# Non-local contribution from small scales in galaxy-galaxy lensing:
Comparison of mitigation schemes

J. Prat1,2, G. Zacharegkas2, Y. Park,3 N. MacCrann,4 E. R. Switzer,5 S.
Pandey,6 C. Chang,1,2 J. Blazek,7 R. Miquel,8,9 A. Alarcon,10 O. Alves,11 A.
Amon,12,13 F. Andrade-Oliveira,11 K. Bechtol,14 M. R. Becker,10 G. M.
Bernstein,6 R. Chen,15 A. Choi,16 H. Camacho,17,18 A. Campos,19 A. Carnero
Rosell,20,18,21 M. Carrasco Kind,22,23 R. Cawthon,24 J. Cordero,25 M.
Crocce,26,27 C. Davis,28 J. DeRose,29 H. T. Diehl,30 S. Dodelson,19,31 C.
Doux,6,32 A. Drlica-Wagner,1,30,2 K. Eckert,6 T. F. Eifler,33,34 J. Elvin-
Poole,35,36 S. Everett,34 X. Fang,37,33 A. Ferté,34 P. Fosalba,26,27 O.
Friedrich,13 M. Gatti,6 G. Giannini,9 D. Gruen,38 R. A. Gruendl,22,23 I.
Harrison,39 W. G. Hartley,40 K. Herner,30 H. Huang,33,41 E. M. Huff,34 M.
Jarvis,6 E. Krause,33 N. Kuropatkin,30 P.-F. Leget,28 J. McCullough,28 J.
Myles,42,28,43 A. Navarro-Alsina,44 A. Porredon,35,36,45 M. Raveri,6 R. P.
Rollins,25 A. Roodman,28,43 R. Rosenfeld,46,18 A. J. Ross,35 E. S.
Rykoff,28,43 C. Sánchez,6 J. Sanchez,30 L. F. Secco,2 I. Sevilla-Noarbe,47 E.
Sheldon,48 T. Shin,49 M. A. Troxel,15 I. Tutusaus,50,26,27 T. N.
Varga,51,52,53 B. Yanny,30 B. Yin,19 Y. Zhang,54 J. Zuntz,45 M. Aguena,18 S.
Allam,30 J. Annis,30 D. Bacon,55 E. Bertin,56,57 S. Bocquet,38 D. Brooks,58 D.
L. Burke,28,43 J. Carretero,9 M. Costanzi,59,60,61 M. E. S. Pereira,62 J. De
Vicente,47 S. Desai,63 I. Ferrero,64 B. Flaugher,30 D. W. Gerdes,65,11 G.
Gutierrez,30 S. R. Hinton,66 D. L. Hollowood,67 K. Honscheid,35,36 D. J.
James,68 M. Lima,69,18 F. Menanteau,22,23 J. Mena-Fernández,47 A. Palmese,37
M. Paterno,30 F. Paz-Chinchón,22,12 A. Pieres,70 A. A. Plazas Malagón,71 M.
Rodriguez-Monroy,72,47 E. Sanchez,47 M. Schubnell,11 M. Smith,73 M. Soares-
Santos,11 E. Suchyta,74 M. E. C. Swanson,72 G. Tarle,11 C. To,35 N.
Weaverdyck,11,29 and J. Weller52,53 (The DES Collaboration)
Author affiliations are listed at the end of the paper E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent cosmological analyses with large-scale structure and weak lensing
measurements, usually referred to as 3$\times$2pt, had to discard a lot of
signal-to-noise from small scales due to our inability to accurately model
non-linearities and baryonic effects. Galaxy-galaxy lensing, or the position-
shear correlation between lens and source galaxies, is one of the three two-
point correlation functions that are included in such analyses, usually
estimated with the mean tangential shear. However, tangential shear
measurements at a given angular scale $\theta$ or physical scale $R$ carry
information from all scales below that, forcing the scale cuts applied in real
data to be significantly larger than the scale at which theoretical
uncertainties become problematic. Recently there have been a few independent
efforts that aim to mitigate the non-locality of the galaxy-galaxy lensing
signal. Here we perform a comparison of the different methods, including the
Y-transformation, the Point-Mass marginalization methodology and the Annular
Differential Surface Density statistic. We do the comparison at the
cosmological constraints level in a combined galaxy clustering and galaxy-
galaxy lensing analysis. We find that all the estimators yield equivalent
cosmological results assuming a simulated Rubin Observatory Legacy Survey of
Space and Time (LSST) Year 1 like setup and also when applied to DES Y3 data.
With the LSST Y1 setup, we find that the mitigation schemes yield $\sim$1.3
times more constraining $S_{8}$ results than applying larger scale cuts
without using any mitigation scheme.

###### keywords:

cosmology: large-scale structure of Universe – gravitational lensing: weak –
cosmology: theory – cosmology: cosmological parameters

##  1 Introduction

When the light of background (source) galaxies passes close to foreground
(lens or tracer) galaxies it gets perturbed, distorting the image of the
source galaxies we observe. Galaxy-galaxy lensing refers to the cross-
correlation between source galaxy shapes and lens galaxy positions. The amount
of distortion is correlated with the properties of the lens sample and the
underlying large scale structure it traces. In the case of a spherical
distribution of matter, the shear at any point will be oriented tangentially
to the direction toward the center of symmetry. Then, the tangential component
of the shear will capture all the gravitational lensing signal produced by a
spherically symmetric distribution of mass. Because of this, the tangential
shear averaged over many source-lens galaxy pairs is one of the typical
measurements that is done to detect this correlation.

Galaxy-galaxy lensing has had a wide range of applications since it was first
detected in Brainerd et al. (1996). It has been extensively used to probe the
galaxy-matter connection at small scales, e.g. Choi et al. (2012), Yoo &
Seljak (2012), Kuijken et al. (2015), Clampitt et al. (2017), Park & Krause et
al., (2016), Zacharegkas et al. (2022) or Luo et al. (2022); to extract
cosmological information using the well-understood large scales in combination
with other probes such as galaxy clustering and/or CMB lensing as in
Mandelbaum et al. (2013), Baxter et al. (2016), Joudaki et al. (2018), van
Uitert et al. (2018), Prat & Sánchez et al., (2018), Singh et al. (2020), Lee
et al. (2022), Prat et al. (2022); to obtain lensing shear geometric
constraints e.g. Jain & Taylor (2003), Prat & Baxter et al., (2019),
Hildebrandt et al. (2020), Giblin et al. (2021), Sánchez & Prat et al.,
(2022), and also recently in Leauthaud et al. (2022) to assess the consistency
of lensing across different data-sets and to carry out end-to-end tests of
systematic errors.

Moreover, galaxy-galaxy lensing is a standard part of the so-called
3$\times$2pt analyses that combine large-scale structure and weak lensing
measurements to extract cosmological information. The 3$\times$2pt stands for
the combination of three two-point correlation functions: the autocorrelation
of the positions of galaxies (galaxy clustering), the cross-correlation of
galaxy shapes and galaxy positions (galaxy-galaxy lensing) and the
autocorrelation of galaxy shapes (cosmic shear). This combination was
originally proposed in Hu & Jain (2004), followed by Bernstein (2009) and
Joachimi & Bridle (2010) and has since then been applied to different galaxy
survey data sets, such as to KiDS data, as in Heymans et al. (2021), and to
DES data, e.g. in Abbott et al. (2022).

These kind of analyses commonly use the well-understood large scales, placing
stringent scale cuts to remove the parts of the data vector that currently add
too much uncertainty in the model due to non-linearities in the matter power
spectrum, galaxy bias and baryonic effects, amongst others (e.g. Krause et al.
2021). However, the galaxy-galaxy lensing signal is non-local in nature, that
is, the predicted signal at a given separation between a source and a lens
galaxy (at the redshift of the lens galaxy) depends on the modeling of all
scales within that separation, including the non-linear small scales. This can
be appreciated expressing the tangential shear of a single lens-source galaxy
pair separated by an angular distance $\theta$ as a function of the excess
surface mass density $\Delta\Sigma$:

$\gamma_{t}\,(\theta,z_{l},z_{s})=\frac{\Delta\Sigma\,(\theta)}{\Sigma_{\mathrm{crit}}(z_{l},z_{s})},$
(1)

where $\Delta\Sigma$ can be expanded as the difference between the mean
surface mass density below a certain angular scale ($<\theta$) and the surface
mass density at this given scale $\theta$:

$\Delta\Sigma\,(\theta)=\overline{\Sigma}\,(<\theta)-\Sigma\,(\theta),$ (2)

where the non-locality of the tangential shear quantity becomes apparent,
since the tangential shear defined at some $\theta$ value will always carry
information of all the scales below this value. $\Sigma_{\mathrm{crit}}$ is
just a geometrical factor that depends on the angular diameter distances to
the lens galaxy $D_{l}$, the one between the lens and the source $D_{ls}$ and
the one to the source galaxy $D_{s}$, and is defined as:

$\Sigma_{\mathrm{crit}}^{-1}(z_{\rm l},z_{\rm s})=\frac{4\pi
G}{c^{2}}\frac{D_{ls}\,D_{l}}{D_{s}}\qquad\mathrm{if}z_{s}>z_{l},$ (3)

and zero otherwise. In the equation above $G$ is the gravitational constant
and $c$ is the speed of light.

Therefore, our inability to model the small scales accurately enough given the
measurement uncertainties impacts the lensing signal at all scales. Expanding
on this, in order to predict the lensing signal, an accurate prediction for
the galaxy-matter correlation function $\xi_{\rm{gm}}(r)$ is required for some
range of physical (3D) scales $r$, see e.g. MacCrann et al. (2020). At large
scales, we expect linear theory to hold and thus we can relate the galaxy-
matter correlation function with the matter correlation through a linear
galaxy bias factor: $\xi_{\rm{gm}}(r)=b\xi_{\rm{mm}}$, see Desjacques et al.
(2018) for a galaxy bias review. At smaller (nonlinear) scales we do not
currently have a precise theory to model $\xi_{\rm{gm}}(r)$. Therefore, it is
crucial to ensure that the tangential shear measurement is only sensitive to
scales in $\xi_{\rm{gm}}(r)$ where we know that the modelling is sufficiently
accurate. Since the galaxy-galaxy signal receives a non-local contribution
that depends on scales in $\xi_{\rm{gm}}(r)$ that are much smaller than the
separation at which the measurement is made (i.e. the impact parameter in the
lens redshift), this non-locality can then force the scale cuts applied in
real data to be significantly larger than the scale at which theoretical
uncertainties become problematic. For example, due to this reason, the scale
cuts in the DES Y1 3$\times$2pt cosmological analysis (Abbott et al., 2018)
were higher for the galaxy-galaxy lensing part (12 Mpc/$h$) than for the
galaxy clustering part (8 Mpc/$h$).

To help with this issue, there have been a few independent efforts to mitigate
the non-locality of the galaxy-galaxy lensing signal. The first that was
proposed was the annular differential surface density estimator by Baldauf et
al. (2010), which consists of modifying the data vector on all scales in a way
that removes information about the lens mass distribution below a chosen scale
$R_{0}$, based on the measured data vector around $R_{0}$. Later on MacCrann
et al. (2020) proposed a methodology to take into account the non-locality by
analytically marginalizing over an enclosed point-mass directly when
performing the cosmological parameters inference. Finally another estimator-
based methodology was proposed by Park et al. (2021), which achieves the
localization of the galaxy-galaxy lensing signal by performing a linear
transformation of the tangential shear quantity. In the recent DES Y3
3$\times$2pt work (Abbott et al., 2022), the point-mass marginalization
methodology was applied to remove the information from small scales above a
certain scale, which resulted in being able to model the galaxy-galaxy lensing
until 6 Mpc/$h$, a much smaller scale cut than the one used in DES Y1 of 12
Mpc/$h$, even with smaller measurement uncertainties, while the galaxy
clustering scale remained the same as in the Y1 analysis (8 Mpc/$h$).

In this paper we perform a thorough comparison of these different proposals
with the main goal of understanding which of them is advantageous to use in
future cosmological analyses. First, we use simulated data vectors with
uncertainties mimicking the LSST Y1 settings to perform such a comparison and
then apply it to DES Y3 data.

##  2 Theory

Eq. (2) can be rewritten as a function of physical scale $R=\theta D_{l}$ in
the small angle approximation as:

$\Delta\Sigma(R)=\overline{\Sigma}\,(0,R)-\Sigma\,(R).$ (4)

Then, expanding each of the terms, the surface density at a given transverse
$R$ scale between the lens galaxy and the source light can be expressed as the
integral of the three-dimensional galaxy-matter correlation function
$\xi_{gm}(r)$ over the line-of-sight distance $\Pi$, with
$r=\sqrt{\Pi^{2}+R^{2}}$:

$\Sigma(R)=\overline{\rho}_{m}\int_{-\infty}^{\infty}\mathrm{d}\Pi\left[1+\xi_{gm}\left(\sqrt{\Pi^{2}+R^{2}}\right)\right]\,,$
(5)

where $\overline{\rho}_{m}$ is the mean matter density. The other term in Eq.
(4) is the mean surface density between two transverse positions, which can be
generally expressed as:

$\overline{\Sigma}(R_{1},R_{2})=\frac{2}{R^{2}_{2}-R^{2}_{1}}\int_{R_{1}}^{R_{2}}\Sigma(R^{\prime})R^{\prime}\mathrm{d}R^{\prime}.$
(6)

For $R_{1}=0$ and $R_{2}=R$ it simplifies to:

$\overline{\Sigma}(0,R)=\frac{2}{R^{2}}\int_{0}^{R}\Sigma(R^{\prime})R^{\prime}\mathrm{d}R^{\prime}.$
(7)

This term is the one containing the information from all scales down to $R=0$,
including the one halo regime for which we do not have an accurate model for
$\xi_{gm}$. Assuming we can model $\xi_{gm}$ accurately only down to some
minimum scale $r_{\mathrm{min}}$, we do not want $\Delta{\Sigma}(R)$ to depend
on $\xi_{gm}$ below $r_{\mathrm{min}}$. From the equation above it becomes
clear that $\overline{\Sigma}(0,R)$, and thus $\Delta{\Sigma}(R)$, are only
sensitive to the total mass enclosed inside $R$ (i.e. to some integral of
$\xi_{gm}$), but not to how the mass is distributed (i.e. to the shape of
$\xi_{gm}$). For instance, $\Delta{\Sigma}$ at scales larger than $R$ will be
the same for a point-mass distribution as for an NFW profile. Also, as shown
by Eq. (7), the contribution from the total enclosed mass will scale as
$1/R^{2}$. This is the key fact that all the estimators described below rely
on to remove the dependency of the enclosed mass (or non-locality) of the
galaxy-galaxy lensing measurements. Below we summarize each of these currently
existing methodologies. We also visualize the modified data vectors for each
of the methodologies in Fig. 1.

###  2.1 Annular Differential Surface Density $\Upsilon$

The Annular Differential Surface Density statistic, $\Upsilon(R)$, is defined
in the following way (Baldauf et al., 2010):

$\displaystyle\Upsilon(R;R_{0})$
$\displaystyle\equiv\Delta\Sigma(R)-\frac{R_{0}^{2}}{R^{2}}\Delta\Sigma(R_{0})=$
(8)
$\displaystyle=\frac{2}{R^{2}}\int_{R_{0}}^{R}\mathrm{d}R^{\prime}R^{\prime}\Sigma(R^{\prime})-\frac{1}{R^{2}}\left[R^{2}\Sigma(R)-R_{0}^{2}\Sigma(R_{0})\right]\,.$
(9)

From the integration limits it is clear this estimator does not include
information from scales below $R_{0}$, because $\Delta\Sigma(R_{0})$ contains
the same small-scale contribution as $\Delta\Sigma(R)$, just rescaled by
$R^{2}/R_{0}^{2}$. The second line follows from the first one by substituting
$\Delta\Sigma(R)=\overline{\Sigma}\,(0,R)-\Sigma\,(R)$ and using the
definition

$\overline{\Sigma}(R_{1},R_{2})=\frac{2}{R_{2}^{2}-R_{1}^{2}}\int_{R_{1}}^{R_{2}}\mathrm{d}R^{\prime}R^{\prime}\Sigma(R^{\prime}).$
(10)

In this estimator and in the ones below that also involve a transformation of
the data vector, the model is transformed in the same way as the measurements.
Moreover, the Annular Differential Surface Density statistic can be
equivalently defined for the tangential shear quantity using angular scales:

$\Upsilon_{\gamma_{t}}(\theta;\theta_{0})\equiv\gamma_{t}(\theta)-\frac{\theta_{0}^{2}}{\theta^{2}}\gamma_{t}(\theta_{0})\,,\\\
$ (11)

using the small-angle approximation to go from $R$ to $\theta$. In realistic
scenarios, where each lens tomographic bin has a non-negligible width, a given
value $\theta_{0}$ will mix a range of physical scales $R_{0}$. In this work
we use the Annular Differential Surface Density statistic based on the
tangential shear quantity throughout the paper (instead of the $\Delta\Sigma$
one), and use $\theta_{0}$ cuts computed with the mean $z_{l}$ for each
redshift bin, given a specified $R_{0}$ cut. In this paper we use values for
$R_{0}$ of 6 Mpc$/h$ and of 8 Mpc$/h$, depending on the data set and its
constraining power.

The covariance of $\Upsilon$ will also generally need to be modified. Given
that we can write the transformation as $\Upsilon=\gamma_{t}-X$, with $X$
being a constant, then
$\mathrm{Var}[\Upsilon]=\mathrm{Var}[\gamma_{t}]+\mathrm{Var}[X]-\mathrm{cov}[\gamma_{t},X]$.
In the case that $X$ is noiseless, the covariance of $\Upsilon$ will be
identical to the $\gamma_{t}$ one. In our implementation we always modify the
covariance of the $\Upsilon$ statistic to include the noise of
$\gamma_{t}(\theta_{0})$.

###  2.2 Y-transform

The $Y$-transformation derived in Park et al. (2021) is a localizing linear
transformation that utilizes the local quantity $\Sigma(R)$ underlying the
galaxy-galaxy lensing observable $\Delta\Sigma(R)$. By inverting the
$\Delta\Sigma(R)$–$\Sigma(R)$ relationship, the $Y$ quantity defined as

$\displaystyle Y(R)$ $\displaystyle\equiv\Sigma(R)-\Sigma(R_{\text{max}})$
$\displaystyle=\int^{R_{\text{max}}}_{R}\mathrm{d}\ln
R^{\prime}\left[2\Delta\Sigma(R^{\prime})+\frac{d\Delta\Sigma(R^{\prime})}{d\ln
R^{\prime}}\right]$ (12)

recovers the local $\Sigma(R)$ up to an overall additive constant
$\Sigma(R_{\text{max}})$. To treat the discretized observables most frequently
used in real analyses, this relation is also discretized to a linear transform
given by

$\mathbf{Y}=\left(2\mathbf{S}+\mathbf{S}\mathbf{D}\right)\mathbf{\Delta\Sigma}=\mathbf{T}\mathbf{\Delta\Sigma},$
(13)

where the trapezoidal summation matrix $\mathbf{S}$ representing the log
integral and the finite differences matrix $\mathbf{D}$ representing the log
differentiation are used to define the final transformation matrix
$\mathbf{T}$. The linear format of this transformation allows a further direct
application to a $\gamma_{t}$ vector, as $\gamma_{t}$ is proportional to
$\Delta\Sigma$. Thus, by transforming an observed galaxy-galaxy lensing vector
$\bm{\gamma}_{t}$ and its covariance $\mathbf{C}_{\gamma}$ as

$\displaystyle\mathbf{Y}_{\gamma}$ $\displaystyle=$
$\displaystyle\mathbf{T}\bm{\gamma}_{t},$ (14)
$\displaystyle\mathbf{C}_{Y_{\gamma}}$ $\displaystyle=$
$\displaystyle\mathbf{T}\mathbf{C}_{\gamma}\mathbf{T}^{\mathbf{T}},$ (15)

we achieve a likelihood analysis with a localized galaxy-galaxy lensing
observable. It is notable that the $\mathbf{T}$ matrix nulls out components in
$\mathbf{\Delta\Sigma}$ ($\bm{\gamma}_{t}$) proportional to $1/R^{2}$
($1/\theta^{2}$), which can also be seen from Eq. (2.2) when a term
proportional to $1/R^{2}$ is added to $\Delta\Sigma(R)$. Another way to see
this is that if $\Delta\Sigma(R)$ has a $1/R^{2}$ shape, the integral from Eq.
(2.2) vanishes. Note that this is also true for the $\Upsilon$ statistic of
section 2.1, i.e. adding a term proportional to $1/\theta^{2}$ makes no
difference to the estimator (see Eq. (11)).

###  2.3 Point-mass marginalization

MacCrann et al. (2020) proposed to analytically marginalize over the
contribution from within the small-scale cut by treating it as a point-mass
(PM) contribution scaling as $1/R^{2}$. This point-mass term can be expressed
as an addition to the tangential shear model for a given lens redshift bin $i$
and source redshift bin $j$:

$\gamma_{t}^{ij}\,(\theta)=\gamma_{t,\rm
model}^{ij}\,(\theta)+\frac{A^{ij}}{\theta^{2}}\,,$ (16)

where we use the small-angle approximation to go from $R$ to $\theta$. Here
the $\gamma_{t,\rm model}^{ij}$ is a prediction based on a model for the 3D
galaxy-matter correlation function $\xi_{\text{gm}}(r)$ that is correct for
scales $r>r_{\text{min}}$, but can be arbitrarily wrong for
$r<r_{\text{min}}$, and $A^{ij}$ is some unknown constant that we can
marginalize over. The simple form of this contamination model makes it
suitable for an analytic marginalization approach given that the scale
dependence is not dependent on cosmology or the lens galaxy properties. We
want to marginalize
$P(\gamma_{t,\text{obs}}\,(\theta)|\gamma_{t,\text{model}}\,(\theta),A)$ over
the unknown constant $A$ in order to obtain the likelihood we are interested
in, namely:

$P(\gamma_{t,\text{obs}}\,(\theta)|\gamma_{t,\text{model}}\,(\theta))=\int\mathrm{d}A\,P(A)P(\gamma_{t,\text{obs}}\,(\theta)|\gamma_{t,\text{model}}\,(\theta),A).$
(17)

In the case that the $\gamma_{t,\text{model}}$ is Gaussian distributed with
covariance matrix $\mathbf{C}_{\gamma}$, and we have a Gaussian prior on $A$
with mean zero and width $\sigma_{A}$, one can show that (Bridle et al., 2002)
$P(\gamma_{t,\text{obs}}\,(\theta)|\gamma_{t,\text{model}}\,(\theta))$ is also
Gaussian distributed with a covariance matrix

$\mathbf{N}=\mathbf{C}_{\gamma}+\sigma^{2}_{A}\vec{x}\vec{x}^{\text{T}},$ (18)

where $\vec{x}$ has elements $x_{n}=(\theta_{\text{min}}/\theta_{n})^{2}$, and
$\theta_{\text{min}}$ represents the scale cut. This means that in order to
marginalize over the free parameter $A$, we only need to add this term to the
original covariance rather than explicitly sampling over possible values of
$A$ in e.g. an MCMC chain. In this work we use an infinite prior for
$\sigma_{A}$. In this case, the extra term is added to the inverse covariance
directly, following the procedure described in MacCrann et al. (2020) and in
Prat et al. (2022).

####  2.3.1 Point-mass marginalization using geometric information

The amplitudes $A^{ij}$ can be written as

$\displaystyle
A^{ij}=\int\mathrm{d}z_{l}\int\mathrm{d}z_{s}n^{i}_{l}(z)n^{j}_{s}(z)B^{i}(z_{l})\Sigma^{-1}_{\rm
crit}(z_{l},z_{s})D^{-2}(z_{l})$

where $B^{i}$ represents the total mass enclosed within $\theta_{\text{min}}$
for the $i$-th lens bin, $n^{i}_{l}(z)$ is the redshift distribution of each
lens bin, $n^{j}_{s}(z)$ for each source bin, and $D(z_{l})$ is the angular
diameter distance to the lens redshift $z_{l}$. If we assume that $B^{i}$
evolves slowly across the width of the lens bins we can drop the $z_{l}$
dependence and define the parameters $\beta_{ij}$ in the following way:

$A^{ij}\approx
B^{i}\int\mathrm{d}z_{l}\int\mathrm{d}z_{s}n^{i}_{l}(z)n^{j}_{s}(z)\Sigma^{-1}_{\rm
crit}(z_{l},z_{s})D^{-2}(z_{l})\equiv B^{i}\beta_{ij}.$

The parameters $\beta_{ij}$ are purely geometrical (sometimes also called
shear-ratio information), and thus can be exactly computed analytically given
the input redshift distributions111Note that since the second term of the RHS
of Eq. (16) only accounts for the unmodeled enclosed mass caused by the
mismodeling of the halo-matter correlation function, other effects such as IA
or magnification would be a part of the first term of the RHS of Eq. (16), and
thus do not need to be included in the computation of the $\beta_{ij}$
factors.. Then, the predicted $\beta_{ij}$ factors can be used to reduce
freedom in the model by fixing the relative scales between the source bins
sharing the same lens bin and only marginalizing over $B^{i}$ instead of over
a free-form $A^{ij}$.

We label this variant of the model as "Point-mass (free per $z^{i}_{l}$)",
since in this case there is only one free parameter per lens bin. When this
approximation is not used (i.e. only using Eq. 16) we label the model as
"Point-mass (free per $z^{i}_{l}-z_{s}^{j}$)". In the DES Y3 analysis, where
the point-marginalization was used, the variant of the model using geometrical
information was employed. In this paper, we also explore the differences,
advantages and caveats of these two variants of the point-mass marginalization
method.

Figure 1: Visualization of the data vectors for each of the galaxy-galaxy
lensing (GGL) estimators that we compare in this paper to localize the
original tangential shear measurements ($\gamma_{t}$), corresponding to the
second lens redshift bin and the highest source bin. We do not add the point-
mass marginalization case since it does not involve a modification of the data
vector, only of the (inverse) covariance. The triangle point for the proj-out
case represents a negative point.

###  2.4 Mode projection: “Project-out” estimator

To further illustrate the similarities and/or differences among the above
estimators, we also construct a new estimator that we refer to as the
“project-out” estimator. This estimator is designed such that it follows the
philosophy of the MacCrann et al. (2020) approach, namely by focusing on the
point-mass $1/R^{2}$ mode within the observed galaxy-galaxy lensing data
vector, while it follows the implementation used in Park et al. (2021), namely
by finding a suitable linear transformation and using it to transform the data
vectors and covariances. In Appendix A we show that the "Proj-out" method is
actually equivalent to the point-mass marginalization method when the prior of
the point-mass is infinitely wide. Also, this basic equivalence between
marginalizing over a parameter and projecting out a given mode had already
been pointed out in Appendix A from Seljak (1998) for a general scenario.

The core idea behind this estimator, thus, is to identify the projection of
the observed galaxy-galaxy lensing data vector onto the $1/R^{2}$ mode, and
then to remove it from the original vector. The projection operator
$\mathbf{P}$ is given by (Aitken, 1936; Tegmark et al., 1998):

$\mathbf{P}=\mathbf{A}\left(\mathbf{A}^{\mathbf{T}}\mathbf{C}_{\gamma}^{-1}\mathbf{A}\right)^{-1}\mathbf{A}^{\mathbf{T}}\mathbf{C}_{\gamma}^{-1}.$
(19)

where $\mathbf{A}$ has columns spanning the subspace onto which we wish to
project. In our case, $\mathbf{A}$ has only one column given by
$\\{1/R_{i}^{2}\\}$, or in practice $\\{1/\theta_{i}^{2}\\}$, with
$\theta_{i}$ being the representative angular separation of the $i$-th bin in
our data vector. Note that instead of the “vanilla” projection operator
$\mathbf{A}(\mathbf{A}^{\mathbf{T}}\mathbf{A})^{-1}\mathbf{A}^{\mathbf{T}}$ we
use the generalized least squares definition of the projection operator to
properly account for the covariances in the data vector.

The “cleaned” data vector is then defined as

$\displaystyle\gamma_{t,\text{proj-out}}$ $\displaystyle=$
$\displaystyle\gamma_{t}-\mathbf{P}\gamma_{t}$ (20) $\displaystyle=$
$\displaystyle\left[\mathbf{I}-\mathbf{A}\left(\mathbf{A}^{\mathbf{T}}\mathbf{C}_{\gamma}^{-1}\mathbf{A}\right)^{-1}\mathbf{A}^{\mathbf{T}}\mathbf{C}_{\gamma}^{-1}\right]\gamma_{t}$
$\displaystyle\equiv$ $\displaystyle\mathbf{M}\gamma_{t},$

and its covariance is given by

$\mathbf{C}_{\text{proj-
out}}=\mathbf{M}\mathbf{C}_{\gamma}\mathbf{M}^{\mathbf{T}}$ (21)

where $\mathbf{C}_{\gamma}$ is the original covariance matrix. The inversion
of $\mathbf{C}_{\text{proj-out}}$ becomes problematic, as $\mathbf{M}$ is not
a full-rank matrix. We thus follow Tegmark (1997) to define a pseudoinverse of
the transformed covariance matrix as

$\tilde{\mathbf{C}}_{\text{proj-
out}}^{-1}=\mathbf{M}\left(\mathbf{C}_{\text{proj-
out}}+\eta\mathbf{A}\mathbf{A}^{\mathbf{T}}\right)^{-1}\mathbf{M}^{\mathbf{T}},$
(22)

where $\eta$ is a constant whose value does not affect the performance of the
pseudoinverse222We have used $\eta=10^{-3}$ for the runs with DES Y3 data. We
have checked that using a different value, e.g. $\eta=10^{3}$, does not change
the results.. This can be understood intuitively as adding back in the lost
mode to $\mathbf{C}_{\text{proj-out}}$, inverting, and removing the added mode
at the end. With $\gamma_{t,\text{proj-out}}$ and
$\tilde{\mathbf{C}}_{\text{proj-out}}$ in hand, a full likelihood analysis can
be defined using this new estimator.

##  3 Methodology

The question we are aiming to answer is whether the different methodologies to
localize the galaxy-galaxy lensing measurements are consistent under the
precision of current and future surveys. In order to do so, we perform a
combined 2$\times$2pt analysis using each of the methods to localize the
tangential shear measurements and compare their performances at the
cosmological posterior level. We choose to do the comparison in a 2$\times$2pt
analysis instead of a whole 3$\times$2pt analysis to maximize the impact that
the galaxy-galaxy lensing part of the data vector has on the cosmological
parameter posteriors, thus maximizing potential differences between the
localizing estimators. We perform the comparison using two different setups:
1) First we assume the characteristics of a future survey to test the
differences under the smallest covariance. In particular, we choose the
specifications of a LSST Y1-like survey, since that will become relevant in
the near-future, and it will already be significantly more constraining than
current generation surveys. 2) Secondly we apply the comparison of the methods
to DES Y3 data, a noisy realistic scenario.

For the LSST Y1 simulated case, we generate the input data vectors using a
contaminated model that includes baryonic effects and non-linear galaxy bias
contributions that mostly affect small scales (see Sec. 3.2), and analyze it
using the simpler fiducial model that does not take into account these
contributions with a linear galaxy bias model and a dark matter only power
spectrum (see Sec. 3.1). We find the appropriate set of scale cuts that allow
us to recover unbiased cosmology in each case following the prescription
described in Sec 3.3.

###  3.1 Fiducial model

Here we summarize the baseline or “fiducial” theory that we will use to model
the observed tangential shear and galaxy clustering quantities. This is the
same one used in the DES Y3 3$\times$2pt analysis. In particular, for the LSST
Y1 simulated analysis described below we assume the model presented in more
detail in Krause et al. (2021) and for the DES Y3 data analysis the one
defined in Abbott et al. (2022)333The two models are in essence identical, but
differ in some of the input parameters such as the lens magnification
parameters and the redshift distributions (which were measured later in the
data analysis), and in the priors of the nuisance parameters..

The tangential shear $\gamma_{t}$ and angular clustering $w(\theta)$
observables can be expressed as various real space projections of angular
power spectra. In particular we model $\gamma_{t}$ as the following curved sky
projection of the galaxy-matter angular power spectra and of other terms that
encapsulate observational effects such as intrinsic alignments, lens
magnification and their cross-talk, which add up to the total observed
$C^{ij}_{\delta_{\mathrm{obs}}\,E}$:

$\gamma_{t}^{ij}(\theta)=\sum_{\ell}\frac{2\ell+1}{4\pi\ell(\ell+1)}P^{2}_{\ell}(\cos\theta)C^{ij}_{\delta_{\mathrm{obs}}\,E},$
(23)

and we model $w(\theta)$ as:

$w^{ij}(\theta)=\sum_{\ell}\frac{2\ell+1}{4\pi}P_{\ell}(\cos\theta)C^{ij}_{\delta_{\mathrm{obs}}\,\delta_{\mathrm{obs}}}.$
(24)

We refer the reader to Krause et al. (2021) for a detailed definition of
$C^{ij}_{\delta_{\mathrm{obs}}\,E}$ and of
$C^{ij}_{\delta_{\mathrm{obs}}\,\delta_{\mathrm{obs}}}$. $P_{\ell}$ and
$P^{2}_{\ell}$ are the Legendre polynomials and the associated Legendre
polynomials respectively. For the matter power spectrum we use the dark matter
only Halofit prescription from Takahashi et al. (2012) and assume a linear
galaxy bias to relate the galaxy and matter density fluctuations.

We obtain the LSST Y1 covariances with the publicly available CosmoCov code
(Krause & Eifler, 2017; Fang et al., 2020), using the number densities and
noise specified in Tab. 1. We include Gaussian and non-Gaussian terms computed
using a halo model. We assume an area of 12300 deg2 for LSST Y1, consistently
with the specifications from the The LSST Dark Energy Science Collaboration
(2018).

| LSST Y1
---|---
Lens Sample |
Number density [arcmin-2] | (2.29, 3.97, 6.06, 3.07, 2.62)
Galaxy bias | (1.7, 1.7, 1.7, 2.0, 2.0)
Source Sample |
Number density [arcmin-2] | (2.50, 2.50, 2.52, 2.48)
Total shape noise | 0.3677
Table 1: LSST Y1 lens and source sample specifications in the setup of this
work. The listed shape noise is the total one including both ellipticity
components. These values are taken from The LSST Dark Energy Science
Collaboration (2018), which specifies a lens number density of 18 arcmin-2,
which we split in 5 redshift bins, and a source number density of 10 arcmin-2
which we split in 4 redshift bins.

We use a $\Lambda$CDM model with 5 (6) free cosmological parameters for the
simulated (data) case: $\Omega_{m}$, $A_{s}$, $n_{s}$, $\Omega_{b}$, $h$,
($\Omega_{\nu}$). We also free additional nuisance parameters to marginalize
over uncertainties related to photometric redshifts – both for the lens and
source samples, intrinsic alignments and shear calibration. The full list of
free parameters and their respective priors can be found in table II444The
only difference between the values that we use and the ones from table II from
Krause et al. (2021) is that we fix the neutrino density parameter
$\Omega_{\nu}h^{2}$ to a null value. This is because CosmoCov is not able to
generate a model for the covariance that takes into account neutrinos.
However, as shown in figure 2 from Krause et al. (2021), the impact of
marginalizing over neutrino density is small for the DES Y3 3$\times$2pt
analysis, so we do not expect this choice to affect any of the conclusions of
this work. from Krause et al. (2021) for the simulated analysis and in table I
from Abbott et al. (2022) for the DES Y3 data analysis. For the simulated LSST
Y1 analysis we assume the same redshift distributions that were used in the
methodology paper of the DES Y3 analysis (Krause et al., 2021), which are an
early estimate of the DES Y3 redshift distributions555LSST is expected to use
a different redshift binning with respect to the one we choose in this work.
However, since we are matching the rest of the settings to Krause et al.
(2021) we decided to also match the input redshift distributions for
simplicity, also given the fact that the redshift distributions that we assume
in this work have a comparable binning and redshift range than the one
predicted in The LSST Dark Energy Science Collaboration (2018) for the LSST Y1
sample.. For the DES Y3 data chains, we use the same settings and priors as in
Abbott et al. (2022), except that we do not include the shear-ratio likelihood
and only combine the tangential shear and galaxy clustering measurements,
since for the 2$\times$2pt case it does not significantly change the results
(Sánchez & Prat et al.,, 2022).

We use the CosmoSIS (Zuntz et al., 2015) framework to generate the data
vectors and perform the 2$\times$2pt analysis. We use MultiNest (Feroz &
Hobson, 2008; Feroz et al., 2009, 2019) to sample the parameter space and
obtain the parameter posteriors, with the following accuracy settings:
live_points=500, efficiency=0.3, tolerance=0.01, constant_efficiency=F. For
the DES Y3 data, we use Polychord with the same high-accuracy settings used in
Abbott et al. (2022).

###  3.2 Contaminated input model

The baryonic contamination is obtained by rescaling the non-linear matter
power spectrum with the baryonic effects from OWLS (OverWhelmingly Large
Simulations project, Schaye et al. 2010; van Daalen et al. 2011) as a function
of redshift and scale, comparing the power spectrum from the dark matter-only
simulation with the power spectrum from the OWLS AGN simulation, following
Krause et al. (2021). For the non-linear galaxy bias contamination we utilize
a model that has been calibrated using $N$-body simulations and is described
in Pandey et al. (2020, 2022). Overall we use the same procedure which is used
in Krause et al. (2021) to contaminate the fiducial data vector with these
effects. Note that while the scale cuts and constraining power for each setup
will depend on these choices (both on the contaminated and fiducial model),
the comparison of the estimators will be independent of it since we use the
same input contamination for all the different localization methodologies.

We generate both the fiducial and contaminated data vectors at the same
cosmological and nuisance parameters that were used to define the scale cuts
in the DES Y3 3$\times$2pt cosmological analysis. In Fig. 2 we display the
differences between the contaminated and fiducial data vectors for the
tangential shear in the top and angular galaxy clustering in the bottom.

Figure 2: Difference between the contaminated data vector and the fiducial
ones for the tangential shear (top) and angular clustering (bottom), for the
redshift bin combination corresponding to the second lens bin and the highest
source bin. This redshift bin combination corresponds to one of the higher S/N
ones. Green error bars represent the uncertainties for DES Y3 and LSST Y1. The
gray regions mark the scale cuts that are needed to obtain unbiased
cosmological results from this contamination, which have been determined to be
$w>8$ Mpc/$h$ and $\gamma_{t}>6$ Mpc/$h$ for DES Y3 and $w>12$ Mpc/$h$ and
$\gamma_{t}>8$ Mpc/$h$ for LSST Y1 when using one of the methods to localize
the tangential shear measurements (otherwise the scale cuts would need to be
larger for the tangential shear quantity, as shown in Appendix C)

.

###  3.3 Procedure to obtain the scale cuts

Here we describe how we obtain the scale cuts that we can use for LSST Y1 that
yield unbiased cosmological results given our input contamination data
vectors. We compute the differences of the posteriors in the 2D
$S_{8}$–$\Omega_{m}$ plane between results using either the fiducial or the
contaminated input data vectors. Specifically, we use the maximum a posteriori
point (MAP) to compute the 2D offsets. We choose to use the "Point-mass (free
per $z_{l}^{i}$)" model to perform this exercise, although we do not expect
this choice to impact the results for the scale cuts. If the difference is
above 0.3$\sigma$, it does not pass our criteria, following the same procedure
as in Krause et al. (2021). We have tested the following set of scale cuts:
$w>8$ Mpc/$h$, $\gamma_{t}>8$ Mpc/$h$; $w>12$ Mpc/$h$, $\gamma_{t}>6$ Mpc/$h$;
$w>12$ Mpc/$h$, $\gamma_{t}>8$ Mpc/$h$. Only this last set of scale cuts meets
the criteria. See Appendix B for the plots showing these differences.

##  4 Results

Figure 3: Cosmological parameter posteriors obtained from an input galaxy
clustering and galaxy-galaxy lensing data vector (2$\times$2pt) with non-
linear bias and baryonic contamination, LSST Y1 covariance, 8Mpc/$h$ scale
cuts for galaxy-galaxy lensing and 12 Mpc/$h$ for galaxy clustering. This
figure demonstrates that all the methodologies to localize the galaxy-galaxy
lensing measurements perform similarly at the cosmological posterior level
with LSST Y1 uncertainties. The 2D contours represent 1$\sigma$ and 2$\sigma$
confidence regions. The shaded area under the 1D posteriors represents the
1$\sigma$ confidence level in 1D.

Using the LSST Y1 setup described in the previous section, we find that all
the estimators perform in a similar way. In Fig. 3 we show the results for the
simulated $2\times 2$pt analysis, combining galaxy-galaxy lensing and galaxy
clustering for all the methodologies that we want to compare that localize the
tangential shear measurements. We also add the result without applying any
mitigation method, to illustrate the importance of using one of these
methodologies to obtain unbiased cosmological constraints. All these results
are applying the fiducial scale cuts that passed the criteria defined in Sec.
3.3: $w>12$ Mpc/$h$ and $\gamma_{t}>8$ Mpc/$h$. In Appendix C we also show
that $\gamma_{t}>40$ Mpc/$h$ cuts would be needed in order to recover unbiased
cosmological constraints if we do not apply any mitigation scheme.

We find that all the methodologies are able to properly mitigate the impact of
the input contamination and recover very similar uncertainties on the most
constrained cosmological parameters of a 2$\times$2pt analysis, that is,
$\Omega_{m}$ and $\sigma_{8}$. In the comparison we also include the new
estimator that we have developed that only projects out the $1/R^{2}$ mode,
without doing any extra transformation as in some of the other methodologies,
labelled as "Proj-out" in the plot. The fact that all the estimators agree
with each other, and also with this new estimator, indicates that projecting
out this mode is the only thing that has any effect in all the mitigation
methodologies at the cosmological posterior level.

Instead we have found that differences between the methods arise from input
assumptions. In particular, we observe the biggest difference is between the
two different variants of the point-mass marginalization. The method labelled
as "Point-mass (free per $z^{i}_{l}-z_{s}^{j}$)" does not use any extra
information with respect to the other estimators and can be compared directly
to them. On the other hand the one labelled as "Point-mass (free per
$z^{i}_{l}$)" uses the approximation that the mass enclosed below the minimum
scale used does not evolve within the redshift range of the lens bin, and
moreover uses geometrical "shear-ratio" information to constrain the scaling
between different sources sharing the same lens bin. We find that the
posteriors for this case are slightly more constraining as expected since they
use more information, but also slightly more biased with respect to the input
true cosmology. Thus, we recommend that when applying the point-mass
marginalization case using geometrical information to LSST Y1 or a more
constraining data set, the assumption that the point mass evolves slowly
within a lens redshift bin should be tested for a given lens sample.

Figure 4: Application to DES Y3 data in a 2$\times$2pt analysis for the MagLim
sample of each of the different methodologies to localize the tangential shear
measurements. This figure demonstrates that all the methodologies to localize
the galaxy-galaxy lensing measurements perform similarly at the cosmological
posterior level with DES Y3 uncertainties and the presence of noise. The 2D
contours represent 1$\sigma$ and 2$\sigma$ confidence regions. The shaded area
under the 1D posteriors represent the 1$\sigma$ confidence level in 1D.

In Appendix D we also compare the intrinsic alignment parameter posteriors. We
find that all the estimators perform similarly, except for the point-mass
using geometric information to constrain the scaling between the lens redshift
bins that gives a much tighter (and still unbiased) posterior on the galaxy
bias of the source sample $b_{TA}$ parameter and a slightly tighter constraint
for the $A_{2}$ and $\alpha_{2}$ parameters, which are the parameters related
to the tidal torque contribution to the model.

###  4.1 Application to DES Y3 data

After testing on noiseless simulated data vector with an LSST Y1 setup we
apply all the methodologies to localize the galaxy-galaxy lensing measurements
to the DES Y3 data, in particular to the 2$\times$2pt setup with the MagLim
lens sample presented in Porredon et al. (2022) but without the shear-ratio
likelihood. The scale cuts we use are the same as in Porredon et al. (2022),
that is, 6 Mpc/$h$ for galaxy-galaxy lensing and 8 Mpc/$h$ for the angular
galaxy clustering. In Fig. 4 we show the results of this comparison. Note that
the "Point-mass (free per $z^{i}_{l}$)" case corresponds to the fiducial DES
Y3 2$\times$2pt result presented in Porredon et al. (2022). We find that all
the methodologies give consistent results, even in this noisy and more
realistic scenario, which presents non-linearities at the small scales and
includes all the effects from the real Universe. In Appendix E we compare the
constraining power of DES Y3 and LSST Y1 to give a sense of scale.

Moreover, we compare the posteriors on the TATT intrinsic alignment parameters
for DES Y3 data and find similar conclusions as in the simulated case, as
shown in Appendix D.

###  4.2 Performance differences

Regarding performance differences between the methods, we find the "Proj-out"
estimator to be less numerically stable than the other approaches in its
current CosmoSIS implementation. This is because the "Proj-out" method
requires an input arbitrary $\eta$ value to obtain the pseudoinverse of the
transformed covariance matrix, as defined in Eq. (22). While we have checked
that a considerable wide range of $\eta$ values yield the same results under
the conditions of this analysis, outside a certain range that is no longer the
case. Thus, robustness against different $\eta$ values might need to be
revisited in other settings. Moreover, we find the point-mass marginalization
method to be the simplest to use as currently implemented in CosmoSIS, since
it does not require modifying the input data vector or covariance matrix
files. Finally, we also compare the running time of the different estimators.
Using the LSST Y1 setup, the point-mass marginalization without geometrical
information took 7h, the $\Upsilon$ statistic took 7h 45 min, the
Y-transformation took 12h and the proj-out estimator took 16h 40 min, using
the MultiNest sampler with the settings defined above and using the same
number of cores. On DES Y3 data and the Polychord sampler with high-accuracy
settings, the point-mass marginalization without geometrical information took
45h, the $\Upsilon$ statistic took 49h, the Y-transformation took 54h and the
proj-out estimator took 70h. We summarize these findings in Table 2.

| Modify data vector? | Numerical Stability | Computational expense
---|---|---|---
Point-mass marginalization | No | Excellent | Fastest
Annular differential surface density ($\Upsilon$) | Yes | Excellent | Fast
Y-transformation | Yes | Excellent | Slow
Proj-out | Yes | Poor | Slowest
Table 2: Comparison of the process and performance of each methodology to
localize the galaxy-galaxy lensing measurements. The computational expense
estimates are based on the current implementation of the CosmoSIS code. More
details about the performance differences can be found in Sec. 4.2.

##  5 Summary and conclusions

In this paper we compare three existing methodologies to localize the galaxy-
galaxy lensing measurements: the Annular Differential Surface Density
estimator ($\Upsilon$) presented in Baldauf et al. (2010), the
Y-transformation derived in Park et al. (2021) and the point-mass
marginalization described in MacCrann et al. (2020). We compare them at the
cosmological posterior level, performing a 2$\times$2pt analysis which
combines projected angular clustering and tangential shear measurements. We
find that all these methods are able to mitigate the impact of small scale
information when using a LSST Y1 setup with noiseless simulated data vectors,
and that they are all performing in a very similar manner, yielding equivalent
posteriors on the cosmological parameters.

To further illustrate the similarities and/or differences amongst the above
listed estimators, we also construct a new estimator that we refer to as
"project-out". The "project-out" method identifies the projection of the
observed galaxy-galaxy lensing data vector onto the $1/R^{2}$ mode, and then
it removes it from the original vector, following a similar procedure to the
Y-transform methodology, but in this case only removing this mode. Then, we
proceed to compare the posteriors obtained with the "project-out" method to
the other ones, finding it yields equivalent results. Therefore, we conclude
that the removal of the $1/R^{2}$ mode is the only relevant transformation
that is needed to localize the tangential shear measurements and that the rest
of the modifications in the other estimators are not producing any significant
differences at the cosmological posterior level.

We also compare two different variations of the point-mass marginalization
methodology, one that uses exactly the same information as the other
estimators and one that uses extra geometrical information to constrain the
scaling of the point-mass between different lens and source bin combinations,
by assuming that the enclosed mass does not evolve with redshift within the
width of the lens bin. We find that the point-mass marginalization using
geometric information yields slightly more constraining but also slightly
biased results on the cosmological parameters in the LSST Y1 simulated case,
due to the approximation it makes. Thus, the assumption going into this point-
mass variant should always be tested before applying it to more constraining
data sets. Notably, we also find that the extra geometrical information
significantly improves the precision (while keeping the accuracy) of the
intrinsic alignment parameters of the tidal alignment and tidal torque (TATT)
model. In particular, we find the biggest difference in the posterior for the
galaxy bias of the source sample $b_{\text{TA}}$ and in the parameters
controlling the tidal torque part of the TATT model.

We also compare the results obtained using any of the mitigation schemes with
the case of not applying any mitigation scheme but applying larger scale cuts.
With the LSST Y1 setup, we find that the mitigation schemes yield $\sim$1.3
times more constraining $S_{8}$ results than applying larger scale cuts
without using any mitigation scheme.

Finally, we apply all the methods to DES Y3 data, reaching very similar
conclusions as in the simulated case. However, even if the DES Y3 data has
larger uncertainties than the simulated LSST Y1 case, this exercise is still
meaningful since it provides an input data vector with the non-linearities and
baryonic effects of the real Universe, together with any other other
unforeseen contamination that is not present in our fiducial model. It also
tests the methods in the presence of noise. In this case we still find that
all the methodologies perform in a similar manner.

## Acknowledgments

Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy
Survey.

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

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

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

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

## Data availability

The data underlying this article are available in the DES Y3 Cosmology Data
Release page at https://des.ncsa.illinois.edu/releases/y3a2/Y3key-products.
The code developed for this project is integrated within the CosmoSIS
framework and can be shared upon request.

## References

  * Abbott et al. (2018) Abbott T. M. C., et al. 2018, Phys. Rev. D, 98, 043526
  * Abbott et al. (2022) Abbott T. M. C., et al. 2022, Phys. Rev. D, 105, 023520
  * Aitken (1936) Aitken A. C., 1936, Proceedings of the Royal Society of Edinburgh, 55, 42–48
  * Baldauf et al. (2010) Baldauf T., et al. 2010, Phys. Rev. D, 81, 063531
  * Baxter et al. (2016) Baxter E., et al. 2016, MNRAS, 461, 4099
  * Bernstein (2009) Bernstein G. M., 2009, The Astrophysical Journal, 695, 652
  * Brainerd et al. (1996) Brainerd T. G., Blandford R. D., Smail I., 1996, ApJ, 466, 623
  * Bridle et al. (2002) Bridle S. L., et al. 2002, MNRAS, 335, 1193
  * Choi et al. (2012) Choi A., et al. 2012, ApJ, 759, 101
  * Clampitt et al. (2017) Clampitt J., et al. 2017, MNRAS, 465, 4204
  * Desjacques et al. (2018) Desjacques V., Jeong D., Schmidt F., 2018, Phys. Rep., 733, 1
  * Fang et al. (2020) Fang X., Eifler T., Krause E., 2020, MNRAS, 497, 2699
  * Feroz & Hobson (2008) Feroz F., Hobson M. P., 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz F., Hobson M. P., Bridges M., 2009, MNRAS, 398, 1601
  * Feroz et al. (2019) Feroz F., et al. 2019, The Open Journal of Astrophysics, 2, 10
  * Giblin et al. (2021) Giblin B., et al. 2021, A&A, 645, A105
  * Heymans et al. (2021) Heymans C., et al. 2021, A&A, 646, A140
  * Hildebrandt et al. (2020) Hildebrandt H., et al. 2020, Astronomy & Astrophysics, 633, A69
  * Hu & Jain (2004) Hu W., Jain B., 2004, Phys. Rev. D, 70, 043009
  * Jain & Taylor (2003) Jain B., Taylor A., 2003, Phys. Rev. Lett., 91, 141302
  * Joachimi & Bridle (2010) Joachimi B., Bridle S. L., 2010, A&A, 523, A1
  * Joudaki et al. (2018) Joudaki S., et al. 2018, MNRAS, 474, 4894
  * Krause & Eifler (2017) Krause E., Eifler T., 2017, MNRAS, 470, 2100
  * Krause et al. (2021) Krause E., et al. 2021, arXiv e-prints, p. arXiv:2105.13548
  * Kuijken et al. (2015) Kuijken K., et al. 2015, MNRAS, 454, 3500
  * Leauthaud et al. (2022) Leauthaud A., et al. 2022, MNRAS, 510, 6150
  * Lee et al. (2022) Lee S., et al. 2022, MNRAS, 509, 2033
  * Luo et al. (2022) Luo R., et al. 2022, A&A, 668, A12
  * MacCrann et al. (2020) MacCrann N., et al. 2020, MNRAS, 491, 5498
  * Mandelbaum et al. (2013) Mandelbaum R., et al. 2013, MNRAS, 432, 1544
  * Pandey et al. (2020) Pandey S., et al. 2020, Physical Review D, 102
  * Pandey et al. (2022) Pandey S., et al. 2022, Phys. Rev. D, 106, 043520
  * Park et al. (2016) Park Y., et al. 2016, Physical Review D, 94
  * Park et al. (2021) Park Y., Rozo E., Krause E., 2021, Phys. Rev. Lett., 126, 021301
  * Porredon et al. (2022) Porredon A., et al. 2022, Phys. Rev. D, 106, 103530
  * Prat et al. (2018) Prat J., et al. 2018, Phys. Rev. D, 98, 042005
  * Prat et al. (2019) Prat J., et al. 2019, MNRAS, 487, 1363
  * Prat et al. (2022) Prat J., et al. 2022, Phys. Rev. D, 105, 083528
  * Sánchez et al. (2022) Sánchez C., et al. 2022, Phys. Rev. D, 105, 083529
  * Schaye et al. (2010) Schaye J., et al. 2010, Monthly Notices of the Royal Astronomical Society, 402, 1536
  * Seljak (1998) Seljak U., 1998, ApJ, 503, 492
  * Singh et al. (2020) Singh S., et al. 2020, MNRAS, 491, 51
  * Takahashi et al. (2012) Takahashi R., et al. 2012, ApJ, 761, 152
  * Tegmark (1997) Tegmark M., 1997, Phys. Rev. D, 55, 5895
  * Tegmark et al. (1998) Tegmark M., et al. 1998, ApJ, 499, 555
  * The LSST Dark Energy Science Collaboration (2018) The LSST Dark Energy Science Collaboration 2018, arXiv e-prints, p. arXiv:1809.01669
  * Yoo & Seljak (2012) Yoo J., Seljak U., 2012, Physical Review D, 86
  * Zacharegkas et al. (2022) Zacharegkas G., et al. 2022, MNRAS, 509, 3119
  * Zuntz et al. (2015) Zuntz J., et al. 2015, Astronomy and Computing, 12, 45
  * van Daalen et al. (2011) van Daalen M. P., et al. 2011, MNRAS, 415, 3649
  * van Uitert et al. (2018) van Uitert E., et al. 2018, MNRAS, 476, 4662

## Affiliations

1 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA
2 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
4 Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, UK
5 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
6 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
7 Department of Physics, Northeastern University, Boston, MA 02115, USA
8 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
9 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
10 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
11 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
12 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
13 Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
14 Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390
15 Department of Physics, Duke University Durham, NC 27708, USA
16 California Institute of Technology, 1200 East California Blvd, MC 249-17,
Pasadena, CA 91125, USA
17 Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo,
Brazil
18 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
19 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA
20 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
21 Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain
22 Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA
23 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
W. Green Street, Urbana, IL 61801, USA
24 Physics Department, William Jewell College, Liberty, MO, 64068
25 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
26 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
27 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
28 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
29 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA
94720, USA
30 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
31 NSF AI Planning Institute for Physics of the Future, Carnegie Mellon
University, Pittsburgh, PA 15213, USA
32 Université Grenoble Alpes, CNRS, LPSC-IN2P3, 38000 Grenoble, France
33 Department of Astronomy/Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA
34 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
35 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA
36 Department of Physics, The Ohio State University, Columbus, OH 43210, USA
37 Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA
38 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679 Munich, Germany
39 School of Physics and Astronomy, Cardiff University, CF24 3AA, UK
40 Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290
Versoix, Switzerland
41 Department of Physics, University of Arizona, Tucson, AZ 85721, USA
42 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford,
CA 94305, USA
43 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
44 Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,
13083-859, Campinas, SP, Brazil
45 Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK
46 ICTP South American Institute for Fundamental Research
Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
47 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
48 Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA
49 Department of Physics and Astronomy, Stony Brook University, Stony Brook,
NY 11794, USA
50 Département de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland
51 Excellence Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany
52 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,
85748 Garching, Germany
53 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany
54 Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile
55 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
56 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
57 Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France
58 Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
59 Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo
11, I-34131 Trieste, Italy
60 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
61 Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
62 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029
Hamburg, Germany
63 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
64 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
65 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
66 School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
67 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
68 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
69 Departamento de Física Matemática, Instituto de Física, Universidade de São
Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
70 Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
71 Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
72
73 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
74 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831

##  Appendix A Equivalence between "Project-out" and Point-mass
marginalization

In this appendix we show that the "Project-out" and the Point-mass
marginalization methods are mathematically equivalent if the prior for the
point-mass is infinitely wide. Let us define $\mathbf{m}$ to be the model
prediction and $\mathbf{d}$ to be the data vector.

For the PM marginalization case we have:

$\chi^{2}_{\rm
PM}=(\mathbf{m}-\mathbf{d})^{T}\mathbf{N}^{-1}(\mathbf{m}-\mathbf{d})\;.$

However, in the infinite-prior case it is:

$\mathbf{N}^{-1}=\mathbf{C}_{\gamma}^{-1}-\mathbf{C}_{\gamma}^{-1}\mathbf{A}(\mathbf{A}^{T}\mathbf{C}_{\gamma}^{-1}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{C}_{\gamma}^{-1}=\mathbf{C}_{\gamma}^{-1}(\mathbf{I}-\mathbf{P})\;,$

which results in

$\displaystyle\chi^{2}_{\rm
PM}=(\mathbf{m}-\mathbf{d})^{T}\mathbf{C}_{\gamma}^{-1}\mathbf{M}(\mathbf{m}-\mathbf{d})\;.$
(25)

So, $\mathbf{N}^{-1}(\mathbf{m}-\mathbf{d})$ exactly simply removes the
contribution from the $1/\theta^{2}$ mode from the model and data vectors,
like the project-out approach. The project-out method is by construction only
removing that mode from the data vector and does nothing else. Therefore, the
two methods are equivalent in the above limit.

##  Appendix B Scale cuts

Figure 5: $S_{8}-\Omega_{m}$ plane showing the differences in the posteriors
using either a fiducial input data vector or a contaminated one with baryonic
and non-linear galaxy bias effects. The dashed gray lines mark the input
fiducial cosmology. Comparing the contaminated and the baseline posteriors
using different sets of scale cuts we have determined that $w>12$ Mpc/$h$,
$\gamma_{t}>8$ Mpc/$h$ cuts are sufficient for the LSST Y1 setup to recover
unbiased results. Specifically, these cuts produce a difference of 0.09
$\sigma$ in the $S_{8}-\Omega_{m}$ plane, which is below the threshold of
$0.3\sigma$ following the criteria from Krause et al. (2021)

.

In Fig. 5 we show the result of the scale cuts tests for the successful
scenario. We have found that scales cuts of $w>12$ Mpc/$h$ and $\gamma_{t}>8$
Mpc/$h$ are sufficient in the LSST Y1 setup to recover unbiased results when
inputting a contaminated data vector with non-linear galaxy bias and baryonic
effects. We obtain that in this case the difference in the $S_{8}-\Omega_{m}$
plane between the fiducial and contaminated data vectors is 0.09$\sigma$ which
is below the threshold of $0.3\sigma$, following the same procedure as in
Krause et al. (2021). We compare posteriors using the fiducial vs. the
contaminated data vectors (instead of with the input cosmology) to remove any
projection effects666Projection effects are residual differences between the
input cosmology and the posteriors under ideal conditions (when the input data
vector and model are the same) due to having broad parameter spaces. impact on
this test. We have also tried using $w>8$ Mpc/$h$, $\gamma_{t}>8$ Mpc/$h$
scale cuts, which produced a difference of $0.46\sigma$ and $w>12$ Mpc/$h$ and
$\gamma_{t}>6$ Mpc/$h$ cuts, which produced a difference of 0.93$\sigma$ in
the same $S_{8}-\Omega_{m}$ plane. Note that these combinations of scale cuts
are an arbitrary choice, and applications on actual data might want to
optimize these choices.

##  Appendix C Mitigation schemes vs. no mitigation

In this appendix we address the following question: How much constraining
power do we gain by applying one of the mitigation schemes vs. not applying
any of them and using fewer scales? To perform this comparison we choose the
point-mass case that includes geometrical information since that is what we
used to define the fiducial scale cuts, as described in Sec. 3.3. We perform
this comparison for the LSST Y1 simulated analysis. We show the results in
Fig. 6. There we compare the posteriors between using the point-mass
mitigation scheme and without applying any mitigation with the following two
sets of scale cuts:

  1. 1.

$w>12$ Mpc/$h$, $\gamma_{t}>8$ Mpc/$h$: the same scale cuts needed to recover
unbiased constraints when applying the point-mass marginalization scheme
including geometrical information.

  2. 2.

$w>12$ Mpc/$h$, $\gamma_{t}>40$ Mpc/$h$: scale cuts needed to recover unbiased
cosmological constraints without using any mitigation scheme, following the
same criteria described in Sec. B. To obtain them, we have chosen to keep the
galaxy clustering scale cut unchanged and increase the tangential shear cut
until we recover unbiased results. Under this setup, we find a 0.17$\sigma$
difference in the $S_{8}-\Omega_{m}$ plane, while we find a 0.35$\sigma$
difference if we use $\gamma_{t}$>32 Mpc/$h$ instead, which does not meet the
criteria.

We find that $S_{8}$ is $\sim$1.3 times more constraining when using the
point-mass marginalization scheme vs. when not using any mitigation scheme and
using larger scale cuts.

Figure 6: We compare the constraining power when applying a mitigation scheme
(in this case the point-mass marginalization) vs. when not applying any scheme
and using less scales for the galaxy-galaxy lensing probe. We conclude that
using a marginalization scheme yields $\sim$1.3 more constraining power on the
$S_{8}$ parameter assuming a LSST Y1 simulated scenario.

##  Appendix D Effect on the intrinsic alignment parameters

In Fig. 7 we show the posteriors for the tidal alignment and tidal torque
(TATT) 5-parameter intrinsic alignment model, at the top for the LSST Y1
simulated case and at the bottom applied to DES Y3 data, in both cases for a
2$\times$2pt analysis without including the shear-ratio likelihood. Using the
simulated data in the LSST Y1 setup, we find that using the extra geometric
information in the point-mass marginalization, i.e. the case labelled as
"Point-mass (free per $z^{i}_{l}$)" and described in Sec. 2.3.1 is beneficial
to constrain the intrinsic alignment parameters. In particular we find the
parameter describing the galaxy bias of the source sample $b_{TA}$ is more
constrained, as well as the parameters affecting the tidal torque part of the
model ($a_{2}$ which describes the amplitude of the IA effect and $\alpha_{2}$
that modulates its redshift evolution). On the Y3 data, we find that the
biggest gain in constraining power is in the $a_{2}$ parameter.

Figure 7: Top: Intrinsic alignment TATT parameter posteriors obtained from an
input galaxy clustering and galaxy-galaxy lensing data vector (2$\times$pt)
with non-linear bias and baryonic contamination with a LSST Y1 covariance,
comparing the different methodologies to localize the tangential shear
measurements. Bottom: Analogous comparison applied to DES Y3 data, for the
MagLim 2$\times$pt analysis without the shear-ratio likelihood.

##  Appendix E Comparison between DES Y3 and LSST Y1

Figure 8: Comparison of the constraining power between LSST Y1 and DES Y3 for
a 2$\times$2pt simulated analysis using the fiducial model data vectors. For
LSST Y1 we use scales $w>12$ Mpc/$h$, $\gamma_{t}>8$ Mpc/$h$, and for DES Y3
$w>8$ Mpc/$h$, $\gamma_{t}>6$ Mpc/$h$. We also include the data results from
DES Y3 2$\times$2pt.

In Fig. 8 we compare the constraining power between the DES Y3 and LSST Y1
setups, which besides being interesting on its own also provides some basic
validation of the LSST Y1 covariance that we compute with CosmoCOV. First, we
compare the fiducial LSST Y1 simulated analysis with the fiducial scales of
$w>12$ Mpc/$h$ and $\gamma_{t}>8$ Mpc/$h$, with the scales used in the DES Y3
data of $w>8$ Mpc/$h$ and $\gamma_{t}>6$ Mpc/$h$. We observe that the
degradation in the constraining power coming only from the differences in the
scales is quite small. Then, we compare the size of the LSST Y1 contours with
the simulated DES Y3 analysis, which was used in Krause et al. (2021) to
determine the scale cuts for that case. The only two differences between the
contours labelled "LSST Y1 sim, with DES Y3 scales" and the ones labelled "DES
Y3 sim" are the covariances and the input redshift distributions, as described
in Sec. 3.1. Finally we also compare the DES Y3 simulated analysis with the
actual final data DES Y3 2$\times$2pt results, which use a different set of
priors as the rest of the chains, had an updated covariance accounting for the
best-fit parameters and are not centered at the same cosmology.