# Unleashing cosmic shear information with the tomographic weak lensing PDF

Lina Castiblanco1,⋆ Cora Uhlemann1,2 Joachim Harnois-Déraps1 Alexandre
Barthelemy3 1 School of Mathematics, Statistics and Physics, Newcastle
University, Herschel Building, NE1 7RU Newcastle-upon-Tyne, U.K.  2 Fakultät
für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
3 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität
München,
Scheinerstraße 1, 81679 München, Germany

###### Abstract

In this work, we demonstrate the constraining power of the tomographic weak
lensing convergence PDF for StageIV-like source galaxy redshift bins and shape
noise. We focus on scales of $10$ to $20$ arcmin in the mildly nonlinear
regime, where the convergence PDF and its changes with cosmological parameters
can be predicted theoretically. We model the impact of reconstructing the
convergence from the shear field using the well-known Kaiser-Squires
formalism. We cross-validate the predicted and the measured convergence PDF
derived from convergence maps reconstructed using simulated shear catalogues.
Employing a Fisher forecast, we determine the constraining power for
$(\Omega_{m},S_{8},w_{0})$. We find that adding a 5-bin tomography improves
the $\kappa-$PDF constraints by a factor of $\\{3.8,1.3,1.6\\}$ for
$(\Omega_{m},S_{8},w_{0})$ respectively. Additionally, we perform a joint
analysis with the shear two-point correlation functions, finding an
enhancement of around a factor of $1.5$ on all parameters with respect to the
two-point statistics alone. These improved constraints come from disentangling
$\Omega_{\rm m}$ from $w_{0}$ by extracting non-Gaussian information, in
particular, including the PDF skewness at different redshift bins. We also
study the effect of varying the number of parameters to forecast, in
particular we add $h$, finding that the convergence PDF maintains its
constraining power while the precision from two-point correlations degrades by
a factor of $\\{1.7,1.4,1.8\\}$ for $\\{\Omega_{\rm m},S_{8},w_{0}\\}$,
respectively.

††thanks<EMAIL_ADDRESS>
##  1 Introduction

Stage IV galaxy surveys such as Euclid (Laureijs  _et al._ , 2011) or the Vera
C. Rubin Observatory Legacy Survey of Space and Time (Željko Ivezić  _et al._
, 2019, LSST), SPHEREx (Doré  _et al._ , 2014), Dark Energy Spectroscopic
Instrument (DESI, Aghamousa  _et al._ (2016)), The Nancy Grace Roman Space
Telescope (Spergel  _et al._ , 2015), promise unprecedented insight into the
physics and evolution of the universe. While the good redshift resolution of
spectroscopic surveys provides access to 3D galaxy clustering, photometric
surveys can map a larger number of galaxies including their shapes to probe
the overall matter distribution. With tomographic techniques, photometric
surveys can also track the growth of structure and thus probe signatures of
dark energy. These next-generation of telescopes will provide a large volume
and quality data that will significantly improve our understanding of dark
energy, dark matter and the initial conditions of the universe.

Gravitational weak lensing is one of the most powerful and accurate probes of
the large-scale structure of the universe (Kilbinger, 2015). Weak
gravitational lensing traces the projected distribution of dark matter through
the gravitational effect on the light emitted by distant source galaxies,
which translates into cosmic shear (the deflection in the shape and size of
the observed galaxies, see Kaiser and Squires, 1993; Bartelmann and Schneider,
1999). Standard weak lensing provides a 2D map of the second derivatives of
the projected gravitational potential, from which it is not possible to fully
recover the time evolution of the density field. Weak lensing tomography
recovers some of the lost information by dividing the galaxy distribution into
different redshift bins (see e.g. Munshi  _et al._ , 2014; Jee  _et al._ ,
2016; Martinet  _et al._ , 2021). The weak lensing signal at different source
redshifts is sensitive to the nonlinear growth of structure influenced by dark
energy (Sipp  _et al._ , 2021) and massive neutrinos, which leave a strong
signal on small scales (Liu and Madhavacheril, 2019). Therefore, cosmic shear
observations provide great precision in constraining the parameters that
describe the content and evolution of the universe.

Analysing upcoming high-quality cosmological data to its fullest potential
represents a significant challenge, especially when aiming to capture the non-
Gaussian late time information stored in small scales. Extracting the weak
lensing signal relies on statistical measurements of correlations between
galaxy shapes. Traditionally, the shear two-point correlation function (2PCF)
or its Fourier transform, the lensing power spectrum, are employed for this
(Joachimi  _et al._ , 2008; Kilbinger  _et al._ , 2013; Hamana  _et al._ ,
2020; Asgari  _et al._ , 2021; Amon  _et al._ , 2022; Doux  _et al._ , 2022).
Both provide valuable information about the spatial distribution and
clustering properties of matter in the universe. However, the non-Gaussian
information encoded in the small scales are not captured by the 2PCF. To
unlock the full potential of Stage IV surveys and improve the constraints on
cosmological parameters higher order statistics (HOS) are crucial. The HOS
capture additional information providing a significant enhancement on the
constraints when combined with the 2PCF (Mellier  _et al._ , 2024). Some
examples of HOS applied to lensing data are: Higher-order moments of the
convergence or aperture mass (Van Waerbeke  _et al._ , 2013; Porth and Smith,
2021; Gatti  _et al._ , 2024), shear/convergence peak counts (Marian  _et al._
, 2009; Dietrich and Hartlap, 2010; Kacprzak  _et al._ , 2016; Martinet  _et
al._ , 2018; Harnois-Déraps  _et al._ , 2021; Zürcher  _et al._ , 2022;
Marques  _et al._ , 2024), Minkowski functionals (Ajani  _et al._ , 2023a;
Vicinanza  _et al._ , 2019; Petri  _et al._ , 2015; Kratochvil  _et al._ ,
2012), Betti numbers (Parroni  _et al._ , 2021; Wilding  _et al._ , 2021),
persistent homology (Heydenreich  _et al._ , 2021), scattering transform
coefficients (Cheng and Ménard, 2021; Gatti  _et al._ , 2024) and the one-
point probability distribution function (Barthelemy  _et al._ , 2020a; Boyle
_et al._ , 2021; Barthelemy  _et al._ , 2023). The constraining power of this
set of HOS has been recently investigated for the Euclid survey in Ajani  _et
al._ (2023b). The authors performed a non-tomographic Fisher forecast for the
matter density parameter $\Omega_{\rm m}$ and the amplitude of fluctuations
$\sigma_{8}$ and demonstrated the potential of each of the HOS over the 2PCF
due to their ability to extract non-Gaussian, small scale information. Most of
the HOS improve over the 2PCF constraints with slightly different degeneracy
directions and partial redundancy between different HOS.

In this work, we focus on the weak lensing convergence one-point probability
distribution function (PDF) which unlike many HOS can be predicted
theoretically. The weak lensing convergence, $\kappa$, offers a direct way to
trace the projected matter distribution along the line-of-sight unaffected by
how galaxies trace the underlying matter distribution. The kappa-PDF offers
valuable information beyond two-point analyses, potentially improving
constraints on cosmological parameters. It can be easily measured from
simulated and observed $\kappa$-maps and can be modelled with high accuracy on
mildly non-linear scales. The theoretical model of the PDF based on the large
deviation theory (LDT) has been applied on 3D density fields to describe dark
matter and halo densities and showed strong potential to constrain different
$\Lambda$CDM parameters (Bernardeau and Reimberg, 2016; Uhlemann  _et al._ ,
2016; Codis  _et al._ , 2016; Uhlemann  _et al._ , 2018a, 2020) as well as the
total neutrino mass $M_{\nu}$ (Uhlemann  _et al._ , 2020), the primordial
skewness $f_{NL}$ (Uhlemann  _et al._ , 2020; Friedrich  _et al._ , 2020) and
dark energy $(w_{0},w_{a})$ and modified gravity parameters (Cataneo  _et al._
, 2022; Gough and Uhlemann, 2022).

The LDT has been also applied for projected fields in 2D for different probes,
including the galaxy weak lensing convergence (Barthelemy  _et al._ , 2020a),
the CMB lensing convergence (Barthelemy  _et al._ , 2020b), the aperture mass
field Barthelemy  _et al._ (2021), the projected galaxy density (Uhlemann  _et
al._ , 2018a; Friedrich  _et al._ , 2022) and density-split statistics
(Friedrich  _et al._ , 2018; Gruen  _et al._ , 2018; Burger  _et al._ , 2023).
The key physics component over which the LDT is built is the spherical
(Valageas, 2002; Bernardeau  _et al._ , 2014) and cylindrical (Bernardeau and
Valageas, 2000) gravitational collapse for 3D and 2D fields, respectively.
Boyle  _et al._ (2023) validated the theoretical framework presented in
(Barthelemy  _et al._ , 2020a), extended it to include massive neutrinos and
performed a non-tomographic Fisher analysis. This study finds that focusing on
the bulk of the PDF achieves good constraining power surpassing the 2PCF while
maintaining a Gaussian likelihood enabling Fisher forecasts. LDT provides a
direct theoretical model for all non-Gaussian contributions to the cumulant
generating function (CGF) from which the PDF is obtained through an inverse
Laplace transformation. Boyle  _et al._ (2023) take the first steps into
studying the cosmological information contained in the weak lensing
convergence CGF, showing it is an alternative statistics to access the non-
Gaussian information at mildly non-linear scales.

The weak lensing convergence is not a direct observable, it has to be
reconstructed from the shear field using mass mapping techniques such as the
Kaiser-Squires (KS) inversion method (Kaiser, 1992). This method uses the
relationship between the spin-2 shear and the spin-0 convergence which is only
explicit in harmonic (Fourier or spherical harmonics) space. However, when
applied to real survey data with missing information (due to masks), the KS
method presents some limitations, as it assumes a periodic and continuous
space. When applied to finite and masked regions, it generates spurious
frequencies at the borders, which leads to a loss of power on the
reconstructed field as well as the loss of fundamental properties of the
lensing fields with the creation of non-physical B-modes.

This paper investigates the power of tomographic weak lensing convergence PDF
for constraining cosmological parameters. We use the method proposed and
validated in Barthelemy  _et al._ (2023) that follows a novel approach by
incorporating the impact of KS-reconstruction directly into the modelling of
the $\kappa-$PDF. We predict the $\kappa-$PDF for a StageIV-like source galaxy
distribution and include shape noise. We employ equipopulated redshift bins,
following the optimisation study performed in Sipp  _et al._ (2021). We
propose an optimised tomographic strategy that matches the number of cross-
correlation bins while combining consecutive bins to suppress shape noise. We
show the accuracy of the theoretical predictions by validating them with the
maps reconstructed from the SLICS shear catalogues (Harnois-Déraps  _et al._ ,
2018). Using a Fisher forecast, we assess the constraining power of the
tomographic $\kappa-$PDF for the set of parameters, $(\Omega_{\rm
m},S_{8}=\sigma_{8}\sqrt{\Omega_{\rm m}/0.3},w_{0})$ finding a large
improvement compared with non-tomography analysis. Following the same
procedure with the shear 2PCF we demonstrate the superiority and
complementarity of the $\kappa-$PDF over the 2PCF in constraining these set of
parameters. In addition, we investigate the effect of varying the number of
parameters to forecast. We find that the $\gamma$-2PCF loses constraining
power when forecasting more than two parameters, while the kappa-PDF maintains
its power, providing better constraints.

Alternative theory and simulation methods have already shown the strong
potential in the tomography $\kappa-$PDF to obtain valuable cosmological
information. Thiele  _et al._ (2020) uses an analytical description based on a
halo-model approach, which is accurate at small scales but not at mildly non-
linear scales. A forecast for the neutrino masses $M_{\nu}$, base on
simulations, is performed in Liu and Madhavacheril (2019). Their method is
able to probe the internal structure of halos by accessing the one-halo
regime. Giblin  _et al._ (2023) uses a method based on Gaussian process
regression to perform a full cosmological parameter estimation analysis using
simulated lensing PDF measurements. Additionally, simulation-based methods
combined with real data have also been applied to infer the value of $S_{8}$
(Thiele  _et al._ , 2023) or to study the effect of baryons (Grandón  _et al._
, 2024). This method allows including highly non-linear scales. Our goal is to
demonstrate the accuracy and precision of our method, which is valid for a
larger range of scales compared to other theoretical methods and less
computationally expensive than simulation-based methods.

This work is structured as follows: in section 2 we present the theoretical
framework of the summary statistics we use, i.e. the $\kappa-$PDF and the
$\gamma-$2PCF. In section 3 we describe the SLICS simulations and use them to
validate our theoretical predictions. In section 4 we describe the cosmology
behaviour of $\kappa-$PDF derivatives and the structure of the data covariance
and show our results on the Fisher forecast. We conclude and provide an
outlook on future work in section 5. We provide additional test in the
Appendix section, in particular we include and study the effect of intrinsic
alignments on the predicted $\kappa-$PDF in Appendix E.

##  2 Theoretical framework for weak lensing statistics

Weak gravitational lensing by large scale structures gives rise to a
phenomenon known as cosmic shear, which manifests as a distortion in the
apparent shapes of observed galaxy images. The deflection on the galaxy images
is quantified in terms of three quantities: the shear components
$(\gamma_{1},\gamma_{2})$ and the convergence $\kappa$, all three quantities
are sourced by the projected lensing gravitational potential
$\psi(\chi\bm{\theta},\chi)$111The projected gravitational potential along the
line-of-sight is defined as
$\psi(\bm{\theta},\chi)=\frac{2}{c^{2}}\int^{\chi}_{0}d\chi^{\prime}\frac{f_{K}(\chi-\chi^{\prime})}{f_{K}(\chi)f_{K}(\chi^{\prime})}\Phi(f_{K}(\chi^{\prime})\bm{\theta},\chi^{\prime})\,,$
where $\Phi$ is the Newtonian gravitational potential, $f_{K}(\chi)$ is the
comoving angular distance. We consider a flat universe, therefore $\chi(z)$ is
equivalent to the comoving angular distance $f_{K}(\chi)$.. In Born’s
approximation these three quantities are given by Kilbinger (2015)

$\gamma_{1}=\frac{1}{2}(\partial_{1}\partial_{1}-\partial_{2}\partial_{2})\psi\,\quad\gamma_{2}=\partial_{1}\partial_{2}\psi\,\quad\kappa=\nabla^{2}\psi\,,$
(1)

with $(\partial_{1},\partial_{2})$ the partial derivatives with respect to
$\bm{\theta}=(\theta_{1},\theta_{2})$, the angular coordinates of the
orthogonal planes to the light-of-sight $\bm{\hat{n}}$.

The weak lensing convergence $\kappa$ is defined as the line-of-sight
projection of the matter density distribution between the observer and the
galaxy source as

Figure 1: Upper: StageIV-like source redshift distribution probing up to
source redshifts of $z_{s}\sim 3$. The redshift distribution is sliced into 5
equipopulated redshift bins. Ranges for each redshift bin are presented in the
legend. Lower: Graphical representation of the convergence field in a fixed
aperture as a superposition of independent layers of the 3D matter density
along the line-of-sight. The contributions of the different slices are
weighted according to the weak lensing kernel (3).

$\kappa(\bm{\theta})=\int_{0}^{\infty}\,d\chi\,w(\chi(z))\delta(\chi(z)\bm{\theta},\chi(z))\,$
(2)

where $\chi(z)$ is the comoving distance to redshift $z$. For a population of
galaxy sources the convergence is also weighted by its redshift distribution
$n(z)$, such that the weak lensing projection weight function $w(\chi(z))$ is
given by

$w(\chi)=\frac{3\Omega_{\rm
m}H_{0}^{2}}{2c^{2}}\int_{z}^{\infty}\\!\\!\\!\\!dz^{\prime}\frac{\left[\chi(z^{\prime})-\chi(z)\right]\chi(z)}{\chi(z^{\prime})}(1+z)n(z^{\prime})\,$
(3)

with $H_{0}$ the Hubble parameter, $\Omega_{\rm m}$ matter density fraction
and $c$ the speed of light. For the tomographic analysis, the redshift
distribution of source galaxies $n(z_{s})$ is divided into different
tomographic bins, as shown in the upper panel of figure 1. The weak lensing
convergence in a fixed aperture can be conceptualised as the superposition of
independent layers of the 3D matter density field in a cone as illustrated in
the lower panel of figure 1. For correlators of the convergence this
corresponds to the so-called Limber approximation.

The convergence quantifies changes in apparent size of the image (and the
flux) while the shear components measure the change in the shape. Weak lensing
affects both the shape and size of the observed galaxy images by a few per
cent such that $\kappa\ll 1$ and $|\gamma|\ll 1$. The shear components can
only be estimated from taking the average over a large sample of observed
ellipticities $\langle\epsilon_{1,2}\rangle\approx\gamma_{1,2}$. Subsequently,
the convergence field can be reconstructed from the shear by performing a KS-
inversion as we will explain in subsection 2.2.3.

In the following subsections, we present the theoretical framework for the
summary statistics we use in the tomographic analysis, the probability density
function (PDF) of the convergence field, $\kappa$-PDF, and the shear two-point
correlation function, $\gamma-$2PCF.

###  2.1 Shear two-point correlation function

The real space two-point correlation function ($\gamma-$2PCF) is the most
widely used statistic to quantify the shear correlations. It is estimated by
multiplying the ellipticities of galaxy pairs and averaging over all pairs
separated by an angular separation $\theta$. As the shear is analogous to a
polarisation tensor, it is mathematically convenient to write the shear as a
complex quantity, $\gamma=\gamma_{1}+i\gamma_{2}=|\gamma|\exp(2i\varphi)$ with
$\varphi$ the polar angle between the two shear components.

The two components of the $\gamma-$2PCF are defined as

$\xi_{\pm}(\theta)=\langle\gamma\gamma^{*}\rangle=\langle\gamma_{1}\gamma_{1}\rangle(\theta)\pm\langle\gamma_{2}\gamma_{2}\rangle(\theta)\,.$
(4)

As the shear components and the convergence are proportional to second
derivatives of the 2D gravitational potential, the two components of the
$\gamma-$2PCF are uniquely determined by the angular convergence power
spectrum in the following way

$\xi_{\pm}(\theta)=\int\frac{\ell
d\ell}{2\pi}J_{0,4}(\ell\theta)C_{\kappa\kappa}(\ell)$ (5)

where $J_{i}$ are the spherical Bessel functions, $i=0,4$ for $\xi_{+}$ and
$\xi_{-}$ respectively. The convergence power spectrum is given by

$C_{\kappa\kappa}(\ell)=\int^{\infty}_{0}d\chi\,w(\chi(z))^{2}P_{\rm
NL}(k=\ell/\chi,z)\,,$ (6)

and $P_{\rm NL}(k,z)$ is the non-linear matter power spectrum.

###  2.2 Convergence one-point PDF

This section details the theoretical framework for calculating the weak
lensing convergence probability distribution function (PDF) (Barthelemy  _et
al._ , 2020a; Boyle  _et al._ , 2021). We consider a set of source galaxies
with a given redshift distribution.

####  2.2.1 Predicting $\kappa-$PDF from the matter CGF

In practice, the probability density function (PDF) is predicted via the
cumulant generating function (CGF) as detailed in Appendix A. Both the CGF and
PDF are determined within the framework of large deviation theory (LDT) which
predicts the distributions of matter density in the limit of small variances
as opposed to small density contrasts. The convergence PDF $P(\kappa)$ and the
CGF $\phi(\lambda)$ are related through an inverse Laplace transformation in
the following way

$P(\kappa)=\int^{+i\infty}_{-i\infty}\,\frac{d\lambda}{2\pi
i}\,e^{\phi(\lambda)-\lambda\kappa}\,.$ (7)

The CGF is the logarithm of the moment generating function and is given by

$\phi(\lambda)=\sum_{l=1}^{\infty}\,\langle\kappa^{l}\rangle_{c}\frac{\lambda^{l}}{l!}\,,$
(8)

where the convergence cumulants are defined as

$\langle\kappa^{l}\rangle_{c}=\int_{0}^{\chi_{s}}\,\left(\prod_{i=1}^{l}d\chi_{i}\,w(\chi_{i})\right)\langle\delta_{1}\cdots\delta_{l}\rangle_{c}\,.$
(9)

The cumulants are computed in the small angle approximation which is valid
when the transverse distances $\chi|\bm{\theta}_{i}-\bm{\theta}|$ are much
smaller than the radial distance $\chi$. Since the correlation length is much
smaller than the Hubble scale $c/H(z)$ the variable $\chi_{i}$ can be written
as $\chi_{i}=\chi_{1}+r_{i}$ with $|r_{i}|\ll c/H(z)$ such that222This is the
Limber approximation which allows to relate the angular and spatial
correlation functions by considering averages over cylinders of the same size.

$\displaystyle\langle\kappa^{l}\rangle_{c}$
$\displaystyle=\int_{0}^{\chi_{s}}\,d\chi_{1}\,w(\chi_{1})^{l}$ (10)
$\displaystyle\times\int_{-\infty}^{\infty}\left(\prod_{i=2}^{l}\,dr_{i}\right)\langle\delta(\chi_{1}\bm{\theta}_{1},\chi_{1})\cdots\delta(\chi_{l}\bm{\theta}_{l},\chi_{1}+r_{l})\rangle_{c}\,.$

Therefore, assuming
$\delta(\chi_{l}\bm{\theta}_{l},\chi_{1}+r_{l})\simeq\delta(\chi_{l}\bm{\theta}_{l},\chi_{1})$
and considering filtering effects, the cumulants of the smoothed convergence
field are related to $\delta_{\text{cyl}}$, the 3D density contrast average
over a cylinder of transverse size $R_{\theta}=\chi\theta$ and length $L$, in
the following way

$\langle\kappa_{\theta}^{l}\rangle_{c}=\int_{0}^{\chi_{s}}\,d\chi\,w(\chi)^{l}\,\langle\delta_{\text{cyl}}^{l}\left(R_{\theta},L\right)\rangle_{c}\,L^{l-1}\,.$
(11)

The density field appearing in this equation is smoothed with a top-hat filter
in cylindrical coordinates as (Uhlemann  _et al._ , 2018b)

$\delta_{\text{cyl}}(\bm{x})\\!=\\!\\!\int\\!\\!d^{3}\bm{x}^{\prime}\,W_{\rm
2D}(|\bm{\theta}^{\prime}-\bm{\theta}|,R_{\theta})W_{\rm
1D}\left(|\chi^{\prime}-\chi|,\tfrac{L}{2}\right)\delta(\bm{x}^{\prime})\,,$
(12)

with $\bm{x}=\chi(\bm{\theta},1)$, for an infinite long cylinder
$L\rightarrow\infty$ the filter reduces to $W_{2D}$. Hence, the convergence
CGF is related to the CGF of the density contrast in cylinders as described in
Barthelemy  _et al._ (2020a); Boyle  _et al._ (2021)

$\phi_{\theta}(\lambda)=\int^{\chi_{s}}_{0}\,d\chi\,\lim_{L\rightarrow\infty}\phi_{\theta,\text{cyl}}\left(Lw(\chi)\lambda\right)L^{-1}\,.$
(13)

We briefly review the procedure to obtain the CGF of density contrast in
cylinders using LDT in appendix A.

####  2.2.2 Including shape noise

Observed galaxy shapes results mainly from two contributions: cosmic shear due
to gravitational weak lensing, and intrinsic galaxy ellipticity. The latter, a
primary source of noise (shape noise), follows a Gaussian distribution with
variance $\sigma_{\epsilon}^{2}$. The effect on the PDF due to shape noise is
included by performing a convolution with a Gaussian distribution centred at
zero and with a variance $\sigma_{\rm SN}^{2}$ as in Clerkin  _et al._ (2017);
Boyle  _et al._ (2021)

$P_{\text{noisy}}(\kappa)=\frac{1}{\sqrt{2\pi}\sigma_{\rm SN}}\int
d\kappa^{\prime}\exp\left(-\frac{(\kappa-\kappa^{\prime})^{2}}{2\sigma_{\rm
SN}^{2}}\right)P(\kappa^{\prime})\,.$ (14)

$\sigma_{\rm SN}^{2}$ is the shape noise variance at a given smoothing scale
$\theta$ weighted by the source galaxy density as

$\sigma_{\rm SN}^{2}=\frac{\sigma_{\epsilon}^{2}}{n_{g}\Omega_{\theta}}\,,$
(15)

with $\Omega_{\theta}$ the solid angle (in arcmin2).

For equi-populated redshift bins, the shape noise contribution is the same
across all redshift bins. Hence, the effect on the convergence PDF depends on
the lensing projection kernel (equation 3) for each source redshift bin, i.e
on how much the light from the source galaxies is affected by weak lensing. As
the projection kernel and its amplitude increase with the redshift of the
sources, higher source redshift bins exhibit a stronger lensing signal.
Consequently, the impact of shape noise on the PDF is strongest for sources at
the lowest redshifts. For a Gaussian shape noise, decorrelated from the
signal, it affects only the variance of the PDF such that the variance of the
noisy $\kappa-$ PDF is equal to the variance of noise-free $\kappa-$ PDF plus
the shape noise variance in the aperture of size $\theta$

$\sigma_{\text{noisy}}^{2}=\sigma_{\text{noise\\-free}}^{2}+\sigma_{\rm
SN}^{2}\,,$ (16)

such that the inclusion of shape noise leaves all the higher-order cumulants
unchanged and thus produces a more Gaussian convergence field.

Additional systematic effects, like photometric redshift errors, shape
calibrations or galaxy intrinsic alignments (IA) caused by external
gravitational tidal fields, can be incorporated into the modelling of the
$\kappa-$PDF. See Appendix E where we include and validate the impact of IA as
described by non-linear alignment models (NLA, Hirata  _et al._ , 2007; Bridle
and King, 2007) into the $\kappa-$PDF predictions.

####  2.2.3 Convergence reconstruction: Kaiser-Squires

As the convergence field is not a direct observable, it needs to be
reconstructed from the shear estimations. To obtain the $\kappa$-maps from the
shear ones, we perform a Kaiser-Squires inversion (KS) (Kaiser and Squires,
1993). To explain the mathematical framework behind the KS inversion, it is
convenient to decompose the convergence field into its electric-like and
magnetic-like modes as $\kappa=\kappa^{E}+i\kappa^{B}$. The shear and the
convergence fields are related thought a linear combination of equations (1)
in Fourier space as

$\displaystyle\hat{\gamma}$
$\displaystyle=\hat{P}\hat{\kappa}\,,\qquad\qquad\hat{\kappa}=\hat{P}^{*}\hat{\gamma}\,,$
(17)

where $\hat{P}$ is a linear operator in Fourier space given by

$\displaystyle\hat{P}=\frac{k_{1}^{2}-k_{2}^{2}+2ik_{1}k_{2}}{k_{1}^{2}+k_{2}^{2}}$
(18)

with ${k_{1},k_{2}}$ the frequencies associated with the coordinates
$\bm{\theta}$ on the orthogonal plane to the light-of-sight. After applying
the operator $\hat{P}$ on the shear field, we get the convergence field by
inverting back to real space. As the convergence field is a scalar quantity,
we expect the $B$-mode to vanish. However, for finite and masked fields the KS
inversion produces spurious frequencies which generates a leakage of
information from the $E$-mode to the $B$-mode and produces border effects on
the convergence maps. Also, of relevance direct to us, this effect reduces the
variance in the convergence field.

Several techniques have been developed to mitigate the impact of
reconstruction, such as the inpainting method (Elad  _et al._ , 2005; Pires
_et al._ , 2009). This technique fills the masked regions by extrapolating the
information from the observed data. It iteratively finds a sparse
representation of data and fills the gaps using this information. Similar
methods have been developed, for example Remy  _et al._ (2023) uses an
iterative Wiener filtering. While these methods minimise the impact of
reconstruction, they can not fully recover all lost information, and it is not
clear what is the best method since the effectiveness depends on the specific
scientific question (Jeffrey  _et al._ , 2021).

We can model the loss of information effect into the theory framework. To do
this we follow the procedure in Barthelemy  _et al._ (2023) where the
theoretical model for the convergence PDF is modified to include the effect of
a survey mask. The idea is to consider that we measure the shear values on a
fraction of the sky, where the rest of the sky is masked. For masked fields
the shear and convergence correlation functions in Fourier space are affected
due to the mixing of frequencies and leakage from E to B modes. This reduction
in the power spectrum impacts the shear and convergence variances. In the LDT
model, because of the Born and Limber approximations, the convergence is
treated as a superposition of independent layers of the matter density field,
seen as a succession of lensing events, along the line-of-sight. Therefore,
the reconstruction effect can be included at each redshift slice by scaling
the non-linear variance of the matter density field according to the
modulation. We achieve this by computing the ratio between the true variance
and the reconstructed variance of the (fictitiously masked) matter density
field. This factor is then used to scale the non-linear variance within the
rescaled cumulant generating function (see equation 43). This method ensures
preserved scaling relations between cumulants of the density.

The non-linear variances of the true and the KS-reconstructed density field
are given by (Barthelemy  _et al._ , 2023; Gatti  _et al._ , 2022)

$\langle\delta_{true}^{2}\rangle=\sum_{\ell}\frac{2l+1}{4\pi\chi(z)^{2}}W_{\ell}(\chi(z)\theta)^{2}P_{\rm
NL}(\ell^{\prime}/\chi(z),z)\,,$ (19)
$\langle\delta_{E}^{2}\rangle=\frac{1}{f_{\rm
mask}}\sum_{\ell}\frac{2l+1}{4\pi\chi(z)^{2}}W_{\ell}(\chi(z)\theta)^{2}f_{\ell}^{-1}\\\
\sum_{\ell^{\prime}}M_{\ell\ell^{\prime}}^{EE,EE}P_{\rm
NL}(\ell^{\prime}/\chi(z),z)f_{\ell^{\prime}}\,,$ (20)

where $M_{\ell^{\prime}\ell}^{EE,EE}$ are the elements of the mode-mixing
matrix that contribute to the transfer of information from the unmasked to
masked shear $E$-modes.
$f_{\ell}=\left[(\ell+2)(\ell-1)\right]/\left[\ell(\ell+1)\right]$ is the
normalisation factor in harmonic space for the full sky reconstruction and
$W_{\ell}$ is the smoothing filter in harmonic space. $f_{\rm mask}$ is a
purely geometrical factor defined as the ratio, at a given smoothing scale, of
the reconstructed convergence variances respectively taking into account all
the pixels on the reconstructed full-sky and only pixels in the field of view.
As detailed in Barthelemy  _et al._ (2023), this factor is a priori
independent of cosmology, has a slight smoothing-scale dependence but is
directly measurable on actual data. Nevertheless, its value is by construction
very close to the sky fraction, $f_{\text{sky}}\simeq A/4\pi$ with $A$ the
area of the unmasked region. We thus fix $f_{\rm mask}=f_{\text{sky}}$ in the
remainder of this work. This will be shown to be accurate enough while having
no influence on the forecasts we perform.

###  2.3 Tomographic strategy

For our analysis, we consider a StageIV-like redshift distribution, plotted in
figure 1. In principle, we divide the redshift distribution into 5 redshift
bins. To mitigate the effect on shape noise in the lower redshift bins and to
have a fair comparison between the $\kappa$-PDF and the $\gamma$-2PCF our PDF
data vector includes the information of 14 redshift combinations. Besides the
$\kappa$-PDF for the full redshift distribution and for the individual
redshift bins we combine consecutive redshift bins into quartets, triplets and
pairs. In table 1 we show the 15 redshift combinations for the $\kappa-$PDF we
can form following this strategy, we do not use the lowest redshift bin as it
is not accurately predicted by theory in the noise-free case and is highly
affected by shape noise. The names on the second column bin$ijkl$ stand for
the redshift bins that has been combined for the $\kappa-$PDF. For the
$\gamma-$2PCF we consider standard auto and cross-correlations. In figure 2 we
show the normalised weak lensing projection kernel (equation 3) for the
different redshift sources we consider.

| $z_{P(\kappa)}$ | $n_{g}\left[\text{arcmin}^{-2}\right]$ |  $\sigma^{2}$ $[10^{-5}]$
---|---|---|---
$\sigma_{\rm SN}^{2}$ | $\sigma_{\text{noisefree}}^{2}$ | $\sigma_{\text{noisy}}^{2}$
$1$ | All bins | 30 | $0.319$ | $2.42$ | $2.74$
$2$ | bin$2345$ | 24 | $0.398$ | $3.52$ | $3.91$
$3$ | bin$1234$ | 24 | $0.398$ | $1.66$ | $2.06$
$4$ | bin$345$ | 18 | $0.531$ | $4.71$ | $5.24$
$5$ | bin$234$ | 18 | $0.531$ | $2.57$ | $3.10$
$6$ | bin$123$ | 18 | $0.531$ | $1.14$ | $1.68$
$7$ | bin$45$ | 12 | $0.797$ | $6.28$ | $7.08$
$8$ | bin$34$ | 12 | $0.797$ | $3.46$ | $4.25$
$9$ | bin$23$ | 12 | $0.797$ | $1.96$ | $2.76$
$10$ | bin$12$ | 12 | $0.797$ | $0.70$ | $1.50$
$11$ | bin5 | 6 | $1.59$ | $8.98$ | $10.60$
$12$ | bin4 | 6 | $1.59$ | $4.47$ | $6.07$
$13$ | bin$3$ | 6 | $1.59$ | $2.70$ | $4.30$
$14$ | bin2 | 6 | $1.59$ | $1.45$ | $3.05$
$15$ | bin$1$ | 6 | $1.59$ | $0.30$ | $1.89$
Table 1: Slicing of the StageIV-like redshift distributions. Number density of
galaxies on each redshift bin combinations. Shape noise variance, variance of
the Noise-Free PDF and variance of the Noisy PDF for a smoothing scale
$\theta=15$ arcmin.  Figure 2: Normalised weak lensing projection kernel as a
function of the comoving distance for different redshift combinations of the
StageIV-like source redshift distribution.

##  3 Theory validation with SLICS simulations

In this section, we validate the theoretical frameworks for the summary
statistics, the $\kappa-$PDF and $\gamma-$2PCF, with measurements from
numerical weak lensing simulations. We describe these here, and detail our
methods to measure the shear and convergence statistics from the corresponding
shear and convergence maps. Finally, we compare the theoretical predictions
against measurements for our summary statistics.

###  3.1 Weak lensing simulations

We use the SLICS (Scinet LIghtcone Cosmological Simulations) (Harnois-Déraps
_et al._ , 2018) and cosmo-SLICS (Harnois-Deraps  _et al._ , 2019) weak
lensing simulations to estimate the covariance matrix and validate our
theoretical framework. Both are dark matter-only simulations obtained by using
the gravity solver cubep3m (Harnois-Deraps  _et al._ , 2013). The comoving
side length of the simulation boxes is $L_{\text{box}}=505h^{-1}\text{Mpc}$,
from which light-cones are extracted with an area of $A_{\rm
SLICS}=100\text{deg}^{2}$. The SLICS simulation box contains $1536^{3}$
particles each with mass $m_{p}=2.88\times 10^{9}h^{-1}\text{M}_{\odot}$
resolving dark matter halos with masses below $10^{11}h^{-1}\text{M}_{\odot}$
and accurately simulates structure formation well into the non-linear regime.
The SLICS cosmology is determined by the best-fitting WMAP9+BAO+SN parameters
(Hinshaw  _et al._ , 2013) as presented in Table 2. The cosmo-SLICS suite is
available for 25 distinct cosmologies, organised in a Latin hypercube
configuration, varying $\Omega_{\rm m}$, $h$,
$S_{8}=\sigma_{8}\sqrt{(\Omega_{\rm m}/0.3)}$ and $w_{0}$ and with the
particle mass varying with $\Omega_{\rm m}$ and $h$ in the range
$[1.42,7.63]\times 10^{9}\text{M}_{\odot}$. For each of these cosmologies,
there are a total of $10\times 2$ realisations, 10 pairs with initial
conditions carefully selected to ensure that the initial power spectrum of
each pair averages to the theoretical prediction (Harnois-Deraps  _et al._ ,
2019), effectively minimising the impact of cosmic variance (Harnois-Deraps
_et al._ , 2019).

A shape noise field $\bm{\epsilon}$ is infused on the shear catalogues by
sampling a Gaussian distributions with standard deviation $\sigma_{\epsilon}$
(per component), constructing observed ellipticities $\bm{\epsilon}_{\rm obs}$
as:

$\displaystyle{\bm{\epsilon}}_{\rm
obs}=\frac{\bm{\gamma}+\bm{\epsilon}}{1+\bm{\gamma}^{*}\bm{\epsilon}}\,,$ (21)

where bold symbols refer to complex notation, i.e. $\bm{\gamma}=\gamma_{1}$ \+
i$\gamma_{2}$. We use the SLICS and cosmo-SLICS StageIV-like simulation suites
for which the source galaxy number density ($n_{g}=30\text{arcmin}^{-2}$),
galaxy shape noise ($\sigma_{\epsilon}=0.26$), and redshift distribution (FIG
1) are representative of upcoming lensing surveys.

$\Omega_{\rm m}$ | $\Omega_{\Lambda}$ | $\Omega_{b}$ | $h$ | $\sigma_{8}$ | $n_{s}$ | $w_{0}$
---|---|---|---|---|---|---
$0.2905$ | $0.7095$ | $0.0473$ | $0.6898$ | $0.826$ | $0.969$ | -1.0

Table 2:  Cosmological parameters of the SLICS simulations

.

###  3.2 Shear and convergence maps

We use the SLICS shear catalogues available for $N_{\rm SLICS}=923$
realisations and the cosmo-SLICS catalogues for the 20 realisations to
generate noise-free and noisy shear maps. Each shear catalogue contains
information about the positions $(x,y)$, the redshift $z$, shear values
$(\gamma_{1},\gamma_{2})$, and observed ellipticities $(e_{obs,1},e_{obs,2})$.
We obtain the shear and observed ellipticity maps by projecting the shear or
ellipticity values for each galaxy onto a $2D$ Cartesian grid. To ensure we
have at least one galaxy per pixel, each map has a size $600^{2}$ pixels. The
value on each pixel is the average shear or ellipticity assigned to the
corresponding pixel333We use the Pylians libraries (Villaescusa-Navarro, 2018)
https://pylians3.readthedocs.io/en/master/index.html.

From the shear maps we reconstruct the convergence maps by using the KS-
inversion presented in section 2.2.3444We use bin2d, ks93 functions from
lenspack https://github.com/CosmoStat/lenspack/tree/master/lenspack. In Figure
3 we present the weak lensing maps for the non-tomographic case. The upper
panel shows the map obtained by combining the two components of the shear as,
$\gamma=\sqrt{\gamma_{1}^{2}+\gamma_{2}^{2}}$ and the lower panel is the KS-
reconstructed kappa map $\kappa_{E}$. In both cases the colour represents the
size of the fluctuation field in units of the field standard deviation, e.g.
$\nu_{\gamma}=(\gamma-\langle\gamma\rangle)/\sqrt{\langle\gamma^{2}\rangle_{c}}$.

Figure 3: Weak lensing maps. Upper: Absolute value of the shear field. Lower:
KS-reconstructed convergence map. The background colour represents the
fluctuation field in units of the field standard deviation, e.g
$\nu_{\gamma}=(\gamma-\langle\gamma\rangle)/\sqrt{\langle\gamma^{2}\rangle_{c}}$.

###  3.3 $\kappa-$PDF validation

We measure the $\kappa-$PDF on the 923 KS-reconstructed SLICS convergence maps
for three smoothing scales, $\theta=\\{10,15,20\\}\text{ arcmin}$ obtained by
performing a convolution with a top-hap filter with a radius equal to
$\theta$. We measure the PDF by computing a histogram of the smoothed
$\kappa-$maps, for each redshift combination, and we average over the
realizations to obtain the measured $\kappa-$PDF.

Theoretical predictions presented in section 2.2 have been implemented in the
public code CosMomentum 555https://github.com/OliverFHD/CosMomentum (Friedrich
_et al._ , 2020). We use CosMomentum to obtain the theoretical predictions for
the SLICS and cosmo-SLICS cosmologies and the redshift distribution for each
of the 15 redshift combinations. We include the effect of shape noise by
convolving the PDF with a Gaussian random field with $\sigma^{2}_{\rm SN}$
variance corresponding to each redshift bin, as explained in section 2.2.2.
The non-linear power spectrum is modelled using the fitting function Halofit
Takahashi  _et al._ (2012). The variance for the KS-reconstructed convergence
field is obtained by using the public code L2DT 666
https://github.com/AlexandreBarthelemy/L2DT (Barthelemy  _et al._ , 2023).

In figure 4, we compare $\kappa-$PDF measurements with theoretical predictions
for $5$ selected redshift combinations for (a) the noise-free convergence, and
(b) the convergence in the presence of shape noise. On the upper panels the
SLICS measurements are presented with data points, we observe that the
theoretical $\kappa-$PDF (solid and dashed lines) predict the measurements
with high accuracy. On the lower panels, we plot the residuals between the
measured SLICS and the predicted $\kappa-$PDF. The error bars represent the
standard deviation across realisations, which is of the order of $10\%$777The
error on the mean is a factor of $\sqrt{N_{\rm sim}-1}\approx 30$ smaller..
When rescaled to the full survey area, the error bars shrink to around $1\%$
precision. For the noise-free $\kappa$, the PDF is less accurate for the
smaller redshift bins, partially due to an overprediction of the variance that
leads to a lower peak height. This is a direct consequence of the fact that we
can’t directly measure $f_{\rm mask}$ with only simulated small patches, and
that we thus fixed it to the $f_{\rm sky}$ value. This effect is mixed with
inherent inaccuracies of the non-linear matter power spectrum recipe that we
use, i.e. Halofit. Additionally, smaller redshift bins probe smaller non-
linear scales due to the mixing of scales inherent to weak lensing for which
the mildly nonlinear predictions are not ideally suited. Nevertheless,
predictions within the 2-$\sigma$ region around the PDF peak are accurate at
the 5%-level for bin2 and below that for higher redshift bins. We find that,
including the effect of the KS-reconstruction by rescaling the non-linear
variance at the CGF level is enough to obtain good agreement between the
theoretical predictions and measurements, in both the noise-free and noisy
cases.

Figure 4: $\kappa-$PDF theory validation with SLICS simulations for
$\theta=15$ arcmin. (a) Noise-free. (b) Noisy. Upper panels: $\kappa$-PDF
comparison between the theory (solid and dashed lines) and the SLICS
measurements (with the error bars given by standard deviation of the mean) for
4 auto redshift bins and all bins combined. Lower panels: Residuals between
the measure SLICS and the predicted $\kappa$-PDF within the $2\sigma$
information around the peak, the error bars represent the standard deviation
over realizations. Other tomographic bin combinations show a similar agreement
(see figure 5).

In Figure 5 we validate the full set of theoretical predictions (solid lines)
for the noisy $\kappa$-PDF by comparing them with measurements (dots) from the
noisy SLICS KS-reconstructed convergence field, for the 15 redshift
combination from our tomographic set. We find good agreement between theory
and measurements for all redshift combinations. In order to see the effect of
shape noise on different redshift combinations, we also include theoretical
predictions for the noise-free $\kappa-$PDF (dashed lines). We can see clearly
that lower redshift bins are more affected by shape noise. On the lower panel,
we present the residuals between the SLICS measurements and the noisy
theoretical predictions, here we have normalised the $\kappa-$values by the
width $\sigma$ of the $\kappa-$PDF. Again, the percentage errors are within
the $5\%$ around the peak, within the $2\sigma$ of the PDF, for all redshift
combinations. We can see from the residuals outside the $2\sigma$ region
around the peak that the theoretical predictions for the tails of the PDF are
less accurate. In consequence, for our Fisher analysis (next section) we cut
the tails by keeping the information within the $2\sigma$ of the PDF, that
still has enough non-Gaussian information.

Figure 5: Noisy $\kappa$ PDF validation with SLICS simulations for the 15
redshift combinations in our tomographic strategy. Upper panel: comparison
between the theory predictions (solid line) and SLICS measurements (dots with
error bar given by the standard deviation of the mean). Theory predictions for
the noisy-free $\kappa-$PDF (dashed lines) are also shown. Lower panel:
Residuals between the SLICS measuremets and the theory predictions, the shaded
bands represent the $5\%$ (dark) and the $10\%$ (light) errors. Same colour
scheme from upper panel.

###  3.4 $\gamma-$2PCF validation

We measure the $\gamma-$2PCF $\xi_{\pm}(\theta)$ in 10 logarithmically spaced
bins from separations $\theta$ from 1 to 600 arcmin using
TreeCorr888https://github.com/rmjarvis/TreeCorr (Jarvis  _et al._ , 2004;
Jarvis, 2015). The tomographic data vector is built from the usual auto-
correlation of individual redshift bins and cross-correlations between two
redshift bins. Theoretical predictions for the $\gamma-$2PCF are obtained with
pyccl 999https://github.com/LSSTDESC/CCL (Chisari  _et al._ , 2019) using the
described source redshift distribution $n(z)$ and corresponding cosmologies.
We use the CAMB setting with the ‘original’ Halofit (Smith  _et al._ , 2003)
to ensure compatibility with CosMomentum currently relying on that
incarnation. We have checked consistency between the predictions obtained from
pyccl and CosmoSIS (Zuntz  _et al._ , 2015).

Figure 6 shows the comparison between the theory and SLICS measurements for
the two components of the $\gamma-$2PCF, $\xi_{\pm}$ (see equation 5).
Predictions for $\xi_{+}$ are accurate for all scales across all redshift
combinations, while the predictions for $\xi_{-}$ are less accurate at smaller
scales. The difference between the two components $\xi_{\pm}$ is determined by
the Bessel filters in equation 5. For small argument $J_{0}$ is positive while
$J_{4}$ is zero. For the $\gamma-$2PCF the shape noise affects the size of
error bars, and more generally its covariance matrix, while the signal remains
unchanged. Indeed, shape noise as described in equation (15) has no spatial
correlation which equates to a constant noise power spectrum and thus only a
contribution to the zero-point contribution to the shear correlation function.

Figure 6: $\gamma-$2PCF validation with SLICS simulations for the individual
redshift bins and cross-correlations between two redshift bins. Comparison
between theory predictions for the two components of the $\gamma-$2PCF (dashed
and solid lines for $\xi_{+}$ and $\xi_{-}$ respectively) and the
$\gamma-$2PCF SLICS measurements from the noisy shear maps (crosses and dots
for $\xi_{+}$ and $\xi_{-}$ respectively). The error bars are obtained from
the standard deviation over realisations (light dashed and solid dark lines
for $\xi_{+}$ and $\xi_{-}$ respectively). The shaded regions represent the
scales cuts we will use for the Fisher joint analysis (dark and light for
$\xi_{+}$ and $\xi_{-}$ respectively).

##  4 Fisher Forecast

We perform a Fisher forecast to quantify how well the one-point PDF and the
combination with two-point correlations can constrain the cosmological
parameters $p_{i}=\\{\Omega_{\rm m},S_{8},w_{0}\\}$ when adding the
tomographic information. The Fisher information is given by the Fisher matrix
defined as

$F_{ij}=\frac{\partial\bm{D}^{T}}{\partial
p_{i}}C^{-1}_{ij}\frac{\partial\bm{D}}{\partial p_{j}}\,,$ (22)

where $\bm{D}$ is the data vector built from the theoretical predictions for
the $\kappa-$PDF and the $\gamma-$2PCF, and $C$ is the data covariance matrix
estimated from the measurements of the SLICS simulations. The parameter
covariance is obtained from inverting the Fisher matrix.

We will perform various Fisher Forecasts to quantify

  * •

the improvement of constraints by adding tomography in the $\kappa-$PDF, see
figure 11

  * •

the improvement of combining the tomographic $\kappa-$PDF and $\gamma-$2PCF,
see figure 12.

  * •

the effect of varying the number of parameters to forecast, see figures 13 and
14.

For the $\kappa-$PDF we only use the information in the bulk, keeping the
$2\sigma$ around the peak, thus excluding the tails. The 2$\sigma$ region
around the peak corresponds to removing the lowest $1.5\%$ and the highest
$3.5\%$ $\kappa$ bins, which can be easily identified using the cumulative
distribution function as integral of the PDF. We combine the $\kappa-$PDF at
three different smoothing scales, $\theta=\\{10,15,20\\}$ arcmin. We finally
interpolate $N_{s}=10$ data points within the corresponding $\kappa$ range. To
be consistent with the scales we are probing with the $\kappa-$PDF and taking
into account the theoretical accuracy in the $\gamma-$2PCF in the combined
Fisher analysis, for $\xi_{+}$ and $\xi_{-}$ we consider scales $\theta>10$
arcmin and $\theta>50$ arcmin, respectively. An individual Fisher forecast for
just the $\gamma-$2PCF with tomography is shown in Appendix C.

###  4.1 Data vector derivatives

The derivatives of our data vector $\bm{D}$ with respect to the cosmological
parameters are obtained by concatenating the derivatives for each summary
statistics and each redshift combination that we consider in the analysis. We
compute derivatives from the theoretical predictions of the $\kappa-$PDF and
the $\gamma-$2PCF using finite differences as

$\frac{\partial\bm{D}}{\partial
p_{i}}=\frac{\bm{D}(p_{i}^{+})-\bm{D}(p_{i}^{-})}{p_{i}^{+}-p_{i}^{-}}\,.$
(23)

We take variations of the cosmological parameters $p^{\prime}=\\{\Omega_{\rm
m}$,$\sigma_{8}$,$w_{0}\\}$ around the fiducial SLICS cosmology (see table 2).
We increase and decrease the values of $\Omega_{\rm m}$ and $\sigma_{8}$ by
$\sim 10\%$ and for $w_{0}$ by $50\%$ with respect to their fiducial values
with equidistant steps. In table 3 we present the fiducial values, the
increment/decrement ($\pm$) and the corresponding values for the individual
parameters ($p_{i}^{+},p_{i}^{-}$).

| $\Omega_{\rm m}$ | $\sigma_{8}$ | $w_{0}$
---|---|---|---
fid | $0.290$ | $0.826$ | -$1.0$
$\pm$ | $0.030$ | $0.075$ | $0.50$
$p^{-}$ | $0.260$ | $0.751$ | $-1.50$
$p^{+}$ | $0.320$ | $0.901$ | $-0.50$
Table 3: Variation of the cosmological parameters.

To obtain derivatives for the set of parameters $p=\\{\Omega_{\rm
m},S_{8},w_{0}\\}$ from the derivatives of the set $p^{\prime}=\\{\Omega_{\rm
m},\sigma_{8},w_{0}\\}$ we use the chain rule

$\frac{\partial\bm{D}}{\partial p_{i}}=\sum_{j}\frac{\partial\bm{D}}{\partial
p_{j}^{\prime}}\left(\frac{\partial p_{j}^{\prime}}{\partial
p_{i}}\right)_{\text{fid}}\,,$ (24)

where the subscript fid means we evaluate the partial derivative at the
fiducial values. The transformations on the derivatives (24) are equivalent to
a transformation of the Fisher matrix as $F_{ij}=M^{T}F^{\prime}_{ij}M^{T}$
(Coe, 2009), where the matrix transformation is given by (Ajani  _et al._ ,
2023b)

$M=\left(\begin{array}[]{ccc}1&0&0\\\
-\frac{\sigma_{8}}{2\Omega_{m}}&\left(\frac{0.3}{\Omega_{m}}\right)^{1/2}&0\\\
0&0&1\end{array}\right)\,.$ (25)

Figure 7 shows the $\kappa-$PDF derivatives for the $14$ redshift combinations
(same colour-code as in figure 2, low redshifts in red to high redshifts in
blue) and a smoothing scale of $\theta=15$ arcmin. We show the derivatives of
the $\kappa-$PDF normalised by the width $\sigma_{\rm fid}$ of the fiducial
$\kappa-$PDF, such that we can visualise the $2\sigma$ information around the
bulk for the different redshift bins. $w_{0}$ derivatives are scaled by a
factor of $5$ for a better visualisation. The upper (a) and bottom (b) panels
shows derivatives for the noise-free and noisy $\kappa-$PDF respectively,
where the dotted vertical line represent the most probable $\kappa-$value for
the fiducial cosmology for the non-tomography redshift distribution (“All
bins”).

Figure 7: PDF derivatives. The upper and bottom panels show derivatives in the
noise-free and noisy cases respectively. We show $\Omega_{\rm m}$ (solid
lines), $S_{8}$ (dashed lines) and $w_{0}$ (dotted lines) derivatives, where
the derivatives for $w_{0}$ are enhanced by a factor of $5$ for better
visualisation. The black dotted vertical line represent the maximum PDF value
for the fiducial cosmology for the “All bins” case.

We observe that derivatives with respect to $S_{8}$ and $w_{0}$ are almost
symmetric around the dotted vertical line. This behaviour is caused because
variations with respect to $\sigma_{8}$ and $w_{0}$ mainly impact the variance
of the PDF. Note that the signatures of $S_{8}$ and $w_{0}$ in the
$\kappa-$PDF are significantly less degenerate than they are for the
$\gamma-$2PCF derivatives shown in figure 17. Increasing (decreasing) the
value of $\sigma_{8}$ increase (decrease) the non-linear variance, which
translates in a PDF with larger (smaller) variance and then a lower (higher)
peak. Additionally, increasing (decreasing) the variance through $\sigma_{8}$
also shift to the left (right) with respect to the fiducial PDF, as the
skewness increases with the variance. Those two effects set the $S_{8}$
derivatives that are only rescaled by a factor $\sqrt{0.3/\Omega_{m,\rm fid}}$
compared with $\sigma_{8}$ derivatives. For $w_{0}$ increasing (decreasing)
$w_{0}$ decreases (increases) the variance which is opposite to the effect of
$\sigma_{8}$ as replicated in the $\gamma-$2PCF derivatives in figure 17. In
the noise-free case, the changes in $S_{8}$ and $w_{0}$ occur in an almost
redshift independent way, we observe a redshift dependent behaviour on the
noisy case due to the shape noise contribution to the variance, which is
larger for the lower redshift bins. We also observe that for various redshift
combinations, the $\Omega_{\rm m}$ derivatives are asymmetric with respect to
this peak location, which is an indication that variations on $\Omega_{\rm m}$
affect the variance and the higher order cumulants.

To understand the derivative behaviour, we study the cosmology dependence of
the PDF that is encoded in the $\kappa-$cumulants (see equation (11)). First,
to gain an intuition for the changes on the cumulants with cosmology, we write
equation (11) in terms of the reduced cumulants of the matter field in an
infinitely long cylinder with transverse angular radius
$R_{\theta}=\chi\theta$ as

$\langle\kappa_{\theta}^{l}\rangle_{c}=\int^{\chi_{s}}_{0}\,d\chi\,w(\chi)^{l}\sigma_{\text{cyl}}^{2(l-1)}(z,R_{\theta})S^{2D}_{l}(R_{\theta})\,.$
(26)

We will discuss the changes in the variance $(l=2)$ and the skewness $(l=3)$.
In the linear regime and a $\Lambda$CDM universe, the variance is proportional
to the square of the linear growth factor
$\sigma_{\text{cyl}}^{2}(z,R_{\theta})\propto D^{2}(z)\sigma_{\rm
L}^{2}(R_{\theta})$, which changes with $\Omega_{\rm m}$ and $w_{0}$.

In an EDS universe the reduced skewness is affected by the scale-dependent
variance through its logarithmic derivative (Bernardeau, 1994; Boyle  _et al._
, 2021) as

$\displaystyle
S^{2D}_{3}(R_{\theta})=\frac{\langle\delta^{3}(R_{\theta})\rangle_{c}}{\langle\delta^{2}(R_{\theta})\rangle_{c}^{2}}\approx\frac{36}{7}+\frac{3}{2}\frac{d\log\sigma^{2}_{\rm
L}(R_{\theta})}{d\log R_{\theta}}\,.$ (27)

Changes on $\Omega_{\rm m}$ of $\pm 10\%$ change the variance in a scale
dependent way by $\mp 3\%$. Note that the logarithmic derivative is redshift
independent and the $\sigma_{8}$ dependence cancels. The variance and skewness
are also affected by the information encoded in the weak lensing kernel.
Hence, for $\Omega_{\rm m}$ derivatives, the redshift dependence comes from
the weak lensing kernel contribution. The highest redshift bin (bin$5$) probes
the largest scales (see FIG 2) where the effect on the skewness is a few
percent compared with the effect on the variance, therefore for this bin and
combinations that include it, changes in $\Omega_{\rm m}$ mostly affect the
variance and then the derivatives are more symmetric around the origin. Note,
that the previous explanation holds for $\Omega_{\rm m}$ derivatives at fixed
$\sigma_{8}$. In our case, the $\Omega_{\rm m}$ derivatives at fixed $S_{8}$
present a larger asymmetry due to an extra subtraction of the $\sigma_{8}$
derivatives that appear from the transformation (24). For the non-tomographic
case, we reproduce the results of figure 5 in Boyle  _et al._ (2021) noting
that their $\Omega_{\rm m}$-derivatives are at fixed $\sigma_{8}$. In the non-
tomographic case, the integration along the line of sight combines all the
scales within the redshift range, obscuring the growth of structure
information. This information is recovered in the tomographic case by
considering the effect in the skewness at different physical scales. The lower
redshift bins are highly affected by the shape noise. As a consequence, the
effect on the skewness is erased for the lower redshift bins. The skewness
contribution is still significant for a few redshift combinations that include
intermediate redshift bins for which the effect of shape noise is sufficiently
small.

We validate the derivatives used in our analysis by using measurements from
maps of the cosmo-SLICS catalogues. For each parameter derivative, we choose
the cosmology nodes in the cosmo-SLICS hypercube for which the other
parameters change the least w.r.t the cosmo-SLICS fiducial cosmology. For each
cosmology node, we measure and predict the $\kappa-$PDF following the
procedures presented in section 3.3. In table 4 we show which cosmo-SLICS
model we use for each parameter (first column) and the corresponding values
for the cosmological parameters. The derivatives are computed by taking finite
differences with the fiducial cosmo-SLICS cosmology (Model fid). We present
derivative validations for $\\{\Omega_{\rm m},\sigma_{8}\\}$ in figure 8, we
find good agreement between the predicted and measured derivatives. For the
dark energy parameter, $w_{0}$ we could not find a suitable validation model
within cosmo-SLICS, but the theory has been previously validated in the non-
tomographic case (Boyle  _et al._ , 2021). Generally, the theoretical
framework for the $\kappa-$PDF including shape noise successfully recovers the
cosmological dependence of cosmo-SLICS (as well as the signal from intrinsic
alignment as shown in Appendix E).

| $\Omega_{\rm m}$ | $\sigma_{8}$ | $w_{0}$ | $h$
---|---|---|---|---
Model fid | $0.2905$ | $0.8364$ | $-1.0$ | $0.6898$
Model 04 $(\Omega_{m})$ | $0.3759$ | $0.8028$ | $-0.9741$ | $0.6034$
Model 18 $(\sigma_{8})$ | $0.2784$ | $0.6747$ | $-1.0673$ | $0.6747$
Table 4: cosmo-SLICS cosmologies for derivative validation.

Figure 8: Noisy PDF derivative validation for $\Omega_{\rm m}$ at fixed
$\sigma_{8}$ (upper panel) and $\sigma_{8}$ (lower panel). (Solid lines)
Theoretical derivative predictions for the cosmo-SLICS cosmology. (Dots)
derivative from measurements of the cosmo-SLICS $\kappa-$maps. The error bars
represent the error of the mean over the $20$ cosmo-SLICS realizations. The
colour scheme follows the one of FIG 4. The light lines and dots show the
comparison between the predicted and the measured $\kappa-$PDF for the
corresponding cosmo-SLICS model (See TAB. 4).

###  4.2 Data vector covariance

The covariance used for the Fisher analysis is estimated from the $\kappa-$PDF
and $\gamma-$2PCF measurements from the SLICS $\kappa-$maps and shear maps
respectively. The covariance is computed as

$C_{ij}=\big{\langle}(D_{i}-\bar{D_{i}})(D_{j}-\bar{D_{j}})\big{\rangle}\,\qquad\bar{D_{i}}=\langle
D_{i}\rangle,$ (28)

where the ensemble average is over the $N_{\rm sim}=923$ SLICS realizations,
and $\\{i,j\\}$ run over the elements of the data vector $\bm{D}$.

In order to study the correlation among the bins in the data vector and how
shape noise affect these correlations, we plot the cross correlation matrix
defined as

$r_{ij}=\frac{C_{ij}}{\sqrt{C_{ii}C_{jj}}}\,.$ (29)

In FIG 9 we show the correlation matrix for the noise-free and noisy summary
statistics, on the lower and upper diagonal triangles respectively, for the
redshift bin2. The data vector in each case is the concatenation of the
$\kappa-$PDF for three smoothing scales $\theta=\\{10,15,20\\}$ arcmin and the
two components of the $\gamma-$2PCF, $\xi_{\pm}$. Each of the blocks on the
$\kappa-$PDF zone contains $10$ $\kappa$ bins and shows correlations among
them and correlations between different smoothing scales. We first summarise
the main features of an individual $\kappa$-PDF covariance, for which a
detailed explanation can be found in Uhlemann  _et al._ (2022). Around the
diagonal, consecutive $\kappa$ bins are strongly correlated due to the
overlapping of the circular apertures (generated during the Top-Hat
smoothing). We remove the mean convergence in our individually simulated small
patches to more closely resemble the case of cutting patches from a large
portion of the full sky. This makes the super-sample covariance term causing
underdense and overdense bins to be negatively correlated subdominant,
revealing a more complex 3x3 tiling pattern caused by the next-to-leading
order term. We also observe correlations among the $\kappa-$PDF at different
smoothing scales, since the circular apertures on the larger scales partially
capture similar information present in the smaller ones. As expected, the
cross-correlations decrease with an increasing difference in the smoothing
scales. As shape noise has a stronger effect on the lower redshift bins, when
comparing the lower and upper triangles we can see a slightly smaller
difference for the correlations. This is less evident for the higher redshift
bins as is shown in the next figure (figure 10). In the presence of shape
noise, the diagonal of $\kappa-$PDF covariance is mainly affected by the Top-
Hat filtering which induces correlations between the shape noise values
(Uhlemann  _et al._ , 2022).

The $\gamma-$2PCF zone involve correlations among the logarithmic spaced bins
for each of the components $\xi_{\pm}$ with the previously discussed scale
cuts. We observe strong positive correlations in $\xi_{+}$, forming a square,
among small angular scales. The two components $\xi_{\pm}$ are strongly
correlated too, but with a shift in the angular scales. In the presence of
shape noise, all the correlations are diminished, with the correlations in
$\xi_{-}$ being more affected. Additionally, we observe small correlations of
$~{}8\%$ between the $\kappa-$PDF and the $\gamma-2$PCF because of the
correlation of the amplitude of the $\gamma-2$PCF with the $\kappa-$variance
that sets the shape of the $\kappa-$PDF.

Figure 9: Cross-correlation matrix from SLICS simulations bin$2$. $\kappa-$PDF
for the smoothing scales $\theta=\\{10,15,20\\}$ arcmin and
$\xi_{\pm}(\theta)$. The blue dashed lines indicate the boundaries between
different segments of the data vector

.

The correlation matrix in FIG 10 consider correlations among the highest 4
auto redshift bins ($\rm bin5,\rm bin4,\rm bin3,\rm bin2$) for the
$\kappa$-PDF including the 3 smoothing scales and for the two components of
the $\gamma-$2PCF, $\xi_{\pm}(\theta)$. We observe in the $\kappa-$PDF zone
that correlations between non-consecutive redshift bins are smaller than the
ones between consecutive redshift bins. Correlations among different redshift
bins are diminished because of shape noise, which affects lower redshift bins
more strongly. Correlations among different $\xi_{+}$ measured bins are larger
for the lower redshift bins, as these bins capture smaller scales compared
with the highest redshift bins (Joachimi  _et al._ , 2008). Correlations
between the $\kappa-$PDF and the $\gamma-$2PCF range from $~{}-8\%$ to
$~{}8\%$ across all redshift bins.

Figure 10: Cross-correlation matrix from SLICS simulations for the case of
single redshift bins and only auto-correlation functions for 4 redshift bins
($\rm bin5,\rm bin4,\rm bin3,\rm bin2$).

###  4.3 Results

Now we present our main results from the Fisher forecast for the different
cases we numbered at the beginning of this section.

To mimic a StageIV-like survey area of $A_{\rm survey}=$15,000deg2, the data
covariance matrix is scaled as $A_{\rm SLICS}/A_{\rm survey}$, where $A_{\rm
SLICS}=$ 100deg2 is the area of the SLICS simulations. In order to mitigate
the sampling noise in the covariance estimation, that may affect the parameter
covariance (Hartlap  _et al._ , 2007; Dodelson and Schneider, 2013), we use
the Sellentin-Heavens prescription (Sellentin and Heavens, 2016) where the
inverse data covariance is rescaled by the factor

$\alpha=\frac{N_{\rm sim}-N_{s}+N_{p}+1}{N_{\rm sim}-1}$ (30)

with $N_{s}$ the length of the data vector $D$, $N_{\rm sim}$ is the number of
realisations for the covariance estimate and $N_{p}$ is the number of
parameters to estimate.

The predicted marginalised 1-$\sigma$ uncertainties on the set of parameters
$p_{i}=\\{\Omega_{\rm m},S_{8},w_{0}\\}$ are determined from the Fisher matrix
from equation (22), as

$\Delta p_{i}=\sqrt{(F^{-1})_{ii}}\,.$ (31)

We summarise the forecasted parameter uncertainties for the different cases in
Table 5.

| $\Delta\Omega_{m}\left(10^{-3}\right)$ | $\Delta S_{8}\left(10^{-3}\right)$ | $\Delta w_{0}\left(10^{-2}\right)$
---|---|---|---
$P(\kappa)$ No Tomography | $9.79$ | $3.29$ | $3.85$
$P(\kappa)$ Tomography | $1.88$ | $2.11$ | $1.65$
$P_{{\rm noisy}}(\kappa)$ No Tomography | $10.74$ | $3.92$ | $4.36$
$P_{{\rm noisy}}(\kappa)$ Tomography only auto | $4.32$ | $3.53$ | $3.19$
$P_{{\rm noisy}}(\kappa)$ Tomography | $2.79$ | $2.99$ | $2.65$
$\xi(\theta)$ Tomography | $3.47$ | $4.85$ | $3.40$
$P_{{\rm noisy}}(\kappa)$ \+ $\xi$ Tomography | $2.08$ | $2.72$ | $2.34$
Table 5: Fisher forecast. $1\sigma$ predicted uncertainties (equation 31) for
the set of parameters $\\{\Omega_{\rm m},S_{8},w_{0}\\}$.

In the following, we investigate the constraining power of the tomographic
analysis with the $\kappa-$PDF and the $\gamma-$2PCF. We start by studying the
effect of tomography on the constraints compared with the non-tomographic
case.

##### Tomographic enhancement of the $\kappa-$PDF constraints

We compare the constraints obtained by performing a tomographic and non-
tomographic analysis with the $\kappa-$PDF. In FIG 11 we show Fisher contours
obtained with the noisy $\kappa-$PDF for: the non-tomography case, tomography
including the 4 higher individual redshift bins along with the non-tomographic
case, and the tomography with 14 redshift bin combinations. We clearly see how
the tomography tightens constraints. In all cases we combine three smoothing
scales $\theta=\\{10,15,20\\}$ arcmin. As illustrated, in the presence of
shape noise, the constraints on $\\{\Omega_{\rm m},S_{8},w_{0}\\}$ are
improved by a factor of {3.8,1.3,1.6}. For the noise-free $\kappa-$PDF the
constraints are improved by a factor of $\\{5.2,1.6,2.3\\}$. The significant
enhancement of the constraints on $\Omega_{\rm m}$ is due to the additional
information provided for the skewness and the higher order moments at
different redshift bins, as we explained in section 4.1. By considering
different redshift bin contributions, it is possible to recover the
information of the growth of structures at different scales. Hence, tomography
allows disentangling $\Omega_{\rm m}$ from $w_{0}$, by distinguishing the
growth effect of $w_{0}$ from the combined growth and skewness effect in
$\Omega_{\rm m}$.

The noise-free and noisy $\kappa-$PDF constraints are degraded by
approximately a factor of $1.5$ for all parameters when adding shape noise.
Degradation in the constraints is caused by a reduction in the amplitude of
the noisy PDF derivatives of about $30\%$ as shown in figure 7, with the lower
redshifts combinations being the more affected, and because the shape noise
increases the diagonal of the data covariance (Uhlemann  _et al._ , 2022).
Those two effects reduce the signal-to-noise only degrading the constraints,
keeping the orientation of the contours almost unchanged.

Figure 11: $1\sigma$ Fisher forecast constraints for $\\{\Omega_{\rm
m},S_{8},w_{0}\\}$ from the $\kappa-$PDF. Tomography including 14 redshift
combinations (solid violet contour). Tomography including only single and “All
bins” redshift bins (dashed teal contour). No tomography (only “All bins”,
dotted coral contour). In all cases we combine three smoothing scales
$\theta=\\{10,15,20\\}$ arcmin.

##### Joint analysis $\kappa-$PDF and $\gamma-$2PCF

In figure 12 we present our main result where we combine the $\kappa-$PDF and
$\gamma-$2PCF, including both components $\xi_{\pm}$. We include the
$\kappa-$PDF for the three smoothing scales $\\{10,15,20\\}$ arcmin (see
appendix D for a study with a single smoothing scale) and consider scales
$\theta>10$ arcmin and $\theta>50$ arcmin for $\xi_{\pm}$, respectively (see
the predicted $\xi_{\pm}$ derivatives in FIG 17). The $\kappa-$PDF outperforms
the $\gamma-$2PCF by approximately $23\%$ for $\Omega_{\rm m}$, $57\%$ for
$S_{8}$ and a $53\%$ for $w_{0}$. The improvement stems from the non-Gaussian
information encoded in the skewness and higher order moments within the
$\kappa-$PDF which breaks the degeneracy between $w_{0}$ and $\sigma_{8}$ that
is pronounced for the $\gamma-$2PCF as illustrated in figure 17. Combining
both summary statistics leads to an improvement in the constraints by a factor
of $\\{1.7,1.8,1.5\\}$ for $\\{\Omega_{\rm m},S_{8},w_{0}\\}$ respectively
compared with the $\gamma-$2PCF.

Figure 12: Joint analysis: $1\sigma$ Fisher forecast constraints from the
$\kappa-$PDF and $\gamma-$2PCF. Noisy $\kappa-$PDF (dashed violet contour)
combining three smoothing scales $\theta=\\{10,15,20\\}$ arcmin.
$\xi=\xi_{+}+\xi_{-}$ (dotted red contour), for scales $\theta>~{}10$ arcmin
and $\theta>~{}50$ arcmin for $\xi_{\pm}$ respectively. Combined $\kappa-$PDF
and $\xi-$2PCF are shown with the solid blue contours.

We have observed that the tomographic $\kappa-$PDF holds strong potential for
constraining the parameter set $\\{\Omega_{\rm m},S_{8},w_{0}\\}$. This is
because it can track the growth of structure and capture non-Gaussian
information across different scales. We now investigate how these conclusions
are affected when forecasting for a different number of parameters. We begin
by including the Hubble parameter, considering $~{}10\%$ steps around the
fiducial value, $h^{-}=0.6198$, $h^{+}=0.7598$. Changing $h$ primarily affects
the variance in a redshift-dependent manner and impacts the skewness without
modifying the growth factor like $\Omega_{\rm m}$. Both $\Omega_{\rm m}$ and
$h$ change the variance and skewness across the redshift bins, while $S_{8}$
and $w_{0}$ do not (significantly). Therefore, constraints with varying $h$
benefit significantly from the additional non-Gaussian information at
different scales, leading to improved constraints as shown in figure 13. In
this case, the joint analysis with the $\kappa-$PDF and the $\gamma-$2PCF
offers a constraint improvement of about a factor of 2.5 for $\\{\Omega_{\rm
m},S_{8},w_{0},h\\}$ compared with the $\gamma-$2PCF alone. This is because
the constraints from the $\gamma-$2PCF are degraded by a factor of
$\\{1.7,1.4,1.8\\}$ for $\\{\Omega_{\rm m},S_{8},w_{0}\\}$ respectively when
adding $h$, while the constraints from $\kappa-$PDF only degrade by less than
$2\%$.

Figure 13: Fisher forecast for $1-\sigma$ constraints from the joint
$\gamma-$2PCF and $\kappa$-PDF tomography as in figure 12 including $h$.

We now focus on forecasting just two parameters, holding the other two fixed.
In figure 14 we show the $1-\sigma$ fisher contours for $\Omega_{\rm m}$ and
$S_{8}$ at fixed $w_{0}$ (for other two-parameter cases, see figure 19 in
Appendix D). When constraining only two cosmological parameters, the
$\kappa-$PDF and the $\gamma-$2PCF provide similar constraints with slightly
different degeneracy directions between the parameters. This helps to improve
the constraint when combining both summary statistics. Hence, when considering
more than two parameters, the $\gamma-$2PCF loses constraining power due to a
degeneracy between the parameters, different from the $\kappa-$PDF that is
able to disentangle all parameters obtaining better constraints.

Figure 14: Fisher forecast for $1-\sigma$ constraints from the joint
$\gamma-$2PCF and $\kappa$-PDF tomography as in figure 12 but for fixed
($w_{0}$, $h$).

##### Relation to previous works

Our findings regarding the strong constraining power of the tomographic
$\kappa-$PDF across mildly nonlinear scales and its complementarity with the
$\gamma-$2PCF are consistent with previous works. For example, the non-
tomographic PDF from Boyle  _et al._ (2021), which is based on the LDT,
demonstrated the power of the $\kappa-$PDF when combining different smoothing
scales and the complementarity for 2PCF and CMB data, in particular for dark
energy. We find that tomography not only preserves but enhances this
information. Thiele  _et al._ (2020) employed a halo-model formalism,
originally formulated for the thermal Sunyaev-Zel’dovich effect, to compute
the one and two-point convergence PDFs, thus including the PDF covariance.
This analytical model accurately describes the positive tail of the PDF
describing large convergence values dominated by massive halos. However, it is
strongly sensitive to small-scale effects and is less accurate on mildly
nonlinear scales and negative convergence values. The authors perform a Fisher
forecast for key cosmological parameters like the total neutrino mass, matter
density, and the amplitude of density fluctuations. Their analysis showed that
the convergence PDF matches or outperforms constraints from the power spectrum
and that by combining both statistics constraints improve. Liu and
Madhavacheril (2019) performed a tomographic analysis that combines the
convergence PDF and power spectrum to constrain neutrino masses. As the impact
of neutrinos is stronger on small scales the authors use cosmological massive
neutrino simulations that provides PDF measurements valid in the one-halo
regime. They found that tomography has the potential to significantly improve
the constraints, obtaining tighter constraints with the PDF than with the
power spectrum. Finally, Giblin  _et al._ (2023) used $\kappa-$PDFs
measurements from the cosmo-SLICS simulations to train a Gaussian process
regression emulator and carry out an MCMC analysis to constrain
$\\{\Omega_{\rm m},S_{8},h,w_{0}\\}$. The authors demonstrate the constraining
power of the tomographic convergence PDF at three highly non-linear scales
(2,7 and 13 arcmin) and demonstrate that combining different smoothing scales
leads to improve the cosmological parameter constraints.

##  5 Conclusion

##### Summary

In this paper, we have studied the constraining power of the tomographic one-
point PDF for the weak lensing convergence field. We consider a StageIV-like
galaxy distribution which we have sliced into $5$ equipopulated redshift bins.
We predict the $\kappa-$PDF using the public CosMomentum code (Friedrich  _et
al._ , 2020) based on the large deviation theory (LDT). We reconstruct the
convergence field from the shear components by performing a KS inversion and
forward model its impact on the non-linear variance following the procedure of
Barthelemy  _et al._ (2023). To account for shape noise in $\kappa-$PDF
predictions, we convolve with a Gaussian (proportional to intrinsic
ellipticity variance). Since shape noise impacts the lower redshift bins more,
we propose a tomographic strategy by considering the combinations of
consecutive redshift bins (quarters, triples, pairs) to minimise noise.

We validated the theoretical predictions with measurements from numerical
cosmic shear simulations sampling multiple $w$CDM cosmologies. We found good
agreement between the predictions and measurements, with errors below $5\%$ in
the 2-sigma around the peak of the PDF for both the noise-free and noisy cases
for all redshift bins. In the presence of shape noise, the predictions become
more accurate as the Gaussian shape noise conceals nonlinear effects from
small physical scales at low redshifts. For comparison, we also measured
$\gamma-$2PCF from the SLICS shear maps and validated them with pyccl
predictions.

We have performed a Fisher forecast to constrain $\\{\Omega_{\rm
m},S_{8},w_{0}\\}$. We compute the derivatives from finite differences of the
theoretical predictions and estimate the covariance from measurements on the
fiducial SLICS maps. The tomographic analysis with the $\kappa-$PDF enhanced
the constraining power compared to the non-tomographic case. In particular,
there is a significant enhancement on the $\Omega_{\rm m}$ constraints due to
the additional information on the growth of structure and the skewness at
different redshift bins which allows disentangling $\Omega_{\rm m}$ from
$w_{0}$. As expected, the $\kappa-$PDF outperforms the $\gamma-$2PCF,
especially on $S_{8}$ and $w_{0}$ by capturing additional non-Gaussian
information from higher order moments. A joint analysis of the $\kappa$-PDF
and the $\gamma-$2PCF improves constraints by around a factor of $1.5$
compared with the $\gamma-$2PCF alone.

In summary, the tomographic weak lensing PDF is a powerful tool to extract new
information from mildly nonlinear scales. The $\kappa-$PDF is not only easy to
measure, it is also accurately predicted on mildly non-linear scales. It shows
strong potential in constraining cosmological parameters by accessing the non-
Gaussian and growth structure information across different redshift bins.
Thus, moving beyond two-point correlation functions will be the key to
increase the precision in the inference of cosmological parameters in the era
of Stage-IV surveys.

##### Outlook.

For measuring the $\kappa-$PDF in a realistic survey mask, we can select a
coverage threshold above which smoothed cells are included. For stringent
thresholds close to 100% as suggested in Barthelemy  _et al._ (2023)
considering only observed pixels is sufficient. For smaller coverage
thresholds around $80\%$, we could apply the same approach as used for
density-split statistics (Gruen  _et al._ , 2018) to ensure an equal masking
fraction across all cells. Our theoretical model for the tomographic
$\kappa$-PDF can be adapted to include further weak lensing systematics as
discussed in Barthelemy  _et al._ (2023). In particular, intrinsic alignments
can be included for the nonlinear-alignment model, which effectively modifies
the weight function by adding a term proportional to the source redshift
distribution. In Appendix E we study and include this effect in our model for
the $\kappa-$PDF and show that it reproduces the cosmo-SLICS measurements
including IA with excellent accuracy. We will use this model for upcoming data
analyses in future work. Additional systematic effects can be incorporated
into the analysis. Photometric redshift errors affecting the $n(z)$ and hence
the lensing weight could be propagated through the theoretical pipeline.
Multiplicative and additive shear biases can be added, and perturbative
recipes have indicated that the $\kappa-$PDF can help break degeneracies
present in the $\gamma-$2PCF (Patton  _et al._ , 2017). On our mildly
nonlinear scales, baryonic effects can likely be captured through their impact
on the variance (as also adopted in the HSC Y1 analysis Thiele  _et al._ ,
2023; Barthelemy  _et al._ , 2023; Grandón  _et al._ , 2024). The simulated
covariances can be replaced by mock catalogues obtained from shifted log-
normal fields that have proven highly successful in covariance modelling for
density-split statistics (Friedrich  _et al._ , 2018) and the $\kappa$-PDF
(Boyle  _et al._ , 2023; Uhlemann  _et al._ , 2022), for which analytical
recipes are available. The tomographic $\kappa$-PDF could be enhanced through
an extension to joint PDFs of multiple redshift bins, most efficiently
captured through a nulling procedure that localises the weighting kernels thus
simplifying the correlation structure (Barthelemy  _et al._ , 2022)

## Acknowledgements

LC, CU and JHD acknowledge funding from the STFC Astronomy Theory Consolidated
Grant ST/W001020/1 from UK Research & Innovation. CU’s research was in part
funded by the European Union (ERC StG, LSS_BeyondAverage, 101075919). JHD is
supported by an STFC Ernest Rutherford Fellowship (project reference
ST/S004858/1). AB’s work is supported by the ORIGINS excellence cluster. The
authors thank the members of the Euclid work package for higher-order weak
lensing statistics and the Rubin LSST:DESC higher-order statistics topical
team for useful discussions. CU thanks the German Centre for Cosmological
Lensing for the integration into the national lensing community.

##  Appendix A CGF of density in cylinders from Large deviation theory

The large deviation theory (LDT) is a mathematical method to quantify the
probability of observing rare events on the tails of a probability
distribution that deviates from typical values (Dembo and Zeitouni, 2009). As
the large scale structure of the universe is non-Gaussian on middle and small
scales, the LDT allows quantifying the asymptotic exponential shape of the PDF
for the density field $\rho$ as the variance $\sigma^{2}$ goes to zero
(Bernardeau and Reimberg, 2016). The PDF satisfy the large deviation principle
if the rate function $\Psi_{\rho}(\rho)$, that characterises the exponential
decay of the PDF, satisfy the following condition

$\Psi_{\rho}(\rho)=-\lim_{\sigma^{2}\xrightarrow{}0}\sigma^{2}\log\left(P(\rho)\right)\,.$
(32)

The LDT uses the contraction principle, which gives the most likely map
between the initial and final density. For line-of-sight projected fields, the
initial conditions have cylindrical symmetry, hence, the most likely mapping
between the initial and final density contrast is the cylindrical collapse
(Valageas, 2002). For Gaussian initial conditions, the rate function for the
linear density contrast $\delta_{\rm L}$ in cylinders is then given by

$\Psi_{\delta_{\rm L}}(\delta_{\rm L})=\frac{\sigma_{\rm
L}^{2}(\chi\theta,z)}{2\sigma_{\rm L}^{2}(r_{\rm ini},z)}\delta_{\rm
L}^{2}\,,$ (33)

where $\sigma_{\rm L}^{2}$ is the linear cylindrical variance in the
Lagrangian radius $r_{\rm ini}$. $r_{\rm ini}$ is related to the dimension and
surface density $\rho$ of the cylinder through mass conservation (assuming
long cylinders) as

$r_{\rm ini}=\chi\theta\rho^{1/2}\,.$ (34)

The time evolution of the initial fluctuation $\zeta_{\rm SC}(\delta_{\rm L})$
and the linear density contrast $\delta_{\rm L}$ is driven by the following
differential equation (Bernardeau, 1992, 1995)

$\zeta_{\rm SC}\delta_{\rm L}^{2}\zeta_{\rm
SC}^{\prime\prime}-c\left(\delta_{\rm L}\zeta_{\rm
SC}^{\prime}\right)^{2}+\frac{3}{2}\zeta_{\rm SC}\zeta_{\rm
SC}^{\prime}-\frac{3}{2}\zeta_{\rm SC}^{2}(\zeta_{\rm SC}-1)=0\,,$ (35)

where the primes stand for derivatives with respect to $\delta_{\rm L}$.
$c_{2D}=\frac{3}{2}$ and $c_{3D}=\frac{4}{3}$ for $2D$ and $3D$, respectively.
In an EDS background, the solution can be written as

$\zeta(\delta_{\rm L})=\left(1-\frac{\delta_{\rm
L}}{\nu}\right)^{-\nu}\,,\qquad\delta_{\rm
L}=\nu\left(1-\rho^{-\frac{1}{\nu}}\right)\,,$ (36)

this is a fitting function that respects asymptotic behaviour in the
$\delta_{\rm L}$ goes to $-\infty$ regime. The value of $\nu$ depends on the
dimension and is set to reproduce the tree-order reduced skewness $S_{3}$,
which captures the behaviour for small and intermediate values of $\delta_{\rm
L}$. For cylindrical symmetry and considering infinitely long cylinders the
skewness in an EDS universe is given by (Uhlemann  _et al._ , 2018b)

$\displaystyle S_{3}^{\rm
2D}(R)=\frac{\langle\delta^{3}(R)\rangle_{c}}{\langle\delta^{2}(R)\rangle_{c}^{2}}=\frac{36}{7}+\frac{3}{2}\frac{d\log\sigma^{2}_{\rm
L}(R)}{d\log R}\,,$ (37)

such that $\nu_{2D}=1.4$.

Within the LDT theory, the scaled cumulant generating function (SCGF)101010The
CGF and the reduced cumulant generating function SCGF are defined as
$\phi_{X}(\lambda)=\sum_{n=1}^{\infty}\langle\delta^{n}\rangle\frac{\lambda^{n}}{n!}\,,\quad\varphi_{X}(\lambda)=\lim_{\langle\delta^{2}\rangle\rightarrow
0}\sum_{n=0}^{\infty}S_{n}\frac{\lambda^{n}}{n!}\,,\quad
S_{n}=\frac{\langle\delta^{n}\rangle}{\langle\delta^{2}\rangle^{n-1}}\,,$ is
predicted from Varadhan’s theorem and is given by the Legendre-Fenchel
transformation of the rate function $\Psi_{\rho}(\rho)$ as

$\varphi(\lambda)=\sup_{\rho}\left[\lambda\rho-\Psi_{\rho}(\rho)\right]\,,$
(38)

or as a simple Legendre transformation if $\Psi_{\rho}(\rho)$ is convex

$\varphi(\lambda)=\lambda\rho-\Psi_{\rho}(\rho)\,,\qquad\lambda=\frac{\partial\Psi_{\rho}}{\partial\rho}\,.$
(39)

The SCGF of the 3D density contrast in a cylindrical filter of transverse size
$\chi\theta$ and long length $L$ is given at leading order by

$\varphi^{\text{l.o}}_{\text{cyl}}(\lambda)=\sup_{\rho}\left[\lambda\delta_{\text{cyl}}-\frac{\sigma_{\rm
L}^{2}(\chi\theta,L,z)}{2\sigma_{\rm
L}^{2}(\chi\theta(1+\delta)^{1/2},L,z)}\delta_{\rm L}^{2}(1+\delta)\right].$
(40)

The SCGF and the CGF are related in the zero variance limit as

$\varphi_{\text{cyl}}^{\rm l.o}(\lambda)=\sigma^{2}_{L}\phi^{\rm
l.o}\left(\frac{\lambda}{\sigma_{\rm
L}^{2}}\right),\quad\phi_{\text{cyl}}^{\rm
l.o}(\lambda)=\frac{1}{\sigma^{2}_{L}}\varphi^{\rm
l.o}\left(\lambda\sigma_{\rm L}^{2}\right).$ (41)

The LDT formalism gives the cumulants of the non-linear density contrast at
the leading order in perturbation theory. As a consequence, the variance given
by the CGF is not valid in the mildly non-linear regime and beyond. It is
convenient to rescale the CGF by the non-linear variance
$\sigma_{\text{cyl}}^{2}=\langle\delta_{\chi\theta,\text{cyl}}^{2}\rangle$
such that the rate function on cylinder becomes

$\Psi_{\text{cyl}}(\rho)=\frac{\sigma_{\rm
L}^{2}(\chi\theta,z)}{\sigma_{\text{cyl}}^{2}(\chi\theta,z)}\frac{\delta_{\rm
L}^{2}(\rho)}{2\sigma_{\rm L}^{2}(\chi\theta\rho^{1/2},z)}\,.$ (42)

and then the following equations (41) the CGF is given by

$\phi(\lambda)_{\text{cyl}}=\frac{\sigma_{\rm
L}^{2}}{\sigma_{\text{cyl}}^{2}}\phi^{\text{l.o}}_{\text{cyl}}\left(\frac{\sigma_{\text{cyl}}^{2}}{\sigma_{\rm
L}^{2}}\lambda\right)\,$ (43)

The non-linear variance can be obtained from prediction of the power spectrum,
for example from Halofit (Smith  _et al._ , 2003; Peacock and Smith, 2014) or
HMcode(Mead  _et al._ , 2016, 2020).

##  Appendix B Gaussianity test for $\kappa-$PDF bins

In figure 15 we demonstrate the Gaussianity of the distribution of the lowest
(solid) and highest (dashed) used bins of the noisy $\kappa-$PDF for three
different smoothing scales for the low redshift bin2. For each SLICS
realisation we compute the fluctuation of the respective bin with respect to
the mean and we normalise them with the measured variance. We then make
histograms from the fluctuations and observe that they closely follow a
Gaussian distribution. We perform the test for other redshift bin combinations
and PDF bins obtaining similar results. For our combined tomographic
$\kappa-$PDF data, we perform a $\chi^{2}-$test to verify the Gaussianity
along with the quality of the inverse data covariance (proposed in Friedrich
and Eifler, 2018). If the inverse data covariance matrix is accurate enough,
it should follow a $\chi^{2}$ distribution, which is demonstrated in figure
16. For comparison, we draw a reference sample from a data vector build from a
Gaussian distribution with the same mean and covariance from SLICS
measurements. Both data vectors ($D$ and $D$ Gaussian) deviate slightly from
the $\chi$-squared distribution due to sampling noise. In the Fisher forecast,
we mitigate this by rescaling the inverse data covariance matrix following the
Sellentin-Heavens prescription (see equation 30).

Figure 15: Distribution of realisations around the mean for bin2 including
three smoothing scales. Solid black like represent the Gaussian distribution
with mean zero and unit variance.

Figure 16: $\chi^{2}$ test for the inverse covariance. $\chi^{2}-$test for the
noisy $\kappa-$PDF data vector (Light blue). $\chi^{2}-$test for the data
vector build from a multivariate normal distribution with same mean and
covariance as our data vector $D$ from the SLICS measurements (dark blue). The
data vector, $D$, includes three smoothing scales and 14 redshift combinations
with 10 PDF bins each

.

##  Appendix C $\gamma$-2PCF derivatives and forecast

In figure 17 we show the $\xi_{\pm}$ derivatives for the set of parameters
$p=\\{\Omega_{\rm m},S_{8},w_{0}\\}$. Note that in this case we rescale the
$w_{0}$ derivatives by a factor of $10$ for better visualisation.

Figure 17: $\gamma-$2PCF derivatives. Upper panel: $\xi_{+}$ derivatives.
Bottom panel: $\xi_{-}$ derivatives. The left panel show the derivatives with
respect to $\Omega_{\rm m}$ and the right panel show the derivatives for
$S_{8}$ and $w_{0}$. Note that $w_{0}$ has been amplified by a factor of 10
for better visualization.

In figure 18 we present the $1-\sigma$ fisher contours for $\xi_{\pm}$ and
their combination $\xi$ when adding tomography. Here we use scales
$\theta>~{}5$ arcmin and $\theta>~{}30$ arcmin (scales larger than the scales
in the darker shaded region in figure 17) for $\xi_{+}$ and $\xi_{-}$
respectively. Note that in this analysis, we are including smaller scales that
were neglected in the combined analysis with the $\kappa$-PDF. Uncertainties
given by using the $\xi_{+}$ and $\xi_{-}$ statistics separately are presented
in table 6. The constraining power of the $\gamma-$2PCF is dominated by the
constraints from $\xi_{+}$. Combining both components of $\gamma-$2PCF
provides an enhancement of the constraints of $~{}7\%$ with respect to
$\xi_{+}$. By neglecting the smaller scales in our analysis in section 4 and
the comparison with the $\kappa-$PDF we are loosing most of the constraining
power of the $\gamma-$2PCF. By adding the smaller scales, there is an
improvement in the constraints given by the $\gamma-$2PCF of $~{}12\%$.

| $\Delta\Omega_{m}\left(10^{-3}\right)$ | $\Delta S_{8}\left(10^{-3}\right)$ | $\Delta w_{0}\left(10^{-2}\right)$
---|---|---|---
$\xi_{+}$ | $3.52$ | $4.60$ | $4.33$
$\xi_{-}$ | $4.73$ | $6.59$ | $6.25$
$\xi$ | 3.27 | $4.26$ | $4.14$
Table 6:  Fisher 1-$\sigma$ uncertainties from the $\gamma-$2PCF tomography.
Figure 18: $1-\sigma$ Fisher forecast constraints from the $\gamma-$2PCF
tomography. Constraints from $\xi_{+}$ along (dark blue contours) for scales
$\theta>~{}5$ arcmin, constraints from $\xi_{-}$ along (dark green) for scales
$\theta>~{}30$ arcmin. Combined constraints $\xi$ (red contours).

##  Appendix D Additional joint $\gamma$-2PCF and $\kappa-$PDF forecast

Here we present extra joint $\gamma$-2PCF and $\kappa-$PDF Fisher contours.

Figure 19 shows the extra cases when fixing a parameter at the time, i.e. for
fixed $\\{S_{8},h\\}$ and $\\{\Omega_{\rm m},w_{0}\\}$. The $\kappa-$PDF and
the $\gamma-$2PCF give similar constraints. The $\kappa-$PDF has a slightly
extra power in constraining $S_{8}$ and $h$ while the $\gamma-$2PCF performs
slightly better in constraining $\Omega_{\rm m}$, and both provide similar
constraints for $w_{0}$, although this similarity is broken due to the
degeneracy between $w_{0}$ and $S_{8}$. As we explain in section 4.3 the
$\gamma-$2PCF loses constraining power when forecasting more parameters while
the $\kappa-$PDF is able to hold the constraint power outperforming the
$\gamma-$2PCF.

Figure 19: Fisher forecast for $1-\sigma$ constraints from the joint
$\gamma-$2PCF and $\kappa$-PDF tomography as in figure 12 but for fixed
$\\{S_{8},h\\}$ (right pannel), and $\\{\Omega_{\rm m},w_{0}\\}$ (left pannel)

Finally, we investigate the impact of employing a single smoothing scale for
$\kappa-$PDF. In section 4.3 we combine the $\kappa-$PDF for three smoothing
scales: $\\{10,15,20\\}$ arcmin. As the smoothing scale increases, the
smoothed field becomes more Gaussian. Consequently, the $\kappa-$PDF at $10$
arcmin retains more non-Gaussian information compared to larger scales. This
translates to a degradation of constraints on cosmological parameters derived
from the Fisher analysis at larger smoothing scales. However, the scale
dependence of derivatives with respect to $\Omega_{\rm m}$ (and $h$)
introduces slight variations in the degeneracy between $\Omega_{\rm m}$ and
$S_{8}$ and $w_{0}$ in the Fisher contours at different scales. This allows to
partially break these degeneracies and achieve tighter constraints by
combining information from multiple smoothing scales.

Figure 20 compares the $1-\sigma$ Fisher contours for $\\{\Omega_{\rm
m},S_{8},w_{0}\\}$ using the $\kappa-$PDF with a single smoothing scale
$\theta=10.0$ arcmin, the $\kappa-$PDF with three combined scales
$\\{10,15,20\\}$ and the $\gamma-$2PCF. When employing the $\kappa-$PDF at a
single smoothing scale for forecasting three parameters, both statistics offer
comparable constraints and become complementary, leading to improved overall
constraints when combined. However, if an additional parameter is included in
the forecast, as mentioned earlier, the $\gamma-$2PCF loses constraining power
such that the $\kappa-$PDF at a single smoothing scale outperforms the
$\gamma-$2PCF. Combining the information from three smoothing scales tighter
the constraints by a factor of $\\{1.5,1.6,1.4\\}$ for $\\{\Omega_{\rm
m},S_{8},w_{0}\\}$ respectively with respect to the single scale.

Figure 20: Fisher forecast for $1-\sigma$ constraints from the $\kappa$-PDF
considering a single smoothing scale $\theta=10$ arcmin (blue contour), and
the $\kappa$-PDF combining three smoothing scales $\\{10,15,20\\}$ arcmin
(violet contour) and the $\gamma-$2PCF and as in figure 12 (red contour).

##  Appendix E Intrinsic alignment effects on the PDF

Besides the intrinsic shape of the source galaxies (shape noise), intrinsic
alignment (IA) is also a major source of systematic noise in the shear signal.
Intrinsic alignment arises from external gravitational tidal fields (galaxies
are influenced by their local environment) that generate physical correlations
between the shapes of neighbouring galaxies. The physics of this effect is not
fully understood, and it is galaxy-type dependent, making it a significant
challenge to interpret the shear signal. There are multiple models to describe
the IA, as for example, the linear and non-linear alignment models (LA/NLA,
Hirata  _et al._ , 2007; Bridle and King, 2007), or the Tidal Alignment and
Tidal Torquing model (TATT, Blazek  _et al._ , 2019). A summary of current IA
models can be found in Lamman  _et al._ (2023).

Here, we study the impact of IA as described by the non-linear alignment model
(NLA) on the $\kappa-$PDF. We include the effect of IA on the theoretical
predictions following the prescription given in section 6.1 of Barthelemy  _et
al._ (2023) and we validate the model with the measurements from a set of
cosmo-SLICS that has been infused with IA signal.

The NLA model assumes that galaxies preferably align with the surrounding
tidal fields in a redshift dependent way, such that the contributions of the
IA to the ellipticity are given by Harnois-Déraps  _et al._ (2022)

$\displaystyle\epsilon_{\rm IA,1}$ $\displaystyle=A(z)(s_{11}-s_{22})\,,$ (44)
$\displaystyle\qquad\epsilon_{\rm IA,2}$ $\displaystyle=2A(z)s_{12}\,,$ (45)

where $s_{ij}=\partial_{i}\partial_{j}\Phi(\bm{x})$ are the Cartesian
components of the tidal tensor. $A(z)$ encodes the redshift-dependent scaling
as

$A(z)=-A_{\rm IA}\left(\frac{1+z}{1+z_{0}}\right)^{\alpha_{\rm
IA}}\frac{\bar{C}\bar{\rho}(z)}{D(z)}\,,$ (46)

with $A_{\rm IA}$ the strength of the tidal coupling and $\alpha_{\rm IA}$
that describes the redshift evolution, $z_{0}$ is a pivot redshift,
$\bar{\rho}(z)$ is the mean density at redshift $z$ and $\bar{C}$ is a
calibrated constant chosen for historical reasons (see Brown  _et al._ ,
2002). This formalism is valid in the linear regime, this assumption extends
to the non-linear case by preserving the functional form of the effect and
using a non-linear potential.

The infusion of the IA signal into the simulation catalogues relies on the
computation of the trace-free projected tidal field. The projected tidal field
maps are constructed for each of the mass sheets in the light-cones of the
simulation and interpolated at each galaxy position, such that the observed
ellipticity is given by

$\displaystyle{\bm{\epsilon}}_{\rm obs}=\frac{\bm{\gamma}+\bm{\epsilon}_{\rm
int}}{1+\bm{\gamma}^{*}\bm{\epsilon}_{int}}\,,$ (47)

where $\bm{\epsilon}_{\rm int}$ is the intrinsic galaxy ellipticity composed
by the shape noise and the IA signal as

$\bm{\epsilon}_{\rm int}=\frac{\bm{\epsilon}_{\rm
IA}+\bm{\epsilon}}{1+\bm{\epsilon}_{\rm IA}\bm{\epsilon}}\,,$ (48)

we choose $A_{\rm IA}=1$ and $\alpha_{\rm IA}=0$.

In the NLA model, as the tidal fields are related to the matter density
contrast, the IA field $\delta_{\rm IA}$ can be written as a bias effective
expansion as

$\delta_{\rm IA}(\chi\bm{\theta},\chi)=A(z)\delta(\chi\bm{\theta},\chi)\,,$
(49)

where the bias factor is given in equation (46). The contribution to the
convergence field is then written as

$\kappa_{\rm
IA}(\chi\bm{\theta},\chi)=\int^{\infty}_{0}\,dz\frac{d\chi}{dz}\,n(z)\delta_{\rm
IA}\left(\chi\bm{\theta},\chi\right)\,,$ (50)

such that we can include the effect of IA in the predicted PDF by modifying
the projection kernel in equation (3) as

$w(\chi)\rightarrow{}w(\chi)+A(z)n(z)\,.$ (51)

We now just re-use the framework described in section 2.2 and Appendix A to
predict the IA-infused $\kappa-$PDF with our modified lensing kernel.

The proposed method to include the NLA model at the PDF level allows to
consistently infuse the IA effect on the full hierarchy of the high-order
cumulants. Nevertheless, it is interesting to note that this is approximately
equivalent to a modification of only the standard deviation in the CGF,
although with a different scaling than what we traditionally expect from tree-
order perturbation theory and equivalently our LDT formalism. Indeed we find
that no matter the sources distribution, the PDF of $\kappa/\sigma$ is mostly
unchanged by the inclusion of the IA effect. This can be seen by writing

$\frac{\kappa}{\sigma}=\frac{\int{\rm d}\chi\,(\omega(\chi)+A(z(\chi))\cdot
n(z(\chi)))\delta}{\sqrt{\int{\rm d}\chi\,(\omega(\chi)+A(z(\chi))^{2}\cdot
n(z(\chi)))\langle\delta^{2}\rangle_{c}}}.$ (52)

From there and given the ‘peaked’ nature of the lensing kernels with or
without IA, it is possible to approximate both integrals in the numerator and
denominator by their integrand values at the peak of the kernels. This leads
to
$\frac{\kappa}{\sigma}\approx\frac{\delta}{\sqrt{\langle\delta^{2}\rangle_{c}}}|_{\chi\,\text{where}\,\omega^{\prime}(\chi)=0}$.
Finally, given that the lensing kernel peaks roughly halfway between the
observer and the mean source, the NLA correction typically does not change the
peak location so that this type of modification of the kernel mostly leaves
$\kappa/\sigma$ unchanged.

Figure 21 shows the residual between the $\kappa-$PDF with and without infused
IA, for both the NLA theory predictions and the cosmo-SLIC measured PDF
averaged over 10 realizations (we exclude shape noise). As predicted by the
NLA regime, the lowest redshift bin (bin1) shows the largest impact, affecting
all combinations that include it. Hence, we only show results for the redshift
combinations that include bin1. The proposed model to include IA in the
$\kappa-$PDF predicts the measured $\kappa-$PDF with infused IA with high
accuracy.

Figure 21: Effect of IA on the $\kappa-$PDF. (Dashed lines) Residual between
the $\kappa-$ PDF measured from cosmo-SLICS simulations with and without
infused IA. All measurements are normalized by the variance of the measure PDF
without IA. (Solid lines) Residuals between the NLA model in the predicted
$\kappa-$PDF and theoretical predicted $\kappa-$PDF without IA, normalized by
the variance of the theory PDF without IA. We show results for the redshift
combinations that include bin1, which is the most affected by IA. The shaded
area represents the standard deviation of the mean across realizations.

## References

  * Laureijs  _et al._ (2011) R. Laureijs _et al._ (EUCLID), (2011), arXiv:1110.3193 [astro-ph.CO] .
  * Željko Ivezić  _et al._ (2019) Željko Ivezić _et al._ , The Astrophysical Journal 873, 111 (2019).
  * Doré  _et al._ (2014) O. Doré _et al._ (SPHEREx), (2014), arXiv:1412.4872 [astro-ph.CO] .
  * Aghamousa  _et al._ (2016) A. Aghamousa _et al._ (DESI), (2016), arXiv:1611.00036 [astro-ph.IM] .
  * Spergel  _et al._ (2015) D. Spergel _et al._ , (2015), arXiv:1503.03757 [astro-ph.IM] .
  * Kilbinger (2015) M. Kilbinger, Rept. Prog. Phys. 78, 086901 (2015), arXiv:1411.0115 [astro-ph.CO] .
  * Kaiser and Squires (1993) N. Kaiser and G. Squires, Astrophys. J. 404, 441 (1993).
  * Bartelmann and Schneider (1999) M. Bartelmann and P. Schneider, Astron. Astrophys. 345, 17 (1999), arXiv:astro-ph/9902152 .
  * Munshi  _et al._ (2014) D. Munshi, P. Coles, and M. Kilbinger, JCAP 04, 004 (2014), arXiv:1112.0495 [astro-ph.CO] .
  * Jee  _et al._ (2016) M. J. Jee, J. A. Tyson, S. Hilbert, M. D. Schneider, S. Schmidt, and D. Wittman, Astrophys. J. 824, 77 (2016), arXiv:1510.03962 [astro-ph.CO] .
  * Martinet  _et al._ (2021) N. Martinet, T. Castro, J. Harnois-Déraps, E. Jullo, C. Giocoli, and K. Dolag, Astron. Astrophys. 648, A115 (2021), arXiv:2012.09614 [astro-ph.CO] .
  * Sipp  _et al._ (2021) M. Sipp, B. M. Schäfer, and R. Reischke, Mon. Not. Roy. Astron. Soc. 501, 683 (2021), arXiv:2002.12695 [astro-ph.CO] .
  * Liu and Madhavacheril (2019) J. Liu and M. S. Madhavacheril, Phys. Rev. D 99, 083508 (2019), arXiv:1809.10747 [astro-ph.CO] .
  * Joachimi  _et al._ (2008) B. Joachimi, P. Schneider, and T. Eifler, Astron. Astrophys. 477, 43 (2008), arXiv:0708.0387 [astro-ph] .
  * Kilbinger  _et al._ (2013) M. Kilbinger _et al._ , Mon. Not. Roy. Astron. Soc. 430, 2200 (2013), arXiv:1212.3338 [astro-ph.CO] .
  * Hamana  _et al._ (2020) T. Hamana _et al._ , Publ. Astron. Soc. Jap. 72, Publications of the Astronomical Society of Japan, Volume 72, Issue 1, February 2020, 16, https://doi.org/10.1093/pasj/psz138 (2020), [Erratum: Publ.Astron.Soc.Jap. 74, 488-491-491 (2022)], arXiv:1906.06041 [astro-ph.CO] .
  * Asgari  _et al._ (2021) M. Asgari _et al._ (KiDS), Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Amon  _et al._ (2022) A. Amon _et al._ (DES), Phys. Rev. D 105, 023514 (2022), arXiv:2105.13543 [astro-ph.CO] .
  * Doux  _et al._ (2022) C. Doux _et al._ (DES), Mon. Not. Roy. Astron. Soc. 515, 1942 (2022), arXiv:2203.07128 [astro-ph.CO] .
  * Mellier  _et al._ (2024) Y. Mellier _et al._ , Euclid: 1. Overview of the Euclid mission (2024).
  * Van Waerbeke  _et al._ (2013) L. Van Waerbeke _et al._ , Mon. Not. Roy. Astron. Soc. 433, 3373 (2013), arXiv:1303.1806 [astro-ph.CO] .
  * Porth and Smith (2021) L. Porth and R. E. Smith, Mon. Not. Roy. Astron. Soc. 508, 3474 (2021), arXiv:2106.04594 [astro-ph.CO] .
  * Gatti  _et al._ (2024) M. Gatti _et al._ (DES), Phys. Rev. D 109, 063534 (2024), arXiv:2310.17557 [astro-ph.CO] .
  * Marian  _et al._ (2009) L. Marian, R. E. Smith, and G. M. Bernstein, Astrophys. J. Lett. 698, L33 (2009), arXiv:0811.1991 [astro-ph] .
  * Dietrich and Hartlap (2010) J. P. Dietrich and J. Hartlap, Mon. Not. Roy. Astron. Soc. 402, 1049 (2010), arXiv:0906.3512 [astro-ph.CO] .
  * Kacprzak  _et al._ (2016) T. Kacprzak _et al._ (DES), Mon. Not. Roy. Astron. Soc. 463, 3653 (2016), arXiv:1603.05040 [astro-ph.CO] .
  * Martinet  _et al._ (2018) N. Martinet _et al._ , Mon. Not. Roy. Astron. Soc. 474, 712 (2018), arXiv:1709.07678 [astro-ph.CO] .
  * Harnois-Déraps  _et al._ (2021) J. Harnois-Déraps, N. Martinet, T. Castro, K. Dolag, B. Giblin, C. Heymans, H. Hildebrandt, and Q. Xia, Mon. Not. Roy. Astron. Soc. 506, 1623 (2021), arXiv:2012.02777 [astro-ph.CO] .
  * Zürcher  _et al._ (2022) D. Zürcher _et al._ (DES), Mon. Not. Roy. Astron. Soc. 511, 2075 (2022), arXiv:2110.10135 [astro-ph.CO] .
  * Marques  _et al._ (2024) G. A. Marques, J. Liu, M. Shirasaki, L. Thiele, D. Grandón, K. M. Huffenberger, S. Cheng, J. Harnois-Déraps, K. Osato, and W. R. Coulton, Mon. Not. Roy. Astron. Soc. 528, 4513 (2024), arXiv:2308.10866 [astro-ph.CO] .
  * Ajani  _et al._ (2023a) V. Ajani, J. Harnois-Déraps, V. Pettorino, and J.-L. Starck, Astron. Astrophys. 672, L10 (2023a), arXiv:2211.10519 [astro-ph.CO] .
  * Vicinanza  _et al._ (2019) M. Vicinanza, V. F. Cardone, R. Maoli, R. Scaramella, X. Er, and I. Tereno, Phys. Rev. D 99, 043534 (2019), arXiv:1905.00410 [astro-ph.CO] .
  * Petri  _et al._ (2015) A. Petri, J. Liu, Z. Haiman, M. May, L. Hui, and J. M. Kratochvil, Phys. Rev. D 91, 103511 (2015), arXiv:1503.06214 [astro-ph.CO] .
  * Kratochvil  _et al._ (2012) J. M. Kratochvil, E. A. Lim, S. Wang, Z. Haiman, M. May, and K. Huffenberger, Phys. Rev. D 85, 103513 (2012), arXiv:1109.6334 [astro-ph.CO] .
  * Parroni  _et al._ (2021) C. Parroni, E. Tollet, V. F. Cardone, R. Maoli, and R. Scaramella, Astron. Astrophys. 645, A123 (2021), arXiv:2011.10438 [astro-ph.CO] .
  * Wilding  _et al._ (2021) G. Wilding, K. Nevenzeel, R. van de Weygaert, G. Vegter, P. Pranav, B. J. T. Jones, K. Efstathiou, and J. Feldbrugge, Mon. Not. Roy. Astron. Soc. 507, 2968 (2021), arXiv:2011.12851 [astro-ph.CO] .
  * Heydenreich  _et al._ (2021) S. Heydenreich, B. Brück, and J. Harnois-Déraps, Astron. Astrophys. 648, A74 (2021), arXiv:2007.13724 [astro-ph.CO] .
  * Cheng and Ménard (2021) S. Cheng and B. Ménard, Mon. Not. Roy. Astron. Soc. 507, 1012 (2021), arXiv:2103.09247 [astro-ph.CO] .
  * Barthelemy  _et al._ (2020a) A. Barthelemy, S. Codis, C. Uhlemann, F. Bernardeau, and R. Gavazzi, Mon. Not. Roy. Astron. Soc. 492, 3420 (2020a), arXiv:1909.02615 [astro-ph.CO] .
  * Boyle  _et al._ (2021) A. Boyle, C. Uhlemann, O. Friedrich, A. Barthelemy, S. Codis, F. Bernardeau, C. Giocoli, and M. Baldi, Mon. Not. Roy. Astron. Soc. 505, 2886 (2021), arXiv:2012.07771 [astro-ph.CO] .
  * Barthelemy  _et al._ (2023) A. Barthelemy, A. Halder, Z. Gong, and C. Uhlemann, (2023), arXiv:2307.09468 [astro-ph.CO] .
  * Ajani  _et al._ (2023b) V. Ajani _et al._ (Euclid), Astron. Astrophys. 675, A120 (2023b), arXiv:2301.12890 [astro-ph.CO] .
  * Bernardeau and Reimberg (2016) F. Bernardeau and P. Reimberg, Phys. Rev. D 94, 063520 (2016), arXiv:1511.08641 [astro-ph.CO] .
  * Uhlemann  _et al._ (2016) C. Uhlemann, S. Codis, C. Pichon, F. Bernardeau, and P. Reimberg, Mon. Not. Roy. Astron. Soc. 460, 1529 (2016), arXiv:1512.05793 [astro-ph.CO] .
  * Codis  _et al._ (2016) S. Codis, C. Pichon, F. Bernardeau, C. Uhlemann, and S. Prunet, Mon. Not. Roy. Astron. Soc. 460, 1549 (2016), arXiv:1603.03347 [astro-ph.CO] .
  * Uhlemann  _et al._ (2018a) C. Uhlemann _et al._ , Mon. Not. Roy. Astron. Soc. 473, 5098 (2018a), arXiv:1705.08901 [astro-ph.CO] .
  * Uhlemann  _et al._ (2020) C. Uhlemann, O. Friedrich, F. Villaescusa-Navarro, A. Banerjee, and S. Codis, Mon. Not. Roy. Astron. Soc. 495, 4006 (2020), arXiv:1911.11158 [astro-ph.CO] .
  * Friedrich  _et al._ (2020) O. Friedrich, C. Uhlemann, F. Villaescusa-Navarro, T. Baldauf, M. Manera, and T. Nishimichi, Mon. Not. Roy. Astron. Soc. 498, 464 (2020), arXiv:1912.06621 [astro-ph.CO] .
  * Cataneo  _et al._ (2022) M. Cataneo, C. Uhlemann, C. Arnold, A. Gough, B. Li, and C. Heymans, Mon. Not. Roy. Astron. Soc. 513, 1623 (2022), arXiv:2109.02636 [astro-ph.CO] .
  * Gough and Uhlemann (2022) A. Gough and C. Uhlemann, Universe 8, 55 (2022), arXiv:2112.04428 [astro-ph.CO] .
  * Barthelemy  _et al._ (2020b) A. Barthelemy, S. Codis, and F. Bernardeau, Mon. Not. Roy. Astron. Soc. 494, 3368 (2020b), arXiv:2002.03625 [astro-ph.CO] .
  * Barthelemy  _et al._ (2021) A. Barthelemy, S. Codis, and F. Bernardeau, Mon. Not. Roy. Astron. Soc. 503, 5204 (2021), arXiv:2012.03831 [astro-ph.CO] .
  * Friedrich  _et al._ (2022) O. Friedrich, A. Halder, A. Boyle, C. Uhlemann, D. Britt, S. Codis, D. Gruen, and C. Hahn, Mon. Not. Roy. Astron. Soc. 510, 5069 (2022), arXiv:2107.02300 [astro-ph.CO] .
  * Friedrich  _et al._ (2018) O. Friedrich _et al._ (DES), Phys. Rev. D 98, 023508 (2018), arXiv:1710.05162 [astro-ph.CO] .
  * Gruen  _et al._ (2018) D. Gruen _et al._ (DES), Phys. Rev. D 98, 023507 (2018), arXiv:1710.05045 [astro-ph.CO] .
  * Burger  _et al._ (2023) P. A. Burger _et al._ , Astron. Astrophys. 669, A69 (2023), arXiv:2208.02171 [astro-ph.CO] .
  * Valageas (2002) P. Valageas, Astron. Astrophys. 382, 412 (2002), arXiv:astro-ph/0107126 .
  * Bernardeau  _et al._ (2014) F. Bernardeau, C. Pichon, and S. Codis, Phys. Rev. D 90, 103519 (2014), arXiv:1310.8134 [astro-ph.CO] .
  * Bernardeau and Valageas (2000) F. Bernardeau and P. Valageas, Astron. Astrophys. 364, 1 (2000), arXiv:astro-ph/0006270 .
  * Boyle  _et al._ (2023) A. Boyle, A. Barthelemy, S. Codis, C. Uhlemann, and O. Friedrich, Open Journal of Astrophysics 6, 10351 (2023), arXiv:2212.10351 [astro-ph.CO] .
  * Kaiser (1992) N. Kaiser, Astrophys. J. 388, 272 (1992).
  * Harnois-Déraps  _et al._ (2018) J. Harnois-Déraps _et al._ , Mon. Not. Roy. Astron. Soc. 481, 1337 (2018), arXiv:1805.04511 [astro-ph.CO] .
  * Thiele  _et al._ (2020) L. Thiele, J. C. Hill, and K. M. Smith, Phys. Rev. D 102, 123545 (2020), arXiv:2009.06547 [astro-ph.CO] .
  * Giblin  _et al._ (2023) B. Giblin, Y.-C. Cai, and J. Harnois-Déraps, Mon. Not. Roy. Astron. Soc. 520, 1721 (2023), arXiv:2211.05708 [astro-ph.CO] .
  * Thiele  _et al._ (2023) L. Thiele, G. A. Marques, J. Liu, and M. Shirasaki, “Cosmological constraints from hsc y1 lensing convergence pdf,” (2023), arXiv:2304.05928 [astro-ph.CO] .
  * Grandón  _et al._ (2024) D. Grandón, G. A. Marques, L. Thiele, S. Cheng, M. Shirasaki, and J. Liu, (2024), arXiv:2403.03807 [astro-ph.CO] .
  * Uhlemann  _et al._ (2018b) C. Uhlemann, C. Pichon, S. Codis, B. L’Huillier, J. Kim, F. Bernardeau, C. Park, and S. Prunet, Mon. Not. Roy. Astron. Soc. 477, 2772 (2018b), arXiv:1711.04767 [astro-ph.CO] .
  * Clerkin  _et al._ (2017) L. Clerkin _et al._ (DES), Mon. Not. Roy. Astron. Soc. 466, 1444 (2017), arXiv:1605.02036 [astro-ph.CO] .
  * Hirata  _et al._ (2007) C. M. Hirata, R. Mandelbaum, M. Ishak, U. Seljak, R. Nichol, K. A. Pimbblet, N. P. Ross, and D. Wake, Mon. Not. Roy. Astron. Soc. 381, 1197 (2007), arXiv:astro-ph/0701671 .
  * Bridle and King (2007) S. Bridle and L. King, New J. Phys. 9, 444 (2007), arXiv:0705.0166 [astro-ph] .
  * Elad  _et al._ (2005) M. Elad, J. L. Starck, P. Querre, and D. L. Donoho, (2005).
  * Pires  _et al._ (2009) S. Pires, J. L. Starck, A. Amara, R. Teyssier, A. Refregier, and J. Fadili, Mon. Not. Roy. Astron. Soc. 395, 1265 (2009), arXiv:0804.4068 [astro-ph] .
  * Remy  _et al._ (2023) B. Remy, F. Lanusse, N. Jeffrey, J. Liu, J.-L. Starck, K. Osato, and T. Schrabback, Astron. Astrophys. 672, A51 (2023), arXiv:2201.05561 [astro-ph.CO] .
  * Jeffrey  _et al._ (2021) N. Jeffrey _et al._ (DES), Mon. Not. Roy. Astron. Soc. 505, 4626 (2021), arXiv:2105.13539 [astro-ph.CO] .
  * Gatti  _et al._ (2022) M. Gatti _et al._ (DES), Phys. Rev. D 106, 083509 (2022), arXiv:2110.10141 [astro-ph.CO] .
  * Harnois-Deraps  _et al._ (2019) J. Harnois-Deraps, B. Giblin, and B. Joachimi, Astron. Astrophys. 631, A160 (2019), arXiv:1905.06454 [astro-ph.CO] .
  * Harnois-Deraps  _et al._ (2013) J. Harnois-Deraps, U.-L. Pen, I. T. Iliev, H. Merz, J. D. Emberson, and V. Desjacques, Mon. Not. Roy. Astron. Soc. 436, 540 (2013), arXiv:1208.5098 [astro-ph.CO] .
  * Hinshaw  _et al._ (2013) G. Hinshaw _et al._ (WMAP), Astrophys. J. Suppl. 208, 19 (2013), arXiv:1212.5226 [astro-ph.CO] .
  * Villaescusa-Navarro (2018) F. Villaescusa-Navarro, “Pylians: Python libraries for the analysis of numerical simulations,” Astrophysics Source Code Library, record ascl:1811.008 (2018), ascl:1811.008 .
  * Takahashi  _et al._ (2012) R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, and M. Oguri, Astrophys. J. 761, 152 (2012), arXiv:1208.2701 [astro-ph.CO] .
  * Jarvis  _et al._ (2004) M. Jarvis, G. Bernstein, and B. Jain, MNRAS 352, 338 (2004), arXiv:astro-ph/0307393 [astro-ph] .
  * Jarvis (2015) M. Jarvis, “TreeCorr: Two-point correlation functions,” Astrophysics Source Code Library, record ascl:1508.007 (2015).
  * Chisari  _et al._ (2019) N. E. Chisari _et al._ (LSST Dark Energy Science), Astrophys. J. Suppl. 242, 2 (2019), arXiv:1812.05995 [astro-ph.CO] .
  * Smith  _et al._ (2003) R. E. Smith, J. A. Peacock, A. Jenkins, S. D. M. White, C. S. Frenk, F. R. Pearce, P. A. Thomas, G. Efstathiou, and H. M. P. Couchman, Monthly Notices of the Royal Astronomical Society 341, 1311–1332 (2003).
  * Zuntz  _et al._ (2015) J. Zuntz, M. Paterno, E. Jennings, D. Rudd, A. Manzotti, S. Dodelson, S. Bridle, S. Sehrish, and J. Kowalkowski, Astron. Comput. 12, 45 (2015), arXiv:1409.3409 [astro-ph.CO] .
  * Coe (2009) D. Coe, (2009), arXiv:0906.4123 [astro-ph.IM] .
  * Bernardeau (1994) F. Bernardeau, Astron. Astrophys. 291, 697 (1994), arXiv:astro-ph/9403020 .
  * Uhlemann  _et al._ (2022) C. Uhlemann, O. Friedrich, A. Boyle, A. Gough, A. Barthelemy, F. Bernardeau, and S. Codis, Open J. Astrophys. 6, 2023 (2022), arXiv:2210.07819 [astro-ph.CO] .
  * Hartlap  _et al._ (2007) J. Hartlap, P. Simon, and P. Schneider, Astron. Astrophys. 464, 399 (2007), arXiv:astro-ph/0608064 .
  * Dodelson and Schneider (2013) S. Dodelson and M. D. Schneider, Phys. Rev. D 88, 063537 (2013), arXiv:1304.2593 [astro-ph.CO] .
  * Sellentin and Heavens (2016) E. Sellentin and A. F. Heavens, Mon. Not. Roy. Astron. Soc. 456, L132 (2016), arXiv:1511.05969 [astro-ph.CO] .
  * Patton  _et al._ (2017) K. Patton, J. Blazek, K. Honscheid, E. Huff, P. Melchior, A. J. Ross, and E. Suchyta, Monthly Notices of the Royal Astronomical Society 472, 439–446 (2017).
  * Barthelemy  _et al._ (2022) A. Barthelemy, F. Bernardeau, S. Codis, and C. Uhlemann, Phys. Rev. D 105, 043537 (2022), arXiv:2106.11632 [astro-ph.CO] .
  * Dembo and Zeitouni (2009) A. Dembo and O. Zeitouni,  _Large deviations techniques and applications_ , Vol. 38 (Springer Science & Business Media, 2009).
  * Bernardeau (1992) F. Bernardeau, Astrophys. J. 392, 1 (1992).
  * Bernardeau (1995) F. Bernardeau, Astron. Astrophys. 301, 309 (1995), arXiv:astro-ph/9502089 .
  * Peacock and Smith (2014) J. Peacock and R. Smith, Astrophysics source code library , ascl (2014).
  * Mead  _et al._ (2016) A. Mead, C. Heymans, L. Lombriser, J. Peacock, O. Steele, and H. Winther, Mon. Not. Roy. Astron. Soc. 459, 1468 (2016), arXiv:1602.02154 [astro-ph.CO] .
  * Mead  _et al._ (2020) A. Mead, S. Brieden, T. Tröster, and C. Heymans,  (2020), 10.1093/mnras/stab082, arXiv:2009.01858 [astro-ph.CO] .
  * Friedrich and Eifler (2018) O. Friedrich and T. Eifler, MNRAS 473, 4150 (2018), arXiv:1703.07786 [astro-ph.IM] .
  * Blazek  _et al._ (2019) J. Blazek, N. MacCrann, M. A. Troxel, and X. Fang, Phys. Rev. D 100, 103506 (2019), arXiv:1708.09247 [astro-ph.CO] .
  * Lamman  _et al._ (2023) C. Lamman, E. Tsaprazi, J. Shi, N. N. Šarčević, S. Pyne, E. Legnani, and T. Ferreira,  (2023), 10.21105/astro.2309.08605, arXiv:2309.08605 [astro-ph.CO] .
  * Harnois-Déraps  _et al._ (2022) J. Harnois-Déraps, N. Martinet, and R. Reischke, Mon. Not. Roy. Astron. Soc. 509, 3868 (2022), arXiv:2107.08041 [astro-ph.CO] .
  * Brown  _et al._ (2002) M. L. Brown, A. N. Taylor, N. C. Hambly, and S. Dye, Mon. Not. Roy. Astron. Soc. 333, 501 (2002), arXiv:astro-ph/0009499 .