additional omission of the high mis-centering tail $f=0$ (dashed red), and the
posteriors of the WL bias in the four snap-shots. hydro-dynamical modelling
uncertainties are dominant at low redshift. When considering our
multiplicative shear bias modelling choices, the bias is consistent with zero,
while the presence of the highly mis-centered tail does not affect the WL
bias.  Table 2:  Calibration parameters with their priors, for the Monte Carlo
marginalisation of the WL bias and scatter determination. Cluster Member
Contamination (Section 3.3.1)
---
$\log_{10}c=4.83\pm 0.22$ | $B_{\text{cl}}=0.467\pm 0.017$
$\ln\sigma_{\text{reg}}=-0.61\pm 0.43$ | $A_{0}=-4.75\pm 0.11$
$A_{1}=-4.38\pm 0.05$ | $A_{2}=-3.68\pm 0.04$
$A_{3}=-3.25\pm 0.04$ | $A_{4}=-3.04\pm 0.04$
Mis-centering Parameters (Section 4.2)
$\sigma_{0}[\text{arcseec}]=4.83\pm 0.22$ | $k=0.308\pm 0.007$
$\alpha=0.443\pm 0.012$ | $\ln f=-0.76\pm 0.11$
$\ln\beta=-0.18\pm 0.04$ | $\sigma_{\text{intr}}=0.104\pm 0.016$
$\ln\lambda=-0.18\pm 0.04$ |
Richness Mass relation (Chiu et al., 2022, app. A)
$A_{\lambda}=36.2\pm 3.6$ | $B_{\lambda}=0.881\pm 0.083$
$C_{\lambda}=-0.46\pm 0.52$ | $\sigma_{\lambda}=0.274\pm 0.066$
photo-$z$ and shape calibrations
$\mathcal{H}\sim\text{Int}(0,999)$∗ |  $\alpha_{\text{NL}}=0.6\pm 0.4$ †
hydro-dynamical modelling uncertainties
$\Delta b=0.02$, $\Delta b_{\text{M}}=0.018$ |  $\Delta s=0.25$, $\Delta s_{\text{M}}=0.59$
Note that the parameters for the cluster member contamination and the mis-
centering are actually drawn from the respective posterior fits to conserve
correlations among the parameters. Further notes: ∗ $\text{Int}(a,b)$ is a
uniform distribution over the integers $a\leq i\leq b$ (cf. Section 3.3.2); †
non-linear multiplicative shear bias (Grandis et al., 2021a, Section 2.1.7)
Table 3:  Calibration of WL bias and scatter for DES Y3 WL on eRASS1 clusters. $z$ | $0.24$ | $0.42$ | $0.68$ | $0.95$
---|---|---|---|---
$\mu_{\text{b}}$ | $-0.066$ | $-0.075$ | $-0.101$ | $-0.092$
$\delta_{\text{b1}}$ | $-0.013$ | $-0.018$ | $-0.034$ | $-0.092$
$\delta_{\text{b2}}$ | $-0.019$ | $-0.022$ | $-0.023$ | $0.015$
$\mu_{\text{s}}$ | $-3.094$ | $-3.232$ | $-2.925$ | $-1.833$
$\delta_{\text{s}}$ | $0.272$ | $0.270$ | $0.275$ | $0.283$
| $b_{\text{M}}=1.030\pm 0.021$ | $s_{\text{M}}=-0.919\pm 0.608$
Numerical values for the WL bias ($\mu_{\text{b}}$, $\delta_{\text{b1}}$,
$\delta_{\text{b2}}$) and scatter ($\mu_{\text{s}}$, $\delta_{s}$) calibration
at different redshift $z$, and their global mass trends $b_{\text{M}}$, and
$s_{\text{M}}$, respectively.

###  4.5 WL bias and scatter

At the end of the synthetic shear profile production and WL extraction, we get
a bias and scatter $(b^{\text{WL}}_{l},\,\sigma^{\text{WL}}_{l})$ for each
snap shot $l$, as well as a mass trend for both the bias
($b_{\text{M}}^{\text{WL}}$), and the scatter ($s_{\text{M}}$). They are
obtained by fitting the relation

$\displaystyle\bigg{<}\ln\frac{M_{\text{WL}}}{M_{0}}\bigg{|}M,z_{\text{cl}}\bigg{>}=b^{\text{WL}}_{l}+b_{\text{M}}^{\text{WL}}\ln\left(\frac{M}{M_{0}}\right)$
$\displaystyle\ln\sigma_{\text{WL}}^{2}=s_{l}+s_{\text{M}}\ln\left(\frac{M}{M_{0}}\right),$
(35)

with pivot mass $M_{0}=2\times 10^{14}\leavevmode\nobreak\ M_{\odot}$, and
$s_{l}=\ln(\sigma^{\text{WL}}_{l})^{2}$. The fit of the relation includes
cluster-by-cluster weights corresponding to the probability of the mis-
centering of the individual synthetic halo, as discussed in Grandis et al.
(2021a). The mean relations resulting from the fit at each redshift are shown
in Fig 13 as black lines.

The uncertainty on these quantities is obtained by re-running the synthetic
shear profile creation and WL mass extraction $\mathcal{O}(1000)$ times with
slightly perturbed input parameters. Specifically, we vary the hyper parameter
of the DES Y3 source redshift distributions $\mathcal{H}\in(0,999)$ ( Myles &
Alarcon et al. 2021, and as outlined in Section 3.3.2), the parameters of the
richness–mass relation $\boldsymbol{p}_{\lambda\text{M}}$ within the posterior
reported by Chiu et al. (2022), the parameters of the cluster member
contamination $\boldsymbol{p}_{\text{cl}}$ within the posterior determine in
Section 3.3.1, the parameters of the mis-centering distribution
$\boldsymbol{p}_{\text{mct}}$ within the posterior determine in Section 4.2,
as well as a wide prior on the non-linear shear biases. To these uncertainties
we add a hydro-dynamical modelling uncertainty of $2\%$ on the WL bias, and
corresponding values for the other parameters, as determined by Grandis et al.
(2021a, Table 2).

The resulting parameter posterior distributions are presented in the contour
plots in Fig. 14. We omit parameters that do not (anti)correlate with the WL
bias and scatter parameters. As can be seen, the WL bias in the highest
redshift slot $b^{\text{WL}}_{4}$ anticorrelates tightly with the critical
surface density
$\Sigma_{\text{crit},4}=\left\langle\Sigma_{\text{crit},s,l=z_{4}}^{-1}\right\rangle_{P^{\text{snth}}_{z_{4}}(z_{\text{s}})}^{-1}$,
defined as the inverse lensing efficiency, reported in units of $10^{15}$
$h$M⊙ Mpc-2. This indicates that the uncertainty on this bias parameter is
dominated by the uncertainty on the lensing efficiency, as induced by the
photo-$z$ uncertainty.

Given these correlations, we compress the posterior on the WL bias into mean
values $\mu_{\text{b},l}$, and two principal components $\delta_{\text{b1},l}$
and $\delta_{\text{b2},l}$, scaled by their variances. This means that a
realisation of the WL bias at an arbitrary redshift $z_{\text{cl}}$ can be
constructed by interpolating the mean and principal components and drawing two
random standard normal variates $A_{\text{WL}}$, $B_{\text{WL}}$, such that

$b(z_{\text{cl}})=\mathcal{I}(z_{\text{cl}}|z_{l},\,\mu_{\text{b},l})+A_{\text{WL}}\mathcal{I}(z_{\text{cl}}|z_{l},\,\delta_{\text{b1},l})+B_{\text{WL}}\mathcal{I}(z_{\text{cl}}|z_{l},\,\delta_{\text{b2},l}),$
(36)

where $\mathcal{I}(z_{\text{cl}}|z_{l},\,\mu_{\text{b},l})$ is the
interpolation of the data vectors $(z_{l},\,\mu_{\text{b},l})$ at redshift
$z_{\text{cl}}$. The resulting posterior on the WL bias as a function of
cluster redshift is shown in the insert of Fig. 14 in blue bands for the 1
$\sigma$ region, and faded lines for the 2 $\sigma$ regions, together with the
results for the 4 snapshots as black data points. For the natural logarithm of
the variance, whose posterior is dominated by the uncertainty from hydro-
dynamical modelling, we employ a single principal component (see below, Eq.
45).

The numerical value of our WL bias is somewhat low, ranging from $-0.05$ to
$-0.1$. A contributing factor is the response of the WL bias to our treatment
of the multiplicative shear bias. While we include the multiplicative shear
bias in the synthetic shear profiles, we do not include this bias in the model
so as to reduce the number of independent calibration product we need to
communicate to the cosmological number counts analysis (Ghirardini et al.,
A&A, subm.). To gauge the impact of this modelling choice on the numerical
value of the bias, we perform a synthetic shear and mass extraction run at the
mean values of all parameters, but artificially set the multiplicative shear
bias $m=0$, that is we normalise the synthetic source redshfit distributions
to 1. This results in a shift, shown as the red line in the insert of Fig. 14
of the order of $-0.03$. It is two sigma away from our inferred bias values
(black data points). This indicates that our extraction model is a reasonable
representation of the synthetic data, once corrected for the above described
modelling mis-match. Slightly negative WL biases can be explained by the fact
that the density contrast of a halo is reduced by the projection of nearby
correlated structure, resulting in a WL mass that is lower than the halo mass
(Oguri & Hamana, 2011; Becker & Kravtsov, 2011; Bahé et al., 2012).

We also test the sensitivity of the WL bias to the presence of the highly mis-
centered tail in the eROSITA digital twin (cf. Section 4.2). A similar large
mis-centering tail is found by Bulbul et al. (A&A, subm.), Section 5 and Fig.
9, left panel, when comparing the centers obtained from the source detection
algorithm and the X-ray post-processing. While the origin of this tail is
still under investigation, we explore the hypothesis that the large mis-
centering tail is completely suppressed by the X-ray post-processing. Setting
the fraction $f=0$ fully suppresses the high exponential tail, but only mildly
alters the WL bias, as can be seen by the difference between the red solid and
dashed line. Our WL bias numbers, and therefore the mass anchoring of the
number counts experiments, are stable w.r.t. the presence of this highly mis-
centered tail, which might be an artefact of the eROSITA digital twin
processing, and be corrected by the X-ray post-processing.

In summary, the two main contributions to uncertainty of the WL bias are the
systematic floor resulting from the comparison of different hydro-dynamical
simulations established by Grandis et al. (2021a), and, at high redshift
$z_{\text{l}}>0.6$ also the uncertainty of the DES Y3 photometric redshift
distributions, shown in Fig. 2.

##  5 Mass constraints

WL measurements around galaxy clusters provide a highly accurate (see Section
4) mass proxy. They have, however, a quite low precision at the individual
object basis, considering that the approximately 2.2k clusters we consider
collectively only reach a signal-to-noise of 65 (cf. Section 3.4). A natural
approach to counter-act this is to perform the mass measurement on stacked
lensing signals (e.g. ; , ). While this simplifies the mass measurement task,
it significantly complicates the modelling in a cosmological context by
introducing scale dependent selection bias effects (Sunayama et al., 2020;
Dark Energy Survey Collaboration et al., 2020; Wu et al., 2022; Sunayama,
2023). In this work, we therefore follow the Bayesian Population Modelling
approach pioneered by Mantz et al. (2015); Bocquet et al. (2019); Grandis et
al. (2019), which predicts the distribution of expected shear profiles given
the selection observable, as a function of the parameters governing the
scaling between halo mass and selection observables. Complete derivations of
this analysis framework can be found in Ghirardini et al. (A&A, subm.);
Bocquet et al. (2023) in the context of eROSITA–selected clusters and
SPT–selected clusters, respectively. Our results here are based on the
implementation described in Ghirardini et al. (A&A, subm.), Section 4.

###  5.1 Constraints of the count rate–mass relation

In the following we discuss the likelihood setup used for the mass calibration
as well as the resulting mass calibration constraints. In general, WL mass
calibration relies on statistically determining the scaling relation between
the observables and halo mass (Mantz et al., 2016; Dietrich et al., 2019;
Schrabback et al., 2021; Zohren et al., 2022, for instance). In our case, we
expand this standard treatment, to accommodate for the residual contamination
of our sample by random superpositions of X-ray candidates and optical
systems, and, more importantly, mis-classified active galactic nuclei (see
Bulbul et al. (A&A, subm.); Kluge et al. (A&A, subm.); Ghirardini et al. (A&A,
subm.)). Both mis-classified AGNs and random superpositions are expect to
produce far less WL signal compared to clusters, which inhabit massive halos.
We therefore employ a _mixture model_ , a weighted sum of the different sub-
populations, to perform the mass calibration in presence of non-negligible
contamination.

####  5.1.1 Mixture model

For each individual cluster, we construct the probability density function
(PDF) of its shear profile $g_{\text{t}}$ given the cluster observables count
rate $\hat{C}_{\text{R}}$, redshift $z_{\text{cl}}$ and sky position
$\hat{\mathcal{H}}$ as

$\begin{split}P(\hat{g}_{\text{t}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})&=f_{\text{C}}(\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})\,P(\hat{g}_{\text{t}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}},C)\\\
\
&+f_{\text{RS}}(\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})\,P(\hat{g}_{\text{t}}|\text{RS})\\\
&+f_{\text{AGN}}(\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})\,P(\hat{g}_{\text{t}}|\text{AGN},z_{\text{cl}}),\end{split}$
(37)

a weighted sum of the pdf of the shear profile for clusters (C), random
superpositions (RS), and mis-classified active galactic nuclei (AGN), with
their respective frequencies
$f_{\tau}(\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})$ as a function of count
rate, redshift and sky position, where $\tau\in$ (C, RS, AGN). Naturally,
$\sum_{\tau}f_{\tau}(\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}})=1$, where
the sum runs over $\tau\in$ (C, RS, AGN). As outlined in Bulbul et al. (A&A,
subm.); Kluge et al. (A&A, subm.); Ghirardini et al. (A&A, subm.), the shape
of the AGN and the random superpositions contributions are calibrated
partially on X-ray image simulations, partially with optical follow up of
random positions and point sources, and have amplitudes ($f_{\text{RS}}$,
$f_{\text{AGN}}$) that are fitted for using the richness distribution of the
sample at a given count rate, redshift and sky position. They find
$f_{\text{RS}}=0.0061\pm 0.0023$ and $f_{\text{AGN}}=0.0462\pm 0.0038$, which
we adopt as priors.

For the WL modelling, we assume that the density contrast of a random source
is zero, resulting in their shear profile pdf being

$\ln
P(\hat{g}_{\text{t}}|\text{RS})=-\frac{1}{2}\sum_{k}\frac{\hat{g}_{\text{t},k}^{2}}{\delta
g_{\mathrm{t},k}^{2}},$ (38)

where the sum runs over the radial bins $k$, with the measured shear profile
$\hat{g}_{\text{t},k}$ defined in Eq. 22, and its shape noise error $\delta
g_{\mathrm{t},k}$ defined in Eq. 23. In practice, all objects in our lens
sample have a richness $\hat{\lambda}>3$. At the lens redshift, the random
source will thus have a few red galaxies along their lines of sight. Their
density contrast is however expected to be small compared to cluster density
contrasts.

For AGNs we assume that their density contrast $\Delta\Sigma_{\text{AGN}}(R)$
follows the profile measured by Comparat et al. (2023) for AGN in eFEDS using
HSC WL data. We multiply this with the average lensing efficiency
$\Sigma_{\text{crit,c}}^{-1}=\langle\Sigma_{\text{crit,cs}}^{-1}\rangle_{P_{\text{c}}(z_{\text{s}})}$,
evaluated as a function of cosmology for the source redshift distribution
$P_{\text{c}}(z_{\text{s}})$ constructed in Eq. 25. The pdf thus reads

$\ln
P(\hat{g}_{\text{t}}|\text{AGN},z_{\text{cl}})=-\frac{1}{2}\sum_{k}\frac{1}{\delta
g_{\mathrm{t},k}^{2}}\left(\hat{g}_{\text{t},k}-\Sigma_{\text{crit,c}}^{-1}\Delta\Sigma_{\text{AGN}}(R=D_{c}\theta_{k})\right)^{2},$
(39)

where $D_{c}$ is the angular diameter distance to the cluster, and
$\theta_{k}$ the average angular distance to the sources in the respective
radial bin $k$, as computed in Eq. 24. In practice, the density contrast of
AGN is much smaller than that of galaxy clusters, given the lower host halo
mass.

For galaxy clusters, the pdf of the shear profile depends on our extraction
model $g_{\text{t}}^{\text{mod}}$, and the WL mass $M_{\text{WL}}$ defined by
this model (cf. Section 4.3). It therefore reads

$\ln
P(\hat{g}_{\text{t}}|\text{C},z_{\text{cl}},M_{\text{WL}})=-\frac{1}{2}\sum_{k}\frac{1}{\delta
g_{\mathrm{t},k}^{2}}\left(\hat{g}_{\text{t},k}-g_{\text{t}}^{\text{mod}}(R=D_{c}\theta_{k}|M_{\text{WL}})\right)^{2},$
(40)

with the extraction model depending on a host of cluster-specific observables,
as outlined above. Note that in all the three pdfs we omitted the
normalization term, which is the same for all three, and just depends on
$\delta g_{\mathrm{t},k}$, thus being constant w.r.t. the parameters of the
inference.

The pdf of the shear profile for clusters needs to be marginalised over the
distribution of WL masses $M_{\text{WL}}$ compatible with the clusters count
rate $\hat{C}_{\text{R}}$, redshift $z_{\text{cl}}$, and sky position
$\hat{\mathcal{H}}$, that is
$P(M_{\text{WL}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}},\boldsymbol{p}_{\text{SR}})$,
whose construction and dependence on model parameters
$\boldsymbol{p}_{\text{SR}}$ we shall discuss below. The pdf of the shear
profile for the cluster component is thus

$\displaystyle
P(\hat{g}_{\text{t}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}},\text{C})=\int\text{d}M_{\text{WL}}$
$\displaystyle P(\hat{g}_{\text{t}}|\text{C},z_{\text{cl}},M_{\text{WL}})$
$\displaystyle
P(M_{\text{WL}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}},\boldsymbol{p}_{\text{SR}}).$
(41)

The distribution of WL masses given the count rate, redshift and sky position
is proportional to

$\displaystyle
P(M_{\text{WL}}|\hat{C}_{R},z_{\text{cl}},\hat{\mathcal{H}},\boldsymbol{p}_{\text{SR}})\propto$
$\displaystyle\iint\text{d}M\text{d}C_{R}P(M_{\text{WL}},C_{R}|M,z_{\text{cl}},\boldsymbol{p}_{\text{SR}})$
$\displaystyle\quad
P(I|C_{R},z_{\text{cl}},\hat{\mathcal{H}}_{i})P(\hat{C}_{R}|C_{R})P(M,z_{\text{cl}}),$
(42)

with $P(\hat{C}_{R}|C_{R})$ modelling the observational uncertainty on the
count rate, $P(M,z_{\text{cl}})$ being proportional to the halo mass function,
and $P(I|C_{R},z_{\text{cl}},\hat{\mathcal{H}}_{i})$ the X-ray incompleteness
due to the extent selection. This ensures that we correctly model Eddington
bias, which in cluster studies manifests as more low mass objects scattering
up to a given observable value, than high mass objects scattering down to same
observable values. This happens even when the observable–mass scatter is
modelled as symmetric on account of the fact that there are just many more low
mass than high mass objects. This fact is quantitatively expressed by the halo
mass function. The expression given above is normalized to be a pdf in
$M_{\text{WL}}$ via integration in $M_{\text{WL}}$ and subsequent re-scaling.
See Ghirardini et al. (A&A, subm.) for an alternative, but equivalent,
derivation of the same likelihood.

####  5.1.2 Scaling relation

Given their deep potential wells, cluster observables scale tightly with the
host halos mass. Leveraging this strong physical prior, we model the
distribution of WL mass $M_{\text{WL}}$ and intrinsic (that is noise-free)
count rate $C_{\text{R}}$ at given halo mass $M$ and redshift $z$ as a bi-
variate log-normal, reading

$P(M_{\text{WL}},C_{R}|M,z_{\text{cl}},\boldsymbol{p}_{\text{SR}})=\mathcal{N}(\bar{\mu},\bar{\Sigma}),$
(43)

with mean and variance

$\bar{\mu}=\left[\left\langle\ln
C_{R}|M,z_{\text{cl}}\right\rangle,\left\langle\ln
M_{\text{WL}}|M,z_{\text{cl}}\right\rangle\right]$
$\bar{\Sigma}=\begin{bmatrix}\sigma_{\text{X}}^{2}&\rho_{\text{WL,X}}\sigma_{\text{X}}\sigma_{\text{WL}}\\\
\rho_{\text{WL,X}}\sigma_{\text{X}}\sigma_{\text{WL}}&\sigma_{\text{WL}}^{2}\end{bmatrix}$

where $\left\langle\ln C_{R}|M,z_{\text{cl}}\right\rangle$ and
$\left\langle\ln M_{\text{WL}}|M,z_{\text{cl}}\right\rangle$ are the scaling
relations between true mass and count rate and WL mass, respectively,
$\sigma_{\text{X}}$ and $\sigma_{\text{WL}}$ their intrinsic scatters, and
$\rho_{\text{WL,X}}$ the correlation coefficient between the two intrinsic
scatters. Inclusion of the latter is crucial to marginalise over possible
selection effects, like more concentrated halos _simultaneously_ having a
higher count rate and WL mass compared to other objects at their mass and
redshift, or any other effect that might similarly correlate the intrinsic
scatters.

As calibrated on dedicated simulations in Section 4, the mean WL mass – halo
mass relation is given by

$\bigg{<}\ln\frac{M_{\text{WL}}}{M_{\text{p}}}\bigg{|}M,z_{\text{cl}}\bigg{>}=b(z_{\text{cl}})+b_{\text{M}}\ln\left(\frac{M}{M_{\text{p}}}\right),$
(44)

with the pivot value for the mass $M_{\text{p}}=2\cdot 10^{14}M_{\odot}$, the
redshift dependent amplitude of the WL bias given by equation 36, and the mass
trend
$b_{\text{M}}=\mu_{b_{\text{M}}}+C_{\text{WL}}\cdot\sigma_{b_{\text{M}}}$,
with mean and standard deviation derived in Section 4.5 and reported in Table
3.

The scatter is similarly modelled as

$\sigma_{\text{WL}}^{2}=\exp\left\\{\mathcal{I}(z_{\text{cl}}|z_{l},\,\mu_{\text{s},l})+D_{\text{WL}}\mathcal{I}(z_{\text{cl}}|z_{l},\,\delta_{\text{s},l})\right\\},$
(45)

where $\mu_{\text{s},l}$ and $\delta_{\text{s},l}$ are the mean and standard
deviations of the WL scatter in the simulation snapshot $l$, derived in
Section 4.5 and reported in Table 3, and
$\mathcal{I}(z_{\text{cl}}|z_{l},\,\mu_{\text{s},l})$ is the interpolation of
the data vectors $(z_{l},\,\mu_{\text{s},l})$ to the redshift $z_{\text{cl}}$.
Given the tight correlations between the systematic uncertainties in the WL
scatters of the different redshift seen in Fig. 14, we opt to model the WL
scatter with a single principal component. We also set the mass trend of the
scatter to $s_{\text{M}}=0$ as in this work we are not interested to explore
mass trends in the intrinsic scatter of the other observables.

The count rate–mass relation is modelled as

$\bigg{<}\ln\frac{C_{R}}{C_{R,\text{p}}}\bigg{|}M,z_{\text{cl}}\bigg{>}=\ln
A_{\text{X}}+b_{\text{X}}(M,z_{\text{cl}})\cdot\ln\frac{M}{M_{\text{p}}}+e_{\text{X}}(z_{\text{cl}})$
(46)

where $C_{R,p}=0.1\text{ s}^{-1}$ is the pivot value for the count rate,
$M_{\text{p}}=2\cdot 10^{14}M_{\odot}$ is the pivot value for the mass.
$b_{\text{X}}(M,z)$ expresses the mass-redshift dependent slope of the scaling
relation, given by

$b_{\text{X}}(M,z_{\text{cl}})=\bigg{(}B_{\text{X}}+C_{\text{X}}\cdot\ln\frac{M}{M_{\text{p}}}+F_{\text{X}}\cdot\ln\frac{1+z_{\text{cl}}}{1+z_{\text{p}}}\bigg{)}$
(47)

where $B_{\text{X}}$ is the classic standard single value mass slope,
$C_{\text{X}}$ allows the slope to be mass dependent, and $F_{\text{X}}$
allows the slope to be redshift dependent, and $z_{\text{p}}=0.35$ is the
pivot value for the redshift. As shown in Grandis et al. (2019), Appendix B,
$F_{\text{X}}$ needs to be sampled to retrieve unbiased cosmological results
from the number counts. Chiu et al. (2023) explored the impact of allowing for
a mass dependence in the slope in a cosmological analysis of the eFEDS sample,
finding that it is not necessary to give the scaling relation that freedom. We
thus set $C_{\text{X}}=0$.

The redshift evolution $e_{\text{X}}(z_{\text{cl}})$ of the X-ray scaling
relation is

$e_{\text{X}}(z_{\text{cl}})=D_{\text{X}}\cdot\ln\frac{d_{\text{L}}(z_{\text{cl}})}{d_{\text{L}}(z_{\text{p}})}+E_{\text{X}}\cdot\ln\frac{E(z_{\text{cl}})}{E(z_{\text{p}})}+G_{\text{X}}\cdot\ln\frac{1+z_{\text{cl}}}{1+z_{\text{p}}}$
(48)

where $E_{\text{X}}=2$ and $D_{\text{X}}=-2$ are the values predicted by the
self-similar model. The last term, with $G_{\text{X}}$, quantifies the
deviation from the self-similar model.
$E(z_{\text{cl}})=H(z_{\text{cl}})/H_{0}$ is the dimensionless Hubble factor.
The intrinsic scatter of our scaling relation is represented by a single
value, $\sigma_{\text{X}}$, independent of mass and redshift.

####  5.1.3 Likelihood setup and priors

Table 4: Priors on the parameters used in the fitting of the count rate–mass
relation.

Parameter | Units | Prior
---|---|---
Cosmology
$\Omega_{\mathrm{M}}$ | - | $\mathcal{N}(0.331,0.038)$
$\log_{10}A_{\text{S}}$ | - | Fixed to $-8.635$
$H_{0}$ | $\frac{\frac{\text{km}}{\text{s}}}{\text{Mpc}}$ | Fixed to 68.06
$\Omega_{\textrm{b},0}$ | - | Fixed to 0.048595
$n_{\text{S}}$ | - | Fixed to 0.9707
$w_{0}$ | - | Fixed to $-1$
$w_{\text{a}}$ | - | Fixed to 0
$\sum m_{\nu}$ | eV | Fixed to 0
$\Omega_{\textrm{k},0}$ | - | Fixed to 0
X-ray scaling relation
$A_{\text{X}}$ | - | $\mathcal{U}(0.01,3)$
$B_{\text{X}}$ | - | $\mathcal{U}(0.1,5)$
$C_{\text{X}}$ | - | Fixed to 0
$D_{\text{X}}$ | - | Fixed to $-2$
$E_{\text{X}}$ | - | Fixed to 2
$F_{\text{X}}$ | - | $\mathcal{U}(-5,5)$
$G_{\text{X}}$ | - | $\mathcal{U}(-5,5)$
$\sigma_{\text{X}}$ | - | $\mathcal{U}(0.05,2)$
WL mass calibration
$A_{\text{WL, DES}}$ | - | $\mathcal{N}(0,1)$
$B_{\text{WL, DES}}$ | - | $\mathcal{N}(0,1)$
$C_{\text{WL, DES}}$ | - | $\mathcal{N}(0,1)$
$D_{\text{WL, DES}}$ | - | $\mathcal{N}(0,1)$
$\rho_{\text{WL,X}}$ | - | $\mathcal{U}(-0.9,0.9)$
Contamination modeling
$f_{\text{AGN}}$ | - | $\mathcal{TN}(0.0412,0.0040,0,\infty)$
$f_{\text{RS}}$ | - | $\mathcal{TN}(0.0064,0.0024,0,\infty)$
555With $\mathcal{U}(\text{min},\text{max})$ we indicate a uniform
distribution between ‘min’ and ‘max’. With $\mathcal{N}(\mu,\sigma)$ we
indicate a normal distribution centered on $\mu$ and with standard deviation
$\sigma$. With $\mathcal{TN}(\mu,\sigma,\text{min},\text{max})$ we indicate a
truncated normal distribution centered on $\mu$ and with standard deviation
$\sigma$ bounded between ‘min’ and ‘max’.

As outlined in Ghirardini et al. (A&A, subm.), the total likelihood of our
shear profiles $\hat{g}_{\text{t},\varkappa}$ of several clusters, given their
count rate $\hat{C}_{\text{R},\varkappa}$, redshift $z_{\text{cl},\varkappa}$
and sky position $\hat{\mathcal{H}}_{\varkappa}$, where $\varkappa$ runs over
the cluster catalog, is given by

$\displaystyle\ln\mathcal{L}_{\text{WL}}=\sum_{\varkappa}\ln
P(\hat{g}_{\text{t},\varkappa}|\hat{C}_{\text{R},\varkappa},z_{\text{cl},\varkappa},\hat{\mathcal{H}}_{\varkappa}).$
(49)

In light of the derivations above, this is a function of the parameters of the
WL scaling relation
$\boldsymbol{p}_{\text{WL}}=(A_{\text{WL}},\,B_{\text{WL}},\,C_{\text{WL}},\,D_{\text{WL}})$.
To represent the WL anchoring performed in Section 4, we place independent
Gaussian priors on these. The parameters of the count rate–mass relation
$\boldsymbol{p}_{\text{X}}=(A_{\text{X}},\,B_{\text{X}},\,F_{\text{X}},\,G_{\text{X}},\sigma_{\text{X}})$,
instead, are the true target of our analysis. These are sampled within wide
flat priors, reported in Table 4. Also the correlation coefficient between the
WL and count rate intrinsic scatters is let free, but within the bounds
$\rho_{\text{WL,X}}\in(-0.9,\,0.9)$. We purposefully do not sample all the way
to $\pm$1, as the covariance of intrinsic scatters becomes singular in that
limit.

We place priors on the random source fractions and AGN ($f_{\text{RS}}$,
$f_{\text{AGN}}$) from our optical follow-up observations. We do not expect
the WL to significantly inform these fractions, as the individual shear
profiles are hardly significantly different from zero themselves. As such, at
an individual object basis, WL just lacks the statistical power to tell
clusters apart from AGN or random sources.

Figure 15: 1- and 2-d marginal posterior on the parameters of the count
rate–mass scaling relations from this work in blue, compared to the 1-d
marginals of the analysis of HSC WL around eFEDS clusters and groups (Chiu et
al., 2022) in black. We find good agreement on the mass trend $B_{\text{X}}$,
and the deviation from self-similar redshift scaling $G_{\text{X}}$, while
constraints on the amplitude $A_{\text{X}}$, and the redshift evolution of the
mass slope $F_{\text{X}}$ are shifted by 1- to 2-sigmas, consistent with
statistical fluctuations. Contrary to the eFEDS-HSC analysis, we do not place
an informative prior of the intrinsic scatter $\sigma_{\text{X}}$, and get
only weak constraints on it.

We fix all cosmological parameters to their Planck values values, except
$\Omega_{\text{M}}=0.331\pm 0.038$ from the first Dark Energy Survey
Supernovae Type Ia results (Dark Energy Survey Collaboration et al., 2019).
Several elements of our likelihood, like angular diameter distances and the
extraction model, depend on cosmology via the background evolution. The exact
cosmological dependence of the WL calibrated number counts is accounted for by
re-running the mass calibration presented here together with the number
counts. In this work, we wish to provide mass calibration results marginalised
over a reasonable range of cosmologies. Our $\Omega_{\text{M}}$-prior allows
us to do this in a physically motivated way, while remaining independent of
CMB results, with which the cosmological number counts analysis might be in
tension.

Figure 16: Posterior predictive distribution of the mean count rate–mass
relation (the band ranges from the 16th to the 84th percentile at each mass),
for three different redshifts (color coded). Filled bands represent the
analysis in this work, hatched bands represent the eFEDS HSC result from Chiu
et al. (2022). The predictions diverge for low mass, high redshift systems.

####  5.1.4 Scaling relation parameters constraints

We sample the total likelihood for the measured shear profiles with the priors
using ultranest (Buchner, 2021) to create a sample of the resulting posterior.
We find that all parameters of interest, the parameters of the count rate–mass
relation, are well constrained by our analysis (shown as blue contours in Fig.
15), while all the calibration and other nuisance parameters have posteriors
consistent with their priors. The find a constraint for the amplitude of the
scaling relation $A_{\text{X}}=0.88\pm 0.20$, when considering the mean and
standard deviation computed on the posterior sample. The mass slope of the
relation is constrained to $B_{\text{X}}=1.62\pm 0.14$, the redshift trend of
the mass slope to $F_{\text{X}}=-0.85\pm 0.93$. No significant deviation from
the self-similar redshift evolution is detected, as $G_{\text{X}}=-0.32\pm
0.69$ is consistent with 0. For the intrinsic scatter we find
$\sigma_{\text{X}}=0.61\pm 0.19$. Noticeably, the intrinsic scatter can be
constrained, despite the low precision of the individual lensing data, in line
the predictions by Grandis et al. (2019). While the intrinsic scatter values
is somewhat large, it matches the count rate mass scatter we find in the
digital twin simulations (Comparat et al., 2020; Seppi et al., 2022). To
better visualize our scaling relation results, we create a posterior
predictive distribution for the mean count rate as a function of mass and
redshift. The 1 sigma region of the predicted relation at three redshifts is
shown in Fig. 16 in filled bands.

Figure 17: Stacked measured reduced tangential shear profiles with measurement
uncertainty in black, for different count rate (rows) and redshift (columns)
bins. As a green line the best-fit shear profile model and with filled green
regions the propagation of the posterior uncertainty on the mean model. The
chosen model fits the data well at all count rates and redshifts. We report
also the median WL mass of each bin, in units of $M_{\odot}$.

###  5.2 Goodness of fit

To assess the goodness of fit of our mass calibration, we resort to binning
the cluster sample in redshift, count rate bins, with edges
$(0.10,0.19,0.27,0.35,0.48,0.8)$, and $(0.010.16,0.25,0.30,0.64,20)$, in
redshift and count rate respectively. For all clusters $\varkappa$ falling in
each of the bins, we stack the tangential reduced shear profiles by defining
the stacking weights $W_{\varkappa}=\sum_{b=2,3,4}w^{b}\sum_{i\in
b_{\varkappa}}w^{\mathrm{s}}_{i}\mathcal{R}_{i}$, computing

$\hat{g}_{\text{t}}^{\text{stack}}=\frac{\sum_{\varkappa}W_{\varkappa}\hat{g}_{\text{t},\varkappa}}{\sum_{\varkappa}W_{\varkappa}},$
(50)

where $i\in b_{\varkappa}$ symbolizes the sources $i$ associated to the
cluster $\varkappa$ in the tomographic bin $b$. Combining this expression with
Eq. 22 shows that it is equivalent to the raw tangential reduced shear
estimator in Eq. 8. Using the same weighting we also derive the error on the
stacked reduced shear

$\left(\delta
g_{\text{t}}^{\text{stack}}\right)^{2}=\frac{\sum_{\varkappa}W_{\varkappa}^{2}\delta
g_{\text{t},\varkappa}^{2}}{\sum_{\varkappa}W_{\varkappa}^{2}},$ (51)

and the mean angular separation of the source galaxies

$\theta^{\text{stack}}=\frac{\sum_{\varkappa}W_{\varkappa}\theta_{\varkappa}}{\sum_{\varkappa}W_{\varkappa}}.$
(52)

The resulting stacked tangential reduced shear profiles are shown as black
points with their error-bars in Fig. 17 given by the square root of the
variance calculated above.

For the model prediction, we draw $N_{\text{MC}}=1000$ points
$\boldsymbol{p}_{\text{SR}}^{v}$ from the posterior sample. For each cluster
$\varkappa$, and for each posterior point $v$, we compute its best-guess WL
mass

$M_{\text{WL},\varkappa}^{v}=\underset{M_{\text{WL}}}{\mathrm{argmax}}\leavevmode\nobreak\
P(M_{\text{WL}}|\hat{C}_{R,\varkappa},z_{\varkappa},\hat{\mathcal{H}}_{\varkappa},\boldsymbol{p}_{\text{SR}}^{v}),$
(53)

as the maximum of the pdf of WL masses, given the count rate, redshift and sky
position, evaluated for our specific posterior draw
$\boldsymbol{p}_{\text{SR}}^{v}$. We report the median mass of the clusters in
each stacking bin in Fig. 17. For each WL mass $M_{\text{WL},\varkappa}^{v}$
we evaluate our extraction model $g^{\text{mod},v}_{\text{t},\varkappa}$,
given in Eq. 33 using the source redshift distribution computed on the real
data, Eq. 25, and the mean positions of the sources $\theta_{\alpha}$ from Eq.
24. For each cluster and posterior point, we thus construct the predicted
reduced shear profile as

$g^{\text{pred},v}_{\text{t},\varkappa}=(1-f_{\text{RS},\varkappa}^{v}-f_{\text{AGN},\varkappa}^{v})g^{\text{mod},v}_{\text{t},\varkappa}+f_{\text{AGN},\varkappa}^{v}g^{\text{AGN}}_{\text{t},\varkappa},$
(54)

with $f_{\text{RS},\varkappa}^{v}$ and $f_{\text{AGN},\varkappa}^{v}$ the
random source and AGN fractions at the count rate, redshift and sky position
of the cluster $\varkappa$ given the model parameters
$\boldsymbol{p}_{\text{SR}}^{v}$.

The average reduced shear model in the bin is then given by stacking the
predicted reduced shear profile with the same weights as the data, and taking
a mean over the posterior points,

$g^{\text{pred}}_{\text{t}}=\frac{1}{N_{\text{MC}}}\sum_{v}\frac{\sum_{\varkappa}W_{\varkappa}g^{\text{pred},v}_{\text{t},\varkappa}}{\sum_{\varkappa}W_{\varkappa}},$
(55)

and variance

$\left(\delta
g^{\text{pred}}_{\text{t}}\right)^{2}=\frac{1}{N_{\text{MC}}}\sum_{v}\frac{\sum_{\varkappa}W_{\varkappa}\left(g^{\text{pred},v}_{\text{t},\varkappa}-g^{\text{pred}}_{\text{t},\varkappa}\right)^{2}}{\sum_{\varkappa}W_{\varkappa}},$
(56)

propagating the uncertainty in the model parameters expressed by their
posterior to the reduced shear profile prediction. The mean shear profile and
its systematic uncertainty are shown as green lines in Fig. 17. The visual
impression of a good fit is corroborated by considering that the chi-squared
between stacked data and predicted model is $\chi^{2}=180.0^{+45.8}_{-30.4}$
for 150 data points and 5 parameters that are effectively constrained. The
upper and lower errors for the $\chi^{2}$ result from the difference induced
when evaluating it with the model $g^{\text{pred}}_{\text{t}}\pm\delta
g^{\text{pred}}_{\text{t}}$, instead of the mean model. Within the errors
propagated from the posterior, we attain a very good chi-squared.

##  6 Discussion

###  6.1 Comparison to previous work

The only previous work that calibrated the eROSITA count rate – halo mass
relation was performed by Chiu et al. (2022), measuring, calibrating and
analysing the WL signal around 313 eFEDS selected clusters and groups with HSC
data. Their results are shown as black lines in Fig. 15. We agree on all
parameters within 2-sigma, with larger than 1-sigma deviations on the
amplitude, the redshift trend of the slope and the scatter. When considering
the predicted mean count rate as a function of mass and redshift, their
results predict higher count rates for low mass, high redshift objects
(hatched bands in Fig. 16). The difference is however not statistically
significant. The constraining power of this work is only marginally better
than the calibration derived by Chiu et al. (2022).

The similar constraining power of eFEDS-HSC analysis to this work, in spite of
the significantly smaller number of objects (313 compared to 2201) is
explained by the higher source density in HSC, as well as the cleaner
background selection. Indeed, the signal-to-noise ratio of an individual
cluster is proportional to

$\frac{\text{S}}{\text{N}}\propto\beta\sqrt{n_{\text{eff}}}\sqrt{1-f_{\text{cl}}},\text{
with
}\beta=\Bigg{\langle}\frac{\min(0,D_{\text{ls}})}{D_{\text{s}}}\Bigg{\rangle}_{P(z_{\text{s}})},$
(57)

where $\beta$ is the lensing efficiency, which is 0 for sources at the lens
redshift, and converges to 1 for infinitely distant sources. $n_{\text{eff}}$
is the effective source density, and $f_{\text{cl}}$ the cluster member
contamination. When considering these elements, each cluster in the HSC
analysis has a WL signal-to-noise approximately a factor of 2.3 larger then
the clusters considered in this work, assuming they have the same mass, and
ignoring cluster member contamination. Considering the overall signal-to-noise
gain stemming from the total number of clusters, $\sqrt{2201/313}=2.6$, we
find that it just about compensates the much lower individual clusters signal-
to-noise. Note also that in the eFEDS-HSC analysis, a tighter
$\Omega_{\text{M}}$, and an informative $\sigma_{\text{X}}$ prior where used.
Taken together, this explains the similar constraining power of the two works.

###  6.2 Cluster line of sight anomalies

We find several cluster line of sight anomalies, that we shall discuss here in
a cohesive manner. Firstly, it is somewhat anomalous that the cluster member
contamination estimates based on the effective number density of sources
display a larger contaminant fraction than the estimate based on the
decomposition of the source redshift distribution. The latter should be
impervious to masking and obstruction effects, as argued by Gruen et al.
(2014); Varga et al. (2019). Estimations of the cluster member contamination
via the effective number density of sources are biased low because of masking
and obstruction by foreground and bright cluster sources, as shown by
Kleinebreil et al. (in prep.). It is therefore surprising that we find the
number density estimate to be larger than the source redshift distribution
estimate, despite it potentially being biased low. A possible source of error
here is our very lenient background selection and the possible performance
loss of the source redshift decomposition method if the cluster redshift lies
too close to the peak of source redshift distribution of the field. This might
be hinted at by the imperfect subtraction signatures discussed in Section
3.3.1 and seen in Fig. 6.

Also interesting are the decrement of the effective shape noise and the
increment of the smooth shear response towards the cluster center. The effect
is rather constant in lens redshift, as opposed to the cluster member
contamination. Indeed, we see this anomaly also in the lowest source redshift
bin, $0.1<z_{\text{l}}<0.19$, where we have practically no cluster member
contamination. This makes it implausible that cluster member contaminants have
significantly different responses or intrinsic shapes. It is worthwhile noting
in Eq. 10 that a larger shear response leads to a smaller estimated effective
shape noise. Also, the shear response used in this work is a smooth function
of the source’s signal-to-noise and its size w.r.t. the PSF. Noticeably, both
these quantities are magnified by the cluster’s potential – a secondary effect
of WL. Such magnification effects might thus alter the inferred smooth shear
response. In general, shape measurement and photo-$z$ calibration in cluster
lines of sight might be effected to second order not only by magnification,
but also by increased blending (Hernández-Martín et al., 2020) and the intra-
cluster light (Zhang et al., 2019). Exploitation of future wide optical and
NIR surveys might necessitate dedicated assessments of the performance of
shape and photo-$z$ measurement techniques in cluster lines of sight.

Figure 18: 1-d and 2-d marginal contours of the amplitude of the count
rate–mass scaling relation $A_{\text{X}}$, the intrinsic scatter around that
relation $\sigma_{\text{X}}$, and the correlation coefficient between the WL
and count rate scatter $\rho_{M_{\text{WL}},C_{R}}$, labelled as
$\rho_{\text{WL,X}}$ for brevity in the plot. While the correlation
coefficient remains unconstrained, it leads to significant degeneracies with
the amplitude and scatter. Astrophysical processes like cluster age, merger
state, environment, orientation, triaxiality, etc., might impact both
observables and lead to selection biases. We agnostically marginalize over the
entire range of such effects, thereby weakening our amplitude and scatter
measurements, but making the mass calibration impervious to such astrophysical
effects.

###  6.3 Accounting for cluster selection biases

Much of the astrophysically driven conversation about the viability of galaxy
clusters as a cosmological probe, and the limits of our ability to effectively
calibrate their masses, focus on so called _selection biases_. When taking a
forward modelling approach using Bayesian population models, these concerns
can be naturally accounted for. We shall discuss in the following the most
(in)famous ”selection bias” as an example: the ”cool core bias” (Rossetti et
al., 2017). The cooling time in the center of massive galaxy clusters can fall
below the Hubble time, especially if they have been undisturbed for a long
time. Effective cooling boosts the central luminosity, resulting in very
peaked X-ray surface brightness profiles. Conversely, the SZe selection,
sensitive to the pressure, is not effected by this. As discussed already
above, taking a population modelling approach, it immediately results that any
sample selected on an observable that scatters around the halo mass will
preferentially contain up-scattered objects compared to down-scattered
objects, on account of the much higher frequency of the former. If a specific
physical property drives the scatter in some observable at fixed mass, then a
sample selected in that observable would show more objects with that physical
property, compared to a sample selected on another observable. X-ray selected
samples will preferentially contain objects that scattered up in X-rays, for
whatever astrophysical reason.

For cosmological inference the complication arises if the scatter in the
follow-up observable used for the mass calibration correlates with scatter in
the selection observable. A glaring case which is being much investigated, is
clusters selected via the number of photometric member galaxies, and the WL
signal of these objects. The scatter of both depends on the concentration,
orientation and triaxiality of the halo, as well as the neighboring
environment, resulting in strong selection biases hampering cosmological
analyses on those samples (Dark Energy Survey Collaboration et al., 2020; Wu
et al., 2022). Analogously, X-ray-selected clusters have been reported to live
preferentially in nodes of the cosmic web, as compared to filaments or sheets,
as found by Popesso et al. (2023) when comparing eFEDS-selected halos with
spectroscopically selected ones. This means that at fixed halo mass, the
cluster properties may depend on secondary halo properties like environment or
concentration – an effect analogous to the _assembly bias_ of galaxies (Wu et
al., 2008; Wechsler & Tinker, 2018). Our scale cuts on the tangential reduced
shear are designed to ensure that we measure the WL signal only in the 1-halo
regime (Grandis et al., 2021a), that is scales dominated by the halo’s own
density profile, and not by neighboring structures. This already limits
significantly the impact of assembly bias. Still, the eRASS1 selection might
correlate with some property that impact also the WL mass, such as
concentration, triaxiality, or dynamical state. To account for any such
possible effect, we introduce the correlation coefficient $\rho_{\text{WL,X}}$
among the scatters in count rate and WL mass. Fig. 18 shows the posterior on
the correlation coefficient, together with its degeneracies with the amplitude
of the scaling relation and intrinsic scatter at fixed halo mass and redshift.
The prior range for the correlation coefficient is naturally given by
$(-1,1)$, though we sample $(-0.9,0.9)$ to avoid numerical instabilities for
singular covariance matrix. Marginalising over the correlation coefficient
makes our analysis impervious to selection bias effects, but deteriorates our
ability to calibrate the observable–mass relation. Given that the numerical
value of the amplitude of the scaling relation directly impacts our
cosmological results, and crucially their agreement with external experiments,
foregoing the use of strong astrophysical priors on the selection effects at
the cost of a less stringent mass calibration is undoubtedly the more prudent
choice.

##  7 Conclusions

In this work, we perform the measurement and calibration of the WL signal
around eRASS1 clusters and groups in the DES Y3 data. To this end, we define a
background sample of DES galaxies by weighting tomographic redshift bins
defined and calibrated for the cosmic shear experiment (; ) as a function of
the photometric redshift of our clusters. After querying sources around our
lenses, out to an angular distance corresponding to $15h^{-1}$Mpc, we validate
and calibrate that measurement. We fit the lens richness, redshift and source-
lens distance dependence of the effective source density and the source
redshift distribution to determine the contamination of our source sample. The
different methods employed agree with each other within the systematic
uncertainties of the source redshift and shape measurement.

We fit a lens redshift trend to the effective shape noise, allowing us to
define a semi-analytical covariance metric for the shape noise. The profiles
of the B-mode-like cross component are consistent with zero, as expected for a
well-calibrated WL signal sourced by gravity. Investigation of the selection
and shear response of our tangential reduced shear estimators show that the
former can be ignored, as it induces a correction much smaller than the
systematic uncertainty induced by source redshift and shape measurements. On
small cluster-centric distances we find interesting anomalies in the shear
response and effective shape noise, that might point at cluster line of sight
specific calibration issues, which arguably are too small to impact our
subsequent analysis steps.

We derive cosmology-ready data products, used as inputs for the mass
calibration of eRASS1 selected clusters, both in this work, and self-
consistently also in the subsequent cosmological analysis by Ghirardini et al.
(A&A, subm.). These products are:

  * •

the tangential reduced shear profile in angular bins, corrected for the shear
response,

  * •

a semi-analytical estimate of the shape noise on that profile, and

  * •

an estimate of the field source redshift distribution from the local
background,

for each source with DES Y3 lensing data. This results in WL data for 2201
eRASS1 selected clusters, with a raw signal-to-noise ratio of 92, reduced to
65 after the our radial scale cuts are applied.

Exploitation of these data for mass calibration and number counts cosmology
requires the anchoring of the mapping between halo mass and WL signal via
$\mathcal{O}(1000)$ realisations of synthetic shear profile simulations,
following the scheme presented by Grandis et al. (2021a). These simulations
include:

  * •

realistic 2d projected surface mass density maps around massive halos,
extracted from the cosmological hydro-dynamical TNG300 simulations (Pillepich
et al., 2018; Marinacci et al., 2018; Springel et al., 2018; Nelson et al.,
2018; Naiman et al., 2018; Nelson et al., 2019),

  * •

a distribution of the off-set between the observational centers used for the
tangential reduced shear profile measurements, constructed from X-ray images
of the Magneticum Pathfinder simulations (Dolag et al., in prep.), and the
‘eROSITA digital sky twin’ (Seppi et al., 2022),

  * •

a painting of the richness using the richness–mass relation calibrated by Chiu
et al. (2022),

  * •

dilution of the tangential reduced shear signal based on the degree of cluster
member contamination we calibrated,

  * •

a noise contribution from the uncorrelated large-scale structure, and

  * •

realistic DES Y3 source distributions reflecting our background selection
based on weighted sums of the DES Y3 tomographic redshift bins (Myles &
Alarcon et al. 2021), together with a prescription for non-linear shear bias.

These synthetic shear profiles are analysed with the same shear profile model
that we use in the subsequent mass calibration. It features a simplified mis-
centered NFW with fixed concentration–mass relation, and accounts for the mean
cluster member contamination, as advocated by Grandis et al. (2021a). It thus
has one free parameter, the _WL mass_. Extracting the WL mass on our synthetic
shear profile simulations allows us to establish the relation between WL and
halo mass, summarized in the WL bias and scatter. Monte-Carlo realisations of
the synthetic shear profile simulation, obtained by varying all the inputs
within our expectation, result in an uncertainty estimate on the WL bias and
scatter, that acts as a calibration prior for the subsequent mass calibration.
We find that at low redshift our mass extraction uncertainty is dominated by
the effect of hydro-dynamical modelling, leading to a systematics floor of
$2\%$. At higher redshift, DES photo-$z$ uncertainties become incrementally
more important.

We calibrate the count rate – halo mass relation by using a Bayesian
population model to forward-model the probability density function of each
cluster’s shear profile given its count rate and redshift. Improving on
previous work (Mantz et al., 2016; Dietrich et al., 2019; Bocquet et al.,
2019; Chiu et al., 2022), we take explicit account of contaminants via a
mixture model. We find that we are able to measure 5 parameters of the count
rate–mass relation: the amplitude, the slope and its redshift dependence, the
deviation from a self-similar redshift evolution, and the intrinsic scatter
around the relation. We assess the quality of this population model based fit
on stacked signals in count rate, redshift bins, and find excellent agreement
among the data and the model. Our analysis finds comparable, and marginally
tighter constraints than the calibration of the eROSITA count rate–mass
relation performed by Chiu et al. (2022) with HSC data on eFEDS-selected
clusters.

This analysis will serve, along with similar analyses on HSC and KiDS, as the
main mass calibration for the number counts of eRASS1 selected clusters
(Ghirardini et al., A&A, subm.), which constrain different cosmological
parameters like the present-day matter density, the amplitude of matter
fluctuations, the equation of state of Dark Energy, and the sum of the
neutrino masses. Methodologically, we demonstrate how photometric data from
wide surveys, calibrated specifically for cosmic shear experiments, can be
effectively used the perform a WL mass calibration of cluster number counts.
These two experiments are highly complementary. Cosmic shear probes the linear
and mildly non-linear scale of the large-scale structure, while WL calibrated
number counts probe the most non-linear regions, massive galaxy clusters.
These probes have very different sensitivities to astrophysical and
observational systematics, only sharing a joint dependence on the source
photo-$z$ and shape calibration. In light of upcoming deep and wide surveys,
tailored mainly for cosmic shear, like Euclid666http://www.euclid-ec.org/,
LSST777https://www.lsst.org/, or Roman888https://roman.gsfc.nasa.gov/, this
work provides a demonstration on how to leverage these data to inform cluster
number counts, enabling independent and competitive cosmological constraints.

###### Acknowledgements.

This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE). The development and
construction of the eROSITA X-ray instrument was led by MPE, with
contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU
Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. The eROSITA data shown here were processed
using the eSASS software system developed by the German eROSITA consortium.
Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities, the Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy
Survey. The Collaborating Institutions are Argonne National Laboratory, the
University of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University. Based in part on observations
at Cerro Tololo Inter-American Observatory, National Optical Astronomy
Observatory, which is operated by the Association of Universities for Research
in Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. E.B., A.L., V.G., X.Z., and acknowledges financial support from
the European Research Council (ERC) Consolidator Grant under the European
Union’s Horizon 2020 research and innovation programme (grant agreement CoG
DarkQuest No 101002585). LK acknowledges support by the COMPLEX project from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program grant agreement ERC-2019-AdG 882679. N.O.
acknowledges JSPS KAKENHI Grant Number JP19KK0076. The Innsbruck group
acknowledges support from the German Research Foundation (DFG) under grant
415537506, as well as 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.

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##  Appendix A Effective shape noise and number density in presence of shear
responses

The aim of this appendix is to present a derivation for the expression of the
effective shape variance, Eq. 10 and the effective number density, Eq. 9, in
presence of a shear response $\mathcal{R}$.

Friedrich et al. (2021) demonstrate that the variance of a reduced shear
profile $\hat{g}_{\text{t}}$ is given by

$\text{Var}\left[\hat{g}_{\text{t}}\right]=\frac{\sigma^{2}_{\text{eff}}}{N_{\text{pair}}}\text{
with
}\sigma^{2}_{\text{eff}}=\frac{1}{2N_{\text{s}}}\sum|e|^{2}w_{\text{n}}^{2},$
(58)

where $N_{\text{pair}}$ is the number of source lens pairs, $N_{\text{s}}$ the
number of sources, $|e|^{2}=e_{\text{t}}^{2}+e_{\text{x}}^{2}$ the squared
modulus of the ellipticity when interpreted as a complex number, and
$w_{\text{n}}$ normalized weights.

To generalize this expression to un-normalized weight $w$, just consider that
such un-normalized weights can always be rescaled to normalized weights with
the ansatz $w_{\text{n}}=Cw$. The normalisation condition then readily leads
to an expression for $C$, reading

$\frac{1}{N_{\text{s}}}\sum w_{n}=1=\frac{C}{N_{\text{s}}}\sum
w\Leftrightarrow C^{-1}=\frac{1}{N_{\text{s}}}\sum w.$ (59)

Moving into the regime of individual cluster lensing, where
$N_{\text{pair}}=N_{\text{s}}$, combining all the above equations, we find

$\text{Var}\left[\hat{g}_{\text{t}}\right]=\frac{1}{2}\frac{1}{\left(\sum
w\right)^{2}}\sum|e|^{2}w^{2},$ (60)

for generic, un-normalized weights.

When considering shear data with shear responses $\mathcal{R}$, Friedrich et
al. (2021), Appendix D, suggest to transform the above equations by re-scaling
both the ellipticities $e$, as well as the weight $w$, as follows

$\tilde{e}=\frac{e}{\mathcal{R}}\text{ and }\tilde{w}=\mathcal{R}w,$ (61)

thus correcting the ellipticity for the shear response, and adjusting the
weights for the necessary re-scaling. Note that the summand in the numerator
of Eq. 60 is invariant under this re-scaling,
$|\tilde{e}|^{2}\tilde{w}^{2}=|e|^{2}w^{2}$, and we thus only have to modify
the summand in the denominator. Given that the above expression is valid for
un-normalized weights, no further corrections are required.

Dropping the tildes, we can readily write the variance of a tangential reduced
shear profile in presence of shear response as

$\text{Var}\left[\hat{g}_{\text{t}}\right]=\frac{1}{2}\frac{1}{\left(\sum
w\mathcal{R}\right)^{2}}\sum|e|^{2}w^{2}.$ (62)

Similarly, this justifies also why the source redshift distribution needs to
be shear response weighted.

Interpreting this expression as the ratio between an effective shape
dispersion $\sigma_{\text{eff},\alpha}$ for $\alpha\in$ (t, x), and an
effective number of source $N_{\text{eff}}$ we can recover the above
expression by using

$N_{\text{eff}}=\frac{\left(\sum w\mathcal{R}\right)^{2}}{\sum
w^{2}\mathcal{R}^{2}}\text{ and }\sigma^{2}_{\text{eff},\alpha}=\frac{\sum
e^{2}_{\alpha}w^{2}}{\sum w^{2}\mathcal{R}^{2}}\text{ for
}\alpha\in(\text{t},\,\text{x}),$ (63)

which are the expressions that we also use for our shape noise modelling.

##  Appendix B Parametrisation of cluster member contamination

We shall justify in the following the parameterisation used for the cluster
member contamination in Section 3.3.1. As shown by Hennig et al. (2017),
cluster member galaxies follow an NFW profile, with the total number of
galaxies enclosed in the virial region correlating tightly with mass, while
the concentration varies depending on the galaxy type. Red-sequence galaxies
have a larger concentration, more similar to the concentration of the dark
matter, than non-red-sequence galaxies. We therefore assume that the cluster
member galaxies that contaminated our background sample also follow an NFW
profile.

The fraction of red-sequence galaxies in the cluster member contamination is
unknown, though it stands to argue that their more precise photometric
redshifts make it easier to exclude them with a background selection. We
therefore let the concentration $c$ of the cluster member contaminants profile
as a free parameter.

Similarly, we choose an agnostic approach to the redshift trend of the cluster
member contamination. We simply assume that the ratio between the number
density of field galaxies $n_{\text{field}}(z)$ and cluster member
contaminants is a smooth function of redshift. We therefore employ a non-
parameteric fit of the amplitudes $A_{z_{i}}$ for pivot redshifts $z_{i}$.

To account for possible mass trends we use the richness $\lambda$ as a mass
proxy, given that we have to fit for the cluster member contamination prior to
performing a mass calibration, and therefore do not have WL calibrated mass
estimates at our disposal. For ICM-selected cluster samples, richness is known
to scale approximately linearly with halo mass, $\lambda\sim M$ (Saro et al.
2015; Bleem et al. 2020; Grandis et al. 2020, 2021b). We therefore use the
richness to define an approximate virial radius for the NFW profile of the
cluster member contaminants as
$R_{\text{vir}}\propto\left(\lambda/20\right)^{1/3}$. We also allow the
amplitude of the cluster member contamination to vary with richness.

Taken together, the above consideration lead to the ansatz

$\begin{split}&n_{\text{fcl}}(\lambda,z,R)=A(\lambda,z,R)\,n_{\text{field}}(z)\text{
with}\\\
&A(\lambda,z,R)=A_{j}\left(\frac{\lambda}{25}\right)^{B_{\lambda}}\Sigma^{\text{norm}}_{\text{NFW}}\left(R\Big{|}r_{S}=c^{-1}\left(\frac{\lambda}{20}\right)^{1/3}\right).\end{split}$
(64)

The total number density results from the sum of the cluster member
contaminants number density and the field number density, as

$n_{\text{tot}}=n_{\text{fcl}}(\lambda,z,R)+n_{\text{field}}(z)=\left[A(\lambda,z,R)+1\right]n_{\text{field}}(z),$
(65)

while the fraction of cluster member contaminants is

$f_{\text{cl}}(\lambda,z,R)=\frac{n_{\text{fcl}}}{n_{\text{tot}}}=\frac{A(\lambda,z,R)}{A(\lambda,z,R)+1},$
(66)

thus justifying the parametrisation chosen in Eq. 12. This parametrisation has
the benefit, that for $A(\lambda,z,R)>0$, $0<f_{\text{cl}}(\lambda,z,R)<1$, as
already noted by Grandis et al. (2021b) in the context of the richness-
dependent contamination of photometrically selected cluster samples.

Finally, one can show that $1+A=(1-f_{\text{cl}})^{-1}$, thus also motivating
Eq. 14.

## Affiliations

  1. 1.

Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr.
25/8, 6020 Innsbruck, Austria

  2. 2.

Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748
Garching, Germany

  3. 3.

Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1,
81679 Munich, Germany

  4. 4.

Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany

  5. 5.

Linköpings Universitet, Institutionen för Systemteknik, Linköpings
Universitet, Linköping 581 83, Sweden

  6. 6.

Kavli Institute for Cosmology, University of Cambridge, Mading- ley Road,
Cambridge CB3 0HA, United Kingdom

  7. 7.

Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, United Kingdom

  8. 8.

Physics Department, University of Wisconsin-Madison, Madison, WI 53706, USA

  9. 9.

Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA

  10. 10.

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia,
PA 19104, USA

  11. 11.

Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA

  12. 12.

Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain

  13. 13.

Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino
77, Rio de Janeiro, RJ - 20921-400, Brazil

  14. 14.

Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain

  15. 15.

Center for Astrophysical Surveys, National Center for Supercom- puting
Applications, 1205 West Clark St., Urbana, IL 61801, USA

  16. 16.

Department of Astronomy, University of Illinois at Urbana- Champaign, 1002 W.
Green Street, Urbana, IL 61801, USA

  17. 17.

Physics Department, William Jewell College, Liberty, MO, 64068, USA

  18. 18.

Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA

  19. 19.

Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA

  20. 20.

Department of Physics, Duke University Durham, NC 27708, USA

  21. 21.

Department of Physics, National Cheng Kung University, 70101 Tainan, Taiwan

  22. 22.

NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Green- belt, MD 20771,
USA

  23. 23.

IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France

  24. 24.

Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

  25. 25.

School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia

  26. 26.

Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berke- ley, CA 94720,
USA

  27. 27.

Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA

  28. 28.

NSF AI Planning Institute for Physics of the Future, Carnegie Mellon
University, Pittsburgh, PA 15213, USA

  29. 29.

Université Grenoble Alpes, CNRS, LPSC-IN2P3, 38000 Grenoble, France

  30. 30.

Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

  31. 31.

Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Dr., Pasadena, CA 91109, USA

  32. 32.

SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

  33. 33.

Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science
and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain

  34. 34.

Department of Physics and Astronomy, Pevensey Building, University of Sussex,
Brighton, BN1 9QH, UK

  35. 35.

School of Physics and Astronomy, Cardiff University, CF24 3AA, United Kingdom

  36. 36.

Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290
Versoix, Switzerland

  37. 37.

Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA

  38. 38.

Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, United Kingdom

  39. 39.

Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA
94305, USA

  40. 40.

Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,
13083-859, Campinas, SP, Brazil

  41. 41.

Physics Program, Graduate School of Advanced Science and Engineering,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526,
Japan

  42. 42.

Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1
Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

  43. 43.

Core Research for Energetic Universe, Hiroshima University, 1-3-1, Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8526, Japan

  44. 44.

Department of Physics, University of Genova and INFN, Via Dodecaneso 33,
16146, Genova, Italy

  45. 45.

Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA

  46. 46.

Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain

  47. 47.

Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA

  48. 48.

Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794, USA

  49. 49.

Département de Physique Théorique and Center for Astroparticle Physics (CAP),
University of Geneva, 24 quai Ernest Ansermet, 1211 Genève, Switzerland

  50. 50.

Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de
Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France

  51. 51.

Department of Physics, Boise State University, Boise, ID 83725, USA

  52. 52.

McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon
University, Pittsburgh, PA 15213, USA

  53. 53.

Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

  54. 54.

Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11,
I-34131 Trieste, Italy

  55. 55.

INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy

  56. 56.

Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy

  57. 57.

Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA

  58. 58.

Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831

  59. 59.

Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg,
Germany

  60. 60.

Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India

  61. 61.

Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

  62. 62.

Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway

  63. 63.

Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049
Madrid, Spain

  64. 64.

Department of Physics, The Ohio State University, Columbus, OH 43210, USA

  65. 65.

Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA

  66. 66.

George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, and Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA

  67. 67.

Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil

  68. 68.

School of Physics and Astronomy, University of Southampton, Southampton, SO17
1BJ, UK


A_{t}(x)|}G_{s-t}((A_{s}(x))^{-1}(y-x))\right|$ $\displaystyle+$
$\displaystyle\frac{1}{|\det
A_{t}(x)|}\left|G_{s-t}((A_{s}(x))^{-1}(y-x))-G_{s-t}((A_{t}(x))^{-1}(y-x))\right|$
$\displaystyle+$ $\displaystyle\frac{1}{|\det
A_{t}(x)|}\left|G_{s-t}((A_{t}(x))^{-1}(y-x))-\tilde{G}_{s-t}((A_{t}(x))^{-1}(y-x))\right|$
$\displaystyle=$ $\displaystyle\text{I}_{3}+\text{I}_{4}+\text{I}_{5}.$

It remains to justify $\sup_{y\in\mathds{R}^{d}}\text{I}_{i}\leq
cG_{s-t}(0)(s-t)^{\varepsilon}$ and $\int_{\mathds{R}^{d}}\text{I}_{i}\,dy\leq
c(s-t)^{\varepsilon}$ for $i\in\\{1,2,3,4,5\\}$ and some $c>0$ for some
$c=c(\kappa(\tau))$ in the general case or $c=c(\bar{h})$ if
$t<s\leq\tau_{0}$.

By Lemma 5.8 we get such estimates for $\text{I}_{1}$. By (8), (9) and (11) we
get the estimates for $\text{I}_{3}$. By Lemma 5.9 and (9) we obtain such
estimates for $\text{I}_{4}$. Analogous estimates of $\text{I}_{5}$ follow
from Lemma 4.8 and definitions of $G_{s-t}$, $\tilde{G}_{s-t}$. ∎

##  Appendix A Estimates for Example 2.4 and Example 2.7

In this section we prove the two inequalities, which were stated and used in
Example 2.4 and Example 2.7.

###### Proof of (18).

For $\rho_{k+1}<|y|\leq\rho_{k}$ we have $\rho_{k}\leq c^{-1}|y|$, hence

$\displaystyle h(r)$ $\displaystyle=r^{-2}\sum_{k:\rho_{k}\leq
r}\rho_{k}^{2}\nu(\rho_{k+1}<|y|\leq\rho_{k})+\sum_{k:\rho_{k}>r}\nu(\rho_{k+1}<|y|\leq\rho_{k})$
$\displaystyle\leq
c^{-2}r^{-2}\int(1\wedge(|y|^{2}r^{-2})\nu(dy)=c^{-2}\frac{4c_{\alpha}}{\alpha(2-\alpha)}r^{-\alpha},$

which proves the upper bound in (18). Similarly, we have

$\displaystyle h(r)$
$\displaystyle\geq\int_{|y|\leq\rho_{1}}(1\wedge(|y|^{2}r^{-2})\nu(dy)\geq
Br^{-\alpha},\quad r\in(0,1],$

which proves the lower bound. ∎

###### Proof of (19).

Without loss of generality we can take $t=0,x=0,$ then

$X_{s}=\int_{0}^{s}A(r)\mathbf{e}_{1}\,dZ_{r}^{1}+\int_{0}^{s}A(r)\mathbf{e}_{2}\,dZ_{r}^{2}$

The characteristic function of $X_{t}$ has the form

$\phi_{s}^{X}(z)=\exp\left\\{-\int_{0}^{s}(|(A(r)z)_{1}|^{\alpha_{1}}+|(A(r)z)_{2}|^{\alpha_{2}})\,dr\right\\},$

and thus the distribution density equals

$p_{s}^{X}(x)=\frac{1}{(2\pi)^{2}}\int_{\mathds{R}^{2}}\exp\left\\{-ix\cdot
z-\int_{0}^{s}(|(A(r)z)_{1}|^{\alpha_{1}}+|(A(r)z)_{2}|^{\alpha_{2}})\,dr\right\\}\,dz.$

In particular,

$p_{s}^{X}(0)=\frac{1}{(2\pi)^{2}}\int_{\mathds{R}^{2}}\exp\left\\{-\int_{0}^{s}(|(A(r)z)_{1}|^{\alpha_{1}}+|(A(r)z)_{2}|^{\alpha_{2}})\,dr\right\\}\,dz,$

below we will show that the latter integral exists.

Lets estimate from below

$\int_{0}^{s}(|(A(r)z)_{1}|^{\alpha_{1}}+|(A(r)z)_{2}|^{\alpha_{2}})\,dr\geq\int_{0}^{s}|(A(r)z)_{2}|^{\alpha_{2}}\,dr$

We recall that

$(A(r)z)_{2}=r^{\gamma}z_{1}+z_{2},\quad z=(z_{1},z_{2}),$

and perform case study.

_Case 1:_ $|z_{2}|>\frac{s^{\gamma}}{2}|z_{1}|$. Then

$|r^{\gamma}z_{1}+z_{2}|\geq\frac{1}{4}|z_{2}|\hbox{ if
}r\in[0,\frac{s}{4^{1/\gamma}}]\hbox{ and }|r^{\gamma}z_{1}+z_{2}|\geq 0\hbox{
othwerwize,}$

which gives

$\int_{0}^{s}|(A(r)z)_{2}|^{\alpha_{2}}\,dr\geq cs|z_{2}|^{\alpha_{2}}.$

_Case 2:_ $|z_{2}|\leq\frac{s^{\gamma}}{2}|z_{1}|$. Consider two intervals
$I(s)=[\frac{s}{2},\frac{3s}{4}),J(s)=[\frac{3s}{4},s].$ At least one of these
intervals is free from the roots of the function
$r\mapsto|r^{\gamma}z_{1}+z_{2}|$, and this function depends on $v=r^{\gamma}$
linearly with the slope $\pm|z_{1}|$. Since the values of this function in the
endpoints are positive, this yields that, at least on the half of the
interval,

$|r^{\gamma}z_{1}+z_{2}|\geq cs^{\gamma}|z_{1}|,$

which gives

$\int_{0}^{s}|(A(r)z)_{2}|^{\alpha_{2}}\,dr\geq
cs^{1+\alpha_{2}\gamma}|z_{1}|^{\alpha_{2}}.$

Now we can complete the estimate of $p^{Y}(0)$. We have

$\displaystyle p_{s}^{X}(0)$
$\displaystyle\leq\frac{1}{(2\pi)^{2}}\int_{\mathds{R}^{2}}\exp\left\\{-\int_{0}^{t}|(A(s)z)_{2}|^{\alpha_{2}}\,ds\right\\}\,dz$
$\displaystyle\leq\frac{1}{(2\pi)^{2}}\int_{|z_{2}|>\frac{s^{\gamma}}{2}|z_{1}|}\exp\left\\{-cs|z_{2}|^{\alpha_{2}}\right\\}\,dz$
$\displaystyle+\frac{1}{(2\pi)^{2}}\int_{|z_{2}|\leq\frac{s^{\gamma}}{2}|z_{1}|}\exp\left\\{-cs^{1+\alpha_{2}\gamma}|z_{1}|^{\alpha_{2}}\right\\}\,dz=:I_{1}+I_{2}.$

Since

$I_{1}=\frac{4}{(2\pi)^{2}s^{\gamma}}\int_{\Re}|z_{2}|\exp\left\\{-cs|z_{2}|^{\alpha_{2}}\right\\}\,dz_{2}=\Big{|}_{s^{1/\alpha_{2}}z_{2}=v}=Cs^{-\gamma-{2/\alpha_{2}}},$
$\displaystyle I_{2}$
$\displaystyle=\frac{s^{\gamma}}{(2\pi)^{2}}\int_{\Re}|z_{1}|\exp\left\\{-cs^{1+\alpha_{2}\gamma}|z_{1}|^{\alpha_{2}}\right\\}\,dz_{1}=\Big{|}_{s^{(1+\alpha_{2}\gamma)/\alpha_{2}}z_{1}=v}$
$\displaystyle=Cs^{\gamma}\cdot
t^{-2(1+\alpha_{2}\gamma)/\alpha_{2}}=Cs^{-\gamma-{2/\alpha_{2}}},$

this completes the proof of (19).∎

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