KiDS-1000 uncertainty and the yellow area to the Euclid uncertainty. The
correlation function is binned here in 10 logarithmic bins between $None$ and
$None$.

Though the EuclidEmulator has better accuracy than the HMCode model, it is
calibrated on a smaller range of cosmological parameters than the HMCode.
Therefore, we must assume tight priors for the inference in the Euclid-like
setup. Furthermore, we cannot use the EuclidEmulator for the KiDS-like setup
since the required tight prior range would bias the cosmological inference.

We further include the impact of intrinsic alignments on the cosmic shear
power spectrum via the non-linear alignment (NLA) model (Bridle & King 2007;
Kirk et al. 2012; see Equations 13–16 in Joachimi et al. 2021), which gives us
the free parameter $A_{\mathrm{IA}}$ characterizing the strength of intrinsic
alignments.

For the cosmological inference, we also require a covariance estimate. We
model the covariance analytically, using the code described in Reischke et al.
(in prep), which follows the approach in Joachimi et al. (2021). To summarise,
we first calculate the three parts of the covariance of the lensing power
spectrum, which are the Gaussian part, depending only on the matter power
spectrum, the dispersion of intrinsic galaxy shapes and source galaxy number
density; the intra-survey non-Gaussian part, depending on the matter
trispectrum at modes smaller than the survey; and the super-sample covariance,
depending on modes larger than the survey. The sum of these terms is then
converted to a covariance estimate for the COSEBIs by convolution with a
suitable kernel function following the approach by Asgari et al. (2021).

For the covariance estimate, we use the component-wise shape noise
$\sigma_{\epsilon}=0.28,0.27,0.28,0.27$, and $0.28$ for each bin of the KiDS-
like setup. For the Euclid-like setup, we assume $\sigma_{\epsilon}=0.21$ for
all bins, which is the same as was assumed in Euclid Collaboration: Blanchard
et al. (2020) and Euclid Collaboration: Deshpande et al. (2024).

###  4.2 Cosmological parameter estimation

Our cosmological parameter estimation is conducted using `cosmosis` (Zuntz et
al. 2015) and uses the priors in Table 3. For the KiDS-1000-like setup, we
sample the posterior for five cosmological parameters
($\Omega_{\mathrm{c}}\,h^{2}$, $\Omega_{\mathrm{b}}\,h^{2}$, $S_{8}$, $h$,
$n_{\mathrm{s}}$) and six nuisance parameters ($A_{\mathrm{IA}}$ and the
redshift distribution shifts for each tomographic bin). We use the same priors
as Asgari et al. (2021, see Table 3), except for the parameter
$\Omega_{\mathrm{b}}\,h^{2}$, for which the true value in the MICE, $0.255$,
is very close to the prior boundary in Asgari et al. (2021), $0.26$. We,
therefore, extend the prior on the right-hand side.

For the Euclid-like case, we sample over the power spectrum normalisation
$A_{\mathrm{s}}$ instead of $S_{8}$ to use the fiducial parameter of the
EuclidEmulator. We also include two parameters for the equation of state $w$
of dark energy, namely $w_{0}$ and $w_{a}$ defined by

$w(z)=w_{0}+w_{a}\frac{z}{1+z}\,,$ (27)

and the redshift distribution shifts of eight more tomographic bins. Due to
the limited range of the EuclidEmulator, we use the tighter priors in Table 3
for the Euclid-like setup. We also change the prior for $\delta z$. While for
the KiDS-like analysis, we assume a correlated prior on $\delta z$, using the
correlation matrix by Hildebrandt et al. (2021), for Euclid, we assume an
uncorrelated Gaussian prior. For the Euclid priors, we choose a width of
$0.001(1+\bar{z})$.

For both analyses, we assume a Gaussian likelihood and use the nested sampler
`nautilus` (Lange 2023). We also perform cosmological parameter estimation
where we fix all nuisance parameters to zero. This tests whether the parameter
constraints are biased due to SLC if we strictly assume no intrinsic
alignments to occur. From the nested sampler we obtain estimates for the
parameter values that maximize the marginal posterior distributions and their
$1\sigma$ uncertainties. However, these generally differ from the parameter
values that maximize the full (high-dimensional) posterior, the so-called
Maximum A-Posteriori (MAP). We therefore run an optimization procedure using
the Nelder–Mead algorithm (Nelder & Mead 1965) to find the MAP for each
analysis, following the same approach as in Asgari et al. (2020).

Table 2: Priors for sampling parameters.

Parameter | KiDS-like setup |  Euclid-like setup
---|---|---
$S_{8}$ | $\mathcal{U}(0.1,1.3)$ | –
$\ln(A_{\mathrm{s}})$ | – | $\mathcal{U}(-20.1,-19.8)$
$\Omega_{\mathrm{c}}\,h^{2}$ | $\mathcal{U}(0.051,0.255)$ | $\mathcal{U}(0.107,0.126)$
$\Omega_{\mathrm{b}}\,h^{2}$ | $\mathcal{U}(0.019,0.026)$ | $\mathcal{U}(0.0214,0.0223)$
$h$ | $\mathcal{U}(0.64,0.82)$ | $\mathcal{U}(0.61,0.73)$
$n_{\mathrm{s}}$ | $\mathcal{U}(0.84,1.1)$ | $\mathcal{U}(0.92,1.0)$
$w_{0}$ | – | $\mathcal{U}(-1.3,-0.7)$
$w_{a}$ | – | $\mathcal{U}(-0.7,0.7)$
$A_{\mathrm{IA}}$ | $\mathcal{U}(-6,6)$ | $\mathcal{U}(-6,6)$
$\delta z$ | $\mathcal{N}(\mu,\mathsf{C})$ | $\mathcal{N}[0,0.001(1+\bar{z})]$
333We sample in $S_{8}$ for the KiDS-like and in $\ln(A_{\mathrm{s}})$ for the
Euclid-like setup. For the KiDS-like setup the priors are the same as in
Asgari et al. (2021) except for $\Omega_{\mathrm{b}}\,h^{2}$, which we extend
to higher values. For the Euclid-like setup the priors are given by the
parameter range of the EuclidEmulator. The $\bar{z}$ denotes the average
redshift of the tomographic bin

##  5  Results

###  5.1 KiDS-1000-like setup

Figure 4: Measurement for the KiDS-like setup. The $x$-axis denotes the COSEBI
order $n$, each panel shows the measurement for one combination of tomographic
bins, whose numbers are given in the upper right corner of each panel. Upper
right: COSEBI $E$-modes $E_{n}$ measured for the original MICE galaxies (black
points), the clustered catalogue (red plusses) and the unclustered catalogue
(blue crosses) with the KiDS-1000 uncertainty (grey band). Lower left:
Difference between $E_{n}$ for the original and clustered sources (black
points) and between the clustered and unclustered sources (blue crosses),
divided by the KiDS-1000 uncertainty. The original sources have shear and
position taken directly from the MICE galaxy catalog, while the clustered
sources have their position taken from the catalogue and the shear assigned
from the shear maps. The unclustered sources have uniformly distributed
positions and shear assigned from the shear maps.

We first consider the KiDS-1000-like setup. In the upper right corner of Fig.
4, we show the $E_{n}$ measured for the MICE simulations using the original,
clustered, and unclustered source galaxy catalogues, together with the
KiDS-1000 uncertainties. We stress that these uncertainties are not error
estimates for the measurements in the mock catalogues, since they were
obtained from a different survey area and did not include shape noise. We also
note that the $B$-modes $B_{n}$ were consistent with zero for all three cases.
We consider in the lower left half of Fig. 4 the difference between the
clustered and the unclustered $E_{n}$, and between the original and
unclustered $E_{n}$ normalized by the KiDS-1000 uncertainty.

The COSEBIs for the original and clustered source catalogueare identical. This
suggests that our method of assigning shears from the MICE shear map to galaxy
positions agrees with the original creation of the MICE catalog.

The impact of SLC is given by the difference between the clustered and the
unclustered $E_{n}$ normalized by the KiDS-1000 uncertainty. The effect shows
a dependence on tomographic bins. While SLC decreases the $E_{n}$ for the
lowest tomographic bin, it increases them at higher bins. This increase is
stronger for larger $n$. The deviation between the $E_{n}$ for the clustered
and unclustered sources is strongest for the lowest tomographic bin (up to
$20\%$). However, since the uncertainty is also the largest there, the ratio
of deviation to uncertainty peaks for the combination of the third and fourth
bin. For all tomographic bins, the SLC has an effect of less than
$0.5\,\sigma$.

Using the COSEBIs, we first perform a cosmological inference with all nuisance
parameters. In Table 3 we report the maxima of the marginalised posteriors
together with their $1\sigma$ uncertainty and the MAP values. We also give the
shifts in the maxima, normalized by the $1\sigma$ uncertainties and the shifts
in the MAP due to the inclusion of SLC in the datavector. Figure 5 shows the
corresponding posteriors for $S_{8}$, $\sigma_{8}$, $\Omega_{\mathrm{m}}$ and
$A_{\mathrm{IA}}$ in dark blue and red. The posteriors for all varied
parameters are given in Appendix C.

Table 3: Maxima of marginal posteriors with $1\sigma$ uncertainties, MAP, and
shifts due to SLC for the KiDS-like setup with varying nuisance parameters

Parameter | Truth | max. + marginal $\sigma$ | MAP
---|---|---|---
|  | clustered | unclustered | $\Delta/\sigma$ | clustered | unclustered | $\Delta_{\mathrm{MAP}}$
$\Omega_{\mathrm{c}}\,h^{2}$ [$10^{-3}$] | $101$ | $73_{-19}^{+37}$ | $77_{-25}^{+41}$ | $-0.14$ | $76$ | $79$ | $-3$
$\Omega_{\mathrm{b}}\,h^{2}$ [$10^{-4}$] | 216 | $387_{-211}^{+75}$ | $391_{-131}^{+59}$ | $-0.041$ | $231$ | $246$ | $-15$
$h$ | $0.700$ | $0.739_{-0.071}^{+0.061}$ | $0.734_{-0.077}^{+0.054}$ | 0.076 | $0.704$ | $0.701$ | $0.003$
$n_{\mathrm{s}}$ | $0.95$ | $0.896_{-0.047}^{0.085}$ | $0.897_{-0.050}^{0.083}$ | $-0.015$ | $0.874$ | $0.884$ | $-0.010$
$S_{8}$ [$10^{-3}$] | $730$ | $731\pm 17$ | $728\pm 18$ | $0.18$ | $726$ | $725$ | $1$
$\sigma_{8}$ [$10^{-3}$] | $800$ | $850_{-120}^{+110}$ | $830_{-130}^{+110}$ | $0.17$ | $884$ | $879$ | $5$
$A_{\mathrm{s}}$ [$10^{-10}$] | $21$ | $36_{-25}^{+44}$ | $30_{-24}^{+41}$ | $0.17$ | $50$ | $47$ | $3$
$\Omega_{\mathrm{m}}$ [$10^{-3}$] | $250$ | $220_{-58}^{+74}$ | $228_{-64}^{+87}$ | $-0.12$ | $205$ | $209$ | $4$
$A_{\mathrm{IA}}$ | 0 | $0.28_{-0.44}^{+0.42}$ | $0.11_{-0.46}^{+0.44}$ | $\mathbf{0.39}$ | $0.21$ | $-0.03$ | $0.24$
$\delta z_{1}$ [$10^{-3}$] | 0 | $-1_{-12}^{+11}$ | $0_{-12}^{+11}$ | $-0.087$ | $-1$ | $0$ | $-1$
$\delta z_{2}$ [$10^{-3}$] | 0 | $5\pm 12$ | $4_{-11}^{+12}$ | $0.083$ | $6$ | $6$ | $0$
$\delta z_{3}$ [$10^{-3}$] | 0 | $-2\pm 12$ | $-3\pm 12$ | $-0.083$ | $2$ | $1$ | $1$
$\delta z_{4}$ [$10^{-3}$] | 0 | $3\pm 9.1$ | $0_{-10}^{+9.1}$ | $\mathbf{0.33}$ | $-3$ | $-7$ | $4$
$\delta z_{5}$ [$10^{-3}$] | 0 | $-1_{-11}^{+10}$ | $0_{-11}^{+10}$ | $0.083$ | $-1$ | $0$ | $-1$
444$\Delta/\sigma$ is the difference between the maxima of the marginal
posterior for clustered and unclustered sources, divided by the mean of the
upper and lower 1$\sigma$ uncertainty of clustered sources. Bold values
indicate differences we consider significant, i.e. $|\Delta/\sigma|>0.25$.
$\Delta_{\mathrm{MAP}}$ is the difference between the MAP for clustered and
unclustered sources.

Figure 5: Parameter constraints for the KiDS-like setup using all parameters
(lower corner) and with fixed nuisance parameters (upper corner), either for a
data vector including SLC (dark blue and cyan) or without SLC (red and pink).
Constraints for all sampled parameters can be found in Tables 3 and 4.

Similarly to Euclid Collaboration: Deshpande et al. (2024), we consider shifts
of more than $0.25\sigma$ significant, since then, if the marginalized
posteriors were Gaussian, the contours with and without shift overlap less
than 90%. We find that, when including the nuisance parameters, none of the
cosmological parameters show a significant shift due to SLC. The largest shift
occurs for $\sigma_{8}$, which is shifted by $0.19\sigma$ towards higher
values. This likely occurs to compensate for the signal increase in higher
tomographic bins due to SLC.

The MAP values are shifted by similar absolute amounts as the maxima of the
marginalized posteriors. Furthermore, all shifts have the same sign. This
indicates that SLC does not have a strong influence on the degeneracy
direction between parameters and simply shifts the multidimensional posterior
to new values, since otherwise the marginalized posteriors would be affected
differently from the MAP.

While the cosmological parameters are relatively unaffected by SLC, the
nuisance parameters $A_{\mathrm{IA}}$ and $\delta z_{4}$ are shifted
significantly ($0.39\sigma$ and $0.33\sigma$, respectively) to higher values.
These shifts compensate for two different redshift regimes. A larger
$A_{\mathrm{IA}}$ decreases the cosmic shear signal, in particular at low
redshifts, which is similar to the decrease in $E_{n}$ for the lowest
tomographic bin due to SLC. A larger $\delta z_{4}$ increases the signal for
$E_{n}$ including the fourth tomographic bin, similar to the increase in
signal due to SLC for the higher tomographic bins. Therefore, these two
nuisance parameters ‘neutralise’ the effect of SLC on the cosmological
parameters, even though their physical meaning is completely unrelated to SLC.

This raises the question of whether the cosmological analysis would still be
insensitive to SLC if we excluded the nuisance parameters. This scenario is
particularly interesting because future cosmic shear analyses are expected to
use informative priors for effects like intrinsic alignment, constrained from
independent observations (e.g. Johnston et al. 2019; Fortuna et al. 2021).
These priors are significantly tighter than the broad flat prior used here. To
study whether SLC is still unimportant for the analyses when the nuisance
parameters are not varied, we perform the second cosmological inference with
$A_{\mathrm{IA}}$ and $\delta z_{i}$ fixed to zero. The resulting maxima of
the marginalized posteriors and the MAP values are given in Table 4.

Table 4: Maxima of marginal posteriors with $1\sigma$ uncertainties, MAP, and
shifts due to SLC for the KiDS-like setup with fixed nuisance parameters

Parameter | Truth | max. + marginal $\sigma$ | MAP
---|---|---|---
|  | clustered | unclustered | $\Delta/\sigma$ | clustered | unclustered | $\Delta_{\mathrm{MAP}}$
$\Omega_{\mathrm{c}}\,h^{2}$ [$10^{-3}$] | $101$ | $72_{-18}^{+29}$ | $80_{-25}^{+36}$ | $\mathbf{-0.34}$ | $98.0$ | $107.3$ | $-9.3$
$\Omega_{\mathrm{b}}\,h^{2}$ [$10^{-4}$] | 216 | $391_{-119}^{+72}$ | $408_{-130}^{+61}$ | $-0.18$ | $252$ | $252$ | $0$
$h$ | $0.700$ | $0.749_{-0.064}^{+0.047}$ | $0.734\pm 0.055$ | $\mathbf{0.27}$ | $0.660$ | $0.654$ | $0.06$
$n_{\mathrm{s}}$ | $0.95$ | $0.893_{-0.047}^{+0.084}$ | $0.892_{-0.046}^{+0.084}$ | $0.015$ | $0.906$ | $0.905$ | $0.01$
$S_{8}$ [$10^{-3}$] | $730$ | $734_{-12}^{11}$ | $729_{-14}^{+13}$ | $\mathbf{0.42}$ | $735$ | $732$ | $3$
$\sigma_{8}$ [$10^{-3}$] | $800$ | $877_{-102}^{+94}$ | $840_{-120}^{+100}$ | $\mathbf{0.38}$ | $921$ | $862$ | $59$
$A_{\mathrm{s}}$ [$10^{-10}$] | $21$ | $43_{-27}^{+41}$ | $30_{-23}^{+36}$ | $\mathbf{0.38}$ | $81$ | $57$ | $24$
$\Omega_{\mathrm{m}}$ [$10^{-3}$] | $250$ | $206_{-44}^{+53}$ | $221_{-50}^{+69}$ | $\mathbf{-0.31}$ | $191$ | $216$ | $-25$
555Columns defined as in Table 3.

Figure 6: COSEBIs for KiDS-like setup when different SLC effects are included,
either the full SLC (red crosses), the estimator bias (EB) and intrinsic
clustering (IC) (orange triangles), only the estimator bias (EB) (green
triangles) and no SLC (blue dots). Upper right: COSEBIs. Lower left:
Fractional difference to COSEBIs for unclustered sources, i.e., without SLC.
Figure 7: Parameter constraints for KiDS-like setup, for a data vector
including the full SLC, the estimator bias (EB) and intrinsic clustering (IC),
only the estimator bias (EB) and without any SLC.

The posteriors for $S_{8}$, $\sigma_{8}$, and $\Omega_{\mathrm{m}}$ are shown
in Fig. 5 as pink and cyan contours and the full posteriors are in Appendix C.
Neglecting SLC would now lead to shifts of $0.42\sigma$ in $S_{8}$ and
$0.31\sigma$ in $\Omega_{\mathrm{m}}$, with $S_{8}$ shifted to higher and
$\Omega_{\mathrm{m}}$ shifted to lower values. These increases in the effect
of SLC are partly due to the decreased parameter uncertainties compared to the
analysis with nuisance parameters, which increase the significance of the
parameter shifts. However, the absolute values of the shifts also increase,
for example $S_{8}$ shifts by $0.7\%$ for fixed nuisance parameters and
$0.4\%$ for variable nuisance parameters. This confirms that SLC would bias
the cosmological inference stronger if nuisance parameters were fixed.

Both with and without nuisance parameters, the MICE simulation’s true
cosmological parameters are within the cosmological inference’s uncertainties.
However, in both cases, the values for the unclustered data vector is closer
to the true values. This is unsurprising, as the unclustered data vector
corresponds to the model assumption of unclustered source galaxies. The
remaining difference between the inferred parameters for the unclustered data
vector and the true cosmology is likely due to the fact that the simulation is
a single realization of the cosmology and the limited accuracy of the model.

We now investigate the impact of the different SLC effects discussed in Sect.
2.2. For this, we show in Fig. 6 the COSEBIs when different parts of the SLC
are included, as well as the fractional difference to the signal for
unclustered sources. The largest impact on the signal is due to the estimator
bias (EB), which suppresses the signal for all tomographic bins. The IC and TC
effects partially counteract each other. While the IC increases the signal, as
expected from Eq. (17), the contamination by lower redshift galaxies decreases
it. The scale dependence of the SLC effects is investigated using shear
correlation functions in Appendix B.

To see how the individual SLC effects affect the cosmological parameter
constraints, Fig. 7 shows the corresponding parameter constraints. For $S_{8}$
the strongest impact occurs due to EB, which lowers the estimated value. This
is in line with the effect on the COSEBIs - lowering the cosmic shear signal
is equivalent to lowering $S_{8}$. The IC slightly increases $S_{8}$ again, as
expected since it increases the cosmic shear signal. Interestingly, $S_{8}$
increases even more, once the TC effect is included. This seems
counterintuitive since TC generally lowers the shear signal. However,
simultaneously to increasing $S_{8}$, the effect also increases the intrinsic
alignment parameter $A_{\mathrm{IA}}$, which lowers the signal. Since
$A_{\mathrm{IA}}$ mainly causes a decrease in the signal at low tomographic
bins, where the suppression due to TC is strongest, while $S_{8}$ mainly
increases the signal at higher redshifts, these two parameters together can
match the total effect by the contamination.

###  5.2 Euclid-like setup

Finally, we consider the Euclid-like setup. Figure 8 shows the measured
COSEBIs for the clustered and unclustered setup, as well as the differences
for a subset of the tomographic bins. The full measurements for all
tomographic bins are provided in Fig. 19. The measurements for the clustered
and original catalogue agree perfectly, which confirms that the creation of
the two mock catalogues is consistent. The difference between the clustered
and unclustered case is smaller for the Euclid-like setup than for the
KiDS-1000-like setup. This is likely due to the narrower redshift
distributions $p_{z}(z)$, with shorter tails and fewer outliers. Consequently,
there is less overlap between source galaxies in a tomographic bin and the
matter structures lensing other galaxies in the same tomographic bin, which
reduces the impact of SLC. Nevertheless, since the statistical uncertainty for
Euclid is much smaller than for Stage III surveys like KiDS, the difference
between the two cases approaches $1\sigma$ for the highest tomographic bins,
while it remains below $0.5\sigma$ for KiDS-1000.

Figure 8: Measurement for the Euclid-like setup. Upper right: COSEBI $E$-modes
$E_{n}$ measured for the original FS2 galaxies (black points), the clustered
catalogue(red plusses) and the unclustered catalogue(blue crosses) with the
Euclid uncertainty (grey band) for a subset of the 13 tomographic bins. Lower
left: Difference between $E_{n}$ for the original and clustered sources (black
points) and between the clustered and unclustered sources (blue crosses),
divided by Euclid uncertainty. The full measurement for all tomographic bins
is in Fig. 19.

Figure 9 shows the parameter constraints for the most important cosmological
parameters (red and blue constraints), while the maxima of the marginalized
posteriors and the MAP values for all parameters are given in Table 5. We find
lower shifts in parameter estimates due to SLC than for the KiDS-like setup,
which is likely due to the narrower redshift bins. This provides an additional
motivation for analyzing Euclid with more than ten tomographic bins, as
already suggested by Euclid Collaboration: Pocino et al. (2021).

Figure 9: Parameter constraints for the Euclid-like setup using all parameters
(lower corner) and with fixed nuisance parameters (upper corner), either for a
data vector including SLC (dark blue and cyan) or without SLC (red and pink).
Constraints for all sampled parameters can be found in Table 5 and 6. Table 5:
Maxima of marginal posteriors with $1\sigma$ credible intervals, MAP, and
shifts due to SLC for the Euclid-like setup with varying nuisance parameters

Parameter | Truth | max. + marginal $\sigma$ | MAP
---|---|---|---
|  | clustered | unclustered | $\Delta/\sigma$ | clustered | unclustered | $\Delta_{\mathrm{MAP}}$
$\Omega_{\mathrm{c}}\,h^{2}$ [$10^{-3}$] | 120.5 | $120.8_{-6.2}^{+4.1}$ | $121.8_{-6.0}^{+4.1}$ | $-0.10$ | 120.3 | 120.6 | $-0.3$
$\Omega_{\mathrm{b}}\,h^{2}$ [$10^{-4}$] | 220.0 | $217.7_{-4.0}^{+1.8}$ | $217.4_{-2.1}^{+3.7}$ | $0.10$ | $219.8$ | $219.7$ | $-0.1$
$h$ | 0.670 | $0.674_{-0.028}^{+0.029}$ | $0.670_{-0.027}^{+0.030}$ | $0.14$ | $0.670$ | $0.669$ | $0.01$
$n_{\mathrm{s}}$ | 0.960 | $0.980_{-0.025}^{+0.017}$ | $0.982_{-0.024}^{+0.016}$ | $-0.095$ | $0.968$ | $0.965$ | $0.003$
$A_{\mathrm{s}}$ [$10^{-10}$] | 21.0 | $21.7_{-2.3}^{2.2}$ | $21.4_{-2.4}^{+2.3}$ | $0.13$ | $21.5$ | $21.1$ | $0.4$
$w_{0}$ | $-1.000$ | $-1.04_{-0.16}^{+0.18}$ | $-1.06_{-0.151}^{+0.19}$ | $0.12$ | $-1.00$ | $-1.00$ | $0$
$w_{a}$ | 0.000 | $-0.04_{-0.39}^{+0.34}$ | $-0.06\pm 0.36$ | $0.05$ | $-0.05$ | $-0.08$ | $0.03$
$S_{8}$ [$10^{-3}$] | 836.5 | $841_{-16}^{+12}$ | $843_{-15}^{+13}$ | $-0.14$ | $841$ | $844$ | $-3$
$\sigma_{8}$ [$10^{-3}$] | 813.0 | $825_{-25}^{+27}$ | $820_{-27}^{+26}$ | $0.19$ | $822$ | $818$ | $4$
$\Omega_{\mathrm{m}}$ [$10^{-3}$] | 317.6 | $313_{-24}^{+25}$ | $316\pm 24$ | $-0.12$ | $318$ | $320$ | $-2$
$A_{\mathrm{IA}}$ | 0 | $0.022_{-0.082}^{0.081}$ | $-0.009_{-0.080}^{+0.079}$ | $\mathbf{0.39}$ | 0.027 | $-0.002$ | $0.029$
$\delta z_{1}$ [$10^{-4}$] | 0 | $2.6_{-9.9}^{+5.4}$ | $0.5_{-8.5}^{+6.8}$ | $\mathbf{0.27}$ | 7.4 | 2.8 | 2.8
$\delta z_{2}$ [$10^{-4}$] | 0 | $-0.5_{-7.0}^{+8.0}$ | $-1.2_{-7.0}^{+8.2}$ | 0.093 | 5.6 | 5.0 | 0.6
$\delta z_{3}$ [$10^{-4}$] | 0 | $-1.2_{-6.5}^{+8.1}$ | $-2.0_{-5.8}^{+8.9}$ | $0.11$ | $-4.0$ | $-4.6$ | $0.6$
$\delta z_{4}$ [$10^{-4}$] | 0 | $3.1_{-8.5}^{+5.9}$ | $1.0_{-8.0}^{+6.5}$ | $\mathbf{0.29}$ | 5.3 | 3.0 | 2.3
$\delta z_{5}$ [$10^{-4}$] | 0 | $-0.2_{-7.3}^{+7.4}$ | $1.9_{-8.6}^{+6.1}$ | $-0.23$ | $-2.9$ | $-0.9$ | $-2.0$
$\delta z_{6}$ [$10^{-4}$] | 0 | $0.8_{-8.0}^{+6.8}$ | $2.4_{-8.7}^{+6.0}$ | $-0.22$ | $-2.5$ | $-1.3$ | $-1.2$
$\delta z_{7}$ [$10^{-4}$] | 0 | $-1.6_{-6.7}^{+8.3}$ | $-0.8_{-6.8}^{+8.3}$ | $-0.11$ | $-1.9$ | 0.1 | $-2.0$
$\delta z_{8}$ [$10^{-4}$] | 0 | $0.0_{-7.4}^{+7.7}$ | $-1.8_{-6.3}^{+8.8}$ | 0.24 | 0.5 | $-2.2$ | $2.6$
$\delta z_{9}$ [$10^{-4}$] | 0 | $-2.2_{-6.0}^{+9.4}$ | $-1.1_{-6.6}^{+8.8}$ | $-0.14$ | $-1.0$ | $-3.9$ | 2.9
$\delta z_{10}$ [$10^{-4}$] | 0 | $0.6_{-8.2}^{+7.0}$ | $-1.5_{-6.5}^{+8.8}$ | $\mathbf{0.27}$ | $4.0$ | $2.3$ | 1.7
$\delta z_{11}$ [$10^{-4}$] | 0 | $0.7_{-8.4}^{+7.0}$ | $2.5_{-9.5}^{+5.9}$ | $-0.23$ | $-1.6$ | $-0.3$ | $-1.3$
$\delta z_{12}$ [$10^{-4}$] | 0 | $1.4_{-9.1}^{+6.4}$ | $1.8_{-9.3}^{+6.0}$ | $-0.052$ | $4.6$ | $5.0$ | $-0.4$
$\delta z_{13}$ [$10^{-4}$] | 0 | $-1.0_{-10.4}^{+5.1}$ | $-3.0_{-5.1}^{+10.3}$ | $\mathbf{0.26}$ | $-1.9$ | $-5.1$ | 3.2
666Columns defined as in Table 3.

Similarly to the KiDS setup, the nuisance parameters are more strongly
affected than the cosmological ones. Several redshift bin shifts $\delta z$
vary by more than $0.25\sigma$ between the cases with and without SLC.
Consequently, these parameters again account for the SLC, while the more
physical parameters remain stable.

The MAP values show similar absolute shifts due to SLC as the maxima of the
marginalized posteriors, which also show the same sign. The shift in $S_{8}$
is slightly larger for the MAP than for the marginalized posterior
($-2.5\times 10^{-3}$ compared to $-2.0\times 10^{-3}$) while the shift in
$\Omega_{\mathrm{m}}$ is slightly smaller ($-1.7\times 10^{-3}$ compared to
$-3.0\times 10^{-3}$). Overall, the similarity of the absolute shifts suggests
that, similar to the KiDS-setup, SLC mainly moves the full posterior to a
different parameter set and does not strongly influence the degeneracy
directions between parameters.

We investigate the effect of neglecting the nuisance parameters in Fig. 9 as
pink and cyan contours. Here, we fix the $\delta z$ and $A_{\mathrm{IA}}$
parameters to zero. In this setup the shifts due to the unaccounted SLC
increase, with the shift for $\sigma_{8}$ and $\Omega_{\mathrm{c}}\,h^{2}$
grazing the significance threshold of $0.25\sigma$ and the shift in $w_{0}$
becoming larger than this threshold. This increase in significance is partly
due to the strongly decreased parameter uncertainties. However, for example
for $w_{0}$, neglecting the nuisance parameters also increases the absolute
value of the shift, from $0.02$ to $0.04$. Consequently, for Euclid, it is
vital to either use nuisance parameters to account for unmodelled biases or to
correctly include SLC in the model.

Table 6: Maxima of marginal posteriors with $1\sigma$ credible intervals, MAP,
and shifts due to SLC for the Euclid-like setup for fixed nuisance parameters

Parameter | Truth | max. + marginal $\sigma$ | MAP
---|---|---|---
|  | clustered | unclustered | $\Delta/\sigma$ | clustered | unclustered | $\Delta_{\mathrm{MAP}}$
$\Omega_{\mathrm{c}}\,h^{2}$ [$10^{-3}$] | 120.5 | $120.5_{-6.0}^{+4.2}$ | $121.3_{-6.4}^{+4.5}$ | $-0.24$ | 121.1 | 121.4 | 0.3
$\Omega_{\mathrm{b}}\,h^{2}$ [$10^{-4}$] | 220.0 | $217.4_{-2.0}^{+3.8}$ | $217.8_{-2.4}^{+3.4}$ | $-0.14$ | 219.1 | 218.4 | 0.7
$h$ | 0.670 | $0.667_{-0.018}^{+0.019}$ | $0.664\pm 0.019$ | 0.16 | 0.672 | 0.665 | 0.007
$n_{\mathrm{s}}$ | 0.960 | $0.975_{-0.020}^{+0.018}$ | $0.977_{-0.020}^{+0.017}$ | $-0.11$ | 0.962 | 0.965 | $-0.003$
$A_{\mathrm{s}}$ [$10^{-10}$] | 21.0 | $21.3\pm 1.8$ | $21.2_{-1.6}^{+1.8}$ | 0.056 | $21.4$ | $21.2$ | 0.2
$w_{0}$ | $-1.000$ | $-0.96_{-0.12}^{+0.13}$ | $-1.00_{-0.13}^{+0.14}$ | $\mathbf{0.32}$ | $-0.96$ | $-0.98$ | 0.02
$w_{a}$ | 0.000 | $-0.10_{-0.36}^{+0.35}$ | $-0.08_{-0.38}^{+0.36}$ | $-0.056$ | $-0.04$ | $-0.02$ | $-0.02$
$S_{8}$ [$10^{-3}$] | 836.5 | $841.9_{-7.7}^{+7.0}$ | $841.3_{-9.3}^{+8.5}$ | 0.082 | 843.4 | 849.3 | $-0.6$
$\sigma_{8}$ [$10^{-3}$] | 813.0 | $819\pm 15$ | $815\pm 16$ | 0.2 | 819 | 816 | 3
$\Omega_{\mathrm{m}}$ [$10^{-3}$] | 317.6 | $321_{-13}^{+12}$ | $319\pm 13$ | 0.16 | 325 | 318 | 7
777Columns defined as in Table 3.

Figure 10: Parameter constraints for Euclid-like setup, for a data vector
including the full SLC, the estimator bias (EB) and intrinsic clustering (IC),
only the estimator bias (EB) and without any SLC.

Finally, we also test the impact of the different SLC effects for the Euclid-
like setup. Figure 10 shows the obtained parameter constraints for $S_{8}$,
$\sigma_{8}$, $\Omega_{\mathrm{m}}$ and $A_{\mathrm{IA}}$ when excluding parts
of the SLC effects. The effects qualitatively induce the same changes in the
parameter constraints as for the KiDS-like setup. The EB effect decreases
$S_{8}$, while the IC and TC effects increase it. The intrinsic alignment
parameter remains stable under the EB and IC effects but shifts to higher
values for the TC effect. Again, the EB has the strongest impact on the
$S_{8}$ parameter.

##  6  Discussion

In this work, we investigated the impact of the clustering of source galaxies
with lensing matter structures, the so-called source-lens clustering (SLC), on
cosmological parameter inference with second-order statistics of cosmic shear.
We considered realistically clustered mock galaxies in the MICE and FS2
simulations, as well as uniformly distributed sources, and measured the
COSEBIs for a KiDS-1000-like and a Euclid-like setup. We then performed a
cosmological inference for both clustered and unclustered sources using nested
sampling of the posterior and including nuisance parameters for intrinsic
alignments and shifts of the source redshift distributions. In summary, we
found that if nuisance parameters are taken into account, SLC has only a minor
impact on cosmological inference for Stage III, making it an unlikely
candidate to resolve the ‘$S_{8}$-tension’.

For KiDS-1000, we found little impact of SLC on cosmological parameters if
nuisance parameters are taken into account. The SLC causes a small decrease
(within $0.5\sigma$) in the cosmic shear signal for the lowest tomographic bin
and a similarly small increase for the higher tomographic bins. These shifts
are ‘absorbed’ by changes in the intrinsic alignment parameter
$A_{\mathrm{IA}}$ and the redshift distribution biases $\delta z$, leading to
cosmological parameters, such as $S_{8}$ and $\Omega_{\mathrm{m}}$, to be
shifted by less than $0.2\sigma$. Thus, it is valid for Stage III surveys to
neglect SLC in the modelling as long as nuisance parameters for intrinsic
alignments and shifts of the tomographic bins are included. We thus confirm
the expectation outlined in Krause et al. (2021) that SLC has only a minor
impact on cosmological inference with Stage III surveys and can be neglected
in their modelling.

This picture changes if nuisance parameters are fixed and not varied with the
analysis. In this case, we see that $S_{8}$ is shifted by $0.42\sigma$ and
$\Omega_{\mathrm{m}}$ by $0.31\sigma$ for the KiDS-1000-like setup.
Consequently, when neglecting nuisance parameters, SLC should be included in
the modelling of the cosmic shear signal. Furthermore, we advise against
interpreting parameters like $A_{\mathrm{IA}}$ as ‘quantifying intrinsic
alignment’. As shown here, effects completely unrelated to intrinsic
alignments can affect this parameter substantially. A further example of this
was found by Li et al. (2021), who found that the intrinsic alignment nuisance
parameter is strongly correlated with the redshift uncertainty for low-
redshift tomographic bins and fixing the assumed redshift uncertainty to zero
could lead to an unphysically high inferred $A_{\mathrm{IA}}$ for blue
galaxies. Furthermore, Leonard et al. (2024) found that the redshift scaling
of the intrinsic alignment amplitude is strongly degenerate with the mean and
variance of the redshift distribution. The interplay between redshift
distribution uncertainties and intrinsic alignment constraints is further
dependent on the considered intrinsic alignment model, and can be affected by
the considered priors (Fischbacher et al. 2023).

The fact that unmodelled systematic effects such as SLC change the meaning of
the nuisance parameters can be an issue when deciding on priors for these
parameters. Ideally, effects like intrinsic alignment could be constrained
from observations using physical models (e.g. Fortuna et al. 2021). These
constraints would provide tight priors on the nuisance parameters. However,
this is no longer straightforward if the parameters are meant to also capture
other effects like SLC. Consequently, if observational motivated priors for
nuisance parameters are used, effects like SLC need to be carefully included
in the modelling.

The parameter uncertainties we find for the KiDS-1000-like setup are of the
same order as those found in Asgari et al. (2021) except for
$\Omega_{\mathrm{b}}h^{2}$, whose uncertainty is more than twice the one from
Asgari et al. (2021). This is due to the larger prior for this parameter we
assume here, as $\Omega_{\mathrm{b}}$ cannot be constrained from a
KiDS-1000-like survey. Since cosmic shear is almost insensitive to the
baryonic matter density $\Omega_{\mathrm{b}}$ the constraints are completely
prior-dominated. Our larger prior thus also leads to a larger posterior. In
general, our confidence intervals are slightly larger than those by Asgari et
al. (2021), which is likely because we are using the sampler `nautilus` to
sample the posterior, while they use `multinest` (Feroz et al. 2019), which
can underestimate posteriors (Lemos et al. 2023). Furthermore, we also do not
vary the parameter $A_{\mathrm{b}}$ for baryonic feedback, which in the
simulation is exactly zero.

For KiDS, we investigated the impact of different parts of the SLC effect. We
found that the largest effect is due to the estimator bias, which occurs due
to the neglect of the source galaxies angular correlation function and
decreases the measured signal. The intrinsic clustering of the sources and the
contamination of tomographic bins due to photometric redshift errors partly
counteract each other, leaving the estimator bias as the most important
contribution.

For the Euclid-like setup, we see a similar effect as for the KiDS-1000-like
setup. However, the absolute shift in the signal due to SLC is smaller. This
is likely due to the narrower $p_{z}(z)$ with shorter tails. These lead to
less scattering between tomographic redshift bins, thus reducing the
tomographic bin contamination (TC) effect. The $p_{z}(z)$ are still broader
than the typical galaxy correlation length, which is of the order of 1 Mpc, so
the intrinsic clustering remains.

We also find that the impact on the cosmological parameter analysis for
Euclid, aside from the nuisance parameters, is small. $S_{8}$ is shifted only
by $0.14\sigma$ and $w_{0}$ by $0.12\sigma$ if nuisance parameters are taken
into account. This indicates that SLC might even be neglected for a Stage IV
survey. Again, the picture would change if nuisance parameters were fixed in
the analysis. Then, the shift in $w_{0}$ becomes significant ($0.32\sigma$)
and shifts in $\sigma_{8}$ and $\Omega_{\mathrm{m}}$ also increase.

This small effect of SLC on cosmological parameters is in apparent conflict
with Euclid Collaboration: Deshpande et al. (2024), who found that neglecting
SLC can cause biases of more than $1\sigma$ on $\Omega_{\mathrm{m}}$ and
$\sigma_{8}$. However, their analysis relied on modelling SLC analytically,
while we directly measured the effect using realistically clustered sources.
The analytic modelling naturally included simplifying assumptions, such as a
modelled redshift distribution, linear galaxy bias and idealized photometric
redshift uncertainties. Our analysis did not require these and could be seen
as a more realistic extension of their analysis.

We note that our uncertainties on the cosmological parameters are larger than
those quoted in Euclid Collaboration: Deshpande et al. (2024). This is likely
due to three reasons. First, we perform a full sampling of the posterior
instead of a Fisher analysis as in Euclid Collaboration: Deshpande et al.
(2024). Fisher analyses can underestimate a posterior’s width, leading to
tighter parameter constraints (Wolz et al. 2012). Second, we include more
parameters in our analysis: the intrinsic alignment parameter
$A_{\mathrm{IA}}$ and the $p_{z}(z)$-shifts for the tomographic bins.
Including nuisance parameters reduces the constraining power on the
cosmological parameters, leading to larger uncertainties. Third, we use a
different observable, namely COSEBIs, instead of the lensing power spectrum.
Consequently, our analysis uses other (and differently weighted) length
scales, and the constraints are not directly comparable. As shown by Asgari et
al. (2021), the COSEBIs used here are most sensitive to $\ell$ between 10 and
1500, while Euclid Collaboration: Deshpande et al. (2024) considered $\ell$ up
to 5000. COSEBIs also provide a separation into $E$\- and $B$-modes, while an
analysis using the $C(\ell)$ directly, implicitly assumes vanishing $B$-modes,
which leads to tighter parameter constraints.

We note that the photometric redshift estimates for our simulated Euclid-like
sample were derived assuming the full depth of the Euclid calibration fields.
These will not be fully available for the first Euclid data releases, so the
photometric redshifts will show larger scatter, increasing the TC effect.
However, the statistical uncertainty for the first data release will also be
significantly larger due to the smaller survey area, so a subtle effect like
SLC decreases in importance.

Another confounding factor to the SLC that has not been considered here is the
impact of blending. Since galaxies have a finite extent on the sky, galaxies
that are close by might not be detected as individual objects, which biases
shear estimates (Euclid Collaboration: Congedo et al., in prep). This bias is
correlated with the distribution of sources, as stronger source clustering
leads to more blended objects, leading to a further correlation between the
estimated shear and the source distribution. Studying the impact of blending
requires not only simulated sources with a realistic spatial distribution,
provided here by the FS2 and MICE simulations but also a realistic (observed)
size distribution obtained from image simulations, as created for example by
Jansen et al. (2024). We leave the investigation of the impact of blending on
SLC to future work.

Accounting for SLC in the theoretical modelling remains a challenge. The
intrinsic clustering (IC) effect can be estimated from first principles, as
discussed in Krause et al. (2021) and Appendix A. However, this is actually
the smallest contributor to the total SLC. Modelling the estimator bias (EB)
and TC effects is more involved, as they require realistic models of the
(three-dimensional) source galaxy distribution
$n(\chi\boldsymbol{\vartheta},\chi)$. In principle, this could be measured
alongside the sources’ shears, but for photometric samples, the distance
estimate will not be accurate enough. Instead, a better understanding of how
the scatter in photometric redshift estimates impacts the number density of
selected sources, conditioned on the overall galaxy number density, could help
in modelling these effects more thoroughly.

Finally, we note that our analysis was concerned with second-order cosmic
shear statistics. Euclid will achieve its tightest constraints on cosmological
parameters from second-order statistics by combining the cosmic shear analysis
with measurements of galaxy clustering and the cross-correlation of shear and
galaxy positions. These constraints will be tighter than for cosmic shear
alone (e.g., about a factor 2 in $\Omega_{\mathrm{m}}$ for a flat $\Lambda$CDM
setting, Euclid Collaboration: Blanchard et al. 2020). Consequently, the
impact of SLC becomes more significant.

Furthermore, higher-order statistics, which could further improve the
cosmological constraints, might be even stronger impacted by SLC. Recently,
Gatti et al. (2024) investigated the impact of SLC on higher-order shear
statistics and found that it can significantly bias measurements and
cosmological inferences. However, this bias is caused partly by the choice of
estimator for the higher-order statistics, which relies on shear or
convergence maps. Since the SLC impacts the number of sources in each pixel of
the map, it also affects the noise of the estimate. The correlation between
the source number and the signal then introduces a correlation of the signal
variance with the signal itself and, thus, an additional third-order
correlation. The effect needs to be modelled for every statistic involving
shear or convergence maps but does not occur for statistics measured directly
from galaxy catalogues. Thus, we expect SLC to have a negligible effect on
higher-order statistics obtained from correlation functions, such as third-
order cosmic shear (Schneider et al. 2005; Heydenreich et al. 2023). We leave
the proper investigation of SLC for third-order shear statistics measured from
correlation functions to future work.

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Agenzia Spaziale Italiana, the Austrian
Forschungsförderungsgesellschaft funded through BMK, the Belgian Science
Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für Luft- und
Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark, the French
Centre National d’Etudes Spatiales, the Fundação para a Ciência e a
Tecnologia, the Hungarian Academy of Sciences, the Ministerio de Ciencia,
Innovación y Universidades, the National Aeronautics and Space Administration,
the National Astronomical Observatory of Japan, the Netherlandse
Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research
Council of Finland, the Romanian Space Agency, the State Secretariat for
Education, Research, and Innovation (SERI) at the Swiss Space Office (SSO),
and the United Kingdom Space Agency. A complete and detailed list is available
on the Euclid web site (www.euclid-ec.org). LL is supported by the Austrian
Science Fund (FWF) [ESP 357-N]. The Innsbruck authors acknowledge support from
the Austrian Research Promotion Agency (FFG) and the Federal Ministry of the
Republic of Austria for Climate Action, Environment, Mobility, Innovation and
Technology (BMK) via grants 899537 and 900565. SU acknowledges support from
the Max Planck Society and the Alexander von Humboldt Foundation in the
framework of the Max Planck-Humboldt Research Award endowed by the Federal
Ministry of Education and Research. H. Hildebrandt is supported by a DFG
Heisenberg grant (Hi 1495/5-1), the DFG Collaborative Research Center SFB1491,
as well as an ERC Consolidator Grant (No. 770935). MA is supported by the UK
Science and Technology Facilities Council (STFC) under grant numbers
ST/Y002652/1 and the Royal Society under grant numbers RGSR2222268 and
ICAR1231094. This work has made use of CosmoHub. CosmoHub has been developed
by the Port d’Informació Científica (PIC), maintained through a collaboration
of the Institut de Física d’Altes Energies (IFAE) and the Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT) and the
Institute of Space Sciences (CSIC & IEEC), and was partially funded by the
”Plan Estatal de Investigación Científica y Técnica y de Innovación” program
of the Spanish government. Based on data products from observations made with
ESO Telescopes at the La Silla Paranal Observatory under programme IDs
177.A-3016, 177.A-3017, 177.A-3018, 179.A-2004, 298.A-5015.

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##  Appendix A Additional term to the convergence power spectrum due to IC
source-lens clustering

This appendix calculates the expected additional term $\Delta C$ to the
lensing power spectrum $C(\ell)$ due to the IC part of source-lens clustering.
As discussed in Sect. 2.2, the effect leads to an additional term
$\Delta\kappa$ to the lensing convergence. Consequently, the convergence
correlation function
$\xi_{\mathrm{\kappa}}(|\boldsymbol{\theta}_{1}-\boldsymbol{\theta}_{2}|)=\expectationvalue{\kappa(\boldsymbol{\theta}_{1})\,\kappa(\boldsymbol{\theta}_{2})}$
contains additional terms and is

$\expectationvalue{\kappa(\boldsymbol{\theta}_{1})\,\kappa(\boldsymbol{\theta}_{2})}=\expectationvalue{\kappa_{0}(\boldsymbol{\theta}_{1})\,\kappa_{0}(\boldsymbol{\theta}_{2})}+\underbrace{\expectationvalue{\kappa_{0}(\boldsymbol{\theta}_{1})\,\Delta\kappa(\boldsymbol{\theta}_{2})}}_{\mathcal{A}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})}+\expectationvalue{\Delta\kappa(\boldsymbol{\theta}_{1})\,\kappa_{0}(\boldsymbol{\theta}_{2})}+\underbrace{\expectationvalue{\Delta\kappa(\boldsymbol{\theta}_{1})\,\Delta\kappa(\boldsymbol{\theta}_{2})}}_{\mathcal{B}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})}\;.$
(28)

We are first considering $\mathcal{A}$, which is

$\displaystyle\mathcal{A}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})=b\int_{0}^{\infty}\differential{\chi_{1}}\int_{0}^{\infty}\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\differential{\chi^{\prime}_{1}}\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{2}}n(\chi^{\prime}_{1})\,n(\chi^{\prime}_{2})\,W(\chi_{1},\chi_{1}^{\prime})\,W(\chi_{2},\chi_{2}^{\prime})\,\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}\;.$
(29)

This term vanishes under the Limber approximation. This can be seen
qualitatively by considering the three-point correlation function. The
correlation function will only give a significant contribution if the
densities are evaluated at similar distances. Therefore,
$\chi_{1}\simeq\chi_{2}\simeq\chi^{\prime}_{2}$. However, for
$\chi_{2}=\chi^{\prime}_{2}$, the lensing efficiency
$W(\chi_{2},\chi^{\prime}_{2})$ vanishes. Consequently, $\mathcal{A}$ becomes
zero.

To see this more clearly, we are Fourier transforming $\mathcal{A}$ to
$\tilde{\mathcal{A}}$, which is

$\displaystyle\tilde{\mathcal{A}}(\boldsymbol{\ell}_{1},\boldsymbol{\ell}_{2})$
$\displaystyle=\int\differential[2]{\theta_{1}}\int\differential[2]{\theta_{2}}\mathrm{e}^{-\mathrm{i}\boldsymbol{\theta}_{1}\cdot\boldsymbol{\ell}_{1}-\mathrm{i}\boldsymbol{\theta}_{2}\cdot\boldsymbol{\ell}_{2}}\mathcal{A}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$
(30)
$\displaystyle=b\,\int_{0}^{\infty}\differential{\chi_{1}}\int_{0}^{\infty}\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\differential{\chi^{\prime}_{1}}\,\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{2}}p(\chi^{\prime}_{1})\,p(\chi^{\prime}_{2})\,W(\chi_{1},\chi^{\prime}_{1})\,W(\chi_{2},\chi^{\prime}_{2})$
$\displaystyle\quad\times\int\frac{\differential{k_{1z}}}{2\pi}\int\frac{\differential{k_{2z}}}{2\pi}\,\mathrm{e}^{\mathrm{i}\,k_{1z}(\chi_{1}-\chi^{\prime}_{2})}\,\mathrm{e}^{\mathrm{i}\,k_{2z}(\chi_{2}-\chi^{\prime}_{2})}\,(2\pi)^{2}\delta_{\mathrm{D}}\left(\frac{\boldsymbol{\ell}_{1}}{\chi_{1}}+\frac{\boldsymbol{\ell}_{2}}{\chi_{2}}+\frac{\boldsymbol{\ell}_{2}}{\chi^{\prime}_{2}}\right)$
$\displaystyle\quad\times
B\left[\sqrt{\frac{|\boldsymbol{\ell}_{1}|^{2}}{\chi^{2}_{1}}+k^{2}_{1z}},\sqrt{\frac{|\boldsymbol{\ell}_{2}|^{2}}{\chi^{2}_{2}}+k^{2}_{2z}},\sqrt{\frac{|\boldsymbol{\ell}_{2}|^{2}}{{\chi^{\prime}_{2}}^{2}}+(k_{1z}+k_{2z})^{2}},\chi_{1},\chi_{2},\chi^{\prime}_{2}\right]\;,$

where $B$ is the (three-dimensional) matter bispectrum. Under the Limber
approximation, the bispectrum simplifies as

$B\left[\sqrt{\frac{|\boldsymbol{\ell}_{1}|^{2}}{\chi^{2}_{1}}+k^{2}_{1z}},\sqrt{\frac{|\boldsymbol{\ell}_{2}|^{2}}{\chi^{2}_{2}}+k^{2}_{2z}},\sqrt{\frac{|\boldsymbol{\ell}_{2}|^{2}}{{\chi^{\prime}_{2}}^{2}}+(k_{1z}+k_{2z})^{2}},\chi_{1},\chi_{2},\chi^{\prime}_{2}\right]\rightarrow
B\left(\frac{\ell_{1}}{\chi_{1}},\frac{\ell_{2}}{\chi_{2}},\frac{\ell_{2}}{\chi^{\prime}_{2}},\chi_{1},\chi_{2},\chi^{\prime}_{2}\right)\;.$
(31)

Then, $\tilde{\mathcal{A}}$ becomes

$\displaystyle\tilde{\mathcal{A}}(\boldsymbol{\ell}_{1},\boldsymbol{\ell}_{2})$
(32)
$\displaystyle=(2\pi)^{2}b\int_{0}^{\infty}\differential{\chi_{1}}\int_{0}^{\infty}\differential{\chi_{2}}\,\int_{\chi_{1}}^{\infty}\differential{\chi^{\prime}_{1}}\,\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{2}}p(\chi^{\prime}_{1})\,p(\chi^{\prime}_{2})\,W(\chi_{1},\chi^{\prime}_{1})\,W(\chi_{2},\chi^{\prime}_{2})\,\delta_{\mathrm{D}}(\chi_{1}-\chi^{\prime}_{2})\,\delta_{\mathrm{D}}(\chi_{2}-\chi^{\prime}_{2})\,\delta_{\mathrm{D}}\left(\frac{\boldsymbol{\ell}_{1}}{\chi_{1}}+\frac{\boldsymbol{\ell}_{2}}{\chi_{2}}+\frac{\boldsymbol{\ell}_{2}}{\chi^{\prime}_{2}}\right)$
$\displaystyle\quad\times
B\left(\frac{\ell_{1}}{\chi_{1}},\frac{\ell_{2}}{\chi_{2}},\frac{\ell_{2}}{\chi^{\prime}_{2}},\chi_{1},\chi_{2},\chi^{\prime}_{2}\right)$
$\displaystyle=(2\pi)^{2}b\,\int_{0}^{\infty}\differential{\chi_{2}}\,\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{1}}\,p(\chi^{\prime}_{1})\,p(\chi_{2})\,W(\chi_{2},\chi^{\prime}_{1})\,W(\chi_{2},\chi_{2})\,\delta_{\mathrm{D}}\left(\frac{\boldsymbol{\ell}_{1}+2\boldsymbol{\ell}_{2}}{\chi_{2}}\right)\,B\left(\frac{\ell_{1}}{\chi_{2}},\frac{\ell_{2}}{\chi_{2}},\frac{\ell_{2}}{\chi_{2}},\chi_{2},\chi_{2},\chi_{2}\right)$
$\displaystyle=0\;,$ (33)

where we used in the last step that $W(\chi_{2},\chi_{2})=0$.

Consequently, the only term that can impact the convergence power spectrum is
$\mathcal{B}$. This term is given by

$\displaystyle\mathcal{B}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})=b^{2}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\differential{\chi_{1}}\int_{0}^{\infty}\\!\\!\\!\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\\!\\!\\!\differential{\chi^{\prime}_{1}}\int_{\chi_{2}}^{\infty}\\!\\!\\!\differential{\chi^{\prime}_{2}}n(\chi^{\prime}_{1})\,n(\chi^{\prime}_{2})\,W(\chi_{1},\chi_{1}^{\prime})\,W(\chi_{2},\chi_{2}^{\prime})\,\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})\,\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}\;.$
(34)

The four-point correlation function can be decomposed into its connected and
unconnected part,

$\displaystyle\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})\,\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1}\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}$
$\displaystyle=\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})\,\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}_{\mathrm{c}}$
(35)
$\displaystyle+\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}\expectationvalue{\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})}$
$\displaystyle+\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})}\expectationvalue{\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}$
$\displaystyle+\expectationvalue{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})}\expectationvalue{\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}\;.$

Using the same arguments as above, only the last summand can contribute to
$\mathcal{B}$, since the lensing efficiency suppresses all correlations with
$\chi_{a}\simeq\chi^{\prime}_{a}$. Consequently,

$\displaystyle\mathcal{B}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})=b^{2}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\differential{\chi_{1}}\int_{0}^{\infty}\\!\\!\\!\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\\!\\!\\!\differential{\chi^{\prime}_{1}}\int_{\chi_{2}}^{\infty}\\!\\!\\!\differential{\chi^{\prime}_{2}}n(\chi^{\prime}_{1})\,n(\chi^{\prime}_{2})\,W(\chi_{1},\chi_{1}^{\prime})\,W(\chi_{2},\chi_{2}^{\prime})\langle{\delta(\chi_{1}\boldsymbol{\theta}_{1},\chi_{1})\,\delta(\chi_{2}\boldsymbol{\theta}_{2},\chi_{2})}\rangle\langle{\delta(\chi^{\prime}_{1}\boldsymbol{\theta}_{1},\chi^{\prime}_{1})\,\delta(\chi^{\prime}_{2}\boldsymbol{\theta}_{2},\chi^{\prime}_{2})}\rangle\,.$
(36)

This is the full impact of source-lens clustering on the convergence
correlation function under the assumptions of the Limber approximation.

To find the impact $\Delta C(\boldsymbol{\ell})$ on the convergence power
spectrum $C(\ell)$, we Fourier-transform $\mathcal{B}$, so

$\displaystyle\Delta
C(\boldsymbol{\ell}_{1})(2\pi)^{2}\delta_{\mathrm{D}}(\boldsymbol{\ell}_{1}+\boldsymbol{\ell}_{2})$
$\displaystyle=\int\differential[2]{\theta_{1}}\int\differential[2]{\theta_{2}}\mathrm{e}^{-\mathrm{i}\boldsymbol{\theta}_{1}\cdot\boldsymbol{\ell}_{1}-\mathrm{i}\boldsymbol{\theta}_{2}\cdot\boldsymbol{\ell}_{2}}\mathcal{B}(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$
(37)
$\displaystyle=b^{2}\int_{0}^{\infty}\differential{\chi_{1}}\int_{0}^{\infty}\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\differential{\chi^{\prime}_{1}}\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{2}}n(\chi^{\prime}_{1})\,n(\chi^{\prime}_{2})\,W(\chi_{1},\chi_{1}^{\prime})\,W(\chi_{2},\chi_{2}^{\prime})$
(38)
$\displaystyle\quad\times\int\frac{\differential[2]{k_{\perp}}}{(2\pi)^{2}}\,\int\frac{\differential[2]{k^{\prime}_{\perp}}}{(2\pi)^{2}}\int\frac{\differential{k_{z}}}{2\pi}\int\frac{\differential{k^{\prime}_{z}}}{2\pi}P\left[|(\boldsymbol{k}_{\perp},k_{z})|\right]\,P\left[|(\boldsymbol{k}^{\prime}_{\perp},k^{\prime}_{z})|\right]\,\mathrm{e}^{\mathrm{i}k_{z}\,(\chi_{1}-\chi_{2})+\mathrm{i}k^{\prime}_{z}\,(\chi^{\prime}_{1}-\chi^{\prime}_{2})}$
$\displaystyle\quad\times(2\pi)^{4}\,\delta_{\mathrm{D}}(\boldsymbol{k}_{\perp}\,\chi_{1}+\boldsymbol{k}^{\prime}_{\perp}\,\chi^{\prime}_{1}-\boldsymbol{\ell}_{1})\,\delta_{\mathrm{D}}(-\boldsymbol{k}_{\perp}\,\chi_{2}-\boldsymbol{k}^{\prime}_{\perp}\,\chi^{\prime}_{2}-\boldsymbol{\ell}_{2})\;.$

Under the Limber approximation, we replace
$P\left[|(\boldsymbol{k}_{\perp},k_{z})|\right]$ by $P(k_{\perp})$, so we can
execute the $k_{z}$ integrals, leading to

$\displaystyle\Delta
C(\boldsymbol{\ell}_{1})(2\pi)^{2}\delta_{\mathrm{D}}(\boldsymbol{\ell}_{1}+\boldsymbol{\ell}_{2})$
$\displaystyle=b^{2}\int_{0}^{\infty}\differential{\chi_{1}}\int_{0}^{\infty}\differential{\chi_{2}}\int_{\chi_{1}}^{\infty}\differential{\chi^{\prime}_{1}}\int_{\chi_{2}}^{\infty}\differential{\chi^{\prime}_{2}}n(\chi^{\prime}_{1})\,n(\chi^{\prime}_{2})\,W(\chi_{1},\chi_{1}^{\prime})\,W(\chi_{2},\chi_{2}^{\prime})\,\delta_{\mathrm{D}}(\chi_{1}-\chi_{2})\,\delta_{\mathrm{D}}(\chi^{\prime}_{1}-\chi^{\prime}_{2})$
(39)
$\displaystyle\quad\times\int\frac{\differential[2]{k_{\perp}}}{(2\pi)^{2}}\,\int\frac{\differential[2]{k^{\prime}_{\perp}}}{(2\pi)^{2}}P(k_{\perp})\,P(k^{\prime}_{\perp})\,(2\pi)^{4}\,\delta_{\mathrm{D}}(\boldsymbol{k}_{\perp}\,\chi_{1}+\boldsymbol{k}^{\prime}_{\perp}\,\chi^{\prime}_{1}-\boldsymbol{\ell}_{1})\,\delta_{\mathrm{D}}(-\boldsymbol{k}_{\perp}\,\chi_{2}-\boldsymbol{k}^{\prime}_{\perp}\,\chi^{\prime}_{2}-\boldsymbol{\ell}_{2})$
$\displaystyle=b^{2}\int_{0}^{\infty}\differential{\chi}\int_{\chi}^{\infty}\differential{\chi^{\prime}}n^{2}_{\mathrm{s}}(\chi^{\prime})\,W^{2}(\chi,\chi^{\prime})\,\frac{1}{\chi^{\prime
2}}\int\frac{\differential[2]{k_{\perp}}}{(2\pi)^{2}}\,P(k_{\perp})\,P\left(\frac{|\boldsymbol{\ell}_{1}-\boldsymbol{k}_{\perp}\,\chi|}{\chi^{\prime}}\right)\,(2\pi)^{2}\,\delta_{\mathrm{D}}(-\boldsymbol{\ell}_{1}-\boldsymbol{\ell}_{2})$
(40)
$\displaystyle=(2\pi)^{2}\,\delta_{\mathrm{D}}(\boldsymbol{\ell}_{1}+\boldsymbol{\ell}_{2})\,b^{2}\int_{0}^{\infty}\differential{\chi}\int_{\chi}^{\infty}\differential{\chi^{\prime}}\int\frac{\differential[2]{L}}{(2\pi)^{2}}\,\frac{n^{2}_{\mathrm{s}}(\chi^{\prime})}{\chi^{\prime
2}}\frac{W^{2}(\chi,\chi^{\prime})}{\chi^{2}}\,P\left(\frac{L}{\chi}\right)\,P\left(\frac{|\boldsymbol{\ell}_{1}-\boldsymbol{L}|}{\chi}\right)\;,$
(41)

where $\boldsymbol{L}=\boldsymbol{k}_{\perp}\,\chi$. Comparing this expression
to Equations (47) and (48) of Krause et al. (2021) shows that

$\Delta C(\boldsymbol{\ell})=\Delta C_{EE}(\boldsymbol{\ell})+\Delta
C_{BB}(\boldsymbol{\ell})\;,$ (42)

where $\Delta C_{EE}$ and $\Delta C_{BB}$ are their additional terms to the
cosmic shear $E$\- and $B$-mode power spectra due to source-lens-clustering
without magnification bias. Consequently, we confirm the expression for the
source-lens clustering effect in Krause et al. (2021). This is different from
the expression given in Euclid Collaboration: Deshpande et al. (2024). There,
they modelled the angular position-dependent
$n(\chi\,\boldsymbol{\theta},\chi)$ by

$n(\chi\,\boldsymbol{\theta},\chi)=n(\chi)\left[1+b\,\kappa(\boldsymbol{\theta})\right]\;,$
(43)

which is different from our Eq. (13).

##  Appendix B Impact of individual SLC effects on shear correlation functions

The scale dependence of the SLC effects is easier considered when using shear
correlation functions than COSEBIs, since they directly give the dependence on
angular scales. We show in Fig. 11 the shear correlation functions $\xi_{+}$
when different parts of the SLC are included, as well as the fractional
difference to the $\xi_{+}$ for unclustered sources. Similarly to the COSEBIs,
the estimator bias (EB) has the largest impact on the signal, which suppresses
the signal across all scales. It has the strongest effect at small scales for
the lowest tomographic bin, where source galaxy positions are show the
strongest correlation. The intrinsic clustering (IC) effect, which partially
cancels the EB effect is also strongest at small scales, so the combination of
EB and IC is roughly scale independent. The TC effect has a similar strength
across all scales.

Figure 11: Shear correlation function $\xi_{+}$ for KiDS-like setup when
different SLC effects are included, either the full SLC (red crosses), the
estimator bias (EB) and intrinsic clustering (IC) (orange triangles), only the
estimator bias (EB) (green triangles) and no SLC (blue dots). Upper right:
Shear correlation functions. Lower left: Fractional difference to $\xi_{+}$
for unclustered sources, i.e., without SLC.

##  Appendix C Additional figures

For completeness, we report in this section the posteriors for all
cosmological and nuisance parameters, which were varied in the cosmological
inference, as well as the measured COSEBIs for all tomographic bins in the
Euclid-like setup.

Figure 12: Cosmological parameter constraints for the KiDS-like setup with
nuisance parameters. Figure 13: Nuisance parameter constraints for the KiDS-
like setup. Figure 14: Cosmological parameter constraints for the KiDS-like
setup without nuisance parameters. Figure 15: Cosmological parameter
constraints for the Euclid-like setup with nuisance parameters. Figure 16:
Nuisance parameter constraints for the Euclid-like setup. Figure 17: Further
nuisance parameter constraints for the Euclid-like setup. Figure 18:
Cosmological parameter constraints for the Euclid-like setup without nuisance
parameters. Figure 19: Measurement for the Euclid-like setup. Upper right:
COSEBI $E$-modes $E_{n}$ measured for the original FS2 galaxies (black
points), the clustered catalogue(red plusses) and the unclustered
catalogue(blue crosses) with the Euclid uncertainty (grey band). Lower left:
Difference between $E_{n}$ for the original and unclustered sources (black
points) and between the clustered and unclustered sources (blue crosses),
divided by Euclid uncertainty.