11institutetext: Université Paris-Saclay, Université Paris Cité, CEA, CNRS,
AIM, 91191, Gif-sur-Yvette, France 22institutetext: CAS Key Laboratory for
Research in Galaxies and Cosmology, Department of Astronomy, University of
Science and Technology of China, Hefei, Anhui 230026, China 33institutetext:
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China 44institutetext: Department of Physics and
Astronomy, University of Utah, Salt Lake City, Utah 84102, USA
55institutetext: NRC Herzberg Astronomy & Astrophysics, 5071 West Saanich
Road, British Columbia, Canada V9E2E7 66institutetext: Ruhr University Bochum,
Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German
Centre for Cosmological Lensing, 44780 Bochum, Germany

# Point-Spread Function errors for weak lensing - density cross-correlations.
Application to UNIONS.

Ziwen Zhang , 0000-0002-9272-5978 11 2 2 3 3<EMAIL_ADDRESS>Martin
Kilbinger , 0000-0001-9513-7138 11<EMAIL_ADDRESS>Fabian Hervas
Peters 11 Qinxun Li  0000-0003-3616-6486 44 Wentao Luo  0000-0003-1297-6142
2233 Lucie Baumont  0000-0002-1518-0150 11 Jean-Charles Cuillandre
0000-0002-3263-8645 11 Sébastien Fabbro  0000-0003-2239-7988 55 Stephen Gwyn
0000-0001-8221-8406 55 Alan McConnachie  0000-0003-4666-6564 55 Anna Wittje
0000-0002-8173-3438 66

(Received; accepted)

###### Abstract

Aims. Calibrating the point spread function (PSF) is a fundamental part of
weak gravitational lensing analyses. Even with corrected galaxy images,
imperfect calibrations can introduce biases. We propose an analytical
framework for quantifying PSF-induced systematics as diagnostics for cross-
correlation measurements of weak lensing with density tracers, e.g., galaxy-
galaxy lensing. We show how those systematics propagate to physical parameters
of the density tracers. Those diagnostics only require a shape catalogue of
PSF stars and foreground galaxy positions.

Methods. We consider the PSF-induced multiplicative bias, and introduce three
second-order statistics as additive biases. We compute both biases for the
weak-lensing derived halo mass of spectroscopic foreground galaxy samples, in
particular, their effect on the tangential shear and fitted halo mass as a
function of stellar mass. In addition, we assess their impact on the recently
published black-hole - halo-mass relation for type I Active Galactic Nuclei
(AGNs).

Results. Using weak-lensing catalogues from the Ultraviolet Near Infrared
Optical Northern Survey (UNIONS) and Dark Energy Survey (DES), we find the
multiplicative biases in the tangential shear to be less than $0.5\%$. No
correlations between additive bias and galaxy properties of the foreground
sample are detected. The combined PSF systematics affect low-mass galaxies and
small angular scales; halo mass estimates can be biased by up to 18$\%$ for a
sample of central galaxies in the stellar mass range 9.0 $\leq$ log $M_{*}/\rm
M_{\odot}$ ¡ 9.5.

Conclusions. The PSF-induced multiplicative bias is a subdominant contribution
to current studies of weak-lensing - density cross-correlations, but might
become significant for upcoming Stage-VI surveys. For samples with a low
tangential shear, additive PSF systematics can induce a significant bias on
derived properties such as halo mass.

###### Key Words.:

cosmology: observations – gravitational lensing: weak – catalogues – galaxies:
halos – methods: statistical

##  1 Introduction

Light from distant galaxies on its way to the observer is affected by
gravitational fields along the line of sight, distorting the light
distribution of the galaxies we observe. Weak gravitational lensing refers to
the typical few per cent distortions of the galaxy image due to foreground
large-scale structure (Jarvis et al., 2016). Weak lensing enables us to
measure the distribution of the total foreground mass, which consists of
baryonic and dark matter. Therefore, it is a powerful tool for studying
cosmology (Kilbinger, 2015; Mandelbaum, 2018).

Among the main applications of weak lensing is galaxy-galaxy lensing, which is
the correlation between the shapes of background galaxies and the positions of
foreground galaxies. Many studies have used this method to estimate the (dark-
matter) halo mass, establishing various relations between the halo mass and
galaxy properties (Mandelbaum et al., 2006; Leauthaud et al., 2012; Velander
et al., 2014; Viola et al., 2015; Luo et al., 2018; Zhang et al., 2021, 2022,
2024).

There are a number of potential systematics that can bias weak-lensing
measurements (Jarvis et al., 2016), such as the contamination from cosmic rays
and satellite trails, among others, thus hindering the measurement of the
galaxy brightness distribution; observed galaxy intensity profiles that may be
contaminated by the flux from nearby galaxies or stars; and the observed
images of galaxies that are blurred due to atmospheric refraction or
turbulence and optical imperfections. The combined effect of image blurring
due to the atmosphere and the optical system is known as the point spread
function (PSF; see for a recent focused review Liaudat et al., 2023).

The PSF can strongly smear the weak-lensing shear information in the observed
galaxy shapes (Jarvis et al., 2021). In addition, the PSF is difficult to
estimate because it varies with the field of view (FOV) and across individual
exposures in multi-epoch observations. Therefore, a central part of weak-
lensing analyses and a formidable challenge is to correct for the PSF on
observed galaxy images. Nevertheless, even after PSF correction systematic
errors can remain, induced by imperfections in the PSF model and the
interpolation process (Jarvis et al., 2016).

Various methods exist to quantify PSF-induced systematics for weak lensing.
For cosmic shear, the $\rho$-statistics (Rowe, 2010; Jarvis et al., 2016) are
additive biases to the shear two-point correlation function and directly
propagate to cosmological parameters. A generalisation is the so-called
$\tau$-statistics Gatti et al. (2021a) allow to separately estimate the impact
of PSF leakage and PSF model errors on galaxy shapes. PSF leakage quantifies,
via the multiplicative parameter $\alpha$, how much the PSF ellipticity
influences the PSF-corrected galaxy ellipticity.

For galaxy-galaxy lensing two kinds of null-test estimators have been
developed. The first is the tangential shear around positions that are not
correlated with the density tracers in an ideal setting. These can be random
positions (Mandelbaum et al., 2005), stars, or points fixed to a CCD-frame
coordinate system (Gatti et al., 2021a). The second type of null test is the
cross-component shear measured around the foreground sample, which is expected
to vanish if parity is conserved.

So far, to the best of our knowledge, no PSF systematic diagnostics have been
devised for lensing around foreground density tracers that directly propagate
to parameters of the foreground population. Here, we developed a set of three
galaxy-PSF cross-correlation functions, which we dub “$\lambda$\- statistics”,
that are additive terms to the tangential shear around arbitrary density
tracers, e.g. galaxies, galaxy clusters, filaments, or voids. The
$\lambda$-statistics for weak lensing of density tracers are the analogue of
the $\rho$-statistics for cosmic shear.

In the study of galaxy evolution, a common approach is to relate galaxy
properties to their halo mass (scaling relation), thus revealing the
evolutionary paths of different galaxies in their dark-matter environment
(Posti et al., 2019; Zhang et al., 2021, 2022, 2024). We examined two scaling
relations in this paper, namely the stellar-mass – halo-mass relation and the
black-hole-mass – halo-mass relation. Various studies investigated these
relations in terms of analytic models (Bower et al., 2017), abundance matching
(Shankar et al., 2020; Moster et al., 2010; Yang et al., 2009), and weak
lensing (Zhang et al., 2024; Li et al., 2024). Of these, weak lensing is the
most direct observational method for probing halo mass. The accuracy of
scaling relations is affected by the accuracy of the halo mass estimates. In
this paper, we developed a method to quantify the impact of PSF-induced
systematics to the weak lensing signal and also the halo mass estimates.

The outline of this paper is as follows. We describe the weak-lensing
catalogues and the foreground sample selections in Sect. 2. Section 3
introduces the $\lambda$-statistics, and presents the methods to quantify PSF-
induced multiplicative and additive biases for the weak-lensing tangential
shear. This section also briefly reviews the measurement of tangential shear,
and the connection of shear to halo mass. In Sect. 4, we show the results for
PSF-induced multiplicative and additive biases. In addition, we investigate
the impact of PSF-induced systematics on halo mass estimation. We discuss the
implications of the $\lambda$-statistics in Sect. 5 and summarize our results
in Sect. 6. Throughout this paper, we assume the Planck cosmology (Planck
Collaboration et al., 2016): $\Omega_{\rm m}=0.307$, $\Omega_{\rm b}=0.048$,
$\Omega_{\Lambda}=0.693$ and $h=0.678$.

##  2 Observational data

###  2.1 Weak lensing catalogues

####  2.1.1 UNIONS

We use the weak-lensing shear catalogue from the Ultraviolet Near-Infrared
Northern Sky Survey (UNIONS). Started in 2018, UNIONS is an ongoing survey
that targets 4,800 deg2 in the Northern Hemisphere and covers the footprint of
the Euclid survey (Euclid Collaboration et al., 2022). UNIONS combines multi-
band photometric images from different telescopes. These are the Canada-France
Hawai’i Telescope (CFHT) providing $u$-and $r$-band images; this part of
UNIONS is called the Canada-France Imaging Survey or CFIS; the Panoramic
Survey Telescope and Rapid Response System (Pan-STARRS) for the $i$-and
$z$-band; and Subaru, which takes images in the $z$-band in the framework of
WISHES (Wide Imaging with Subaru HSC of the Euclid Sky), and the $g$-band
Waterloo Hawai’i IfA Survey (WHIGS). Here, we only use the UNIONS $r$-band
data to calculate the weak-lensing shear. Shape measurement was performed with
ShapePipe (Farrens et al., 2022). A first version of the ShapePipe catalogue
was presented in Guinot et al. (2022). In this paper, we use v1.3 of the
catalogue, which contains $83,812,739$ galaxies covering $3,200$ deg2 of
effective sky area, which was the available data in 2022 at the time of
processing. The PSF was modelled with MCCD (Liaudat et al., 2021), which
builds a non-parametric multi-CCD model of the PSF over the focal plane. To
obtain the parameters of the PSF model, stars are selected on the individual
exposures. The star sample is selected on the stellar locus in the size -
magnitude diagram. They are split into a training sample ($80\%$) and a
validation sample ($20\%$). The PSF model is obtained by optimization using
the training sample.

####  2.1.2 DES

We also use the DES Y3 weak lensing catalogue in our analysis (Gatti et al.,
2021a). The catalogue contains $100,204,026$ galaxies, covering $4,139$ square
degrees in the sky. Images were taken in the $g$-, $r-$, $i$-, $z-$, and
$Y$-band and has a weighted source number density of $5.59$ arcmin-2. The
corresponding shape noise is $0.261$.

The DES star catalogue contains 56 million objects, which are identified as
stars using the stellar locus in the magnitude range $[16.5,22.0]$. The PSF is
modelled and interpolated using the PIFF algorithm (PSFs In the Full FOV,
Jarvis et al. (2021)). The parametric model for a star i is fitted iteratively
using neighbouring stars, rejecting outliers until convergence is reached.
Each of the $riz$ bands is fitted separately. PIFF is intended to support
fitting to the entire focal plane; however, the DES Y3 catalogue is restricted
to single CCD modelling.

###  2.2 Foreground galaxy catalogues

The following subsections describe two samples of foreground galaxies, which
are used for weak-lensing cross-correlation analysis in this paper.

####  2.2.1 Central galaxy samples

We use galaxies from the New York University Value Added Galaxy Catalogue
(NYU-VAGC111http://sdss.physics.nyu.edu/vagc/, Blanton et al., 2005) of the
Sloan Digital Sky Survey Data Release 7 (SDSS DR7, Abazajian et al., 2009).
Three selection criteria were applied to these galaxies: 1.) $r$-band
Petrosian apparent magnitude $r\leq 17.72$; 2.) spectroscopic redshift in the
range $0.01\leq z\leq 0.2$; 3.) redshift completeness $C_{z}>0.7$. We only use
the central galaxies in our analysis, they are defined as the most massive
galaxy in a galaxy group, which are identified by the group catalogue (Yang et
al., 2005, 2007)222 https://gax.sjtu.edu.cn/data/Group.html. NYU-VAGC provides
the measurements of stellar mass ($M_{*}$). We cross-match our central
galaxies with the MPA-JHU DR7 catalogue333https://wwwmpa.mpa-
garching.mpg.de/SDSS/DR7/, which provides galaxy star-formation rates (SFR,
Brinchmann et al., 2004) derived from both spectroscopic and photometric data
of SDSS.

In the overlapping sky area between SDSS and UNIONS, there are $126,675$
central galaxies. We divide these galaxies into three stellar-mass bins:
$9.0\leq\log M_{*}/\rm M_{\odot}<9.5$, $10.0\leq\log M_{*}/\rm M_{\odot}<10.5$
and $11.0\leq\log M_{*}/\rm M_{\odot}<11.5$. We further divide each stellar-
mass bin into star-forming and quenched subsamples by using the demarcation
line from Bluck et al. (2016).

####  2.2.2 Active Galactic Nuclei samples

The AGNs used here correspond to the type I sample described in Li et al.
(2024). They are collected from two catalogues. The first one is the SDSS DR16
Quasar Catalog (Lyke et al., 2020). The total number of quasars is $750,414$
with redshifts in the range $0.1<z<6$. Their black-hole masses were estimated
by Wu & Shen (2022) using the FWHM of H$\beta$, Mg II, and C IV broad emission
lines from spectroscopic observations. The black-hole masses used here
correspond to the H$\beta$ emission lines (see Wu & Shen (2022) for more
details).

The second catalogue is the SDSS DR7 AGN catalogue (Liu et al., 2019),
including both quasars and Seyfert galaxies. There are $14,584$ AGNs with
redshifts less than $0.35$. Liu et al. (2019) used H$\alpha$ and H$\beta$
emission lines to measure their corresponding black-hole masses. We also adopt
the black-hole mass based on the H$\beta$ emission line.

We merged the two catalogues with duplicates removed, selecting AGNs in the
redshift range $0.05<z<0.6$. The final catalogue of type I AGNs in the joint
SDSS-UNIONS footprint contains $14,649$ objects. We divide these AGNs into
low-, medium- and high-black-hole mass bins, which are $\log M_{\rm BH}/\rm
M_{\odot}<7.9$, $7.9<\log M_{\rm BH}/\rm M_{\odot}<8.5$ and $\log M_{\rm
BH}/\rm M_{\odot}>8.5$, respectively.

Note that the host galaxies of the selected AGNs include both central and
satellite galaxies.

##  3 Methods of analysis

This section describes the methods to estimate the impact of PSF-induced
systematics on galaxy ellipticity and tangential shear $\gamma_{\textrm{t}}$.
We introduce the $\lambda$-statistics as additive PSF systematics to the
tangential shear. We discuss our measurement and modelling methodology and
review how physical galaxy properties such as halo mass are derived from the
measured tangential shear.

###  3.1 PSF error propagation

In general, the observed ellipticity $\varepsilon^{\textrm{obs}}$ of a galaxy
is not an unbiased estimator of shear $\gamma$ at that position. The relation
between those quantities is, in complex notation to linear order,

$\varepsilon^{\textrm{obs}}=\varepsilon^{\textrm{s}}+(1+m)\gamma+c+\delta\varepsilon+\alpha\varepsilon^{\textrm{psf}}.$
(1)

Here, $\varepsilon^{\textrm{s}}$ denotes the intrinsic ellipticity of the
galaxy, which is assumed to have random orientation, and consequently has
vanishing expectation value, $\langle\varepsilon^{\textrm{s}}\rangle=0$. $m$
and $c$ are the multiplicative and additive biases, respectively. The fourth
term on the right-hand side, $\delta\varepsilon$, represents the residual in
the PSF at the galaxy position due to errors in PSF measurement, modelling,
and interpolation. The final term quantifies the leakage from the PSF
ellipticity into galaxy ellipticity, which can arise, for example, from an
insufficient PSF correction during galaxy shape measurement. The coefficient
$\alpha$ is the leakage amplitude.

These biases have a variety of origins (Massey et al., 2013): $m$ can arise
from the misestimation of the PSF (Gatti et al., 2021a), but this effect is
typically small (Massey et al., 2013). Larger contributions come from
calibration errors in the shear measurement algorithm. The potential origins
of $c$ are PSF errors or the incomplete application of the charge transfer
inefficiency (van Uitert & Schneider, 2016; Gatti et al., 2021a).

In the model described by Eq. (1), we assume that $m$ and $c$ are not
correlated with the PSF; we exclude PSF-induced effects from these two biases.
The additive bias due to the PSF is shown in the last two terms in Eq. (1). We
will show below how the PSF uncertainty induces an additional multiplicative
bias that is not included in the above equation. Since we are primarily
focusing on PSF errors, we assume that the shear measurements have been
accurately calibrated for $m$ and $c$, and set $m=c=0$.

Assuming ellipticity to be measured via second moments of the light
distribution, the PSF residual has been derived via Gaussian error propagation
in Paulin-Henriksson et al. (2008) as

$\delta\varepsilon=\left(\varepsilon^{\rm
obs}-\varepsilon^{\textrm{psf}}\right)\frac{\delta
T^{\textrm{psf}}}{T}-\frac{T^{\textrm{psf}}}{T}\delta\varepsilon^{\textrm{psf}},$
(2)

where $T$ is the galaxy’s intrinsic (PSF de-convolved) size, and
$T^{\textrm{psf}}$ is the PSF size. The PSF residual is induced by the error
of the PSF model, which is denoted as the difference between the measured and
modelled PSF in their size, $\delta T^{\textrm{psf}}$, and ellipticity,
$\delta\varepsilon^{\textrm{psf}}$. Unfortunately, we can not directly measure
$\delta T^{\textrm{psf}}$ and $\delta\varepsilon^{\textrm{psf}}$ at galaxy
positions. However, the PSF can be estimated using stars, and PSF residuals
can be obtained at star positions. Therefore, we write eq. (2) as

$\delta\varepsilon=\left(\varepsilon^{\textrm{obs}}-\varepsilon^{\textrm{psf}}\right)\frac{T^{\textrm{psf}}}{T}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}-\frac{T^{\textrm{psf}}}{T}\delta\varepsilon^{\textrm{psf}},$
(3)

and measure $T^{\textrm{psf}}/T$ at galaxy positions, and $\delta
T^{\textrm{psf}}/T^{\textrm{psf}}$ at star positions.

###  3.2 PSF-induced systematics for tangential shear

A sample $S$ of objects that trace matter in the large-scale structure induces
a tangential shear $\gamma_{\textrm{t}}^{S}$ on background galaxies. This
tangential shear can be written as second-order correlation between background
shear and foreground number density $n$. Therefore, an average tangential
shear caused by a foreground sample $S$ is

$\gamma_{\textrm{t}}^{S}=\left\langle\gamma_{\textrm{t}}\,n\right\rangle.$ (4)

An estimator of this tangential shear is

$\hat{\gamma}_{\textrm{t}}\equiv\gamma_{\textrm{t}}^{\rm
obs}=\left\langle\varepsilon_{\textrm{t}}^{\textrm{obs}}n\right\rangle.$ (5)

This estimator is the average of the observed tangential ellipticity of
background galaxies $\varepsilon_{\textrm{t}}^{\textrm{obs}}$ around
foreground positions.

We can additionally define the cross-component of shear, $\gamma_{\times}$,
rotated by $45$ degrees with respect to $\gamma_{\textrm{t}}$. These two
components can be combined to form the complex shear
$\gamma=\gamma_{\textrm{t}}+\textrm{i}\gamma_{\times}$. The tangential shear
$\gamma_{\textrm{t}}$ is identified as $E$-mode, induced via gravitational
lensing, whereas the cross-component $\gamma_{\times}$ indicates the parity-
odd $B$-mode.

We denote $\delta\gamma_{\textrm{t}}$ with

$\delta\gamma_{\textrm{t}}=\gamma^{\rm
obs}_{\textrm{t}}-\gamma^{S}_{\textrm{t}},$ (6)

the difference between the estimated and true shear. Inserting Eq. (1) into
Eq. (5), we find for the residual tangential shear component to be

$\displaystyle\delta\gamma_{\textrm{t}}=\left\langle\varepsilon^{\textrm{s}}_{\textrm{t}}n\right\rangle+$
$\displaystyle\left\langle\frac{T^{\textrm{psf}}}{T}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\gamma_{\textrm{t}}n\right\rangle-\left\langle\frac{T^{\textrm{psf}}}{T}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle$
$\displaystyle-\left\langle\frac{T^{\textrm{psf}}}{T}\delta\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle+\left\langle\alpha\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle.$
(7)

The correlators can be further expanded under some assumptions, as follows.
First, we can safely assume that the intrinsic galaxy ellipticity is
uncorrelated with foreground number density. Next, we assume that the PSF
model uncertainties are not correlated to the shear. Third, we separate out
the prefactors $T^{\textrm{psf}}/T$ and $\alpha$, following Rowe (2010). This
yields

$\displaystyle\delta\gamma_{\textrm{t}}=$
$\displaystyle\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left\langle\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right\rangle\gamma^{S}_{\textrm{t}}-\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left\langle\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle$
$\displaystyle-\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left\langle\delta\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle+\alpha\left\langle\varepsilon_{\textrm{t}}^{\textrm{psf}}n\right\rangle.$
(8)

We define three new cross-correlation functions in analogy to the
$\rho$-statistics introduced for cosmic shear in Jarvis et al. (2016). These
functions are

$\displaystyle\lambda_{1}$
$\displaystyle=\left\langle\varepsilon_{\textrm{t}}^{\textrm{psf}}\,n\right\rangle;$
$\displaystyle\lambda_{2}$ $\displaystyle=\left\langle\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\varepsilon_{\textrm{t}}^{\textrm{psf}}\,n\right\rangle;$
$\displaystyle\lambda_{3}$
$\displaystyle=\left\langle\delta\varepsilon_{\textrm{t}}^{\textrm{psf}}\,n\right\rangle.$
(9)

With that, we write the shear difference as

$\delta\gamma_{\textrm{t}}=\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left\langle\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right\rangle\gamma^{S}_{\textrm{t}}+\alpha\lambda_{1}-\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left(\lambda_{2}+\lambda_{3}\right).$
(10)

The first term on the right-hand side, the prefactor of $\gamma_{\rm t}^{S}$,
is a PSF-induced multiplicative bias. The remaining three terms are PSF-
induced additive biases, expressed as the $\lambda$-statistics, i.e., the
correlations between the PSF and foreground positions. We use the
TreeCorr444https://pypi.org/project/TreeCorr/ package (Jarvis, 2015) to
calculate the $\lambda$ statistics. Their error bars are estimated using the
jackknife method.

In the case of cosmic shear, the multiplicative term is usually ignored, and
only the additive terms to the shear two-point correlation function $\xi_{+}$
are kept. In the following, we will consider both contributions using two
different approaches. The first approach is indicated in Eq. (10), which
assumes the knowledge of $\gamma^{S}_{\textrm{t}}$. However,
$\gamma^{S}_{\textrm{t}}$ is not a direct observable. Therefore, we adopt a
theoretical model to derive $\gamma^{S}_{\textrm{t}}$ given a sample of
foreground tracers $S$ (see Sect. 3.3 for details). For the second approach,
we calculate $\delta\gamma_{\textrm{t}}$ from the observation by combing Eqs.
(6) and (10). Consequently, $\delta\gamma_{\textrm{t}}$ can be written as

$\delta\gamma_{\textrm{t}}=\gamma_{\textrm{t}}^{\rm
obs}-\frac{\gamma_{\textrm{t}}^{\rm
obs}-\alpha\lambda_{1}+\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left(\lambda_{2}+\lambda_{3}\right)\
}{1+\left\langle\frac{T^{\textrm{psf}}}{T}\right\rangle\left\langle\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right\rangle},$ (11)

where $\gamma_{\textrm{t}}^{\rm obs}$ is measured by using the shear catalogue
around foreground tracers $S$ (see Section 3.4 for the calculation of
$\gamma_{\rm t}^{\rm obs}$). In the limit where PSF-induced multiplicative
bias vanishes, this expression reduces to Eq. (10).

We refer to the two expressions Eqs. (10) and (11) as the theory-based and
observation-based approach, respectively. We denote the corresponding residual
tangential shear as $\delta\gamma_{\rm t}^{\rm theory}$ and $\delta\gamma_{\rm
t}^{\rm obs}$, respectively.

###  3.3 Theory-based approach and its impact on constraining halo mass

The key ingredient of the theory-based approach, Eq. (10), in calculating the
residual tangential shear $\delta\gamma_{\rm t}^{\rm theory}$ is the
derivation of the model prediction $\gamma^{S}_{\textrm{t}}$. We use an
analytical method based on three assumptions. All galaxies from the foreground
sample described in Sect. 2.2.1 are central galaxies. We assume that these
galaxies follow the stellar mass - halo mass relation (SHMR) from Kravtsov et
al. (2018). We calculate the average stellar mass of the galaxy sample and
interpolate the SHMR to infer the average halo mass of the sample. We also
assume that the halo is described by the Navarro–Frenk–White (Navarro et al.,
1997, NFW) density profile and define the halo mass as the total mass within a
spherical region of radius $r_{200{\rm m}}$, inside which the mean mass
density is equal to $200$ times the mean matter density of the Universe. We
further model the halo mass-halo concentration relation with the fit from
Bhattacharya et al. (2013). With these assumptions, we can use the analytical
equations from Yang et al. (2006) to derive the excess surface density,
$\Delta\Sigma_{\textrm{t}}$.

The relation between $\Delta\Sigma_{\textrm{t}}$ and $\gamma^{S}_{\textrm{t}}$
is given by the following equations

$\displaystyle\Delta\Sigma_{\textrm{t}}$ $\displaystyle=\Sigma_{\rm crit}\
\gamma^{S}_{\textrm{t}};$ $\displaystyle\Sigma_{\rm crit}(z_{\rm l},z_{\rm
s})$ $\displaystyle=\frac{c^{2}}{4\pi G}\frac{D_{\rm s}}{D_{\rm l}D_{\rm
ls}},$ (12)

where $c$ is the speed of light, $G$ is the gravitational constant, $D_{\rm
l}$, $D_{\rm s}$ and $D_{\rm ls}$ are the angular diameter distances to the
lens, to the source and between the lens and source, respectively. We adopt
the mean redshift of the foreground galaxy sample to calculate $z_{\rm l}$ and
the corresponding $D_{\rm l}$. Instead of using a single source redshift value
for $z_{\rm s}$, we use the redshift distribution $n(z)$ of the UNIONS
catalogue to derive $\Sigma_{\rm crit}$ via

$\Sigma^{-1}_{\rm crit}(z_{\rm l})=\int_{z_{\rm l}}^{z_{\rm lim}}\Sigma_{\rm
crit}^{-1}(z_{\rm l},z_{\rm s})\,n(z_{\rm s})\,{\rm d}z_{\rm s},$ (13)

where $z_{\rm lim}$ represents the maximum redshift value in the redshift
distribution of the UNIONS catalogue.

Taken together, we can derive $\gamma^{S}_{\textrm{t}}$ and hence
$\delta\gamma_{\rm t}^{\rm theory}$. The above approaches were referred to as
the Yang-halo model. Finally, we combine $\delta\gamma_{\rm t}^{\rm theory}$
and $\gamma^{S}_{\textrm{t}}$ to infer $\hat{\gamma}^{\rm obs}_{\rm t}$, which
represents the estimator of the observed tangential shear after the true
tangential shear has been contaminated by PSF-induced systematics.

To investigate the influence of the $\delta\gamma_{\rm t}^{\rm theory}$ on the
estimated halo mass, we constrain the halo mass by fitting $\hat{\gamma}^{\rm
obs}_{\rm t}$ using a modified version of the Yang-halo model: instead of
using the halo mass - halo concentration relation from Bhattacharya et al.
(2013), we set these two parameters as the free parameters in the model. The
priors of these parameters are chosen to be flat, with the logarithm of the
halo mass (log$M_{\rm h}$/M⊙) in the range of [10.0, 16.0] and halo
concentration in the range of [1.0, 16.0]. To constrain these parameters, we
use $emcee$555https://emcee.readthedocs.io/en/stable/ (Foreman-Mackey et al.,
2013) to run a Markov Chain Monte Carlo (MCMC). The likelihood function is set
to be

$\ln\mathcal{L}=-{1\over 2}\left(\hat{\gamma}^{\rm
obs}_{\textrm{t}}-\gamma^{\rm
model}_{\textrm{t}}\right)^{\textrm{T}}C^{-1}\left(\hat{\gamma}^{\rm
obs}_{\textrm{t}}-\gamma^{\rm model}_{\textrm{t}}\right),$ (14)

where $\gamma^{\rm model}_{\textrm{t}}$ is the model prediction and $C^{-1}$
is the inverse of the covariance matrix.

In the following, the constrained halo masses are the medium values of the
posteriors, and the error bars correspond to $16\%$ and $84\%$ of the
posterior distributions.

###  3.4 Observation-based approach and its impact on constraining halo mass

To apply the observation-based approach to study the PSF-induced systematics,
we need to obtain the observed tangential shear $\gamma^{\rm obs}_{\rm t}$ for
a galaxy sample, as shown in Eq. (11). We use the UNIONS shear catalogue to
calculate $\gamma^{\rm obs}_{\rm t}$ based on galaxy samples with different
properties. The ellipticities of galaxies in UNIONS are calibrated, therefore,
the following equation is accurate enough to perform the calculation

$\gamma^{\rm obs}_{\textrm{t}}=\frac{\sum_{i}^{N}w_{i}\,\varepsilon_{i,\rm
t}}{\sum_{i}^{N}w_{i}},$ (15)

where $w_{i}$ and $\varepsilon_{i,\textrm{t}}$ are the weight and the
ellipticity along the tangential direction around the lens of the source
galaxy with index $i$, respectively. The sum is carried out over all suitable
source galaxies. We calculate $\varepsilon_{\textrm{t}}$ as

$\varepsilon_{\textrm{t}}=-\varepsilon_{1}\cos 2\theta-\varepsilon_{2}\sin
2\theta,$ (16)

where $\varepsilon_{\textrm{1}}$ and $\varepsilon_{\textrm{2}}$ are two
ellipticity components of the source galaxy, and $\theta$ is the angle between
the line connecting the lens and the source and the direction of increasing
right ascension.

The source galaxies in UNIONS do not have photometric redshifts, and we use
all sources in our calculations. Some source galaxies are thus in front of the
“foreground” galaxies and will dilute the observed tangential shear. Eq. (13)
models this dilution correctly.

Based on the UNIONS shear catalogue and a given foreground galaxy sample, we
apply TreeCorr to calculate its $\gamma^{\rm obs}_{\textrm{t}}$, the error
bars are estimated using the jackknife method.

We further quantify the impact of the observation-based approach on halo mass
estimates. More specifically, we investigate the impact of PSF-induced
systematics on black-hole-mass - halo-mass relation. Li et al. (2024) recently
investigated this relation for type I AGNs using the UNIONS shear catalogue;
we adopt the method in Li et al. (2024) to calculate $\gamma^{\rm
obs}_{\textrm{t}}$ and therefore $\Delta\Sigma_{\textrm{t}}$. Then, we derive
$\delta\gamma_{\rm t}^{\rm obs}$ by Eq. (11). The
$\hat{\gamma}^{S}_{\textrm{t}}$ was obtained by combining $\gamma^{\rm
obs}_{\textrm{t}}$ and $\delta\gamma_{\rm t}^{\rm obs}$, which represents the
estimator of the true tangential shear without PSF-induced systematics.

We use the halo model introduced in Guzik & Seljak (2002) and applied in Li et
al. (2024) to fit the excess surface mass density $\Delta\Sigma_{\textrm{t}}$
and $\Sigma_{\rm crit}\hat{\gamma}^{S}_{\textrm{t}}$ of the AGN sample,
respectively. Our model contains contributions of central host halo,
$\Delta\Sigma_{\textrm{cen}}$, and satellite host halo,
$\Delta\Sigma_{\textrm{sat}}$. It further includes a baryonic term from the
galaxy, $\Delta\Sigma_{\rm b}$, and a two-halo term, $\Delta\Sigma_{\rm 2h}$.
The overall model is

$\Delta\Sigma_{\textrm{t}}=(1-f_{\rm sat})\ \Delta\Sigma_{\rm cen}+f_{\rm
sat}\ \Delta\Sigma_{\rm sat}+\Delta\Sigma_{\rm b}+\Delta\Sigma_{\rm 2h},$ (17)

where $f_{\rm sat}$ describes the fraction of satellite galaxies in the
sample.

We treat the baryonic contribution of the galaxy as a point mass and consider
the stellar mass as a free parameter in the model. The two-halo term is
calculated by combining the halo bias model from Tinker et al. (2010) and the
linear matter-matter correlation function from COLOSSUS (Diemer, 2018). The
model contains a total of three free parameters, which are the halo mass
($M_{\textrm{h}}$), the stellar mass ($M_{\ast}$), and the satellite fraction
($f_{\rm sat}$), respectively. We constrain these parameters by running MCMC
with the $emcee$ package. The likelihood function is the same as Eq. (14). We
refer to the above model as the AGN-halo model. See Li et al. (2024) for more
details on calculating and modelling the weak lensing signals of the AGN
samples.

###  3.5 Halo mass change and tangential shear residuals

The average halo mass of a galaxy sample derived from the tangential shear
$\hat{\gamma}^{\rm obs}_{\rm t}$ as described in Sect. 3.3 might be biased by
PSF-induced systematics. To visualize this potential change in halo mass, we
define a halo mass $M_{\textrm{h}}^{P}$ modified by a percentage $P$ as

$M_{\textrm{h}}^{P}=M_{\textrm{h}}\cdot(1+P).$ (18)

where $M_{\textrm{h}}$ corresponds to the halo mass derived from the SHMR. In
this way, we can evaluate the percentage change in halo mass corresponding to
PSF-induced residual tangential shear $\delta\gamma_{\textrm{t}}(\theta)$ at
different angular scales $\theta$. As before, we use the Yang-halo model to
generate the tangential shears corresponding to different
$M_{\textrm{h}}^{P}$.

##  4 Results

In this section, we present our results on the impact of PSF-induced
multiplicative and additive biases on galaxy-galaxy lensing.

###  4.1 PSF-induced multiplicative bias

As introduced in Eq. (10), the multiplicative bias associated with the PSF
manifests itself as a prefactor. Table 1 shows the average prefactors for both
the UNIONS and DES catalogues. The PSF leakage $\alpha$ is typically computed
via ratios of galaxy- and PSF auto- and cross-correlations. We quote its
values from Guinot et al. (2022) and Gatti et al. (2021a) for the UNIONS and
DES catalogues, respectively.

Table 1: Prefactors and PSF leakage $\alpha$ for different catalogues. Parameter |  UNIONS |  DES
---|---|---
$\left\langle\frac{T^{\rm psf}}{T}\right\rangle$ |  $1.3189$ |  $1.014$
$\left\langle\frac{\delta T^{\rm psf}}{T^{\rm psf}}\right\rangle$ |  $-0.0032$ |  $0.0003$
$\left\langle\frac{T^{\rm psf}}{T}\right\rangle\left\langle\frac{\delta T^{\rm psf}}{T^{\rm psf}}\right\rangle$ |  $-0.0042$ |  $0.0003$
$\alpha$ |  $0.033$ |  $0.001$

The PSF-induced multiplicative biases in both catalogues are lower than 1$\%$,
indicating that their PSFs are well calibrated. This bias is thus smaller than
the typical residual multiplicative bias of current weak-lensing surveys,
which is of order $1$–$2\%$ (e.g. Giblin et al., 2021; MacCrann et al., 2022).
We conclude that the PSF uncertainty is not a major contributor to the
multiplicative bias.

###  4.2 PSF-induced additive bias

To examine the overall intrinsic bias levels of the UNIONS and DES catalogues,
we randomly generate $2,000,000$ positions in the survey footprints. We use
these random samples as the position catalogue $n$ in Eq. 9. This gives us an
indication of a correlation between the PSF and the footprint mask.

Figure 1: Comparison of PSF-induced additive bias for the DES and UNIONS
catalogues based on random positions. The bias for DES (UNIONS) is shown in
red (blue). The error bars correspond to the 1$\sigma$ uncertainty using the
jackknife method.

We show the residual tangential shear (additive terms of Eq. 10) in Fig. 1.
The PSF-induced additive biases for UNIONS and DES are broadly consistent with
zero within the margin of error. The overall amplitude of PSF-induced additive
biases is below $2\cdot 10^{-5}$ for both surveys, with DES displaying a lower
level of systematic contributions. The consistency with zero on most angular
scales indicates an accurate PSF correction for both surveys. However, in
regimes where the tangential shear induced by a density tracer is below a few
times of $10^{-6}$, the gravitational lensing shear might be significantly
affected by PSF errors.

Previous weak-lensing-based studies have investigated the relation between
galaxy stellar mass and halo mass and have found a positive correlation
between these two quantities. The dispersion in this relation is however
large. In addition, at fixed stellar mass, quenched galaxies tend to reside in
more massive halos compared to star-forming galaxies (Mandelbaum et al., 2016;
Bilicki et al., 2021; Zhang et al., 2021, 2024). Here, we investigate whether
PSF-induced systematics can bias the relation between galaxy properties and
halo mass.

Galaxy samples have selection functions that depend on their properties, e.g.
size, magnitude or star-formation rate. The galaxy selection can vary with
observing conditions such as seeing, image quality, local star and galaxy
density. Since these factors also depend on the PSF, the galaxy-galaxy lensing
PSF systematics discussed in this work can vary between different foreground
samples.

We calculate the dependence of PSF-induced additive bias on the foreground
galaxy samples with different stellar masses as well as SFR (see Sect. 2.2.1
for the sample selections). We focus on the UNIONS catalogue and its large
overlap with SDSS/eBOSS.

Figure 2: The PSF-induced additive bias for the central galaxy samples with
different stellar mass and SFR. The three panels correspond to the three
stellar-mass bins. In each panel, black circles show all galaxies in the
stellar-mass bin, while the red diamonds (blue stars) correspond to the
quenched (star-forming) subsample. Weak-lensing measurements use the UNIONS
data.

The PSF-induced additive biases based on different stellar-mass samples are
shown in Fig. 2. In each stellar-mass bin, PSF-induced bias is consistent with
zero on most angular scales; on small scales (below $1.3$ arcmin) there is a
large scatter. This indicates that PSF-induced systematics do not have a
significant effect on the galaxy-galaxy lensing of these samples.

Also shown are PSF-induced additive biases for subsamples with different SFR.
Overall, the biases of the quenched and star-forming subsamples are consistent
within the error bars. We do not find a significant dependence of PSF-induced
additive bias on stellar mass and SFR.

###  4.3 The overall PSF-induced systematics on weak-lensing measurements

We now turn towards the overall effect of PSF-induced biases on weak lensing
measurements and halo mass estimates. We used stellar-mass samples described
in section 4.2 as density tracers and the UNIONS star and galaxy catalogues as
shape catalogues.

We first applied the theory-based approach introduced in Sect. 3.3 to predict
the theoretical tangential shear $\gamma_{\textrm{t}}^{S}$ of the foreground
galaxy sample. With that we calculated the residual tangential shear
$\delta\gamma_{\rm t}^{\rm theory}$ using Eq. (10). The observed shear
$\hat{\gamma}_{\rm t}^{\rm obs}$ was computed by combining
$\gamma_{\textrm{t}}^{S}$ and $\delta\gamma_{\rm t}^{\rm theory}$, which now
includes PSF-induced systematics. We then fitted to the $\hat{\gamma}_{\rm
t}^{\rm obs}$ by the model described in Sect. 3.3. By comparing the best-fit
halo mass with the corresponding halo mass of $\gamma_{\textrm{t}}^{S}$, we
can quantify the difference in halo mass due to PSF-induced systematics. The
results are shown in the upper panels of Fig. 3.

Figure 3: The effect of PSF-induced systematics in UNIONS weak lensing by the
theory-based and observation-based approaches. Different columns correspond to
the results of galaxy samples in different stellar mass bins. In each upper
panel, the blue solid line is the theoretical $\gamma_{\textrm{t}}^{S}$, while
the red circles with error bars and the red dashed line are the ’observed’
tangential shear ($\hat{\gamma}_{\rm t}^{\rm obs}$) and model fitting,
respectively. Their corresponding halo masses are also labelled in the panel.
In each lower panel, the blue and red symbols with error bars correspond to
the observed tangential shear ($\gamma_{\textrm{t}}^{\rm obs}$) calculated
from the UNIONS shear catalogue and the estimator of the true tangential shear
($\hat{\gamma}_{\textrm{t}}^{S}$), respectively.

The tangential shear is most affected by the PSF in the lowest stellar mass
bin, and mainly on small scales. The corresponding halo mass is biased by up
to 18$\%$. For higher stellar masses, the measured halo masses deviate by
11$\%$ and 1$\%$, respectively.

Next, we applied the observation-based approach. We calculated the observed
tangential shear $\gamma^{\rm obs}_{\rm t}$ for the above galaxy samples,
computed $\delta\gamma_{\rm t}^{\rm obs}$ using Eq. (11) and
$\hat{\gamma}^{S}_{\rm t}$ from Eq. (6). The comparisons are shown in the
lower panels of Fig. 3. Similar to the theory-based approach, PSF-induced
systematics mainly affect the results in the low stellar mass range at a level
which is smaller than the statistical errors.

We compared PSF-induced systematics from the two approaches and also the
residual tangential shear due to halo mass change (For calculating the halo
mass-induced residual tangential shear, see Sect. 3.5) in Fig. 4. The theory-
and observation-based approaches yield similar results. As discussed in Sect.
4.2, PSF-induced multiplicative bias is very small so that the residual
tangential shear $\delta\gamma_{\rm t}$ is dominated by the $\lambda$
statistics (PSF-induced additive bias). We further compared the halo mass-
induced residual tangential shear with the PSF-induced residual tangential
shear. On individual angular scales, PSF-induced systematics can impact the
tangential shear corresponding to a change in halo mass significantly
exceeding the average bias of the halo mass. Such residuals can be removed
using the methods described in this work.

Figure 4: Comparison of $\delta\gamma_{\rm t}^{\rm theory}$ and
$\delta\gamma_{\rm t}^{\rm obs}$ and the halo mass-induced residual tangential
shears from UNIONS. Different panels correspond to the results in different
stellar mass bins. In each panel, the blue and red symbols with error bars
correspond to the $\delta\gamma_{\rm t}$ from the theory-based and
observation-based approaches, respectively. Shaded regions of different colors
correspond to the different degrees of the halo mass-induced shear residuals.
The subplot in each panel is the zoomed-in view of the large-scale signal.

Our analysis suggests that the weak-lensing measurements for the low-mass
galaxies are facing potential challenges from PSF-induced systematics. In
addition, the galaxy number density of these galaxies is low. The current
survey depths are insufficient to detect more low-mass central galaxies.
Together, these drawbacks make the accuracy of the weak-lensing measurements
at the low-mass end always low. The error bars in the lensing signals are more
significant than the PSF-induced systematics we detect here, making PSF-
induced systematics less important. However, our work may inspire future
studies using deeper data, for which correcting for PSF-induced systematics
may become important.

###  4.4 Impact of PSF-induced systematics on black-hole-mass - halo-mass
relation

Recently, Li et al. (2024) studied the black-hole-mass - halo-mass relation
based on SDSS AGNs as foreground samples and UNIONS as background shear
catalogue. We now quantify the impact of PSF-induced systematics on the halo
mass estimates for the type I AGN samples.

We begin by constructing the foreground samples as in Li et al. (2024) (see
Section 2.2.2 for the sample selections). We then calculate the residual
tangential shears for these samples using the observation-based approach (Eq.
(11)). The results with and without the inclusion of PSF-induced systematics
are shown in Fig. 5.

The weak-lensing tangential shears in the three black-hole mass bins are
weakly affected by PSF-induced systematics on scales larger than $2$ arcmin.
The largest differences appear in the low-black-hole-mass sample. We use the
AGN-halo model (see Section 3.4) to measure the halo masses for the three AGN
samples, their black-hole-mass - halo-mass relations are shown in the lower-
right panel of Figure 5. Ignoring PSF-induced biases leads to a slight but
systematic underestimation of the halo masses in the low and medium-black-hole
mass samples, with the degree of deviation of their halo masses being $15\%$
and $11\%$, respectively. These halo mass deviations are however smaller than
the statistical errors.

Figure 5: The effect of PSF-induced systematics on the black-hole-mass - halo-
mass relation as measured with UNIONS. The upper-left, upper-right, and lower-
left panels correspond to the low, medium, and high-black-hole mass AGN
samples, respectively. In each panel, blue (red) symbols show the raw measured
$\gamma_{\rm t}^{\rm obs}$ (PSF-induced systematics corrected
$\hat{\gamma}_{\rm t}^{S}$) tangential shear; lines correspond to the best-
fit, with the corresponding best-fit halo mass indicated in the panel. The
lower-right panel shows the black-hole-mass - halo-mass relation using the raw
measured (PSF-induced systematics corrected) halo masses in blue (red).

##  5 Discussion

###  5.1 Motivation for the $\lambda$-statistics

In analyses of weak-lensing correlations with density tracers, e.g. weak
lensing of clusters or galaxy-galaxy lensing, null tests and diagnostics of
additive biases are used routinely. These are, for example, the mean
tangential shear around non-tracers such as random points, stars, or
coordinates relative to the CCDs, and the cross-component of shear around
tracers or non-tracer points Mandelbaum et al. (2005). These correlations are
either found to be consistent with zero or very small and then discarded. Or
they are subtracted from the tangential shear to remove this potential
systematic effect from the data.

The $\lambda$-statistics Eq. (10) introduced here are motivated in a similar
way. They extend those previous works by quantifying the contribution of PSF-
induced systematics for lensing by foreground density tracers. In particular,
our formalism allows the propagation of PSF errors to derived physical
quantities. Such quantities are, for example, the average halo mass of
foreground galaxy samples, as illustrated in this paper. Other examples not
studied here are the halo concentration, galaxy bias parameters, cluster mass
calibration for cosmology, the intrinsic galaxy alignment amplitude, and void
density profiles to name just a few.

We argue that residuals quantified by the $\lambda$-statistics are present in
all weak-lensing measurements of density tracers. This is due to the spatially
varying PSF within the field of view, a phenomenon that can lead to a
correlation between the PSF and sky position.

Some of the PSF residuals can in principle be removed from the tangential
shear from a density tracer by subtracting the shear around random points.
However, this does now remove correlations between PSF and tracer number
density. This was noted in Mandelbaum et al. (2005). The $\lambda$-statistics
quantify this correlation, and correspond to the term $\langle\delta
n\,\gamma_{\textrm{sys}}\rangle$ in Eq. (26) of Mandelbaum et al. (2005).

###  5.2 Interpretation of the $\lambda$-statistic terms

We now provide an intuitive interpretation of the three $\lambda$-statistic
terms. The function $\lambda_{1}$ is qualitatively different from the other
two, $\lambda_{2}$ and $\lambda_{3}$, which can be grouped together.

The first function, $\lambda_{1}$, quantifies the correlation between
tangential PSF ellipticity and foreground sample position. The function
$\lambda_{1}$ is proportional to the PSF ellipticity, not its residual, and
can be of order $10\%$. For example, the requirement on the PSF ellipticity
for the Euclid space mission is $0.15$ (Laureijs et al., 2011); in UNIONS the
focal-plane PSF ellipticity averaged over atmospheric fluctuations is of order
$0.05$ (Guinot et al., 2022). Multiplied with a PSF leakage $\alpha$ at the
percent-level (Giblin et al., 2021; Gatti et al., 2021b, e.g.) yields an
additive PSF-induced bias, nal̈ively of order $10^{-4}$ – $10^{-3}$, which is
indeed in the range of sought-after weak-lensing shear correlations.

For the $\lambda_{1}$-term to impact the measured tangential shear, the PSF
ellipticity needs to display a spatial pattern that is correlated to the
positions of the foreground sample. An example is the commonly observed
circular PSF pattern in the focal plane and the foreground sample being a
cluster (sample) near the image centre. Note that the $\lambda_{1}$ term is
independent of the quality of the PSF model.

The second function, $\lambda_{2}$, weighs the PSF ellipticity-foreground
position correlation by the relative PSF size residual. This function shows
some similarity to
$\rho_{5}=\langle\varepsilon^{\textrm{psf}}(\varepsilon^{\textrm{psf},\ast}\delta
T^{\textrm{psf}}/T^{\textrm{psf}})\rangle$ for cosmic shear, see App. A. The
third function, $\lambda_{3}$, is the correlation of PSF ellipticity residuals
with foreground positions, having a resemblance in
$\rho_{2}=\langle\varepsilon^{\textrm{psf}}\delta\varepsilon^{\textrm{psf},*}\rangle$.
Both the second and third $\lambda$-statistics contribute to galaxy-galaxy
lensing in the presence of PSF residuals that are spatially correlated to the
positions of the foreground sample. Such correlations can be induced where
detection and selection of the density tracer populations depend on the PSF.
Examples where PSF residuals can affect the tracer number density are:

  * •

seeing and depth variations;

  * •

star-galaxy separation, cross-contamination of both samples;

  * •

detector effects such as charge transfer inefficiency (CTI)

  * •

survey strategy, e.g. scan direction, fiber placement

  * •

photometry and photometric redshifts;

  * •

local object density (crowded fields) and extinction

In all those cases, spurious correlations between PSF residuals and tracer
positions might be introduced that will be captured by $\lambda_{2}$ and
$\lambda_{3}$. Some of those effects such as varying seeing will only give
rise to PSF - number density correlations if the lensing and density tracer
catalogues originate from the same data. Others such as crowded fields are
intrinsic. The contamination of the PSF sample by density tracers objects and
vice versa can induce systematic biases studied here even if they are selected
from different surveys if the target sample consists of small objects with
size close to the PSF in either survey.

###  5.3 Comparison to the $\rho$-statistics for cosmic shear

Contrary to the $\rho$-statistics that were introduced to quantify PSF
systematics for cosmic shear, the $\lambda$-statistics do not involve
correlations between PSF uncertainties. Instead, they quantify the correlation
of PSF residuals with respect to foreground positions.

The conditions for non-vanishing $\lambda$-statistics are stronger than for
$\rho\neq 0$. In both cases, we need a non-perfect PSF model or PSF leakage
that displays a spatial pattern. In addition, for the $\lambda$-statistics to
be significant, the PSF residual pattern also needs to be correlated to the
positions of the density tracer in question.

Related to this is the second difference between $\lambda$ and $\rho$: Cosmic
shear involves second-order correlations of the statistical homogeneous and
isotropic galaxy shear field. The $\rho$-statistics are evaluated not at
galaxy but at star positions. The star or PSF ellipticity field can be assumed
to be statistically homogeneous and isotropic. This allows for the addition of
the measured $\rho$ and $\xi_{+}$.

Galaxy-galaxy lensing however is the cross-correlation between two correlated
fields, background shear and foreground density. The second-order correlation
estimators are not invariant under translation or rotation of only one of the
fields. To capture PSF systematic correlations we need to use the actual
galaxy positions.

##  6 Summary

This paper introduces PSF-induced systematics for weak-lensing cross-
correlations with foreground density tracers. We developed a theoretical
framework for quantifying PSF-induced systematics which contributes to both
the multiplicative and the additive bias in weak-lensing tangential shear. In
particular, we introduce three scale-dependent ellipticity-position
correlation functions dubbed “$\lambda$-statistics” to characterize PSF-
induced additive biases. These correlation functions can be computed from the
information about ellipticities and size residuals of the PSF, and the
positions of the foreground sample. In this framework, PSF-induced systematics
propagate to physical parameters of the density tracer sample measured from
weak lensing.

The PSF-induced multiplicative bias is a prefactor in the residual tangential
shear. We use the UNIONS and DES weak-lensing catalogues to calculate PSF-
induced multiplicative bias, which we find to be $-0.0042$ and $0.0003$,
respectively. This is a subdominant contribution to the overall multiplicative
bias of current weak-lensing surveys.

We quantify the impact of PSF-induced systematics on the halo masses of two
cases of the foreground galaxy samples. The first case is a sample of central
galaxies from the Yang group catalogue (Yang et al., 2005, 2007), which we
split into subsamples by stellar mass and star-formation rate. The PSF-induced
bias acts mainly on small angular scales and low stellar masses. The largest
resulting bias in the weak-lensing derived halo mass is $18\%$ for the
subsample in the stellar mass range $9\leq\log M_{\ast}/M_{\odot}<9.5$.

The second case is the type I AGN sample, used in Li et al. (2024) to estimate
the black-hole-mass - halo-mass relation. We calculate the impact of PSF-
induced systematics on the halo mass estimation. Similar to the previous case,
PSF-induced systematics is most important on small scales and absent on large
scales. Without accounting for PSF-induced systematics, the halo mass is
underestimated at low black-hole masses.

Our proposed framework can be used for quality-checking of weak-lensing -
density cross-correlations. It is straightforward to extend the formalism to
weak-lensing-like observables and estimators such as intrinsic alignments of
galaxies.

###### Acknowledgements.

This work was made possible by utilizing the CANDIDE cluster at the Institut
d’Astrophysique de Paris, which was funded through grants from the PNCG, CNES,
DIM-ACAV, and the Cosmic Dawn Center and maintained by S. Rouberol. We are
honored and grateful for the opportunity of observing the Universe from
Maunakea and Haleakala, which both have cultural, historical and natural
significance in Hawai’i. This work is based on data obtained as part of the
Canada-France Imaging Survey, a CFHT large program of the National Research
Council of Canada and the French Centre National de la Recherche Scientifique.
Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and CEA Saclay, at the Canada-France-Hawaii Telescope (CFHT) which is operated
by the National Research Council (NRC) of Canada, the Institut National des
Science de l’Univers (INSU) of the Centre National de la Recherche
Scientifique (CNRS) of France, and the University of Hawaii. This research
used the facilities of the Canadian Astronomy Data Centre operated by the
National Research Council of Canada with the support of the Canadian Space
Agency. This research is based in part on data collected at Subaru Telescope,
which is operated by the National Astronomical Observatory of Japan. Pan-
STARRS is a project of the Institute for Astronomy of the University of
Hawai’i, and is supported by the NASA SSO Near Earth Observation Program under
grants 80NSSC18K0971, NNX14AM74G, NNX12AR65G, NNX13AQ47G, NNX08AR22G,
80NSSC21K1572 and by the State of Hawai’i. This work was supported in part by
the Canadian Advanced Network for Astronomical Research (CANFAR) and Compute
Canada facilities. This work was funded by the China Scholarship Council
(CSC).

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##  Appendix A $\rho$-statistics for cosmic shear

Two correlation functions between PSF ellipticity and its residuals were
introduced in Rowe (2010). The purpose of those diagnostics was to distinguish
between different PSF models in a quantitative way In particular, the second
correlation function, $D_{2}$, corresponding to $\rho_{2}$ in the Jarvis et
al. (2016) notation, is an indication of over-fitting. In that case the PSF
model fits part of the noise, which creates a correlation between the observed
PSF ellipticity and the model (residual).

These correlations were generalised in Jarvis et al. (2016). This work
rederived the two original Rowe (2010) diagnostics and three additional
functions by considering the two-point correlator of Eq. (1), which is an
estimator of the shear two-point correlation function,

$\hat{\xi}_{+}(\theta)=\left\langle\varepsilon^{\textrm{obs}}\varepsilon^{\textrm{obs},\ast}\right\rangle(\theta),$
(19)

into which Eq. (3) is inserted. The $\rho$-statistics are defined as second-
order correlation functions

$\displaystyle\rho_{1}(\theta)$
$\displaystyle=\left\langle\delta\varepsilon^{\textrm{psf}}\,\delta\varepsilon^{\textrm{psf},\ast}\right\rangle(\theta);$
$\displaystyle\rho_{2}(\theta)$
$\displaystyle=\left\langle\varepsilon^{\textrm{psf}}\,\delta\varepsilon^{\textrm{psf},\ast}\right\rangle(\theta);$
$\displaystyle\rho_{3}(\theta)$
$\displaystyle=\left\langle\left(\varepsilon^{\textrm{psf}}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right)\left(\varepsilon^{\textrm{psf},\ast}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right)\right\rangle(\theta);$
$\displaystyle\rho_{4}(\theta)$
$\displaystyle=\left\langle\delta\varepsilon^{\textrm{psf}}\left(\varepsilon^{\textrm{psf},\ast}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right)\right\rangle(\theta);$
$\displaystyle\rho_{5}(\theta)$
$\displaystyle=\left\langle\varepsilon^{\textrm{psf}}\left(\varepsilon^{\textrm{psf},\ast}\frac{\delta
T^{\textrm{psf}}}{T^{\textrm{psf}}}\right)\right\rangle(\theta).$ (20)

With some prefactors that depend on the ratio between PSF and galaxy size and
the leakage parameter $\alpha$, the five $\rho$-statistic functions are
written as additive terms to the cosmological shear two-point correlation
function $\xi_{+}$, estimated by the correlation of observed galaxy
ellipticities.

The choice to leave the size ratio and leakage parameters out of the
$\rho$-statistics is conveniently done such that the quantities to correlate
only depend on quantities that are available at star positions. In Rowe (2010)
the size ratio and leakage parameter are computed independently from the
$\rho$-statistics.

The ensemble average of the $\rho$-statistics is thus estimated using star
position, whereas the two-point correlation function $\xi_{+}$ is estimated by
averaging over galaxy positions. Both can be added if both stars and galaxies
randomly sample the underlying PSF and cosmic-shear fields, respectively.

The case of galaxy-galaxy lensing is more complex since the
$\lambda$-statistics involve star - galaxy cross-correlations.