scenarios
considered in Section 4.3 do have opposite-sign $A_{1}$ and $A_{2}$, we see no
systematic difference between same-sign and opposite-sign cases. The $2\times
2$pt bias is typically smaller than the $1\times 2$pt bias in both.

One alternative hypothesis is that although IAs contribute to $\gamma_{t}$
measurements at a significant level, the scale and redshift dependence of that
signal (after scale cuts) is simpler than in cosmic shear. If we refer back to
the equations in Section 2.3, we see that $\xi_{\pm}$ is sensitive to two IA
components, GI and II (or three if we count the II B-mode spectrum
separately), each of which can deviate significantly from NLA. On the other
hand, $\gamma_{t}$ responds only to the GI term – Eq. (13) and (14) depend on
$P_{\rm GI}$ only. Ultimately we only need to solve for one IA power spectrum,
to good approximation. It is possible that the unmodelled TATT signal in
galaxy-galaxy lensing is just closer to NLA, and so more easily absorbed. Not
only this, but scale cuts in $2\times 2$pt analysis tend to more cleanly
remove small physical scales than those in cosmic shear (see the discussion in
Section 4.2.3). According to this argument, the worst of the higher order TATT
contribution is removed by the cuts on $\gamma_{t}$, and the surviving signal
is relatively simple, allowing NLA to adjust in $2\times 2$pt more easily than
$1\times 2$pt. In practice, however, there is at least some evidence against
this, at least as a full explanation. We see in Figure 3 that $A_{1}$ shifts
significantly in response to the unmodelled TATT terms (implying the presence
of at least some higher order signal to be absorbed after the scale cuts), but
the details of the IA model used do not seem to be important. As we saw in
Section 4.2.2, reducing the IA model complexity worsens the $\chi^{2}$ but
does not greatly increase the bias in cosmology. In Figure 5 we found that
even removing the IA model completely does not shift our fiducial case by more
than a fraction of a $\sigma$. We should note that it is still possible to
achieve this if _both_ the higher-order TATT terms are well approximated by
NLA and some form of cancellation leaves the overall IA amplitude close to
zero (i.e. by chance there is near-perfect cancellation). This seems somewhat
unlikely, however, given our other observations. We can see that are some non-
trivial $\theta$ and redshift scalings in the TATT contributions in Figure 10.
In the bins where there is contamination, it is not clear that NLA is
approximating the TATT signal any better here than it is in the cosmic shear
case in Figure 9. This is reflected in the roughly comparable $\chi^{2}$ per
degree-of-freedom for the two probes, as quoted in Section 4.1. We should also
note that we see a similar pattern across our 21 IA scenarios. We find the
$\chi^{2}$ per degree of freedom from our $2\times 2$pt chains is frequently
at least as large as that from a cosmic shear analysis of the same IA
scenario. This, again, suggests that higher-order TATT terms are not simply
well approximated by NLA.

Our final explanation, which we believe is the most significant factor, is
that internal degeneracy breaking means that IA error in $2\times 2$pt simply
does not translate into a bias in cosmological parameters in the way it does
for shear. To see this, compare Figures 9 and 10. Whereas $\xi_{\pm}$ is
affected by IAs throughout the whole data vector, in $\gamma_{t}$ the signal
is relatively cleanly contained to certain redshift bin pairs. As illustrated
in Figure 9, the TATT model can quite easily produce a significant II signal,
especially in the auto-bin pairs, accompanied by a GI signal with quite
different redshift and scale dependence. IAs appear in almost all parts of the
$\xi_{\pm}$ data vector at a level between $\sim 10\%$ and $40\%$ of the
cosmological signal. If we now consider Figure 10, we see that the IA signal
is similarly non-negligible in $\gamma_{t}$. In fact, in several bins it
accounts for at least as large a fraction as, or even larger than, in
$\xi_{\pm}$ (up to around $90\%$ on intermediate scales in certain bins – this
in consistent with the strong, if biased, constraint on $A_{1}$ shown in
Figure 3). Crucially, however, the IA signal does not appear uniformly across
the $\gamma_{t}$ data vector. Unlike in the case of cosmic shear, there are
pairs of redshift bins where the lens-source overlap is very small, and hence
in these bins there is almost no sensitivity to IAs (e.g. the lower left
corner of Figure 10). It is likely that this allows a degree of internal
degeneracy breaking – whereas a change in $S_{8}$ or $\Omega_{\mathrm{m}}$
enters all $\gamma_{t}$ bins in roughly the same way, an unmodelled IA signal,
clearly, can only appear in the bin pairs that have some sensitivity to IAs.
This explanation is backed up by the observations in Figure 5 (i.e. we are not
reliant on any degree of flexibility in the IA model to absorb error). It is
also supported by the fact that switching from redMaGiC to MagLim lenses (i.e.
allowing the lens redshift distributions to shift/widen, and thus complicating
the clean distinction between bins affected by IAs or not) is one of the few
changes seen to increase the IA-induced bias in $2\times 2$pt. Finally, when
considering the ensemble of IA scenarios in Section 4.3, we see no significant
correlation between the level of contamination and cosmological bias for
$2\times 2$pt (compare the black and purple points in Figure 8; see also
Figure 4 of Campos et al. 2023, which similarly shows a strong correlation for
$1\times 2$pt). The cases with the worst $\chi^{2}$ are not the ones with
highest parameter bias, and the scenario that gives largest bias does not have
a noticeably larger contamination than most of the other samples. This, again,
points to a lack of degeneracy between the model error and cosmology as the
driving factor, rather than e.g. cancellation or error being absorbed by the
IA model.

In summary, then, we have identified four potential factors that could all
lead to $2\times 2$pt analyses being less sensitive to IA modelling error than
cosmic shear: (a) galaxy-galaxy lensing simply being less sensitive to IAs
than cosmic shear; (b) cancellation of different TATT contributions, leading
to a smaller higher-order IA signal; (c) a less complex TATT signal, that can
be more easily approximated by a relatively simple model; (d) degeneracy
breaking between IAs and $S_{8}$ in $\gamma_{t}$, enabled by the fact that IAs
are confined to certain bin pairs. Considering the evidence for each
explanation in turn, we conclude that (d) is likely the most significant
effect. It is worth noting, however, that they are (mostly) not mutually
exclusive. For example, is possible that some level of overall cancellation
occurs in TATT scenarios with opposite sign $A_{1}$ and $A_{2}$. Likewise the
differences in sensitivity to physical scales will likely have some impact on
the magnitude of higher order terms and how a particular TATT IA power
spectrum translates into bias. These effects exist and tend to act towards the
same end, even if they are not the primary factor.

###  4.5 Extension to future surveys

Figure 11: Predicted cosmological constraints from a mock Rubin Observatory
LSST Year 1 like analysis. The left hand panels show the results from a data
vector with a relatively extreme input IA scenario (the same as in Section
4.1). The right shows a less extreme case with the input TATT parameters
$(A_{2},\eta_{2})$ reduced. Both sets of input parameters are consistent at
the $<2\sigma$ level with the DES Y3 results of Secco, Samuroff et al. (2022);
Amon et al. (2022). The upper and lower panels show $\Lambda$CDM and $w$CDM
analyses respectively. In each case, purple contours represent cosmic shear
only and black are galaxy-galaxy lensing + galaxy clustering. See Section 3.4
for details about the mock LSST data and analysis choices. As in previous
figures, the black star marks the input cosmology.

The next question we consider in this section is how specific these findings
are to our DES Y3 like setup – i.e. can we extrapolate our conclusions to
future data sets. To explore this, we use a set of Rubin LSST Y1 like
simulations (see Section 3.4 and Appendix C). Compared with DES Y3, the
constraining power is expected to increase significantly, reflecting an
increase in the depth (i.e. number density) and area of both the source and
lens samples (see e.g. DESC Collaboration 2018). We follow the DESC SRD by
setting up our mock data with 5 source and 5 lens bins (shown in Appendix C),
all of which have associated redshift uncertainties. We should note here that
although these capture the rough gain in depth in LSST Y1, they are opimistic
in the sense that they assume Gaussian uncertainties. In reality, redshift
distributions estimated from real data tend to have non-trivial structure and
tails, particularly at low redshift (Figure 1; see also Myles, Alarcon et al.
2021; Giannini et al. 2023; Campos et al. 2024). These features increase bin
overlap, and so tend to increase the sensitivity to IAs. Given this, we expect
our LSST simulated tests to represent a lower bound on the impact of IAs on
any future $2\times 2$pt and cosmic shear analyses.

We choose two IA scenarios to focus on. The first is the same as in Section
4.1 above. This has not been ruled out by cosmic shear data to date, but is on
the extreme end of what is consistent with previous results. In addition, we
consider a less extreme case. For this, we keep $A_{1}$, $\eta_{1}$ and
$b_{\rm TA}$ fixed, but scale down the tidal torque parameters to
$A_{2}=-0.55$, $\eta_{2}=1.5$. This represents a reduction by roughly a factor
of 5 in the amplitude of the quadratic IA term at the pivot redshift
$z_{0}=0.62$; it also reduces the relative signal at high redshift, with an
increase in $A_{2}(z)$ over the range $z=0.62-2.0$ of $\sim 2.5$ compared with
$\sim 5.6$ in the more severe case. Note there is still a significant amount
of uncertainty in the behaviour of IAs at higher redshifts, and these two
values of $\eta_{2}$ are both entirely consistent with existing data. In the
LSST setup, we run both $\Lambda$CDM and $w$CDM analyses. These are identical,
except for the addition of a dark energy equation of state parameter $w$,
varied with a flat prior over the range $w=[-2,-0.333]$. Our findings are
summarised in Figure 11.

Unsurprisingly, we see a significant gain in the constraining power of each of
the analyses compared with our DES Y3 like simulations (compare the contours
in Figures 3 and 11). In $\Lambda$CDM, like before, there is a significant
bias in the NLA cosmic shear results (purple, labelled “$1\times 2$pt NLA") in
the presence of IA modelling error. In the more extreme case (upper left
panel), there is a shift of several $\sigma$ relative to the input in the high
$S_{8}$ direction, very similar to the bias seen for our DES like setup in
Figure 3. Note that the overall reduction in the size of the contours means
there is less distortion due to prior edge effects – we are no longer hitting
the edge of the allowed parameter space in the $S_{8}-\Omega_{\mathrm{m}}$
plane. In the less extreme scenario (upper right), the bias is small, at the
level of $\sim 0.5\sigma$ in the high $S_{8}$ low $\Omega_{\mathrm{m}}$
direction. As before, we see that the galaxy clustering + galaxy-galaxy
lensing results exhibit less bias than cosmic shear. Even in the case with
larger unmodelled TATT parameters, the offset in the 2D
$S_{8}-\Omega_{\mathrm{m}}$ plane is below $1\sigma$ for $2\times 2$pt. In the
low TATT case, $S_{8}$ is almost unbiased, though there a small shift in
$\Omega_{\mathrm{m}}$. We note that for both IA scenarios shown here,
rerunning these chains with TATT results in unbiased constraints (with
projection effects at the level of $\sim 0.17\sigma$ in the 2D
$S_{8}-\Omega_{\mathrm{m}}$ plane in $\Lambda$CDM).

In the $w$CDM fits (the lower two panels of Figure 11) we see a similar
pattern. Interestingly, the trends here are even stronger than in the
$\Lambda$CDM. In the more extreme case (left), the $1\times 2$pt posterior is
shifted towards high $S_{8}$ and $w\neq-1$ by many $\sigma$. Here we _are_
beginning to hit the $w<-1/3$ prior bound (this is the reason for the
difference in the size of the purple contour in the lower left and lower right
panels). In contrast the $2\times 2$pt constraint recovers the input to
$<1\sigma$. Again in the weaker TATT scenario, there is a slight bias in the
high $S_{8}$ \- low $w$ direction, but it is sufficiently weak that the probes
appear consistent.

Unlike in our DES Y3 like runs, we note that there is a very notable
difference in the goodness-of-fit in these analyses. For the (we stress,
noiseless) analyses of the extreme TATT scenario, we find
$\chi^{2}_{\rm\Lambda CDM}=334.8$ and $\chi^{2}_{w\mathrm{CDM}}=312.2$ for
cosmic shear. For $2\times 2$pt, the equivalent best-fit values are much
better at 10.6 and 6.8. We compute the number of effective degrees of freedom
in the same way as in Section 4.1, and find 277.6 and 277.5 for our
$\Lambda$CDM and $w$CDM cosmic shear analyses respectively (286 data points).
For $2\times 2$pt, the equivalent values are 359.1 and 358.6 (378 data
points). This gives typical $p-$values of $p(>\chi^{2})\sim 10^{-27}$ for
$1\times 2$pt and $p(>\chi^{2})\sim 0.3$ for $2\times 2$pt. That is, given the
precision of the Stage IV data, it seems clear that the goodness-of-fit would
flag a problem in the $1\times 2$pt chains. As we discussed in Section 4.1,
this was not the case for the same IA scenario at DES Y3 precision. In the
less extreme case we find $\chi^{2}_{\rm\Lambda CDM}=3.8$ and
$\chi^{2}_{w\mathrm{CDM}}=3.8$ for cosmic shear and $\chi^{2}_{\rm\Lambda
CDM}=7.9$ and $\chi^{2}_{w\mathrm{CDM}}=6.9$ for $2\times 2$pt.

Interestingly, we note that our LSST $2\times 2$pt setup experiences a weaker
overall IA signal at the data vector level than the DES Y3 version of the same
IA scenario. This is reflected in the relatively good $\chi^{2}$ values listed
above (despite the improved precision of the data). We put this down to a
reduction in the fraction of the $\gamma_{t}$ data strongly affected by IAs,
due to the highly idealised analytic redshift distributions in Appendix C.
There are fewer bins with a strong IA signal compared with our real Y3
$n(z)$s, and so the fit (and the resulting $\chi^{2}$) is dominated by
lensing.

Our overall conclusion from this exercise is that our qualitative findings
from the previous sections hold, even with the enhanced statistical precision
allowed by Rubin. That is, for a cosmic shear analysis in the presence of an
unmodelled IA signal, model error tends to translate into potentially large
bias in cosmological parameters. For an $2\times 2$pt data vector affected by
the same IA signal, in contrast, the posteriors are relatively well centred
and the best fits are reasonably good.

###  4.6 Exploiting internal tension as a sign of IA model error

Figure 12: The relation between inter-probe “tension" and cosmological bias in
our 21 IA scenarios. Tension here is the distance between the peaks of the
marginalised posteriors from cosmic shear and $2\times 2$pt analyses of the
same IA scenario. Bias is the offset of each from the input values. The
diagonal line shows the 1-1 correspondence between the two. For illustrative
purposes we also show the $3\times 2$pt case for our fiducial IA scenario (the
black star), which falls roughly halfway between the shear and $2\times 2$pt
results. As in Figure 7, the fiducial IA scenario is indicated as an unfilled
star.

Finally, we will briefly consider how IA driven differences between cosmic
shear and galaxy-galaxy lensing might be used in practice. As we have seen in
Section 4.1, it is typically easier for model-data residuals from an
insufficient IA model to translate into cosmological bias in a shear only
analysis compared with $2\times 2$pt. In Figure 7 we see that IA scenarios
that cause of the order of several $\sigma$ bias in cosmic shear can leave
$2\times 2$pt virtually unbiased. We should note that in practice, since the
source sample used in shear and $\gamma_{t}$ measurements are generally the
same, discrepancies of this magnitude are unlikely due to random fluctuations.

One consequence of this observation is that cosmological bias from IA model
insufficiency can manifest itself as an apparent tension between probes. This
could potentially provide a way of flagging an effect which can otherwise be
difficult to detect using conventional goodness-of-fit and model comparison
statistics (at least without calibration; see Campos et al. 2023). To
illustrate this, Figure 12 shows the bias (i.e. the offset from the input
truth in the $S_{8}-\Omega_{\mathrm{m}}$ plane) plotted against the naive
tension between shear and $2\times 2$pt analyses (i.e. the distance between
the peaks of the two posteriors). We show all 21 IA scenarios considered in
Section 4.3. As we saw before, the $2\times 2$pt bias falls consistently below
the $1\times 2$pt equivalent. Particularly in the high bias regime
($>1\sigma$) the “tension" tracks the cosmic shear bias fairly well.

As referred to above, an offset in terms of number of $\sigma$ cannot be
easily interpreted in terms of a probability when the data are correlated.
There is typically some significant correlation between $\xi_{\pm}$ and
$\gamma_{t}$, as measured from the same survey. In order to make a rigorous
prediction one would need to use a metric designed to gauge these sorts of
internal tensions Doux, Baxter et al. (2021). That said, we might expect a
more rigorous treatment of tension to shift the x-axis in Figure 12 to smaller
values, but not necessarily to change the relation with bias. It is thus
useful to consider this as a demonstration of the concept.

##  5 Conclusions

In this paper we have presented a series of simulated cosmological analysis,
designed to shed light on the behaviour of joint weak lensing and clustering
analyses in the presence of intrinsic alignment modelling error. Such joint
analyses have become common in the field, and offer significant gains in terms
of constraining power and self-calibration of uncertainties. In addition to
those benefits, our results suggest that differences between probes in a shear
+ galaxy-galaxy lensing + galaxy clustering analysis may also provide a useful
diagnostic. That is, we find that galaxy-galaxy lensing and cosmic shear have
notably different sensitivities to IA modelling error. In the presence of a
significant and insufficiently modelled IA signal, this can lead to apparent
internal tension within the joint $3\times 2$pt data vector. In more detail,
our results are as follows:

  * •

At the precision of DES Y3, we find clustering + galaxy-galaxy lensing
($2\times 2$pt) analyses to be relatively unbiased in the presence of a range
of input IA scenarios ($<1\sigma$ in the $S_{8}-\Omega_{\mathrm{m}}$ plane in
almost all scenarios considered). This is true even when using a simple
1-parameter IA model. The same IA scenarios produce considerably larger biases
in the equivalent cosmic shear analyses. Both the more-or-less unbiased
$2\times 2$pt fit and the considerably more biased $1\times 2$pt one return
acceptable $\chi^{2}$ values ($p-$values > 0.05).

  * •

This robustness is not the result of insensitivity to IAs in galaxy-galaxy
lensing: we find a significant contribution at the level of simulated data
vectors. Rather, it points to a level of degeneracy breaking within the
$2\times 2$pt data vector, allowing even an overly-simple IA model to absorb
the error rather than cosmological parameters.

  * •

We find some dependence on the choice of lens sample. The differential
sensitivity is most striking with a redMaGiC like sample with well controlled
photometric redshifts. Our conclusions also hold with a MagLim like sample
(greater number density, but wider priors on redshift nuisance parameters). We
see slightly larger shifts in response to IA error, but still reliably under
$1\sigma$ and much smaller than the equivalent shifts in cosmic shear.
Similarly, including more small scale information in a $2\times 2$pt analysis
tends to increase the sensitivity to IA modelling error. Again, however, for
all plausible setups tested, the bias was $<1\sigma$.

  * •

Considering the range of IA scenarios tested in Campos et al. (2023), we find
$>90\%$ are biased by less than $1\sigma$ in a DES Y3 like $2\times 2$pt
analysis (see Figure 7). The corresponding biases in cosmic shear span a range
from $\ll 0.1\sigma$ to $>5\sigma$. This suggests that the differences are
general, and down to a generic difference in sensitivity, rather than anything
specific about our choice of fiducial input parameters.

  * •

Extending our analysis to a mock LSST Y1 like setup, we find the same IA model
error can, again, lead to differences of several $\sigma$ between cosmic shear
and galaxy-galaxy lensing + clustering analyses. These differences are more
pronounced in $w$CDM than $\Lambda$CDM. We see a similar pattern as before,
with much smaller biases in $2\times 2$pt, and significant internal tension
between $1\times 2$pt and $2\times 2$pt analyses based on the same survey.

Considering a range of simulated tests and other evidence, we conclude that
the most likely explanation for most of the differential sensitivity is
internal degeneracy breaking, or self-calibration. Unlike shear, $\gamma_{t}$
measurements cleanly separate into bin pairs that are either strongly affected
or more or less unaffected by IAs. This separation effectively breaks the
degeneracy between $S_{8}/\Omega_{\mathrm{m}}$ and IA parameters. Other
factors (e.g. signal cancellation and the greater control over the physical
scales used in $\gamma_{t}$) likely contribute but are not thought to be the
primary mechanism at work here.

In the wider context of the field, our results have a number of implications.
Tests for internal tension have become a standard part of the unblinding
process for many contemporary cosmology experiments (Doux, Baxter et al.,
2021). Although conventionally thought of as a test for calibration errors and
observational systematics, our results suggest they may carry information
about modelling error as well. That is, the absence of internal tensions may
be a sign that our choice of IA model is sufficient (at least at the level of
$<\sim 1\sigma$).

Additionally, our results may help simplify the practical work of implementing
data-driven model selection of the type proposed in Campos et al. (2023) (at
least for IAs). If in general, cosmic shear is significantly _more_ sensitive
to IA error than $2\times 2$pt, it may be sufficient to perform model
selection tests with the former. A calibrated goodness-of-fit that guarantees
that our $1\times 2$pt analysis is robust to modelling error at $<X\sigma$
(where we can choose our tolerance $X$), by extension will mean the same for
$2\times 2$pt.

Another implication is for the interpretation of IA constraints themselves.
While the high S/N of a $\gamma_{t}$ measurement provides a potentially
powerful constraint on IA parameters, that IA constraint is only as good as
the model behind it. As we have seen, IA parameters can quite easily absorb
modelling errors – which, while useful for recovering cosmological parameters
does tend to break the link with the physics of intrinsic alignments. Although
this is true to an extent for cosmic shear as well, the tendency for IA errors
to be absorbed by whichever IA parameters are available in $2\times 2$pt
analyses may make this more of an issue. As discussed in Appendix D, a similar
thing can happen when shear ratios are included in a cosmic shear analysis,
with model error being absorbed by the available parameters.

In terms of the lens sample selection itself, our results offer a new angle.
Although robustness to IA is unlikely to ever be the main driving factor, it
is perhaps one consideration when constructing a lens catalogue – that greater
redshift precision translates into a $2\times 2$pt analysis that is more
robust to uncertainties in the IA model. Further along the pipeline, our
results are also a motivation for including both very well-separated and
overlapping source-lens bin pairs in the eventual cosmology analysis. As we
have seen, the combination can provide enough information to disentangle IAs
from cosmology relatively effectively.

We should also state that we are not presenting these results as a solution to
the DES Y3 $X_{\rm lens}$ question (Pandey et al., 2022; DES Collaboration,
2022). Tests have suggested $X_{\rm lens}$ was likely the result of selection-
related systematics in the Y3 redMaGiC lens sample, which can be avoided by
changing the selection threshold. Although superficially similar to the sort
of model-driven differences described here, the tension there was between bias
values derived from $\gamma_{t}$ and $w(\theta)$. There are also features of
$X_{\rm lens}$ (e.g. redshift dependence, the greater impact on redMaGiC than
MagLim) that do not match well with the effects we describe in this paper.

Finally, we highlight the importance of IA measurements of all sorts. There
are several complementary ways to learn about IAs. Cosmological surveys
contain more information about IAs than might be assumed. Most obviously,
comparing the marginalised constraints from cosmological analyses provides a
direct way to study the behaviour of IAs in real lensing data. Although this
comparison can be done by reviewing the published literature, a deeper
understanding can be obtained from analysing different lensing data sets in a
homogenised model framework (Chang et al., 2019; DES & KiDS Collaboration,
2023). Further information can be gleaned about the sufficiency and relative
favourability of different models from goodness-of-fit statistics (Campos et
al. 2023; Secco, Samuroff et al. 2022). In addition to simultaneous
constraints, direct measurements (e.g. Singh et al. 2015; Johnston et al.
2019; Fortuna et al. 2021b; Samuroff et al. 2023) and tomographic IA-lensing
separation (e.g. Blazek et al. 2012) are also key to fully understanding IAs
in future lensing data sets. At present the conclusions we can draw from such
measurements are limited by the data available. Future experiments such as the
Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration 2016), the
Physics of the Accelerating Universe Survey (PAU; Serrano et al. 2023) and the
Wide-Area VISTA Extragalactic Survey (WAVES; Driver et al. 2019), as well as
subsets of data from LSST, Euclid and Roman, will offer us a way to test our
models and begin to derive informative priors for IAs in lensing surveys.

This paper provides one additional piece in the puzzle. We have shown that,
since constraints from different probes respond differently to modelling
error, a comparison of individual probe constraints can provide information
that would not otherwise be obvious from the joint analysis of those same
probes. In other words, there is value in analysing and presenting different
probes separately. It is finally worth noting that, although we have attempted
to test the robustness of our findings as much as possible, there are many
factors at work. Dramatic changes in the source/lens samples or redshift
binning, say, could affect our results. Likewise, an analysis with very
different (or even no) scale cuts in either cosmic shear or galaxy-galaxy
lensing (e.g. Aricò et al. 2023) may respond differently. Taken at face value,
the addition of very small scales in $\xi_{\pm}$ is likely to make one more
sensitive to unmodelled higher order IA terms (and so increase the differences
between shear and $2\times 2$pt). Such analyses, however, tend to also include
extra parameters to account for baryonic feedback, and it is difficult to
predict how these will interact with higher order IA terms.

As data sets get deeper, we are also entering a regime where IAs are not well
understood. As we have discussed in this paper, current data sets are
primarily sensitive to IAs at low redshift. Upcoming surveys will reach much
higher redshifts than any of the present generation, potentially significantly
changing the sensitivity of the data to IA. Understanding the dynamics of IA
constraints (i.e. the degree of potential self-calibration from different
measurements, but also the possibility for greater sensitivity to error and
for the failure of common parameterisations) is an important topic, and will
be the focus of future work.

## Acknowledgements

Thanks to Noah Weaverdyck and Johannes Lange for their help setting up the
Nautilus sampler. SS and JB are partially supported by NSF grant AST-2206563.
AC acknowledges support from Department of Energy grant DE-SC0010118 and the
NSF AI Institute: Physics of the Future, NSF PHY-2020295. AP acknowledges
support from the UK Science and Technology Facilities Council (STFC) under
grant numbers ST/V000594/1 and from the European Union’s Horizon Europe
program under the Marie Skłodowska-Curie grant agreement 101068581. The
contour plots shown in this paper were made using GetDist (Lewis, 2019). This
work was completed in part using the Discovery cluster, which is supported by
Northeastern University’s Research Computing team.

## Data Availability

All simulated data vectors and chains used in this work are available on
request.

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